1.OURRIS. PENINSULAM.ANONAM
CIRCUMSPICE
1872 VILL
SCIENTIA
ARTES
VERITAS
LIBRARY
OF THE
UNIVERSITY OF MICHIGAN
QA
5
S88
OK
1743
HERITAHINMAINI
IIIIIII
muuhun
HUMOkulttuun
umid
NON
CIRCULATING
به مسی
ASE
SHELF
l[
NI.al,
SOIT
HON
SE ..؟
Im
وتی
لمزرد /
LA
5
.5 88
1743
A
NEW MATHEMATICAL
Wherein is contain'd, not only the
DICTIONARY:
EXPLANATION
OF THE
BARE TERMS,
But likewiſe an
H H I STORY
OF THE
Rife, Progreſs, State, Properties, &c. .
T H I N GS,
OF
BOTH IN
PURE MATHEMATICS,
AND
NATURAL PHILOSOPHY,
So far as theſe laſt come under a Mathematical Confideration.
The SECOND EDITION, with Large Additions.
By E. STONE, F. R. S.
Καθαρμοί ψυχής λογικής εισιν αι μαθηματικαι επισήμαι.
Matheſis mentis expurgatio.
HIEROCL,
L O N D ON:
Printed for W. INNYS, T. WOODWARD, T. LONGMAN, and M. SENEX.
M. DCC. XLIII.
Wir emera
Lianat -
8-do-43
48359
( iii )
!
TO THE
R E A D E R.
T
8. 42.BHP
1
HE firſt Impreſſion of my Mathematical Dic-
tionary being long ſince ſold off, and the Books
fellers having been from time to time frequent-
ly applied to, for the ſame, and that to no pur-
W poſe; and conſidering the Uſefulneſs of ſuch a Work, not
I only to thoſe, who may be fomewhat ſtopt in their
07 Reading of Authors for want of being acquainted with
O the Significations of ſome Words they may poſſibly meet
with in theſe Arts and Sciences, or delight in the Hi-
ſtorical Knowledge of Mathematicks, or elſe want a Guide
to direct them to ſuch as treat upon theſe Subjects : But
likewiſe to thoſe who are learned herein, by furniſhing
them with a convenient and neceſſary Repoſitory of Rules,
Propoſitions, and Properties, of the moſt notable and
eminent Terms defin'd, to which they may have imme-
diate Recourſe, as often as occaſion offers, either through
Forgetfulneļš
, or Want of Books ready at band : Theſe
Inducements ſet me upon publiſhing this ſecond Impref-
fion, with Corrections, Alterations, conſiderable Improve-
ments, and Additions; the whole either itſelf fufficiently
anſwering the Reader's preſent Expectations and Pur-
poles, or at leaſt pointing out ſuch Authors as will
.
Particularly amongſt the many Words berein ranged and
orderly
A 2
iv
PREFACE.
orderly explained, are to be found the following ſelect
ones, and their relative Appendages, viz.
Acceſlible Altitude, How to meaſure the ſame, and
how to meaſure the inacceſſible Depth of a Well.–Addi-
tion Algebraical, How to perform it.—Addition of whole
Numbers, How to perform and prove it.--Addition of
vulgar and decimal Fractions, How to perform it
Ædipile, It's Uſe, and ſome Authors that mention it,
seith an extreordinary Accident that happened upon ſet-
ting one of them upon too great a Fire. Æther, Sir
Ifaac Newton's Queries, relating to the Effects thereof,
&c.-Age of the Moon, How to find it.----Air, Its:
Gravity, Denfty, Elaſticity, Expanſión, Height, &c.--
Air-Pump.--Adjutage, The Laws of the Motion of
Water through them.Alternations of Quantities, The
Rules to find them.- Altitude Inacceſſible, The beſt and
moft uſual Ways of meaſuring ſuch.----Altitude of the
Cone of the Earth's Shadow._Amplitude, How to
find that of the Sun or Stars.--Analemma, Its firſt In-
ventor, and ſome Writers concerning it.-- Angle of
Contact, Some Properties thereof. Angle refracted,
How to find the Law of Refraction out of Air into
Glaſs. Apparent Diameter and Magnitude, How to
find them.----Annuities, Theorems relating to the fame.
Aſtrolabe. —Aſtronomy, Its Antiquity, and ſome of
the chief Writers concerning it.-
Afymptotes, some
Properties of them, and how to find them for geome-
trical Curves.-Axis in Peritrochio, The Proportion of
the Power to the Weight raiſed by it.-
-Azimuth
Compaſs.
Back-Staff, or Sea-Quadrant.–Balance, Its Proper-
ties.Barometer, Rules to judge of the Weather by it.
Binomial Theorem of. Sir Iſaac Newton, its Uje in
the
V
PRE FA C E.
the Extraction of Binomial Roots, &C. Bi quadratic
Equation, Its Formation, Reduction to a Cubic, Solution,
and Conſtruction.am-Biquadratic Parabola, Several Sper
cies thereof, with ſome new Ovals expreſſed by its Equa-
tion...-Bombs, The firſt Uſe of them, &c. -Burning-
Glaſſes, or Speculums.
Calculus Differentialis, or Fluxions, Sir Iſaac Newton's
own Account of bis Invention thereof. Catacauſtic
Curves.- Catenaria, Its Nature and the Manner of find
ing Points throwbich it paſſes.- Centre of Gravity,
How to find the ſame by Fluxions, and where that of
ſeveral Magnitudes falls; alſo a Way how to find the
Areas of Surfaces, and Solidity of Solids by means
thereof Centre of Oſcillation and Percuffion, How to
find it, and where that of ſome Magnitudes fall.
Centripetal and Centrifugal Force, Somé Properties
thereof, and ſome Writers upon the ſame.- Characters,
The ſeveral Characters uſed in Algebra, Aſtronomy, and
Muſic.Circle, Many of the principal Properties of the
Circle ; amongſt which are ſome rare and uncommon
ones, with Vieta's very elegant Solutions of the Pro-
blems of Tactions, viz. the Deſcription of a Circle to
paſs through one or more Points to touch one or more
right Lines given in Poſition, and one or more Circles ;
alſo how to cut a given Circle into two Segments, that
Mall have a given Ratio. ---Ciffoid, Its Generation and
Equation.-Clock, The firſt Inventor, and ſeveral Wri-
ters upon the ſame.-Colours
, Some Account tbereof from
Sir Iſaac Newton.--- Combinations of Quantities, The
different Ways they may be varied.---Comets, Some Ac-
count of them, and Writers upon them.--- Sea-Compats,
A Dejiription, and the firſt invention thereof.--- Com-
pound Intereſt, How to find the ſame.---Concave-Glaſs,
3
Tbe
vi
PREF A C E.
;
The Quantity of the Diminution of an Object ſeen through
one of them.---Conchoid, Its Equation, and three Species
thereof.---Cone, Its Generation, fome Properties, and
the Fluxion of the Surface of an oblique one.--- Conic
Sections, Some Writings upon the ſame.--- Conſtruction
of Equations, How to perform the ſame by the Interfec-
tion of two Loci.---Convex-Glaſs, How to find its Fo-
čus, and the Magnitude of an Image ſeen through it.---
Cubic Equation, Some Properties of it, and how to ex-
tract the ſame, or find its Roots.---Cubic Parabola, Its
Equation and Deſcription, and ſome Properties of it.---
Curves, Some Writers concerning them. — Cycloid, Its
Deſcription, Equation and Hiſtory.--- Cylinder, Some
Properties of it.
Decimal Fractions, The firſt Inventors of them.----
Declination of the Sun, How to find it.--- Departure,
How to find it.--- Deſcent of heavy Bodies, The Laws
thereof.--- Dioptrics, Some Account thereof, and Writers
thereupon.--- Direct Erect Eaſt and Weſt Dials, How to
draw them.--- Diacauſtic Curve.--- Dials, Some Writers
concerning them.--- Direct Erect South or North Dials,
Their Manner of Deſcription.--- Diviſion of Numbers
and Fractions, How to perform the ſame.--- Duplication
of the Cube, How to perform the ſame.
.
Earth, Its Magnitude, Figure, and various Opinions
about its Figure, &c.- Eclipſes, Of the Sun and Moon,
various Theorems relating to them, the Data neceſſary
to compute them, and ſome Writings concerning them.
Elaſticity, The Rules of the Congreſs, of perfectly Elaf-
tick Spherical Bodies. Ellipſis, Its Generation (vaa
rious ways) and Properties. Equation, Its Nature,
and Generation. Equilateral Hyperbola, Its Equa-
tion.
1
*
)
PRE FACE vii
tion. Erect declining Dials
, How to draw them.
Evolute Cůrvės Their Defcriptions, and ſome Properties
of them.-Exponential Curve.--Extermination, of the
unknown Quantity from an Equation, with Rules how to
perform the ſame. Extraction of Roots, Söme Rules
to perform the fame, and the Writers upon it.
Fibres, Some Properties of Elaſtick Fibres, Figure
of the Secants.-- Figure of the Sines. Figure of the
Tangents, Some Account of them. Fix'd Stárs; Some
Account of them, and thoſe who have made Catalogues of
them.-Fluents
, How to find them in various Caſes.---
Fluids, Several Laws of their Gravitation and Motion.
Fluxion, How to find the fame. Fortification,
The Maxims thereof, and ſome of the Writings upon the
Jame. Fractions, The Properties of them. Fruſtum
of a Pyramid or Cone, How to find the Solidity 'therea,
of.
Gauging, How to find the Contents of a Caſk in Ale or
Wine - Gauging-Rod, A Deſcription thereof.- Geo-
metrical Curves, Some Account of their ſeveral Orders,
Species, and Equations, and particularly thoſe of the re-
cond, wherein you have two new Curves of this Order,
not taken' notice of before ; as alſo the ſeveral particu--
lar Equations, that the general one of all Curves of the
third Order is divided into. Geometry, Some Ac-
count of its origin, and the Writings upon it.- Gra-
vitation, An Account thereof, and its probable Cauſe.
Gunter's Quadrant and Scale, their Defcription..
Heat, Some Properties thereof.—Helécoid Parabo. .
la; or Parabolic Spiral. Heterogeneal Surds, How to,
reduce them to one common radical Sign. Homoge-,
neal Surds. Horizon, Its Ulos.----- Horizontal Dial,
How
Vint
P R E FACE.
How to deferibe the fame - Hydraulics, Some of the
Writings thereon.--Hygroſcope, Its Deſcription.
Hyperbola, Various Ways of deſcribing the fame, and
fome of its general Properties.
Imaginary Roots, How to find them, according to Sir
Ifaac Newton. Inclination of the Orbits of the Pia-
nets. -- Inclin'd Plane, The Proportion of the Weight
fuſtain’d by it, to the Power.- Indetermind Problems,
Some Account of them, and the Way to reſolve them.
miten Index of a Quantity, The Nature and Doctrine of
Exponents. Inflection Point of a Curve, How to find
it in any Curve. Ionic Order in Architecture, The
Proportions of its Pillar.--Iſoperimetrical Figures,
Some Properties of them. – Jupiter, bis periodic Time,
Magnitude, comparative Diſtance from the Sun, and
Earth, &c.
Level, With a Table for the Correction of the Sta-
țions taken in Leveling. Levér, Its Properties.
Libration of the Moon. Light, Some of its Pro-
perties.com Locus, How to find which of the Conic,
Sections is the Locus of a given Equation of tw Di-
menhons, with ſome of the Writings upon this Subject.
Logarithms, Their firſt Invention, Nature, and
Conſtruction. Logarithmic Curve, Its Deſcription,
Nature, and Properties, and ſome Writings upon it.
Logarithmic Spiral. Longitude in Navigation, The
ſeveral ways how to find it.
Magnet, or Load-ſtone, Several Properties of its
attračtive Force or Virtue, and the firſt Invention of
it. Maps, The firſt Inventors and Conſtructors from
time to time afterwards. Mars, His Periodic Time,
Magnitude, Comparation, Diſtance, &c. Mathe-
matics,
ix
PREFACE.
matics, A ſhort Hiſtory of it.--Maximis & Minimis,
or Methodus de Maximis & Minimis, How to find the
fame. Mean Motion, Of the Sun, and Moon.
Meaſure, Uſeful Tables of different Meaſures.
Mechanics, Several of the Writings thereon.- Mere
cator's Chart, or Projection, Its Nature and Deſcription,
with the ſeveral Caſes, and their Proportions, in Mer-
cator's Sailing -- Mercury, His Periodic Time, Com-
parative Magnitude and Diſtance, &c. -— Meridian
Line, How to draw the ſame upon an Horizontal
Plane. Micrometer, Some Writings thereon.
Microſcope, Aphort Account thereof. Middle Lati-
tude Sailing, Its Uſe in Navigation. Moon, Her
Periodic Time, and Diſtance from the Earth, Magni-
tude, Motion, Attraction, &c. Motion, Several
Properties thereof. Multiplication, How to perform
the ſame in Numbers and Fractions, Vulgar, Decimal
,
and Algebraical. Mufick, Afoort Account thereof,
with ſome Writings upon it..
Nocturnal, The Deſcription and Uſe thereof.-
Opticks, Some Writings concerning them. Order
of Curve Lines, The general Properties of thoſe of
the ſecond Kind. Organical Deſcription of Curve
Lines, How to deſcribe thoſe of the firſt Order, by a
continued Motion, and by means of Points. Ortho-
graphic Projection of the Sphere, Its LawOſcil-
lation, The Proportion of the Time of Performance of
the fame in the Archs of Cycloids and Circles, with
the Length a Pendulum muſt have, that performs its
finall Vibrations in one Second of Time.- Oval, How
to deſcribe what the Workmen call by this Name, and
an Account of twelve remarkable Species of Ovals ex-
preljed,
a
PREF A CE.
preſſed by the Equation py
+ dx te..
ax* + bx: fo.cx*
Parabola, Its Generation, and some of its principal
Properties.- Parabolic Conoid, Its Solidity and Sur-
face. --- Parabola Carteſian, its moſt fimple Equation,
Deſcription, and Uſe. Parabola Diverging, The ſe-
veral Species, moſt fimple Equations, and Ways of find-
ing Points, through wbich they muſt paſs, and how the
ſeveral Sections of a Solid, generated by the Rotation of
a Semicubic Parabola, exhibits them all.-- Parabolic
Space, A Hint at its Quadrature, from a Pyramid's
being į of a Parallelepipedon of the ſame Baſe, and
Altitude, &c. Parallel Ruler, the Uſe thereof in
reducing any Multangular Figure to a Triangle.
Parallelogram, Some Properties thereof... Pendulum,
the firſt Inventor, and the Uſe of them. Perfect
Numbers, How to find them, by common Algebra.--
Perpetual Motion, the Impoſſibility of it. Perſpec-
tive, Some Writers thereupon. Plain Angle, tbe ſe-
veral Equations for dividing it into two, three, four,
&c. equal Parts.-- Polar Dial, its Nature, and ſome
Theorems, by means of which it may be deſcribed.
Polygon, Some Properties thereof, with a Table of E-
quations, for deſcribing them in a Circle. Polygonal
Numbers, The Rules for dumming them up. Pofi-
tion, or Rule of Fallé, How to perform the fame.-
Priſm, Some of its Properties.- Progreſſion Geome-
triçal, Sqmie of the Properties thereof: Projectiles,
The Lines of Motion that they deſcribe, in Vacuo,
and Air, Projection of the Sphere, Some Writings
co cerning the ſame.- Projection Monſtrous, How to
deſcribe fuch.
Proportion, The Nature, and ſome
Properties of proportional Quantities, &c. Protractor,
its Deſcription and Uje.- Ptolemaic Syſtem, A Mort
Account
3
PREFACE.
xi
Account thereof.-Pully, the Ratio of the Weight to
be raiſed by it, to the Power raiſing it. Pyramid,
Some Properties thereof, with an Algebraic Inveſtiga-
tion of the Rules of finding the Solidity and Surface of
a Fruſtum thereof, as alſo an eaſy Way of finding the
Area of a Parabola, by the Method of Indiviſibles.
Quadratic Equation, The feveral Forms thereof.
Quadratrix, Its Generation, and ſome Properties of it.
Radius of the Curvature of a Curve, The way of find-
ing the fame. Rainbow, Some Account, with an Ex-
planation of the Cauſe thereof. Ratio, Some Account
thereof, with the Inveſtigation of the Rule of finding a
Numerical Ratio in ſmaller Numbers, the neareſt ap-
proaching a given Ratio in greater Numbers, whoſe
Terms are Prime to each other. Regular Poly-
gon, a Trigonometrical Examination of the Truth of
the general Rule, which fome have given to inſcribe
them in Circles.-- Reſiſtance of a Medium, The Propor-
tions thereof to different Figures moving in it:- Rhombs,
Some Propofitions of Uſe in the Theory of Navigation,
with their Demonſtrations.
Right-angled Tri-
angle, How to find Series's of whole or mixed Numbers,
accurately expreſſing the three sides of a Right-angleá
Triangle.- Rule of Three, How to perform the fame.
Satellites of Saturn, and Jupiter, Their Comparative
Diſtances, Periodic Times,' &c. Sector, Age-
neral Account thercof.
Segment of a Circle,
An Approximating Rule to find the Area of it.
Segment of a Sphere, How
How to find its Solidity.
Semicubical Parabola, Its moſt fimple Equation,
Series, The Doctrine of Increments. Solid of
the leaſt Reſiſtance, Its Nature. Sound, The Caule,
and
xii
(
PREF A CE.
*
and fome Properties thereof. Spheroid, Its Satin
dity, and Surface with the Solidity of its ſecond "Ség-
ment, expreſſed in an Approximating Series. - Spiral
Line of Archimedes, Its Generation, and ſome Proper-
ties thereof. Stentoreophonic Tube, or Speaking
Trumpet, The firſt Inventor, and thefe Writers who
have mentioned the fame. Stereographic Projection
of the Sphere, Its general Properties. Subtangent
of a Curve, A general Rule to determine it, in Geome-
trical Curves. Subtraction, How to perform the ſame
in whole Numbers and Fractions. -- Sun, Several Par-
ticulars relating to its Comparative Magnitude, Denſity,
Motion, &C... Surd Roots, Some Account of them.
Teleſcope Reflecting, A Mort Deſcription there-
of: A very large one of Mr. Jackſon's the Mathe-
matical Inſtrument-Maker. Tide, Several Par
ticulars relating to it. -- Trapezium, Several
Properties thereof, amongſt which are five new ones, or
at leaft, ſuch as are not mentioned in any Writings
which I have ſeen. Triangle, Many curious
and uſeful Properties thereof, among which is Hono-
ratus Fabri's Propoſition about the three ſhorteſt Lines
drawn from a Point within a Triangle to the three
Angles, with a Geometrical Demonſtration thereof.
Trigonometry Plane, The Canons, or Properties for the
Solution of the ſeveral Caſes thereof - Trigonometry
Spherical, The ſeveral Affections of Spherical Triangles,
with the Canons or Properties, by help of which all their
Cafes may.be folved.
Venus, Her Periodick Time, Comparative Diſtance,
Diameter, &c.
I
mm
A
MATHEMATICAL
DICTIONARY.
A
E
T
I11
A
BACUS, in Architec- circular Arches, ab, bc, cd, ad, be
ture, the upper Part or drawn from Centres that are the Ver-
Member of the Capital tex's of Equilateral Triangles, whoſe
of a Column
In the Tuſcan, Dorick,
be
and Iunick Orders, it is moſt commonly
C
ſquare, (ſpeaking in the Workman's
Phraſe) that is, every Section of it,
parallel to the Horizon, is a Square.
Some make it round, others make the
Sides quite plain, without any Ogee,
and ſome make it have a Fillet in-
B
ſtead of an Ogee. The Height of
it, in the Tuſcan and Dorick Orders,
a
is of that of the Capital; in the Sides are each equal to the side of the
Jonick of that of the Capital. Square; and if the Ends of the Ar-
In the Corinthian and Compoſite ches be cut off by the equal Lines
Orders, the figure of the Abacus AB, CD, EF, GH, at right Angles
differs from that of the other Orders, to che Diagonals of the Square, each
the four Faces being circular, and Line being to of the Side of the
hollow'd inwards, having a Roſe on Square, any Section of the Abacus,
the Middle of each, and the four parallel to the Horizontal Plane of
Corners cut off.
the Baſe, will be alike to the mixted
If the Square a b c d be equal to Line Figure ABCDEFGH.
the Plinth of the Baſe, and four equal In the Corinthian Order, the
B
Heighs
1
A
1
4
an
}
A B A
A BS
Height of the Abacus is generally Digits; and thereby to aſſiſt thoſe
of that of the whole Capital: who are about to learn Multiplica-
Vitruvius makes it ; ſome make it tion, to fix theſe Products in their
leſs, others greater, as or
Memory. The thing is done by
In the Compoſite Order, its Height ſeeking one of the Digits at the
is z of that of the lower Part of the Head, and the other in the firſt up-
Column.
right Column; then the Number un-
Thoſe who have a mind to have der that Digit at the Head, falling in
a more particular Account of this the fame Horizontal Row as that o-
Member, may conſult the Writings ther Number is in, will be the Pro-
upon the five Orders of Architecture ; duct of the Multiplication of thoſe
amongſt which Mr.Perrault's is a ve- two Digits,
ry good one.
In ſome Books of Arithmetick, I
ABACUS, of Pythagoras, in Arith- have ſeen a Table of Multiplication
metick, is the common Table of a good deal more compendious than
Multiplication, conſiſting of 81 Num- this of Pythagoras, having no Num-
bers, within a Square or Oblong, dif- ber above a Digit twice, wherein
tributed into nine upright Columns, ore of the Rows of the Digits runs
and nine Lateral or Horizontal ones; diagonally. The Table is this which
the nine Digits 1, 2, 3, &c. orderly is here annex’d.
proceeding in the firit Horizontal
Column, and the firſt upright one;
the Numbers in each of the Horizon-
tal Columns in order being the Pro-
2 413
ducts of the Multiplication of each
Digit of the upper Horizontal Co-
3161914
lumn, firſt by 2, then by 3, and ſo
on till the laſt is multiplied by 9. 4812116 5
3 4 5
6
7
8
9
5 10 15 20 25
6
4
6 8 10 12 14 16 18
612 18 24 30 367
36
9
12 15 18. 2124 27
7 14 21 28 351421498
4 18 12 16 20 24 28 32 36
8 16 24132 40 481561649
5 10 15 20 25 30 35 40 45
911827136|45|5463172 81
6 12 18 24 30 36 42 48 54
ABSCISS, ſtrictly ſpeaking, is a
Part AP of the Diameter of a
7 14 21 28 35 42 495663
Curve Line, intercepted between the
8 16 24 32 40 48 56 64 72
Vertex A of that Diameter, and the
M
911827 36' 45154163172181
I
2
I
2
1 -
1
6
2
1
I
1
4
101
1
1
1
The Uſe of this Table is to ſhew
by Inſpection, the Product of the
Multiplication of any of the nine
-3
А
P
1
Point
f
1
A B S
A BS
Point P, where any Ordinate or Se- and ſuch Curves ; except this Word
mi-Ordinate MP to that Diameter be taken in a larger Sanſe, or ſome
falls
other Word be uſed, we ſhall be at a
M
lofs to expreſs compendiouſly the Na-
ture of ſuch Curves which have no
Diameter, or even of thoſe that have,
when we would mention their Na.
ture by the Relation of Lines drawn
A P
B parallel from Points of the Curve
M (viz. Ordinates) to Points of a
Itreight Line given in Pofition, with-
in or without, or partly within or
B
A А
partly without, to the Part or Parts
P of this Line, intercepted between a
given Point in it, and the Points where
the ſaid Parallels do cut it. There-
fore; in my Opinion, it may not be
1. Hence there are an infinite amiſs to define an Abſciſs more gene-
Number of variable Abfciſes in the ral, in ſaying it is the Part AP of
fame Curve, as well as an infinite a right Line given in Poſition, taken
Number of Ordinates.
M
2. If the Curve be the common
Parabola, one Ordinate P M has but
one Abſciſs AP: if an Ellipfis, it has A B P,
two Abſciſes, AP,PB, falling contra-
ry ways. And if an Hyperbola, con-
M
fiſting of two parts or Curves, one
Ordinate PM'has alſo two Abſciſes
AP, BP, both falling the ſame
B/
way.
3. If a Curve be one of the fe-
A P
cond kind, one Ordinate may have
three Abſciſſes: If the Curve be one
from a given Point A in that Line to
of the third kind, one Ordinate
the Point P, where a right Line PM
may
have four Abfciffés, and ſo on; the drawn from any Point M in a Curve
greateſt Number of Abſciſes being al- Line BM C, in a given Angle MPC,
ways one inore than the Order of the cuts it.
Curve.
4. For Method's fake, an Abſciſs
M
in a Curve is uſually mark'd with the
B
Capital Letters A and P, Band P, C
and P, &c. the Point P being where
A
P
the Ordinate falls, or elſe with the
: ſmall Letters, x or z.
6. In Mechanical Curves there are
5. As this Name was invented for no Abfciffes, properly ſpeaking, unleſs
more eaſily ſpeaking and convey- you will have iurv'd-lined ones; to
ing an Idea of the Nature of a be ſuch as in the common Cycloid are
Curve, by the Relation of an Abſciſs the Arches of the generating Circle
to its correſpondent Ordinate or taken from the Vertex of the Figure,
fhewing Properties of them in ſuch when the generating Cirele is in ſuch
C С
В 2
А СА
A CC
a Situation, as to have the Abſciſs pitals of this Order. But moſt com-
coincide with the Diameter.
monly in the antique Buildings, they
ABSOLỤTE, that which is inde- are Olive Leaves raffled into five.
pendent upon, or has no relation to Some divide them only into four,
any thing elſe.
and others into three : They are
ABSOLUTE Equation, Number, ſometimes unequal in Height, the
Motion, Quantity, Space, and Time. undermoſt being talleft: Sometimes
See reſpectively Equation, Number, the ſecond Range are higheſt, and
Motion, Quantity, Space, and Time. ſometimes they are equal. The Ribs
ABSTRACT Mathematicks, Num- in the middle are very often raffled
ber, and Quantity. See reſpectively on both ſides ; ſometimes they are
Mathematicks, Number, and Quan. not cut at all: And the firſt Row
tity.
commonly ſwell out towards the bot-
ABUNDANT Number. See Num- tom, but more in ſome Buildings
ber.
than others.
ACANTHUS. The Herb Bear's. AcceLERATION. Swiftneſs in-
foot, whoſe Leaves are repreſented creaſing perpetually.
in the Capital of the Corinthian Or ACCELERATION of Motion or
der of Architecture. There are two Velocity. The ſame with accelera-
forts of them; the one wild, and ar ted Motion, or Velocity; which
med with Prickles; the other ſmooth, ſee.
and cultivated in Gardens. The for ACCESSIBLE Altitude, or Height,
mer of which is repreſented in Go- in Practical Geometry, ſignifies the
thick Buildings, and the latter in Altitude or Height of any Object; as
thoſe of the ancient Greeks and Ro- of a Tower, Steeple, Tree, &c.
mans. Vitruvius, Book 4. Chap. I. which may be either mechanically
fays, The Leaves of this Herb are meaſured, by applying a Meaſure to
ſaid to have been the firſt Occaſion it, or elſe, whoſe Baſe or Foot may
of this Ornament being invented by be approached to, from a remote
Callimachus, a famous Statuary at A- Station (uſually on the Ground) with-
thens, upon ſeeing this Plant ſpread out any Obſtacle in the Way; as a
ing itſelf around a Baſket that had River, Wood, Houſe, &c. to hinder
been placed upon the Tomb of a thë ſpeedy Menſuration from this
young
Corinthian Lady, and cover'd Station to the Foot of the Altitude.
1. THe molt uſual way of meaſu-
Vitruvius, Serlio, Barbaro, and ring anacceſſible Altitude or Height,
Cataneo, uſe theſe Leaves on the Ca- AB, when its Baſe or Foot A can be
.
up with a Tile.
B В
663
33°36
A
100
only
1
A CC
only approached, is by meaſuring which may be either performed by
the Diſtance from fome Station C, Scale and Compaſs, or arithmetical-
on the Ground to the Foot A of the ly, by ſaying, as the Cofine of C:
Altitude ; and then at the Station C, Sine of C:: AC: A B.
taking, with a Quadrant, or fome 2. Another way of doing this, is,
ſuch like Inftrument, the Number of by placing a Bowl or Pail of Water,
Degrees and Minutes contained in the or a Looking-glaſs horizontally, at
angle C, formed by the Horizontal fome convenient meaſured Diſtance
Line CA, and the Viſual Ray CB, C from the Foot A of the Altitude,
going from C to the Top B of the and moving backwards or forwards,
Altitude : For when this is done, in till the Top B of the Altitude is per-
the imaginary Triangle ACB,right- ceived by Reflexion in the middle of
angled at A, you have the Baſe CA the Surface of the Water or Look-
given, and the Angle C, to find the ing-glaſs. Then if a be the Place of
Length of the Perpendicular AB; Station when this happens, and the
B
300ANTIITINTIIKID,
48
1
bi
Waidrint
a 50
40 A
!
3. Prob.
Diſtance from a to C, as alſo the of Inſtances in Treatiſes of Practical
Height ab of the Eye be given, and Geometry, under the Title of Alti-
you ſay as a C=5 Feet): CA metry. The good old Clavius, in his
(40 Feet) :: ab 6 Feet): BA Practical Geometry, Lib. 3. 39.
48 Feet) ; this will be the Mlea publiſhed in the Year 1606, was the
fure of the Altitude A B:
firſt who ſhewed how to find an ac-
The Reaſon of this follows from ceſſible Altitude by means of a re-
the Similarity of the right-angled flecting Surface, in the manner as de-
Triangles a b C, ABC; which are livered above.
fuch from the Equality of the Angles ACCESSIBLE Depth. A Depth
6 Ca of Inſidence and Reflexion the Perpendicular of which may be
ACB.
come at, and mechanically meaſu-
Note, The Height of the Eye a- red. .
bove the place whereon you ſtand, Sir Iſaac Newton, in his Univer-
muſt be added to the Perpendicular fal Arithmetick, propoſes a very in-
found, in order to have the true genious Way of meaſuring the Depth
Height above the horizontal Plane. of a Well, from the Sound of a Stone
Thoſe who have a mind to be ſtriking againſt the Bottom of it, and
more fully informed how to find an meaſuring the Time elapſed from the
acceſſible Altitude, will have plenty Moment the Stone is let fall until its
B 3
Sound
i
C C
ACH
Sound be heard : For if + be that bed by the Fall of Bodies are to ono
Time, a a given Space that a Body another, as the Squares of the Times
freely deſcending fallsfrom the Begin- of Deſcent; and the Spaces deſcri-
ning of its Motion, and b the Time bed by Sound are as the T'imes. This
in which it falls : Alſo, if d be the Depth may be more eaſily compu-
Time in which the Sound moves that ted, after the Manner of the excel-
given Space: then the Depth of the lent Hugh de Merick, in his Analyſis
adt tabb ab
Well will be
Geometrica, than after Sir Iſaac's Me-
dd
add
thod; as you may thus fee. Let
AX be the Depth, let A B bea,
BC=b, and BD=d; and icon-
Which Equation is gained from theſe ceive ACY to be part of a Parabola
woTheorems, viz.the Spaces deſcri- deſcribed through the Vertex A, and
A
i vb6+407
D
B
Y
X
Z
the Point C: Alſo ſuppoſe the right Becauſe the Space moved by Sound
Line A DZ drawn through A and D; in one Second of Time is about 1140
then, from the Nature of the Para- Feet, and a Body falls in that Time
bola, any Ordinate, YX, repreſents but about 161 Feet ; the Quantity
the Time of the Fall from A to X; BD (d) will be very ſmall in regard
and if A D be continued down to Z, to CB, b. And ſo in finding the
XZ will repreſent the correſpondent Depth of a Well, both B D d, and
Time in which the Sound moves that XZ, may be rejected without any
Space : Therefore Y Z, the Sum of great Error ; and the Depth will be
the Times, will be =t, viz. given. had, by making AX: AB (a) :;
Make LBC:: BC:AB. Then YX (+2): CB? (62).
LxAX=XY? Wherefore L: In the Supplement of the A Et a
XY::XY:AX. Alfo DB:AB:: Eruditorum for the Year 1713, pag.
XZ: AX. Make L:M::DB: 317,
317, and 339, an anonymous Author
A B. Then L:M::X2: AX, and has largely explained and comment-
L:XY:: XY: AX. Therefore ed upon this ingenious Problem .
LXAX-MXXZ=XY; confe-
ACHRONICAL, a Word of very
little uſe now-a-days ; in vogue a-
quently M: XY::XY:YZ-XY;
that is (calling XY, Z) M:z:: the Time of the riſing and ſetting
mongſt the ancient Poets, regarding
Therefore
of the Stars with reſpect to thoſe of
x=V Mxt+IMM-M=XY, the Sun : As a Star is ſaid to riſe or
and when you have this, it is eaſy ſet achronically, when it riſes or ſets,
when the Sun fets. But Ptolemy,
to get AX, fince YX is=AX.
Kepler, and other Aftronomers, will
L
have it, chat a Star or Planet is ſaid
zitz,
10
1
ļ
1
A CR
A CU
to be achronical when it is oppoſite ACCIDENTAL Point in Perſpec-
to the Sun, and ſhines all Night. tive, is that Point C wherein a right
And fo a Star or Planet is ſaid to riſe Line OC, drawn from the Eye 0,
achronically, when it riſes when the parallel to one (Line A B) or more
Sun ſets ; and ſets achronically, when given parallel right Lines, meets the
it fets while the Sun is riſing. perſpective Plane DE.
D
0
c
А
B
1. The Repreſentations of all pa. Anno Dom. 1306. wherein it is de-
rallel Lines will, if continued, all termined how many Perches in
meet upon the perſpectivePlane, in Length and Breadth ſhall make an
the accidental Point; and all Paral- Acre, that they had in thoſe Days
lels to the geometrical Plane, when very indifferent Geometricians; when
not ſo to the perſpective Plane, have it is there ſaid, That when an Acre
their accidental Point in the hori of Land contains 10 Perches in
zontal Plane. The manner of find. Length, it ſhall contain 16 in
ing this point is ſhewn in moſt Books Breadth; when 11 in Length, its
of Perſpective.
Breadth ſhall be 14 Perches one half,
Accord, a Term in Muſick, to and of a Foot; when 12 in Length,
be found in Ozanam's Mathematical its Breadth ſhall be 13 Perches si
Diftionary, fignifying either a Con- Feet, &c. and ſo on to a Length
cord or a Diſcord.
of 40 Perches, and the reſpective
ACHERNER, a Star of the firſt Breadths. For all theſe Lengths and
Magnitude in the Conſtellation E. Breadths, except the firſt, might
ridanus, whoſe Longitude is 10°. 31'. have been very well omitted.
of Piſces, and Latitude 59º. 18/. ACUBENE, a Name given by
ACRE, a ſuperficial Meaſure for ſome to a Star on the Southern Claw
Land, containing 160 ſquare Per- of Cancer.
ches: So that the Side of a ſquare ACUTE- Angle. See Angle.
Acre will be nearly equal to 12.4691 ACUTE-ANGLED CONE, is ſuch
Perches.
a right Cone, whoſe Axis makes an
One would imagine, from the Or- acute Angle with its Side. Pappus,
dinance for meaſuring of Land, made in his Mathematical Colle&tions, ſays,
anno 33 and 34 of Edward I. and this Name was given to ſuch a
B 4
Cone
:
AD, D
A DD
called, ang'!
Act
2; of
bi
Cone by Euclid, and ithe Ancients = 1; of io.and + 8 will be
before Apolontis, & Time; and they
- Sam and + Scars will be o.
CUTE-ANGLED Seation of a
ADDITION of whole Numbers, or
Cone, an Ellipſis, made by a Plane's Integers, is an expeditious Way of
cutting an acute-angled Cone; they finding one Number equal to two
not knowing that ſuch a Section
or more Numbers taken all together,
could be generated from any Cone
or finding the moſt fimpleExpreſſion
whatſoever, till Apollonius did.
of a Number, according to the eſta-
ACUTE-ANGLED Triangle, ſuch bliſhed Notation, containing as ma-
a one whoſe three Angles are acute. ny Units as are in all the given
ADDITION, the uniting or put- Numbers taken together, the Num-
ting of two or more things toge- ber found being called their Sum.
ther.
1. The Rule for Addition is to
ADDITION, Algebraical, or of place all the Numbers of like kind
Algebra, is the Connexion or put- under one another, that is, the U-'
ting together of all the Letters or
nits under Units, Tens under Tens,
Numbers to be added, with their and Hundreds under Hundreds, &c.
proper Signs + and, and uniting and adding up the Units; and if their
into one Sum thoſe that can be fo Sum be under ten, ſetting that Sum
united; as the Sum of a and b, is
atb; that of a and —b, isa-b; under the Units : But if a good to
that of a and -- b, is
that of za and Sa'is = 3a + så ten or tens, ſetting the Excers,
8; that of.ba and com 296a
underneath, and for every ten, car-
za is =40; that of a, b, c,
rying a Unit to the next Place to the
a+b+cmd; and fo of others. left hand, and foon ; as if I ſhould add
The Order in which they are ſet
down being of no great conſequence, 343 } that is { 300I 401
to 513
500 +3
though it may not be amiſs to ſet
them down according to the Order
Sun 855
800 + 50+5
of the Letters, writing a before b, 2. The Demonſtration of the Rule
b before c; and ſo on.
of Addition is very eaſy, and depends
1. When a negative Number or entirely upon the Notation in uſe,
Quantity is to be added to an affir- and Euclid's Axiom, to wit, that the
mative one, the Sun is the Diffe- whole is equal to all the Parts taken
rence remaining, by taking away together.
the negative Number or Quantity 3. The addition of Numbers may
from the affirmative one; as i and be performed, by beginning to add
i is o; 4. and 3 is 1 ; 7ac and up the firſt Column to the left hand,
- 3ac is 4ac; and ſo of others But and then the other Columns in Or-
when the negative Number or Quan- der from the left to the right, accord-
tity is greater than the affirmative ing to the Rule above; and when
ore, the Sum will be equal to what all the Columns are gone through,
remains, by taking away the atir their Sums will give the Sum of the
mative Number or Quantity from Numbers to be added. And this may
the negacive one : But will be reg?- be a very good way of proving Ad-
tive ; as the Sum of — 2 and to iis dition.
96057
-d, is
!
}
A D D
96057
Example
7025
Sum II 2168
9086
ADD
This is obſerved (whether firft I
know not by. One Defaguliers,
formerly a Profeſſor of Mathema
ticks at Amſterdam, in a Treatiſe
of his de Scientia Numerorum: "As
alſo by Dr. Wallis in his Aritb-
metick.
9
22
O
15
18
112168
1
1
t
cauſe a wrong
4. Addition may be proved ſeve- Exceſs or Number leſs than
9,
laft
ral ways; firſt, by adding all the marked ; if not, your Work has
Numbers together, and afterwards been wrong. For example,
diſtributing the Numbers into Par-
cels of 10 or 12 in a Parcel, (I ſpeak
274317
of a great many Numbers to be ad--
4678167
ded together;) and then adding to-
53651
gether each Part by itſelf, and after-
127861 6
wards their Sums into one total Sum,
and ſeeing whether the ſame total
Though this Proof of Addition
Sum comes to be the ſame in each
be not quite to be relied upon, be-
Sum
way. Secondly, by cafting away of
may
ſometimes
9's, (which I believe was firſt done appear true from it; yet the Pro-
by Dr. Wallis in his Arithmetick, bability of its being true, to that of
publiſhed anno 1657) which is thus:
And
Take
each of the given Numbers Truth of any Sum proved this way.
ſo we may be pretty ſecure of the
ſeparately, and add all their Figures
together as ſimple Units, and in do- mentioned, was the firft who ſhewed
Dr. Wallis, in the Treatiſe above
ing ſo, when you have made a Sum
equal to 9, or greater than
this laſt way of proving Addition,
but
9,
leſs than 18, neglect the 9, taking will alſo find it in Mr, Malcolm's A-
with the Reaſon of the ſame. You
what is over, and add to the next
rithmetick,
Figure ; and go on fo till you have
i gone through them all, and mark
ADDITION of Frations, is find.
what is over or under', at the laſt ing a Fraction equal to two or more
Figure : But if the Sum of all the given Fractions.
Figures be leſs than 9, fet down that
This is done by reducing all the
Sum. Do the ſame with each of the given Fractions (to fimple Fractions
Numbers, ſetting all theſe Exceſſes of one Unit, if they be numerical
of 9 together in a Column; then fum Fractions) to one Denomination, if
them up the fame way, making the they be not fo already : Then the
Exceſs of 9, as before, or what the sum of the Numerators being made
Sum is leſs than 9. Laſtly, Do the
a Numerator to the common Deno-
ſame with the total Sum, and what minator, makes the fractional Sum
is under 9, or over any Number of
ſought; which may be further re-
9's in this, muſt be equal to the duced as the Caſe requires.
1
EX.
Æ o
OL
+
Æ OL
EXAMPLES
ato; ab ad abe.
alco +
abe-fracd.
6 + is=2743
acd
1S
ce
to
e
ce
ce
3 14 15
is ng
7 35 35
35
62
6
s =
m. 8 som
7
t
ADDITION of Decimal Fraktions, der Pipe opening into the Ball
is finding a Decimal Fraction equal which, if fcrewed on, is the beſt
to two or more given Decimal Frac- way, becauſe then the Cavity may
tions.
be more eaſily filled with Water.
The Rule to do it is; Whether the This Inſtrument, of more Curio-
Numbers given be pure or mixed fity than real Uſe, is for repreſent-
Decimals, or ſome of them whole ing a kind of artificial Wind; and
Numbers, write them down under that after the following manner :
one another, in ſuch Order that the Fill it almoſt full of Water, which
Decimal Points on the left ſtand all you may eafily do if the Neck un-
in a Line, or under one another; ſcrews ; if not, you may heat the
and the Figures all in diſtinct Co- Ball red-hot, and throw it into a Vef-
lumns, in order as they are removed fel of Water, which will be fucked
from the Point either on the right or in through the ſmall Hole, if it be
left: Then, beginning at the Column kept immerged. This done, if the
on the right hand, add the Figures Æolipile be put upon, or before the
in every Column together, as in Fire, ſo that the Water and it be
whole Numbers, placing a Point in very much heated, a vapourous Air
the Sum, under the Points of the will fly out through the Pipe, with
given Numbers.
great Noiſe and Violence ; but by
EXAMPLE S.
Fits, and not with a conſtant and
.24 .004
36.24 -uniform Blaſt.
-378 015 450.058
This Inſtrument is ancient, being
.057 .367,8 378.72
mentioned by Vitruvius, Lib. 1.
.9356 .291
42.005
Cap. 6. Deſcartes too ſpeaks of it
.6827
.6778
in his Meteor. Cap. 4. It is alſo
mentioned in ſeveral other Authors,
2.2933
amongſt whom Father Merſennus,
ADDITION of Ratio's, the ſame Prop. 29. Phænom. pneumat. uſes it to
with ſome of the modern Writers, as weigh the Air, by firſt weighing the
Compoſition of Ratio's. Which ſee. Inſtrument when red-hot, and having
ADERAIMIN, or ALDERAMIN, no Water in it; and afterwards
is a Star upon the left Shoulder of weighing the fame when it becomes
Cepheus.
cold. But the Conclufion gained
Adhil, is a ſmall Star of the ſixth from this Operation cannot be very
Magnitude, upon the Garment of accurate, fince there is ſuppoſed to be
Andromeda, under the laſt Star in her
no Air in the Ball when it is red-hot.
Foot.
Varenius alſo, in his Geogr. cap. 19.
ADJACENT ANGLE. See Angle
. Sect. 6. paragr. 10. uſes it to ſhew the
ÆOLIPILE, a round hollow Ball Air's Rarefačtion by Fire.
of Iron, Braſs, or Copper, having a There is one thing I would have
Neck in, which there is a very ſlen.
obſerved
$
Æ T H
• Ε Τ Η
obſerved in the Uſe of this Inſtru- them; and fo exceedingly thin, as
ment; and that is, that you take not to cauſe any ſenſible Reſiſtance
care it be not ſet upon too violent a in the Motions of the Planets in ena-
Fire, with too little Water in it, for ny thouſand Years.
fear left it ſhould burſt and do mil-
All this is from Sir Iſaac Newton's
chief; which may ſometimes be the Queries, at the latter part of his Op-
Caſe, as once happened to my know- ticks. It is pity we have not Expe-
ledge.' A Perſon ſetting one of theſe riments fufficient to thew there is ſuch
Copper Inſtruments upon too great a a Fluid, and ſomewhat ſurpriſing
Fire in a Tavern Drinking-room, it this great Man himſelf, who was the
burſt with a Noiſe like a Cannon, in- moſt likely of any Mortal to diſco-
to ſeveral pieces ; which flew about ver any ſuch Fluid, ſhould make
the Room, and cauſed ſuch a vio- Queries about the Effects of it, be-
lent Concuſſion of the Air, as not fore he was aſſured of its real Exi-
only put out the Candles upon a ſtence; eſpecially if he had no other.
Table, and threw down the Bottles Proof than the Experiment we find
and Glaſſes, but broke moſt of the at the latter end of the Scholium, at
Panes of Glaſs, in Number 12 or 14, Sect. 6. Lib. 2. Princip. Mathem.
of a Sky-light, being the only Win- Philoſoph. Natur. The Subſtance of
dow in the Room.
which is, That he made a Pendulum
ÆRA, the ſame as Epocha; which of a deal Box, of about 11 Foot long;
ſee.
and having raiſed up the Box to a
ÆTher, a very thin elaſtick and noted Place, fix Foot from the Per-
active Fluid, readily pervading the pendicular, and then having let it
Pores of all Bodies, and by its elaf- go, he marked three other places to
tick Force expanded thro' all the which it returned, at the end of the
Heavens. Much rarer within the firſt, ſecond and third Oſcillations.
Pores of denſe Bodies, as thoſe of After which he filled the Box with
the Sun, Stars, Planets and Comets, Lead, and other heavy Metal, having
than at Diftances from them,growing firſt weighed the empty Box, the part
denſer and denſer perpetually, as the of the Thread the Box was hung to,
Diſtance increaſes ; cauſing the Gra- which was wrapt about it, one half
vity of thoſe Bodies towards one an the Thread, and as much Air as the
other, and of their Parts towards the Capacity of the Box took up; and the
Bodies; the Reflexion and Refracti- whole Weightwas about partofthe
on of the Rays of Light; the Du Box of Metal. Then having ſome-
ration of the Heat of hot Bodies ; what ſhortened the Thread, by rea-
the Communication of their Heat to ſon of the Box of Metal's ſtretching
cold Bodies ; performing Viſion by it, ſo that the Pendulum had the
its Vibrations, excited in the Bot- fame Length as at firſt; he drew up
tom of the Eye by the Rays of Light, the Box to the Place firſt obſerved,
and propagated through the folid, and letting it fall, numbered about
pellucid and uniform Capillamenta of 77 Swings before the Box returned
the Optick Nerves into the Place of to the ſecond place marked, and as
Senſation ; and animal Motion ex- many afterwards before it returned
cited in the Brain by the Power of to the third Place, and ſo alſo before
the Will, and propagated from thence it returned to the fourth Place.
through the ſolid pellucid Capilla- From whence he concluded, that the
menta of the Nerves into the Muf- whole Reſiſtance of the Box when
cles, for contracting and dilating full, to that when empty, had not a
greater
1
t
1
1
A FR
A FR
t
greater Ratio than 78 to 77: For if tempted to cut thro', to open a Paffage
the Reſiſtance of both of them were from the Red Sea tothe Mediterranean,
equal, the full Box, by reaſon of the but in vain : And Cleopatra thought
inactive Force of its Matter, being to have hoiſted her Fleet over it
78 times greater than that of the from the Mediterranean to the Red
empty Box, ought to have preſerved Sea, to get clear of the Romans. Be-
its ſwinging Motion ſo much the tween the Channel of the River Nile,
longer; and ſo it muſt have return and the Red Sea, that Iſthmus is but
ed to the four marked Places always nine Miles. This Country is ſome-
when 78 Swings had been perform- what of a triangular Figure. The
ed: But it returned to the fame when Baſe may be reckoned at Tangier,
77 Swings had been compleated. from whence to the Iſthmus, it is about
Therefore, ſays he, if A be the Re- 1920 Miles broad; but from the
fiftance of the external Surface of the Vertex of the Triangle,to the north-
Box, and B the Refiftance of the ermoft Part of the Baſe, 4155 Miles';
empty Box in the internal Parts ; and being much leſs than Aſia, and about
if the Reſiſtances of equally ſwift three times as big as Europe. A great
Bodies in the internal Parts, be as the part of it is ſituate under the Torrid
Matter or Number of Particles which Zone, and croſſed by the Equator.
is refifted, 78 B will be the Refi- The furthermoſt ſouthern Bound be-
itance of the full Box in its internal ing the Cape of Good Hope, in about
Parts : And ſo the whole Reſiſtance 34
Deg. of South Latitude; and the
A+B of the empty Box, will be to moſt northern Extreme is about Bar-
the whole Reſiſtance A+77B of the bary, in the Lat. of 37 Deg. North.
full Box, as 77 to 78. And there A great Part of this Country was
fore A is to B as 5928 to 1 ; that is, unknown to the Ancients; and even
the Reſiſtance of the empty Box in now the Inland Parts thereof are
the internal Parts, is about five thou- not well diſcovered. The general
fand times leſs than its Reſiſtance in Hiſtorians thereof are, Leo, Mar-
its external Superficies. And all this mol, Metellus, Gramaye, M. Li-
could not happen but from the Actic vio Sanuto, Le Croix, and Dapper ;
on of ſome ſubtle Fluid included which laſt is reckoned the beft ex-
within the Metal, or elſe by ſome tant, and abridged by Mr. Ogilby
other unknown Cauſe.
in an Engliſh Folio Edition. There
AFRICA, one of the four great are alſo many Travellers to parti-
Continents, or general Parts of the cular Parts : as Paul Lucas up the
Earth, containing Egypt, Barbary, Nile, as far as the Cataraits ; Don
Bildulgerid, Zaara, Negroe-Land, John de Caſtro's Voyage up the Red-
Guinea, Nubia, and Æthiopia ; and ſea to Sues, in the Year 1540,; and
the moſt remarkable Iſlands thereof Chardin, Le Brun, and Vanfleb to
are the Canaries, Maderas, Mada. Egypt. For the Defarts of Arabia
gaſcar, and Cape Verde Iſlands. It is and Meſopotomia, we have De la
bounded on the Eaſt with the Red Valle, Teixira, Thevenot, Vertoman,
Sea and Arabia, on the Weſt by the and Sir Henry Middleton. But in
Atlantick, on the North by the Me. all theſe we have ſcarcely any Ac-
diterranean, and on the South by counts of the Inland Parts of Ara-
the Æthiopick Oceans. It is joined to bia Fælix ; nor has any body de-
Aſia by an Iſthmus, of 40 German ſcribed the Inland Parts of Barba-
Miles broad, which ſome Kings of E- ry, Zaara, Bildulgerid, and the
gypt, and Sultans, had a Deſign and at- lower Æthiopia.
The
A
1
or two.
the Moon's Age.
A.GE
A IR
The Travellers to the upper Æ 1. By taking the mean Place of
thiopia, are Bermuda, Almeida, Pe- the Moon from
that of the Sun at the
ter Pais, Ludolphus, the Jeſuits Let- given Time (a whole Circle being
ters, Poncet,&c. but the beſt Account added when neceffary) which Dife
of áll is, the Hiſtoria del Æthiopia ference will be the Moon's mean
per Telles; being a Collection of all Elongation from the Sun ; and di-
the Authors aforeſaid, except one viding this by the mean Diurnal
Elongation, being the Difference
To Morocco, there are Moquet, between the Sun and Moon's Diur-
Movett, St. Olon, L' Eſtat de "Roy- nal mean Motions ; and the Quo-
aumes de Barbary, Frejus to' Mauri- tient is the mean Age of the Moon,
tania, Janiquin to Libia, &c. Job- that is, the Time elapſed from the
ſen's Voyage to the River Gambia. laft new Moon.
Bofman's Deſcription of Guinea, is 2. Or more eaſy ; by adding to
the beſt I am told for that Country,
the Radix of the mean new Moons
and likewiſe Tenrhyne's for the Cape (ſuppoſe that of the Year 1700,
of Good Hope. There are many being 21d. 135.5m. 34.)the Epacts
other Authors who have deſcribed of the given Years, Months, Days,
particular Parts of Africa, which I Hours and Minutes ; and from the
cannot mention, becauſe I have not Sam taking compleat fynodical
feen them.
Months, one of which is 29d. 12h,
AFFIRMATIVE Quantity. See 44m. 3.8. and the Remainder will be
Quantity.
AFFIRMATIVE Sign, in Alge The Moon's Age may alſo be
bra, is this, +
found, by turning the Difference
This Sign before and between two between the Time given, and the
or more Numbers, or literal Expref- known Time of any paft Conjunc-
fions of them, or Quantities, implies tion, or of an Eclipſe of the Sun,
their Sum, Addition, or putting to into Days, Hours, &c. and after-
gether, being an elegant Mark to uſe wards multiplying the ſame by 10d.
inſtead of the Words plus, more, or 15h. iim. 388. and then dividing
added to, and of more Aſſiſtance to
the Product by 29d. 12h. 44m.
the Imagination; as +5, +7, or
3./. and the Remainder, after the
plus 5, plus 7, or more 5, more 7, Diviſion, will be the Moon's Age.
or 5 added to 7, fignifies the Sum of Note. At the End of every !9
5 and 7, viz. 12; ſo alſo toa, tb, Years, the Moon's Age will return
or plus a, plus b, or more a, more b, upon the ſame Day of the Month,
or added to b, is the Sum of a and but will fall ſhort of the preciſe
b. So alſo in Geometry, the Lines' Time by a ſmall quantity.
+ AB + CD + EF, fignifies the
I have ſeen in ſome Books of Na.
Sum or Aggregate of the
Lines AB, vigation, under what is called the
CD, and EF; that is, theſe Lines Julian Calendar, (ſuch as Atkin-
put together.
ſon's Epitome, &c.) a very eaſy
Age, of the Moon, is the Time way to find the Moon's Age: But it
elapſed, or the Number of Days (al- is not exact enough, and ſo ſhall ſay
ways leſs than 30) from any propo- nothing of it.
ſed mean Conjunction, or new Moon, Air, or ATMOSPHERE, an in-
to the next, and is to be had from mott viſible, compreſſible, dilatable, ela-
of our common Almanacks. But ſtick fluid Body, in which we breathe
yet, without theſe, may be found, and live, encompaſſing the whole
Earth
AIR
AIR
Earth to a great Height, being hard a Glaſs Ball of about 283 Inches
ly perceivable by our Senſes ; but Capacity, weighed 100 Grains.
that manifeſts itſelf by its Reſiſtance But it is found that no two equal
to Bodies moved in it, and by its Quantities taken at the ſame time,
ftrong Motion againſt other Bodies, but at different Heights, were ever
at which time it is called Wind, be- found of equal weight, the lower
ing abſolutely neceſſary for the Vi- Air always outweighing the upper.
tality of Animals and vegetables, Even in the ſame place, an equal
the Collection, Preſervation, Direc. Quantity of Air will ſcarcely ever
tion and Augmentation of Fire. be found to be of the ſame Weight.
Some of the moſt noted Proper 3. The common Air near the
ties of the Air are as follow: Surface of the Earth, as well as the
1. It contains various kinds of Surface and all Bodies upon it, are
Corpuſcles ſwimming in it; neither continually preſſed by the Weight
can it be deprived of its Fluidity by of the Atmoſphere, or of the up-
the utmoſt Cold or Compreſſion, per Parts upon the lower; and this
nor made viſible to the Eye by the Weight is greateſt
, the nearer Bo-
beſt Microſcopes; and all Bodies dies are to the Center of the Earth,
have more or leſs Air contained and leſſer the higher you go : which
within them : And tho' the Particles Weight, upon every ſquare Inch
of this Fluid are exceeding ſmall, near the Surface of the Earth, is a-
yet they cannot make their way bout 15 Pounds Avoirdupois. Mr.
through Metals, Glaſs, Wood, or Boyle ſays, he found it to be 181
good Paper, which even thoſe of Pounds Troy. But it may be ob-
Wine, Water, sc. will do.
ſerved, that the Weight of the At-
2. Galileus, in his Mechanical moſphere in our Climate is conſtant-
Dialogues, was the firſt who diſco. ly changing, which Change is ob-
vered that the Air was heavy; for, fervable upon the Alteration of
by thruſting it into a hollow Ball Weather. And by repeated Expe-
by means of a Syringe, he found riments of about 86 Years, we come
the Weight of the Ball augmented; at length to know, that in Europe
and, upon opening the Ball, found it the greateſt Weight of the Atmo-
to have the ſame Weight as at firſt. ſphere is ballanced by a Column of
Torricellius, the Florentine Geo- Mercury of 31 Inches in Height,
metrician, Anno 1643, firſt attempt- and the leaſt by one of 28. Alſo
ed to weigh the Air; and after him the Atmoſphere's Preſſure upon the
Otto Guerick, a German, and then fame Bodies in the ſame Places is
Burcher de Volder, (inQuæftionibus A- variable, which Variation notwith-
cademicis de Aeris gravitate) whollanding is never found in the ſame
fays, that the weight of a Cubick Place to exceed of the whole.
Foot of Air is one Ounce and 27 Moreover, Air preſſes upon every
Grains; and this by ſuch nice Scales, Side of Bodies with an equal Force.
that if 25 or 30 Pounds was put into 4. The Air has an elaſtick Pro-
each,amanifelt Preponderation would perty, that is, all known Air oc-
enſue, upon' putting in, or taking a- cupying any certain Space, and be-
way one or two Grains from one ing confind there ſo that it cannot
ſide or other, Mr. Boyle fays, a- eſcape, will, when preffed by a de-
bout of a Pint of Air weighs terminate Weight, reduce itſelf into
one Grain and í Part; and Mr. a leſs Space, which will be always
S'Graveſande found, that the Air in reciprocally proportional to the
com-
2
1
1
1
A IR
AIR
compreſſed Force, and the Denſity on the contrary, by Cold it is con-
proportional to it ; and when that tracted into a ſmaller Space, and be-
Weight is removed, the Air will of comes denſer, as appears by the
itſelf be reſtored to the Space it had Thermometer ; conſequently the
loſt.
Height of the Atmoſphere perpe-
Mr. Boyle ſays, that two poliſh'd tually varies, being greateſt at Noon,
Marbles, which would in open Air and leaſt at Midnight. Its Denſity
ſuſtain a weight of 80 Pounds, be- is alſo greater in Winter than Sum-
fore they would fall afunder, would mer, being always in a Ratio com-
do ſo in the exhauſted Receiver with pounded of the direct Ratio of the
a Pound, and ſometimes half a Heights of the Mercury in the Ba-
Pound weight. And the ſame Phi- rometer, and the reciprocal Ratio of
loſopher ſays alſo, that the Weight the Diviſions made to the Degrees
of a Cylindrical Column of one of the Thermometer.
Inch in Diameter, is 14 Pounds, 8. Mr. Krukius, in his Meteoro-
2 Ounces, and 3 Drams Troy. And logical Tables, has ſhewn, that there ·
Mr. S'Graveſande fays, when the falls upon, and exhales from, the
Air. was drawn out of two equal Earth in one Year's time, about the
Braſs Hemiſpheres well joined toge height of 30 Inches of Water ; fo
ther, of 3 Inches in Diameter, it that it follows from hence, that there
would require a weight of 140 is a great Quantity of Water always
Pounds to pull them aſunder. ſuſpended in the Air, under the
5. Air may be condenſed by Art, Form of Fog, Rain, Dew, Hoar-
ſo as to take up but the both part of froſt, Snow, &C.
the Space it did before, as has been 9. If Altitudes of the Air be ta-
done by ſeveral, and which may be ken in the ſame arithmetical Pro-
feen in the Philofoph. Tranfaat. N° greffion increaſing, the Denſities
182. It is very hard to reduce the thereof will be in a geometrical
common Air into a Space 64 times Progreſſion decreaſing. But this is
leſs than it naturally takes up; and on the Suppoſition that the Den-
fince it is probable, that the doo ſity of the Air condenſed by Com-
part of the common Air at leaſt con preſſion is as the compreſſive Force,
fifts of aqueous, ſpirituous,oily, faline, or,which is the fame thing, the Space
and other Particles ſcattered thro' taken up by the Air reciprocally as
it, it is likely that common Air can that Force. Dr. Halley, I believe, is
never be reduced into a ſpace 1000 the firſt who publiſhed a Demonftra-
times leſs than it uſually takes up, tion of this in Philos. Tranfa&t. Nº
without becoming folid.
181. by means of an aſymptotical
6. The elaſtick Power of any hyperbolical Space. Dr. Gregory
Portion of Air, can by the Air's too, in his Aftronom. Prop. 3: Lib. 5.
Expanſion repel the 'Bodies that has fhewn the Truth thereof by the
compreſs it, with the ſame Force as Logarithmick Carve.
that which is exerted by the whole Dr. Jurin, in his Append, ad Geogr.
Body of the Air.
Vareni, has compendiouſly de
7. When Air is condenſed in a inonſtrated the fame, by a Method
certain determinate Degree by the not at all different from that of
Application of Heat, it acquires a Sir Iſaac Newton, in Lib. 2. Princip.
greater Power of Expanſion every Mathem. prop. 2.
way than it had before ; that is, it From this Theorem, it is eaſy to
is rariſied, or becomes thinner: and find the Denfity of the Air at any
Moreover,
given
A IR
AIR
1
1
&
given Height above the Earth's it muſt be at the Diſtance of the
Superficies ; for let the right Line Earth's Semi-diameter from the
AB be 33 Feet,
Earth, will fill all the planetary Re-
viz, the Alti-
gions as far as, and much beyond,
tude of a Co-
ЕНЕ
the Sphere of Saturn.
lumn of Wa-
9. Theſe Theorems are founded
ter of the ſame
upon the Suppofition, that the Air,
Weight with a
as you go higher and higher, is of
Column of Air
the ſame Nature with that near the
quite up to the
Earth. But Experiments ſhew it to
Top of the At-
to be otherwiſe ; for Mr. Caſini,
moſphere; and
C
and Picart, when meaſuring the
let BDF be
D. Heights of ſeveral Mountains, dili.
perpendicular
gently obſerved the ſeveral Altitudes
to AB; aſſume
of the Mercury of the Barometer,
BD equal to
and by that means found, that the
850 Feet, be
Proportion of the Rarity of the Air
ing the Num-
was not according to Dr. Halley's
ber of times
Theorem, but much greater than
that the weight
what ought to ariſe from the faid
of a Quantity A
B
Proportion. See Hift. de l'Acad.
of Water ex-
Roy. Anno
1703
1705. Moreover
ceeds that of the fame Quantity of Dr. Halley, and the Academy del
Air: that is, a Column of Air 850 Cimento, affert, that the Reduction
Feet high, of the ſame Weight with of Air into Spaces proportional to
a Column of Water one Foot high: the compreſſive Weights, does not
Then if DC be drawn perpendicular hold good beyond that Space, which
to BD, and made equal to 32 Feet, is 850 times leſs than that which is
it will repreſent the Denfity of the taken up by the common Air.
Air at the Height of 850 Feet. This 10. It is likely that the Height of
done, if a Logarithmic Curve ACE, the Atmoſphere is indefinitely ex-
be ſuppoſed to paſs thro' the Points tended many (perhaps thouſand)
A,C; the right Line BDF will be its Miles above the Surface of the Earth.
Afymptote, and any Ordinate EF Tho' ſeveral Authors will have it to
will be as the Denſity of the Air at be of a ſmall limited Height : But
the correſpondent Altitude BF: fo about this Height they differ very
that if the Denſity EF of the Air at much. Poſidonius makes the Height
a given Height BF be wanted, fay 12 German Miles; Alhazen and vi.
as BD is to BF, ſo is the Difference tellio, 13 ; Clavius and Nonius, 11 ;
between the Logarithms of AB and Tycho Brahe, 12; Gaſſendus, 10;
CD, to a fourth number, which will Ricciolus, 191 when loweſt, and 16
be the Difference of the Logarithms when higheit ; Varenius makes it
of AB and EF; and ſince the Loga- of a German Mile, from two obſer-
rithm of AB is given, you will have ved Altitudes of a Star at two Alti-
that of EF, and ſo EFitſelf. Sir Iſaac tudes, at Prop. 30. Sect. 6. Cap. 19.
Newton, towards the End of Lib. 3. Geogr and in Prop. 38. he makes it
Princip, Mathem. concludes from a I German Mile. Mr. Boyle makes
Computation of this kind, that a it 7 Engliſh Miles; but, upon a Sup-
Globe of our Air of the Diameter poſition it is every where of the ſame
of one Inch, if rarefied ſo much as Denſity.
Denſity. Harris and ſome others
will
.
8 or 9.
A IR
A IR
will have it to be about 41 Engliſh according to Mr. Haukſbee, in the ſaid
Miles. But all theſe, in my opinion, Tranſ. N. 305, as I to 885, (which is
are little better than mere Suggeſti- eſteemed the neareſt to the Truth).
ons, computed from uncertain, erro 14. Sir Iſaac Newton (in Schol. fub
neous Principles, chiefly grounded fin. Sect. 9. lib. 2. Princip.) fays, If
upon the Obſervations of the Twin the Particles of the Air be ſuppoſed
light, which is obſerved commonly nearly of the ſame Denfity with Par-
to begin and end when the Sun is 18 ticles of Water or Salt, and the Ra-
Deg. below the Horizon ; as appears rity of the Air ariſes from the Di-
from what Varenius ſays at Prop. 37, ſtance of the Particles, the Diameter
38. Sect. 6. Cap. 19. of his Geogr.
of a Particle of Air will be to the Di-
II. The Preſſure of the Air, near ſtance between the Centre of the Par-
the Surface of the Earth, upon any ticles, as about 1 to 9 or 10; and the
Baſe, is ballanced by a Column of Diſtance between the Particles as I to
Water of the fame Baſe, of about
33
Feet in Height when that Preffure is Moreover, in the Schola gener.
greateſt, and of about 30 Feet when Sect. 6. lib. 2. of the ſame Book, he
that Preſſure is at a Mean. From fays, he found by Experiments with
whence, and by the Theorem at N. 8. Pendulums, that the Reſiſtance of the
it follows, that the Expanſion of the Air is as the Square of the Velocity
Air will be 4096 times more than at of a Projectile moving in it.
the Surface of the Earth; and at that Thoſe who have a mind to be
Height, the Altitude of the Mercury more fully informed of the nature of
in the Torricellian Tube, will be but the Air, may conſult the ſeveral
about one hundredth part of an Inch. Writings of Mr. Boyle, Marriotte, Paf-
12. The firſt who obſerved the Bal- chal, our Philoſophical Tranſactions,
lance of the Air with Water, was a the Hiſtory and Memoirs of the
Gardener of Florence, who, wonder- : Royal Academy of Paris, Wolfius's
ing that he could not raiſe Water in Aerometry, the ingenious Dr. Hales's
a Pump, higher than to 18 Cubits, Vegetable Staticks, Boerhaave's Che-
communicated the unexpected Phæ- miftry, and others.
nomenon to Gallileo, who himſelf Air Pump, a Machine by means
did not then know any thing of it; of which the Air contained in any
as you find in his Mechanical Dia- proper Veſſel may be drawn out.
logue's, i p. m. 15, 16. firſt publiſhed There have been ſeveral ſorts of
about the Year 1638. After him ſe- Air Pumps contrived and conſtruct.
veral others experienced the ſame ed from time to time from the firſt
thing, amongſt whom was Mr. Mar- Invention, moſt of thoſe firſt made
riotte, a Frenchman, who found that confiſting of but one Barrel, or hol.
Water would not riſe higher than low Cylinder of Metal, uſually Braſs,
32 Paris Feet. And Torricellius, a with a Valye at the Bottom opening
Scholar of Gallileo, uſing Mercury inwards, and a Pifton (with a Valve
inſtead of Water, found it would be at the Top opening upwards) ſo ex.
fufpended at about 30 Inches. actly fitted to the Cavity of that Bar-
13. The Weight of any Quantity rel, and moving therein, that when
of Air, to the fame Quantity of Was it is drawn up from the Bottoin of
ter, near the Earth's Surface, acord- the Barrel (by means of an indented
ing to Merſennus, is as I to 1356; Iron Rod or Rack affixed to it, and
according to Mr. Boyle, as i to 1000; an Handle turning a ſmall indented
according to Dr. Halley, in the Phi- Wheel, playing in the Teeth of that
lofoph. Tranſ. N. 181. as i to 800 ; Rod) all the Air will be excluded
с
from
AIR
A IR
t
i
1
from the Cavity thereof; and having great, that the Power required to
alſo a ſmall Pipe at its Bottom, by raiſe the other is not much more
means of which the Barrel may have than what exceeds the Friction of
a Communication with any proper the Parts in motion; whereas in o-
Veſſel to be exhauſted of Air, the thers, the nearer the Cavity of the
whole being affixed to a convenient Receiver approaches a Vacuum, the
Frame of Wood-Work, where the greater is the Labour of working
End of the Pipe turns up into an ho- them: And the ſecond, That it per-
rizontal Plate or Diſh, upon which forms its Buſineſs in half of the time.
ſuch a Veſſel is placed.
The Air's Elaſticity is the Foun-
Mr. Boyle's Air-Pump, deſcribed dation of this Machine : For when
in his New Phyſico-Mechanical Ex a Piſton is thruſt down to the Bot-
periments about the Gravity and Spring tom of its Barrel, and then it be
of the Air, publiſhed anno 1660, has raiſed up, the Air in the Receiver
but one Barrel ; fo alſo has that will expand itſelf, and part of it
which was firſt uſed by Mr. Papin, will enter into the Barrel; ſo that
as likewiſe that deſcribed by Wole the Air in the Receiver and the
fius, in Elem. Aerom. But Mr. Boyle Barrel will have the fame Denſity,
was the firſt who contrived and ap- which will be to the first Denſity,
plied a mercurial Gage or Index for as the Capacity of the Receiver is to
meaſuring the Degrees of the Air's the Capacity of the Barrel and Re-
Rarefaction or Quantity of Exhau- ceiver together. And by thruſting
ftion out of a given Veſſel ; whoſe down the Piſton a ſecond time, and
Deſcription he gives at the Begin- drawing it up, the Denſity of the
ning of his firſt and ſecond Phy,aco- Air in the Receiver and Barrel will
Mechanical Continuations. The a. again be leſſened in the Ratio afore-
foreſaid Mr. Papin moreover was the ſaid; and repeating the Motion of
firſt who contrived an Air Pump the Pifton, the Air in the Recei-
with two Barrels; as you may ſee in ver will be reduced to the leaſt
Mr. Boyle's Contin. ſecond. Experim. Denſity, but can never be drawn all
Nov. Phyſico-Mechan. in Pref. & out: And if m be the Capacity of
Iconiſ. 2.
the Receiver, and n that of the
But the double Barrel Air Pump Barrel, d the Denſity of the Air in
of Mr. Haukſee's, publiſhed in his the Receiver, before the Pump be-
Phyfico-Mechan. Exper. anno 1709, gins to work: Then n tmin::d:
which is now commonly uſed in
England, far exceeds any that were = Denſity of the Air in the
ever made before, and is equal to,
n+m
and, I believe, may exceed thoſe of Receiver at the End of the firſt draw-
{ome Foreigners, ſuch as Leopold, ing up of the Piſton. And n-tom:n::
s'Graveſande, Muſchenbroeg, &c. dn dn2
that have come after him, and had
= to its Denſity at
each a mind to be Sharers in the Im-
ntom
the End of the ſecond Lifting up.
provement of this Machine.
This double Barrel Pump is pre-
dn²
d n3
ferable to any other made before, in And n tm: n::
ntm
two things; the firſt is, That when
the Receiver comes to be nearly ex its Denſity at the End of the
haufted of its contained Air, the third Lifting up of the Pifton; and
Preſſure of the outward Air upon ſo on. Wherefore if s be the
the deſcending Piſton is nearly fo Number of Strokes of the Pifton,
the
dn
:
not m²
2
3
+
nt m3
1
nt mis
A IR
A JU
the Denfity of the Air in the Re- of them obliged the world with
ceiver at the End of thoſe Number their Labours herein. By this In-
dns
ſtrument moſt of the Concluſions,
of Strokes will be that is, but now mentioned under the Word
Air, they have verified and con-
in Words, The Denfity of the natural firmed, as well as a great multitude
Air in the Receiver is to its Denſity of others, which are really very
after any Number of Railings up of curious, and wonderfully ſurpriſing.
the Pifton, as the capacity of the Re- Too many to relate in this place.
ceiver and the Cavity of the Barrel AIR-GUN. See Wind-Gun.
together, raiſed to a Power having AIR, in Mufick, is a Name given
the Number of Liftings up of the Pi- by fome to any ſhort Piece of Muſick.
Hon for it's Index, is to the like Power of theſe there are Sets compoſed by
of the Capacity of the Receiver alone. Mr. Handel, Dr. Pepufch, &c.
Which is the Theorem given by
AJUTAGE, a French Word for
Mr. Varignon, in the Memoirs de the Spout of the Stream of Water in
Mathemat. & Phyſ. for Dec. 1703.
any Fountain. Here follow. fome
Wolfus, in his Aerometry, fays, Obſervations and Concluſions rela-
that the firft Inventor of the Air ting to Ajutages, and the Spouts of
Pump was Otto de Guericke, a Bur- Water moving through them.
gomaſter of Magdeburg, who per 1. A Fluid Spouting upwards
formed ſeveral Experiments with it through any Adjutage, would aſcend
at Ratisbon, in the Year 1654. be- to the fame Altitude as the upper
fore the Emperor, and ſeveral other Surface of the Fluid in the Veſſel,
illuſtrious Perfons. Be this as it were it not for the Reſiſtance of the
will, Mr. Boyle ſoon after having Air, the Friction near the sides of
taken the Hint from Schottus's Trea- the Adjutage, and ſome other Cau-
tife, entitled, Mechan. Hydraulica- fes in the notion of the Fluid itſelf,
Pneumatica, publiſhed in the Year whereby Defects from that Altitude
1657..(tho he himſelf, in his Phy- do always arife; which are nearly
fico-Mechan. Exper. fays he did not in the ratio of the Square of the Al-
fee the Book) directed Dr. Hook, citude of the Fluid above the Adju-
and another Perſon, to contrive à tage, and is to be underſtood of
newer and better Air-Pump than ſmall Heights only.
Otto de Guericke's, which he heard 2. It is found by experience, that
was defective, it requiring the. La if the Direction of the Adjutage be
bour of two ſtrong Men for more ſomewhat inclined, the Fluid will
than two Hours to get the Air out aſcend higher than when it is exact-
of glaſs Veſſels, plunged under Wa- ly' upright ; and an even poliſhed
round Hole at the End of the Pipe,
The Air-Pump is a very uſeful or Tube, will give an higher ſpout
Inſtrument, which from time to time of Fluid than when the Adjutage is
has employed the Thoughts and cylindrical or conical : Which laft
Pains of ſeveral very ingenious and is the uſual Figure, and indeed beta
diligent Philoſophers, (ſuch as Mr. ter than the cylindrical one.
Boyle, Mr. Papin, Mr. Haukefbee, Dr. 3. It is found by experience, that
Hales, Father Merſennus, Mr. Mar- the Bignefs of the Adjutage muſt be
riotte, &c.) in making Experiments inlarged where the Height of the Ci-
concerning the Nature and Proper- ftern is, and that the Pipes convey-
ties of Air, and its Effects upon na ing the Water muſt be wide with re-
túral Bodies, who have every one gard to the Adjutage; and amongſt
ter.
C 2
the
A JU
A JU
1
!
the ſeveral Diameters of Ajutages, by experience, that the Air's Refift-
there is a ſtated Length in order to ance, and the Friction of the Water
give the greateſt Height of the Spout againſt the sides of the Ajutage, do
poffible, which muſt not exceed id fomewhat. difturb this Ratio, the
Inch. Likewiſe the Height of the Quantity of Water being always leſs
Spout of Water has its Limits, which than what ſhould ariſe from it: But
is not much above 100 Feet. in Altitudes under go Feet, the De-
4. If AG be a Ciſtern of Water, viation is not very great,
and the Side A B be biſfected in C, 6. The times in which cylindri-
and about the Centre C, with any cal Veſſels of Water of the ſame
Diameter and Height are emptied
through Holes or Ajutages, are in-
verſely as thoſe Ajutages. And when
theſe Veſſels are unequal, but the
A
Heights and Ajutages equal, the
times of emptying will be as the Ba-
D
E
ſes of the Cylinders: Therefore, in
any cylindrical Veffels, the times of
emptying are in the Ratio com
pounded of the Baſes, the inverſe
Ratio of the Diameters of the Ajuta-
е e
ges, and the ſquare Roots of the
Heights.
7. If the Side of a cylindrical
FA
B
Veſſel, beginning from the Baſe, be
divided into Lengths, which are as
1, 2, 4, 9, 16, &c. viz. the Squares
Radius CE, a Semi-circle be deſcrib- of the natural Numbers, 1, 2, 3, 4,
ed; and if E be an Hole or Ajutage &c. the Surface of the Water (run-
in the Side of the Ciſtern, and ED ning out through an Hole at the
be drawn perpendicular to AB, the Bottom) will deſcend from every of
Water will run out from E to F in thoſe Diviſions to the next in the
the horizontal Plane, the Diſtance fame time.
BF, which will be = ? the Perpen 8. If the Heights of two Veſſels
dicular E D: So that the Water run- continually, full of Water be une-
ning at the Centre C will go to the qual, and the Ajutages alſo unequal,
greateſt horizontal Diſtance poſſible. the Quantities of Water running out
And if Ce be=CE, the Water in the fame time, are in the Ratio
running out at e will go to the ſame compounded of the ſimple Ratio of
Diſtance BF, as when running out the Ajutages, and the ſub-duplicate
at E. This Theorem is demonftra- Ratio of the Heights.
ted by ſeveral hydroftatical Writers, 9. If the Heights of two Veſſels
amongſt which ſee Mr.s'Graveſande's continually full of Water be equal,
Inftit. Philof. Newton, cap. 7. the Water will run out through A-
5. The Squares of the Quantities jutages any-how unequal, with the
of Water running through Ajuta- fame Velocity.
ges in any Directions whatſoever, in 10. If the Height of Veſſels con-
Ciſterns kept conſtantly full, are in tinually full of Water, and their A-
the ratio of the Heights of the Sur- jutages be unequal, the Velocities of
face of the Water in the Ciftern a the Water ſpouting out are in the
bove the Ajutage i tho it is found ſub-duplicate Ratio of the Heights.
Moit
A LG
A L G
Moſt of theſe Concluſions are de- any other Symbols would do, but
monſtrated by Mr. Marriotte, in his not fo conveniently) with Marks fig-
Traitè du Mouvement des Eaux; as nifying Sums, Differences, Products,
alſo in the aboveſaid Book of Mr. Quotients, Rectangles, &c. deduced
s'Graveſande's.
from ſtated Rules; which (from ſome
Alcove, a Term in Architec- fort of Analogy they have to thoſe of
ture, fignifying a part in ſome Cham- Addition, Subtraction, Multiplica-
bers, higher than the others, having tion, and Diviſion in common Arith-
an arch-like or other Figure, and ſe- metick) are therefore called Alge-
parated by Pilaſters, and other Or. braick Addition, Subtraction, Mul-
naments ; in which is placed a Bed tiplication and Diviſion, and chiefly
of State, or elſe Seats for Entertain- founded upon Euclid's Axioms about
ment. I have heard, that there are the Addition or Subtraction of equal
ſeveral Alcoves at the Seats of the Numbers or Quantities, to or from
Nobility in Spain, Italy, France, and equal or unequal ones ; as alſo upon
ſome very good ones in our own the like Axioms of the Equalities or
Country, contrived and made by our Inequalities of the Rroducts or Quo-
celebrated Architects, ſuch as Gibbs, tients of Numbers.
Campbell, &c. But do not find in the This Art is ſurpriſingly uſeful in
Treatiſes of Architecture that I have Arithmetick and Geometry, being
ſeen any thing ſaid about then, ex one of the moſt general, extenſive,
cept in that of Daviler's Cours d' Ar- ſhort and eaſy Helps of diſcovering
chiteft. Tab. 16. p. 177.
and proving mathematical Truths
ALDERAIMIN, a Star of the that has been hitherto invented, or
third Magnitude on the right Shoul- perhaps ever will. By this the So-
der of Cepheus.
lution of innumerable arithmetical
ALDĦAPHR A, a Star of the third Queſtions, which one of ever ſo
Magnitude.
much Skill in common Arithmetick
ALDEBARAN, a Star of the firſt would never be able to effect, with-
Magnitude on the Head of Taurus, out the utmoſt Pains and Trouble,
and uſually called the Bull's Eye. and perhaps not at all, is but a mere
Its Longitude for the Year 1700 was Play; and the Reaſons of all the
5° 49' 30", of Gemini, and Lati- Rules of common Arithmetick, ſuch
tude 56. 27. 30". South, according as Addition, Subtraction, Multipli-
to Mr. Flamſtead's Catalogue. cation, Diviſion, Extraction of Roots,
ALEGRO, a Term in Mufick, fig. Fractions, &c. do fo evidently ap-
nifying that that part over which it pear, and naturally flow from it, that
is placed muſt be ſung or play'd whoever ſhould go about to ſeek for
ſwiftly.
others, would be ſaid to do little elſe
ALGEBRA, an univerſal Arith- than abuſe Time. This is the
gene-
metick, or certain kind of Logick ral Analyſis that alone does allilt us
or way of Reaſoning in the Soluti. in finding the different Species, Fi-
on, Invention, and Proof of Propo- gures and Properties of geometrical
fitions, regarding the Equality or Curve lines, eſpecially thoſe that ex-
Inequality of Numbers, or any kinds ceed the firſt Order, which by any
of Quantity in pure or mixed Ma- other known Means would be plain-
thematicks; and that by means of ly impoſſible to come at (and that
artful Diſpoſitions, Connections and for want of other fufficient Elements;
Combinations of Numbers, or the which I believe we ſhall never have,
Letters of the Alphabet, (repreſents becauſe of our Shortneſs of Life,
ing Numbers or Quantities, though ſmall Extent of Knowledge, Nar-
rowneſs
C3
A LG
ALG
1
rowneſs of Conception, and the Di Diophantus was the firſt Greek
minution of Inclination, uſually hap- Writer of Algebra. About the Year
pening when our Advances in theſe 800 he wrote thirteen Books, only
Studies become great enough to make fix of which were publiſhed in Latin
us capable of making ſuch, and by Xylander in the Year 1575; and
rightly applying them.) In a word, afterwards, viz, anno 1621, in Greek
by this numberleſs Problems may be and Latin by Monſieur Bachet and
ſolved, which could not any other. Fermat, with Additions of their own.
ways be effected ; and often times This Algebra of Diophantus's only
more Theorems are expreſſed in one extends to the Solution of arithme
Page than could be expounded and tical indeterminate Problems.
demonſtrated in whole Volumes, af Before Diophantus's came out, Lu-
ter any other Method.
cas Paciolus, or Lucas de Burgo, a
The Word Algebra is derived from minorite Friar, publiſhed a Treatiſe
the Arabick, to which the firſt Euro- of Algebra in Italian, printed at Ve-
pean Writers have aſcribed various nice anno 1494. He may be ſaid to
Names, as Reſtorationis & oppofiti be the moſt ancient European Writer
onis Regula ; that is, the Method of on this Art.
Reſtoration and Oppoſition. Regula The Title of the Book is Summa
Rei & Zenſus; that is, the Doctrine Arithmeticæ & Geometriæ; and he
of the Root, and of the Square ( Rei ſays he explains it ſuch as he recei-
in Italian fignifying a Root, and ved it from the Arabians, but goes
Zenſus its Square;) the Cofick Art, no further than ſimple and quadra-
from the Italian Word Cofa, a Root; tick Equations : Nor does Stifelius,
1 Arte Maggiore, or the Great Art; in his Arithmetica Integra, publiſhed
others, more modern, Arithmetica
anno 1544; Hemiſchius, in his Arith-
Speciofa, or Specious Arithmetick; metica Perfecta, and others, make
Logiſtica Specioſa; Elementa Mathe- any farther Advances. But Scipio
feos univerſa, univerſal Elements of Ferreus added Rules for reſolving
Mathematicks; the Art of ſolving cubick Equations, (though indeed
Queſtions by Equation.
not general ones) firſt publiſhed anne,
It is highly probable, that the In- 1545, by Cardan, in Arte Magna.
dians or Arabians firſt invented this Ludovicus Ferrarienſis, or Lewis of
Art, for it may reaſonably be con- Ferrara, ſhews a way how to reduce
jectured, that the ancient Greeks biquadratick Equations, which Ra,
knew nothing of it, becauſe Pappus, phael Bombelli publiſhed anno 1579.
in his Mathematical Colle£tions, in his in his Algebra : But this is imper-
Enumeration of their Analyſis, makes feet.
no mention of any thing like it; and Tartalia was alſo another ancient
beſides, ſpeaks of a local Problem Italian Writer upon Algebra. About
begun by Euclid, and continued by the Year 1590, Franciſcus Vieta, a
Apollonius, which none of them could Frenchman, found out the Literary
fully reſolve, which doubtleſs they Arithmetick, and applied it to Al-
might eaſily have done, had they gebra ; and has given a very inge-
known any thing of Algebra. Nei- nious way of extracting the Roots of
ther does the Greek way of numerical any Equation by Approximation,
Notation ſeem at all adapted to the and explaining their Nature from
Purpoſes of ſuch an Art, nor their Proportions.
ſmall Knowledge in Arithmetick and Mr. Ougbtred, in his Clavis Ma-
the Properties of Numbers, imply thematica, firſt publiſhed anno 163?,
they had any ſuch thing.
followed and improved the ſpecious
Alge
1
0
1
A LG
AL G
Algebra of Vieta ; and invented fe- lineal Conſtructions of cubick, bi-
veral compendious Characters to ex- quadratick, and ſome higher Equa-
preſs Sums, Differences, Rectangles, tions, by means of the Conick Sec-
Squares, &c. But goes no further tions, &c. which Deſcartes firſt
than quadratick Equations. fhews how to do, may be ſaid to be
Before Vieta, fome algebraick fuch.
Writers uſed the four Letters R, Q, Dr. Pell reviſed and altered a
C, S, ſignifying the Root, (or un Piece of Algebra, firſt publiſhed in
known Number) Square, Cube, Sur- high Dutch at Zurick, anno 1659;
de-ſolid ; or alſo for R they put N, and afterwards tranſlated into Eng-
that is, a Number. Others uſed the life by Mr. Branker, under the Ti-
Characters 2, 3, regro,which ariſe title of An Introduction to Algebra,
from the Letters r, Z, C, s, fignifying and publiſhed anno 1668. In this
(the Root, or unknown Quantity) Dr. Pell gives us a particular Me-
Rem, Zenfum, (its Square) its Cube, thod of his own for applying Alge-
and its Surde-folią. But Vieta, Ough bra to Problems of various kinds,
tred, and others, put the Letter A at and introduced the way of keeping
pleaſure for any Root, or unknown a Regiſter of the whole Proceſs in
Quantity; and for the ſeveral Pow- the Margin, that fo you may fee
ers thereof they join the Letter 9 how any Quantity or Equation in
and c as Aq, the Square of A, Ac the large Column towards the right
its Cube, A99 the ſquared Square, Hand is produced ; as alſo invented
or fourth Power of A, and ſo on. ſeveral other uſeful Things.
About this Time, or ſome few Dr. Wallis, anno 1664, publiſhed a
Years before, there were ſeveral o Treatiſe of Algebra, both Hiſtorical
ther algebraick Writers; as Nonnius, and Practical, containing ſeveral
Ramus, Clavius, Girard, &c. All good Things ; but not many Im-
of which, together with thoſe al- provements, unleſs it be the finding
ready mentioned, are very deficient, the Roots of a Cubick Equation uni-
when compared with the Treatiſe of verſally, and ſome other things a-
Algebra wrote by Mr. Harriotte, who bout Combinations, Alterations, and
died at London anno 1621, and pub- aliquot Parts.
liſhed by Mr. Warner anno 1631 ; Mr. Kerſey, anno 1671, publiſhed
wherein he uſes the ſmall Letters in- a Folio Treatiſe of Algebra, ex-
ſtead of the capital ones of Vieta and plaining the Nature of Equations,
Oughtred, ſhews the true Nature and and illuſtrating his Precepts with
Conſtitution of Equations; and gives plenty of Examples. He explains
many uſeful Theorems relating to Diophantus throughout, and gives
them and their Roots, not taken no many things out of Marinus Ge-
tice of by any before him, moſt of thaldus de Refolutione & Compofitione
which are contained in the Geome- Mathematica. Monſieur Preſlet, a
try of Deſcartes, firſt publiſhed in Frenciman, publiſhed much ſuch an-
French, anno 1637, which was af- other Treatiſe anno 1694. Alſo
terwards tranſlated into Latin by Monſieur Ozanam, publiſhed Ele-
Van Schooten, a mathematical Pro- ments of Algebra anno 1703, in French,
feſſor at Leyden, and publiſhed anno wherein, beſides the literal Calculus,
1659, with a prolix Commentary and the Doctrine of Equations,
upon it, and ſome other algebraical he wonderfully illuſtrates the Dio-
Pieces of other Perſons annexed ; the phantean Doctrine of reſolving nu-
whole affording none, or very little merical Problems, in which this
Improvement to the Art, except the Author has chiefly excelled. And a
Com
A L G
A LG
Compendium of Mr. Preſet was Synopſis, publiſhed anno 1706, does
printed anno 1704, by Mr. Lamy, likewiſe neatly treat of Algebra.
with the Title, Elemens de Mathe. There is alſo Ward's Algebra, well
matique ; which, by reaſon of its enough for ſome to learn from.-
Perfpicuity, may be eſteemed a fit But amongſt all the pieces on this
Piece for Learners, though he does Subject, the Univerſal Arithmetick
not touch upon the Diophantean of Sir Iſaac Newton, which were
Doctrine.
Lectures formerly read by him at
Monſieur Ozanam, in his French Cambridge, when he was Lucaſian
Treatiſe of Geometrick Loci, as alſo Profeffor, and publiſhed by Mr.
Monſieur De la Hire and Guiſnee, have Whifton, anno 1707, is by far the
applied Algebra to Geometry, as beſt ; and would be a complete
well as the Marquis de l'Hospital, in thing, if to the ſame were added
his Conick Sections, and ſeveral o Sir Iſaac's Method of extracting the
ther Authors, too many to mention. Roots of Equations by infinite Se-
But we muſt not paſs by a neat Piece ries, which we have in the Commer-
of Algebra, publiſhed in Dutch, anno cium Epiftolicum, and the Fragmenta
1661, by Mr. Kinckhuyſen, wherein Epiſtolarum, publiſhed by Mr. Jones,
the Rules of Algebra are perſpicu- anno 1711 ; and alſo the Artifice
ouſly explained, but without Ex- of managing unlimited Problems,
amples; the chief Properties of the which no doubt he would have done,
Conick Sestions algebraically inveſti- had he ever deſigned it for the Pub-
gated, and many elegant Conſtruc- lick.– What is eminent in this Trea-
tions of geometrical Problems found tiſe, and no where elſe to be met
out by Algebra are laid down. The with before, are his excellent Choice
worth of this Tract will pretty evi- of Problems, uncommon Skill in
dently appear, if we conſider that their Solution, and great Dexterity
Sir Iſaac Newton, formerly, when in ſome of their Conftructions : Al-
he was Profeſſor of Mathematicks fo his Rules for finding Diviſors to
at Cambridge, thought it not be compound Quantities ; - for redu-
neath his Pains to complete and a- cing radical Expreſſions to more fim-
dorn it, and add to the fame his ple ones, by the Extraction of Roots ;
Method of Infinite Series and Fluxi- --for exterminating unknown Quan-
ons, which he had almoſt prepared tities from two or more compound
for the Preſs; as we learn from Mr. Equations ; – for making Choice of
Collins's Letters to Mr. Borellus and ſuch and ſuch Lines, rather than
Mr. Vernon, to be found in the Com- others, in the Solution of geometri.
mercium Epiftolicum de varia re Ma cal Problems, to get the moſt ſimple
thematica, publiſhed by order of the Equation ; --- for finding the imagi-
Royal Society.
nary Roots of an Equation; - for
In Degraave's Courſe of Mathema- finding the Limits of the Roots ;-
ticks in Dutch, there is a pretty Piece for finding whether an Equation of
of Algebra. There is alſo Baker's four, fix, or more Dimenſions, may
Geometrical Key, containing the Con not be reduced : And his Method
ſtructions of cubick and biquadra- of applying Algebra to the Deſcrip-
tick Equations. - Mr. Ralphſon's U- tion of the Conick Sections through
niverſal Analyſis of Equations. - Rey- given Points, and to touch given
nau's Algebră, publiſhed anno 1707, Lines, are all what no one elſe
contained in his Analyſe demontré, a could ever give, and are perfectly
heavy, tedious Piece, though con. correſpondent to the Genius of that
raining ſome good Things.--7ones's wonderful Perſon.
There
+
i
A LI
ALL
There are many other Treatiſes ALIQUOT Part of a Number,
of Algebra, ſuch as Rohnane's, con- is ſuch an one as will exactly mea-
taining many good Things, all mere ſure it without any Remainder, as
Collections. Wolfius's, in his Ele. 2 is an aliquot Part of
4, 3 of 9,
ments of Mathematicks, s'Grave- 4 of 16, &c.
fande's, and others, too numerous to All the aliquot Parts of any Num-
mention, as well as unneceſſary. ber may be found by the following
You have alſo ſeveral little Diſcour- Rules: Divide the given Number
ſes here and there diſperſed in the by its leaſt Diviſor, and the Quo-
Philoſ. Tranſ. of London, Paris, Ber- tient by its leaſt Divifor, until you
lin, Peterſburgh, as ſo many At- get a Quotient that cannot be fur-
tempts to improve and bring Alge- ther divifible, and you will have all
bra to its utmoſt Perfection.
the prime Divifors, or aliquot Parts
ALGEBRA Numeral, is that which of that Number; then, if every two,
gives the Solution of arithmetical three, four, &c. of theſe Diviſors be
Problems, in Numbers only ; ſuch as multiplied into themſelves, the Pro-
that of Diophantus, Lucas de Burgo, ducts will be the ſeveral conjoined
Steifel, and others of the Ancients. Diviſors, or aliquot Parts of that
ÁLGEBR A. Specious, is that which Number. As ſuppoſe you want all
formed by the Letters of the Al- the aliquot Parts of 60, divide it by
phabet, firſt introduced by Vieta and 2, and the Quotient 30 by 2, and
Harriot ; and is far more general the Quotient is by 3, and there re-
than numerical Algebra, being no mains the indiviſible Quotient 5:
ways limited to any certain fort of Therefore all the prime aliquot
Problems: And no leſs uſeful in find- Parts are 1, 2, 2, 3, 5; and the com-
ing out any kind of Theorems, than pound ones from the Multiplication
in diſcovering the Solutions and De- of every 2, are 4, 6, 10, 15, and
monftrations of Problems; as may from that of every three, 12, 20, 30.
be ſeen in Treatiſes upon this Subject. In like manner, the aliquot Parts of
ALGEBRAICK Curve. See Curve. 360 will be found to be 1, 2, 3, 4,
ALGENE B, a fixed Star of the ſe- 5, 6, 8, 9, 10, 12, 15, 18, 20, 24,
cond Magnitude, on the right Side 30, 36, 40, 45, 60, 72, and 180 ;
of Perſeus.
for all the prime aliquot Parts are
ALGOL, a fixed Star of the third
1, 2, 2, 3, 3, 5; and thoſe from
Magnitude, alſo called Meduſa's the Multiplication of every 2 of
Head, in the Conſtellation Perſeus. theſe are 4, 6, 9, 10, 15 i thoſe
ACCORITHM, the four chief from every 3 are 8, 12, 18, 20,
Rules of Arithmetick, viz. Addi. 30, 45; thoſe from every 4; 24, ,
tion, Subtraction; Multiplication, 36, 40, 60, 90; and thoſe ftom e-
and Diviſion.
very 5; 72, 120, 180.
ALIDADA, an Arabick Name for ALLIGATION, one of the Rules
the Label or Ruler which is move-
move- in Arithmetick, being ſo called from
able about the Centre of an Aſtro- the Numbers being bound or con-
labe, Quadrant, &*c, and carries the nected together by circular Lines,
Sights of a Teleſcope.
relating to the Mixture of Corn,
ALIQUANT PART, is that Num- Wine, Sugar, Metals, or any other
ber which cannot meaſure any Num- Things of different Prices; ſhewing
ber exactly without ſome Remain- how to find ſuch Quantities of given
der, as my is an aliquant Part of 16; Prices, that when mixed, any given
for twice 7 wants 2 of 16, and three Quantity of the Mixture ſhall have
times 7 exceeds 16 by 5.,
a given intermediate Price. As ſup-
poſe
A L L
A L T
poſe a Mixture of 100 Pounds of Thoſe who have a mind to ſee more
Sugar, was required, which ſhould of the Rule of Alligation, with its
be worth 12 Pence a Pound, and Demonſtrations, may conſult Dr.
that Mixture was made up of four Wallis, Taquet in his Arithmetick,
forts of Sugar, at 6, 10, 15, and and particularly the ingenious Mr.
17 Pence per Pound; to find how Malcolm's Syſtem of Arithmetick.
much of each kind of Sugar is necef ALMAGEST, an Arabick Name
fary to that Compoſition.
of a Treatiſe of Aſtronomy written
The Rule is, place all the Prices by Ptolemy : As alſo of another
(except the main one) one under Piece upon the fame Subject by
another, and let every Number leſs Ricciolus.
than the mean one, be linked to ALMACANTOR,
is a Circle
one greater, then take the Dife of the Sphere paſſing thro' the
ference of each Number from the Centre of the Sun or a Star parallel
mean Price, and place this Dif- to the Horizon, being the fame as
ference againſt the Number it is a Parallel of Altitude. , Which fee.
linked to alternately : But every The Word is Arabick. Some call it
Number linked to more than one, Almicanter, and others Almucanter.
muſt have all the Differences of the ALMICANTER's Staff, an In-
Numbers it is linked to, ſet againſt ſtrument (of no great Account) for-
it. This done, as the Sum of all merly uſed by fome at Sea, being
the Differences is to the whole given made of Pear-Tree or Box, con-
Mixture, fo is any Difference to a taining an Arch of 15 Degrees,
fourth Number ; being the required ſerving to obſerve the Degrees of
Quantity of that Thing which Itands the Sun's Amplitude at Sea.
againſt that Difference. Thus in the ALMANACK, an Arabick Word
Cafe above.
for ſeveral annual Books, or Sheets
1
of Paper, publiſh'd under various
27
Names, with various Matters con-
6
54
tain'd. In moſt of which you
have
the Days of the Month, the E-
5 45
18
clipſes, the Age of the Moon,
Times of high Water, riſing and
16 144Pounds.
ſetting of the Sun, Feſtivals, &c.
that is, 27 Pound of that of 6 Pence, ALTERNATB Ratio,is the Ratio
of that of 15 Pence, 45 of that of Antecedent to Antecedent, as
of 10 Pence, and 18 of that of 17 Conſequent to Conſequent, in any
Pence.
Proportion. As if it be as A:B::
Note, as there may be ſeveral C:D, then will the Ratio of A to
Varieties of Linkings to the ſame C, equal to the Ratio of B to D be
given Prices, there will ariſe from alternate; ſo that this Sort of Ratio
the Rule ſo many ſeveral Anſwers only takes place when the Quan-
to the fame Queſtion. But in Re- tities in a Proportion are of the ſame
ality all the Queſtions within the kind.
Bounds of this Rule, are unlimited, ALTERNATE ANGLE. See
being capable of an Infinite Number ANGLE.
of Anſwers ; and the eaſieſt and ALTERNATION, of Quantities,
plaineſt way of reſolving all ſuch is the Number of ways that they
is by common Algebra, which any may be varied, changed,or differently
one of but a very ſlender Skill in the placed. As ſuppoſe the Quantities
fans, will eaſily know how to do. were a, b, c, &c. then will all their
Varictics,
و/ی
1 2
Pounds of each
10V
19)
2
54
!
2
A L T
A LT
Varieties of Order be a b c, acb, Number of Quantities, the Num.
bac, bca, cabecba, viz. 6:-And ber of Alternations will be
if n be the Number of Quantities, nx71 X-2*1*3&c. as above.
the Number of Alterations will be in like manner is deduced the ge-
nx n–1*1–2 x 1-3, &c.to nulla, neral Rule when the ſame Quantity
that is, it will be had by the conti-' is more than once repeated.
nual Multiplication of the Number ALTIMETRY, à Name given
of Things by the ſeveral natural by ſome to that Part of practical
Numbers gradually decreaſing from Geometry which thews how to
it to unity. As ſuppoſe it be re- meaſure the Heights of Objects ;
quired to find the Changes of 12 ſuch as Towers, Steeples, Hills,
Bells, the ſame will be 12 x 11X10 Clouds, &c. both acceſſible and in-
*9*8*7*6* 5 * 4 * 3 * 2 X I acceſſible,
479001600. But if the faine ALTITUDE or HEIGHT, of any
Quantity occurs feveral Times Point of a terreſtrial Object, is a
ſuppoſe it be repreſented by n, then Perpendicular let fall from
that Point
will the Number of Variations be to the Plane of the Horizon.
11—1XN-2 X 13 x 14, &c.
ALTITUDE INACCESSIBLE, of
an Object, is ſuch an one as cannot
Mo I XM-2Xm-3Xm-4, &c.
that is, continuing on the series, be approach'd by reaſon of fome
Impediment.
until the continual Subtraction of !
from n and m leaves o.
This may be found ſeveral ways;
the beſt and moſt uſual of which
This Rule is given by many Writers
are from two Stations on the hori.
in Algebra or Arithmetick; as Dr. zontal Plane,and by means of the Ba-
Wallis, Wolfius, Jones, Malcolm, &c.
and the Invention is inferred from and uſeful.
rometer : Both of which are pleaſant
an Induction of the ſubordinate par-
ticular Cafes, as when there are 2
Suppoſe it were required to
meaſure the Altitude or Height AB
Quantities a and b, they may of a ruinated Tower. To do this,
either be wrote ab or ba; ſo that I make choice of two Stations
the Number of Variations will be
When there are three
B
Quantities a, b, and c, it is evident
that one as c, may be combin'd firſt
with ab, and then with ba; ſo
that the Number of Variations or
Alternations will be 3 x 2x1=6.
Okolice for the
If there be four Quantities, every
one of them may be combined with
any Order of three of them, ſo that
the Number of the Alternations
will be 6*4=4 3 * 2 X=24.
So alſo if there be five Quantities,
every one of them join' with any D
А.
Order of four of thoſe Quantities, D and C, in the ſame right Line
produces 5 Variations ; wherefore with A, and whoſe Diſtance DC
the Number of all the Alternations is ſuch, that the Angle CB A be
will be 24 * 55* 4 X 3 X 2 X 1 not too ſmall, nor the Station C too
5120: And ſo generally if n be the near to AB; then I meaſure the
ftationary
22 XI.
lei
THE
METEPE
À LT
AL T
}
AM
Aationary Diſtance DC, and the
Angles BDA, BCA. This done
in the Triangle B DC, there are
given the Side DC, and the adjacent
Angles at D and C, to find the Side
CB ; and then in the right angled
Triangle CBA, the Hypotheneuſe
CB is given, and the Angle B A C to
find the perpendicular AB ; to which,
if the Height of the Eye be added,
you will have the true Height of
the Tower.
A little otherwiſe. When the fta-
tionary Diſtance CD is not in the
C С
E
B
B
D
Quadrants or other proper Inftru-
ments the Meaſures of the Angles
ABE, ADE, ACE. This done,
from the three given Angles, and
the given Stationary Diſtances BC,
CD, BD, the deſir'd Altitude E A
may be thus found. If A E be the
Radius, BE, CE, DE, will be the
D
А
I
c
1
S
H
C
fame Plane with the Altitude A B,
let the Angles ADC, ACD, be
not very acute and nearly equal.
Take the Quantity of the Angles
BDC, BCD,
as alſo of
the
Angle ACB; then as before in
the Trianglé CDB, there are given
T
the ſtationary Diſtance DC, and the
Angles at D and C, to find the Side
E
CB, and ſo in the right angled Tri-
angle ACB you have given the
B
F D
Angle ACB and the Hypotheneufe
CB, to find the perpendicular Co-Tangents of the given Angles of
AB.
Obſervation A BE, ACE, ADE,
The Altitude EA of an Object and ſo theſe Co-Tangents are given,
any how moving in the Air; as and the Ratios of BE, E C, ED, are
ſuppoſe of the Cloud A, may be given. Divide BD in the Point F
found from three Stations, B, C, D, in the given Ratio of the Co-Tan-
apon the horizontal Plane, after the gent of the Angle A B E, to the Co-
following Manner.
Tangent of A DE, and continuing
Let three Perſons at the Stations out BD, make as FG:FD::BF:
B, C, D, take at the ſame time with BF-FD, and from G deſcribe the
circulas
A L T.
A LT,
circular Arch FES: Alfo divide DC Proportion to the Semidiameter iof
in the Point H, in the given Ratio of the Earth, ſuch as, for Inſtance, is
the Co-Tangent of the Angle A DE that of the Moon, the Methods
to the Co-Tangent of the Angle aforeſaid will be ineffectual, becauſe
ACE, and continuing out DC, make the ſtationary Diſtance here being ſo
HI:HC::HD:HD-HC, and great to cauſe a due Difference in
from I deſcribe the Arch HET, the ſtationary Angles, (I ſpeak of
interſecting the former Arch in the the firſt Method) becomes the Arch
Point E. Then if right Lines B E, of a Circle of the Earth, inſtead of a
CE, DE be drawn, and with either right Line, as indeed is any ſtation-
of them as a Baſe, and with the cor- ary Diſtance ; but then when it is.
reſpondent Angle of Obſervation, you hort, the circular Arch, of which
make a right angled Triangle, the it is a Part, may be taken for a
perpendicular of that Triangle will right Line.
be the Altitude required.
The way of taking conſiderable
The Meaſures of the Lines CE, terreſtrial Altitudes, of which thoſe
or E D, may be computed thus, of Mountains are the greateſt, by
Draw the Lines EI, IG. In the means of the Barometer, is very
Triangle DIG, there are given two pretty and expeditious. This is
Sides DG, DI, and the included done by obſerving on the Top of
Angle IDG, viz. the Complement the Mountain, how many Inches
of the given Angle B DC to two and Parts of an Inch the Mercury
right Angles. Therefore find the has fell below what it was at the
third -Side GI, and the Angle Foot of the Mountain. When this
DIG; then in the Triangle GEI, is done, you will have its Altitude in
there are given the three Sides Engliſh Feet, by means of the
GE, EI, IG, to find the Angle Table of Mr. Cafini. (See Hift
. de
EIG. When you have the Meaſure l'Acad. Roy. 1703, and 1705,) which
of this Angle, take it from that of he founded upon very accurate Men-
the Angle DIG, and you will get furations of the Altitudes of ſeveral
the Angle EIC. This done in the Mountains.
Triangle EIC, there are given two There are other ways of mea-
Sides, EI, CI, and the included furing Altitudes by having given the
Angle EIC, to find the third Side Degrees
Degrees of Diſtance, that the ſame
E Č; after which in the right ang firſt becomes in fight (uſually at
led Triangle ACE, (Fig. 1.) right Sea,) and the Semidiameter of the
angled at E, you have given the Earth, of which, if you pleaſe, you
Baſe CE, and the Angle A CE, to may have an Account in Books of
find the perpendicular A E.
Geography, ſuch as Varenius at
Note, If B F be greater than FD, Chap. il. Part 1. You have alſo
or D H than HI; the Centres G, I, a pretty Diſcourſe in the Philofo-
muft be taken towards the Points phical Tranſactions, N. 405, by
B and D, and if BF be-FD, and Mr. Schutzer upon the Altitudes of
DH-HC; inſtead of the circular Mountains, and the Ways of find-
Arches FES, HET, you muſt ing them by the Barometer, where
draw right Lines from the Points you have Tables of three different
F, H, perpendicular to BD, CD, Perſons, viz. Mr. Marriotte, Caſini,
and their Interſection will give the and himſelf, for that purpoſe.
Point E, as before.
ALTITUDE, of a Cylinder or
Note, When the Height of an Ob- Priſm, is a perpendicular Line
ject is ſo great as to have a ſenſible drawn from one Bife to the other.
ALTITUDE
1
1
1
A LT
AL T
ALTITUDE of a Figurt, is the The finding the Meridian Alti-
Perpendicular, drawn from the tude of the Sun or Stars, is the Bafis
Vertex of the Baſe, as the right of all aſtronomical Obſervations, and
cannot be made with tão much
B
Care and Exactneſs. It is uſually
done with large Quadrants, Sex-
tants, &c. Some of the Ancients,
and Moderns too, have uſed high
Poles for this purpoſe. Ricciolus, in
A D
his Aſtron. Reform. fays, that Ulugh
Line BD, drawn from the Vertex about the Year 1437, uſed a Pole
Beigh, a King of Parthia and India,
B, of the Triangle ABC, perpen- above 180 Roman Feet high, and
dicular to the Baſe AC, is the Al- Mr. Caffini, in the Church of St.
titude of the Triangle.
Petronius at Bononia (in the Year
The heights of Figures muſt be
1655) another of 20 Feet.
known, in order to have their Areas
ALTITUDE Apparent, of any
and Solidities.
Point in the Heavens, is the Arch
ALTITUDE of the Sun, Star, Pla- of a vertical Circle contain'd be-
net, or any Point in the Heavens,
tween the ſenſible Horizon, and the
is an Arch of a vertical Circle, paf- vertical Circle, in which that Point
ſing thro' the Centres of the Sun, is. As let CD be the true Horizon,
Star, Planet or Point, contained be-
tween the Horizon and their
୧
Centres.
Theſe are found by large Qua-
drants, Sextants, or Gnomons. See
Hevelius Machin. Cæleft. Tom. 1.
De la Hire's Tab. Aftron. Bion on H
Mathem. Inftruments. Wolfius's Elem.
0
M
Aftron. and other Authors.
ALTITUDE Meridian of the
с
.D
Sun, Star or a Planet, is an Arch
of the Meridian, intercepted between
the Horizon and the Centre of the
and the ſenſible Horizon HO, a
Sun, Star, or Planet. As let H ZRN
vertical Circle DR, whoſe Centre is
C, the Centre of the Earth ; and let L
Z
be any point in the Heavens ; let H
be the Place of Obſervation, and
*S
LM an Arch of a Circle, drawn
thro' L about the Centre H; then is
LM the apparent Altitude of the
Н
R
Point L, which is always leſs than
the
TRUE Altitude, which is the
Arch QD of a vertical Circle,
whoſe Centre is the Centre C of the
N
Earth.
be the Meridian, HR the Horizon, The True Altitudes of the Sun,
and let there be a Star at S; then is fixed Stars and Planets, do differ but
RS the Meridian Altitude of that a very ſmall Matter from their ap-
Star.
parent Altitudes, by reaſon of their
great
1
1
ALT
A MB
great Diſtances from the Centre of may be found in Books of Afro-
the Earth, and the ſmallneſs of the nomy, amongſt which, fee Wolfius's
Semidiameter of the Earth, when Elem. Affron. N. 211, 212.
compared thereto. But the true and
ALTITUDE of the Eye, in Per-
apparent Altitudes of the Moon do fpective, is a right Line let fall
differ, and that about
52
Minutes. from the Eye, perpendicular to the
ALTITUDE of the Cone of the geometrical Plane, being the Point
Earth's Shadow, is found when the from whence the principal Ray pro-
Sun is at a mean Diſtance, by ſaying ceeds.
as the apparent Semidiameter of
AMBIENT, encompaſſing round
the Sun, viz. about 16 is to Radius, about. As the Bodies that are placed
ſo is the Semidiameter of the Earth, about any other Body, are called
to a fourth Proportional 214.8 Se- Ambient Bodies, and fometimes Cir-
midiameters of the Earth, which cum-ambient Bodies; and the whole
will be the Altitude fought for. Body of the Air, becauſe it encom-
But when the Earth is moft diftant paſſes all things on the face of the
from the Sun, its apparent Semi- Earth, is call'd the Ambient Air.
diameter will be 15' 50", and then AMBIGENAL HYPERBOLA, a
the Altitude of the Cone will be 217 Name given by Sir Iſaac Newton,
Semidiameters of the Earth.
in his Enumeratio Linearum Tertii
The Altitude of the Cone of the Ordinis, to one of the Triple Hy-
Earth's Shadow, is to that of the Sha-
dow of the Moon, as 10 to 28, which A
is the Ratio of the Diameter of the
Earth to that of the Moon.
E
ALTITUDE, or Elevation of the
Pole, is an Arch of the Meridian in-
tercepted between the Horizon and
either of the Poles of the World.
G
This is equal to the Latitude of
the Place, and may be found from
D
the Meridian Altitudes of the Pole-
Star, it being the Diſtance of theſe
F
Altitudes added to the leſſer Altir perbola's of the ſecond Order, hav-
tude, or elſe by Means of the Sun's ing one of it's infinite Legs falling
Altitude, and Declination.
within an Angle form’d by the A-
ALTITUDE OF Elevation of the ſymptotes, and the other falling
Equator, is the Arch of a Meri- without that Angle ; as let A C,
dian intercepted between the Ho- CD be two Aſymptotes, and EGF
rizon and the Equator, being always one of theſe Hyperbola's ; then if
equal to the Complement of the La- the infinite Leg GE falls within
titude.
the Angle A CD, and the Infinite
ALTITUDE of the nonagefimal De- Leg GF without that Angle, the
gree, is the Altitude of the nonage- ſaid Hyperbola is calld Ambigenal.
ſimal Degree, reckon'd from the AMBIT of any Figure in Geome-
Point at which it riſes : or it is the try, is the Line or Lines by which
Complement to a Quadrant of the the fame is bounded.
Diſtance of the nonagefimal Degree AMBLIGONAL, among the an-
from the Vertex of
any
Place. cient Geometricians, fignifies Obtul-
The manner of finding this at a angular ; as a Triangle or other
given Time, in a given Latitude, plain Figure, that has one obtuſe
Angle,
A M E
A M E
Angle, is ſaid to be Obtuf-angular. Iſland ; to whom they had given an
AMERICA, one of the four Parts, Account of their Voyage, and the
of the Earth, greater than the other country they had diſcovered. Go-
three. It lies in Length from South lumbus applied for Afiftance to-dir-
to North, under the Shape of two cover this Country to Alphonfus the
vaft Peninſulas, join'd together by 5th, King of Portugal, and Henry
the Streights of Panama, where the the 7th, King of England, who re-
Land is not above 17 Leagues from jected his Propofal, thinking, it a
Sea to Sea. It is bounded on the mere Dream. In the Year 1486,
Weſt by the Pacifick Ocean, on the he communicated his Deſign to fe-
Eaſt by the Atlantick Ocean, and veral Perſons of the Spaniſ Court,
on the South by the Streights of Ma- they too thought his Requeſt to be
gellan. But its northern Bounds are vain and extravagant; till at laft
not yet diſcovered, at leaſt beyond Alphonſus De Quintavile, Great
Davis's Streights, nor is it known Treaſurer of Spain, and Cardinal
whether it joins to the North Parts Gonzales de Mendofa, Archbiſhop
of Europe, or is feparated from them. of Toledo, making a favourable Re-
It's utmoſt ſouthern Bound is Capelation of his Affair to the King and
Horn, in the Latitude 57° 30: Queen of Spain, promiſed to aſſiſt
It's leaſt Diſtance from Afia is about him in it ſo ſoon as the War which
the Streight of Anian; from Groene the Spaniards had with the Moors
Land about Davis's Streight; and was ended. And accordingly he be-
from the ſouthern Land, about the gan his Voyage the third Day of
Streight of Magellan. And its neareſt Auguft, in the Year 1492, and on
Diſtance from the weſtern Part of the uth of Otober he diſcovered the
England, is about 950 Leagues ; the Iſland of Jamina, one of the Sugar-
ſouthern Parts thereof are Panta- Iſlands, afterwards called Cuba, and
gonia, and Brazil, belonging to the landed in the Iſland called the Spa-
Portugueze ; Peru, Mexico, Chili, &c. miſh Iſland.' Returning into Spain,
and many Iſlands to the Spaniards ; he was very well received, and made
and the more northerly Parts, as Ca- Admiral of all thoſe Seas. In the
rolina, New England, &c. belong to Year 1493, he went a ſecond time
the Engliſh. There are alſo innu- from Spain with 18 Sail, and found
merable Ilands belonging to it. out the Iſland of Deſire, all the
America was unknown to the An- northern Coaſt of the Iſland of Cuba,
tients ; the following ſhort Account the Iſlands of Jamaica, and Bori-
of the Diſcoveries of it, and its Parts, guen, and other ſmall neighbouring
tho' a little foreign to our Deſign, Illands. In the Year 1497, he made
take as follows.
another Voyage, in which he dif-
A Portugueze Veſſel, going to the covered the Gulph of Paria, about
Eaf-Indies, was by ſtreſs of Weather 450 Leagues off the Coaſt to Cape
drove upon the Coaſt of Ponant, and de Vela, and the Iſland of Cubaga,
the found her ſelf near this Country, famous for the great Quantities of
All the Crew periſhed through Hun- Pearls found therein. In the ſame
ger and Want, except one Pilot and Year Sebaſtian Cabot diſcovered
four Sailors, who being return'd New-England. In the Year 1499,
to a Port of the Iſland of Madeira, Pierre Alphonſo Nigno, a Spaniard,
full of Fatigue and Miſery, died in diſcovered the Countries of Cumana
a little tine after, at the Houſe of and Curiana. The ſame Year Diego
one Chriſtopher Columbus, a Genoeſe Lopez a Spaniard, diſcovered the
by Birth, who was a Sailor in that Coaft from the Mouth of the Ama.
Zons,
1
In 1534
and I
A ME
A MÊ
brons, to Cape St. Auguſtine. In the New Granada ; and Francis Pizarra,
Year 1500, Vincent Yanes Pinſen, a found out the Coaſt of Peru. In
Spaniard, diſcovered the ſeveral In- 1528, Ambroſe Dolphinger, diſco-
lets of the great River of the Ama- vered the coaſt of Venefæcla, in the
zons ; and Gafper Cortel Real, a Name of the Emperor Charles the
Portugueze, the Gulph of St. Lat. V. In 1531 Ferdinand Cortez, found
rence, and the Iſland of Terra Nova.
out Chiametlan, Xaliſco, Cineloa,
In the Year 1522, Columbus diſco- and Culiacan.
15359
vered the Coaſt from the River James Quartier of St. Malo, diſco-
Hiquras to Nombre de Dios, and the vered all the Coaſt of Canada,
Coaſt of Veragua. Alſo the ſame reaching from the Mouth of the
Year Roderic de Battidas a Spa- northern Inlet of the great River
niard, diſcovered 200 Leagues of the of Canada, to the River of Iroquois,
Coaſt from Cape de Vola to the and from the eaſtern Coaſt to the
Gulph of Uraba. In the Year Gulph de Chateaux, In 1535, Pierrà
1508, Diego Niqueſa, a Spaniard, de Mendoza, a Spaniard, found out
diſcovered about 90 Leagues of the a great part of the Inlets into Rio
Coaſt, from Nombre de Dios, to the de Plata : And Almagro found out
Rocks of Darien. In 1512, John the Coaſt of Chili. In the Year
Dias de Solis, a Portugueze, found 1538 Marké de Niza, a Spaniard,
out the Coaſt of Brazile, from Cape diſcovered the Coaſt of Cirola, and
St. Auguſtine, to the River De Plata. California. In 1541, Francis Val-
In the ſame Year, John Ponce de ques found out the Province of
Leon, found out the Coaſt of Florida. Quirini.
In 1513, the 25th of September, Vaſco The Hiſtorians of this Country
Nugnes de Vaſcoa, a Spaniard, diſ are very numerous, ſome of them
covered the South Sea, and after- are, Acojta, Hrrero, De Laet, Diaz,
wards the weſtern Coaſt of Golden Gage's Survey, Antonio de Solis's Ac-
Caftile. In 1517, Francis Har. count of the Conqueſt of Mexico,
mandies de Corduba, a Spaniard, Alexander Vrſina, Caſas, Conqueſta
found out the Coaſt of Jutican, and del Peru by Aguſtine de Zorata ;
John de Grailva the Coaſt of Ta- Vega's, Cieza's, and Acarete's De-
baſco, to St. John de Ulna ; alſo fcription of that Country, Seppe's,
Francis Garay, a Spaniard, diſco- and del Techo's Voyage to Paraguay,
vered the Coaſt from Florida to Pa Alonſo de Ovale's Hiſtory of Chili,'
In 1519, Francis Magellan, Ogilby's America, all relating to the
a Portugueze; diſcovered the Streight Spanish Poffeffions. And for the
of that Name. In 1520, Lucas Vafo French Settlements in North Ame-
ques a Spaniard diſcovered the Coaſt rica, you have Clamplain, Geuxius,
between Cape St. Helen and the Ri- and Mont, to Canada ; Fernand,
ver of Jourdan. In 1521, Ferdi- Soto, and Navraez, to Florida ;
nand Magellan found out the Iſlands de la Salle, de la Hotein, and Fa-
of Ladrones. In 1523 and 1524, ther Henepin's Travels into North
John Verazan, a Florentine, diſco- America. Alſo Newho's Deſcrip-
vered the Coaſt from Florida to the tion of Brazil, is well enough for
40th Degree of Latitude, in the the Dutch and Protugueze Acquifi-
Name of the King of France : The tions there. For the Britiſh Part,
fame Year Roderic de Battidas, a fee the Britiſh Empire in Ame-
Spaniard, found out the Country of rica, Smith's Account of the firſt
St. Martha. In 1525, Gonzales Engliſh Plantations in Virginia, Le-
Ximenes, a Spaniard, diſcovered derer's Diſcovery from Virginia to
D
nuco,
tre
A MI
A M P
1
។
I
1
the weſt of Carolina, the preſent Theorem relating to theſe Numbers.
State of Virginia, Accounts of the If you take the Number 2, or any
Diſcovery, and firſt Settlements in other produced from the Multipli-
New-England, New-York, Mary- cation of 2; provided itbe fuch, that
land, Penſilvania, Newfoundland, if i be taken from the Triple
Voyage to Darien, Dampier's thereof, the remainder be a prime
Voyages, Ligon's Hiſtory of Bar. Number, and alſo if i be taken
bados, Sloane's Hiſtory of Jamaica, from 6 times the ſame, the re-
Columbus's, Frobilher's, Sir Walter mainder be a prime Number ; I
Raleigh's, Cavendiſh's, Hudſon's, Da- ſay, that if that Number be ſuch,
vis's, Sparrey's, Monk's Voyages. and this prime Number be multi-
There are alſo many Voyages to plied by thrice the ſame, the Pro-
the South Sea, as Magellan's, Sebald duct will be one of the Amicable
de Weert's, Spilbergen's Corneilſon's, Numbers, and the other will be
Frezier's, Cook's, Wood's, &c. the Product of the firſt and ſecond
AMICABLE Numbers, are ſuch prime Numbers aforeſaid, multi-
that are mutually equal to the Sumplied by the Square of the Number
of one another's aliquot Parts, as are
firſt taken.
theſe Numbers 284, and 220. For It is eaſy to apprehend from the
all the aliquot Parts 1, 2, 4, 5, 10,
nature of theſe Numbers that there
II, 20, 22, 44, 55, 110 of 220, are but a very few of them, at
are equal to all the aliquot Parts leaſt to be ſet down and manageable
1, 2, 4, 71, 142 of the Number by us ; for 284, and 220, are the
284. Theſe two Numbers are alſo two leaft; and the two next greater
Amicable Numbers viz. 18416, are 18416, and 17296. Thoſe who
For the aliquot Parts are curious may find out the next
1, 2, 4, 8, 16, 23, 46, 47, 92, 94, Pair, for I neither know what they
184, 188, 368, 376, 752, 1081, are, or have any Inclination to do
2162, 4324, 8648 of 18416, are it.
equal to the aliquot Parts 1, 2, 4, AMMUNITION. A Name for
8, 16, 1151, 2302, 4604, 9208 of Powder and Ball, and other Im-
17296.
plements of War. Cannon, Mortars,
Van Schouten was the firſt who (I &c. are ſometimes alſo called by
believe) gave this Name to ſuch this Name. The Quantity of Am-
Numbers at Sect. 11. Miſcellan. at munition neceſſary for the Siege of
the end of his Exercitationes Geometr. a Place is ſhewn in the Chevalier de
where he ſhews how to find them Saint Julien's Treatiſe de la Forge
by common Algebra, bringing, out de Vulcain, p. 126, & feqq. where
the firſt Pair above mentioned, by he brings three Examples of his
fuppofing one of the Numbers to be own, ſpecifying particularly how
4%, and the other 4 yz, and making much Ammunition was brought to
an Equality between them and their the Sieges, and how much ſpent.
ſeveral aliquot Parts, and bringing But the Quantity neceſſary to de-
out the ſecond Pair above men fend a Place, you will find in Suirey
tion'd, by ſuppoſing one of the de Saint Remy's Memoires d'Artil-
Numbers to be 16x, and the other lerie, Part 4. p. 292, & feqq,
16yz, and making an Equality Tom. 1.
between them and their aliquot AMPHISCII, Are the Inhabi-
Parts.
tants of the Torrid Zone, which
In the fame Section he tells us, are thus called, becauſe the Shadow
that Deſcartes gives the following of the Sun at Noon, falls at one time
of
1
>
ATM'P
AN A'
of the Year towards the North, and Eaſt or Weſt Point thereof, and the
at 'the other, towards the South., Centre of the Sun, or a Planet or
And when the Parallel that the Sun Star at its riſing or ſetting. As let
moves in, is'equal to the Latitude of HR be the Horizon, AĞ the E-
the Place, and on the ſame fide the quinoctial, Of the true Eaſt or Weft
Equator, the Moon's Shadow falls Point of the Horizon, and S the
neither North nor South. See Va- Centre of the Sun or a Star at its
renius's Geogr. gener. Chap. 27. riſing or ſetting; then the Archos
Prop. 3.
of the Horizon is the Amplirade,
AMPHİPROSTILE, in Architec- which is either ortive er occafive,
ture, is à fort of a Temple of the northern or fouthern..
Ancients, having four Columns in As the Cofine of Latitude : Ra-
the Front, and the fame Number in dius : : Sine of the Sun's or Star's
the hinder Face. Vitruvius gives Declination : Sine of Amplitude.
the Deſcription, C. 1. Lib. 3.
It is of uſe in Navigation,' to find
AmphitheatreA very large the Variation of the Compaſs. See
Building of the Ancients either more in Wolfius's Elemen.' Aftron. §.
round or ovalar, having a Pit, and 196. and his Geogr. §. 299. in the
a great number of riſing Seats fournal des Obſervations Phifiques,
within it, whereon the People uſed Mathematiques & Botaniques, made
to fit to fee barbarous Shews, as the in America from the Year 1707 to
Combats of Gladiators, of wild 1712, by Father Feuillée, at the
Beaſts, &c. There were ſeveral of Command of the King of France.
them, as the Amphitheatre of Ver See alſo Dechales's Mundus Mathe.
pafian, vulgarly called the Coliſeum maticus, Lib. 7. de Navigatione.
at Rome, the Amphitheatre of Tom. 3. Fol. 335. & feqq.
Flavius, the Amphitheatre of Sta AMPLITUDE MAGNETICAL,
tilus Taurus, a Friend of Auguſtus's is an Arch of the Horizon contain'd
at Rome, the Soldiers Amphitheatre between the Centre of the Sun at
at Rome, the Amphitheatre at Ve his riſing or ſetting, and the Eaft
rona, and the Amphitheatre at Niſmes or Weſt Point of the Compaſs. It
in Languedoc in France ; the Re- is found by an Amplitude or azi-
mains of all which are ſtill to be much Compaſs, by obſerving the
feen. See Deſgodetz in his Edifices Sun at his riſing or ſetting, and is
Antiques de Rome. Overbeke's Re- always equal to the Difference be-
fi quia antiquæ urbis Romæ. Mont tween the true Amplitude, and the
faucon's Antiquities. Trattato degli Variation of the Compaſs.
Amfiteatri del Marchefe Scipione
ANABI BA 2ON.
The northern
Maffei
. Fontana del Amfiteatri Node of the Moon is ſometimes fo
Flavio.
called.
AMPLITUDE, is an Arch of the ANACAMPTICKS. A Name given
by the Ancients to that part of
Opticks which treats of Reflexion,
being the ſame which we now call
H Н
Catoptricks.
ANACHRONISM.
SR
A Miftake in
Chronology.
ANACLATICKS. An ancient
G
Name for that Part of Opticks
which treats of Refraction, being
Horizon, intercepted between the the ſame we now-a-days call Diop-
cricks.
A
D2
1
Α Ν Α
1
A NA
tricks. See the Compendium of Am- until at length we come to ſome
broſius Rhodius, a mathematical Pro- known or evident Truths, or ſome
feffor at Wittenburg, in Lib. 3. Op Impoſſibility, of which the firſt Pro-
ticæ, p. 384. & feqq.
poſition is a neceſſary Conſequence,
ANALEMMA. An aſtronomical thereby to conclude the Truth or.
Inſtrument, being a circular Plate Impoſſibility of that Propoſition,
either of Braſs or Wood, containing which may afterwards be demon-
a Projection of the Circles of the ſtrated by Compofition, from the Re
Sphere, from an Eye placed at an aſſumption of the Reaſonings where
infinite Diſtance in that Diameter by it was performed and finiſhed.
of the Sphere which paſſes thro' the The Analyſis of the ancient Geo-
Eaſt and Weſt Points of the Ho- metricians, which may be called
rizon, wherein the Solftitial Colure, Geometrical Analyſis, .conſiſted in a
and all Circles parallel to it, will judicious Application of the Propo-
be concentrick Circles. All Circles ſitions of ſeveral Books, (ſuch as
oblique to the Eye will be Ellipſes, Euclid's Data, Apollonius de séati-
and all Circles whoſe Planes paſs one Rationis, de Sectione Spatii, de
thro' the Eye, will be right Lines. Tactionibus, de Inclinationibus, de
The uſe of this Inſtrument is to fhew Locis Planis, de Sectionibus Conicis.
the common aſtronomical Problems, Ariſleus de Locis Solidis. Euclid de
whịch it will do very eaſily, but not Locis ad Superficiem, Eratoſthenes de
over and above exact, unleſs it be Medietatibus ; Euclid's Poriſms ; and
very large.
other Books, to the Number of 31,
The Inſtrument is very ancient, as we learn from Lib. 7. of Pappus's
being handled ſo long ago as by Pta Collectiones Mathematica) proceed-
lemy himſelf in a peculier Treatiſe, ing Step by Step from one known
which was afterwards publiſhed Truth to another, till they arrived
with a Commentary upon it by at laſt to that required. Examples
Frederick Commandine. The beſt of which may be ſeen at Prop. 107,
Treatiſe (at leaſt of the Conſtruc- 117, 155, 204, 205, of the faid 7th
tion) of this Inſtrument, is in Agu- Book of 'Pappus. The Ingenious
lonius's Opticks, Lib. 6. See alſo Hugo d'Omerique too, in his Analyſis
Taquet's Optic. Lib. 3. c. 7 f. 208 Geometrica, has endeavoured to‘re-
Witty, in his Treatiſe of the Sphere, ſtore this Analyſis of the Ancients ;
Harris's Lexicon, under the where he has ſet an Example wor-
Word Analemma, and Dechales, Lib. thy the Imitation of all thoſe who
2. de Aſtrolabiis f. 127, & feqq. have at heart the true and genuine
Tom. 4. Mundi Mathem.
Way of ſolving Geometrical Pro-
ANALOGY.. The ſame as Pro- blems, tho' it muſt be confeſs'd, that
portion ; which fee.
Algebra, which may be called an
ANALYSIS. This properly is a Arithmetical Analyſis, is the moſt
Reſolution of any thing into its ready, and general Method, (but not
component Principles, or taking always the ſhorteſt and moſt elegant)
its Parts all to pieces, in order to that has been hitherto found out, or
diſcover the thing. And in Mathe- perhaps ever will, for this purpoſe.
inaticks it is the Art of diſcovering ANALYSIS of Infinites, the ſame
the Truth or Falfhood of a Propo- with Fluxions ; which ſee.
ſition, or its Poſibility and Impoſſi ANALYSIS of Powers. The ex-
bility, by ſuppofing the Propoſition traction of Roots, or Refolution of
ſuch as it is, that is, true ; and ex- Powers.
mining what follows from thence, ANALYSIS of Situation. A Bra.
gadocia
C. I.
1
1
She is
100.
A NE
ANG
çadocio Term of Wolfius in his Elem. Houſe, St. James's, my Lord Go-
Mathem. attributed to Mr. Leibnitz, dolphin's, &c. See a more parti.
where he ſays that this latter would cular Deſcription by Vitruvius, Ca-
give the Solutions of Problems by fatus in his Mechanicks, Lib. 5. Cap.
it after a manner quite different' 9. and Ozanam in his Mathem.
from what has been hitherto known Diktionary.
or thought of. But alas ! the Dream Wolfius, in his Mathem. Lexicon,
has not been yet diſcovered to the ſpeaks of an Anemoſcope, conſiſting
Publick,
of a little wooden Man, which by
ANDROMEDA. A ſmall northern its riſing and falling in a Glaſs
Conſtellation, conſiſting of 27 Stars Tube, Thews the Change of the
viſible to the naked Eye, behind Weather, and the Alteration in
Pegaſus, Caſiopeia, and Perſeus. the Gravity of the Air, which was
She repreſents a Woman chained the Invention of Otto Guerick, who
to a Rock. The Poets have many mentions it in Lib. 3. Experimene
Fictions concerning her.
torum Magdeburg. c. 20. f.
called Mulier: Catenata, Perfea, Vir. But makes a Secret of it, which he
go Devota, and by fome Vitulus Ma- would not diſcover. But at length
rinüs Catenatus. Shiller makes her Mr. Comiers Profeſſor of Mathem.
the holy Sepulchre. Harſdorff, will at Ambrun, has diſcovered it in the
have her to be Abigail, 1.Sam. XXX. AEta Eruditorum, Anno. 1684.p. 26,
5. and Weigel changes her into the & feqq. where he would have the
Arms of Heidelburgh.
Homunculus to be moved up and
ANEMOMETER. An Inſtrument down by the riſing and falling of the
to meaſure the force of the Wind, Mercury in the Barometer.
invented by Wolfius in the Year ANGLE, is the Inclination of
1708, and firſt publiſhed by him, two Lines meeting one another. As
Anno 1709, in his Areometry. As let the Line A B, meet the Line
alſo in the Aeta Eruditorum for
1709, and in the Areometry belong-
A
ing to his Elem. Matheſeos : And in
his Mathematical Diktionary. He
fays, he tried the goodneſs thereof;
and tells you that the inward Struc-
ture thereof may be preſerved, even
B В
C
to meaſure the Force of running
Water, or that of Men and Horſes CB in the Point B : Then is their
when they draw.
Inclination or bending towards each
ANEMOSCOPE.
An Inſtrument other, an Angle.
ſhewing at any time which way the Angles are of vaft uſe, not only
Wind blows, that is, from which in Geometry, but almoſt in all
of the 32 Points of the Compaſs it other parts of Mathematicks. The
comes, by means of an Hand or In- nature of Figures cannot be ex-
dex moving about an upright Cir- plained without them. They are
cular Plate; which Index is turn'd half the Subject of Trigonometry,
about by an Horizontal Axis, which and have much to do in Geography
Horizontal Axis is turned about by and Aſtronomy.
an upright Staff, at the top of ANGLE ACUTE, is the Angle
which is the Fane, moved about by ABC, being leſs than a right Angle.
the Wind. Theſe are very common ANGLES ADJACENT, are ſuch
about London, as at Buckingham- that have the ſame Vertex, and one
coninon
:
D 3
ANG
ANG
true
common Side continued out, as the Line to a Superficies, or a Superficics,
Angles ABC, CBD are Adjacent to a Solid ; and that becauſe if it be
never ſo often multiplied, it will
never be equal to, or exceed any.
the leaſt right-lin'd Angle.
But
Dr. Wallis, in a Diſcourſe of the
B
D
Angle of Contact, pabliſhed in
A
his Arithmetick of Infinites, does
Angles, and both taken together, wrongly (as I think) with Pele-
are always equal to two right tarius, ſay it is no Angle at all,
Angles, (13. 1. Eucl.) And if the one Taquet, in hi. Euclid too, at Schol.
be acute, the other will be obtuſe, Prop. 16. Lib. 3 (where he gives
and contrariwiſe.
us Paradoxes about the Angle of
ANGLE of Contact, is the Angle Contact) will not have any Angle
which a right Line that touches a whatſoever to be a Quantity. But
Curve Line makes with it. As let a Mode or Quality only, and ſo ac-
the right Line A B, touch the Circle cording to him the Compariſon of
X in the Point D: Then is the Angles is not as to Equality and In-
equality, but Likeneſs and Unlike-
А.
B neſs. But alas this is a mere Fetch
D
to anſwer his purpoſe : A falſe
Strain to account for his Difficulties.
Angles of Contact are
X
Angles, and may be compared to
one another, tho' they cannot to
right-lin'd Angles; they being in.
finitely ſmaller than theſe ; for the
circular Angles ADF, AD G, of
Angle ADX which the right Line Contact, are to each other in the
A Ķ makes at the Point D of Con- reciprocal fub-duplicate Ratio of
the Diameters DC, DE. And if in-
tact, an Angle of Contact.
Euclid, in Prop. 16. Lib. 3. has
А
B
demonſtrated an admirable thing
regarding a circular Angle of Con-
tact, viz. That it is leſs than any
given right-lin'd Angle. And this
has given riſe to many Diſputes a-
E
mongſt the Geometricians about how
it ſhould happen, and to many fur-
prizing Paradoxes. To account for
C
which, they have involved them-
felves into much Abſurdity and ſtead of Circles, the Curves had been
Error. The good old Clavius, and Parabolas, and the Point of Contact
Peletarius, a Profeſſor of Mathe- D, the "Vertex of their Axes ; the
maticks in France, had a long Dif- Angles of Contact would have been
pute about it, as you may fee in Lib. then reciprocally in the ſub-dupli-
3. of his Euclid, where the former cate Ratio of their Parameters.
afferts, and indeed rightly too, that But in ſuch elliptical and hyper-
an Angle of Contaci, is of a different bolical Angles of Contact, theſe
kind from a right lind Angle, will be reciprocally in the ſub-du-
having the fame regard to it, as a plicate Ratio of the Ratio com-
pounded
1
+
3
A
3,
B P
A
ANG
A N G
pounded of the Ratios of the Para-
thefemi-cubical
meters, and tranſverſe Axes.
Parabola MA
Hence a circular Angle of Con-
Q_expreffed by
tact may be divided into any Num-
the Equation
M
ber of equal Parts by circular
yy33, is, in-
Arches, or into any given Ratio.
finitely greater
IF GBD be a common Parabola,
Р
than any cir-
and EF a Tangent to the Vertex
cular Angle of
at B, and ABC be a cubical Para-
Contact which
bola, which E F touches in B, that
is of the fame
Q
is, if the Abſciſs B P be called and
kind with a pa-
the rectangular Ordinates PQ, PR,
rabolical one.
be called y; if I x y be=xx, in the This wonderful and almoſt in-
comprehenſible Doctrine, was firſt
G
advanced by Sir Iſaac Newton, in
D
Schol. Lem. 2. at the beginning of Lib.
1. Princip. Mathem. as alſo in his
Treatiſe of Fluxions ; but without
any ſort of Proof or Demonſtration,
R
The whole I believe depends upon
Q. F theſe two things, that thoſe Angles
of Contact are infinitely greater
than others, when any one evaneſ-
cent or infinitely ſmall Subſtance of
common Parabola, and 1 xy=x3 the former, is infinitely greater than
in the cubick one, and if other pa- any one of the latter. And when
rabolical Curves were deſcribed to * the Abſciſs of any Curve becomes
the Abfcifs or Tangent EF, being infinitely ſmall, x, **, **, **, &c.
' ſuch that y=x+, y=x5, y=, &c. and x1, xả, x, &c. will be a
then will
the parabolical Angle of Series of Quantities decreaſing,
Contact RBP be infinitely greater whereof any one of the former, will
than the cubical parabolical Angle of be infinitely greater than that next
Contact QBP, and this here Angle following it. And x, x1, xs, xi,
of Contact infinitely greater than
that of the Curve, whole Equation is &c. **, **, xả, &c. will be a Series
y=x4, and that of this latter Curve increaſing in the ſame manner.
infinitely greater than that of the
ANGLE CURV'D LINE, is the
Curve, whoſe Equation is y=xS,
mutual Inclination of two Curve
and ſo on ad infinitum. And more-
Lines, meeting in one Point, in the
over,
between
the Angles of fame Plane, as the Angle ACB
contain'd under the two Curves
Contact of any two of this kind, contain'd under
may other Angles of Contact be BC, AC in the fame Plane meeting
found ad infinitum, that will infi-
B
in the Point
nitely exceed each other, and yet the
C, is a Curve-
greateſt of them are infinitely leſs
Under this
than any the leaſt right-lin'd Angle ;
ſo alſo y2=X3,33 =*4,34x5,&c.
Denomination
denote a Series of Curves, of which
are contain'd
the Curv'd-
every ſucceeding one makes an Angle
with its Abſcils, infinitely greater
D
line Angles,
than the preceding one, where it
Plane from the
may be obſerved that the Angle of
Contact MAP, at the Cuſp A
itereographi-
D 4
ol
!
lind Angle.
made upon a
E
A N G
A N G
cal Projection of the Circles of the Angles CONTIGUOUS, Sec
Sphere, which may all be meaſured Angles adjacent.
by Rules laid down in Treatiſes of ANGLES EQUAL
are ſuch
that Projection.
(right-lin'd ones) where the Arches
If AC, BC, be circular Arches of Circles deſcribed from the an-,
whoſe Radius's are the equal right gular Points, and intercepted be-
Lines DC, EC ; the right-lin'd tween the Sides, are proportional
Angle DCE will be equal to the to their reſpective Radius's : As let
cury'd Line Angle ACB; for ſince the Arches AC, FD of Circles
the Angle DCB is =E CA. If deſcribed from the Centres or an-
from each be taken the common
Angle DCE, there will remain the
A
D
right Line Angle DCE, equal to
the curv'd Line Angle A CB.
E
"ANGLE OF ELEVATION, in Me-
B
chanicks, is the Angle which the
C
F
Line of Direction of a Body (uſually gular Points BE, be proportional
a Ball) projected with any force,
to the Radii BC, DF ; then are
makes with an Horizontal Line, the Angles ABC, D E F, equal.
Gallileus, in his Dialogues of Motion,
ANGLES EQUAL (SOLID,) are
was the firſt who has ſhewn that this ſuch as are contained under equal
Angle muſt be 45 Degrees, to cauſe Numbers of equal Plane Angles.
the projected Body to go to the
ANGLE OF EMERGENCE. That
greateſt Diſtance or Range poſſible, which any Body (moſt commonly
with the fame force, and that at E a Ray of Light) projected from one
levations as much above 45 De Fluid or Medium (as Air) into ano-
grees, as under it, will fly to the ther, makes at its going out of the
fame Diſtance. And on the con-
latter Fluid or Medium (as Water
trary, when the Line of Direction or Glaſs, whoſe Surfaces are parallel
is parallel to the Horizon, the Planes, with a Perpendicular to
Range will be the ſhorteſt poſible. thoſe Planes ; as let AB, CB be
This is demonſtrated by Dr. Keil, parallel Planes bounding Water or
in his Introduction to true Philoſophy : Glaſs, and ſuppoſing a Body pro-
by Mr. Cotes, in his Harmon. Men- jected in the Direction F E, entering
Jirarum; by Wolfius, in his Me- into theſe at E, and goes out at G,
chanicks, and many other Authors.
K, H
But it is grounded upon a Suppoſiti- F
on, that the Projectile ſuffers no Re-
ſiſtance from the Air it moves thro',
А
G B
which it really does; and this cauſes E
the fartheſt horizontal Range not
-B
to happen from an Angle of Eleva- C G
H
tion of 45 Degrees, but from an
K K
Angle ſoinewhat leſs, (See Euler's in the Direction GH. Let G K be
Book de Motu) and that all the perpendicular to A B, CD, then is
Ranges under 45 Degrees of Eleva- the Angle KG H an Angle of E-
tion, are a little greater than when
mergence.
the Elevation are equally diflant
The Sine of the Angle of Emer-
from the 45th Degree above it.
gence, when the projected Body
paſſes
GB
VG
A N G
ANG
paſſes quite thro’ the Medium, is to reflected at the Point of Incidence,
that of Incidence, in a conſtant Ra- without entering at all into the
tio. But when the projected Body Medium.
fies back or out of the Medium, The two Propoſitions aforeſaid,
the ſame way it came in, without are inveſtigated or proved after other
paſſing quite thro'; the Angle of ways by Authors ; particularly
Emergence, will always be equal to Dr. Barrow, in his Lectiones Optice,
the Angle of Incidence, which is deduces them from a very remote
the caſe of Cannon Balls, ſhot ob. Conſideration, where he would
liquely into the Water, or even have a Ray of Light to be of a
light Earth, or flattiſh Stones that cylindrical-Figure, or rather right-
Boys throw into the Water to make angled Parallelogram,and to revolve
Ducks and Drakes, as they fay : about upon its coming to touch
All of which will come out again, the Surface ſeparating the Me-
and perhaps ſeveral times, according diums. See Le£t. 2. Others, as Mr.
to their Velocity, Figure, and the Jones in his Synopſis Palmariorum,
Obliquity of the Incidence.
and Wolfius, in his Dioptricks and
Sir Tjaac Newton, Seet 14. Lib. 1. Catoptricks, ſhew how this happens
Princip. Mathem. Philos. Natur. has by inquiring after a point in the Sur-
given moft ingenious Demonftra- face ſeparating the Mediums, ſuch,
tions of theſe two uſeful and funda- that the time a Ray paſſes from one
mental Propoſitions, by conſidering given Point in one Medium to
the immerging Mediums to conſiſt another given one in the other Me-
of Particles that uniformly attract dium, ſhall be a Minimum (to ſhew
the immerged Body in its Paſſage, the conſtant Ratio of the Sines of the
and from thence concludes the Line Angles of Emergence and Incidence)
EG, thro' which it paſſes, to be a and that the Aggregate of the
Parabola ; and then ſhews how Lines drawn from one given Point
the firſt Propoſition follows from in the upper Medium to another in
the Nature of the Parabola. And the ſame, ſhall be a Minimum (to
as to the ſecond, he gathers that ſhew the equality of the Angles of
from a Propofition founded upon Reflexion and Incidence.) See in
the firſt, viz. That the Velocity of Wolfius $. 35. Diop. and 24 Catop.
the Body before the Incidence, is to See alſo Dr. Gregory's Opticks, &c.
that after it is emerged, as the Sine Sir Iſaac Newton, in his Opticks,
of the Angle of Emergence is to Prop. 6. Part 1. has ſhewn the truth
that of Incidence; and from this he of the conſtant Ratio of the Sines
deduces the ſaid Propofition, to of the Angles of Incidence and
gether with the following one; that Emergence ; and Dr. Keil, in his
the Motion before the Emergen- Vera Phyfica the truth of the Equa-
cy, muſt be greater than that af- lity of the Angles of Incidence and
ter it, to cauſe the Body to be re- Reflexion ; both after a different
flected.
manner than thoſe hinted at before,
Hence the famous catoptrical by the Reſolution of the Motion
Propoſition, that the Angle of In- of the Body into two, the one pa-
cidence is equal to the Angle of rallel, and the other perpendicular
Reflexion, follows as a Corollary, to the Surface of the Fluid or Me-
viz. by ſuppoſing the Depth or dium, &c. See Mr. Graveſande
Way of the Emergence to become alfo, in his Infiitutiones Philos.
infinitely ſmall, or the Body to be Newtonian. Lib. z. Part 2. C. 6.
ANGLE
ANG
A N G
ANGLE OF INCIDENCE; is that that Ray refracted. Alſo let N H be
which the Line of Direction of a the Continuation of the Ray of In-
ftriking Body (as a Ray of Light, cidence, then is 'FNH the Angle
&c.) makes at the 'Point where it of Refraction.
firſt touches or comes at the Body ANGLB REFRACTED, is the
it ſtrikes againſt, with the Perpendi- Angle which a refracted Ray makes
cular to the Surface of the Body it with a Perpendicular to the refract-
ftrikes againſt.
ing Surface ; as let GN be perpen-
ANGLE OF REFLEXION, is that dicular to the reſracting Surface
which is made by the Line of di- DE, then is GNF the refracted
rection of a Body rebounding after it Angle.
has ſtruck againft another Body, at 1. The Ratio of the Sine of the
the Point of contact, from whence Angle of Incidence to the Sine of
with a Perpendicular at that point the refracted Angle, is found to be
of Contact it rebounded : as let a invariable.
invariable. If the Refraction be
Body moving in the Direction AB, from Air into Glaſs, it will be
ftrike againſt the Surface D E in the greater than 114 to 76, but leſs
than 115 to 76 ; that is, nearly as
C
3 to 2, as Mr. Huygens has ſhewn
in his Dioptricks, p. 5. Sir Iſaac
Newton too, in his Opticks, Part
3. Lib. 2. agrees with Mr. Huygens,
viz. That the Ratio of the Sine of
the Angle of Incidence, is to that
D
B
of the refracted Angle as 31 to 20,
Point B, and by that means be that is, nearly as 3 to 2, which is a
reflected or driven back again in very proper Ratio to explain the
the Direction 'BC, and let 'BF be Refraction in Glaſs Lens's.
perpendicular to DE; then is ABF 2. Deſcartes in Trat. de Meteoris,
the Angle of Incidence, and FBC C. 8. $. 10. p. m. 222. found that
in Rain Water the Ratio
the Angle of Reflexion; and upon
the Equality of theſe two Angles, boveſaid was as 250 to 187, or
the whole Science of Catoptricks is nearly, as 4 to 3; to whom Sir
entirely founded.
Ifaac Newton, in his Opticks, agrees ;
ANGLE of RefracȚION, in where he ſays, it is as 529 to 396,
Dioptricks, is the Angle which a but in Spirit of Wine he makes it
as 100 to 73
Ray of Light refracted makes with a
Ray of Incidence, continued out be-
3. If one Angle of Incidence be
yond the refracting Superficies. As given, and the correſpondent re-
let D E be the refracting Superficies, fracted Angles be obſervd by Ex-
MN a Ray of Incidence, and NF periment, it will be eaſy to compute
the refracted Angles anſwering to
M
every Angle of Incidence. Kircher
(in Arte Magna Lucis et Umbre,
D
E
Lib. 8. Part 1. c. 2) and Zahan
(in Oculo Artific. Fund. 2. Syn, 1.
c. 2. f. 228, & ſeq.) ſay, when the
Angle of Incidence is 70°, they found
G.
the refracted Angle to be 38° 50'.
H
When a Ray moves out of Air into
Glaſs, or out of a rarc Medium
info
a-
A NG
A NG
into a denſer one, the refracted not arrive at the preciſe Truth.
Angle is always leſs than the
6. There are feveral ways of ob-
Angle of Incidence'; and when the ſerving the Quantity or Law of Re-
Angle of Incidence is nothing, the fraction (to be found here and there
refracted Angle will be fo too.
in Authors) whereof the following
5. If the Angle of Incidence be one is eaſy ;, ſuppoſe it be from
teſs than 20°, and a Ray moves out Air to Glafs, being that which is
of Air into Glaſs, the refracted moſtly wanted in Dioptricks. Let
Angle will be nearly one third Part FGBC be a well poliſhed Glaſs
of the Angle of Incidence, and this Cube, ſtanding upon a Plane Board
is the Principle that Kepler, and NIPO ; at the end of which
after him moſt other Writers of Op- there is another NABI fix'd at
ticks have uſed to explain the Re- right Angles, having the ſame
fractions in Glaſs ; for imitating height CH with the ſide of the
Alhazen, , and Vitellio, they fought Cube, and ſuppoſe their common
after the Law of Refraction in the breadth I N, to be greater than the
Ratio of the Angles, and ſo could fide I H of the Cube, and the length
B
C
12
***
A
D
*
P
G
I
K
L
1
TAIDI!
M: 2017!
kann!
N
O
M
ON, to be much longer than either; uſe a little Voffel of Water or other
then when theſe Boards and the Liquors, you may obſerve the
Cube upon them cloſe to the up- Law of Refraction in Water or
right one, be turned to the Sun at other Liquors.
different Latitudes above the Ho-
7. The firſt Invention of this fa-
rizon, note the end of the Shadow mous Dioptrick Theorem of the con-
of the fide A B, both within the ſtant Ratio of the Sines of the Angles
Cube at K and without it at L ; then of Incidence and refracted Angles,
fince CK is the refracted Ray, and . upon which the whole Science de-
CL the unrefracted one, HCK will pends, is commonly attributed to
be the refracted Angle,and H CL the Deſcartes, (ſee his Dioptr. c. 2.9. 2. P.
Angle of Incidence ; ſo that if m. 57.) tho' it was well known to
CL be the Radius, HL will Willebrord Snell. (See Huygens's
be the Sine of the Angle of Dioptr. p. 2, and 3.) And Vofius de
Incidence, and HK that of the Natura & Propri, Lucis, p. 36. pub-
refracted Angle ; ſo that if HK liſhed anno 1642, wherein this laſt
and HL be carefully meaſured by ſays it appears from Snell's Papers,
an exact Scale of equal Parts, you which he himſelf had ſeen, that
will have in Numbers the Ratio of Snell had found out that the Pro-
the Sine of the Angle of Incidence, portion between the Seçants of the
to that of the refracted Angle, and Angles, which are the Comple-
if inſtead of a Cube of Glaſs, you ments of the Angle of Incidence,
and
t
A N G
A N G
1
and the refracted Angle to right ones ftances,appears equal,bigger or leffer
is conſtantly the ſame. Kepler alſo than others, as may be ſeen in optical
was very near finding out this Writers ; among which, ſợe Wol-
Theorem, who at Prop. 5, 6. in his fius Cap. 5. Elem. Optic. The an-
Tra&t. called Paralipom. in Vitel. cient Opticians, as Euclid, Ptolemy,
lionem, lays down theſe Secants for Alhazen and Vitellio, formerly uſed
the reſpective Meaſure of Refrac- theſe Angles to , explain how one
tions.
Thing or Object appears great or
ANĠLE MIXT-LIN'D, is that con- ſmall.
tain'd under a right Line and a ANGLE OF COMMUNICATION,
curve Line, as the Angle A BC. in Aftronomy, is the difference
between the true Place of the Sun,
A
ſeen from the Earth, and the Place
of the Planet, when reduced to the
Ecliptick ; as let TF be the Orbit
of the Earth, TAPG the Orbit
B.
of a Planet P, and S the Sun; let B
C C
ANGLE OBLIQUE, is one leſs or
P
greater than a right Angle.
ANGLE OBTUSE, is one that is
B
T
greater than a right Angle, as the
Angle A.
S
G
JA
A
be the Place of the Planet reduced
ANGLE OPTICKOR OF VISION,
is the Angle ABC, which two TS B the Angle of Commutation.
to the Ecliptick ; then is the Angle
Rays A B, CB, iſſuing from the ex-
treme Points A, C of an Object
, mutation, to the
Sine of the Angle of
As the Sine of the Angle of Com-
form at the Centre of the Eye.
Elongation, ſo is the Tangent of the
heliocentrick Latitude of a Planet to
A
that of its geocentrick Latitude.
B
ANGLE RIGHT, is that which is
made by two right
Lines perpendicu-
C
lar to each other,
as the Angle A.
The apparent Magnitude of an A
This always is e-
Object is meaſured by this Angle.
qual to that con-
Thoſe things which are ſeen under tiguous to it, and the meaſure thereof
a greater Angle, appear to be is 90 Degrees.
greater, and thoſe under a leffer, ANGLE PLANE, is the mutual
to be leſs ; and thoſe under an equal Inclination of two Lines in a Plane,
one, to be equal. This fame Angle meeting in one Point.
is alſo uſed in Opticks, to fhew how ANGLE RIGHT-LINE, is that
one Object under given Circum- made by two right Lincs meeting
in
OB
1
4
)
{
ܪ
A N G
A N G
in one Point, viz. whoſe Sides are 4. The Sum of all the internat
right Lines, as the Angle B. or inward Angles of any right-lin'd
Figure, is equal to twice as many
right Angles, excepting 4, as the
Figure has fides ; this follows from
Prop. 32. Lib. 1. Euclid. and the
Sum of all the external Angles,
B
which are the Angles without the
Figure, when all the sides are le-
verally produced, make 4 right
1. The Quantity of a right-lin'd Angles; this follows alſo from Prop.
Angle, is not meaſured by the 13. and 32. Lib. 1. Euclid. See
length of its Sides, it being no ways Clavius, Barrow, and other Ex-
proportional to them, but by the pounders of Euclid.
Arch of a Circle deſcribed within But here we ought to obſerve,
the Angle, intercepted between the that when a right-lin'd Figure has
Legs of the Angle, whoſe Centre is one or more Angles which open
the angular Point ; that is, if there outwardly, as the Angle BCD of
be an Angle given, and you want the Trapezium ABCĎ; what is
to meaſure it, you need only find meant by this Angle in the Propo-
the number of Degrees contained in
B
the Arch of any Circle deſcribed
within the Angle, from the an-
gular Point, intercepted between
C
the ſides of the Angle ; and that
number of Degrees is the meaſure
of the Angle. This follows from
Prop. 33. Lib. 6. and Prop. 1. Lib.
12. Euclid. (ſuppoſing a Circle to
А.
D
be a Polygon of an infinite number fition, is the ſum of the Angles
of Sides.)
ACB, ACD made by drawing the
The Doctrine of right-lin'd Line AC from the oppoſite Angle
Angles, is of great uſe, as weil BAD: for if otherwiſe, you would
in the Theory as Practice of Geo- underſtand the Angle BCD, which
metry, becauſe they are principal according to the Definition of an
parts of all right-lind Figures. Angie, muſt be one Angle of the
2. No Angle can have for its Figure, the Propoſition is falſe.
Meaſure quite 180 Degrees, for in 5. Angles in practical Geome-
this Caſe one ſide will fall into the try, meaſured upon Paper
fame right Line with the other, with a Line of Chords, or Protractor,
that is, they will be both one right and upon Ground or at Sea with a
Line, and ſo cannot form an Angle, Theodolite, Circumferenter, Qua-
they having no Inclination. And drant, Croſs Staff, &c. as may be ſeen
from hence there ſeems to arile an in the uſes of the ſeveral Inſtruments,
odd Paradox, viz. That the Ag 6. A given Angle may be mul-
gregate or Sum of ſeveral Angles tiplied any number of times geo-
ſhall be no Angle at all.
metrically ; but on the contrary,
3. The Sum of all the Angles you cannot divide one geometrically
that can be made at the fame Point, into any number of equal Parts.
conſiſts of 4 right Angles, whole But the Cycolid will aſſiſt us in doing
Meaſure is 360 Degrees.
this thing univerſally.
ANGLES
are
ANG
A N G
!
1
each other in the Point E ; then grees, which Magnitude they will
are the Angles A EC, DEB, and not exceed, as may be ſeen in Frey.
CEB, AED vertical Angles. tag's Fortification all the Moderns
When two right Lines or two make it above 100 Degrees. Seo
great Circles of the Sphere cut each Wolfius's Elem, Architect. Milit. cap.
other, the vertical Angles are equal. 2. Nouvelle Maniere de Fortifier les
The firft is ſhewn by Euclid, Prop. Places, p. : 25. Sturmy too, in his
15. Lib. i. and the other in moſt Keritable Vauban, P: 150, 151.
Treatiſes of pherical Trigonome- makes it obtufe ; and all Ingeniers
try ; amongſt which fee Wolfius S. agree, that this Angle muſt not be
33. Elem. Spheric.
leſs than 60 Degrees, tho' Mr. de
ANGLE OF A BASTION, in For- Ville, in his Fortification, ſays 90
tification, is the Angle BCD which Degrees is the beſt bigneſs for
the two Faces B C, CD of a Baſtion this Angle. See his Reaſons. Some
ABCDE make at the Point of call this Angle a Flank'd Angle.
the Baſtion.
Angle of or at the Centre, (in
In the Dutch Fortification they Fortification) is the Angle CKH,
make this Angle of that of the drawn from two Angles C, H,
Polygon, until it comes to 90 De- (neareſt to each other) to the Centre
C
H Η
JIALISI....
I
D
G
B
A
E
T
41LIWICKAUK.
K
K of a regular Figure. Theſe are
fence CF and the Flank FG of a
found by dividing 360 Degrees by Baſtion.
the number of Sides that the Figure In the ancient Fortification, this
has.
Angle is acute, as may be ſeen in
ANGLE of a Polygon, is the Angle Freytag's Book, and then the Angle
which one Side of a Polygon makes EF G was a right Angle. Blondell
with the other. In regular Figures makes it obtuſe ; but on the con-
the Quantity of this Angle is 180° trary De Grave from Pagan, with
360°
moſt of the Moderns, a right Angle;
if n be the number of Sides. which is look'd upon
ANGLE of the Tenaille, is the reaſonable. See Wolfius's Elem. Ar-
Angle CIĦ made before the Cour-chit. Milit. §. 64. Becauſe in this
tain by the two Lines of Defence caſe the Face GH of the Baltion
CF, and EH.
has a ſtronger and better Defence.
ANGLE FLANKING, is the Angle ANGLE re-entring or re-entrant,
CFG made by the Line of De- by the French, is any Angle in For-
tification
as more
I Xa"
XI
1
L Xm
a
xn-
I )
m
ma
ANN
Α' Ν Ν
tification whoſe Point turns inwards So that when any three of theſe
towards the Place; that is, thoſe Legs four Quantities m,n,a r,are given; it
open outwards towards the field. is very eaſy to find the Value of the
It is not eaſy to be fortified, as may fourth.
fourth. But if it be compound
be ſeen in the Writers of irregular Intereſt, and x =1+r) be equal
Fortification, where it is particularly to the Principal and Intereſt of 1
handled ; amongſt which, ſee Wol. Pound, at any given rate; then will
fius's Archit. Milit. S. 187.
ANGLE SORTANT, or SALIANT,
ora
m be =
by the French (in Fortification,) is.
any Angle whoſe Point turns out-
wards, (ſuch as thoſe of Baſtions,
& C.) that is, whoſe Legs open in-
wards towards the Place.
n = L, X ixmta L,
ANGUINEAL. HYPERBOLA, A
L, X
Name given by Sir Iſaac Newton to
four of his Curves of the ſecond
Order, viz. Species 33, 34, 35,
* * *
ta
and 36. exprefled by the Equation L being the Logarithm of $-T
xyy. eyş-ax 3.6x2.c x. d. being and of a.
Hyperbola's of a ſerpentine Figure.
2. If the Diſcount, &c. in buy-
ÁNGULAR. , Any thing belong- ing and ſelling of Annuities, &c. at
ing to, or which has Angles.
fimple Intereſt be wanted, let
ANGULAR MOTION, in Afro-
nomy, is the increafing or decreaſing
natnu 12 xar
bes, then
1+nr
Angle made by two Lines drawn
from a central Body, (as the Sun or
2+2 nr xs
Earth) to the apparent places of two will a be
Planets in motion.
2nr-r*72
The angular Motions of a Planet
and the Earth at the Sun made in
the fame time, are reciprocally pro- fuppaling 2 sitra--2a=%.
25--an taxn
portional to their periodical Times,
ANIMATED NEEDLE. Some
call a Needle touched with a Load-
z tzzt8sar.
n will be
ſtone by this name. See Compaſs.
ANNUAL EQUATION. See E. But when it is compound Intereft
quation.
s will be sa
ANNUITY A name for any
yearly Income, ariſing from Money
lcnt, Houſes, Lands, Salaries, Pen-
fions, &c. being divided into two
forts, viz. for a Term of Years, or
na 5 X 2
; and
2 r a
**
;
xпxx
I XS;
a
771 -
1
upon a Life,
m
a
1. If the Amount of Annuities
La
L, at sms *
in Arrear at fimple Intereſt be
L,
wanted, and a be the Annuity, r the
rate of 1 Pound per annum, m the
And if r2 be ſuppoſed
Amount thereof, and n the number
of years ; then if a, 1, 1, arc given, to become infinite ; a being the
will be 11 a to
annual Rent, it follows that sa
E
a.
22 22
22
X01*,
2
A N N
Α Ν Ο
:
I
nI
and X
XI
n
D
---
a, ſo that from hence you ANOMALY, mean or ſimple, in
may have Rules for buying and the old Aftronomy, is the diſtance.
ſelling Eſtates in Fee-fimple at com-
of a Planet from the Line of the
1 pound Intereſt.
Apfes according to its mean Motion :
So that if it be required to find. As let ESD be the Sun's Orbit,
how may. Years Purchaſe at com AMNB the Ecliptick, the Earth
pound Intereſt any Annuity is worth
m will be in
x=
B
N
3. All this is from Mr. Jones's
S
Synop. Palmar. Matbefeos, p. 208,&c. M
T
where the Inveſtigation is ſhewn.
As to the Doctrine of Annuities
upon Lives, which is founded upon
Bills of Mortality, fee Dr. Halley's
Diſcourſe in the Philofoph. Trans. N.
196, and Mr. De Moivre's Treatiſe
of Annuities.
at T, the Sun at S, and AB the
ANNULET. In Architecture, is Line of the Nodes ; then is the
a narrow flat Moulding belonging Angle ATM or the Arch AM
to the Capital or Eaſe of a Column, the Sun's mcan Anomaly. Ptolemy
being ſometimes called a Fillet or calls it the Angle of ihe mean Motion.
Lift. Harris from Ozanam calls it But in the new Aftronomy, where
a ſmall ſquare Part, turn'd about a Planet as P deſcribes an Ellipſis
into the Corinthian Capital, under APBA about the Sun, ſituate in
the Echinus, or Quarter Round. the Focus S, it is the Arch, or
E
A
H
11111 «RTII
Zi
former
F
S
B
Angle or Trilineal Area ASP con- and drawing SF perpendicular to
tain'd under the line of the Apſes AB, the Radius QC continued ; the
(viz. the tranſverſe Axis,) and the mean Anomaly may be repreſented
Line SP drawn to P the Planet's by the trilineal circular Area AQS,
Place, which is proportional to the or by the Arch AQSF, as is de-
Time Drawing the Perpendicular montirated by Dr. Keil in his Leet.
QP HI thio' P the Planet's place, Aſtron, and others. The Ancients
call,
}
EXCENTRICK OR
OF
has alſo given
ANO
Α Ν Ο.
call this the mean Anomaly excentrick. trical Solution thereof by means of
It is eaſy to find the mean Ano- the protracted Cycloid. So has Sir
maly ; as may be ſeen in aſtrono- Iſaac Newton too in Prop. 31. Lib.
mical Writers. See Kepler Epitom. 1. Princip: Mathem. Philof. Nat.
Afron, Copernic. Lib. 5. p. 686. and Theſe are ingenious indeed, but not
Wolfius Elem. Aftron. 6.622.
fit for the purpoſe of an Aſtronomer,
ANOMALY
and therefore Sir Iſaac Newton gave
2°HE Centre, is the Arch other Solutions by Series's, one of
AQ of the excentrick Circle A QB, which may be feen in Dr. Wallis's
and the right Line Q H drawn from Works, Vol. 3. p. 625. and in
the Centre of the Planet P, perpen- (Newton's) Fragmenta Epiſtolarum,
dicular to the Line AB of the p. 26. And the other in the Schol.
Apſes. This muſt be given in order to the Prop. above mentioned ;
to find the mean Anomaly, as may which
which laſt is much the beſt,
be ſeen, amongſt others, in Wol being not only fit for the Planets,
but even the Comets whoſe «Orbits
fius's Elem. Affron. . 622.
ANOMALY CO E QUATE OR
are very excentrick. Dr. Gregory
True, is the diſtance of the Sun in his Aftron. Lib.
3.
from his Apogæum, or of a Planet a Solution by a Series, and Reyneau
from its Aphelium, where it is ſeen in his Analyſe Demontre, Lib. 8. p.
from the Sun ; that is, it is the 713, 714. But Dr. Keil, in his
Angle A S P at the Sun, which the Prelection. Apron. p. 375. is much
Planet's diſtance from the Aphelium better than theirs, it converging very
A appears under. Ptolemy calls this faft. He ſays, if the Arch Å N be
the Angle of the true Motion, and the mean Anomaly, and its Sine be
fome the Angle of the Sun.
e, and Cofine f, and the Excentricity
It is not an eaſy Problem to find FC be g, and g e be called , then
directly the true Anomaly from the
will A bea
& C.
mean one given ; or, which is the
ſame thing, to find the Poſition of (luppofing r= 57°. 29578.) =
a right Line SP paſſing thro' one of Degrees in that Arch ; and the firſt
the Focus's S of a given Ellipfis,
Term" will be enough in all the
which ſhall cut off an, Area PSA
by its Motion, being to the Area of Planets, even Mars it felf, where the
the whole Elipfis in a given Ratio, Error will not be more than the
viz, in the Ratio of the periodick 200th part of a Degree; and from
time of a Planet deſcribing the thence it will be eaſy to find the
Ellipfis to another given time ; Angle ASR, and afterwards the
which being found, the Point P Angle AS P. See his Inveſtigation,
or Place of the Planet at that together with the Reaſon of what
time will be had. Kepler, who firſt Sir Iſaac ſays in the Scholium above
propoſed this Problem, expreſly inentioned ; as alſo an Example of
owns that there is no direct way of the Rule.
ſolving it, that is, of finding the The Difficulty of this Problem
Angle PSA from the Area A P $. made Kepler fly to other Suppofi-
But
he did it indirectly by the Rule tions about the Motion of the
of Falſe, as may be ſeen in his Book Planets, where he imagin'd ſome
before mentioned, p. 695. So alſo Point about which the Motion
has Wolfius Elem. Aftron. §. 6:3. would be equable, when in reality
Dr. Wallis firſt gave the geome- there is no ſuch Point. Seth Ward
too,
g 23
za3
E 2
1
Α Ν Τ
A N T
too, in his Aftron. Geometr. takes viz. from Taurus to Aries, &c.
the Angle at the Focus, where the ANTECEDENT, is the firſt of
Sun is not for the mean Anomaly, two Terms of a Ratio, or that which
which indeed will nearly repreſent it is compared with the other ; as in
when the Orbit is not very excen- the Ratio of 2 to 3, or a to b; 2
trick, and then gives a very elegant and a are each Antecedents.
Solution of the Problem. But if Antes. In Architecture, are
the Planet's Orbit be pretty excen- ſquare Pilaſters placed at the Cor-
trick, as is that of Mars ; the So ners of Buildings. See Vitruvius,
lution will not give the true Ano. Lib. 3. C. 1. The French call theſe
maly exact enough, as is ſhewn ſometimes angular Pilaſters, as may
by Bulialdus in the Defence of the be ſeen in Daviler, p. 35. See
Philolaick Aftron. againſt Seth Ward; alſo Goldman's Treatiſe of Archi-
where he fhews from four Places of tecture, Lib. 1. p. 10. and Wolfius's
Mars obſerved by Tycho Brahe, that Elem. Architect. . 75. As likewiſe
in the firſt and third Quadrants of Perrault upon Vitruvius, p. 22, 23,
the Anomaly, the Place of Mars is and 26. m. 62, and 64.
forwarder than it ſhould be, and in ANTEPAGMENTS. Vitruvius in
the ſecond and fourth Quadrants, Lib. 4. C. 6. calls by this Name the
the true Anomaly is too little, and Ornaments of Doors and Windows,
gives a Correction ; but this Cor- from whom Mr. Perrault has tran-
rection is not ſo good as that of Sir flated it in French by the Word
Iſaac Newton at the end of the Chambrantes, and the French fome-
Scholium above mentioned
times uſe it in the fame Senſe with
ANSER. A ſmall Star of the fifth or
Tablette, and the Italians, with il pi-
fixth Magnitude in the Milkey-Way, anazzo, as may be ſeen in Scamozzi.
between the Swan and Eagle, firit ANTICKS, in Architecture, are
brought into order by Hevelius, the Figures of Men, Beaſts, &c.
See his Prodrom. Affron. p. 117. placed for Ornaments to Buildings.
308.and Firmamen. Sobieſcan. Fig. L. ANTÆCI, ' in Geography, are
Anses, or Anfæ, Handles'; the the Inhabitants of the Earth, which
parts of Saturn's Ring, which are live in the fame Semicircle of the
to be ſeen on each ſide the Planet fame Meridian, but on different
when viewed through a Teleſcope, Sides of the Equator, viz. the one
and the Ring appears ſomewhat North and the other South. But
open. See Ring of Saturn. equally diſtant from the Equator.
ANTARES. A Star of the firſt Theſe have Noon and Midnight,
Magnitude in Scorpio. It is call?d and all Hours at the ſame time.
the Scorpion's Heart. Hevelius in his But contrary Seaſons of the Year,
Prodrom. Aſtron. p. 300. makes its that is, when it is Spring to one,
Longitude for the Year 1700 in un it is Autumn to the other ; when
5º. 32.43". and ſouthern Latitude Summer to the one, Winter to the
4º. 271. 1911.
other. The Days of the one are
ANTARCTICK POLE, is the
the equal to the Nights of the other,
ſouthern Pole, or ſouthern End of and vice verſa. See other Affections
the Earth's Axis.
of the Antæci in Varen. Geogr. c. 8.
ANTARCTICK CIRCLE, the Prop. 4 Sect. 6.
ſame with Polar Circk. Which fce. ANTILOGARITHM, is the Lo-
ANTECEDENTIA, or in Antece- garithm of the Co. fine or Co.tangent
dentia. A Planet, Comet or Point or Co-ſecant of any Sine, Tangent,
of the Heavens, is ſaid to be or Secant; which how to find, feu in
in Antecedntia,
when it moves Books of Trigonometry.
contrary to the Order of the Sign.,
ANTI-
f
D
1
mans,
Α Ν Τ.
A PE
ANTIPARALLELs, are thoſeLines, and Winter when we have Summer,
as F E, BG, that make the ſame Day when we have Night, and
Angles AFE, ACB, with the two Night when we have Day. See the
Affections of theſe in Varen. Geogr.
A E
Cap. 28. Prop. 9. Sect. 6.
In former times it was taken for
a great Fable for any one to ſay
F 1
there were People that walked with
their Feet to ours, and the ancient
B
Fathers, St. Auguflin lib. 16.de Civi-
E/ ZG
tate Dei, c. 9. and Lactantius Iaftit,
A I
K B divin. lib. 3. c. 24. ftrenuouſly denied
it as well as others.
C
2
ANTIQUE. A Building or Statue
made when Building and Satuary
F
H
were at the utmoſt Perfection a-
mongſt the ancient Greeks and Ros
Lines A B, AC, cutting them, but
contrary ways,
as parallel Lines ANTISCU, in Geography, are
that cut them. But Mr. Leibnitz, thoſe Inhabitants of the Earth
in the Aeta Erudit. An. 1691. p. which live in two Places on the
279. calls Antiparallels thoſe Lines fame Meridian equally diſtant from
(ſee Fig. 2.) as E F, GH, which cut the Equator, the one on the North,
two Parallels AB, CD; ſo that the and the other on the South Side
outward Angle AIF, together with thereof; the one having Summer
the inward one A KH, is equal to a when the other has Winter, and
right Angle.
contrary wiſe ; and when the Days
When the Sides A B, AC of a of the one are longeſt, thoſe of the
Triangle, as ABC (Fig. 1.) are cut other are ſhorteſt. See more of the
by a Line EF antiparallel to the Affections of theſe in Varon. Geogr.
Raſe BC, the ſaid Sides are cut re- general. as alſo Wolfius's Geogr.
ciprocally proportional by the faid Cap. 6.
Line EF ; that is, AF:BF: APERTURE, in Opticks, is a
EC : AE, the Triangles AFE, round Hole (whoſe Diameter is a
ABC being ſimilar or equiangular. little leſs than that of the Object-
ANTIPODES, in Geography, are Glaſs) in a turn'd bit of Wood or
the Inhabitants of two Places that Plate of Tin, placed within fide of
live diametrically oppoſite to one a Teleſcope or Microſcope near to
another, or that walk Feet to Feet, the Object-Glaſs, by means of
being 180 Degrees diſtant from one which you get an Admittance of
another ; that is, if a Line was con more Rays, and a more diſtinct
tinued down from our Feet quite Appearance of the Object.
thro' the Centre of the Earth till Mr. Huygens, (in his Syſtem of Sa-,
it arrived at the Surface on the turn, p. 82. and Dioptr. Prop. 53.
other Side, it would fall upon the p. 195.) firſt found the uſe of
Feet of our Antipodes, and vice pertures to conduce much to the
verfa. If one was continued in perfection of Teleſcopes ; and in
like manner from their Feet, it his Dioptr. Prop. 56, p. 205.89
would fall upon ours, who are their ſeg. he found, by Experience that
Antipodes. The Antipodes have the beſt Aperture for an Object
Summer when we have Winter, Glaſs of 30 Feet is as 30 to 3, or 10
13
A PH
A PO
48
o 51
IES
8 25
n
о
to 1 ; that is, as ļo to i, fo is the Saturn upon them, move a ſmall
ſquare Root of the focal Diſtance of matter in Conſequentia with reſpect
any Lens multiplied by 30 to its pro to the fixed Stars, and that in the
per Aperture ; and that the focal Di. fefquiplicate Ratio of the Diftance of
ſtance of the Eye Glaſſes are propor- theſe Planets from the Sun.
tional to the Aperture. It has alſo Kepler places the Aphelia for the
been found by Experience, that Year 1700, as in this Table.
Object Glaſſes will admit of greater
Apertures, if the Tubes be blackened
within fide, and their Paſſage be
h 28
furniſhed with wooden Rings.
3
8 10
40
Mr. Auzout fays, that he found by
51 29
Experience that the proper Aper-
3 .24 27
tures of Teleſcopes ought to be
V
30
nearly in the ſub-duplicate Ratio of
their Length. Whether this be true,
I know not.
But De la Hire, in his Tab. Aſtron.
APHELIUM, or ApheLion, is will have them to be for the ſame
that Point of any Planet's Orbit
, in Year as in this other Table.
which it is at the fartheſt Diſtance
from the Sun; being, in the Coper.
nican Aftronomy, that end of the
29 14 41
17
greater Axis of the elliptical Orbit
14
of the Planet, molt remote from the
35 25
Focus wherein the Sun is.
6 56 10
The times of the Aphelia of the
Yo 1 13 3 40
primary Planets, may be known by
their apparent Diameters appearing And makes the yearly Motions of
leaſt, as alſo by their moving flowelt them to be thus,
in a given time. You will ſee how
1
to find them by Computation in
Wolfius's Elem. Afirox.9.659,667. In
h
Ricciolus's Almag. Nov. lib. 7. Seet.
4
2.f. 543. and foll. and Sect 3. Cap.
7
8. and foll. f. 586. and foll. See
26
allo Street's Aftron.Carolin.p. m. 25.
39
and foll. Dr. Halley too has given
a way to find them in the Philofoph. APOGÆUM. That Point of the
Tranſ. n. 128.and ſo has Dr. Gregory Orbit of the Moon or Sun, (in the
in his Affron. lib. 3. prop. 14. and Dr. old Aftronomy) which is fartheſt
Keil in his Aſtronomical Lectures. from the Earth.
Theſe laſt being the beſt of any. The manner of finding the Apo.
Sir Iſaac Newton, in prop. 14. lib. gæum of the Sun or Moon, is ſhewn
3. of his Princip.as alſo Dr. Gregory by Wolfius in Elem. Aſtron. §. 618.
in his Afronomy prove the Aphelia and by Ricciolus in Almag. Now. lib.
of the primary Planets to be at reſt; 3 cap. 24. alſo by Street in Aſtron.
tho'at the ſame time, in the Scholium Carolin. p. m. 7. You have alſo a
to the ſaid Propoſition, he ſays the Geometrical way of finding the
Planets neareſt to the Sun, viz, Mer- fame by Mr. Caſſini. See Tranſ.
cury, Venus, the Earth, and Mars, Philofoph. n. 57. Žela Hire in Tab.
from the Actions of Jupiter and Apron, p. 15. makes the Apogæum
of
! 1
1
22
I 34
I
I
1
1
i
goes forward
APO
APO
of the Sun to be in 80 71 3011 of $ AB=1 of a Square, and its Diago-
and its annual Motion 1 211 and nal AC-V 2; will be an Apotome,
the Apogæum of the Moon in 6e viz. equal to 1-VZ; fo alſo will
53' 401 of #, and its annual Mo- the Difference between the Side A
tion i fi. 10° 39' 5211.
The Moon's Apogæum moves un-
B
equally ; when ſhe is in the Syzy gy
with the Sun, it goes forwards, and
in the Quadratures, backwards; and
theſe Progreſſions and Regreſſions,
are not equable, but it
flower when the Moon is in the
А
D o
Quadratures
, or perhaps goes back- C=2 of an equilateral Triangle
wards; and when the Moon is in the ABC, and the perpendicular B
Syzygy, it goes forwards faſteſt
of all. See more of the Apogæum
D=V3, be an Apotome, viz.
of the Sun and Moon, in Sir Iſaac. 52V3: And generally if A C be
Newton's Theory of the Moon. a Semi-parabola, whoſe Axis is A B,
APOPHY GE, in Architecture, and Latus Rectum be=1, and if AD
is a concave Part or Ring of a be a Tangent to the Vertex at A,
Column, lying above or below the and this be divided into the Parts
fat Member. "The French call it Le A a 2, Ab=3, A c=5, Ad=6.
Conge d'en Bas,or d'en Haut; the Ita- & c. and Perpendiculars a 1, b 2,
lians, Cavo da Bafo, or di Supra, as <3, d4, 6c, be drawn, there will
alſo Il vivo da Bajo. Amongſt the be (from the nature of the Curve)
ſeveral Authors that tell how to V2,
VV5 V6, &c. reſpec-
deſcribe it, ſee Wolfius's Elem.
C
4 낙
​APOTOMÉ. Euclid in his tenth
Book at Prop. 74. calls an Apotome a B
3
Line BC which is the Difference be-
tween a rational Line AC,and a Line
A B only commenſurable in power
to the whole Line A C, and may be
2
A
B
C
1
Arch. S. 115
a b
cd
I
1
D
A
D
A
tively ; and fo Aa (=1)- 21,
G
will be 1-2; A amb 2 will be
2-V3; Ab- 3 will be 3–15;
Ac-d4 will be 5-V6, &c. .
B
С
Wherefore by this means you will
have an infinite Series of different
expreſſed thus ; ſuppoſing (AC-a
Apotomes.
and A B5,) viz. a-Vb or in Euclid in lib. 10. (ſee his third
Numbers 24V 3. Hence the dif- Definition after Prop. 85 ) diſtin-
ference GC between the side guiſhes Apotomes into firit, fecond,
third,
do
E 4
AP P
A P. P.
NA
third, fourth, fifth, and fixth ; and in the Squares 16 and 3 is 13, and Vi
the Propofitions immediately follow- has not a Ratio in Numbers to 4.
ing, ſhews how to find each of them,
A fifth Apotome, is when the leaſt
being indeed no othes than the Sub- Number is Rational, and the ſquare
ductions of the leſſer Names or Parts Root of the Difference of the Squares
of Binomials from the greater. As in of the two Numbers has not a Ratio
Numbers, if 6+1 20 be a firſt in Numbers to the greateſt. Such is
Binomial, then ſhall 6-V zo be V6-2 where the Difference of
a firſt Apotome, and ſo will 3-V5; the Squares 6 and 4 is 2, and V2
that is, when there are two Num- tovő has not a Ratio in Numbers.
bers ſuch, that the greateſt is a A fixth Apotome, is when both the
rational one, and the Difference Numbers are irrational, and the
between their Squares is a ſquare ſquare Root of the Difference of
Number,
their Squares has not a Ratio in
A fecond Apotome, is when the Numbers to the greateſt. Such is
leaf Number is rational, and the V7-V 2, where the ſquare Root
ſquare Root of the Difference of the
Squares of the two Numbers, has a
4 =2 of the Difference (4) of
Ratio in Numbers to the greateſt the Squares of 6 and 2 has not a
Ratio in Numbers to vā.
Number. Such is V 18–4, for the
The Doctrine of Apotomes in
Difference between the Square 18; Lines, as handled by Euclid, in his
and 16 the Square of 4, is 2, and tenth' Book, is a very curious Sub-
V Z has a numerical Ratio to V18, ject, and worthy to be peruſed and
viz. as I to 3; for VT8 is=3 VZ; improved by all thoſe who would
in like manner 48–6 is a ſecond lay down geometrical Elements,
Apotome ; for the Difference be- from whence might be deduced the
tween 48 and 36, is 12, and V12 Quadratures, of Curve-lineal Figures,
has a numerical Ratio to V 48, viz. and perhaps lineal Solutions of Dio-
as 2 to i, for v T2 is =2V 3, and phantus's Problems, and others of
v2
48= 4V 3.
the like kind, tho' all'the uſe, (one
would think) Euclid himſelf made
A third Apotome, is when the two
Numbers are both irrational, and the of this Book, was only to ſhew the
ſquare Root of the Difference of nature of the five regular Bodies,
their Squares has a Ratio in Numbers which by Plato and his Sett (of
to the greateſt Number. Such for which Euclid's was one)were held in
Example is, V 24,4V 18 for the Kepler (in his Myſteria Coſmogra-
great Efteen. And in the laſt Age
Difference of their Squares 24 and phica) would have the Number of
8, is 6, and V6 has a numerical the Planets, and the Magnitudes of
Ratio to v
viz, as that of i to the Syſtems of the World to ariſe
24
2, for V 24 is=2 V5.
from theſe Bodies, and (in pref: ad
A fourth Apotome, is when the lib. 1. Harmonices Mundi. j. 3.)
greateſt Number is rational, and ſharply reprimands Peter Ramus
the ſquare Root of the Difference of for undervaluing Euclid's tenth
the Sqr.res of the two Numbers: maticarum, p. 252.) Kepler ſays:
Book (in lib. 21. Scholarum Mathe-
has not a Ratio to that. Such is
Veftrum eft carpere, quæ non intelli-
4-V3, where the Difference of gitis
, mihi qui rerum Caufas indago,
præter-
}
APP
APP
-præterquam in decimo Euclidis fe have much greater apparent Dia-
mite ad illas nulla patuerunt. And meters than the reſt of the Planets ;
Ramus ſays, Materies decimo Libro the ways to obſerve the Quantity
propoſito eo modo eft tradita, ut in of theſe are different from thoſe,
humanis literis atque artibus fimi- whereby thoſe of the reſt of the
lem obſcuritatem nufquam deprehen- Planets are had. Ricciolus (in Al-
derimi obfcuritatem dico non ad in- mag. nov. lib. 3. C. 10. f. 16. and fol.)
telligendum, quid præcipiat Euclides, gives five different ways to obſerve
-fed ad perſpiciendum penitus et the apparent Diameter of the Sun,
explorandum quis firis & ufus fit and eight ways for thoſe of the
operi propofitus, que genera, ſpecies, Planets and fix'd Stars, (in lib. 6.c.
differentiæ fint rerum ſubječiarum :
9. f. 422. and fol.) The beſt way
nihil enim unquam tam confufum vel of doing this in general, is by a Mi-
involutum legi vel audivi.
crometer fix'd in the focus of a
Old Oughtred in his mathematical Teleſcope. See Micrometer.
Key has a Declaration of the tenth 1. One way of finding the ap-
Book of Euclid, demonſtrated by parent Diameter of the Sun, is by
Symbols. Dr. Barrow too in his taking the meridian Altitudes of
Xuclid has done the ſame. You his upper and lower Limbs, with
have alſo in Michael Stifel his Arith- a good Quadrant and Teleſcope
metica Integra, lib. 2. c. 13. and fitted to it, and afterwards taking
fol. p. 143. and fol. The aforeſaid their Difference, which will be his
Book of Euclid, and alſo the Doc- apparent Diameter ſeen from the
trine of Apotomes clearly explain'd Earth.
and fully handled at Cap. 23. p. 2. Another way is, by erecting
two perpendicular Threads over the
Apotomes are alſo called reſidual, Meridian Line, and while the Eye
and reſidual Binomials.
is at reſt obſerving the Sun's Paſſage
Apotome, by ſome Writers on
over the Meridian, and noteing the
the Theory of Muſick, is the Dif. Inſtant that the Limb of the Sun
ference between a greater and leſſer comes to the Threads, by an ac-
Semi-tone, being expreſſed by the curate Time-Keeper or Clock, and
Ratio 128 : 125.
the Inſtant that its oppoſite Limb
APPARENT DIAMETER, in leaves them; and the Difference is
Aſtronomy, is the Angle under the Time wherein the Diameter of
which . we ſee the Sun, Moon and the Sun is paſſing over the Meridian:
Stars: As when we ſee the Sun S which if the Sun be in the Equator,
under the Angle DOE; this Time turn'd into Minutes, &c.
S
this Angle is the Ap- of a Degree, will be the Angle under
parent Diameter. The which the Sun appears. But if the
apparent Diameters of Sun be out of the Equator, the
DE
the Sun, Moon, and Arch found is one in that parallel
Planets muſt be known, Circle the Sun moves in, which
in order to compare. muſt be turn'd into Minutes, &c. of
the Bigneſſes of them the Equator.
with each other, to 3. "The Diameter of the Sun,
know how much one is Moon and Planets is not found to
bigger or leſs than be thë fane at all times ; but in
another, and to com each of tliem it increaſes to a cer-
pute the true Magnitude of either tain Liinit ; and then again de-
of chem. Becauſe the Sun and Moon creaſes. And particularly it is found
that
187. and fol.
1
1
APP
APP
that the ſuperior Planets appear much greater when in oppoſition to the
greater when they are in oppoſition Sun, than when it is near a Con-
to the Sun, than when near a.Con- junction ; ſo that in July and Auguſt,
junction ; and the inferior Planets in the Year 1529, it was taken for
appear greater, when their Light a new Star, by reaſon of its pro-
is lefſen'd, than when they ſhine digious. Magnitude. See Kepler.
more bright; and particularly Ric- in Aftronom. Optic. c.
ciolus ſays (in Almageft. Nov. lib.7. 333.
feet.. 6. c. 10. f. 713.) that the Dia 4. The apparent Diameter of the
meter of Mars is almoſt nine times Sun was obſerved by
10. P.
mean.
1
//
-
20
O
0
}
O
IO
1
IO
--
greateſ.
leat.
Ptolemy (in his Almag. lib. 5. c. 14.}-
f. m. 117.)
-33
32 18 3r. 20
Tycho Brahe (in his Progymnafin.
lib. 1. c. 1. p. m. 135.)
- .32 31
30
Kepler (in his Tab. Rudolph. f. 92.)
-31
4 30 30 30
Ricciolus (Afron. Reform. lib. 1. c.
12. f. 38.
-32
8 31 40 30
Callini (ſee in Ricciolus above)-
32 31 40 31 8
De la Hire (in Tab. Aſtron.)-
-32 431 32
31 38
And now-a-days it is obferv'd ons and Oppoſitions, and the other
that the Sun's apparent Diameter in the Quadratures; for the appa-
is leaft when he is in o, and greateſt rent Diameter in thoſe is leſs,
when in 4.
and in theſe greateſt ; and the
5. There is a two-fold Increaſe l'eaſt in thoſe is leſs than the leaſt in
and Decreaſe of the Moon's apparent theſe. In the firſt Caſe we have by
Diameter, the one in the Conjuncti-
greateft. leaſt.
1
Ptolemy (in the Place as above.)
-31
20 35
20
-25 36 28 48
Oppoſition.
-32
36 O
Kepler.
-30 32 44
De la Hire,
- 29 30 1 33 30
Tycho (in the Conjunet
.
}
•
I
B
1
O
-42
1
Tycho Brahe.
In the latter Caſe by
leaft.
greateft.
I
/
Ptolemy.
8
155
-32 32 i 36
6. Hevelius ( in Tractatu de Mer- perior. Planets by different Authors
curio in Sole viſo, f. 101.) exhibits as follows.
the apparent Diameter of the lu-
1
Albategnius
Α Ρ Ρ
A PP
ol:
읽
​0.
18, 24
49.. 46
46
24. 22
leaft.
mean. greatefi.
1
// /
/
1. 29. 13 1.
44. 13
2.
5. 59
1. 34. O 1.
50. O . 12.
0
0. 21. O 0,
25.
O O 38.
0. 46. 0 0.
57
O I.
12.
0
o. 14. 10 16. 2 0. 19. 40
2. 9. 25 2. 36. 40 3.
2.
4. 0 2. 45.
O
3. 59.
. 30. 0 O. 38.
O. 50, O
0. 38. 18
0.
1. 8
14. 36o. 18. 2 0.
0.
54
I. 34.
6. IO, 0
o. 57:
I. 40. 0
O. 54.
0
34.
O
. 10. O
22.
I. 32.
o. 2. 46 . 5.
2
0.
20. 50
I. 49. O
3.
8.
O
1.
52. O 3. 15.
0
4. 40.
2. O 1.
O
7
6.
O.
33. 30 1. 4
4-
8.
O
9. 34 0. 16. 46
5
I.
2. 5.
20 3. 41. 45
1.
29.
o 2. 10. O, 3. 57.
0.
9. 20
0. 13. 48 | o.
0.
4.. '4. O. 6. 3. 0. II. 48
Albategnius B
Tycho
Keplerus
Ricciolus
Hevelius
Albategnius #
Tycho
Keplerus
Ricciolus
Hevelius
Albategnius 1o
Tycho
Keplerus
Ricciolus
Hevelius
Albategnius ?
Tycho
Keplerus
Ricciolus
Hevelius
Albategnius 18
Tycho
Ricciolus
Hevelius
이
​6. 46.
I.
6. 30.
1
16. 42.
48.
I 2
1
58
27. 21
le
25. 12
7. Mr. Huygens, (in Syſtem. Satur. accurately performed. As to the
nino, p. 77. and fol.) has obſerved, apparent Diameters of the fix'd
by the moſt exact Method, the leaſt Stars, by the beſt Infruments that
Diameter of ħ to be 30"; of its have been yet invented, they have
Ring 1'8"; of H, to be 1'. 4"; of , hitherto appeared but as ſo many
to be 30'; of p, to be 1'. 25". He- Points. Even Mr. Huygens ſays, he
velius found the apparent Diameter found the apparent Diamcter of the
of Mercury, when ſeen in the Sun, Dog-ſtar not to be more than 4".
to be not more than 11" 4".
APPARENT DISTANCE, is that
8. The great Difference between Diſtance which we judge an Object
the apparent Diameter, as given to be from us when ſeen afar off,
by the Ancients, from what the being moſt commonly very different
Moderns obſerve them to be, is, from the true Diſtance; becaule we
that (thoſe, ſuch as Albategnius and are apt to think that all very re-
Tycho,) they took them by the mote Objects, whoſe Parts cannot
naked Eyes only; but the Moderns well be diſlinguiſhed, and which
uſe Teleſcopes, by which the falſe have no other Object in vicw near
Light cauſing them to appear bigger them, to be at the ſame Diſtance
then really they are, is removed from us, tho' perhaps they may be
Indeed Ricciolus uſed Teleſcopes ; thouſands of Miles, as in the Caſe
but then he wanted a Micrometer : of the Sun and Moon.
without which the thing cannot be APPARENT Figure, is that
Figure
A PP
APP
no more
Figure or Shape which an Object 1. All things appear the leſs, the
appears under when view'd át a
more remote they are; and it is
Diſtance, being often very different found by Experience when their
from the true Figure. For a ſtreight Diítance becomes ſo great that the
Line, view'd at a Diſtance, may ap- apparent Magnitude is
pear but as a Point ; a Surface, as a than an Angle of one ſecond, they
Line ; and a Solid, as a Surface; and will become ſo fmall as to appear
each of theſe of different Magni- but like a Point, and be no more
tudes, and the two laft of different ſeen.
Figures, according to their Situa 2. Thofe things G F and CH
tions with regard to the Eye. Thus which appear under the ſame Angle
an Arch of a Circle may appear a
G
ſtraight Line, a Square or Oblong
C
a Trapezium, or even a Triangle,
a Circle, an Ellipfis ; ang ular Mag-
E B В
nitudes, round ; a Sphere, a Circle,
&c.
A
Alſo any ſmall Light (as a Candle,
Link, &c.) ſeen at a diſtance in the
D
Dark, will appear magnified, and
farther off than really it is.
CAH, have their Magnitudes pro-
Add to this, that ſeveral Ob- portional to their Diſtances A E,
jects ſeen at a diſtance under Angles AB,
that are ſo ſmall as that each of 3. If the Eye O be placed be-
them is inſenſible, as well as each tween two Parallels AB, CD, theſe
of the Angles ſubtended by any one Parallels will appear to converge
of them, and that next to it ; I ſay
all theſe Objects will appear to be A
E
contiguous, to conſtitute, and feem
but one continued Magnitude.
APPARENT MAGNITUDE of an 0
Object, is the Magnitude of an Ob-
ject as it appears to the Eye, and
its Meaſure is the Quantity of the
C
F
D
Optick Angle; as let DC be an
Object view'd by an Eye at A and
or come nearer and nearer to each
other, the further they are conti-
nued out, and at laſt will appear to
C coincide in that point where the
Sight terminates, which will happen
when the optick Angle BOD be-
E
comes equal to about one Second.
А
4. The apparent Magnitudes of
the fame Object DC, (ſee Fig.above)
ſeen at the Places A and B, that is,
the Angles CAD, and CBD, are
B, then the Angle CAD is the op- in a Ratio leſs than the reciprocal
parent Magnitude of that Object Ratio of the Diſtances AE and BE;
ſeen at A, and the Angle CBD, its but when the Object is very remote,
apparent Magnitude, when view'd
or the optick Angles CAB, CBD
not above one Degree or there
E
abouts,
B
B
)
D
at B.
A
A PP
APP
bouts, they are nearly as the Difu appear of the fame bigneſs. This
tances reciprocally.
is done by defcribing two fimilar
of upon each of
will appear of the fame Magnitude the Lines
, and their points de
from any Point, as C of the Peri- terſection will be the Point fought;
phery; ſo that the beſt Figure for a and indeed the Curve in which all
ſuch Points D do fall, will be one
of the fifth order, as it is eaſy to
find by Computation.
8. If three Situations F, D and
E, of the Eye be wanted, fuch, that
any given Parts A B, BC, of an
}
F
D
А
B
Theatre is the Segment of a Circle,
where the Actors are in a Chord,
F
and the Spectators in the Periphery.
E
.6. The equal Parts of the ſame
Line appear unequal, alſo equal
Objects at the ſame Diſtance, but А. B
ſome more oblique to the Eye than
others ; thoſe will appear to be Object AC, as alſo the whole Ob-
biggeſt that are more direct to the ject ſhall all appear of the fame
Eye.
Magnitude ; it is. but deſcribing
7. To find the Poſition D of the three Semi-circles, or three ſimilar
Eye being ſuch, that viewing the Segments of Circles upon the ſaid
Parts, and the whole 'Object; and
the ſaid Parts and Whole will
appear of the ſame Magnitude from
any Points F, D and E, in the re-
D
ſpective Peripheries of thoſe Circles.
9. And if it were required to find
А.
thé Locus of the Point M, being
B
ſuch that an Eye placed at it ſhall
unequal Objects A B, BC, they ſhall always fee unequal Parts, A B, BC
M
.
1
1
А
B C
of the ſame Ohject, of the ſame M of this Semi-circle, the right Lines
Magnitude, it is but continuing out MA, MB, MC be drawn ; the
B C the leſſer Part to D, ſo that CD Angles A MB, BMC will be e-
be a fourth Proportional to BC, qual.
AC, and deſcribing a Semi circle 10. Altho' the optick Angle be the
upon BD; for if from any Point uſual Meaſure of the apparent Mag-
nitude
APP
Α Ρ Ρ
nitude of an object, yet Cuſtom and nitude ; ſo that the apparent Mag-
the frequent Experience of looking nitude of an Object will be judged
at diftant Objects, by which we to be more than in the Ratio of the
know they are bigger than they optick Angle ; and perhaps this
appear, has ſo far prevail'd upon may be the whole (or at leaft Part
the Imagination, as to cauſe this of the Cauſe) why two Rows of
too, to have ſome ſhare in our Ef- Trees A B, C D placed in two right
timation of the apparent Mag- Lines AB, CD, meeting in the
B
1
A
.
0
1
C
+
D
Point O, will not appearto be parallel or leſs; and ſince in the ſame
to an Eye placed at 0,but too much Perſon the more Light there comes
diverging ; when nevertheleſs,if the from an Object, the leſs will the
optick Angle be the ſole meaſure Pupil of his Eye, looking at that
of the apparent Magnitude, they Object, be ; the optick Angle will
muft appear parallel; and I doubt be leſs too, and ſo will the apparent
not but would do ſo, to one that Magnitude of the Object. I have
fhould look at them juſt after he often experienced the truth of this,
was recover'd from a Blindneſs `by looking at a Perſon with me in
which he always had before. This a Room, (not having ſo much Light
too may be part of the Reaſon why as out of doors) at Noon Day, and
an Object at a confiderable Diſtance afterwards looking at him at the
horizontally appears_bigger, than fame Diſtance when we have been
when at the ſame Diſtance verti- both out of doors in the Sun Shine,
cally ; as the Sun and Moon near for he always to me appear'd bigger
the Horizon appear bigger than in the Room than out of Doors. So
when in the Meridian, and the Ball alſo Objects up in the Air, having
(for Inſtance) of St. Paul's Croſs, more Light coming from them than
which is fix Feet in Diameter, when they are upon or near the
appears leſs when ſeen from the Ground, may appear leſs in the for-
Ground, than if it was placed at the mer than in the latter Caſe.
ſame Diſtance on the Ground.
12. It is ſomewhat extraordinary
11. The apparent Magnitude of the that Epicurus and his Followers (Men
fame Object at the ſame Diſtance, of tolerable Judgmentin many thing)
will be different to different Perſons, ſhould be ſo ſtupid and enormouſly
and different Animals, and even to miſtaken, when they ſay the Sun,
the ſame Perſon, when view'd in Moon, and Stars are no bigger than
different Lights ; all which may be they appear to be. We find Epi-
occaſion’d by the different Mag- curus himſelf afferting this in his
nitudes of the Eye, caufing the op. Epifle to Pythocles, to be fcen in
tick Angle to differ as that is bigger
Epicurusas
>
APP
APP
Epicurus's Life given by Diogenes, lib. 5: de Natura Rerum fings the
Laertius, lib. 1o. Lucretius too in fame in theſe Words,
Nec nimio folis major rota nec minor Ardor
Ele poteft, noftris quam fenfibus effe videtur.
And again,
Lunaque five Notho fertur loco Lumine luftrans,
Sive ſuam proprio jactat de Corpore Lucem,
Quicquid id eft nihilo fertur majore Figura,
Quam noftris oculis quam cernimus effe videtur.
Thus rendered into Engliſh by Creech,
But farther on : The Sun and Moon de bear
No greater Heats, nor Figures than appear.
And thus the Moon,
Whether with borrow'd Rays or with her own,
$he views the World, carries no larger Size,
No fiercer Flames, than thoſe that ſtrike our Eyes.
Our blundering Hobbes too affirms."
o off, that its Diſtance took away
the ſame thing
" any of its Magnitude.
The chief Reaſon they give is, 13. If the Eye be placed in a rare
" That as we retire from any Fire, Medium, and views an Object thro'
“ ſo long as we are within ſuch a a denſer, as
Glaſs or Water,
« Diſtance from it, that we can having plane Surfaces ; that Ob-
perceive its Light and Heat, the ject will appear bigger than it is,
« Fire ſeems no leſs than it does and contrarywiſe. And in each
56 when we are near it ; but we Caſe the apparent Magnitude QH
“ feel the Heat, and perceive the will be to the true Magnitude AB,
“ Light of the Sun ; therefore the in a. Ratio compounded of FL the
“ Sun is of the ſame Magnitude as Diſtance of the Point F, to which
“ it appears to be ; and as to the the Rays from B and A go unrea
“ Moon, we ſee the utmoſt Verge fracted from the refracting Surface,
or and Face of it diſtinctly, which to the Diſtance GL of the Eye
we ſhould not do, if it were ſo far from the fame, and of the Diſtance
D
A
L
G
M
F
E
Б
H
GM of the Eye from the Object, the Object A B be very remote ; it
to FM the Diſtance of the Object will be AB : MH:: GL :FL ;
from the fame : that is QH: AB :: for in this Cale FM will be nearly
FLXGM : GLX FM. And if =GM, and the nearer the Object
is
.
APP
A PP
1
is to the Surface (of the Medium tances AB, A E from the Eye, they
next to it) the bigger it will appear will ſeem to move with the fame
to be, even till it touches that Sur- Velocity. 3. But if the molt remote
face where it is ſeen at its greateſt E, move flower than the neareſt,
Bigneſs; and coming within the the Motion of the neareſt will ap-
Medium, it will again become leſs, pear to be much ſwifter than it
(tho' it will ſtill appear greater than is. And if they both move the ſame
it really is) the nearer it approaches . way, the apparent Velocities are in
the Surface next to the Eye. And a Ratio compounded of the direct
hence it is that Fiſhes or any thing Ratio of the true Velocities, and
elſe, ſeen in the Water from one in the inverſe Ratio of the Diſtances
the Air, appear bigger than when AB, AE from the Eye. 5. The
in the Air.
Object E moving with any Velocity
14. The apparent Magnitude of whatever, will ſeem to be at reſt,
an Object will alſo be augmented, if the Ratio of the Space it really
by looking at it thro' a Globe of deſcribes in one Second of Time, be
Glaſs, or Water, or any convex to the Diſtance thereof from the Eye,
ſpherical Segments of theſe; and as i to 1400, or as even i to 1300.
on the contrary, it will be dimi- For ſince the Motion of the Hour-
niſhed, when view'd thro' a Concave Hand of a Clock, and the Motion
of Glaſs, or Water. See more under of the Stars about the Earth, are not
the Word Lens.
viſible to the Eye, and in one Se-
APPARENT CONJUNCTION. cond of Time, an Arch of 15 Se-
See Conjunction apparent.
conds is paſs’d over, it is evident
APPARENT
HORIZON.
See the
the way moved thro' by a moveable
Horizon.
Body is imperceptible, if it be ſeen
APPARENT MOTION, is either under an Angle of 15 Seconds, and
that Motion which we perceive a much more fo, when it appears under
diftant Body really moving to have a leſs Angle. 6. It is poſſible
(when we perceive it move, or know for the Motion of a Body to be ſo
it does by its change of Place) ſwift, as that throughout the whole
while the Eye is at reſt or in mo Space it deſcribes there ſhall con-
tion, or that Motion which an Ob- ſtantly appear a Solid, as it were
ject at reft ſeems to have, while the generated by the Motion of the
Eye is in motion.
greateſt Section thereof, (which Sec- .
The Motions of Bodies at a great tion is perpendicular to the way
Diſtance, tho' really movirg very moved thro’j ſo that if the Body be
equally, and deſcribing equal Spaces a Sphere, and it moves in a right
in equal Times, may appear to be Line, (not in the Direction of the
very unequal and irregular to the Eye) inſtead of ſeeing the ſaid Sphere,
Eye, which can only judge (as a Sphere,) you will ſee a Cy-
Ee of them by the Mutation of linder, having the Diameter of its
the Angle at the Eye; as Baſe for the Section of the greateſt
particularly, i. If two Ob- Circle of that Sphere ; and if that
jects B and Eat unequal Dif- Sphere revolves in a Circle, inſtead
tance from the Eye A, at reſt, of viewing a Sphere, you will ſee
move with the fame Velocity; a cylindrical or elliptical Ring;
the moſt remote. E will ap- and whether this may not be che,
pear to move the floweſt ; caſe of Saturn's Ring, I leave to
and, 2. If their Velocities others to judge. A ſmall Inſtance
Al be proportional to their Dil- of the Truth of this, will appear
from
1
APP
Α Ρ Ρ
from the boyiſh Performance of Object at reſt at H, will appear
trundling a crooked Stick, one End to move the contrary way, v.Z.
of which is on fire, between your from H to I with the ſame Velo-
Fingers, in a dark Place. For while city the Eye moves. But if that
this is ſwiftly done, you will per- Object moves the ſame way in the
ceive an agreeable Curve of Fire. 7. fame Direction with the fame Ve-
The more oblique the Eye is to the locity that the Eye has, that Object
Line or Plane which a diſtant will ſeem to ſtand ſtill. If the Object
Body moves in, the more will the has lefs Velocity than the Eye, the
apparent Motion differ from the
true Motion. 8. So that if a Body
I
H
revolves equably in the Circumfe-
rence of the Circle ABFCED, de-
ſcribing equal. Arches in equal
times, and the Eye be at 0 in the
0
Plane of that Circle ; it will, when
G
k
А
:
B
Object will appear to go backwards,
with a Velocity equal to the Dif-
ference of their Velocities. But if
the Object has a greater, it will
C
appear to go forwards with that
Difference.
D
If an Object and the Eye move
E
contrary ways in the ſame Direction
with any Velocities, the Object
will
appear to go backward with
the Sum of the Velocities of both,
The truth of all this appears to
any one fitting in a Boat moving
in a River, as alſo in
any
Wheel-
Carriage that is running faſt ; and
viewing Houſes or Trees, & c.
on the Shoar or Road Side, or other
Boats or Wheel Carriages in Mo-
at the Point A, ſeem for ſome tion.
time to ſtand ſtill, and conſtantly APPARENT PLACE of an Ob.
afterwards to move faſter, till it jest, in Opticks, is that in which
gets to the Point F, where the it appears when ſeen thro' or in
Motion appears to become greateſt; Glaſs, Water, or other refracting
after which it appears to decreaſe, Subſtances, being moſt commonly
till the Body comes to C, where it different from the true Place.
will again ſeem to ſtand ſtill; and Thus, 1. The apparent Place of
then again, its apparent Motion an Object ſeen thro” (or in) Glaſs
will increaſe backwards, till the or Water, terminated by parallel
Body arrives at E, where it will Planes, will be brought nearer to
ſeem again to move faſteſt ; after the Eye than its real Place.
which while it is going from E to an Object be ſeen thro' a convex
A, it will appear to decreaſe. Claſs, its apparent Place will be
If the Eye moves directly for- more remote from the Eye than its
wards from G to O, &c. any remote true place. 3. If an Object be ſeen
F
thros
2. If
APP
A PP
!
thro' à concave Glaſs, its appa- Parts of the fame Pupil, be in dif
rent Place will be brought nearer ferent Planes of Reflexion, the
to the Eye than its true Place. Image of the Object will moſtly
APPARENT PLAC# of the Image appear in the Concurrence of the
of an Object, in Catoptricks, is that reflected Rays, with the Perpendi-
where the Image of an Object made cular drawn from the Object to
by the Reflexion of 'a Speculum ap- the Plane of the Speculuin ; yet
pears to be in. The Ancients (as ſomecimes it will appear without
Euclid, in his Catoptricks; Albazen that Perpendicular, viz. when the
and Vitellio, in their Opticks ;) give Eye is very near to the Speculum,
it for a general Rule, that the ap- and the Object be removed from
parent Place of the image of an Ob- it beyond the Centre. Add to this,
jećt ſeen by Reflexion, is where the that if an Object be placed in the
Teflected Rays meet the Perpendicu- Focus, it cannot be ſeen at all. (See
lar drawn from the Object to the Wolfius's Gatoptr. ſ. 51. 188.233,
Plane of the Speculum, (ſo that iſ 234.)
the Speculum be a Plane, the appa The Doctrine of the apparent
rent Place of the image will be at Places of the Images of Objects in
the fame Diſtance behind the Specu- ſpherical Speculums, as well convex
lum as the Eye is before it; if con as concave, is not quite perfect ; for
: vex, it will appear behind the Glaſs becauſe different Planes of Reflexion
nearer to the fame ; but if con-
cannot be conveniently delineated
cavé, it will appear before the Spe- in the fame Plane ; neither can it
culum :) Tho they lay down this be eaſily demonſtrated which of the
Rule as general, and indeed is uni- Rays proceeding from the ſame
verſally true in plain and convex Point in the Speculum, and reflect-
ſpherical Speculums, and moſt ed to different Parts of the Pupil,
commonly too in ſpherical concave meet; it has been hitherto thought fit
Speculums ; yet, there are a few to lay it down as a Maxim, ſatisfying
Cafes in which the true Rule fails, moft Phenomena, that the
apparent
as has been ſhewn long ſince by Place of the Image, is where the re-
Kepler (in his Paralipomena in Vi- flected Rays meet with the Perpen-
tellionem. prop. 18. p. 70. and fol.) dicular drawn from the Object to
One of them is this, that if two Eyes the Plane of the Speculum. In
D and E, be in the fame Plane cylindrical and conical Speculums,
with the Perpendicular AF, drawn it is found, by Experience, that the
from the Object, to the Plane of Image is not far from the Surface ;
but what Lines, are there interſect-
D
ed, where the Image appears, is not
E
yet determined ; no more than in
Speculums of other Shapes, where
the Loci of the images have not
yet been geometrically determined.
H
APPAREN'T Place of a Planet, in
Aftronomy, is that point upon
the Surface of the Sphere of the
F
World, whereat we ſee the Centre of
the Sun, 'Moon or Stars, from the
the Speculum, the Place of the Surface of the Earth.
Image will appear to be at Hon APPLICATE, i a Right-Line,
Ordinate or
this Side the ſaid Perpendicular. otherwiſe called
Moreover, if both Eyes or different Scori-Ordinair. Which fce.
AP-
an
1
A QU
I
ab
ab
A PP
APPLICATION, is ſometimes the actly. There are ſeveral Methods
geometrical Term for Diviſion ; of Approximation laid down by
but Application alſo ſignifies the Dr. Wallis, Mr. Ralphſon, Dr.
fitting or applying one Quantity Halley, Ward, &c. and they are
to another, whoſe Areas, but not all nothing but a Series infinitely
Figures, are the ſame. Thus Eu- converging or approaching ſtill
clid, lib. 6. prop. 28. ſhews how to nearer to the Quantity ſought, ac-
apply a Parallelogram to a Right- cording to the Nature of the Series.
Line given, that ſhall be equal to a If there be any Non-Quadrat or
Right-lin'd Figure given.
Non-Cubick. Number, the former
Apöly. This Word is uſed being expreſs'd by aat-b, and the
three Ways.
latter by aaa tb, where aa andaaa
1. It fignifies to transfer a Line are the greatelt Square and Cube in
given into à Circle, (moſt uſually,) the propoſed Numbers, then will
or into any other Figure ; ſo that
its Ends ſhall be in the Perimeter Vaath = at
and
of the Figure.
2aat 16.
2. It is alſo uſed to expreſs, Dia Vai tb=a+
viſion in Geometry, eſpecially by
3 aaa+b.
the Latin Writers, who as they
fay duc A B in CB, (draw A B into žatvlaaf b nearly.
C B) when they would have AB
3а
multiply'd by CB, or (rather) Theſe will be eaſy and expeditious
have a Right-angled Parallelogram Approximations to the Square and
made of thoſe Lines. So they ſay Cube Root.
applica AB ad CB, (apply AB to
Arron, is a piece of Lead that
CB,) when they would have CB wraps over, or covers the Vent or
divided by AB ; which is thus Touch-Hole of a piece of Ord-
CB
exprefs?d
Ав"
APSIS, is uſed as well for the
3. It fignifies alſo to fit Quan- higheſt Part of an Orbit, to which
tities, whoſe Area's are equal, but when a Planet comes, it is at the
Figures different ; as, when Euclid, greateſt Diſtance from the Sun, as
in his fixth Book, ſhews how on a the loweſt Part of that Orbit, when
Line given to apply a Parallelo- the Planet is in its neareſt Diſtance
gram, equal to a Right-lind Fi. to the Sun.
gure given. . .
The Line of the Apfis or Apfides,
APPROAChes, in Fortification, is a Line drawn from the Apheli-
are Works caſt up on both sides ; un to the Perihelium,
ſo call'd, becauſe the Befiegers, by AQUARIUS, a Conſtellation in
that means, may draw ncar a For- the Heavens, being the eleventh
treſs, without fear of being dif- Sign in the Zodiack, and is com-
covered by the Enemy. Or Ap- monly mark'd with this Character
proaches are all Sorts of Advan- en ons and conſiſts of thirty three
tages, by the Help of which an Scars.
Advancement may be made to AQUEDUCT, is a Conduit of
wards a Place beſieg’d.
Water, and ſignifies an artificial
APPROXIMATION, in Arith- Canal. either running under ground,
metick, or Algebra, is a continual or rais'd above it, and ſerving to
Coming ſtill nearer and nearer to convey Water from one place to
the Root or Quantity fought, another according to their Level,
without expecting to have is ex-
F2
cance.
}
not-
+
A RC
ARC
B
C
IX3
?
+ 2 x 4x6x7
u9&c.
notwithſtanding the Unevenneſs of dius AD, as the Arc B C is to the
the intermediate Ground. The Rois Arc DE, then the Arcs BC and
mans built ſeveral very conſider- DE are ſimilar. If the Radius
able ones in their City : And Ju-
lius Frontinus, who had the Di-
rection of them, tells us of nine
which diſcharg'd themſelves thro'
1314 Pipes of an Inch Diameter ;
and Blaſius upon Livy obſerves, that
AS
theſe Aqueducts brought into Rome
above ſooooo Hogſheads of Wa-
ter, in the Space of twenty-four
Hours.
E E
AQUEOUS HUMOUR, or the AD of any Arch D E be ſuppos'd
watry Humour of the Eye, is the I, and the Sine DF thereof be
utmoſt, being tranſparent, and of callid y, then the Length of the
no Colour; it fills up the Space that Arch D E will be expreſsd by this
lies between the Cornea Tunica infinite Series;
and the Cryſtalline Humour.
IX3X5
AQUILA, or VULTUR VOLANS, 37+3+
a Conſtellation in the - Northern
2x3 2X445 *
Hemiſphere, conſiſting of thirty 1X3X5X79
two Stars.
5+2x4x6x8x9
Ara, the Altar, a Southern Con- And if the firſt Term of this Series
ſtellation containing eight Stars. be called A, the ſecond B, the
ARACHNOIDES, is the Cryſtal- third C, the fourth D, Qc. and the
line Tunic of the Eye ; by ſome fecond be multiply'd by }, the third
called alſo Aranea. Tunica, or Cry by }, the fourth by si &c. then
Pallina, and is that which fur- that'Series will become this:
rounds and contains the Cryſtal
reaſon
_3
thin Contexture, like that of the
Web of a Spider, it has the 5
Name of Aranea. This Coat, by 6x7
+Cy2+%26+%Dy+
8x9
means of the Ciliary Proceſſes,
helps to move the Cryſtalline
&c.
Humours of the Eye nearer to, or
further from, the Retina, and per ARCHES, in Architecture, are
haps alſo to render its Figure more Parts of the inward Support of any
or leſs Convex.
Superſtructure, and they are either
ARCH, or Arc, in general is circular, elliptical, or ſtreight, (as the
any part of a Curve Line; but it Work-men improperly call them.)
is more uſually taken for any Part ARCHES (ELLIPTICK,) were for-
of the Circumference of a Circle. mcrly much uſed inſtead of Mantle-
Arcs (EQUAL) of the ſame trees in Chimneys : They had a
Circle, are ſuch that contain the Key Stone, and Chaptrels, or Im-
fame Number of Degrees.
posts, and confiſted of two Haunſes
ARCS (SIMILAR ;) if the Arc and a Scheme.
BC does contain the ſame Num ARCHES (GOTHICK,) are ſuch
ber of Degrees as the Arc DE; as are uſed in Gothick Buildings,
or if the Radius A B is to the Ra- call’d by the Italians Diterzo &
di
9
IOXII
1
or
A Ř C
ARC
di quarto acuto, or of third and. chitecture are numerous. Some of
fourth Point, becauſe they confift them are, Philander, Barbarus,
of two Arches of a Circle, meeting Salmafius, Baldus, Leo Baptiſta Alber-
in an Angle at the Top, and drawn tus, Gauricus, Demoniofus, Perrault,
from the Diviſion of a Chord into Di P’Orme, Rivius, Sir Henry Wotton,
three, four or more Parts, at plea-' Serlio, Palladio, Strada, Vignola,
ſure.
Scamozzi, Dieul'art, Catanei, Fre-
Arch'd (SKENE, SCHEME,) is a ard, Dé Chambray, Blondel,' Gold-
fiat Arch, leſs than a ſemicircular one. man, Sturmy, Dominicus de Roſi,
Arches (STREIGHT,) as the Defgodetz, Baratteri, Mayer, Gu-
Workmen improperly call them, lielmus. Theſe three laſt treating
which are uſed over Windows and of Water-Architecture. To which
Doors, &c. have plain ſtreight may be added an anonymous French
Edges both upper and under, which Treatiſe concerning the making
are parallel, but both the Ends of Rivers Navigable.
and Joints do all point towards a ARCHITECTURE (Milita-
certain Centre. They are now uſual. RY) inſtructs us in the beſt Ways
ly about a Brick and a half thick; of fortifying Cities, Camps, Sea-
which, when rubb’d, is about Ports, any other places of
twelve Inches. The levelling End Strength. And,
of this Arch is called the Skew Architecture (NAVAL,) is
Back; and the ſeveral Joints be- the building of Ships.
tween the Courſes of Bricks in the ARCHITRAVE
is the princi-
Arch, the Workmen call the Som- pal Beam, or Poitrail in any Build-
mering
ing, and the firſt Member of the
ARCHIPELAGUS, in Geogra- Entablement, being that which
phy, is a part of the Sea, con bears upon the Column, and is
taining many ſmall Iſands one near made ſometimes of a ſingle Sum-
another, and conſequently manymer, as appears in moſt of the an-
little Seas denominated from thoſe cient Buildings, and ſometimes of
Iſlands ; as, the Grecian Archipe- ſeveral Hauntes, as is uſual in the
laro, or Ægean Sea.
Works of the Moderns. It is call'd
ARCHITECT, is one that under- the Reaſon-Piece or Maſter-Beam,
ſtands Architecture, which is the in Timber Buildings; but in Chin-
Art or Science of well Building, neys it is called the Mantlepiece;
that is, of conceiving an Idea of and over the Jaumbs of Doors, and
an Edifice in the Mind, and build- Lintels of Windows, Hyperthyron.
ing it according to the fame, ſo as ARCTICK CIRCLE, is a leſler
to anſwer the End of the Builder ; Circle of the Sphere, or Globe,
and is divided into Civil, Military, parallel to the Equator, and 23
and Naval.
30'. from the North Pole of the
ARCHITECTURE ( Civil,) World, from whence it takes its
teaches how to make any kinds of Name. This, and the Antarctick
Buildings ; as Palaces, Churches, Circle, which is one parallel to the
or private Houſes.
Equator, and at the ſame Diſtance
The molt ancient Writer of Ar- from the South Pole, are call'd
chitecture extant, is Vitruvius, who the tivo Polar Circles.
lived in the Reign of Augufius the ARCTOPHYLAX. See Brötis.
Roman Emperor, to whom he dedi ARCTURUS: A fix'd Star of
cated his ten Books on that Subject ; the firſt Magnitude, placed in the
and lince him the Writers on Ar. Skirt of Arčophylax.
ARBU,
1
E 3
A RI
ARI
1
$
Area, of any ſuperficial Figure, Homberg of Paris in the Memoirs of
n Geometry, is the internal Capa- the French Academy for the Year
city or Space containd within the 1699.
Lines or Line bounding it in the AREOSTYLE, in Architecture, is
ſquare Parts of any Meaſure; as, a fort of Edifice where the Pillars
D are ſet at a great diſtance from one
another.
Argo Navis, a Southern Con-
ſtellation, conſiſting of forty-two
Stars,
ARGUMENT of Inclination, is an
Arc of an Orbit, intercepted be-
tween the Node aſcending, and the
Place of the Planet from the Sun,
being number'd according to the
Succeſſion of Signs.
ARGUMENT of the Moon's Lati-
А
B
tude, is her Diſtance from the Node.
ARIES, a Conſtellation of Stars
ſuppoſe the Side A B of the Paralle- drawn on the Globe in the Figure
logram ABCD to be three Inches, of a Ram. It is the firſt of the
or three Foot, or three Yards, &c. twelve Signs of the Zodiack, and
and the Side A C to be four Inches, mark'd thus r, and conſiſts of nine-
or four Foot, or four Yards, &c. teen Stars.
then the Area or ſuperficial Capaci ARITHMETICK, is the Art or
ty of the ſaid Parallelogram will be Science of Numbers.
twelve Inches, or twelve Foot, or Proclus, in his Commentary, upon
twelve Yards, or will contain twelve the firſt Book of Euclid, ſays, that the
little equal Squares, each of whoſe Phænicians, by reaſon of their Traf-
Sides is one Inch, or one Foot, or fick and Commerce, were thought
to be the firſt Inventors of Arith-
For the Areas of Figures, ſee un metick. Which Pythagoras and his
der their reſpective Names. Followers, as alſo the Egyptians,
AREOMETER, is an Inſtrument Greeks, and Arabians afterwards
to meaſure the Gravity of Liquors ; much improved; as Clavius and o-
and it is uſually made of a tlün fine thers tell us. But if we are to judge
glaſs Ball, with a long taper Neck, of the Knowledge of thoſe Antients
feald at the top, there being firit as in Arithmetick from their Writings
much running Mercury put into it upon the Subject, which have been
as will ſerve to keep it ſwimming in tranſmitted to us, we may ſafely
an exact Poliure.
The Stem, or conclude, that their Advances here-
Neck, is divided into Parts, which in were but very ſhort and ſcanty:
are number'd, that ſo by the depth For ſetting aſide Euclid, who indeed
of its Deſcent into any Liquor, its has given ſeveral very plain and
Lightneſs may be known by thoſe pretty Properties of Numbers in his
Diviſions : for that Flaid or Liquor Elements, and Archimedes in his Ä-
in which it finka lealt, mult be hea. renar, they moſtly confift in dry
vieft ; and that in which it finks diſagreeable Diſtinctions and Divi-
molt, will be lighteſt
fions of Numbers : as may be ſeen,
There is another rewer Inliru- in ſome ſort, in Nichomachus's and
ment of this kind deſcribed by Mr. Boetius's Arithmetick. Nor is the
3
Greck
one Yard.
now.
ARI
A RI
Greek Manner of Numeration, by gicourt, in the Hiſtory of the Royal
the Letters of the Alphabet, at all Academy of Sciences for the Year1703,
fit for the Performance of the prac- p. 105. gives us a Specimen thereof
tical Parts of Multiplication, Divi- about Arithmetical Progreſſionals;
fion, &c. with the Eaſe and Expe- where he fhews, that becauſe in Bi-
dition that they are now-a-days per- nary Arithmetick only two Cha-
formed by the Indian Figures, or racters are uſed, therefore the Laws
nine Digits.
of Progreſion may be the eaſieft of
Dr. Wallis, in his Hiſtory of Alge- all diſcover'd by it.
bra, ſays, that there are at Oxford ARITHMETIck (COMMON.)
two Arithmetical Manuſcripts of This fignifies the practical Rules of
Johannes de Sacro Bofco, who died Addition, Subtraction, Multiplica-
about the Year 1250, wherein the tion, Diviſion, &c. of Numbers,
Operations of Addition, Subtraction, and Decimal Fractions.
Multiplication, Diviſion, and Ex ARITHMETICK (DECADAL,) is
traction of the ſquare and cube Roots the Arithmetick which we uſe by
are performed much the ſame as the nine. Figures and a Cypher,
which is commonly attributed to be
Boetius's Arithmetick was wrote the Invention of the Arabians, and
in the ſixth Century. And in the was, no doubt, taken from the
ninth Century, Pjellius wrote a Number of our Fingers, which is
Compendium of the ancient Arith- ten ; becauſe, in Computations, we
metick in Greek, tranſlated into La- uſe the Fingers before we underftand
tin by Xylander, and publiſhed anng Arithmetick.
1556, at Bafil. Such a Compen ARITHMETICK (DECIMAL) is
dium too was publiſhed by Willi- the Doctrine of decimal Fractions.
chius, an. 1540. Other Writers are ARITHMETICK (INSTRUMEN-
Jordan, (whoſe Arithmetick was TAL,) is the Performance of the
publiſhed an. 1480) Barlaam the Rules of Common Arithmetick by
Monk, Frater Lucas de Burgo, Sti- Inſtruments.
fel, Nicholas Tartaglia, Maurolycus, ARITHMETICK (LOGARITH-
Heniſchius, Andrea Tacquet, Clavius, METICA L.,) is the Doctrine of Lo-
Leotaude, Wells, Metius, Gemma Fri- garithms.
fius, Wingate, Kerſey, Bayer, Hat ARITHMETICK (POLITICAL,)
ton, Cunn (of Fractions,) Weſton, is the Application of Arithmetick to
with a Multitude of others, too ma- Politicks.
ny to ſet down here. But the beſt
ARITHMETICK (SEXA G E S 1-
and moſt abſolute Work of this kind, MAL) is the Doctrine of Sexageſi-
both as to Matter, Order, Clearneſs mal Fractions.
of Expreſſion, and even Language, ARITHMETICK (Specious) is
is the Syſtem of Arithmetick, pub- the ſame as Algebra,
liſhed in Language a few
ARITHMETICK (TETRACTY-
Years ago, at London, by the very CAL,) is that wherein only 1, 2, 3,
ingenious Mr. Malcolm.
and o are uſed.
ARITHMETICK (BINARY,) is There is a Treatiſe of this Arith-
that wherein only Unity, or i and metick written by Mr. Echard W'ci-
o are uſed. This was deviſed by gel, a German. But borh Bisary A-
Mr. Leibnitz, (ſee Miſcellanea Ben rithmetick and this are uleleis' Co-
rolin. p. 336, og feq.) who thews it rioſities, eſpecially with regard to
to be ape for diſcovering the Pro- the practical Part, ſince the Decadal
perties of Numbers; and Mr. Dan. Arithmetick is received by a! Na-
1
our
F 4
tions
1
ARI
ARI
1
tions, and ingrafted in us while Sum of the Extremes is equal to
Children, and ſince the Trouble of the Double of the Mean ; as 2,
learning a new Numeration will 4, 6, are ſo ; whence 2+6=2X4.
not be ballanç'd by the Advantage 2. If there be four Quantities in
gain'd from it ; and laſtly, becauſe continual Arithmetical Proportion,
Numbers may be vaſtly more com the Sum of the Extremes is equal
pendiouſly expreſs’d by Decadal to the Sum of the Means ; as 2, 4,
Arithmetick,than by either of theſe. 6, 8, are fo; whence 2+8=4+6.
ARITHMETICK ( THEORETI 3. If never fo many Quantities
CAL) is the Knowledge or Science are in an Arithmetical Progreſſion,
of the Properties of Numbers, the Sum of the Extremes is always
ARITHMETICK of Infinites, is equal to the Sum of any two Means
the Method of ſumming up a Series, equally diſtant from the Extremes,
or Row of Numbers, conſiſting of or to the Double of the middle
infinite Terms, or of finding the Term, if the Number of Terms be
Ratio's of them.
odd; as ſuppoſe 2, 4, 6, 8, 10, 12,
This Method was firſt invented be an even Number of Terms,
by Dr. Wallis, as may be ſeen in then 2+12=4+10=6+8 ; and if
bis Opera Mathematica, vol. I. 2, 4, 6, 8, 10, be an odd Number,
where he ſhews the Uſe of it in then 2+10=2x6.
Geometry, in finding the Area's of 4. The Sum of any Number of
Superficies, and the Contents of So- Terms of an Arithmetical Progreſ-
lids, and their Proportions. But the fìon, is equal to the Sum of the
Method of Fluxions, which is a Extremes multiplied by half the
univerſal Arithmetick of Infinites, Number of Terms, or half that
performs theſe things much eaſier, Suin multiplied by the whole Num-
and a Multitude of Things can be ber of Terms ; as the Sum of all
perform'd by the latter, that the the Terms in the laſt Progreſſion
former will not touch.
ARITHMETICAL Co M PLE is=2+10 xz] or=5*2+10
30.
MENT of a Logarithm, is what that
5. The Ratio of the Sum of an
Logarithm wants of 10000000; as Arithmetical Progreffion, whether
the Arithmetical Complement of the finite or infinite, whoſe firſt Term is
Logarithm 8.15 4032 is 1.845968 ; 0, is to the Sum of as many Terms
where every Figure, but the latt 8, equal to the greateſt ; as i to 2.
is taken from 9, and that from 10. 6. The Ratio of the Sum of the
ARITHMETICAL PROPORTI: Squares of every Term of an A-
on, or PROGRESSION, is when rithmetical Progreſſion, beginning
Numbers, or other Quantities, do at o, and continued to Infinity, is
proceed by equal Differences, either
as I to 3
increaſing or decreaſing ;
7. The Ratio of the Sum of the
6, 8, 10, &c. or a, 2a, 4a, 6a, Cubes of ſuch a Progreſſion, is to
&c. or 5, 4, 3, 2, 1, or 56, 4a, the Sum of as many Terms, equal
3a, 2a, a ; where the two former
to the greateſt ; as i to 4.
Series are increaſing, and the two
8. And univerſally, if m be the
latter decreaſing, the common Dif- Power that every Term of ſuch a
ference in thoſe being. 2, and in theſe Progreſſion is raiſed to, the Sum of
Here follows fome Properties all thoſe Powers will be to as many
of Arithmetical Progreſſionals. Terms equal to the greateſt ; as I
1. If there are three Quantities to mtr.
in Arithmetical Progrefion, the All thefe Theorems, but the laſt,
2
as 2, 4,
I.
are
1
6
ARM
A SC
are demonſtrated by Sturmy, in his ftone confiſis of two Steel Shells
Matheſis Enucleata.
faften'd to one another by a Joint,
ARITHMETICAL
INSTRU and muſt cover a good Part of the
MENTs, are Inſtruments to per- Convexity of the Stone. This muſt
form Arithmetical Operations with; be alſo filed away by degrees, until
ſuch as Napier's-Bones, and Sli- the Effect of the Loadſtone is found
ding Rules, &c.
to be the greateſt poſſible.
ARITHMETICAL MEAN, is the. It is very wonderful that the Ar-
middle Term of three Quantities in mour of a Loadſtone will ſo much
Arithmetical Progreſſion.
augment its Effect, that good Stones
Ark,the ſame as Arch. Which ſee. after they are arm’d, will lift up
ARK of Direction or Progreſſion, above 150 Times more than before.
in Aſtronomy, is that Arch of the There are indifferent good Load-
Zodiack that a Planet appears to ſtones,which when unarmed weigh a-
deſcribe, when its Motion is pro- bout three Ounces; but when armid,
greſſive according to the Order of will lift up more than ſeven Pounds.
the Signs.
ARMILLARY SPHERE, is when
In the Ptolemaick Syſtem, it is the greater and leffer Circles of the
the Ark of the Epicycle, which a Sphere, being made of Braſs, Wood,
Planet :deſcribes when it is progreſ- Palboard, & c. are put together in
five according to the Order of the their natural Order, and plac'd in a
Signs.
Frame, ſo as to repreſent the true
Ark of Retrogradation, is that Poſition and Motion of thoſe Circles.
which a Planet deſcribes when it is ARTIFICIAL DAY, being the
retrograde, or moves contrary to fame as the Natural Day, is that
the Order of the Signs.
Space of Time elapſed from the
Ark of the firſt and ſecond Station, Riſing of the Sun to the Setting
is the Ark that a Planet deſcribes thereof; whence the Length of the
in the former or latter Semi-circum- Artificial Day, of thoſe inhabiting
ference of its Epicycle, when it ap- under the Equinoctial will always
pears ſtationary.
be twelve Hours ; and to thoſe that
Armin. A Loadſtone is ſaid to are nearer the Poles, the Artificial
be armed, when it is capp'd, cafed, Day is ſo much the longer; ſo that
or ſet in Iron or Steel, in order to the length of the Artificial Day to
make it take up a greater Weight, and thoſe under the Poles, (if there be
alſo to diſtinguiſh its Poles readily. any People there,) will be half Year.
The Armour of a Loadſtone, in ARTIFICIAL Numbers, Sines,
figure of a Right-angled Parallelo- and Tangents, are the Logarithms
pipedon, confifts of two thin Pieces of the Natural Numbers, Sines, and
of Steel or Iron, in figure of a Tangents.
Square, having a Thickneſs propor ASCENDING Node, is that Point
tional to the Goodneſs of the Stone; from whence a Planet runs North-
for if a weak Stone has a ttrong Ar- ward beyond the Ecliptick.
mour, it will produce no Effect ; and ASCENSIONAL DIFFERENCE,
if the Armour of a ſtrong Load- is the Difference between the Right
ftone be too thin, it will not pro- and Oblique Aſcenſion of any Point
duce ſuch an Efect as when thick- in the Heavens ; or it is the Space
er : A convenient Thickneſs for of Time the Sun riſes or ſets before or
the Armour is found by filing it after fix o'Clock; as Co-T. Lat :
thinner and thinner, until you find T. O Decl. ::R:S. of Aſcen-
its Effect to be the greateſt poſſible. fional Difference,
The' Armour of a Spherical Load-
ASCENSION
A S T
AST
ASCENSION (OBLIQUE,) is that nets, cut the Ecliptick in two Points,
Degree and Minute of the Equi- that are fixty Degrees diſtant from
noctial, reckoning from the Begin- one another, then thoſe Planets are
ning of Aries, which riſes with the ſaid to be in a Sextile Aſpect. Un-
Centre of the Sun, or a Star, or derſtand the ſame in others.
which coines to the Horizon at the ASTERISM ; the ſame with Con-
ſame time as the Sun, op a Star, in ftellation, or a Collection of many
an Oblique Sphere.
Stars into one Claſs, or Syſtem,
ASCENSION (RIGHT) of the which is uſually on the Globe re-
Sun, or a Star, is that Degree of preſented by ſome one particular
the Equinoctial, accounted from Image, or Figure, to diſtinguiſh the
the Beginning of Aries, which Stars that compoſe this Conſtella-
riſes with it in a Right Sphere. tion from thoſe of others.
R: Co-s, O's greateſt Decl. : : T. ASTRAGAL, from Aſtragalos in
Dift. from r or a : T. Right Af- Greek, the Bone of the Heel, is a
cenfion.
little round Moulding, which en-
Ascul are the Inhabitants of the compaſſes the Top of the firſt,
Torrid Zone, which twice a Year or Shaft of a Column, and differs
have the Sun (at Noon) in their only from the Torus in Bigneſs,
Zenith, and conſequently then its Height being 1 Module, and
their bodies
caſt
no Shadow.
3
Min.
Whence comes the Name of Aſcii. ASTROLABE. The Name of a
ASPECT, is the Situation of the plain Sphere, or Stereographick
Planets and Stars, in reſpect of one' Projection of the Sphere, either up-
another. Of theſe they commonly on the Plane of the Equinoctial,
reckon five different Sorts.
the Eye being ſuppoſed in the Pole
1. SEXTILE, is when two Pla- of the World, or upon the Plane of
nets, or Stars, are fixty Degrees the Meridian, when the Eye is fup-
from one another.
poſed in the Point of Interſection
2. QUARTILE, when they are of the Equinoctial and Horizon.
ninety Degrees diſtant from one Stofler, Gemma Frifius, and Clavius
another.
bave treated of this Projection.
3 TRINE, when they are diſtant ASTROLABE (SEA) is an In-
120 Degrees.
ftrument for taking the Altitude of
4. OPPOSITION, when they are the Sun or Stars,at Sea ; being a large
18o diftant.
braſsRing of about 15 Inches in Dia-
5.. CONJUNCTION, when they meter, whoſe Limb, or a convenient
are both in the fame Degree. Part thereof, is divided into De-
Kepler added eight new Aſpects grees and Minutes, with a move-
to theſe, viz. the Demi-ſextile of able Index or Label, which turns
30°, the Decile of 36°, the Octile upon the Centre, and carries two
of 45°, the Quintile of 72°, the Sights. At the Zenith is a Ring,
Tredecile of 108°, the Seſquatile of to hang it by in time of Obſerva-
1359, the Biquintile of 144°, and tion, when you need only turn it
the Quincunx of 150.
to the Sun, that the Rays may paſs
All theſe different Poſitions of the freely thro’ both the Sights, and
Planets are reckond in the Eclip- the Edge of the Label cuts the Al-
tick by the ſecondary Circles drawn titude in the Limb. This Inſtru-
thro' the Centres of the Planets; ment, if well niade, (tho' not now
that is, if the ſecondary Circles, much in uſe,) is as good, if not
drawn throʻ the Centres of two Pla- better than any of the other Inftru-
ments
A S T
AST
ments that are uſed for taking the the Contrivance well known to our
Altitude at Sea ; eſpecially for Inſtrument-Makers, of moving the
taking of Altitudes between the Index, by help of a Screw on the
Tropicks, when the Sun comes Edge of the Limb, and of readily
near to the Zenith.
and eaſily directing it, and the Qua-
AsTROLOGY, is an Art that pre. drant upon its Pedeſtal, to any
tends to foretel future Things from deſir'd Phänomena by means of
the Motion of the heavenly Bodies, Screws and dentated Wheels, is a
and their Afpects to one another, ſtill greater Improvement of this
and from imaginary Qualities that Inſtrument.
are ſuppoſed to be in the Planets and Tycho Brahe was the firſt that uſed
Stars affecting Mortals here below. a tolerable Apparatus of Aftronomi-
But as there is nothing of Truth in cal Inſtruments, which are deſcrib'd
this Art, as all diſcerning People in in his Aſtronomia Inſtaurat. Me-
this Age are very well ſatisfy'd of; chanica, printed in the Year 1602.
therefore it will be to little or no But Hevelius's Apparatus defcribd
purpoſe to explain the Terms of it. in his Machina Cæleftis, A. D. 1673.
ASTRONOMICAL KALENDER, are abundantly more ſumptuous,
is an Inſtrument engraved upon and better contriv'd than Tycho
Copper-plates, printed on Paper, Brahe's. Yet theſe, one ſhould think,
and paſted on Board, with a Braſs could not perform Obſervations fo
Slider, which carries a Hair, and exact, as if he had uſed Teleſco-
ſhews, by Inſpection, the Sun's Me- pick Sights ; for he would not uſe
ridian Altitude, Right Afcenfion, them. And that occaſion'd Dr.
Declination, Rifing, Setting, Am- Hooke to write Animadverſions upox
plitude, &c. to a greater Exactneſs Hevelius's Inſtruments, printed in
than our common Globes will ſhew. the Year 1674, wherein he deſpiſes
ASTRONOMICAL HOURS, are them on account of their Inaccu-
the equal Hours : Whereof there racy. But Dr. Halley, at the Deſire
are 24 accounted from the Noon of the Royal Society, went over to
of one natural Day, (or, as ſome Dantzick in the Year 1679, to in-
will have it, from Midnight) to the ſpect his Inſtruments, and did ap-
Noon or Midnight of the next na prove of the Accuracy of tltem,
and of his Obſervations with them.
ASTRONOMICAL QUADRANT, ASTRONOMY. The Knowledge
is a large Quadrant made all of of the Motions, Times, and Cauſes
Braſs, or of Wooden Bars, uſually of the Motions, Diſtances, Magni-
faced with Plates of Iron, having tudes, Gravities, Light, &c. of the
its Limb divided into Degrees and Celeſtial Bodies, viz.the Sun, Mcon
Minutes, and even Seconds if pof- and Stars ; explaining the Cauſes
fible, with plain Sights fix'd to one and Nature of the Eclipſes of the
Side of it, or inſtead thereof a Te- Sun, and Moon; the Conjunctions
leſcope, and an Index moving a and Oppoſitions of the Planets, and
bout the Centre, carrying either any other of their mutual Aſpects,
plain Sights, or a Teleſcope. with the time when any of them did
T'heſe Quadrants are uſed in tak or will happen.
ing Obſervations of the Sun, Planets, ASTRONOMY (SPHERICAL,) is
or
fix'd Stars.
The Ancients uſed the Conſideration of the Univerſe,
only plain Sights; but the Moderns as it offers it felf to our Sight.
have found it of vaſt Benefit to uſe AsTRONOMY (THEORETICA I.)
Teleſcopes inttead of them. And is the Conſideration of the crue
Structure
tural Day
1
AST
AS Y
1
Structure of the Univerſe ; and from their Want of the Knowledge of the
thence the Determination of the Ap. Teleſcope, and the Uſe of the Mi-
pearances thereof.
crometer, and the falſe Syſtem of
ASTRONOMY is very ancient, the World that they ſo ſtrenuouſly.
as we may learn from Porphyry, and adhered to, till Copernicus having
Simplicius in his Comment upon revived the true Pyihagorean Syſtem
Ariſtotle's 2d Book de Coelo, who about the Year 1556, in Libro de
ſay, that when Alexander the Great Revolutione Cæleftium, and after-
took Babylon, Calliſthenes, one of wards Kepler, from the Obſervati.
Ariſtotle's Scholars, by the Deſire of ons of Tycho Brahe, (in' his Com-
Ariſtotle, carried from thence to ment on the Motions of Mars, printed
Greece, Celeſtial Obſervations made in the Year 1609,) having found
by the ancient Chaldeans and Ba out the Laws of the Motions of the
bylonians, of two thouſand Years Planets, Aftronomy then began to
ftanding. And Sir Henry Savil to- gain ground, and ſhine in its true
wards the latter Part of his ad Luſtre ; and at length, by the La-
Lecture upon Euclid, ſpeaking of bours of ſeveral ingenious Perſons,
this, ſays that altho' the common (most our own Countrymen) eſpe-
printed Edition of Simplicius men- cially Sir Iſaac Newton, it is now
tions but two thouſand Years; yet arrived, perhaps, to the greateſt
in his Manuſcript it is thirty-one 'Perfection that Mortals will be ever
thouſand Years ; and Cicero, in lib. able to bring it.
1. de Divinatione, forty-ſeven thou ASYMPTOTES, are properly
ſand Years.
ſtraight Lines, that approach near-
Some of the aſtronomical Writers, er and nearer to the Curve they
are Ptolemy, who has preſerved the are ſaid to be the Afymptotes of ;
Obſervations of the Ancients, a but if they, and their Curve, are in-
'mongſt which are thoſe chiefly of definitely continued, they will never
Hipparchus, in his Almageſt.--- Alba- meet : Or Aſymptotes are Tan-
tegnius, who has given the Obſer- gents to their Curves at an infinite
vations of the Saracens, -Sacro Bof- Diſtance. And two Curves are faid
co, - Regio Montanus, -Purbachius, alſo to be Afymptotical, when they
-Copernicus, -Tycho de Brahe, - continually approach to one ano-
Lansbergius,-Longomontanus,—-Cla. ther; and if indefinitely continu’d,
vius, - Kepler, -Gallilæo, --Bayer, do not meet : As two Parabola's,
-Hevelius,-Dr. Hook ---Ricciolus, that have their Axes placed in the
- Horrocs, — Sir Jonas Moor, - fame ſtraight Line, are Aſympto-
Mr.Huygens, -Tacquet,-Flamſtead, tical to one another.
-Bullialdus,-Seth Ward, Count Of Curves of the firſt kind, that
Pagan,--Wing, -Street, -Mr. De is, the Conick Sections, only the
la Hire,-Newton-Gregory,--Mer- Hyperbola has Aſymptotes, being
cator,-Whijion,-Dr. Halley, - Du two in Number.
Hamel - Dr. Keil, - the two Cal All Curves of the ſecond kind
fini's, both Father and Son.-Mr. have at leaſt one Aſymptote; but
Leadbetter. - Mr. Brent, &c. they may have three: And all Curves
We learn from Ptolemy, that of the fourth kind may have four A-
Tymocarls, and Aryftillus left fe- fymptotes. The Conchoid, Cifroid,
veral Obſervations of the fix'd and Logarithmick Curve have each
Stars about. I 20 Years before Chrift.
one Afymptote.
But the Altronomy of the An The Nature of an Afymptote will
cients was very defective upon ac be very eaſily conceiv'd fiom that
count of their bad Inſtruments, and of the Conchoid: For if CDE be
а.
AS Y
AS Y
a Part of the Curve of the Conchoid, equal, and LlÝMI=AE? And
and A its Pole, and the right Line moreover, any Annulus, or Ring,
MN be ſo drawn, that the Parts made by Mm, or Ll, when the
BC, GD, FE, &c. of right Lines, whole Figure revolves about the
C
Diameter AP, will always be e-
D
qual to a Circle, whoſe Diameter
E
is AE.
M
FAN 3. Again, in the ſecond Figure,
if one of the Aſymptotes be con-
BIG/P /P
tinu'd out to T, and the Line T
SR be drawn parallel to the Dia-
AT
Р
А
el
C D
drawn from the Pole A, be equal
to each other, then the Line MN
P
will be the Afymptote of the Curve,
M
R
becauſe the perpendicular Dis
ſhorter than BC, and EP than DP,
and ſo on; and the Points E, &C.
B.
and p can never coincide.
1. If CP be a Diameter of the
G
G
Hyperbola R AS, and CD be the
Semi-conjugate to it; and if the meter CO, then TRXSR=A C2;
Line F E be a Tangent in the Point and if the Line PM be any where
A, and A E=FA=CD; then, if drawn parallel to the Aſymptote
the Lines CG, CG, be drawn from CS, then CPxPM will be always
the Centre C thro' the Points Eand of the fame Magnitude, that is, al-
ways a ſtanding Quantity.
4. If the Hyperbola GMR be
of
any kind, whoſe Nature, with
regard to the Curve and its Aſymp-
tote, is expreſs'd by this general Ê-
quation im ja am?", and the
M
Right Line PM be drawn any
where parallel to the Afymptote
CS, and the Parallelogram PCOM
G/R
be compleated, then this Paralle-
P
lograin is to the hyperbolick Space
PMGB, contain'd under the de-
F, theſe Lines CG, CG, will be terminate Line PM, the Curve of
the Afymptotes of the Hyperbola the Hyperbola GM indefinitely
RAS. And,
continued towards G, and the Part
2. If any right Line LM be P B of the Afymptote indefinitely
diawn parallel to the Tangent FE, continu'd the fame way, as mon is
(or even not parallel) to cut the ton; and ſo if m be greater than n,
Curve and the Aſymptotes, then the ſaid Space i ſyuarable ; but
will the Parts L by Mm be when 71, as it will be in the
E
F
I
mi
I T
G
S
Coinon
AS T
AS T
or
common Hyperbola, the Ratio of the Abſciſs x is; that is, find the
the foregoing Parallelogram to that Value of y when x is infinite, which
Space is as o to i, that is, the ſaid cannot be generally done without a
Space is infinitely greater than the Series, then will the Ordinate z of
Parallelogram, and ſo cannot be the Afymptote be equal to all the
had ; and when m is leſs than n, firſt Terms of chat Series, which do
then that Parallelogram will be to not decreaſe upon augmenting x;
that Space, as a negative Number to and conſequently the Equation of
a poſitive one, and the ſaid Space is the aſymptotical right Line
ſquarable ; and the Solid, genera- Curve will be had : And if the firſt
ted by revolving the indeterminate Term of the Series, which, upon
Space GMOL about the Aſymp- augmenting x, does grow leſs, be
tote CE, is the Double of the Cy- affirmative, the Alymptote lies
linder, generated by the Motion of between the Curve and the Ab-
the Parallelogram PCOM about ſciſs; but if not, the Curve lies
the Axis CO.
between the Afymptote and the
5. If MS be the Logarithmick Abſciſs : for no Term of the Se-
Curve, and PR an Afymptote, ries becomes equal to that Part of
and PT the Subtangent, and MP the Ordinate intercepted between
an Ordinate, then will the inde- the Leg and its Aſymptote, when x
terminate Space RPMS = PM is infinitely great ; and if ſeveral
x PT; and the Solid, generated by Values of the initial Terms of the
the Rotation of this Curve about Series coincide, ſeveral Afymptotes
the Aſymptote VP, will be of a coincide : But when a Curve has
Cylinder, whoſe Altitude is equal right-lin'd Afymptotes, which are
to the Length of the Sub-tangent, parallel to the Ordinates of that
and Semidiameter of the Baſe equal Curve, theſe cannot be determind
to the Ordinate a V.
by reducing the Value of y into a
IR
Series ; but they may be found by
reducing the Value of the Abſciſs
T
into a Series, confilling of the de-
ſcending Powers of the Ordinate ;
or elſe by ſuppoſing the Ordinate
M
P
to be infinitely great, and taking as
many Values of x as correſpond to
them; and right Lines drawn from
V
the Extremities of thoſe Values, pa-
All Curves that have infinite rallel to the Ordinates, will be A.
Legs, have one or more Afymptotes, fymptotes.
being either right Lines or Curves ; The Inveſtigation of right-lin'd
and to find generally the Nature of Afymptotes may be found for
the right Line or Curve, which is Curves of any Order, without ha-
the Aſymptote of a given Curve, by ving recourſe to Series's, by means
having the Equation of the Curve of the general Equation of that
given : Let z be the Ordinate of Order, thus : Let the Equation be
the Aſymptote, whether a right A y2 + Bxy + C x2 + Dy+E
Line or Curve, then reduce the Va a + FO. Suppoſe y = axt
lue of the Ordinate y of the given 5 + 5*, &c. then will A az ut
Curve into an infinite Series, las Bato che =0; and by extract-
to converge the ſooner, thgrcatering th 2.cots of this lail Equation,
९
WE
1
A TM
A T M
we ſhall have a, and b will be about the Moon, parallel to her
Data
Limb, which he could very well
and content
2a A +B'
perceive not to be a lucid Part of
the Sun ; for the Sun's Splendor not
A 62 +Db+ F
; and if the Equa- only by far exceeded the Silver
2 A at B
Splendor of the Ring, but likewiſe
tion be A 33 + B x y2 + C x2 y-t the lucid ſmall Part of the Sun did
D x3 + E y2 + Fxy +Gx? +H not terminate in the ſame Periphe-
y + Kx +L=0, the Roots of this ry as the Ring; and the Ring ap-
Equation A a3 + Ba? + CatD pear'd more denſe on the obverſe
=o. will give e, and b will be = Side of the Moon, than on the con-
A a? +Ba+c
trary Side, yet notwithſtanding it
and c =
3 Ea? + 2Fato
terminated in the ſame Periphery.
--3862 +Bb? Eab +F6 +HatK veral others, as may be ſeen in the
And this Ring was obſerved by ſe-
3 A a2 + 2 B a to C. Hiſtory of the Academy Royal of
Sciences, for the Year 1706.
Where a is the Inclination of the
2. Mr. De Tſchirnhauſe, at Dref-
Afymptote to the Abſciſs, b is the den, with a Teleſcope of 16 Foot
Diſtance between the Beginning of long, a little before the Beginning
the Abſciſs ; and the point in which of the aforeſaid Eclipſe, did obſerve
the Aſymptote cuts the ſame, and
a Trembling in that Limb of the
Thews on which ſides of the Aſymp- Sun that the Moon firſt obſcurd ;
totes the Legs of the Curve lie.
as he did likewiſe in the laſt Digit,
Right-lind Afymptotes may be at the Inſtant of the Obſcuration .
confider'd as Tangents to Points of Moreover, Kepler, in his Book De
the Curve infinitely diftant; fo that Nova Stella Serpentarii, fays, the
the Doctrine of Afymptotes may be fame Thing was obſerved in the
reduced to that of Tangents. Year 1605, at Antwerp and Naples,
ATMOSPHERE, is all the Air, in October, when the sun was to-
that the Earth is encompaſs’d with, tally hid. And Scheiner, in his Roja
conſider'd together.
Urſina, ſays, That in an Eclipſe of
A very ſenſible Effect of the Preſ- the Sun, in December 1628, there
ſure of the Atmoſphere is ſhewn,
was obſerv'd a Trembling about the
by drawing the Air out of two Limb of the Moon : And Hevelius,
equal Braſs Segments of a Sphere, in his Cometography, ſays, in ſome
whoſe Brims are well poliſhed, of Eclipſes the ſame Phænonena pre-
about three Inches in Diameter; for ſented it felf to him.
when the Air is drawn out of them
3. Mr. Caffini, in the Memoirs de
after they are apply'd to each other, l'Acad. Royal des Sciences, Àn. 1706.
it will require a Weight of about
p. 327. ſays, he has often obſerved
140 Pounds to pull them aſunder.
in the Occultations of Saturn, Ju-
That the Moon has an Atmo- piter, and the Fix'd Stars by the
ſphere, may be gather'd from feve- Moon, that when they come near
ral Obſervations made by Aſtrono- either the enlighten'd'or darken'd
Limb of her, their Figures, from
1. Mr. Wolf, in the Acta Erudi- being Circular, appear Oval, juſt as
torum, for the Year 1706, p. 385. the Sun and Moon, riſing or ſetting
fays, That at the Time of the great in a vaporous Horizon, appear not
Echiple of the Sun, May the ift, Circular, but Elliptical.
17011, he obſerved a lucid Ring
Атом, ,
mers.
AT T
A T T
ment.
ATOM, is ſuch a very ſmall Centre of the Sphere, but without
Particle of Matter, that it cannot the Surface of the Sphere; by a Force
phyſically be cut or divided into lef- proportional to the Square of its
ſer Parts Epicurus and his Fol- Diſtance from the Centre.
lowers firſt called the component
4. And in his Opticks he fhews,
Principals of all Bodies, which they That of thoſe Bodies th it are of the
ſuppoſed to be infinitely ſmall and fame Nature, Kind, and Virtue, by
hard, by this Name of Aloms. how much leſs any Body is than
ATTICK ORDER, is a little Or- another, the greater is its attracting
der, conſiſting of Pilaſters, with a Force, in proportion to its Magni-
Cornice architrav'd for an Entable- tude; as the Magnetical Attraction
is ſtronger in a ſmall Load-ſtone, in
ATTRACTION, is the Drawing proportion to its Weight, than in
of one Thing to another. Whether a larger one : And ſo, ſince the
among the Operations of natural Rays of Light are the ſmalleſt Bo-
Bodies upon one another, there is dies that we know of, they muſt
any ſuch Thing as Attraction, it is needs have the greateſt and ſtrongeſt
hard to determine ; and perhaps attractive Force. Now, the Attra-
moſt of thoſe Effects, that the An- ction of a Ray of Light, with re-
cients not knowing ſo well the gard to its Quantity of Matter, is
Cauſes of, may be ſolved by Pulſion. to the Gravity that any projected
Sir Iſaac Newton, in his Principia, Body has, in proportion to the Quan-
applies every where this Word to tity of Matter in that Body ; in the
Centripetal Forces; and ſays, Seet. Ratio, compounded of the Velocity
11. Lib. 1. That Centripetal Forces of a Ray of Light, to the Velocity
are perhaps rather Impulſes, if we of that projected Body; and of the
Speak phyſically : But he uſes the Flexure or Curvature of the Line,
Word, as being familiar, and eaſier which the Ray deſcribes in the Place
to be underſtood by Mathematicians. of its Refraction, to the Curvature
He demonſtrates, Prop. 58. Cor. 1. or Flexure of the Line that the pro-
1. That if two Bodies mutually jected Body deſcribes. And from
attract cach other, by Forces pro- hence he calculates, that the Attra-
portional to their Diftances, they ction of the Rays of Light is above
will deſcribe both about the com 1,000,000,000,co0,000 Millions of
mon Centre of Gravity, and alſo Millions of Times greater than the
about one another Concentrical El- Force of Gravity on the Earth's
lipfes ; and Cor. 2. Prop. the ſame. Surface, according to the Quantity
2. That if two Bodies attract one of Matter in each, and ſuppoſing
another with Forces proportional to Light to come from the Sun in about
the Squares of their Diſtances, they ſeven or eight Minutes : And in the
will deſcribe both about the com very Point of Contact of the Rays,
mon Centre of Gravity, and alſo a their attracting Force may be much
bout one another Conick Sections, greater.
having their Foci in the Centre, ATTRACTIVE, the ſame with
about which the Figures are deſcri- Attracting.
bed. And in Prop. 73, 74. Lib. 1. ATTRU10N, in Phyficks, is the
3. He demonſtrates, that any rubbing of one thing againſt ano-
Particle of Matter within the Sun ther; as when Ember and other E-
perficies of any Sphere or Globe, is lectrick Bodies are rubbed, to make
attracted by a Force proportional to them attract or emit their Electrick
the Dillance of a Particle from the Force.
AVANT
A X I
A XI
AVANT Foss, or Ditch of the Axis of a Cone, is the ſtraight
Counterſcarpe is a Moat, or Ditch, Line, or Side, about which the
full of Water, 'running round the Right-angled Triangle, forming the
Counterſcarp, on the Out-lide, next Cone, moves ; and ſo only a Right
to the Country, at the Foot of the Cone can properly have an Axis,
Glacis. It is not proper to have becauſe an Oblique Cone cannot be
ſuch a Water-Ditch, where it can generated by the Motion of a plain
be drained dry; becauſe it is a Figure about a ſtraight Line at reft.
Trench ready made for the Beſieg- But becauſe it is plain from the De-
ers to defend themſelves againſt the finition, that the Axis of a Right
Sallies of the Beſieged. Beſides, it Cone is a ſtraight Line, drawn from
hinders putting Succours into the the Centre of its Baſe to the Ver-
Place, or at leaſt makes it difficult tex, therefore the Writers of Conick
ſo to do.
Sections call likewiſe that Line,
AUGE, the ſame as Apogæum. drawn from the Centre of the Baſe
AURIGA, a Conſtellation, con of an Oblique Cone to the Vertex,
fiſting of 23 Stars in the Northern the Axis of the Cone.
Hemiſphere.
Axis of a Conick Section, is a
AUSTRAL, the ſame as Southern. ſtraight Line dividing it into two
As,
equal Parts, and cutting all its Or-
AUSTRAL SIGNs, are the ſix dinates at Right Angles : As, if
laſt Signs of the Zodiack, being A P be drawn fo as to cut the Or-
called thus, becauſe they are on the dinate M N at Right Angles, and
South Side of the Equinoctial. dividing the Section into two equal
AUTOMATA, are Mechanical or Parts, then is the Line AP the
Mathematical Inſtruments, that,
Inſtruments, that, Axis of the Section.
going by Springs, Weights, &c.
ſeem to move themſelves, as a Watch,
A
Clock, & C.
Aux, the ſame with Apogæum.
Ax, or Axe, the ſame with Axis.
N
Which ſee.
Axiom, is ſuch a common, plain,
ſelf-evident, and receiv'd Notion,
PI
that it cannot be made more plain
and evident by Demonſtration, be-
Axis (CONJUGATE, or Second)
cauſe it is itſelf better known than of an Ellipfis, is the Line E F drawn
any thing that can be brought to through the Centre, C, parallel to
prove it; as, That nothing can act
A
where it is not; That a Thing cannot
be, and not be, at the ſame time ;
M
that the Whole is greater than a Part
thereof; that no Bodies can naturally F
F
go into nothing
Axis. This properly ſignifies that
ſtraight Line in a plain Figure at
P
reſt, about which the Figure re-
volves, in order to produce or ge-
the Ordinate M N to the Ax
nerate a Solid.
AP, being terminated by the Curve,
Axis of a Balance, is that Line and is the ſhorter of the two Axes.
about which it moves, or rather
furns about
G
AXIS
1
M
N
or rather And the
Α A XI
A X I
1
Ar
8
Axis (CONJUGATE or SECOND) the Point of Incidence to the refract-
of an Hyperbola, is the Right Line ing Superficies, drawn in the ſame
F F drawn thro' the Centre C, pa- Medium that the Ray of Incidence
rallel to the Ordinates MN, MN, comes from.
to the Axis AP, which cuts the Axis in Opticks, is that Ray,
among all thoſe that are ſent to the
Eye, which falls perpendicularly
M
N
upon it, and which by conſequence
paſſes through the Centre of the
Eye.
A
Axis of Oſcillation, is a Right
TEL
Line parallel to the Horizon, pal-
P
ſing thro' the Centre, about which
a Pendulum vibrates.
N
The Axis of the Parabola is of an
M
indeterminate Length, that is, it
is infinite. The Axis of the El-
lipſis is determinate : And the Axis
Curve in the Points A and P. This of the Hyperbola is of a determinate
Axis (tho' more than infinite) is of a Length, (tho' it is more than infi-
determinate Length, which may be nite.) In the Ellipſis or Hyperbola
there are two Axes, and no more ;
found by this Proportion, as A M X and in the Parabola one. And the
PM: A Pż:: MN2: FF
AXIS in Peritrochio, is a Ma-
Axis (TRANSVERSE, or FIRST, chine for the Raiſing of Weights,
or PRINCIPAL) of an Ellipſis, or conſiſting of a cylindrical Beam,
Hyperbola, is the Axis AP, which which is the Axis lying horizon-
in the Ellipſis is the longeſt, and in tally, and ſupported at each end by
the Hyperbola cuts the Curve in a piece of Timber, and ſomewhere
the Points A and P.
about it has a kind of Tympa-
Axis of a Cylinder, is properly num, or Wheel, which is called the
that Quieſcent Right Line, about Peritrochium, in whoſe Circumfe.
which the Parallelogram, forming rence are Holes made to put in
the Cylinder, revolves. But in both Staves, (like thoſe of a Windlaſs
Right and Oblique Cylinders, that or Capſtan,) in order to turn the
Right Line, joining the Centres of Axis round more eaſily, and thereby
the oppoſite Baſes, is called the Axis to raiſe the Weight requir'd by
of the Cylinder.
means of a Rope, which winds
Axis of the Earth, is a Right round the Axis.
Line, about which the Earth re In this Inſtrument, and all ſuch
volves in the Space of 23 ho. 56 like, as all Crane-Wheels, Mill-
min. and
4
ſec. The Axis of the Wheels, &c. if the Power that is to
Earth always remains parallel to it liſt up any Weight, be to the
ſelf, and is at Right Angles with Weight as the Cicumference of the
the Equator.
Axis, about which the Rope is
Axis of a Glaſs, in Opticks, is a winded, is to the Circumference of
Right Line, joining the iniddle the Tympanum or Peritrochium,
Points of the two oppoſite Surfaces then the Power will ſuitain the
of the Glaſs.
Weight ; and if it be a little aug-
Axis of Incidence, in Dioptricks, mented it, will raiſe it.
is a Right Line perpendicular in Axis of any Planet, is that Line
drawia
A ZI
ВА С
drawn through the Centre, about Index, is faften'd a Thread, to thew
which the Planet revolves.
the Shadow of the Sun upon a Line
All the Planets, and the Sun itſelf, that is on the Middle of the Index.
except Mercury and Saturn, are ob- This Compaſs being thus fitted, is
ſerved to move about their Axes.
hung in ſtrong Brafs Rings, and
Axis of Refraction, is a right the Rings are hung in a Wainſcot
Line drawn from the Refracting Square Box.
Medium, from the Point of Refrac AZIMUTH MAGNETICAL, is
tion, perpendicular to the Refract an Arch of the Horizon contained
ing Superficies.
between the Azimuth Circle the
Axis of a Sphere, is a ſtreight Sun is in and the Magnetical Meri-
Line drawn thro’ the Centre thereof dian ; or it is the apparent Diſtance
from one ſide to another, being of the Sun from the North or South
terminated by the Surface, and is Point of the Compaſs; and is found
| the fame as the Diameter of a by obſerving the Sun by the Azi-
Sphere.
muth Compaſs, either in the Fore-
Axis of the World, is an imagi- noon or Afternoon, when he is a-
nary right Line, conceived to paſs bout five or ten Degrees above the
thro' the Centre of the Earth, from Horizon.
one Pole to the other, about which
the Sphere of the World, in the
Ptolemaick Syſtem, revolves in its
Diurnal Motion.
B.
AZIMUTH of the Sun, or any
, , ABYLONISH HOUR. A Ba.
intercepted between the Meridian byloniſh
and the Vertical Circle the Sun is the Time from the Sun-riſing of
in ; or it is the Complement to a one Day, to Sun-ſetting of the next,
Quadrant of the Ortive and Occa- being reckoned from the Sun-riſing.
five Amplitude. As R: T. Lat. BACK-STAFF, the ſame with the
::T.O's Altit. : Co-S. of the Azi- Sea-Quadrant, Davis's, or the Eng-
muth from the South at the Time liſh Quadrant, as the French call it.
of the Equinox.
It was invented by Captain Davis,
AZIMUTH COMPASS, is a Com a Welchman; and is of good Uſe
paſs that takes its Name from its for taking the Sun's Altitude at Sea,
Ufe, which is principally to find and confits of two concentrick Ar-
the Sun's Magnetical Azimuth at ches of Box-Wood; the Arch of
Sea, and does not much differ from the greater Circle being divided
the common Sea-Compaſſes. into 30 Degrees, and every Degree
It conſiſts of a round Box, ha- into five Minutes, by means of Dia-
ving a Fly and Needle in it; gonals ; and the Arch of the leſſer
and upon that Box is a broad Braſs into 60 Degrees. There are like-
Circle, having one half of the Limb wiſe three Vanes belonging to it ;
thereof divided into:90 Degrees, and that upon the Arch of 30 De.
diagonally divided into Minutes. grees being called the Sight-Vane ;
Upon this Limb there moves an that upon the Arch of 60, the Shade-
Index ; and upon this Index there Vane; and the other Vane, being
is erected a Sight, which for Con- in chc Centre of the Arches, the Ho-
veniency is to fall down with an rizon- Vane.
Hinge; and from the Top of this BACULE, in Fortification, is a
Sight, down to the Middle of the kind of Port-Cullis, or Gate, made
like
Port-C
G%
B A L
B A L
}
Tike a Pitfall, with a Counter poiſe, Equality or Difference of Weights
and fupported by two great Stakes. in heavy Bodies.
It is uſually made before the Corpse The Action of a Weight to move
de-Gard, advanced near the Gates. a Ballance is by ſo much greater,
BACULOMETRY, according to as the point preſſed by the Weight
fome, is the Art of meaſuring ac is more diſtant from the Centre
ceſlible or inacceſſible Lines, by the of the Ballance ; and that Action
help of one or more Staves. follows the Proportion of the Di-
BAKER'S CENTRAL RULE, for ftance of the ſaid Point from the
the Conſtruction of Equations, is a
Centre.
Method of conſtructing all Equati A Ballance is ſaid to be in E-
ons, not exceeding four Dimenſions, quilibrio, when the Action of the
without any previous Reduction of Weights upon each Brachium, to
them, or firſt taking away their fe- move the Ballance, are equal, fo
cond Term by means of a given that they mùtually deſtroy one ano-
Parabola and a Circle, See his ther.
Clavis Geometrica Catholica.
Unequal Weights can equiponde.
BALDACHIN, in Architecture, is rate; for if the Diſtances from the
a Building in form of a Canopy, or Centre be reciprocally as the
Crown, ſupported by Pillars, often Weights, the Ballances will be in
ſerving for the Covering of an Al- Equilibrio ; as one Ounce, at nine
tar. Some alſo call the Shell over Inches diſtance from the Centre,
a Door by this Name, and pro- will equiponderate with three
nounce it Baldaquinin.
Ounces at three Inches diſtance
BÀLL and Socket, is an Inftru- from the Centre: And upon this
ment made of Braſs, with a per- Principle is made thé
petual Screw, to hold a Teleſcope, Roman Ballance, or Steel.rard,
Quadrant, or furveying Inſtrument which weighs every Thing with
on a Staff, for Surveying, Aftrono one Weight, and is a
mical or other Uſes.
of unequal Arms, one of which CA
BALLANCE, or Scales, is one of is extended in length from the Axis
the fix fample Powers in Mecha- of Motion C, (and which ought to be
nicks, and ſerves to find out the the Axis of Equilibrium) ſuppoſe one
Leaver
8 7 6 5 4 3 2 1
A
B
D
P
Inch or leſs; the other Arm CB put farther from the Centre C,
being of a greater length, divided the Number whereat it hangs, will
into Parts, each equal to A C, and ſhew how much it weighs. For ex-
numbered by the Figures 1, 2, 3, ample, if the Weight P, at the
4, 5, 6c. then, if a Body whole Diflance 8, equiponderates the
W Veight e we want to diſcover, Weight Q at A, it muſt follow,
be hang’d on at A, and the given by reaſon the Weights are recipro-
Weight P, moveable on the con- cally to their Diſtances from C, that
trary Arm, be moved towards, or the weight is eight times the
3
weight
!
+
2 P
B: A L
BAL
weight P; that is, if P, be one upon the Arm BC, be divided into
Pound; Q, will be eight Pounds. four equal Parts, theſe, without any
Altho' this be the uſual Deſcription great Êrror may be taken for. Di-
of the Roman Steel-Yard, yet it muſt viſions, for Quarters, halfPounds, and
be falſe; when the Arms, being of one three Quarters ;. provided the Arms
continued Thickneſs, are divided into be throughout of the fame Thick-
the Parts 1, 2, 3, 4, & c. each equal neſs, and uniform Matter. But how-
to AC, unleſs the Arms have no ever convenient the Uſe of this In-
weight, and ſo much the more falſe, ftrument may be, by reaſon of its
the heavier the Matter of which not requiring ſeveral Weights ; yet
they confift, is. For, ſuppoſe the the Uſe of it is not to be too much in-
weight of the Arm CA, to be w, dulged amongſtTrades-People; who,
and that of CB to be W, and biſa thereby may deal out falſe Weight,
ſect CB in D, and C A in E, then that cannot be readily diſcovered by
will D, E, be the Centers of Gra. thoſe who buy their Goods.
vity of the Arms CB, CA. Con Sir Iſaac Newton, in his Univerſal
fequently the Weights P, W, and Arithmetic, Prop. 49. hints at a Bal-
low, will be in Equilibrio a- lance or Steel-Yard, conſiſting of
bout the Center C; it muſt be Strings only, whereby the Weight
{BCⓇW +PxPC=ACxwt of any Body E, may be known by
OXCA.. Wherefore B C will be only one Weight F. What he ſays is
2 Qtwx!AC-WXBC contained in the following Problem.
that
A String ACD B, being divided into
is, ſince 2 P is given ; PC will be al. given Parts AĆ, CD, DB, and its
ways as 2 QtwxAC-BCXW.
B
Or, ſuppofing AC to be =1, P to
be 1, and w to be i then will W
be=CB, and so P will be always
A
a's 2 Qti CB. And ſince this
1)
laſt Expreſſion is not as Q; and
therefore P not as Q; the Diviſions
с
of the Arm CB will not be as the
Weights Q. Much after the fame
way, it will appear that the Divi.
fions will be more unequal, if the
Р
two Arms confifted of a Cone or
Fruſtum of one, or priſmatical Py-
ramid or Fruftum of one. So that
the only true Way of making a Bal-
lance of this fort, is to do it mecha Ends being faſtened to two Pins A, B,
nically, viz. by firſt hanging at a given in Poſition, and if to the Point's
various different Weights, and then of Diviſion C, D, be hung the two
moving the given Weight P back- Weights E and F: To find the weight
wards or forwards along the Arm E, from the given weight F, and the
CB till there be an Equilibrium : Situation of the Point C, D. Con-
and marking down that Number tinue out the Lines A C, DB, until
expreſſing the Weight Q. upon the they meet the Lines DF, CE, in
Place where P hangs: And if the the Points Q, P; then will the
ſeveral Weights Qbe Pounds; and weight E be to the weight F, as
each of the Diſtances thus mark'd DQ to CP.
BAL-
2
1
G 3
B AL
BAR
BALLANCE of a Clock or Watch, BAND, in Architecture, is any
is that part of it which by its Mo- flat. Member that is broad, and not
tion regulates and determines the very deep; and the Word Face is
Beats : The Circular Part of it is ſometimes made to fignify the ſame
called the Rim, and its Spindle, the thing.
Verge. There belongs alſo two BANQUETTe, in Fortification,
Pallets, or Nuts, that play in the is a little Foot-pace or Elevation of
Fangs of the Crown-Wheel. In Earth, in figure of a Step, at the
Pocket-Watches that ſtrong Stud in bottom of a Parapet, or that which
which the lower Pevet of the Verge' the Soldiers get upon to diſcover the
lies, and in the middle of which Counterſcarp, or to fire upon the
one Pevet of the Crown-Wheel runs, Enemy in the Moat, or in the Co-
is called the Potans, or rather the vert-Way. Theſe Banquettes are
Potence; the wrought Piece, which generally a Foot and an half high,
covers the Ballance, and in which the and almoſt three Feet broad.
upper Pevet of the Ballance plays, BAROMETER, or Baroſcope, is an
is the Cock ; and the ſmall Spring Inſtrument for eſtimating the ſmall
in Watches is called the Regulator. Variations of the Weight or Preſ-
BALLANCE (HYDROSTATICAL) ſure of the incumbent Air. From
is a very exact Pair of Scales, for whence we can give a tolerable
making Hydroſtatical Experiments, Judgment of the Weather; and con-
relating to the Gravity of Fluids ; Gifts of a Tube of Glaſs of above
and they differ from coinmon Scales thirty Inches long, hermetically ſeal-
only in having an Hook under each ed at one End, and being filled with
Scale, for ſuſpending ſuch Bodies Quickſilver, according to the Tor-
that are to be immerſed in Liquids. ricellian Experiment is inverted, ſo
BALLANCE, or Lifra, is the as to have the open End of it m-
Name of one of the Twelve Signs merſed in ſtagnant Quickſilver, con-
of the Zodiack; the Character of tain'd in a larger Glaſs under it;
which is w; into the firſt Degree out of which open End, after ſuch
of which when the Sun comes, the Immerſion, the Quickſilver in the
Autumnal Equinox happens, and is Tube being ſufferd to run as much
about the i2th Day of September. as it will into the ſtagnant Quick-
BALLON, in Architecture, is filver, there remains a Cylinder of
taken for a round Globe, or Top Quickſilver ſuſpended in the Tube,
of a Pillar.
that will be always between 28 and
BALLUSTER, is a little Column, 30 Inches in height, above the Sur-.
or Pilaſter, either round or ſquare, face of the ſtagnant Mercury, ac-
adorned with Mouldings, and ſery- cording as the Preſſure of the Air is
ing to form a Reft or Support to more or leſs; and the upper Part
the Arm, and, in ſome meaſure, to of the Tube will be left void of
anſwer the Ends of a Balcony. common Air. This is the common
BALLUSTRADE, in Architecture, Barometer ; but there are others,
is the Continuity of one or more
Rows of Balluſters, made of Marble, BAROMETER, (DIAGONAL, )
Iron, Wood, or Stone, ſerving either where the Mercury, inſtead of riſing
for an Elbow-Rest, as in Windows, three Inches, as in the common one,
Balconies, and Terraſſes, or as a riſes obliquely near thirty Inches,
Ferce, to keep off Things from which is made by bending a Torri-
without. And thus we ſee them celian Tube of more than 58 Inches
around ſome Altars, Fonts, donc. long, at the 2t8h inch above the
Sur-
}
as the
1
BAR
BAR
Surface of the ſtagnant Mercury; above the Surface of the Earth, the
ſo that the encloſed End thereof, lower will the Mercury in the Tube
when the lower part of the Tube fink. This was obſerved firſt by
ſtands upright in the ſtagnant Mer- Mr. Pafchal, in his Treatiſe De É-
cury, is more than thirty Inches quilibrio Liquorum.
above the Surface of the ſtagnant 2. The Motion of the Mercury
Mercury. This Barometer, of all does not exceed three Inches in its
others, is the beſt,
Riſing or Falling in the Barometer
BAROMETER (MARINE,) is an of the common Form.
Inſtrument ſerving for the ſame Uſes 3. The Riſing of the Mercury
at Sea, as the common Barometer preſages, in general, fair Weather,
at Land, and conſiſts of an Air- and its Falling, foul; as Rain, Snow,
Thermometer, and a Spirit-Thermo- high Winds, and Storms.
meter ; for the Mercurial Barometer, 4. In very hot Weather, the
eſpecially the common ones, cannot Falling of the Mercury foreſhews
be uſed at Sea, becauſe it always Thunder.
requires a perpendicular Poſture, 5. In Winter, the Riſing pre-
and the Quickſilver vibrates therein fages Froſt; and in froſty Weather,
with a great Violence, upon any if the Mercury falls three or four
Agitation. See the Deſcription and Diviſions, there will certainly fol-
Uſes of this Inftrument, by Dr. low a Thaw: but in a continued
Halley, in the Philoſophical Tranſ Froſt, if the Mercury riſes, it will
actions, Nº 269. who carried one of certainly ſnow.
them along with him in his laſt 6. When foul Weather happens
Southern Voyage; and he ſaid, that ſoon after the Falling of the Mer-
it never failed to give him early cury, there will be but a little of it;
Notice of a Storm, and of all the and the ſame will happen when the
bad Weather they had.
Weather proves fair, ſhortly after
BAROMETER (PORTABLE,) is the Mercury has riſen.
one that can be conveniently and 7. In foul Weather, when the
ſafely carried about from Place to Mercury riſes much and high, and
Place, without the danger of ſpil- fo continues for two or three Days
ling the Mercury out of the Ciſtern, before the foul Weather is over,
or Vefſel, or letting the Air get in then a Continuance of fair Weather
at the bottom of the Tube; or the follows.
Mercury, included in the Tube, 8. In fair Weather, when the
breaking the Top of it off.
Mercury falls much and low, and
BAROMETER (WHELL,) is a thus continues for two or three Days
common Barometer with an Index, before the Rain comes, then a great
that ſhews the Variation of the Al. deal of Wet, and probably high
titude of the Mercurial Cylinder, Winds follow.
which at moft does not exceed three 9. The unſettled Motion of the
Inches; tho' by this Index it may Mercury denotes uncertain and
be made as diſtinguiſhable as if it changeable Weather.
were three Foot, or three Yards. 10. More Northerly Places have
The Manner of making one of theſe a greater Alteration of the Riſe or
Barometers is ſhewn us by Dr. Hook, Fall of the Mercury than the more
in the Philoſophical Tranſactions, Nº Southerly.
185.
11. Within the Tropics, and neač
1. The higher the Barometer is them, there is little or no Variation
1
G4
of
B A R
BAR
of the Height of the Mercury in all cury of the Barometer, to be open,
Weathers.
as Mr. Wolfe has ſhewn in the Aet a
12. The Words that are graved Eruditorum of Leipfick : For he ſays
near the Diviſions of the Inſtrument, he found by Experience, when it is
are not ſo ſtrictly to be minded, al- every way ſo well encloſed as to
though, for the moſt part, it will admit ſcarcely the leaſt Quantity of
agree with them, as the Riſing and external Air to fall upon the Surface
Falling of the Mercury; for if it of the Mercury, that, notwithſtand-
ſtands at much Rain, and then riſes ing, the Changes in the Height of
up to Changeable, it preſages fair the Mercury, were not in the leaſt
Weather, altho' not to continue ſo altered or diſturbed.
long as it would have done, if the 16. In England, and theſe Parts of
Mercury were higher : and ſo on the World, it has been long ob-
the contrary
ſerved, that the Riſing of the Mer-
13. It is confirmed from Barome- cury foretels_fair Weather after
trical Tables, and the Remarks of foul, and an Eaſterly or Northerly
ſeveral curious Obſervers of this In Wind; and that on the contrary,
Atrument, that the greateſt Rifings the Falling thereof, fignifies Souther-
and Fallings of the Mercury in ly or Weiterly Winds, with Rain,
Places at a good Diſtance from each and ſtormy Winds, or both; and
other, happen commonly on the in a Storm, when the Mercury be-
fame Day, and the Barometers have gins to riſe, it is a certain ſign that
been found to agree in their Mo- it begins to abate : and this has moſt
tion to an Hour, ſo far aſunder as commonly been found to be true -
Townley in Lancaſhire, and Green- in high Latitudes both to the North
wich near London ; fo that it might and South of the Equator. More-
be expected that the Weather would over, in Foggy Weather the Mer-
be the ſame at thoſe diſtant Places. cury is uſually high.
But it is often otherwiſe: And the 17. The moſt rational Account of
Barometrical Alterations of the Air, all theſe Alterations of the Riſing
extend farther than their Effects, as and Falling of the Mercury, is chat
to the Production of Rains,
of Dr. Halley in the Philoſophical
14. The mean Height of the Tranſactions, Nº-187. which, he
Barometer may be apply'd to find ſays, are cauſed by the variable
the reſpective Heights of Places, as Winds, blowing in the temperate
well as their abſolute Height above Zones, and the uncertain Exhala-
the Surface of the Sea. See Dr. tions and Precipitations of Vapours
Scheuchzer's Tables, in the Philo. lodging in the Air, whereby it
Sophical Tranſactions, Nº 405, 406, comes to be at one time, much
where he ſuppoſes the mean Height more crouded than at another, and
at the Surface of the Sea to be conſequently heavier : But theſe lat-
29.993 Inches, and allowing about ter depend upon the former. He
90 Feet for each oth of an Inch ſays, the Lowneſs of the Mercury in
in the Height of the Mercury in rainy Weather, is cauſed by the Air's
ſmaller Altitudes, or in greater, becoming lighter, ſo as not to be
according to the Tables of Dr. able to ſupport the Vapours ſwiin-
Scheuchzer and Dr.Nettleton, N°388. ming in the Air.--That the Mercury's
of the ſaid Tranſactions, you will have being lower at one Time than another,
the Height of each Place pretty near. is cauſed by two contrary Winds
15. It is not neceſſary, for the blowing from the place where the
wooden Veſſel which holds the Mer- Barometer ſtands. --That the great,
en
no
BAR
BAR
er Height of the Mercury in Fair when it has been very low, ſeems to
Weather, is cauſed by two contrary be occaſioned by the ſudden Ac-
Winds blowing towards the Place ceſſion of new Air to ſupply the
whereat the Barometer ſtands, where great Evacuation, which continued
by the Air of other Places is brought Storms make thereof, in thoſe Pla-
there and accumulated, and the Mer- ces, where they happen ; and, by
'cury preſſed up higher in its Tube.
the Recoil of the Air, after the Force
That the extraordinary Sinking of the ceaſes that impell'd it. — That the
Mercury in great Storms, is cauſed Variations of the Barometer in the
by the rapid Motion of the Air in more Northerly Places, ſeem to ariſe
theſe Storms; becauſe the Tract or from the greater Storms, happening
Region of the Earth’s Surface, where in thoſe Places, than in thoſe more
in theſe Winds rage, not extending Southerly, whereby the Mercury
all round the Globe; that ſtagnant ſhould ſink lower in that extreme;
Air which is left behind, as alſo and then the Northerly Winds
on the Sides, cannot come in ſo faſt bringing the condenſed and pon-
to ſupply the Evacuation, made by derous Air from the Neighbourhood
ſo ſwift' a Current, ſo that the Air of the Pole, and that again being
muft neceſſarily be attenuated, when check'd by a Southerly Wind, at
and where the ſaid Winds continue a ſmall diſtance, and ſo heaped,
to blow, more or leſs, according to muft neceſſarily make the Mercury
their Violence .-That the Mercury in ſuch a Caſe ftand higher on the
ſtands higheſt, cæteris paribus, upon other Extreme. That near the E.
Eaſterly or North-Eaſterly Winds, hap- quator, as at Barbadoes, and St.
pens, becauſe that in the Atlan:ic- Helena there is very little or
Ocean, on this ſide the 35th Degree Variation of the Height of the Mer-
of North Latitude; the Weſterly cury, is, becauſe of the ſteady Winds
and South-Weſterly Winds, are al- conftantly blowing in thoſe Parts,
ways Trade-Winds; ſo that, when- nearly upon the ſame Point, viz.
ever the Wind here comes up at E. N.E. at Barbadoes, and E. S. E.
Eaſt, or North-Eaſt, it is check'd at St. Helena, ſo that there being
by a contrary Gale, as ſoon as it
no contrary Current of the Air to
reaches the Ocean, and ſo the Air exhauſt or accumulate it; the At-
muſt be accumulated over this Iſland, moſphere continues much in the
and cauſe the Mercury to ſtand fame State : Altho', indeed ſome-
high. But tho' this be true for our times upon Hurricanes, it has been
Country, it is not a general Rule obſerved to have been very low.
for others, where the Winds are This is the Subſtance of what the
under different Circumſtances.--That Doctor ſays, about the Cauſes of the
the Mercury generally ſtands high in Rifing and Falling of the Mercury
Froſty Weather: is, becauſe it fel- of the Barometer ; which, altho'
dom freezes, but when the Wind not ſatisfactory, perhaps, in ſeveral
is Eaſterly or North, and ſo the Air things, yet we may very well ac-
brought here from the Northern or quieſce therein, till ſomebody gives
North-Eaſterly Countries, which are
us better.
ſubject to almoſt continual Froſt in There are ſeveral Writings about
Winter, is very much condenſed, Barometers,as Deſcartes's, Mr. Boyle's,
and accumulated by the Oppoſition Mr. Huygens's, Mr. Pafchal's, Mr.
of the Weſterly Wind blowing in Dalence's Traittez des Barometres,
the Ocean.--That the faſt Riſing of Thermometres, & Notiomeires, Mr.
the Mercury after very great Storms, De la Hire's, in the Freych Memoirs,
Dr.
BAS
i
BAS
Dr. Hook's in our Tranſactions, Nº imaginary Line, which is drawn '
185, Mr. Saul's, Mr. Amonton's in from the flank'd Angle of a Baltion
Memoirs of the French Academy, for to the Angle oppoſite to it.
the Year 1705, and many others. BASE LINE, in Perſpective, is
BAROSCOPE, the ſame with Ba- the common Section of the Picture,
rometer. Which fee.
and the Geometrical Plane.
BARREL, an Engliſh Veffel for BASE, the lealt Sort of Ordnance ;
Beer, containing 36 Gallons. the Diameter of whoſe Bore is in
BARREL, in Clock-Work, is the Inch, Weight 200 Pound, Length
Cylinder about which the Spring is four Foot, Load five Pound, Shot
wrapped.
1 Pound Weight, and 1 Inch
BARRIERS, in Fortification, are
in Fortification, are Diameter.
great Stakes, about four or five Foot BASE RING of a Cannon, is the
high, placed at the Diſtance of eight great Ring next behind the Touch-
or ten Foot from one another, with Hole.
their Tranſoms, or Overthwart-Raf Base of a Triangle. Any one Side
ters, to ſtop either Horſe or Foot, ofa Triangle may be call'd the Baſe; .
that would enter or ruſh in with but uſually and more properly, that
Violence. Theſe Barriers are com Side that lies the loweit, or is parallel
monly ſet up in the void Space, be to the Horizon, is taken for the Baſe.
tween the Citadel and the Town, And the ſame is to be underſtood of
in Half-Moons.
the Baſe of any other plain Figure.
BARS, in Muſic, are the Spaces BASILIC, a large Piece of Ord-
quite through any Compoſition, ſe- nance, being a forty-eight Pounder,
parated by upright Lines drawn a thoſe of the French being ten Feet
croſs the five horizontal Lines, each long, and thoſe of the Dutch fifteen
of which, either contains the ſame Feet.
Number of Notes, of the ſame kind, BASILIC. This, among the An-
as two Minims, two Crochets; three tients, was a large Hall, with Por-
Minims, three Crochets, &c. or tico's, Iſles, Tribunes, and Tribu-
elſe contains ſo many of various nal; where the Kings themſelves
Kinds, that are in Length of Time, adminiſter'd Juſtice. But the Name
equivalent to the fame Number of is ſomewhat differently applied now-
one Kind.
a-days, being given to Churches and
Base, in Architecture, is the Foot Temples, as alſo to certain ſpacious
of a Pillar, that ſuſtains it, or that Halls in Princes Courts, where the
Part that is under the Body, or lies People hold their Aſſemblies, and
upon the Pedeſtal, or Zocle, when the Merchants meet, and converſe
there is any ; and therefore is not together; as that, for Inſtance, of
uſed for the loweſt Part of a Column, the Palace at Paris.
but for all the ſeveral Ornaments or BASILICUS, Cor Leonis, a fixed
Mouldings that reach from the Apo- Star of the firſt Magnitude in the Cone
pbyges, or Rifing of the Shafts of ftellation Leo. Its Longitude is 145
Pillars to the Plinth.
deg. 21 min. Latitude 26 min. and
BASE of any ſolid Figure, is its Right Aſcenſion 147 deg. 47 min.
lowermoſt plain Side, or that on Ba s, in Muſic, is the loweſt and
which it ſtands; and if the Solid has the fundamental Part thereof, without
two oppoſite, parallel, plain Sides, which any Piece of Muſic is im-
and one of them is the Baſe, then perfect.
the other is alſo called its Baſe. BASSOON, a Wind- Inſtrument be-
BASE, in Fortification, is the ex- ing a Baſs to the Haut Boy.
terior Side of the Polygon, viz. the
BASS
1
1
BAS
BAT
Bass-Viol, a Baſs to the Violin. Diſtance between the Angles of the
BASTIon, in Fortification, is Interior Polygon be double the uſual
now what was antiently called a Length, then a Baſtion is made in
Bulwark; and conſiſts of two Faces, the Middle, before the Curtain. But
and as many Flanks, formerly called it generally has this Diſadvantage,
a Gorge. It is uſually made, at the That unleſs there be an extraordi-
Angles of Forts, of a large Heap of nary Breadth allowed to the Moat,
Earth ; ſometimes lined with Stone, the turning Angle of the Counter-
or Brick, but uſually faced with ſcarp, runs back too far into the
Sods, or Turfs. The Lines ter- Ditch, and hinders the Sight and
minating it are two Faces, two Defence of the two oppoſite Flanks.
Flanks and two Demi-Gorges. The BASTION (REGULAR,) is that
Union of the two Faces makes the which has its due Proportion of
utmoſt Angle, called the Angle of Faces, Flanks, and Gorges.
the Baſtion ; and the Union of the BASTIONS (SOLID,) are thoſe
two Faces to the two Flanks, makes that have their Earth equal to the
the Side-Angles, called the Shoul- Height of the Rampart, without
ders, or Epaules; and the Union of any void Space towards the Centre.
the two other Ends of the Flanks, BASTIONS (VOID or HOLLOW,)
to the two Curtains, forms the An are thoſe that have a Rampart and
gles of the Flanks:
Parapet ranging only round about
Bastion (Compos'D,) is when their Flanks and Faces, ſo that a
the two Sides of the Interior Poly- void Space is left towards the Cen-
gon are very unequal, which makes tre, and the Ground is there fo low,
the Gorges alſo unequal,
that if the Rampart be taken, no
BASTION (Cut,) is that which Retrenchment can be made in the
makes a Re-entering Angle at the Centre, but what will lie under the
Point, and is ſometimes called
Fire of the Beſieged.
BASTION with a Tenaille, whoſe BASTON, a French Word in Ar-
Point is cut off, and makes an An- chitecture, the ſame with Torus.
gle inwards, and two Points out Batten,is theWorkmen's Name
wards. This is done when Water, for a Scantling of Wooden Stuff,
&c. hinders carrying the Baſtion to from two to four Inches broad, and
its full Extent, or when it would about an Inch thick, and of a con-
be too ſharp.
fiderable Length.
BASTION (DEFORMED,) is that BATTERY, in Fortification, is a
which wants one of its Demi-Gor- Place raiſed on purpoſe, where Can-
ges, becauſe one side of the Interior non are planted, from thence to
Polygon is ſo very ſhort.
play upon the Enemy; the Platform
BASTION (DEMI,) has but one on which they are fixed being made
Face and Flank, and is uſually be- of Planks that ſupport the Wheels
fore a Horn-work, or Crown-work of the Carriages, ſo as to hinder the
This is alſo called an Epaulment. Weight of the Cannon from fink-
BASTION (DOUBLE) is that ing them into the Ground; and
which, on the Plane of the great incline a little to the Parapet ſo
Baſtion, hath another Baſtion built as to check the Recoiling of the
higher, leaving 12 or 18 Feet be. Pieces.
tween the Parapet of the lower, and
In all Batteries, the open Spaces,
Foot of the higher.
left to put the Muzzles of the
BASTION (FLAT,) is that which Guns out, are call's Embrazines
is built on a Right Line. If the and the Diſtances beiween co- lm-
great
biazuies,
BE A
BEA
or PAR
brazures, Merlons ; the Guns are BEAD, in Architecture, is a Mould-
generally about 12 Foot diftant one ing, which in the Corinthian and
from another, that the Parapet may Roman Orders, is cut and carved into
be ſtrong, and the Gunners have ſhort Emboſſments, which look like
room to work.
Beads worn in Necklaces ; and fome-
BATTERIES (Cross) are two times an Aſtragal is thus carved.
Batteries, which play athwart one A Bead Plain is ſometimes ſet alſo
another, upon the ſame Thing, form- on the Edge of each Faſcia of an
ing there an Angle, and beating Architrave. Its Convexity is uſually
with more Violence and Deſtructi- about a Quarter of a Circle, and
on, becauſe, what one Battery differs from a Boultine, only in not
thakes, the other beats down.
being ſo large. A Bead is often
BATTERY (DE ENTILADE,) is placed on the Lining-Board of a
one that ſcours or ſweeps the whole Door-Caſe, and on the upper Edges
Length of a ſtraight Line.
of Skirting-Boards.
BATTERY (EN ESCHARP,) is BEAM, in any Building, is a Piece,
that which plays obliquely. of Timber lying acroſs it, and into
BATTERY (JOINT or
which the Feet of the principal
CAMERADE,) is when ſeveral Guns Rafters are framed. No Building
play at the ſame time upon any has leſs than two of theſe Beams,
Place.
viz. one at each Head; and into
BATTERY (DE REVERSE,) or theſe Beams the Girders of the
Murdering Battery, is one that bears Garret-Floor are framed; and if it
upon the Back of any Place, be a Timber Building, into them
BATTERY (Sunk or BURIED,) the Teazle-Tennons of the Pofts are
is when its Platform is ſunk, or let alſo framed.
down into the Ground, ſo that there Beam COMPASS, is an Inftru.
muſt be Trenches cut in the Earth ment conſiſting of a ſquare Woo-
againſt the Muzzles of the Guns, den or Braſs-Beam, having ſliding
for them to fire out at, and to ſerve Sockets, that carry Steel or Pencil
for Embrazures. This ſort of Bat- Points; and they are uſed for de-
tery, which the French call en Terre, ſcribing large Circles, where the
and Ruinate, is generally uſed on common Compaſſes are uſeleſs.
the firſt making of Approaches, to
Bear. There are two Conſtel-
beat down the Parapet of any Place. lations of Stars called by this Name,
BATTLEMENTS, are the Tops the Greater and Leſer Bear, or
of the Walls of Buildings, made in Urſa Major and Minor ; and the
the Form of Embrazures and Mer- Pole-Star is in the Tail of the Leſſer,
lons, in fortify'd Places.
which is never diſtant from the
BAY, a Term in Geography, is North-Pole of the World above two
an Arm of the Sea, coming up into Degrees.
the Land, and terminated in a Nook. BEARER, in Architecture, is a
Iais a kind of leſſer Gulph, bigger Poſt, or Brick-Wall, which is trim-
than a Creek; and is larger in the med up, between the two Ends of a
Middle within, than it is at the En- Piece of Timber, to ſhorten its Bear-
trance into it ; which Entrance is ing, or to prevent its Bearing with
called the Mouth of the Bay. the whole Weight at the Ends only.
BEACONS, are Fires maintained BEARING, in Navigation, figni-
on the Sea Coaſt, to prevent Ship- fies the point of the Compaſs that
wrecks, and to give notice of In one Place bears or Itands off from
vaſions, &c.
another: Or if there are two Places,
1
3
2"ce
2
2n
4n
m 3
+
B I M
BIM
A and B, propoſed, then B is ſaid
to bear from Ā, by the Quantity of
A B 0
an Angle contained under an infi-
nitely (mall Part of a Rhumb-Line,
drawn thro' both the places at the and is called a Firſt Bimedial Line
Place A, and an infinitely finall See Euclid, Lib. 10. Prop. 38.
Part of the Meridian of the Place A.
BINOMIAL Room, is a Root come
BEATS, in a Watch or Clock, pos’d of two Parts or Members, and
are the Strokes made by the Fangs or
no more, connected together by the
Pallets of the Spindle of the Ballance
Sign Plus +. Thus atb, or 2
or of the Pads in a royal Pendulum. + 3 is a Binomial Root, conſiſting of
As the Beats of the Ballance the Sum of thoſe two Quantities.
If it has three Parts, as a+b+,
in one Hour are to the Beats in one
Turn of the Fufy, fo is the Number it is called a Trinomial Root; if it
of the Turns of the Fufy to the Con. has four, a Quadrinomial.
tinuance of a Watch’s going.
Any Root m of the Binomial
2. As the Number of turns of a + b may be extracted, or it may
the Fufy is to the Continuance of a the following Series in Form of a
be raiſed to any given Power mby
Watch's Going in Hours, ſo are the
Beats in one Hour to the Beats of the
Ballance in one Turn of the Fuſy.
Theorem, viz. P+PQn=pat
Bed of the Carriage of a great Gun, m
is that thick Plank which lies im-
AQ+""B2+m-2nd
mediately under the Piece, being as
it were, the Body of the Carriage.
* DQ+c. where P
41
BBD-MOULDING, is a Term uſed + p'e fignifies the Quantity
by Workmen for thoſe Members in whoſe Root, or any Dimenſion,
a Cornice which are placed below or Root of the Dimenſion, is to
the Coronet, or Crown. And now- be found. P is the firſt Term of
a-days, a Bed-Moulding uſually con that Quantity ; Q the next of the
fifts of theſe four Members : 1. An
Ogee. 2. A Lift. 3. A large Boul- Terms divided by the firſt and ***
tinee. 4. Under the Coronet ano-
is the numerical Index of the Di-
ther Lift.
Berme in Fortification, is a little menſions ofP+PQ; whether that
Berme in Fortification, is a little Dimenſion be Integral or Fractional,
Space of Ground, three, four, or
;
five Foot wide, left without,' be that is, repreſents a Power or Root
or whether it be affirmative or ne-
tween the foot of the Rampart, and
the side of the Moat, to receive the gative; as ſuppoſe, in the Binomial
Earth that rolls down from thence, a3 + bxx
and to prevent its falling into the
23
Moat. Sometimes, for more Secu- a3 + bxx1 – ž, will be (in the The-
rity, the Berme is palliſado'd.
BEVEL, an Inſtrument uſed by
Carpenters and Bricklayers for ad orem), +PQ7; P will be =a3.;
jufting of Angles.
Q=
BIMEDIAL. If two Medial Lines,
as A B and BC, commenſurable Letters A, B, C,D,&c. ſtand for the
only in Power, containing a rational Terms already found in the Quo-
Rectangle, are compounded, the
whole Line AC will be irracional,
tient. A for the firſt Term P
B
n
3 ( 76 xxl
2
bbx,
2,
2
3; the
BIM
BIM
1426
a a
72
x4
+
уу
2 Z
20
863
-I,
578
+
7x9
256c",
m
2
aa
a a
th
уу.
2 ee
4.63
5
B for the ſecond AQ; and ſo on. +
8197, &c. For P=y3.Q=-
For Example V 10+ ** =
- 1.1 = 3. A (P" =
=c+
cet **
9:*-->) will be= y that is
6
&c. For
16 c5 128c7
5
BC(=
AQ==x
=-
in this Cafe P= 66, Q=*,
EG°C.
21, n=2, A(=P=0(1) 3y$
Moreover the Cube Root of the
Ec; BE
Bf="AQ) = TE
;C fourth Point of d+, (that is,
="27" BQ=
) is d +
on.
+ &c. For P=de
In like manner
9 de 3 Side
Vs- xs
m= 4, n= 3, A(=P
ņ)
(that is cst ctx-mx by
will be
=d4, &c. Alſo fimple Powers may
=c+
after the ſame Manner be found.
as if the fourth Power of d te were
2 4 g
268 xx + 464x6 - 2 410
wanted ; that is dte or d tel
*:
+86c.
for in this caſe m (in the Theorem) then will P=d, Quý, m= 5,
is = 1, n=5, P=0s, and Q=
c4 x-focs
and n=1. And fo A(=P )
orallo - 45 may be put
64 x tcs
= ds. B= AQI=5 d* e. and
for P, and
for Q, then
C= 10d3 ee. D=10dde3. E =
5de4. Fes and G.
will
V75 +(4xx5
+
c4 x tcs
2c*xx+40°x+2010 (=5" FQ)=0. that is,
+
2539
t&c. The former Caſe being to dt
d7ef =
=ds +5d4 et 10d3 ec
be taken when x is very little, and t 10 dde3 + 5 den tes.
thelatter when it is very great. Again, Even Diviſion, whether it be
N
N |-
fimple or repeated, may be perform’d
by this Theorem, as if
dte
will be = N*
gys
(that is, tel er dtel
c4 xX
xs
5 c4
5
.
259
x5
5
5
5x4
( - اردو - Nx93
✓ 103
1
a
azy
I
294
+
3y3
telt
be
m
n
and A (ponad :)
(
ate=3-áa team
,
1
2
e 3
14,
7
BIM
BIM
be to be expanded into a Series, we “ Arithmetick of Infinites, ſee Prop.
118, 121, of his Algebra, Chap.
th
Q
“ 8.) and confidering the Series of
es univerſal Roots, by the Interpo-
“ lation of which, he exhibits the
-1. n=1. and A (Pmd “ Area of the Circle and Hyper-
“ bola; for inſtance, in this Series
d
AQ
“ of Curves, whoſe common Baſe
or Axis is x, and the reſpective
EIX
and ſo
dd
« Ordinates 1 - x x
- *x[?, 1-xx]},
Eg c. that is,
-- x x17, 1 **
- *|*, 1—***,
24
1 xx}, &c. I obferved that
dd
“ if the Areas of the alternate Curves
66 which are x, X X3, X
&*c.
6 *
+
*?, &c. could be interpo-
From theſe few Examples the
“ lated, we ſhould, by this means,
“ obtain the Areas of the interme-
great Uſe of this wonderful Theo-
“ diate ones; the first of which
rem may, in ſome meafure, appear.
But indeed its Uſes are almoſt infi 1-xx17, is the Area of the
nite ; comprehending the Method
" Circle: in order to this; firſt it
of Indiviſibles, the Arithmetick of
Infinites, the Doctrine of Series's ;
was evident, that in each of theſe
“ Series's the firſt Term was x, chat
and many other Concluſions, where-
" the ſecond Terms x3, x3,
in Diviſion and the Extraction of
Roots are neceſſary.
« 73, 73, &c. were in an Arith-
Our great Sir Iſaac Newton firſt
“ metical Progreſſion, and conſe-
found out this Theorem, and ſent it
quently the two firſt Terms of the
in a Letter, in the Year 1676, to
“ Series to be interpolated muſt be
Mr. Oldenburgh, the (then) Secretary
of the Royal Society, for him to
3
3
communicate to Mr. Leibnitz, as
may be ſeen in a little Book, called
&c.
Commercium Epiſtolicum de varia re
3
mathematica inter celeberrimos pre-
“ Now for the Interpolation of
fentis fæculi mathematicos: But no “ the reſt, I conſidered that the De-
where tells us his manner of inveſti “ nominators 1, 3, 5, 7, &c. were
gating it, nor gives any ſort of proof (in all of them) in an arithmetical
thereof. He ſays, indeed, in his " Progreffion, and conſequently the
next Letter to the above-named Mr. " whole Difficulty conſiſted in find-
Oldenburgh, (to be found in the Book ing out the numerical Coefficients.
but now mention'd) chat the Occa " But theſe in the alternare Areas,
ſion of its Diſcovery was this: " which are given, I obſerved were
“ Not long (ſays he) after I had or the ſame with theFigures of which
“ ventured upon the Study of the “ the ſeveral aſcending Powers of the
“ Mathematicks, whilft I was per s Number u do confift, viz. 11°,
uſing the Words of the celebrated 11', 112, 113, 114, ETC. that is,
“ Dr. Wallis, (viz. the Doctor's “ firſt 1; the ſecond 1,1; the third
***
2 x3
1
73
t
1
B I M
BI M
6
6
1
m I
O
Х
2
X
1
2
X
m be
M
66 riſe 4*
6
1
6
6
ter.
1, 2, 1; the fourth 1, 3, 3, 1; • In the ſame manner, the Areas
“ the fifth 1, 4, 6, 4, 8, &c. to be interpolated of the other
“ I apply'd myſelf therefore (ſays Curves might be produced, as
he) to find out a method by which might alſo the Area of the Hy-
" the two firſt Figures of this Series perbola, and the reſt of the alter-
might be derived from the reſt ; nate Curves in this Series
" and I found, that if for the ſecond
it**]i, i+xx]{, 17x2]},
Figure or numerical Term I put m,
" the reſt of the Terms will be pro 1 + x x?, &C. By the ſame way
“ duced by the continual Multipli-
· likewiſe other Series's might be
cation of the Terms of this Series
interpolated, and that too if they
-3 • ſhould be taken at the diſtance of
X
3 4
two or more Intervals.
4
· This was the way by which I
&c.
' firſt opened an Entrance into theſe
5
“ For inſtance, if the ſecond Term Speculations, which I ſhould not
put
for
have remember'd, but that in turn-
and there will a-
42
ing over my Papers a few weeks
that is 6; which ago, I, by chance, caft my Eyes
upon thoſe relating to this Mat-
" is the third Term. The fourth
• After I had ſo far proceeded, it
"6 Term will be 6 x that is
immediately occurr'd to me, that
3
m 3
« the Terms 1-**7], 1**]},
4. 4 X
I, is the fifth
4
14, 1
- 4
si - x x?, Sg c. that
v Term; and the fixth is
4 X
is, 1, 1- * *, I - 2 x * + 44,
So; which ſhews the ſeries is
3. * x + 3 ** *&c.
" here terminated in this Caſe. might be interpolated in the ſame
· This being found, I apply'd it • manner as I had done the Areas
as a Rule to interpolate the above generated by them; and for this,
( mention'd Series ; and ſince in the o there needed nothing elſe but to
• Series, which will expreſs the Cir leave out the Denominators 1, 3,
• cle, the ſecond Term was found to “ 5, 7, &c. in the Terms that ex-
« preſs the Areas, that is, the Co-
o be Therefore I put m =Ž,
o efficients of the Terms of the
3
« Quantity to be interpolated
• and there was produced the Terms
(T-
1 /
017, or
ra
3 2 niverſally T— **1") will be had
// 3
* by the continual Multiplication of
or + i +
o the Terms of this Series, m x
4
Tżs and ſo on ad infinitum. Hence m-1
• I found that the Area ſought of
3
4.
• the Segment of the Circle is x -
x3
1 / 2 x 5
* Thus for example, 1 -- * *
1 / 2 x² & *4 to *, Ec.
3
7
5
6 and 1 X X
I- { *2
9
6
m
I
* * 쪽
​I
2
I
I
XX
; Or U-
1
2
***
Oro 8
X
2
1 /
I
ti
Х
or
2
х
m 3 , &c.
Х
2
2
747
mange
1
Tigene
1
3
I
1
* *2
5
&c.
ett
XX.
BIN
BIN
- the Terms after theſe in infinitum
*4 + 18x6, &c. and 1- xx1
vaniſhing; and ſo I - *xx
1-
** - **,
X', &c. twice multiplied
• into itſelf produced 1 --- As
« Thus I diſcover'd a general Me " this was a certain Proof of the
thod of reducing radical Quantities Truth of theſe Concluſions, ſo I
into infinite Series by the binomial was thereby naturally led to try
Theorem, which I fent in my laſt " the Converſe of it, viz. whether
· Letter, before I obſerved that the " theſe Series's that now were known
' fame thing might be obtain'd by to be the Roots of the Quantity
• the Extraction of Roots.
- X X, might not be extracted
< But after I had found out that
. thence, by the Rule for Extraction
• Method, this other way could not • of Roots in Arithmetick; and upon
long remain unknown; for, in or ( trial I found it to ſucceed accord.
. der to prove the truth of theſe O-
ing to my Defire.
perations, I multiplied 1 - *2 I ſhall here ſet down the form of
•=*+- 15*0,&c. by itſelf, the Proceſs in Quadratics :
• and the Product is 1 -% x, all
I
.
1-*c (1-{** ***
Tax, &c.
I
xx +
4
x4
*++* +48
*
74 **, &c.
This being found, I laid afide viz. when the Exponent of the Bi-
the Method of Interpolation, and nomial is an whole Number, and
• aſſumed theſe Operations as a more that either by a kind of Induction,
genuine Foundation to proceed deduced from the Obſervation of the
upon. In the mean time I was not Series's of the Co-efficients of the fe-
ignorant of the way of Reduction veral Powers of a Binomial, ſuppoſe
by Diviſion, which was ſo much a tot, and the Doctrine of figurate
• eaſier.' Thus far the Great New- Numbers; or elſe by the Method of
ton: who alſo ſays, in the ſame Let Increments or Fluxions; or ſome
ter, that the diſmal Plague in the other the like obſcure, ſtrained, un-
Year 1665 made him remove from fatisfactory, and unnatural way. See
Cambridge, and think of other Ralphſon's Hiſtory of Fluxions, Jones's
things. This admirable Theorem, Synopſis, Sterling's Enumeration of the
which is put upon his Monument in Lines of the third Order, Wolfius's
Weſtminſter-Abbey, has never been Algebra, Brook Taylor's Methodus In-
yet demonſtrated, although many crementorum, Cunn's Method of In-
able Mathematicians have made va crements, in this Dictionary under
rious Attempts to come at the Rea- the word Series.
fon thereof. But in my Opinion, The Perſon aforeſaid has alſo
they have all faild; for all that ever given two Theorems as Rules for
I ſaw done, on this Subject, amounts reducing Binomials, conſiſting of ra-
to no more than finding the Truth tional and ſurd Quantities, or both
of the Theorem in one Caſe only, furd Quantities, to more ſimple
H
Terms
A
· BIN
BIN
in
2
!
2
n2
gt
r
or
>
2 S
2
2
2c
6
is i.
4
/ 24
2.
Terms where poffible. And this greater than 56; its cube Root 1:2
you will find at the Beginning of his the neareſt Number 4; and ſo r = 4.
Algebra. They are theſe: 1. Īf A ex- Moreover by extracting all that is
presſes the greater part of a Quantity, rational from AVR or V968,
and B the le fer part: then will
it will be 22 v 2: Therefore v
A+VAA BB
the radical Part of it will be s,
be the Square
of the greater part of the Roots and
5
A VAABB the Square of
in the neareſt
✓
the lefer part, to be added to the
greater with the Sign of B. So that integral Numbers is 23 therefore :
if the Binomial be 3 +78; (A = 2. Laſtly, is is 2 V 2.
being = 3, and B ✓ 8) we ſhall
have the ſquare Root of 3 +18 ttss~~ n is 1. and Vē or VT
I+In like manner 32
Sir Iſaac Newton has not thought
fit to lay down a Demonſtration of
18
Se-
theſe two Theorems, or Rules; which
condly, If A+B be a Binomial, are much more elegant and general
whoſe greater Part is A, and the In- than thoſe given us for extracting
dex of the Root to be extraeted f, and the Roots of Binomials, in Van-
n be found to be the leaſt Number, Schouten's Commentary upon Des Carr
whoſe Power n can be divided by tes's Geometry. But Mr. :'Grave-
AA B B, without a Remainder, ſande, at the latter part of his Alge-
and l be the Quotient: And if bra, has been at the pains to give
us a Demonſtration of the latter of
VA+B x Vē be computed in the the ſaid Theorems, judging (I fup-
A + B x V Q be computed in the poſe) the former to be ſo eaſy, as
alareſt integral Numbers, and the
not to ſpend time about evincing its
ſame be called r, and if Avē be Truth." In order to which, he pre-
divided by the greateſt rational Di- miſes eight Lemma's; which are
vifor, and the Quotient bes, and if theſe :
1. If to any Power whoſe Index
rt
is c, be elevated the Binomial a tb,
in the neareſt integral Num- and the Terms of this Power alter-
nately taken, (that is, the ift, 3d, 5th,
bers be t, then will
7th, &c. and the zd, 4th, 6th, 8th,
ts + Vitas
&c.) be united into one Sụm, and ſo
be the Root whoſe the whole Power be divided into two
Parts; the Difference of the Squares
va
of the Parts will be äa -bble.
Index is c, provided the Root can be
extracted. So that the cube Root of whereof a is the greater, and the
2. If a and b repreſent Numbers,
968 + 25 will from hence be 2
Binomial Vātnē be elevated
2+1. AA-- BB being = 343, to the Power c, and this Number be
its Diviſors 7, 77; n = 7, and odd, this Power will be a Binomial
= 1. alſo the Root of the firſt part one of whoſe Members is multiplied
of A + B x V , or V968 +25 by Va, and the other by VT
being extracted, will be a little and theſe Members will be the Parts,
(Lemma
C
2. $
n
7
BIR
BIO
x-d
a
0.
✓7.
(Lemma 1.) of which the greater is be conceiv'd to be generated or pro-
that which is multiplied by va.
duced from the continual Multipli.
3. If the ſame things being ſup- cation of four fimple Equations, (as
poſed, the Number o be even, the if x be = a, x=b, x=1,x=d,
O,
-b=0, x
Power forms a Binomial, one of or x
mom af
whofe Members is rational, and there, do; then will x
other multiplied by vab, the xx — bxx
.6 x x - c x x dobe.
Members will be alſo the Parts men- get a biquadratic Equation) or, from
tion'd in Lem, I.
the Multiplication of one ſimple E-
4. Any Power of a numerical Bi- quation, and a cubic Equation, (as
nomial va tv7 has both its & a XX3 + 6x2 to date=0)
Members pofitive; the Power of a or laſtly, from the Multiplication
Binomial or Apotome va-vī of two quadratic Equations, as
has one Member negative; and the x2 + bx to 22 oto d x + ?
Members themſelves do not differ,
whether it be tonī or
2. Any biquadratic Equation may
5. If a Binomial VatVb be firſt reducing it to another, wanting
be reduced to a cubic Equation ; by
raited to a Power whoſe Index is c, the fecond Term. If the propoſed
the Difference of the Squares of the
one does not want its ſecond Term,
Members of the Power is a bl: and fuppofing this laſt to be pro-
6. The Root of a Binomial whoſe duced by the Multiplication of two
affumed quadratic Equations, and
Index is c, that is v, cannot be then finding the Values of the ſeve-
extracted, unleſs the Difference of ral Co-efficients of theſe laft Equa-
the Squares of the Parts of the given tions, expreſs'd in the known Co.
efficients of the Terms of the bi-
Binomial has vam rational. quadratic Equation; whereby a new
7. If two continual decreaſing Équation will be had, conſiſting of
geometrical Progreſſions have the four Terms, containing only the
middle Term common, the Diffe- fixth, fourth, and ſecond Powers of
rence between the firſt Terms of the the unknown Quantity, and a known
Progreſſion will be greater than the Quantity, which in reality is but
Difference between the laſt.
a cubic Equation, being reducible
thereto by ſubftituting ſome un-
8. The vof a Binomial can- known Quantity for the Square of
not be extracted, if c be an even that in the Equation, wherein the
Number, unleſs the greater Mem- unknown Quantity has six Dimen-
ber of the given Binomial be ratio- fions. But it mult be confeſs'd that
nal.
this Operation is long and trouble-
BIQUADRATIC EQUATION, in ſome in moſt Cafes.
Algebra, is any Equation conſiſting Take the following Example from
of not more than four Terms, and Sir Iſaac Newton's Algebra: Let 24
where the unknown Quantity of one to a x3 + 6*2 focx+d=o be
of the Terms has four Dimenſions : a biquadratic Equation, having all
As *4 to a x3 + 6*2 to cxt its Terms : tranſmute the ſame into
d=, is a biquadratic Equation, another wanting the ſecond; which
becauſe the Term x4 is of four Di- let be 44 fo q * * to po* ts
menſions.
Now let us ſuppoſe this Equation to
1. Any biquadratic Equation may be generated by the Multiplication
1
H 2
of
BIĆ
BIĆ
$4
A to
toeg
*}
f
.ee
2
e
2
rr
e e
of two quadratic Equations xx + = o above given, will be always
* + f = 0, and x x - ext of poſſible.
=o; that is, let it be the ſame as 3. In Deſcartes's Geometry there
tg
g
* + fg finding the Roots of a biquadratic
ef
Equation: and another in Sir Iſaac
= 0; then by comparing the Terms Newron's Algebra much more ele-
ogether, we ſhall have f+g ee gant and general, extending to E-
=q, ég-ef=r, and f g=s. quations of fix, eight, and ten Di-
menſions, which is to find a ſurd Di-
Wherefore g tec=f+8 = = viſor, whereby to try to divide the
Equation into two equal Parts, and
then to get the Roots of the Parts.
+
4. Mr. Deſcartes was the first who
has fewn how to find the lineal
Roots of cubic, and conſequently of
steer
biquadratic Equations, (ſince theſe
- fi
laſt can be always reduced to cubics)
by the Interſection of a Circle and
Parabola ; and after him ſeveral
99 +2eegtet
others have made Improvements in
=fg) this Buſineſs: Amongſt others, ſee
4
Baker's Geometrical Key, Sluſius's
=s; and by Reduction e' + 2qe+ Meſolabium, the Philoſophical Tranſ-
actions, Nº 188, 190. the Marquis de
eerr = 0, put y for e e, l'Hospital's Conic Sections, Wolfius's
For it
and then it will be go3 + 2q9y would be foreign to my Deſign to
Elementa Matheſeos, &c.
+4y=-==-. Find the Root
-gr=. Find the Root extenſive Subje&t here; but the Con-
be ſufficiently particular upon this
45
or Roots of this Equation, and put- ftruction of the following biquadratic
Equations x4 + px?
ate
x 4
p x² at q =o being ſhort,
ting Vj = ,
and perhaps uſeful, may not be dif-
plealing to ſuch who delight in theſe
things. The former of theſe Equa-
ateetan
tions has always two real equal
= 8, and extracting Roots, one affirmative, and the
the Roots of the aſſumed quadratic other two being imaginary: and the
other negative, and no more; the
Equation sxx + ex+f=0, xx
latter has two Pair of equal real
ex + g = 0; their Roots will
give the four Roots of the given bi-
IP
Roots, when
is leſs thang.
quadratic Equation *4 + 9xx +
4
x +so, viz. x =
1. Let it be x4 + D x2
9
Vee-5, and x = {e IV extract the Root, and then will **
And if the four Roots of be= 1p+vatipp; and ma-
8.
the given biquadratic Equation be
poffible, the three Roots of the cubic king : 1p :: 10:V9, we ſhall
Equation y3 + 29yy 450 have x x
Vatrex
+49
45
? = 0, and
ee
-f;
2
>
2
1
e
+1 -1
O,
ce
4
+99
4
4
.
BIR
BIR
✓ ý tre; and fox=
ctv Nato x ~ ✓
9.
Now draw the two right Lines A E,
CF, interſecting each other at right
to get the leſſer affirmative or nega-
tive Root, every thing elſe as before,
only make (fig. 3.) BE (= AH)
E
!
B.
o
А
B D
E
1
F
H
D D
F
A
G
• px
xv
4
و
upon the
nary Roots.
Angles at B, make AB=
92
and BC=-6, join AC, make B & the Difference between AB and DC,
AB + BC, and upon A E de- and BG = AC, and then will be
fcribe a Semicircle cutting C F in
x be the leſſer affirmative or
the Point F, then will BF be = negative Root of the Equation.
#x, being the only real Roots of
5. In all biquadratic Equations,
the Equation.
if the Sign of the laſt Term be ne-
2. Let the Equation be x4
gative, it can have but two imagi-
+9=0; then in like manner, as
nary Roots.In any biquadratic
above, x will be =
Equation wanting the ſecond and
fourth Terms, if the Signs of the
cVT
other three be. all affirmative, its
9
IS four Roots will be all imaginary.
So alſo will a biquadratic Equation,
draw ACE
and
fame deſcribe a Semicircle A B C; (when affirmative) have four imagi-
having only the firſt and laff Terms
E
6. In any biquadratic Equation
having all its Terms, if of the
B
Square of the Co-efficient of the fe.
cond Term be greater than the Pro-
U
C
duct of the Co-efficients of the firſt
А.
and third Terms, or of the Square
F
of the Co-efficient of the fourth
G
Term be greater than the Product
of the Co-efficients of the third and
fifth Terms, or of the Square of
in which apply A B=C, and draw the Co-efficient of the third Term
the right Line EBCG thro' B and greater than the Product of the Co-
C; make ÇG=AC and B E efficient of the ſecond and fourth
A B biſfect £ G in the Point D, and Terms; all the Roots of that Equa.
with the Diſtance D E deſcribe a tion will be real and unequal: and
Semicircle cutting B A (continued) if either of the ſaid Parts of thoſe
in the Point F; then will BF = Squares be leſs than either of thoſe
+ x be the greater affirmative or Products, that Equation will have.
negative Root of the Equation. But imaginary Roots.
za
HS
BIR
BI
7. Being much pleaſed with the
following elegant Conſtruction of a DF=
biquadratic Equation 24 - p%3
2V I join A F, and upon
tazz go z fu s = 0. (whoſe Af deſcribe a Semi-circle ADF,
Roots are all affirmative, by means
of a'Circle and equilateral Hyper- and about the Centre F deſcribe'a
and in the fame apply AG=V9,
bola, which is Van Hadden’s a Dutch Circle paſſing thro' the Point G,
man, as we find in Schouten's Com-
which · Circle will cut or touch the
mentary upon Deſcartes's Geome-
try) I cannot omit laying down his Hyperbola in the fame Number of
Operation, which is thus.
Points' as the Equation has different
Roots; from which, if the Perpendicu-
at right Angles, to one another, and lars H I, hi, hi, be drawn to the Line
at right Angles, to one another, and À C, theſe will be the Roots fought.
in A B, take the Line AD=P Where it muſt be obſerved, that if
and from D draw DF parallel to
A C, and in this find the Point E
A G ſhould be too great to be in-
ſuch, that the Rectangle ADXDE fcribed in the Semicircle deſcribed
upon A F, or the Circle G H h fo
be equal
v
and thro' E deſcribe ſmall as not to cut or touch the Hy-
an equilateral Hyperbola HE habout perbola, it is a Sign that all the Roets
the Afymptotes A B, A C. také of the Equation are imaginary.
S,
TO
and leasinsvavassador
i
1
E
H
I
R
А
D
B
Novo
%,
met
or
-
2V
The Demonſtration is eaſy : for
fuppofing IH=%, and ſince A D
XDE = AIXIH = V, it will
✓
be AI=DK
and KF
2 N
رد
2
and ſo KF
andalen
+
1*
22
4 $
But
BIO
ΒΙΟ.
But KH=-ip, or -x; Quantity drawn into the Abfcifs
A P, equal to the Square of the
and fo KH=z
%%+ pp.
Fig.3
2
Wherefore FH=
.to
B M
2
s
23
1
21
+%%--*+*pp; and ſince
A
P
-2
2
this is = AF- AGPpt
IM
9, by ordering the Equation
45
we fhall have 34 medio p 23 +9
N
32'
go z for so.
correſpondent Ordinate AP, plus the
BIQUADRATICAL PARABOLA, Rectangle, under the Difference of
is a Curve Line of the third Order, two invariable, unequal right Lines
having two infinite Legs AM, AN AB, AC, and the faid Abſciſs, to-
tending the ſame way; being of ſuch gether with the Difference of the
a Nature, that the Cube of ſome in. Rectangle under theſe variable Lines,
variable Quantity, drawn into the all drawn into the Square of the ſaid
Abfcifs AP (ſee fig. 1.) is equal to correſpondent Ordinate, (ſee fig. 3.)
the ſquar'd Square, or fourth Power That is, fuppoſing AP, *, Þ M,,
of the correſpondent Ordinate PM; AB, b, AC, c, and the invariable
Quantity (whoſe Cube is drawn
Fig.
into the Abſciſs) a; the Equations
of the Curves will be a 3x=yt,a3 x
- 24 a 2 x4,23 x **
*4-at-b
X x3
ab * x.
A
P
It is very eaſy to find Points thro'
which one of theſe Parabola's is to
paſs, by common Geometry alone,
N
by firſt reſolving the Equation of the
or elſe the Cube of an invariable Curves into Analogies, and then
Quantity drawn into the Abfcifs aſſuming fourth Proportionals. But
A P equal to the Difference of the more eaſy ftill, by means of two
common Parabola's. The Way of
doing which for the Curve of fig. 1.
(expreſſed by the Equation a3 x=94)
B
being very ſhort, take as follows:
Let AN be a common Parabola,
Р
B
Fig. 2 M
N
MI
Q
N
R
Squares of the correſpondent Ordi-
nate PM, and the invariable Line
AB, or AC, drawn into the Square
of the ſaid Ordinate (ſee fig. 2.) Or
laſtly, the Cube of an invariable
S
A
H 4
P
whoſe
BIQ.
BIR
Fight
n
2
2
whoſe Axis is A Q.land AR another, be taken in the infinite Abſciſs AS
whoſe Axis is AS, at right Angles whether on this fide A, between
to A 0. Take any Point P in SA, A, B ; B, C;C, D; or beyond D; ſo
continued out, and draw PN parallel that it is an eſſential Property of the
to A Q, cutting the Parabola AN Curve.
in the Point N, and from N draw If the Curve has no ſerpentine
N R cutting the Parabola A R in R. Part (as that of (fig. 6.) the Equation
Then, if in PN you take P Mequal
to QR, the Point M will be one
Point thro' which the biquadràtical
Parabola will paſs. And after the
ſame manner may any Number of
рів
Points be found.
A S
IfAS ( fig.4.) be an Abſciſs to this
will then be more fimple; for in
Fig.4
this caſe, it will be pmxa =Apx
T*
M
M
BP
p B. whether the Paint p be taken
А
on this ſide A or beyond B.
EP S
CPD
This Curve is of much uſe in con-
m т
172
Curve, and the right Lines pm, PM, the third Order, determining their
Curves of
at right Angles to them be Ordi- the third Order, determining their
Numbers, different Species and Fi-
nates, and Ap, or AP be called x,
and p m, PM, -y, ty,
Y, + y, and a,
gures. For Example, Let it be
b, c, d, e, f, are invariable Quan- of the different Species and Figures
required to conſtruct and find one
tities; then will the Equationa y = of the Curves expreſſed by the E-
Nature of that Curve, or the Re- quation zx = ax4.6*3.6*2.dx.e.
lation of each correſpondent Ab where % is an Ordinate, x the Ab-
fcifs A p; A P, and Ordinate p m, Quantities
. Let MB MC DM
ſciſs, and a, b, c, d, e, invariable
,
tion of the Curve poffißle ; and the E M (fig. 7.) be a biquadratical
four Lines AB, AC, AD, AE,
are the four Roots of the Equation
M
ob x4. cx3. d x2.ex. f.
If the Beginning of the Abſciſs
M
M M
A (ſee fig. 5.) be taken in the Curve, A
DPE
PN PS
M
Fig.5
T
CP / D
M
S
A B
a
S
in
then will any Ordinate Pm drawn
M Fig. 8
m
K
into the invariable Quantity a of a
G
proper Magnitude be equal to the Parabola, whoſe Abſciſs AS, cats
Product of Ap, into BP, into CP, the Curve in four Points B, C, D, E,
and into Df, wherever the Point and the Relation of any Value of
A
Fig 7
BPC
per
M
in
m.
Canduan
LOL
If
CPU
e
m т.
P
BIO
BIO
AP (*) to the correſpondentValue of 633.682. d x. 6, has its two mid.
PM (y) be expreſſed by theEquation dle Roots equal, the Curve will be
= ax4.693.cx. dx e. p being that of Fig. 9. having a conjugate
a given Quantity; this done, draw
any abſciſtal Line a s parallel to AS,
Fig.
at a convenient Diſtance from it,
and from the Points A, B, C, D, E,
let fall the perpendiculars A a, Bb,
CC, D d, Ee to as, and taking any
Abſciſs AP (in fig. 7.) and corre-
fpondent Ordinate PM; continue
down PM to cut the Abſciſs a s (of
Point P between the oppofite para-
fig. 8.) in p, and make p m both a-
bove and below as, equal to v py, its two letter or greater Roots equal,
bolic Legs. 2. If that Equation has
or as PM (fig. 7.) Then will the and the other two unequal, the
Points m, m, be thoſe of the Curve Curve will be that of fig. 10. con-
required; and thus may an infinite
number of other Points be found.
But becauſe PM (9) between B and
C, and D, E, are negative, and fince
Fig. 10
the ſquare Root of a negative Quan-
tity cannot be taken; it follows that
no Part of the Curve wanted, will
fall between the Points b, c; and d,
é; ſo that the Curve conſiſts of two
oppoſite infinite Parts FBG, IeK, fiſting of a pure Parabola, and a no-
with an Oval c m d m between them, date Parabola. 3. If the two mid-
having the Line a s for a Diameter, dle Roots be imaginary, the Curve
and theſe Parts will be Bell-form or will be that of fig. 11. conſiſting of
diverging Parabola's. And this wi
always be the caſe when the Equa-
tion 0 a x4. b 33. C x2. d x. l,
Fig.11
has four real and unequal Roots, AB,
ĄC, AD, AE, or ab, a c, ad,
1
Sant
ae.
There are five more different pa-
rabolic Curves expreſs’d by the E-
quation aforeſaid, where the greateſt
Term a x4 is Affirmative ; all of two pure Parabola's. 4. If three
which may be conſtructed from a of the Roots be equal, the Curve
biquadratical Parabola, after the will have a Cuſpe or triple Point, as
ſame way as has been ſhewn already that of fig. 12.
for that of kg. 8. The Difference
being only, in the abfcifſal Line AS
Fig.12
cutting the Curve only in two Points ;
touching it in two Points, or cut-
ting it in two ; touching it in one
Point, and cutting it in two; touch-
ing it in three Points, and cutting
it in one; or not touching it at all:
that is, 1. If the Equation on a **,
So
I
BIR
BLO
Fig.13
С
Fig.16
So likewiſe by inverting the bi- Number, as 16 is the biquadratic
quadratical Parabola, (of fig. 7.) or Power of 2; for 2 x 2 is = 4, and
turning the Concavity downwards, 4 x 4 is = 16.
the five Ovals ( fig. 13, 14, 15, 16, BIQUADRATIC Root of a Nume
and 17.) may be had, when the ber, is the ſquare Root of the ſquare
Root of a Number, as the biqua-
dratic Root of 81 is 3; for the ſquare
Root of 81 is 9, and the ſquare Root
of 9 (again) is 3.
BIQUINTILĒ, an Aſpect of the
Planets, when they are 144 Degrees
diſtant from each other.
Fig.14
BISSEXTILE, in Chronology, is
the ſame as our Leap-rear. And
the Reaſon of the Name is, becauſe
in every 4th Year they accounted
the 6th Day of the Kalends of March
twice; for once in four Years the
Fig.15
odd Hours, above 365 Days, made
up just a whole Day, which was in-
ſerted into the Calendar to the 24th
of February
BLACKNESS.
The Colour fo
called, ſeems to ariſe from ſuch a
peculiar Texture and Situation of
the ſuperficial Parts of any black
Body, that it does, as it were, deaden
the Light falling upon it, and reflect
Fig. Z
none, or very little of it outwards
to the Eye.
Sir Iſaac Newton, in his Optics,
Book 2. Obj. 4. 17, and 18, thews,
That for the Production of Black
greateſt Term à x4 is negative. For Colours, the Corpufcles muſt be leſs
when all the Roots of the Equation than any of thoſe that exhibit other
are real and unequal, there will be Colours.
a Pair of Ovals; as at fig. 13. when BLACK SUBSTANCES, of all o-
the two greater or leffer Roots are thers, do fooneſt become hot, and
equal, there will be one Oval, and burn.
á conjugate Point, as at fig. 14. BLINDS, in Fortification, are cer-
when the two middle Roots are tain Pieces of Wood, or Branches
equal, there will be two Ovals join'd of Trees, laid a-croſs, from one
together, in ſhape of a Figure of fide of a Trench to the other, to
Eight, as at fig. 15. when two Roots ſuſtain the Bavins or Hurdles laden
are imaginary, there will be but with Earth; and ſerve to cover the
one Oval, as at fig. 16. and when Pioneers from above; and are com-
three Roots are equal, there will be monly uſed when the Works are
but one Oval, in ſhape of a Pear, as carry'd on towards the Glacis, and
that of fig. 17.
when the Trench is extended in
BIQUADRATIC POWER, is the Front towards the Place.
fourth Power, or ſquared Square of a BLOCKADE, is encompaſſing any
Town,
1
dies :
BO'M
BÓM
Town, or Place, ſo all round with The Uſe of Bombs is not very
arm'd Troops, that it is impoſſible ancient; for altho' we have ſome
for any Supplies to be brought to it; mention in Hiſtory made of certain
and ſo it muſt be ſtarved, or ſurren. Fire-Pots, thrown with Engines into
der: But there is no deſign of ta- the Towns of the Enemy, yet theſe
king it by Attack, &c. And when were quite different from Bombs
any Place is in this Condition, it is filld with Gunpowder, of which
faid to be block'd up, or blockaded. they had not the leaſt Knowledge.
BODY, in Geometry; is that which The firſt which we know of, were
has three Dimenfions, Length, thrown into the City of Watchen-
Breadth, and Thickneſs. As a Line donck in Guelderland, which was
is formed by the Motion of a Point, beſieged by Count Mansfield, under
and a Superficies by the Motion of the Command of the Prince of Par.
a Line; fo a Body is generated by ma, in the year 1588; where the ,
the Motion of a Superficies. But, Bombs in a ſhort time, having ruin'd
Body, in Natural Philoſophy, is all the Lodgments, fo aftonith'd the
uſually definid to be a Subſtance Beſieged, that they were obliged
impenetrably extended, or which to ſurrender.
to ſurrender. Some ſay that an In:
having Partes extra Partes, cannot habitant of Venlo, in the fame Pro-
be in the ſame place with, or pene-
or pene- vince, invented them fome time be-
trate the Dimenſions of other Bo- fore, having uſed them only as Fire-
Which Property Sir Iſaac Balls of Pleaſure to divert the Duke
Newton expreffes by the word so- of Cleves, then at Venlo: and having
tidity; and fo the idea we have of thrown ſeveral in his preſence, one
a Body proceeds from its being ex- by Misfortune fell into a Houſe,
tended, folid, and moveable.
which it fired with fo violent and
Bomb-Chest, is a kind of Cheſt, horrible a Blaze, that the greateſt
which, being filled with Gunpowder Part of the City was confum'd be-
and Bombs, (according to the in- fore any Help could be gotten.
tended Execution) is placed under There are ſome Datch Hiſtorians
Ground, to blow it up into the Air, who relate, that a few Months be.
together with thoſe that ſtand upon fore this Misfortune happen'd, an
it.
Italian Engineer made ſome ſuch
Thefe Bomb-Cheſts are frequently like Experiments at Bergenopzoom,
uſed to drive the Enemy from a Poſt trying to make theſe Bombs eaſy
they lately poffefſed, or whereof they and uſeful in War. But in doing ſo
are about to take Poffeſfion; and are he was miſerably burnt, by the ac-
fet on fire by means of a Sauſage cidental firing off of the Compofition
faften'd at one End.
which he had deſign'd for that pur-
BOMBS, are hollow Balls of Caft- poſe. Be this as it will, it is very
Iron, which are fill'd with whole certain that Bombs were not then to
Powder, and ſometimes Nails, Pieces be found; altho' the Uſe of Mortars
of Iron, &c. along with it. Their may perhaps be as ancient as that of
Ufe is to be ſhot out of Mortar- Cannon themſelves: becauſe there
Pieces into beſieged Towns, to an are to be found Iron ones of a very
noy the Garriſon, fire Magazines, ancient Make; and it is known that
ſeveral of them were uſed in the I-
The largeſt are about ſeventeen talian War of the laſt Age, to fing
Inches in Diameter, two Inches in Stones and red-hot Balls co fet Towns
Thickneſs, carry 48 Pounds of Pow- on fire. Nay, there is even a De.
der, and weigh about 490 Pounds. fign of a Mortar caſting forth a Fire-
E c.
Ball,
BOM
BOM
Ball, amongſt divers other Pieces upon the Parapet of the Redoubt:
of Artillery, mark'd upon the Fron. every body try'd to do the ſame ;
tiſpiece of a Book of Nicolas Tarta- but the Confuſion and Diſorder was
glia the Mathematician, printed in ſo great, that the Bomb burſt with-
the year 1538.
in the Mortar, and broke it into a
The Spaniards and Dutch, in the thouſand Pieces, killing and wound-
long Wars between them, uſed Bombs ing many people.
and Granado's. And they were firſt But at length, this Engineer him-
to be found in uſe amongſt the ſelf was killed at the laſt Siege of
French, in the year 1634, at the Gravelin, by a very extraordinary
Siege of La Motte. Nor is it Truth, Misfortune; he having pitch'd upon
as Caſimir ſays in his Book of the a Poft very near the Counterſcarp
great Art of Artillery, that they of the Enemy, where he deſign'd to
were in uſe at the Siege of Rochelle. puſh his Work as ſoon as it was
Lewis the XIVth of France having dark, and having a deſire to ſhew it
ſent for, from Holland, one Maltus to the General officer, he jump'd
an Engliſh Engineer, who had the up in the Trench to thew him its Si-
chief Direction in uſing them, with tuation ; the Officer himſelf did the
much Succeſs, at ſeveral Sieges; fame after him, but being not ſuffi-
particularly at Cohoure, in the year ciently inform'd of it, he deſired
1642, he threw one which pierced that Maltus would jump up once
thro' the Ciſtern, and obliged the more, that ſo he might have a bet-
Beſieged to ſurrender much fooner ter Knowledge of it: Maltus did fo,
than they would have done, were it and at that Inſtant was ſhot thro' the
not for that Accident. At firſt he Head with a Muſquet-Ball. .
had not all the Experience that he All this perſon's Knowledge con-
acquired afterwards: for at the Siege fifted in pure Experience, being quite
of Landrecy, in the year 1637, his deftitute of Mathematical Helps, or
Battery was in a Redoubt of the any ſort of Science that could inform
Cardinal de la Vallet; where they him of the Nature of the Motion of
were conſtantly coming to him and Bombs, and the Curve Line which
complaining that the Bombs which they deſcribe in their Paſſage thro'
he deſign'd to throw into the Place, the Air, or the Difference of their
flew over, and fell beyond the Town, Ranges according to the different E-
killing a great many people in the levations of the Mortar; ever direct-
Trenches at Mr. De Candale's and ing his Mortar by accident and gueſs,
Meilleray's Attack, on the other ſide or rather by the Eſtimation that he
of the Town Eye at this very made of the Diſtance of the Place
Siege a great Mist. ä e happend to which he had a mind to throw
from him: For once wuen a great the Bomb, according to which he
many General Officers had come gave it a greater or leſs Elevation ;
out of Curioſity to his Battery, he obſerving whether the firſt ſhots
fired off ſeveral Bombs in their pre were juſt or not; and lowering his
ſence. But at length, having ſet Mortar, if its range was too ſhort ;
fire to the Fuzee of a charged Bomb, or raiſing it, when it fell beyond the
and then going to ſet fire to the Mark; uſing a ſort of a Square for
Touch-hole of the Mortar, he found that purpoſe.
his Match gone out; and imme Nicholas Tartaglia the Mathema-
diately giving the alarm, crying tician, in his Treatiſe Concerning a
out for every one to take care of New Science, ſays that a Bomb de-
himſelf that could, he jump'd firſt ſcribes a Curve in its Paffage with
។
BOM
BOM
a Motion partly violent, whoſe Force or falſely related to him, or elſe the
conſtantly decreaſes, and by a na Piece for the ſecond Diſcharge was
tural Motion conſtantly increaſing. loaded with more or better Powder
which is falſe in the Line deſcribed than at the firſt ; becauſe, ſays he,
by Projectiles; becauſe their Velocity Reaſon ſhews that the Range of the
continually decreaſes.-He thought fecond Diſcharge muſt not be ſo
a good deal upon this Subject, and great in proportion to the firſt. And
promiſed to give us the Order and in this, indeed, Tartaglia is in the
Proportion of the Shots of Cannon right: for if the firſt Range made
or Mortars, whereby they increaſe from the Elevation of 45° is 1972
or diminiſh, according to the Ele. Perches, the other from the Elevation
vation of the Piece ; and how to cal- of 30muſt be but 1710 Perches.
culate all the different Diſtances made Don Diego Ufano, a Spanif Cap-
with the ſame Charge of Powder, tain of Artillery (who long ſerved
by knowing and meaſuring only one in the Wars in Flanders, and par-
Diſtance. But he ſays afterwards, ticularly at the Siege of Offend, in
that as the ſaid Science might contri- the year 1611, in a Book of Gunne-
bute to the Ruin and Deltruction of ry publiſhed by him) is the firſt who
Mankind, he was reſolved to ſup- obſerved that the Ranges of Balls or
preſs it; with this Reſerve neverthe- Bombs, ſhot with equal Charges of
leſs, to communicate the ſame viva Powder from Cannon or Mortars, at
voce to thoſe who were deſirous of Elevations equally above or below
Serving it againſt the Infidels.--He 45 Degrees, are equal.--He alſo
was the first who obſerved that it was makes the path of a Ball or Bomb
impoſſible for any part of the Path in its flight to conſiſt of two right
of a Projectile to be a right Line.. Lines and a Curve: for he makes its
That the greateſt Range was at the Motion to be threefold, the firſt of
Elevation of 45 Degrees, and the which he calls violent, is along a
Gunners of his Time thinking that right Line; the ſecond, which he
the greateſt Range was at 30 De- calls mixt, is along a Curve; and
grees, he undeceived them both by the third, which he calls a pure and
his Doctrine and Experience; and a natural Motion, is alſo along a right
Wager was laid about it at Vero- Line; that is, he ſuppoſes the Force
na, in the year 1532, where a 24 of the Powder communicates a Mo-
Pound Culverin, loaded equally with tion to the Bomb, carrying it along
Powder and Ball, was diſcharged at a right Line in the Direction of the
an Elevation of 45 Degrees, and an Mortar, as long as that Force con-
Elevation of 30 Degrees, affirming tinues conſiderable; but when it be
that he was not indeed preſent at the gins to abate, it is ballanced by the
Experiment, but what he ſays of Weight of the Bomb, its Direction
the Length of each Range, was only is alter'd, and becomes a Curve, by
by the Report of others, who told the Mixture of the two Impreſſes.
him that the Range at the Elevation And this Curve deviates into an up-
of 45° was 1972 Perches of Verona, right ſtreight Line, when the Weight
and that at the Elevation of 30° was being overcome, and the Force im
1872 of thoſe Perches ; and makes the preſs'd by the Powder quite loft, it
following Reflection upon it, that in is at liberty to carry the Bomb in a
the Computation of thoſe cwo Num- right Line directly towards the
bers, one of theſe three things muſt Centre of the Earth, and upon
this
happen, viz. that the Meaſures of Sentiment he has calculated a Table
the Ranges were not exactly taken, of the Ranges of Bombs to every
Degree
BOW
BRA
}
Degree of Elevation. But they are Bow, alſo is a Beam of Wood,
not exactly true.
or Braſs, with three long Screws,
Bonnet, in Fortification, is a that govern or bend a Lath of
certain Work raiſed beyond the Wood
or Steel
to any Arch ;
Counterſcarp, having two Faces, and is of great Uſe for drawing
which form a Saliant-Angle, and, Arches, that have large Radii, &c.
as it were, a ſmall Ravelin, without which cannot be ſtručk with Com.
any Trench. The Height of this paſſes.
Fortification is three Foot; and it is BOULTINE, in Architecture, is
environ'd with a double Row of Pa- the Workmen's Term for a Convex
liſadoes, ten or twelve Paces diſtant Moulding, whoſe Convexity is juft I
from each other. It has a Parapet
It has a Parapet of a Circle. This is placed next
three Foot high, and is like a little below the Plinth in the Tuſcan and
advanced Corps de Gard.
Dorick Capital.
BONNET A PRESTRE, or the Box AND NEEDLE, is the ſmall
Prief's-Cap, in Fortification, is an Compaſs of a Theodolite, Circum-
Outwork, having at the Head three ferentor, or Plain-Table.
Saliant-Angles, and two inwards; BOYAU, or Branch of the Trenches,
and differs from the double Tenaille in Fortification, is a particular Ditch
only in this, that its Sides, inſtead ſeparated from the main Trench, ,
of being parallel, are made like a which in winding about encloſes
Swallow's Tail, that is, narrowing, different Spaces of Ground, and runs
or drawing cloſe at the Gorge, and parallel with the Works and Fences
opening at the Head.
of the Body of the Place; ſo that
BOOT ES, the Name of a Northern when two Attacks are made near
Conſtellation of the Fixed Stars; of one to another, the Boyau ſometimes
which one, in the Skirt of his Coat, makes a Communication between
is called Artturus, and is of the firſt the Trenches, and ſerves as a Line
Magnitude. This Conſtellation is of Contravallation, not only to hin-
called Aretophylax, and conſiſts of der the Sallies of the Beſieged, but
thirty-four Stars.
alſo to ſecure the Miners. But when
BOREAL SIGNS, are the fix firſt it is a particular Cut,, that runs from
Signs of the Zodiac, or thoſe on the Trenches to cover ſome Spot of
the Northern Side of the Equinoc- Ground, it is then drawn parallel
tial.
to the Works of the Place, that it
BOSPHORUS, in Geography, is may not be enfiladed, that is, that
a long narrow Sea, running in be- the Shot from the Town may not
tween two Lands, by which two fcour it.
Continents are ſeparated, and by BRACE, in Architecture, is a
which way a Gulph and a Sea, or Piece of Timber framed in with
two Seas, have a Communication Bevil-Joints, and is uſed to keep
one with another, as the Thracian the Building from ſwerving either
Boſphorus, now called the Streights way. When a Brace is fram'd into
of Conſtantinop:e.
the Kindleſſes, and principal Rafters,
Bow, a Mathematical Inſtrument, it is called by ſome à Strut.
made in Wood, formerly uſed by BRACKETS, in Gunnery, are the
Seamen, to take the Altitude of the Cheeks of the Carriage of a Mor-
Sun, but now is out of uſe; and tar. They are made of ſtrong Planks
conſiſts of a large Arch of 90 De- of Wood of almoſt a ſemicircular
grees, three Vanes, and a Shank or Figure, and bound round with thick
Staff.
Iron Plates. They are fixed to the
Bed
3
1
1
BUR
BU R:
Bed by four Bolts, which are called The Breadth of one of theſe
Bed-Bolts; they riſe up on each ſide Concaves, if it be the Segment of
of the Mortar, and ſerve to keep a great Sphere, muſt not exceed an
her at any Elevation, by means Arch of eighteen Degrees; and if a
of fome ſtrong iron Bolts, called Segment of a ſmall Sphere, at moſt,
Bracket-Bolis, which go thro' theſe an Arch of thirty Degrees.,
Cheeks or Brackets.
Kircher, in Arte Magna Lucis &
BRANCH of the Trenches. See Umbra, lib. 10. part 3. c. i. ſays,
Boyau.
That he found by Experience, that
BREACH, in Fortification, is the the beſt Burning Concaves were ſuch
Ruins that are made in any part of that did not exceed an Arch of
the Works of a Town, &c. by eighteen Degrees in their Breadth.
playing Cannon, or ſpringing of If the Segments of a greater and
Mines, in order to form the Place, a leſſer Sphere lie each eighteen
or take it by Affault.
Degrees in Breadth, or even ſome-
BREAK GROUND, in Fortifica- thing greater or leſs, the Number
tion, fignifies to begin the Works of Degrees in both being the ſame,
for carrying on the Siege about a the Effects of the greater Segments
Town or Fort.
will be greateſt.
BreasT SOMMERS, in a Tim Burning Glaſſes, that are Segments
ber Building, are the Pieces in the of a greater Sphere, do burn at a
outward Parts of it, and in the greater Diſtance than thoſe that are
Middle Floors, (not in the Garret Segments of a leſſer Sphere.
and Ground-Floor,) into which the Schottus, in Magia Univerſ. part i.
Girders are fram'd.
lib.
7. ſect. 6. p. 1418. ſays, That
BREAST-WORK, the ſame with one Manfredus Septala, at Milan,
Parapet.
made a parabolic Speculum of this
BRIDGE of Communication, is a kind, that would burn Wood at the
Bridge made over a River, by which Distance of fifteen or fixteen Paces.
two Armies, or Forts, that are fe Mr. Villette, at Lyons in France,
parated by that River, have a free made a metalline Burning Concave
Communication one with another. of a round Figure, thirty Inches
Broken Ray, or Ray of Re- in Diameter, and about a hundred
fraction, in Dioptrics, is a right Pound Weight, the Focus, or burn-
Line, whereby the Ray of Incidence ing Point, being diſtant from the
changes its Rectitude, or is broken Concave about three Foot, and its
in croſſing the ſecond Medium, whe- Bigneſs about half a Louis d'Or.
ther it be thicker or thinner.
This would melt Iron in forty Se-
BURNING GLASSES are convex conds, Silver in twenty-four, Cop-
concave Glaſſes, commonly per in forty-two; and turned Quarry
Spherical, that being expoſed di- Stone into Glaſs in forty-five, and
rectly to the Sun, do collect all the Mortar in fifty-three Seconds; and
Rays of the Sun falling upon them melted a piece of Watch-Spring in
into a very ſmall Space, called the nine Seconds. See the Philoſophical
Focus, diſtant from the Glaſs in the Tranſact. N° 6. pag. 418. and the
Axis thereof, where Wood, or any Diary of the Learned at Paris, Ann.
other combuſtible Matter being put, 1679.
will be ſet on fire. Metalline Con Mr. Villette afterwards made ano-
caves, that produce this Effect by ther of thirty-four Inches in Dia-
Reflection, are called Burning Cone meter, that would melc all ſorts of
caves.
Metals of the thickneſs of a Crown-
or
picce
BUR
BUT
piece in leſs than a Minute, and vi Sir Iſaac Newton preſented a
trify Brick in the ſame time. Phi- Barning-Glaſs to the Royal Society,
lofoph. Tranfa&t. Nº. 49.
conſiſting of ſeven Concave Glaſſes,
In the Philoſoph. Tranf. Nº. 188. fo placed, as that all their Foci join
and the Aeta Éruditorum Ann. 1687. in one phyſical Point. Each Glaſs
P. 52. you have mention'd a Cop- is about eleven Inches and a half
per Burning-Concave, made at Lu- in Diameter: Six of them are plac-
face in Germany, of near three Leip- ed round the ſeventh ; to which
fick Ells in Diameter, and its Focus they are all contiguous, and they
two Ells off, being ſcarce twice ſo compoſe a kind of Segment of a
thick as the back of a common Sphere, whoſe Subtenſe is about
Knife, and whoſe force is incredible; thirty-four Inches and a half; and
for a piece of Wood put in the Fo. the Central-Glaſs lies about an
cus, flames in a moment ſo as it can Inch further in than the reſt. The
hardly be put out by a freſh Wind. common Focus is about twenty-two
A piece of Lead or Tin three Inches Inches and a half diftant, and of
thick, will be melted quite through about half an Inch in Diameter.
in three Minutes time. A piece of This Glaſs'vitrifies Brick or Tile in
Iron or Steel is preſently red hot, a Moment, and in about half a
and ſoon after hath a Hole burnt Minute melts Goid.
through it. Copper, Silver, &c. A certain Artificer of Dreſden is
applied to the Focus, melt, and the ſaid to have made very large Burn-
Iron aforeſaid will melt in five or ing-Concaves of Wood, whoſe Ef-
fix Minutes. Slate, in a few Mi- fects were little inferior to thoſe of
nutes, will be turn'd into black Glaſs. the Burning-Speculums of Mr.
Tiles and Earthen Potſheds, in a lit- Tſchirnhauſe.
tle time, do melt into Glaís. Bones Zahn, in Oculo. Artific. Fundam.
are turn'd into black Glaſs, and 3. Syntagm. 3. cap. 10. f. m. 634!
a Clod of Earth into greeniſh Glaſs. ſays, T'hat one Neruman, in the
Mr. Tſchirnhauſe is ſaid to have Year 1699, at Vienna, made a Burn-
made Convex Burning-Glaſſes of ing-Speculum of tiff Paper and
three or four Feet in Diameter, and Straw glued to it.
whoſe Focus is twelve Feet diftant, And Zacharias Traberus, in Nervo
and of an Inch and a half in Dia- Optic. lib. 2. C. 1 2. prop. 5. cor. 2.
meter ; and to make this Focus yet ſays, That very large Burning-Spe-
Itronger, he contraéts it by a ſe- culums may be made of thirty, for-
cond Lens, placed parallel to, and at ty, or more Concave Speculums, or
a due diſtance from the firſt, and ſo ſquare Pieces of Glaſs, conveniently
makes the Focus but eight Lines in placed in a large turn'd wooden
Diameter, This Glaſs vitrifies Concave, or Diſh, and that their
Tiles, Slates Pumice-ſtones, &c. in effect will not be much leſs than if
a moment. It melts Sulphur, Pitch, the Superficies were contiguous.
and all Roſins, under Water. Any BURNING Zone. See Zone.
Metal expoſed to it, in little Lumps BUTMENTS, in Architecture, are
upon a Coal, melt in a moment, the Maſons and Bricklayers Term
and Iron ſparkles as in a Smith's for thoſe Supports or Props, on or
Forge. All Metals vitrify on a againſt which the Feet of Asches
piece of China Plate, if it be not ſo reft : Alſo little Places taken out of
thin as to melt itſelf; and Gold, in the Yard of the Ground-Plot of a
vitrifying, receives a purple Colour. Houſe for a Buttery, Scullery, &c.
See L'Hiſtoire de l'Academie des are ſometimes called Butments.
Sciences, Ann. 1699.
BUTTRESS,
C
1
Ĉ A
CAL
BUTTRBSS, is an Arch, or Maſs might be the leſs, he invented dif-
of Stone, ſerving to ſupport the ferent Words and Notes from thoſe
șides of a Building, Wall, &c. on in Sir alfaac's Method, as for the
the outfide, and are chieſty uſed in Fluxion of x, he pats dx ; and for
ſuch Buildings as are of the Gothick yn dy; and thefę are uſed by almoſt
kind.
all the Foreigners. Yet even James
BY'QUARTILE, the ſame with Bernoulli
, in the Leipfick Aěts for
Biquartile.
I anuary 1691, owns," that our face
mous Dr. Barrow (before Sir Iſaac,
or Leibnitz either,) had given ſome
Specimens of this Method, above
C:
ten Years before that Date, in his
Geometrical Lectures, and of which
ADENCE or Cloſe, in Mu- all his Apparatus of Propoſitions
fic, is a concluſion of a Piece there contain'd, are ſo many Ex-
of Muſic, in ſome Keys it is not amples. He alſo acknowledges, that
fet in: and in long Pieces of Muſic Mr. Leibnitz's Method of the Cala
there are ſeveral Cadences. The culus Differentialis is founded upon
more there are, the pleaſanter is Dr. Barrow's, and differs from it
the Muſic, provided they are art only in ſome Notes and compendi-
fully diſpoſed.
ous Abridgments.
CAISSON, or Superficial Fourneau, But to give a full and more parti-
is a wooden Caſe, or Cheſt, into cular account of the Origin of this
which three, four, five, or fix Bombs great Invention, take what follows
are put, according to the Execution from Sir Iſaac Newton himſelf, be-
they are to do, or as the Ground is ing part of his Remarks upon Mr.
firmer or looſer. Sometimes the · Leibnitz's Letter to the Abbé Conti;
Cheſt is only filld with Powder : wherein this laſt endeavours to vin-
When the Beſieg'd diſpute every dicate his own Conduct about the
Foot of Ground, this Caiſſon is bu. Invention of the Calculus Differen-
ried under ſome Work the Enemy tialis : which Remarks, together
intends to poſſeſs himſelf of ; and with Letters of Mr. Leibnitz, Sir
when he is Maſter of it, they fire it Iſaac Newton, Dr. Clarke, &c. are
by a Train convey'd by a Pipe, and contain'd in a French Treatiſe en-
fó blow them up.
titled, Recueil de diverſes Pieces fur
CALCULUS DIFFERENTIALIS, la Philoſophie, la Religion naturelle,
is the Arithmetic of the infinitely l'Hiſtoire, les Mathematiques, &c.
ſmall Differences between variable
Sir Iſaac, ſpeaking of Mr. Leiba
Quantities, and is by us in England nitz, mentions, that at his Arrival
call'd Fluxions.
at London from Paris, his firſt Leç-
Mr. Leibnitz, about the Year ter turn'd chiefly upon other Sub-
1676, by moſt of the Foreigners, is jects than Geometry, which laſted
allowd to have firſt invented this till Mr. Huygens had inſtructed him
Doctrine of infinitely ſmall Quanti- in theſe matters ; that he found out
ties, who called it the Calculus Dif the Arithmetical Quadrature of the
ferentialis; but it is plain, from Circle, towards the end of the Year
Sir Iſaac Newton's Papers, that Sir 1673 i that the following Year he
Ifaac was the firſt Inventor of it, began to write thereof to Mr.
who being too free in communicat- Oldenburgh; that a little while after,
ing it to Mr. Leibnitz, he ſtole it he diſcovered the general Method
from him; and that the Sufpicion of Series's from the Affumptions
Ա
of
1
CAL
CAL
O
of an arbitrary one, and the Calcu- Problems relating to the Tangents
lus Differentialis in the Year 1676. and Curvatures of Curves : That
which he deduced from a Series of in another Paper, dated the 16th of
Numbers by conſidering the Diffe- May 1666, there are ſeven Propo-
rences; and that in his Letter of fitions, concerning a general Method
the 27th of Auguſt, 1676, he meant of reſolving Problems relating to
by the Words Certa Analyſ, the Motion, and that the laſt of theſe
Differential Analyſis. But, ſays Sir Problems is the ſame as the Pro-
Ifaac, have not I the ſame liberty blem abovementioned, dated the
to affirm and certify, that I invent- 13th of November 1665. That in a
ed the Method of Series's and Fluxi- little Treatiſe wrote in November,
ons in the Year 1665 ; that I car 1666, the ſaid ſeven Propoſitions
ried them farther in the Year 1666; are again repeated, with this Dif-
that I have now in my hands ſeveral ference, that the ſeventh is carried
Mathematical Papers, wrote in the fo far, as not to be limited by Frac-
Years 1664, 1665, and 1666; fome tions or furd Quantities, or even
of which are dated ? Amongft by what are now call's Tranſcendent
which there is one, dated the 13th Quantities; that an eighth Propo-
of November, 1665. containing the fition is added to this Treatiſe, con-
direct Method of Fluxions, in theſe taining the inverſe Method of Flu-
Words. Prob. There being given xions, as far as I had advanced it at
an Equation, exprelling the Relation that time, viz. ſo far as it can de-
of ſeveral Lines, x, y, z, &c. de- pend upon the Quadrature of curv'd-
ſcribed at the ſame time by two or lin'd Spaces, and the three Rules
more Moveables, A, B, C, &c. to upon which is founded my Analyſ13
find the Relation of their Velocities per Æquationes Numero Terminorum
infinitas, and the moſt part of the
The Solution. • Put all the Terms other Theorems, contained in the
on one ſide of the Equation, ſo Scholium of the tenth Propoſition of
. that they be equal to o; and mul- my Book of Quadratures; that in
tiply each by ſo many times
P the ſaid Treatiſe, when the Area a-
xy riſing from fome one of the Terms
as x has Dimenſions in that Term : of the Ordinate, cannot be expreſ-
' then multiply each Term by asſed by the common Analyſis, it is
9 9 오
​repreſented by writing the Marko
as y has Dimen- before that Term. For Example,
у
* fions in that Term : After this if the Abſciſs be x, and the Ordi.
bi
multiply each Term by as many nate ax - .bt
the whole
ata
6 times as x has Dimenſions in
bb
that Term, &c. and the Sum of Area is { ABC-bx+o
atx
thoſe Products will be =0; which that in the ſaid Treatife I fometimes
Equation gives the Relation of uſe Letters mark'd with one Point
only, to repreſent Quantities that
I
may add (ſays Sir Iſaac) that involve firit Fluxions ; and ſome-
the ſaid Example is therein illuf- times the ſame Letters, mark'd with
trated with ſeveral Examples; that two Points, repreſenting ſecond
it is demonſtrated therein; that it is Fluxions; that a more compleat
there applied to the Solution of Treatiſe, which I wrote in the Year
1671,
P, q, r, &c.
<
many times
at
6
Di q, r, &c.
I
CAL
C A L
1671, and mention'd in my Letter Solid of the leaſt Reſiſtance. But
of 'the 24th of Oktober 1676, is becauſe this Artifice ſuppoſes the
founded upon that little Treatiſe, differential Method as known, and
and begins with the Reduction of that its Extent is ſtill farther; that
finite Quantities into infinite Series's, beſides, it is to this Artifice that
and with the Solution of theſe two Mr. Leibnitz and his Scholars owe
Problems ; 1. The Relation of flowing the Solution of the Problems, which
Quantities to one another being given, he ſo much eſteems; finally, bes
to find the Relation of the Fluxions. cauſe Mr. Leibnitz calls this Artis
2. And an Equation being given, in- fice a Method of the higheſt conſe-
volving the Fluxions of Quantities, quence, and the greateſt extent; it
to find the Relation of the Quantities iß fufficient for me, that he has
between themſelves.
own'd that I am the firſt Perſon,
And when I had wrote that Trea- who, in a publick Work, has made
tiſe, I made my Analyſis ſo general, it appear that I knew of the ſaid
by means of the Method of Series's Artifice.
and the Method of Fluxions con In the Year 1689. Mr. Leibnitz
jointly, that it even extended to publiſh'd as his own, the principal
almoſt all ſorts of Problems ; which Propoſitions of the Principia, in
is what I mention'd in my Letter three different Writings, entitled,
of the 13th of June, 1676. and it Epiftola de Lineis Opticis; Schediaſma
is that very Method which I have de Reſiſtentia Medii & Motu Projecti-
deſcribed in my Letter of the roth lium gravium in medio reſiſtente ; Ego
of December, 1672.
Tentamen de Motuum Cæleftium Cauis:
In the Year 1684. Mr. Leibnitz. pretending that he had found out all
publiſhed only the Elements of the thoſe Propofitions before the Prin-
Calculus Differentialis, which he has cipia appear'd; and in order the
applied to Tôme Queſtions concerns better to appropriate to himſelf the
ing Tangents, and other things re- principal of thoſe Propofitions he
lating to the Method of Maximums thought fit to ſubjoin a Demonſtra-
and Minimums, as Mr. Farmat and tion thereto, which he had found
Gregory had done before ; and has out; but as it was erroneous, he
thewn how to proceed in theſe retracted it himſelf, and ſhewd that
kinds of Queſtions, without taking he did not underſtand how to work
away the irrational Quantities; but with fecond Fluxions. This here
does not meddle with the Problems was the ſecond Effay given to the
of the higher Geometůy. The publick, wherein the Method of
Book of Mathematical Principles Fluxions is applied to the higher
contains the firſt public Specimens Geometry. Hitherto this Method
of the Solutions of the more ele- was but a little known, but in a
vated Problems by this Calculus ; Year or two after it began to ſpread
and it is in this ſenſe I underſtand abroad.
what Mr. Leibnitz fays in the Leip Dr. Barrow publiſhed his Diffed
fick Acts for the Month of May, rential Method for Tangents in the
1700, pag. 206. But Mr. Leibnitz Year 1670. Mr. Gregory, by means
would have it obſerved, that what of this Method compared with
he ſaid then muſt be underſtood of his own, deduced a general Me-
a particular Artifice of Maximums thod for drawing Targents, which
and Minimums, which he owns I was did not require any Calculation
maſter of, by giving in my Prin- and of this he inform’d Mr. Colliäs
cipia, the Figure of the Veffel or by a Letter, wrote the 5th of Sepo
tember
I 2
CAL
CAL
4
tember 1670. and in November in pear'd, I did deduce Mr, Gregory's
the Year 1672. Mr. Slufius inform'd and Slufius's Method of Tangents
Mr. Oldenburgh of a Method of his from my general Méthod. At that
of the ſame nature. In one of my time Mr. Leibnitz was not only ig-
Letters of the roth of Deceínber, norant of the higher Geometry,
1672, I fent a like Method to Mr. but even Algebra itſelf.
Collins, and added, that I men In his Letter of the z7th of Au-
tion'd the ſame to Dr. Barrow, at guſt 1676, is contained this Paſſage:
the time of his publiſhing his Geo- It does not appear to me, what we find
3metrical LeEtures; that I was of o is ſaid, that moſt Dificulties (except
pinion that the Methods of Gregory the Diophantean Problems) may be
and Stufius were the ſame as mine, reduced to infinite Series's ; for there
and that the faid Method was only are many Problems fo very knotty and
a Branch or Corollary of a much intangled, as not to depend upon either
more general Method, which with- Equations or Quadratures, fome of
out any troubleſome Calculation, which ( amongſt many others) are the
extended not only to drawing Tan- Problems of the inverſe Method of
gents, but likewiſe other more ab- Tangents. But when I made anſwer
Itruſe Problems ; ſuch as thoſe re to him, that ſuch kind of Problems
lating to the Curvatures, Areas, were in my power to folve, he re-
Lengths, Centres of Gravity, of ply'd in his Letter of the 2rft of
Curves, &c. and that without any June 1677, that I truly muſt mean
neceffity of freeing Equations from by infinite Series's, but that he
furd Quantities. I added'likewiſe, meant by common Equations ; to
that I had ſubjoin'd the Method of which may be ſeen an Anſwer in
Series's to the ſaid Method, mean- the Commercium Epiftolicum, pag.
ing in the ſaid Treatiſe which I 92.
wrote in 1672.
He fays, that one might judge
Mr. Oldenburgh, in June 3676. that when he wrote his Letter of
nt Copies of theſe two Letters a the 27th of Auguſt 1676, he had
mongſt the Extracts of Gregory's Let- made ſome entrance into the Dif
ter to Mr. Leibnitz ; and Mr. Leib- ferential Calculus, fince in that Lét-
nitz in his Letter of the 21ſt of ter he ſhews, how to ſolve Mr.
June, 1677, fent nothing back in Beaune's Problem by a certain Ana-
exchange, but what had been done lyfts ; but ſays he, how can it be
before, and of which the ſaid Let ſuppoſed to ſolve the ſame by a cer.
ters informed him : His Method tain Analyſes, without the help of
of Tangents, which he ſent at that the Differential Calculus ? For all
time, being only the Method of Dr. the Analyſis in doing this, is only to
Barrow, which he diſguiſed under fuppoſe the Ordinate of the Curve,
a. new Notation, and extended it to to increaſe or decreaſe in a Geo-
Gregory's and Sluſus's Method of metrical Progreſſion, while the Ab-
Tangents, to Equations involving ſciſs increaſes in an Arithmetical
irrational Quantities, and to one one, and conſequently the Abfcifs
of the moſt ſimple Caſes of my and Ordinate have the ſame relation
Quadratures. But I cannot be re to one another, as the Logarithm
proach'd of the ſame thing with to its Number. But, for Mr. Leib-
regard to Dr. Barrow; he law my nitz to infer from hence, that he
Treatiſe of Analyſis in 1669. and had made an entrance into the Cal-
has teſtified that he read it; and be- culus Differentialis, is the very fame
fore his Geometrical beatures ap- thing as to ſay Archimedes had made
advances
1
1
:
1
1
CAL
C AL
advances that way, becauſe he knew yet ſee any valuable Uſes that have
how to draw Tangents to the Spiral, hitherto been made of it.
Square of the Parabola, and found CALCULUS INTEGRALIS, is the
out the Proportion of the Sphere method of finding the proper flow-
and Cylinder, or the ſame thing as ing Quantity of any given Fluxion,
to ſay, that Cavallerius, Fermat, and is the reverſe of the Calculis
and Wallis, had made an entry into Differentialis, which finds the Flu-
it, becauſe theſe have done many xion from the flowing Quantities.
things of the fame nature with thoſe ÇALENDAR, much the ſame as
above-mentioned.
Almanac ; which ſee. The Word
Thus far 'the great Newton. Calendar ſeems to come from the
Thoſe who have a mind to ſee more Calendæ, which, among the Romans,
of the Hiſtory of this Invention, were the firſt Days of every Month,
its various Improvements, and the There have been many Correc-
Uſes thereof, may conſult the Com- tions and Alterations of the Calen-
mercium Epiftolicum (publiſh'd by dar. The firſt was made by Nuna
Order of the Royal Society). Pompilius ; and this afterwards was
The Marquis de l'Hoſpital's Analyſe much improv'd by Julius Cæfar,
des Infiniment Petits (in French or and was by him called the Julian
Engliji). --Mr. Nieuwentiit's Analyſis Account, which, in our Nation, and
Infinitorum, in Latin.- Mr. Craig's ſome other places, is ſtill retain'd,
Calculus Fluentium, in Latin.--- Mr. and called the Old Style.
Carré's Methode pour la Meſure des Pope Gregkry XIII. pretended to
Surfaces, &c. in French.- Hayes's reform it again, and ordered his
Fluxions, in Engliſh.-- Mr. Ditton's Account to be current, as it is itill
Fluxions, in Exgliſh--Mr. Reyneau's in all the Roman Catholick Coun-
Analyſe Demontrée, in French.-- Dr. tries, where it is called the Grega
Cheyne's Methodus Inverſa Fluxionum, rian Calendar; and with us Now
in Latin.-Sir Iſaac Newton's Flu- Style. It begins eleven Days before
xions, in Engliſh; with, or without ours.
Mr. Colfon's Commentaries. Dr. CALENDAR (ASTRONOMICAL)
Harris's Fluxions, in Engliſh-Mr. See Aſtronomical Calendar.
Muller's Mathematical Treatiſe, in CALENDS; fo the Romans callet
Engliſh. Mr. Hudſon's Fluxions, in the firſt Days of every Month,
Engliſh-Mr. anes's Synopfis, in from the Greek Word Calee, to call;
Engliſh.-Mr. Simpſon's Fluxions, in becauſe anciently counting their
Engliſh.-The Philofophical Tranf- Months by the Motion of the Moon,
actions of London, Paris, Leipſick, there was a Prieſt appointed toob-
Petersburgh, &c. and other Writings. ſerve the times of the New Moon;;
CALCULUS EXPONENTIALIS, who, having ſeen it, gave nocice to
is the manner of finding the Flu- the Preſident over the Sacrifices,
xions; and of ſumming up of the and he called the People togethes,
Fluxions of Exponential Quantities. and declared to them how they imut
This Calculus was diſcovered by reckon the Days until the Norius
Mr. John Bernoulli
, and communi- pronouncing the Word Coiles five
cáted to Mr. Leibnitz, who made it times.if the Nones did happer: on
public in the Asta Eruditorum for the 5th Day, ** or ſeven tires if
the Year 1697, pag. 125, & feq. they happened on the 7th Day of
But notwithſtanding the great value the Month.
ſome People may perhaps put upon CALIBER, or CA:LIDER is the
this Invention, yet I could never Bigneſs, or rather Diameter of :2
1
A
)
I
Pi
CAM
CAN
}
Piece of Cannon, or any Fire-Arms The Repreſentations of Objects
at the Mouth,
in this Machine are wonderfully
CALIPERS, is an Inſtrument pleaſant, not only becauſe they ap.
made like a Sliding-Rule, to 'em- pear in the juft Proportions, and
þrace the two Heads of any Cak are endued with all the natural Co-
to find the Length of it. There lours of their Objects, but likewiſe
are alſo Calipers, or Caliper-Com- ſhew their various Motions, which
paſjes, which are uſed by Gunners, no Art can imitate; and a kilful
with crooked or bowing Legs, to Painter, by means of one of theſe
meaſure the Diameters of Bullets Machines, may obſerve many things
and Cylinders of Guns, &c. from the Contemplation of the ap-
CALLIPIC Per 10D, was an Im- pearing of Objects therein, that
provement of the Cycle of Meton of will be an help to the Perfection of
nineteen Years, which Callipus, a the Art of Painting; and even a
famous Grecian Aſtronomer, finding Bungler may accurately enough de-
in reality to contain nineteen of Na- lineate Objects by means of it.
bonafjar's Years, four Days, and Mr. s'Graveſande, at the end of
331
his Perſpective, has given the De-
he, to avoid Fractions, qua- ſcription and Uſe of two Machines
459
drupled the Golden Number, and by of this kind, being the beſt that
that means made a new Cycle of have as yet been made, eſpecially
ſeventy-fix Years; which time be the former.
ing expired, he ſuppoſed the Luna CANCER, one of the twelve
tions, or Changes of the Moon, Signs of the Zodiac, drawn on the
would happen on the ſame Day of Globe in the figure of a Crab,
the Month and Hour of the Day, and thus mark'd , and that Cira
that they were on ſeventy-fix Years cle that is parallel to the Equinoc-
before.
tial, and paſſes through the Begin-
CAMBER-Beam, in Architec- ning of this Sign, is called the Trą.
ture, is a Beam or piece of Timber pic of Cancer, or the Northern Tra-
cut hollow, or arching in the mid- pic; to which Circle when the Sun
dle. They are uſed in Platforms, comes, it makes the Summer Solo
Church-Leads, &c. and are very ftice, and is turning his Courſe
proper where ever is occaſion for back again towards the Equinoctial.
long Beams, being much ſtronger CANIS Major and Minor, the
than Alat Beams of the fame fize ; greater and leſſer Dog, are two
for being laid with the hollow fide Conſtellations of Stars drawn upon
downwards, and having good But- the Globe in figure of this Animal,
ments at the ends, they ſerve for a and the greater of them has in his
kind of Arch.
Mouth that vaſt Star called
CAMERA OBSCURA, is the Name CANICULUS, or the Dog-Star,
of an Optic Machine ; wherein (the which riſing and ſetting with the
Light only coming through a dou- Sun from about the 24th of July
ble Convex-Glaſs,) Objects expoſed to the 28th of Auguſt, gives occa-
to broad Day-light, and oppoſite fion to that time, which is uſually
to the Glaſs, are repreſented invert- very hot and dry, to be called the
ed upon any white Matter, placed Canicular, or Dog-Days.
within the Machine in the Focus of CANNON, a piece of Ordnance.
the Glaſs. The firſt who obſerved See Ordnance.
this Phänomena was Baptiſta Porta, CANNON-ROYAL, is a piece of
lib. 4. c. 2. Magia Naturalis. Ordnance, eight Inches in Diame-
ter
1
1
five paces.
CAN
САР
ter in the Bore, t'velve Foot long, Theſe Bags are ſometimes, upon oc-
weighs eight thouſand Pounds ; its caſion, fill's with Powder.
Charge is thirty-two Pounds of
CAP-SQUARES, are broad Pieces
Powder ; its Ball is forty-eight of Iron on each ſide of the Carriage
Pounds Weight, and ſeven Inches of a great Gun, and lock'd over
and a half in Diameter, and ſhoots the Trunnions of the Piece with an
point-blank one hundred and eighty- Iron Pin. Their Uſe is to keep the
Piece from flying out of the Car-
CANON, in Arithmetic, is a riage when it is ſhot off with its
Rule to ſolve all things of the ſame Mouth lying very low, or, as they
nature with the preſent Enquiry. call it, under Metal.
Thus every laſt Step of an Equation CAPACITY, is the ſolid Content
in Algebra, is ſuch a Canon, and of any Body ; alſo our hollow Mea-
if turn'd into Words, is a Rule to fures for Wine, Beer, Carn, &c. are
ſolve all Queſtions of the ſame na called Meaſures of Capacity.
ture with that propoſed. The Ta CAPE, or Promontory, is any high
bles of Logarithms, artificial Sines, Land, running out with a Point
and Tangents, are called likewiſe into the Sea ; as Cape Verde, Cape
by the Name of Canon.
Horn, the Cape of Good Hope, &c.
CANON, in Muſic, is a Line of CAPELLA, a bright fix'd Star in
any length, ſhewing, by its Divi- the left Shoulder of Auriga, whoſe
fions, how muſical Intervals are Longitude, according to Hevelius
diſtinguiſh'd according to the Ratio's (in his Prodromus Aftronom. for the
or Proportions that the Sounds ter- Year-1700,) is 170.40'. 4611, in II,
minating the Intervals bear the one and Northern Latitude 229.52': 9".
to another, when conſider'd accord CAPITAL of a Baſtion, is a Line
ing to their degree of being acute drawn from the Angle of a Polygon
or grave. As the Diapaſon conſiſts to the Point of the Baſtion, or from
in a double Ratio, the Diapente in a the Point of the Baſtion to the
Seſquialteral, the Diateſſaron in a middle of the Gorge. Theſe Capi-
Seſquitertian, and the Tone itſelf, by tals are from thirty-five to forty
which the Diapente and the Dlateffa- Fathom in length; that is, from the
ron differ, in a Seſquioctave, &c. Point of the Baltion to the Place
CANTALIVERS, in Architecture, where the two Demi-Gorges meet.
are a kind of Modillions; only CAPITAL, or Chapital, or Cha-
thoſe are plain, but theſe are carv'd. piter, fignifies the top of a Pillar ;
They are much the ſame with Car- and this is different, according to the
touzes, and are ſet as Modillions different Orders.
are, under the Corona of the Cor CAPITAL-LINE. See Line.
niſh of a Building.
CAPONNIER E, is a cover'd Lodg-
CANVAS-Bags, or Earth-Bags, ment of about four or five Foot
are Bags holding about a Cubic broad, encompafied with a little
Foot of Earth, and are uſed to Parapet of about two Foot high,
raiſe a Parapet in haſte, or to repair which ferves to ſupport divers
one that was beaten down. They Planks laden with Earth.
are chiefly uſed when the Ground This Lodgment is large enough
rocky, and affords no Earth to to contain fifteen or twenty Soldiers,
carry, on the Approaches : Then and is uſually placed upon the Ex-
are theſe Bags of Earth very necef- tremity of the Counterſcarp, having
fary, which can be fill'd at another ſometimes ſeveral little Embraſures
place, and remov'd at pleaſure. made therein, uſually called Mad-
I 4 neſſes,
o
CAR
CAS
nelles. They are generally on the
CARDINAL-SIGNs, are the Signs
Glacis, or in dry Moats.
of the Zodiac, Aries, Libra, Can- .
CAPRICORN, the Goat, one of cer, and Capricorn.
the Zodiacal Signs, mark'd thus Y. CARRIAGE of a great Gun, is
The Tropic of Capricorn, or the the Frame of Timber, on which a
Southern Tropic, paſſes through piece of Ordnance is laid, fix?d and
the firſt Degree of this Sign, at mounted. The common Propor-
twenty-three Degr. thirty Minutes tion is one and a half of the Length
Diſtance from the Equinoctial. of the Gun for the Carriage; the
: CARACT, is the Part of any Wheels half of the Length of the
Quantity, or Weight ; being a piece in height, and four times the
Word uſed by Minters and Gold Diameter of the Bore of the Gun,
ſmiths, who divide it into four parts, gives the depth of the Planks at
which they call Grains of a Caract; the fore-end, in the middle three and
and one of theſe they ſubdivide into
a half.
Halves and Quarters.
CARTOUCHE, the ſame as Car-
CARAT. A Carat of Gold is pro- tridge.
perly the Weight of twenty-four
CARTRIDGES,
or Cartriages,
Grains, or one Scruple ; ſo that 24 are Caſes of Paper, or Parchment,
Carats make an Ounce.
fitted exactly to the Bore of a Piece
And if an Ounce of Gold be ſo of Ordnance, or Mufquet, and cons
pure, 'that in its Purification with taining its due Charge of Powder.
Antimony, or otherwiſe, it loſes no CARTOUZEs, are Ornaments of
thing at all, it is then ſaid to be carvid Work, of no determinate Fi-
Gold of twenty-four Carats : If it gure, whoſe Uſe is to receive a Motto,
Joſes one Carat, it is then Gold of or Inſcription.
twenty-three Carats: If it loſes two CARYATIDES, from the Greek
Carats, it is called Gold of twenty- Caryatides, a People of Caria.
two Carats, & C.
Theſe in Architecture ſignify certain
A Carat of Diamonds, Pearls, or Figures of captive Women, with
precious Stones, is the Weight of their Arms cut off, cloathed after
four Grains only.
the manner of that Nation, down
CARCASS, is an
Iron Cafe, or to their Feet, and ſerve inſtead of
hollow Capacity, about the Bigneſs Columns to ſupport the Entable-
of a Bomb; ſometimes made all of ments.
Iron, except two or three Holes, CASCABELL, is the hindermoſt
through which the Fire is to blaze; round Knob, or the utmoſt part of
and ſometimes made only of Iron the Breech of a piece of Ordnance.
Bars, or Hoops, and then cover'd CASCADE, an Italian Word, that
over with pitch's Cloth, Hemp, &c. fignifies a Fall of Waters, either na-
and fill'd with ſeveral kinds of Ma- tural or artificial.
terials for firing of Houſes. They CASCAN, in Fortification, is a
are thrown out of Mortar-pieces certain Hole, or hollow Place in
into beſieg'd Places, &c.
figure of a Well, from whence a
CARD. See Chard.
Gallery, dug in like manner under
CARDINAL - WINDS, are the
are the ground, is convey'd to give Air to
South, Weſt, North, and Eaſt Points the Enemies Mine. Some of theſe
of the Compaſs : Alſo the Equi- are more hollow than others, and
noctial and Solftitial Points of the they are uſually made in the Re-
Ecliptic, are called the Four Car- trenchment of the Platform near
dinal-Points.
the Wall.
CASIO
1
+
1
1
1
CAS
CAT
CASEMATE, in Fortification. Cast a Point of Traverſe, in
This ſometimes ſignifies a Well, Navigation, fignifies to prick down
with its ſeveral ſubterraneous on the Chart the Point of the Com-
Branches, or Paſſages, dug in the paſs any Land bears from you, or
Paffage of the Baſtion, till the to find on what Point the Ship
Miner is heard at work, and Air bears at any Inſtant, or what way
given to the Mine. It fometimes the Ship has made.
fignifies
CASTOR, a fix'd Star of the
A Vault of Stone-Work in that ſecond Magnitude in Gemini, whoſe
part of the Flank of a Baſtion being Longitude is one hundred and five
next to the Curtain, on purpoſe to Degrees, forty-one Minutes. Lați-
fire upon the Enemy, and to defend tude ten Deg. two Min.
the Face of the oppoſite Baſtion of Castor and POLLUX, are two
the Moat.
Meteors, that ſometimes, in a great
It ſometimes conſiſts of three Plat- Storm at Sea, appear ſticking to
forms, one above another ; the ſome part of the Ship, in the ſhape
Terre-plan of the Baſtion being the of fiery Balls; and when but one
higheſt. Behind the Parapet that of them is feen, it is called Helena ;
fronts along the Line of the Flank, and both of them are by fome called
there are Guns placed loaded with Tyndaride.
Cartridges of ſmall Shot, to ſcour A Conſtellation of the fix'd Stars
along the Ditch ; and theſe are co. being the ſame with Gemini, one of
ver'd from the Enemies Batteries by the twelve Signs of the Zodiack, is
Earth-Works, faced or lined with called by the Name of Caſtor and
Walls, and are called Orillons, or Pollux.
Epaulments.
CATACAUSTIcs, or Cauſtics by
CASERN, in Fortification, is a Reflection. Theſe Curves are ge-
little Room, Lodgment, or a Build- nerated after the following manner :
ing, erected between the Houſes of If there be an infinite Number of
fortified Towns and the Rampart, Rays, as AB, AC, AD,&c. pro-
ſerving as Apartments, or Lodg- ceeding from the radiating Point A,
ings, for the Soldiers of the Garri- and reflected at any given Curve,
fon, to eaſe the Garriſon. There BDH, ſo that the Angles of Inci-
are commonly two Beds in each dence be ftill equal to thoſe of Re-
Caſern for fix Soldiers to lie in, flection, then the Curve BEG,
three and three in a bed; but the to which the reflected Rays BI,
third part of them being always up- CE, DF, &c. are Tangents con-
on the Guard, there are but four tinually; as in the Points 1, E, F,
left in the Caſern, two in a bed. &c. is called the Cauflic by Re-
CAS E-SHOT, are Muſket-Balls, flection. Or it is the fame thing, if
Stones, old Iron, &c. put into we fay, that a Catacauſtic Curve
Caſes, and ſo ſhot out of great is that form’d by joining the Points
Guns; and they are principally of Concurrence of the ſeveral re-
uſed at Sea, to clear the Enemies flected Rays. And if the reflected
Decks, when they are full of Men. Ray IB be produced to K, ſo that
CASSIOPEA, the Name of one AB BK, and the Curve KL
of the Conſtellations of the fix'd be the Evolute of the Cauſtic BEG,
Stars in the Northern Hemiſphere, beginning at the Point K, then
conſiſting of twenty-five Stars, and the Portion of the Cauſtic BE=
is placed oppoſite to the great Bear, AC - AB +cE - BI conti-
on the other ſide the Pole-ftar, nually. Or if any two incident
Rays,
CAT
1
CAT
A
80, &c.
Rays, as A B, AC, be taken, that Pound weight, to the Diſtance of
Portion of the Caultic that is about Half
a Quarter of a Mile. See
their Deſcription by Vitruvius, Lib.
K
M
10. cap. 15. See alſo Mr. Perrault
N upon Vitruvius, fol. 335. as alſo
Rivius, fol. 597.
B
CATARACT, is a Precipice in the
I
L
Channel of a River, cauſed by Rocks,
or other Obſtacles, hindering the
Courſe of its Stream, from whence
the Water falls with great Impe-
E
tuoſity; as, the River Nile has two;
H Η the River Wologda in Muſcovy; the
River Zaire in the Kingdom of Con-
F
G
Catches, are thoſe parts of a
evolved, while the Ray A B ap- Clock that hold by hooking, and
proaches to a Co-incidence with catching hold of.
AC, is equal to the Difference of CATENARIA, the Name of a
thoſe incident Rays + the Diffe- Carve-Line, form’d by a Rope,
rence of the reflected Rays. hanging freely from two Points of
When the given Curve B D H is Suſpenſion, whether the Points be
a Geometrical one, the Cauſtic will horizontal or not.
be fo too, and the Cauſtic will al The Nature of this Curve was
ways be rectifiable.
ſought after in Galileo's Time; but
The Cauſtic of the Circle is a little was done concerning it, till the
Cycloid, form’d by the Revolution Year 1690 Mr. Bernoulli propoſed it
of a Circle along a Circle.
as a Problem to the Mathematicians
The Cauſtic of the vulgar Semi- of Europe.
Cycloid, when the Rays are parallel This Catenary is a Curve of the
to the Axis thereof, is alſo a vulgar Mechanical kind, and cannot be
Cycloid, deſcribed by the Revolu- expreſſed by a finite algebraic Equa-
tion of a Circle upon the fame Baſe.
tion.
The Cauſtic of the Logarithmic If you ſuppoſe a Line heavy and
Spiral is the ſame Curve.
flexible, firmly fixed to the Points
CATACAUSTICS, or Cataphonics, A, B, the Extremes thereof, then.
is the Science of reflected Sounds; the Weight thereof will bend it into
or that which treats of the Doctrine the Curve ACB, called the Cate-
and Proportions of Echoes.
nary, whoſe fundamental Property
CATADIOPTRICAL TELE. (if DB, dc, be parallel to the Ho-
SCOPE, or Reflecting Teleſcope. See
Telefcope.
d bo
CATAPULTA. A warlike En A
gine of the Ancients, which ſhot
D
B В
Darts, Lances, and long Spears : and
fometimes caſt both Darts and Stones.
Some of theſe Inſtruments were of rizon, CD perpendicular to.A B,
ſuch Force, as to throw Spears, or and B a parallel to CD, and the
rather Beams of eighteen Feet long, Points D and d infinitely near to one
with Iron Heads, and Stones of three another, and a be any given Quan-
Talents, or three hundred and fixty tity) will be this, viz. bc:Bb ::
C
2.
ż.
CA T
CAT
a : C B. The Demonſtration of Po, Pp, be to one another as a , j,
this Property, as alſo of ſeveral o-
thers, may be ſeen in what was Then becauſe the Arch A P is
publiſhed by Dr. Gregory in the year ſuſtaind in Equilibrio, by the Force
1697, for the Month of Auguft: fee of its Weight, whoſe Direction is
alſo its Conſtruction and Nature by parallel to the Line o p, by the
Mr. John Bernoulli in the Alta E. Force of the contiguous Arch AC
ruditorum, for the year 1691, p. drawing according to the Direction
277
of the Tangent at A, parallel to the
But as Mr. Cotes, in his Harmony little Line Po, and by the Force of
of Meaſures, has given a ſhort and the contiguous Arch PB, drawing
neat Account, why may not I lay in the Direction of the Line pP: it
down the ſame here? Let BAC be is evident from Mechanics, that there
a very ſlender Chain, or rather ma- Forces are to one another as op, OP,
thematical Line, flexible throughout pP, or as x, y, z. Therefore if the
by any ſmall Force, which can be Weight of the Arch AP be expreſs'd
neither extended or contracted. This by its Length z, and the given Force
drawing the Arch AC, be expound-
ed by a given Length a, it will be
B
ä :y :: Zia; and fox : Väjj
•; 2:
Vaat zz.
C
P
Therefore a =
Vaatzz
fo a +x=Vaatzą: where-
1
๕ %
; and
2
a a
G
fore z = Vat *
A А
E
2 a x + xx. Wherefore, if the
D
right Line Q A be continued down-
wards to D, ſo that A D be
fuſpended by its Ends B, C, hy the and in the Tangent A E be taken
Force of its own Weight, equally AE to the Arch AP, and DE
diffuſed through all its equal Parti- be join'd: this will be equal to DQ.
cles, is ftretch'd into the Curve B P Wherefore if A E the Length of any
RAC: it is required to find any Arch A P be given, as alſo A Q the
Points of this Curve. If a Plane be Height of the ſaid Arch: there will
ſuppoſed to paſs thro' its Ends B, C, be given AD= a, by joining QE,
perpendicular to the Horizon; it is and biſſecting the ſame at right An-
evident, that all the Points of the gles: for the Perpendicular will paſs
propoſed Curve are ſituated in this thro' the Point D. And A D being
Plane; and ſo, that each will de- once given; from thence will be
ſcend as low as it can. thro' its low- given A E, the Length of any Arch
eſt Point A draw AQ perpendicular AP, whoſe Altitude AQ is given,
to the Horizon, and let P Q drawn by deſcribing a Circle from the
from any Point P, be perpendicular Centre D with the Diſtance QA,
to it, and thro' po being the neareſt which cuts A E in E; and theſe
Point to P poffible, let po be drawn are the mutual Relations of the Pas
parallel to AQ; call A Q, *; rameter AD, the Arch AP, and its
PR, y; and the Arch A P, z; Altitude A Q. Let us now ſee a-
then will the very ſmall Lines Po, bout its Breadth.
From
CA T
САТ
2
From what has been already faid, Surfaces : and particularly, Plane,
a
Spherical, Cònical, and Cylindrical
ỳ =
And ones.
až v za tx
This is a very diverting and uſe-
the Fluent of this laſt Expreſſion ful Part of Knowledge! The Pha-
will be an hyperbolic Space: which nomena ariſing from the Effects of
Space may be ineaſured by the Lo- the Inſtruments that have been in-
garithms. So that P Q will be the vented in this Art, are ſurprizing,
Logarithm of the Ratio between even to thoſe who know the Reaſons
DĚ + EA and AD, or of AP + of the Phænomena they exhibit : But
AQ to AP-AQ, (which Ratio many of thoſe, who are ignorant
is equal to the former) when the thereof, have thought that thofe
Length of the Line A D is 0.43429 wonderful Phænomena were pro-
4481903. So that A D being given duced by Divination. And thoſe
or found, as above; if any Points crafty Knaves, called Conjurers, or
be taken in the Axis Al, fo many Cunning Men, have often had re-
correſpondent Points P of the Curve courſe to catoptric Inſtruments, to
will be had.
help on the Buſineſs of more pro-
CATHETUS. The perpendicular foundly deceiving ignorant People
Leg of a right-angled Triangle, is that came to them, to foretell things,
often called by this Name. Alſo Euclid is ſaid by Proclus, in Lib. 2.
Catbetus, in Catroptrics, is a Line and Marinus in his Preface to Eu-
drawn from the Point of Reflection clid's Data, to have wrote a Trea-
perpendicular to the Plane of the tiſe of Catoptrics, which was tranf-
Glaſs.
lated into Latin by John Pena, and
CATHETUS, in Architecture, is publiſhed in the year 1604. But
taken for a Line ſuppoſed to croſs according to the Opinion of Dr. Gre-
the Middle of a cylindrical Body gory and Sir Henry Savile, it is good
directly, as of a Balliſter, or Co- for little, which makes them believe
lumn. In the Ionic Chapiter it is it to be ſpurious; or, if it was wrote
alſo a Line falling perpendicularly, by him, it has been entirely cot-
and paſſing thro' the Centre or Eye rupted by the Length of Time.
of the Volute.
You have it in Peter Herigon's Courſe
CATHETUS of Incidence, is a right of Mathematics: as alſo in Dr. Gre-
Line drawn from a Point of the Ob- gory's Edition of Euclid's Works.
ject, perpendicular to the reflecting Alhazen an Arabian, compiled a
Line,
large Volume of Optics, wherein he
CATHETUS of Reflektion, or Ca- treats of Catoptrics, about the year
thetus of the Eye, is a right Line, 1100; and after him, Vitellio a Poo
drawn from the Eye, perpendicular lander publiſhed another, in the
to the reflecting Line.
year 1270.-Andrew Tacquet, in his
CATOPTRICs, is that part of Optics, has very well demonſtrated
Optics that treats of reflex Viſion, the fundamental Propoſitions of plane
and explains the Laws and Proper- and ſpherical Speculums.---So alſo
ties of Reflexion; chiefly founded has Dr. Barrow, in his Optical Lec-
úpon this Truth, that the Angle of turesimme
-There is moreover Zachary
Reflection is always equal to the An- Trabe's Catoptrics, David Gregory's
gle of Incidence; and from thence Elements of Catoptrics, Wolfius's Éle-
deducing the Magnitudes, Shapes and ments of Catoptrics, and the learned
Situations of the Appearances of Oh- Dr. Smith's Catoptrics; with ſeveral
jects, ſeen by the Reflexion of poliſh'd others that I do not here mention.
CA
I
1
1
CEL
CEN
CAVALIER, in Fortification, is to be an Affection of Motion, by
á Heap of Earth raiſed in a Fortreſs, which any moveable Body runs thro
to lodge the Cannon for fcouring the a given Space in a given Time.
Field, or oppoſing a commanding CELESTIAL GLOBE. See Globe.
Work. They are ſometimes of a CENTAUR, a Southern Conftel-
round, and ſometimes of a fquare lation, conſiſting of forty Stars.
Figure; and the Top is bordered CENTESM, is the hundredth Part
with a Parapet, to cover the Can- of any Thing:
non mounted in it. There muſt be CENTRAL RULE, is a Rule found
twelve Foot between Cannon and out by Mr. Thomas Baker, and by
Cannon; and if they are raiſed on him publiſh'd, in his Geometrical
the Incloſure of any place, whether Key, in the year 1684; whereby he
in the Middle of the Curtain, or in finds the Centre of a Circle, that is
the Gorge or Baſtion, they are ge- to cut a given Parabola in as many
nerally fifteen or eighteen Foot high Points as an Equation, to be con-
above the Terre-Plane of the Ram- ſtructed, has real Roots : And by
part.
that means he conſtructs all Equae
A Cavalier is ſometimes called a tions, not exceeding Biquadratics,
Double Baſtion; and the Uſe thereof without any previous Reduction or
is to overlook the Enemy's Batteries, Alteration whatſoever.
and to ſcour their Trenches.
Centre of a Circle, is a Point
CAVAZION, in Architecture, is within the fame, from whence all
the Digging or Hollowing away of right Lines, that are drawn to the
the Earth from the Foundation of a Circumference of the Circle, are ea
Building; and this may be one fixth qual to each other.
Part of the Height of the whole CENTRE of a Dial, is that Point
Building.
where the Axis of the World inter-
CAVETTO, is a round Concave feets the Plane of the Dial : And ſo,
Moulding having a quite contrary in thoſe Dials that have Centres, it
Effect to the Quarter-Round. The is that point wherein all the Hour-
Workmen call it a Mouth, when it Lines meet. All Dials have Cen-
is in its natural Situation; and tres, but ſuch as are parallel to the
Throat, when it is turned upſide Axis of the World.
down,
CENTRE of an Ellipfis, is that
CAUKING, in Architecture, is Point thereof, wherein the Diame-
Dove-tailing acroſs.
ters interſect each other; or it iş
CAULICOLI, in Architecture, are that Point biſfecting any Diameter.
the little carved Scrolls, which are The ſame may be ſaid of the Cens
under the Abacus in the Corinthian tre of an Hyperbola.
Order,
CENTRE of the Equant, in the old
CAUSTIC Curves. See Cata. Aftronomy, is a Point in the Linc
cauſtics, and Diacauftics.
of the Aphelion, being fo far diſtant
CAZBRN. See Cafern.
from the Centre of the Excentric,
Cazemate. See Cafemate. towards the Aphelion, as the Sun is
ÇEGINUS, a Fixed Star of the from the Centre of the Excentrica
firſt Magnitude, in the left Shoulder towards the Perihelion.
of Boötes; whoſe Longitude is 194
Centre of Gravity of any Body,
deg. 5 min. Lat. 49 deg. 33 min. is ſuch a Point thereof, that if the
and right Aſcenſion 2 15deg. 39 min. Body be ſupported on it, or fuſpend-
CELERITY, is the Swiftneſs of ed from it, the Body, will reſt in any
any Body, in motion; and is defined given Situation.
CENTRE
1
CEN
CEN
;
CENTRE (COMMON) of Gravity 2 Syx ; whence the Diſtance of the
of two Bodies, is a Point in a Right Centre of Gravity from the Vertex is
Line, joining their Centres, lo po. S y xs
fited, that their Diſtances from it
are reciprocally proportional to the Sy* ; and ſo when you have the
Weights of the Bodies. And if there flowing Quantities of theſe Fluxions
be another Body in the ſame Right yxx and ys, the Centre of Gravity
Line, fo placed, that its Diſtance will be determined.
from fome Point in it be recipro-
3. Every Figure, whether ſuper-
cally, as the Weight of both the ficial or ſolid, which is generated
former Bodies taken together, that by the Motion of a Line or Figure,
Point Thall be the common Centre is equal to the Rectangle under the
of Gravity of all three of the Bo- generating Magnitude, and the Way
dies. Underſtand the ſame of the of its Centre of Gravity, or the
common Centre of Gravity of four, Line which the Centre of Gravity
or more Bodies.
deſcribes.
1. The common Centre of Gra-
The Demonſtration of this moſt
vity of two or more Bodies, does excellent Theorem may be thus:
not change its State of Motion, or
Let us conceive the Weight of the
Reft, by the Actions of Bodies among whole generating Magnitude to be
Reft,
by the Actions of Bodies among collected into the Centre of Gravity
themfelves. And ſo the common
Centre of Gravity of all Bodies, then the whole Weight, produced
mutually acting upon each other, by that Motion, will be equal to the
(all external A&tions and Impedi- Product of the Weight moved into
ments being excluded,) will either the way of the Centre of Gravity ;
reſt, or move uniformly forwards in but ſince Lines and Figures may be
a ſtraight Line.
conſidered as homogeneous Weights,
2. If the Elements, or infinitely their Weights are to one another,
ſmall Parts, as m M N n of any
as their Balks: and ſo the Weight
Figure SAH, be conceived as lo moved is the generating Magnitude,
many Weights hung to the Axis and the Weight produced, the ge-
AE, the Point of Suſpenſion being nerated Magnitude. Wherefore the
in the Vertex A, the Centre of Gra- Figure generated, is equal to the
vity K, in that Axis, will be deter- Product of the generating Magnia
mind by dividing the sum of the tude, drawn into the way of its
Moments of all thoſe ſmall Weights Centre of Gravity.
by the Sum of them all, that is, which can be divided lengthwiſe into
4. In homogeneous Magnitudes,
R-
A
-I
fimilar and equal Parts, the Centre
of Gravity is the ſame as the Centre
of Magnitude. And ſo the Centre
M
of Gravity of any phyſical Right'
Line is in the Middle thereof; as
FK
likewiſe is that of a Parallelogram,
Cylinder, &c. Moreover the Centre
S
H
of Gravity of any equilateral Trian-
if AP=x, MP=y, Pp= x, then gle, regular Polygon, Circle, or El-
is one of the ſmall Weights 2 yx, lipfis, is the ſame as the Centre of
and the Sum of them all 2 S y x, the Magnitude; as is that of a regular
Moment of one of the ſmall Weights Polyhedron, Sphere, and Spheroid,
is 2 yxi, and the Sum of them all is 86,
SH
I
PIN
N
an
n
E
CEN
CEN
5. In any Triangle ABC, if the
mt I
Bale BC be biffected by the Right Ordinate, then will X AD
2 muti
А
A
3
4
12
1
Pt
E
F
P
C D B
C D B
be = AP, the Diſtance of the Cen:
tre of Gravity P from the Vertex A
Line AD, the Centre of Gravity P of the Figure. So that when m=2,
of that Triangle will be in that Line, as in the Apollonian Parabola, AP
at a Diſtance from the Vertex A, will be =AD; if m be
equal to of the biſfecting Line A D. in the cubical Parabola, we ſhall
have
And if the Right Line E F be drawn APS AD; if m= 4, as in the
thro' P, parallel to the Baſe CB, biquadratical Parabola, we ſhall have
dividing the Triangle into two Parts AP=AD; and ſo on. But if
CEFB and EAF, the Part EAF
m be , in which Caſe the Axis
next to the Vertex will be leſs than AD of the Parabola becomes a Tan
the Part CEFB next to the Baſe. gent to the Vertex, we ſhall have
6. If a Trapezium ABCD be di- A D for the Diſtance of the Centre
vided into two Triangles DAB, of Gravity of a double external pa-
rabolical Space from the Vertex; if
B m be AP will be = t; if m
C
be=, AP will be ={ AD; and
ſo on.
G
8. The Diſtance of the Centre of
Gravity of an Arch of a Circle,
E
from the Centre of the Circle, is
to the Radius, as the Chord of that
D
A
Arch is to the Arch itſelf; and in
the Semi-circumference, as the Dia-
DCB, by a Diagonal DB; and if
E be the Centre of Gravity of the
meter is to the Semi-circumference.
Triangle DB A, and F that of the cle, and the Radius A D biſfects the
9. If ABC be a Sector of a Cir-
Triangle DCB: and the Line E F
joining the ſaid Centres, be divided
D
in G, in ſuch manner, that the
whole Line EF be to the Diſtance
FG, as the Trapezium is to the
B
C
Triangle AD B; or the whole Line
EF to the Line EG, as the Trape-
zium is to the Triangle DCB, the
Point G will be the Centre of Gra-
vity of the ſaid Trapezium.
1. IF CAB be any Parabola, whoſe Arch BC, then the Distance AP of
Nature is expreſs'd by the Equation the Centre of Gravity P of that
Ixx = ym, and AD (x) be a Sector, will be to of the Radius
Diameter, and CB=y) a double AD, as the Chord of the Arch BC
مر
to
CEN
}
CEN
2
མ་ས
1
to the Arch BC itſelf: ſo that in a Liné parallel to its Axis, &c. Take
Semicircle,as half the Circumference a few Examples.: 1, To find the Arta
is to its Chord, fo is of the Radius of a Circle ADCE. This may be
to the Diſtance of the Centre of Gra- genérated by the Rộtation of the
yity of a Semi-circle from its Centre. Semidiameter A B above the Centre
Conſequently, by knowing the Cen-
tre of Gravity of the Sečtor ABC
of a Circle, and the Centre of Gra-
D
vity of the Triangle ABC, we
can find the Centre of Gravity of
PS
the Segment BDC of a Circle.
10. The Centre of Gravity of a
B
Pyramid, or Cone, is diſtant' from
the Vertex 4 parts of the Axis.
1. The Centre of Gravity of a
Parabolic Conoid is diſtant from the B. But ſince the Centre P of Gía-
Vertex ſ Parts of the Axis.
vity of A B is in the middle there-
12. In a Segment of a Sphere, of, and this deſcribes the Periphery
it is as three times the Radius leſ- PFG of a Circle, Concentric to
ſen'd by the Altitude, is twice the ADCE, whilft A B is defcribing
Radius lefſen'd by 1 of the Altitude ADCE; therefore the Area of
of the Segment ; fo is the Altitude the Circle A DCE, will be equal
of the Segment to the Diſtance of to the Periphery PFG, (being the
the Centre of Gravity from the way of the Centre of Gravity P)
Vertex. and the Segments of Spheres drawn into IAB, that is, (ſince
and Spheroids having a common the Circumferences of Circles are to
Altitude, have the ſame Centre of each other as their Diameters) equal
Gravity.
to į the Periphery ADCE, drawn
13. In an Hyperbolic Conoid, as into the Radius.
fix times the tranſverſe Axis added
2. To find the Surface of a right'
to four times the Altitude of the Cone A BDE; this Solid may be
Conoid, is to four times the tranſ- generated by the Rotation of the
verſe Axis added to three times the right-angled Triangle ABC, about
Altitude ; fo is the Altitude to the
Diſtance of the Centre of Gravity
A
from the Vertex.
The Theorem above-mentioned at
7. 3. is of excellent uſe in finding
P
F
out the Areas of Surfaces, and the
Solidity of Solids, generated by the
G
Rotation of Curve-lin'd Spaces, a-
E
bout Lines given in poſition, by hav B C
ing their Centres of Gravity given;
D
as that of a Semi.Circle, Semi-El-
lipfis or Semi-Parabola, or Hyper- its Perpendicular AC, and the Sur-
bola, revolving about its Diameter, face thereof generated by the Ro-
or any right Line parallel to it, the tation of the Hypotheneuſe A B.
Segment of a Circle, Ellipfis, Para- Where fince P, the middle of A B,
bola, or Hyperbola about its Baſe, is the Centre of Gravity of AB;
or any right Line parallel to it, or the Rectangle under AB, and the
a whole Ellipſis about any right Circumference of a Circle P. GF,
being
5
.
8r
onumu
=*r* =
C EN
CEN
being the way of the Centre of Rotation of a Semicircle BDC,
Gravity, will be equal to the Sur- about the Diameter B C. Let the
face of the Cone; that is, ſince the
Circumference P GF, is of the
>
Circumference BDE; the Area
of the Surface of the Cone will be
one half the ſlant Height AB,
P.
drawn into the Periphery BDE of
P
c
A
the Baſe.
3. To find the Solidity of a Cone
ABFD; ſuppoſe the Iſoſceles Tri-
E
angle ABC, whoſe Centre of Gra-
vity is P, to revolve about its fide Periphery of the Circle BDCE, be
AC; this will deſcribe a double be called p; its Radii AD, t; and
then (by 9. of this) the Diſtance
AP of the Centre of Gravity P
B
8rr
from A, will be
and fo
3
the way of the Centre of Gravity
P, or Circumference of the Circle
deſcribed by AP, will be
3
And ſince the Semicircle BDC is
G
pr
; the Solidity of the Sphere
4
8r pr
pr
will be
3 4
pr
D
X2 rx
Cone ABCD, the half of which
will be the Cone ABF D, whoſe
x Circle BDCE.
Baſe is the Circle FBGD, and Al. 5. To find the surface of the faid
titude the Line A E, being the half Sphere; the Diſtance AP of the Centre
of the ſide AC of the Iſoſceles of Gravity of the Semicircumference
Triangle A BC. Therefore the So- BDC, deſcribing the Surface of the
lidity of this Cone will be equal to Sphere, in this caſe will be (by 8.
the Area of the Triangle A BE,
4rr
drawn into the Circumference of a
of this)
and the way of
P p
Circle, whoſe Radius is E P, this the ſaid Centre of Gravity will be
being the way of the Centre of Gra-
P rp
vity P; but ſince EP is Š
Therefore
the Periphery deſcrib'd by P will be
ſ of that deſcrib’d by B. Conſe- = 4* Area of the Circle BDCE,
quently the Solidity of the ſaid will be the Surface of the Sphere.
Cone, will be = of the Periphery 6. If the Plane of a Circle
of the Bafe drawn into AEX EB ABGC, whose Centre is P, re-
- Periphery of the Baſe x L EB x volves about the right Line EF, at:
AE=Baſe x } AE.
the Diſtance DP from its Centre,
4. To find the Solidity of a Sphere. thereby generating a Cylindrical
A Sphere may be generated by the Ring 1 The Solidity of that Ring will
K
be
fo
MICRKT
X
2
frx4
2
to e
cs
4in x
4 *
2
2
CEN
CEN
be equal to a Cylinder, whoje Baſe FH,a; EA or AF,b. and the
is the Circle ABGC, and Altitude Periphery of the Circle, whoſe
{
A
B
D
P
C
G
H
PH
G
a; and ſo
D
P
E A F
Radius is E, G, or AD, p. and ſup-
poſe P to be the Centre of Gravity
of the ſaid Trilineal Space AGH,
the Circle whoje Radius is the right then will AP be = 3
Line DP. Thisis evident, becauſe the
Centre of Gravity of the Circle, is the Periphery deſcribed by P, will
the ſame as the Centre of the Circle, be =%DBut ſince the Area of
and the way of the Centre of Gra- the Space GAH is į ab; therefore
vity is the Periphery, whoſe Radius the Solid generated as above will be
is DP.
=*pxab=lap xb, that is
7. And the surface of that Solid is it will be one half of a Cylinder,
equal to the Surface of that Cylinder. whoſe Altitude is GH, and Radius
8. If the ſaid Ring be divided of the Baſe EG: And ſince this
into two parts by a Cylindrical Sur. Solid is the Complement to a Cy-
face paſſing thro' the Circumference linder of the two equal parabolical
of the Circle, deſcribed by the Cen- Conoids generated by the equal Se-
tre P of the generating Circle mi-parabola's AGE, AHF; there-
ABGC, the outermoſt part of that fore it follows, that any parabolic
Ring will be to the innermoſt, as Conoid is of a Cylinder of the
* DPxp-tore is to DP pærr,
ſame Baſe and Altitude.
and the Surface of the one will be 10. If the Parabola BAC re-
to that of the other, as DP p volves about its Baſe BC, or double
ter is to I Dº xp-rr. Both Ordinate BC: to find the Solidity of
theſe Propofitions evidently follow
upon the Suppoſition that the Di-
A
ſtance of the Centre of Gravity of
a Semicircle from the Centre of the
Circle is
and that of the
3 P
P
Semicircle
B D c
9. If AG, AH, be two equal
Parabola's touching one another in the Solid generated thereby. Let us
their principal Vertex A, and the call the Axis AD, a; the Baſe
Irilineal Space GAH revolves a CB, b, and the Circumference
- bout tbe common Axis E AF of the (whoſe Radius is a,) p, then will PD
Parabola's, and it be required to find bea; and the Circumference
the Solidity of the Solid generatid by deſcribed by P,will be p. Therefore
ſuch a Motion. Let us call EG or ab x pisiabpallo will be equal
8 rr
4 149
to
1
8 8 rr
ар
8r
OEN
C
CEN
to the Solid generated as above,
which will be to its circumſcribing PF will be = a - and
3 P
Cylinder as 4 to į or as 8 to 15.
n. To find the Solid generated the Periphery deſcribed by P, will
by the Rotation of the Quadrilineal
be
Wherefore fince the
Space ABFE (contain'd under
3
the Quadrantal Circular Arch AB, Area of the Quadrant AB is =
the Tangent AE, the Perpendicu-
rp
lar EF, equal to the Radius AQ, the Area of the Solid generat-
and the Continuation BF of the
Radius QB) about the right Line ed as above, will be s
ар
FE.
3
It is plain from the Generation,
pr.
rr p
that the Solid thus generated is one
And ſo bea
8
3
hálf of the Solid produced from the
Revolution of the whole Space cauſe is the Solidity of the Cy-
ABSHE about the right Line EH.
aap
app arp
S
H
linder aforeſaid ;
t
8 3
8 ,
8r
X
a PP
аар
2
+
2
Fig3
Q
PBF
A
E
aco
a PP
+
2
11
BAE C
H
will be the Solidity of the Solid re-
Fig.2
quired. (See the Solid at Fig. 2.)
And as p: 4%::
D
А.
8
E
gege p
G
: 2 a à y Larpt 33
3
Square folid of Fig. 3. that may be
And that this laſt Solid is the Com. inſcribed in the ſolid of Fig. 2.
plement of a Cylinder (whoſe Ra- Much after the ſame manner the
dius of the Baſe is AE, and Altitude Surfaces of theſe Solids may be
EF) to the Solid produced by the found, which I leave to be done by
Rotation of the Quadrant AQB thoſe who delight in theſe things.
about the Line EF. Let QF, be
Thus I have given a few Exam.
called a; EF, r; and the Arch ples of the Excellence of our Theo-
of the Circle - whoſe Radius is
rem, in expeditiouſly and entily find-
ing the Areas of Surfaces, and Soli-
AQ, p. Then fince QP
dities of Solids, by means of the
3p) Centre of Gravity. It is mention'd.
8 for
K2
by
CEN
CEN
by Pappus at the latter End of his ceived to be divided into three or
Preface to his ſeventh Book of Ma- more equal Parts, it will.be cut into
thematical Collections ; but Father equal and like Cylinders.
Guldin the Jeſuit, in his ſecond and Centre of Motion of any Body, is
third Book of the Centre of Gra- the Point about which any Body
vity, has more exprefly demonftrat- moves, when faſten'd any ways to
ed it, by an Induction of ſeveral it, or made to revolve round it.
Examples
Centre of Oſcillation, is a Point,
The aforeſaid Father Guldin, in wherein, if all the Gravity of a
his Centrobarica, has ſhewn how to compound Pendulum be collected,
find the Centres of Gravity of Fi- every Oſcillation will ſtill be per-
gures; and fo has Dr. Wallis, in formed in the ſame time as before.
his Mechanics : But their manner Or it is that point of a Compound
of Performance is both tedious; Pendulum, whoſe Diſtance from the
troubleſome, and imperfect. Ca- Point of Suſpenſion is equal to the
fatus too, in his Mechanics, has Length of a ſimple Pendulum, whoſe
fhewn how to find them mechani- Oſcillations are performed in the
cally, or by Trials. But the moſt ſame time as the Oſcillation of the
ready, elegant, and general Help, that Compound ones.
the Nature of the Buſineſs ſeems to If * be the Abſciſs of an oſcillat-
admit of, is the inverſe Method of ing plain Figure, as. A SH, and 2 y
Fluxions. See Carré, Hayes, Wol the correſpondent double Ordinate:
fiús, &c.
then will the Diftance of the Centre
Mr. Borellus, in Lib. de Motu A- O of Oſcillation (from the Axis RI
nimalium, Part 1. Prop. 134. ſays, of Oſcillation) be equal to the fluent
That the Centre of Gravity of a hum of y x2 à divided by the Fluent
man Body, when extended, is be- of yxå. And therefore, if from
tween the Nates and Pubis; and ſo the particular Equation of any gi-
the whole Gtavity of the Body cen ven Figure, the Value of
tres in that place where Nature has preſſed in the Terms of x, and the
allotted the Seat of the Genitals ; Fluents be duly found and divided
which, no doubt, was for facilitating by each other, the Diſtance of the
the Buſineſs of Coition.
Centre of Oſcillation from the Axis
Centre of an Hyperbola, is that will be had in common terms.
Point wherein the Diameters meet; If ſeveral Weights D, H, B, TA
or it is that Point biffecting any being ſuppoſed to gravitate in
Diameter, and is without the Fi the Points D, H,B, do keep
gure, and common to the oppoſite at the fame Diſtance, with
D
Sections.
regard to one another, and
CENTRE of Magnitude of any from the Point of Sufpenfion to
Body, is that Point which is equally A, on the inflexible Rod AB,
remote from its extreme Parts. In and, oſcillating about the Point Н
Homogeneal Bodies, that can be A, do make a Compound
cut into like and equal Parts, ac Pendulum : the Diſtance of O,
cording to their Length, the Centre the Centre of Oſcillation from IB
of Gravity is the ſame as the Con- the Point of Suſpenſion A, will be
tre of Magnitude.
had, by drawing each of the
Such an Homogeneal Body is, for Weights into the Squares of their..
Example, a Leaden Cylinder, that Diſtances, and dividing the Aggre-
can be cut lengthwiſe into like Parts; gate by the Sum of the Moments
for if the Length thereof be con of the ſame Weights.
The
be ex-
.
.
1
AE.
C EN
CEN
The Centre (O) of Oſcillation of had done the thing in a few of the
a ſtraight Line AB will be diſtant moſt eaſy Cafes only, without any
from A, the Point of Suſpenſion ſ ſufficient Demonftration ; and not
of the whole Line. The Centre (O) ſolved Merſennus's Problems only,
of Oſcillation of the Equicrural Tri- but found out many others much
angle ASH, oſcillating about the more difficult, ſhewing a way of
Axis RI, parallel to the Baſe SH finding this Centre in Lines, Super-
will be diſtant from A, the Point of ficies, and Solids.-In the Aeta Ēru-
Suſpenſion, of A E.
ditorum for Leipfick, An. 1691. pag.
317. ad An, 1714. pag. 257. you
А
have this Doctrine handled by the
R-
I
two Bernoulli's ; you have alſo the
fame by Mr. Herman, in his Trea-
tiſe de Motu Corporum Solidorum &
Fluidorum. The fame is to be found
in Treatiſes of the Inverſe Method
of Fluxions: See Hayes, Carré,
Wolfius, &c.
CENTRE of Percuſion, is that
-S E H
Point of a Body in Motion, where-
in all the Forces of that Body are
And if SAH was the common united into one; or it is that Point
Parabola, A being the Vertex, and wherein the Stroke of the Body
AE the Axis, then the Diſtance AO will be greateſt ; and is much the
ſame, with reſpect to the Forces, as
Mr. Huygens, in his Horologium the Centre of Gravity to the
Oſcillatorium, has firſt ſhewn how Weights.
to find the Centre of Oſcillation, The Centre of Percuſſion is the
He tells us at the Beginning of his ſame as the Centre of Oſcillation, if
Diſcourſe on this Subject, that Mer- the ſtriking Body revolves about a
ſennus firſt propoſed the Problem to fixed Point. Whence a Stick of a
him, when he (Huygens) was very Cylindrical Figure, ſuppoſing the
young, even a.Youth, requiring him Centre of Motion at the Hand, will
to ſolve the ſame in Sectors of Cir- ſtrike the greateſt Blow at a diſtance,
cles ſuſpended from their Angles, about of its Length from the
and the Middles of their Baſes; as Hand.
alſo when they oſcillate fide-ways : The Centre of Percuſſion is the
In the Segment of Circles and Tri- fame as the Centre of Gravity, if all
angles, hanging from their Vertex, the Parts of the ſtriking Body are
and the Middles of their Baſes. But, carried by a parallel Motion, or
ſays Huygens, I at firſt, not having move with the ſame Velocity.
found out any thing that would Centre of a Regular Polygon, or
open a Paffage into this Buſineſs, Regular Body, is the ſame as chat
was repulſed at firſt ſetting out, and of the infcrib'd Circle or Sphere.
ſtopt from a further Proſecution of CENTRE of a Sphere, is a Point
the thing ; till at length being in- in the middle thereof, from whence
cited thereto, by the Conſideration all Right Lines, drawn to the Su-
of attempering the Motion of the perficies, are equal to one another. :
Pendulums of my Clock, I conquer.
CENTRIFUGAL Force, is that
ed all Difficulties, going far beyond Force by which all Bodies that move
Deſcartes, Fabry, and others, who round any other Body in a Curve,
1
K 3
do
CE N
CEN
:
do endeavour to fly off in every Point from the Body to that Centre) pro-
of the Curve.
portional to the time. And contra-
CENTRIPETAL Force, is that rywife,
by which a Body is every where im That Body which is moved in any
pelled, or any how tends towards Curve in a Plane, and by a Radius
fome Point, as a Centre. Among drawn to ſome Point at reft, or mo-
which may be reckon'd Gravity, ving uniformly in a right Line, de-
whereby Bodies tend towards the ſcribes Areas about that Point pro-
Centre of the Earth; the magneti- portional to the time, is urged by a
cal Attraction whereby it draw centripetal Force ten ling to that
Iron; and that Force, whatever it Point.
-Theſe are the two famous
be, whereby the Planets are conti- Theorems of Sir Iſaac Newton, firſt
nually drawn back from right-lin'd found out and demonſtrated by him,
Motions, and made to move in as you may ſee at the Beginning of
Curves.
Lib. 1. Princip, Mathem. and upon
The Centripetal and Centrifugal which all the phyſical Aſtronomy is
Force of the ſame revolving. Body founded.
in the fame Point of the Curve that The greater the Quantity of Mat-
it deſcribes, are always equal and ter in any Body is, the greater is its
contrary.
centripetal Force; all things elſe a-
If a Body laid upon a Plane, does like.
at the ſaine time, and about the If a Solid with a Fluid be included
fame Centre revolve with that Plane, in a determinate Space'; if it be
and ſo delcribes a Circle : and if the lighter than the Fluid, it will come
centripetal Force, by which the to the Centre; if heavier, it will
Body is drawn or impelled every recede from that Centre: becauſe
moment towards that Centre, ſhould the heavier Body has the greater
cerſe to act, and the Plane ſhould centrifugal Force.
continue to moye with the fame Ve-
The centrifugal Forces of revol-
locity; the Body will begin to re- ving Bodies, are in a Ratio com-
cede from the Centre, with reſpect pounded of their Quantities of Mat-
to the Plan?, in a Line which paſſes ter; Diſtances from the Centre;
thro’the Piane. The truth of which and the inverſe duplicate Ratio of
will eaſily appear, by faſtening a their periodical Times.
Ball to a Packthread, one End of If a Body moves in an Ellipfis ;
which is fixed to the Centre of a the Law of the centripetal Force,
found Table, moving about that tending to the Centre of the Ellipfis,
Centre, and laying the Ball upon will be directly as the Diſtance froni
the Plane of the Table, ſo as to roll the Centre: but if to the Focus,
round together with the Plane of reciprocally as the Square of the
the Table at the ſame time.
Diſtance. The ſame holds good in
When a Body moves about a Çen- the Hyperbola and Parabola, when
tre, if as it moves it comes nearer the centripetal Force tends to their
to the Centre, its Motion is accele- Foci.
rated; but on the contrary, retard If ſeveral Bodies revolve about a
ed, if it recedes from the Centre. common Centre, and the centripetal
A Body which is kept moving in Force be in the reciprocal duplicate
à curve Line, by a Force tending Ratio of the Places from the Centre,
towards a fixed Centre, deſcribes A- the principal Latus Re£tums of the
reas (contained under Portions of Orbits are in the duplicate Rátio of
that Cuve, and right Lines drawn the Area's, which the Bodies, by
Radij
CEN
CEN
Radii drawn to that Centre, do de. CG be drawn from the Centre C,
ſcribe: alſo the Squares of the pe- parallel to the Ray RP, meeting
riodical Times in Ellipſes are in the the Tangent to the Section at G,
feſquiplicate Ratio of the greater the Law of the centripetal Force
Axes, and drawing right Lines to;
CG3
the Bodies, which there touch the will be as
Orbits, and letting fall Perpendicu-
RPZ
lars from the common Focus to theſe The Doctrine of centrifugal Forces
Tangents; the Velocities of the Bo was firſt mentioned by Mr. Huygens,
dies are in a Ratio compounded of the in his Horologium Oſcillatorium, (at
inverſe Ratio of the Perpendiculars, the end) which was publiſh'd anno
and the direct fubduplicate Ratio of 1673, where he has given a few
the principal Latus Reetums. ealy Caſes in Bodies revolving in the
If a Body P, in revolving about Circumference of Circles, although
the Centres deſcribes the Curve A without any Demonftration. But
PQ, and the right Line P R touches Sir Iſaac Newton, in his Principia,
was the firſt who has fully handled
R
this Matter; at leaſt as far as re-
P
gards the conic Sections. After him
there have been ſeveral other Wri-
T
ters upon this Subject, as Mr. Leiba
nitz, Mr. Varignon, in the Memoirs
S
A
de l'Academie Royale des Sciences;
Dr. Keil, in the Philofophical Tranſ-
the Curve in P, and the Line QR actions ; Mr. Bernoulli, Mr. Herman,
be drawn parallel, and infinitely Mr. Cotes, in his Harmonia Menſu-
near to SP, and Q T be drawn per- rarum; Mr. Maclaurin, in his Geo-
pendicular to SP: then will the cen-
metria Organica; Mr. Euler, in his
tripetal Force in any Point P of the Liber de Matu; wherein this laſt
Curve be reciprocally proportional confiders the Curves deſcribed by a
S P x @ T'
Body acted upon by centripetal
QR
Forces tending to ſeveral fixed Points.
CentrOBARYCAL, is what re-
If the periodic Times of Bodies, lates to the Centre of Gravity:
revolving in Circles, be as any
Cepheus, a Conſtellation in the
Power Rn of the Radii, then the Northern Hemiſphere, conſiſting of
centripetal Force will be recipro- ſeventeen Stars,
Cetus, the Whale, a Southern
cally as the Power R2*-*. And Conſtellation, confifting of twenty.
contrarywiſe,
three Stars,
If the Body P, tending to any CHAIN, an Inſtrument of hard
given Point Ř, moves in the Peri- Wire, diſtinguiſhed into a hundred
equal Parts, called Links, being uſed
to meaſure Lengths in ſurveying of
Land. They are of ſeveral ſorts;
R
1. A Chain of a hundred Foot
long, each Link being one Foot in
Length, and at each tenth Foot there
meter of any given conic Section, is a Plate of Brats, with a Figure
whoſe Centre is C, and if the Line engraved upon it, ſhewing readily
how
to
G
P
as
C
K 4
C Η Α
CHA
how many Links are from the Be- Hand from this laſt Product, the
ginning of the Chain; and for more reſt will be Roods.
eaſe in reckoning, there is, or ſhould Laſtly, if you multiply the five
be a braſs Ring at every five Links, Figures cut off at the ſecond Mul-
that is, one between every two tiplication by 40; and five Figures
Plates.
being cut off
, the reſt will be ſquare
This Chain is moſt convenient for Perches or Poles.
meaſuring of large Diſtances.
CHAIN-SHOT, is two Bullets, or
2. A Chain of fixteen Foot and rather Half-Bullets, faſten'd toge-
a balf in Length, and made ſo as ther with a Chain, their Uſe being
to contain a hundred Links, with chiefly to ſhoot down Mafts, or cut.
Rings at every tenth Link. This the Rigging of a Ship, &c.
Chain is moſt uſeful in meaſuring CHAMBER, is that part of the
ſmall Gardens, or Orchards, by Cavity of a great Gun, where her
Perch or Pole Meaſure.
Carriage lies.
3. A Chain of four Poles, or CHAMBRANLE, an Ornament in.
Perches in Length, (called Gunter's Maſonry and Joiners Work, bor-
Chain) which is fixty-ſix Foot, ordering the three Sides of Doors,
twenty-two Yards; for each Perch Windows, and Chimneys, and is
contains fixteen Foot and a half. different according to the ſeveral
This whole Chain is divided into a Orders, and conſiſts of three Parts,
hundred Links; whereof twenty- viz. the Top, called the Traverſe,
five is an exact Perch or Pole; and and the two Sides the Aſcendants.
for readily accounting, there is uſual CHANDELIERS, in Fortification,
ly a 'remarkable Diſtinction by ſome are wooden Parapets made of two
Plate, or large Ring, at the end of upright Scakes, about fix Foot high,
· twenty-five Links; alſo at the end ſupporting divers Planks laid a-croſs
of every tenth Link it is uſual to
one another, or Bavins filled with
faften a Plate of Braſs with Notches Earth. They are made uſe of in
in it, ſhewing how many Links are Approaches, Galleries, and Mines,
from the Beginning of the Chain; to cover the Workmen, and to hin-
and this Chain, of all others, is der the Beſieged from forcing them
the moſt convenient for Land-Mea- to quit their Labours. Theſe differ
fure.
from Blinds only in this, viz. that
If two Lengths for finding the. the former ſerve to cover the Pio-
Area of any Parallelogram, Tri neers before, and the latter to cover
angle, &c. in Acres, Roods, and them over Head.
Perches, be given in Chains and CHANEL, in the lonic Capital,
Links; and if the Links be above is a part fomewhat hollow under
ten, you ſet the Chains and Links the Abacus after the Liſtel, and lies
down with a Prick of the Pen be- upon the Echinus, having its Con-
tween them; but if. under ten, a tours or Turnings on each side to..
Cipher be ſet before the Links, and make the Voluta's.
you multiply the two Lengths like CHAPITERS, in Architecture,
decimal Fractions. Then if five are the Crowns, or upper Parts of
Figures towards the Right Hand be a Pillar. Thoſe that have no Or-
cut off, the Figures to the Left Hand naments, are called Chapiters with
will be Acres
Mouldings, ſuch as the Tuſcan and
If the five Figures cut off be mul. Doric; the firſt whereof is the moſt
tiplied by 4, and five Figures be ſimple, having its Abacus ſquare,
again cut off towards the Right without any Mouldings; but the
Abacus
1
CH A
CH A
Abacus of the other is crowned with tity to be leſs than 'nothing, and
an Aſtragal, and three Annulețs therefore ſuch Quantities are called
under the Echinus. All thoſe that negative Quantities ; as 5 is a
have Leaves and carv'd Ornaments, negative Quantity, or 5 leſs than
are term'd Chapiters with Sculp- nothing.
tures, and the firſt of them is the This negative Sign is alſo the
Corinthian, which is adorned with Mark of Subtraction, and fignifies,
two Rows of Leaves; as alſo eight that the Quantities on each ſide of
greater, and as many leſs Voluta's, it, are ſubtracted from each other';
placed under a Body called the Tym- as when you ſee a -6, it is read i
panum. Theſe are called uſually leſs b, or 6 ſubtracted from a.
Capitals.
cs, or 1, is the Character ex-
CHAPTRELs, in Architecture, preſſing the Difference between two
are the ſame with Impofts, and fig- Quantities when it is not yet known
nify thoſe Parts on which the Feet which is the greater of the two;
of Arches ftand, and their Height for here the Sign — cannot be uſed,
or Thickneſs is commonly equal to becauſe it ſuppoſes the Quantity
the Breadth of the lower part of the following to be always. leſs than
Key-Stone.
that going before it.
CHARACTERISTICK of a Loga x is the sign of Multiplication;
rithm. See Index, or Exponent. fhewing, that the Quantities on each
CHARACTERS (MATHEMATI. fide the ſame are to be multiplied
CAL,) are certain Marks invented by one another; as a xb, or AB x
by Mathematicians, for avoiding CD, is to be read a multiplied by
Prolixity, and more clearly convey- b, or 'A B multiplied by CD.
ing their Thoughts to Learners, and = is the Mark of Diviſion, fig-
are as follow:
nifying, that the firſt of the two
= is the Mark of Equality, (tho' Quantities between it is divided by
Deſcartes, and ſome others uſe this the latter; as a-b, ſignifies that
D,) and ſignifies that the Quantities a is divided by b.
on each ſide of it are equal to one
o is the Character of Involution,
another; as, a=b, fignifies that a that is, of producing the Square of
is equal to b.
any Quantity, or of multiplying
+ in Algebra, is a sign of real any Quantity into itſelf. In ſome
Exiſtence of the Quantity it ſtands Books of Algebra it is placed in the
before, and is called an affirmative Margin, and thews, that the Step
and poſitive Sign, becauſe it implies of the Equation, againſt which it
the Quantity to be of a poſitive and ſtands, is to be multiplied into it-
real Nature, and is directly contrary ſelf; or if it be a Square already,
to the following Sign --
then to be raiſed to that Power that
This affirmative Sign is alſo the the Index fet after the Character
Mark of Addition, and ſignifies that expreſſes.
the Quantities on each ſide of it are is the Character of Evolution,
added together; as, if you ſee a t-b, that is, of extracting the Roots out
or 3 +5, it implies that a is added of the ſeveral Powers, and is the
to b, or 3 added to 5, and is uſually Reverſe of the laſt-mentioned Sign.
read a more b.
:: is the Mark of Geometrical
- This is the Note of Negation, Proportion disjunct, and is uſually
negative Exiſtence, or Non-entity; placed between two pair of cqual
and whenever it ſtands alone before Ratio's; as 3:6 ::4:8 few's
any Quantity, it ſhews that Quan- that 3 is to 6, as 4 to 8.
CH A
CH A
4
I Gemini.
% Cancer.
2 Leo.
THR Virgo.
no Libra.
ma Scorpio.
Sagittarius.
Yo Capricorn.
more on Aquarius.
* Piſces.
The Characters of the Aſpects are,
o Conjunction,
A Trine.
O Quartile.
Sextile.
Oppofition.
The chief Characters in Muſick are,
3
Semibreve.
Minim.
is the Mark of Geometrical
Proportion continued, and implies the
Ratio to be ftill carried on without
any Interruption ; as 2, 4, 8, 16,
32, 64, .
✓ is the Sign of Radicality, and
Thews (according to the Index of the
Power, that is ſet over or after it,)
that the Square, Cube or other Root,
is extracted, or is to be ſo out of
any Quantity; as ✓ 16, or 16,
orv (2) 16, fignifies the Square
Root of 16, and 16 is the Cube
Root of 16.
C., or, is the Character
of greater. And,
And, the Mark of the
leſler of two Quantities.
1. is the sign for Parallels, and
fignifies that two Lines, or Planes,
are equi-diftant.
A Triangle.
Square.
Od Rectangle.
O Circle, or the Sun.
Ā Equiangular, or similar.
Equilateral.
< Angle.
r Right-Angle.
T Perpendicular.
::: is the Mark for Arithmetical
Progreſſion.
a. b=d. This, by Wolfius, fig.
nifies, that a is to b, as c to d.
The Characters of the ſeven Pla-
nets are,
ħ Saturn.
7 Jupiter.
Mars.
Sol.
Vepus, .
Mercury.
( Luna.
The Characters of the Twelve
Signs are,
r Aries.
8 Taurus.
Crochet.
Quaver.
Semi-Quaver.
B
Demi-Quaver.
Baſe-Cliff.
Treble-Cliff.
Tenor-Clift
Coun-
The
t
CHA
CHE
CHARLES'S-WAIN, ſeven Stars
Counter-Tenor-Cliff.
in the Conſtellation, called Urfa
Major.
CHARTS, are Sea-Maps for the
Uſe of Seamen, having the Sea-
coafts, Sands, Rocks, &c. depicted
Sharp
upon them, and are principally of
two kinds, viz. the plain Chart,
and Mercator's or rather Wright's.
Flat.
Of theſe you will ſee more under
the Words Plain Charts, and Mera
//
cator's Chart.
CHASE of a Gun, is its whole
Shake.
Length.
CHAUSE-TRAPPES, or Coltrops,
in Fortification, are Iron Inftru-
ments with four Spikes about four
Beat.
Inches long, made in ſuch a manner,
that let them fall which way ſoever,
one Point will always lie uppermoft,
like a Nail. They are uſually ſcat-
ter'd and thrown into Moats and
Common Time Now.
Breaches, to gall the Horſes Feet,
and ſtop the hafty Approach of the
Enemy.
Common Time ſwifter. Chemin de Ronds, in Fortifica-
tion, is the way of the Rounds, or a
Space between the Rampart and the '
low Parapet under it, for the Rounds
Minim, or Bar-Reft. to go about the ſame, with the Faufe
Bray.
CHEMISE in Fortification, is a
Wall that lines a Baftion, or any o
ther Bulwark of Earth, for its greater
Crochet-Reſt.
Support; or it is the Solidity of the
Wall from the Talus to the Stone-
Row.
CHERSONESUS, in Geography,
fignifies the ſame with Peninſula,
and is a part of the Land encloſed
all round with Water, except one
Semi-Quaver-Reit.
narrow Neck, by which it joins to
the main Land, that being called an
Iſthmus. Of theſe Cherſones there
are reckon'd up fourteen by Varenius,
Contradict.
in his Geography, Chap. 8. Prop. 10.
CHARGED CYLINDER, is that
CHEVAUX DE Frise, or Friſe.
Part of the Chaſe of a great Gun, land Horſe, is a large Joift
, or Piece
where the Powder and Ball are
of Timber, about a foot in Diame-
placed.
ter, and ten or twelve in Length.
There
HI
Quaver-Reft.
lib. 1.
C HR
CIR
There are driven a great Number which abounds in Semi-Tones, and
of wooden Pins into the sides there. contains only the leaſt diatonical
of, about fix Foot long, croſſing one Degrees.
another, and having their Ends CHRONOLOGY, as it is common-
arm'd with Iron Points, Their ly taken, is the Arithmetical Com-
principal Uſe is to ſtop up Breaches, putation of Time for hiſtorical Uſes ;
or to ſecure the Avenues of a Camp that thereby the Beginnings and
from the Inroads both of Horſe and Endings of Princes Reigns, the Re-
Foot. Theſe are much the ſame volutions of Empires and Kingdoms,
with Turnpikes.
Battles, Sieges, or any other me-
CHILIADS, are the Tables of morable Actions, may be truly
Logarithms; being ſo called, be- ftated.
cauſe they were at firſt divided into CHRONOSCOPE, the ſame as a
Thouſands. Thus, in the Year Pendulum, to meaſure Time with.
1624. Mr. Briggs publiſhed a Table CHRYSTALLINE HE AVENS,
of Logarithms for twenty Chiliads of Theſe, in the Ptolemaic Syſtem, were
abſolute Numbers, and afterwards two: Whereof one ſerved them to
for ten Chiliads more, and then for explain the flow Motion of the fixed
one more, that is, for thirty-one Stars, and cauſed them (as they
Chiliads.
thought) to move one Degree Eaſt-
And, in the Year 1628, Adrian wards, in the ſame ſpace of ſeventy
Vlacque publiſhed this again with a Years.
Supplement of the Chiliads before And the other helped them out
omitted by Mr. Briggs; in all mak- in ſolving a Motion, which they
ing up an hundred and one Chi- called the Motion of Trepidation, or
liads.
Libration; by which they imagined
CHILIOGEN, a regular plain Fi- they ſwag from Pole to Pole.
gure, of a thouſand Sides and An ČIMA, or Cymaiſe, is what we
gles.
call, in Engliſh, an Ogee, Ogive, or
CHORD, in general, is a Right-, barely OĞ; by which we mean a
Line drawn from one Part of an Moulding waved on its Centre, con-
Arch of a Circle to the other. But cave at the top, and convex at the
the
bottom, and which makes the up-
CHORD of an Arch, is a Right- permoſt Member, and, as it were,
Line joining the Extreams of that the Cime or Top of large Cornices.
Arch.
Of theſe there are two kinds : In
1. A Chord is biffected by a Per- the one, that Part which has the
pendicular drawn to it from the greateſt Projecture, is concave, be-
Centre of the Circle.
ing term’d Doucine, or an Upright
2. Chords in the ſame Circle, Ogee. In the other, the convex
whoſe Arches are equal, are like- Part has the greateſt Projecture.
wiſe'themſelves equal.
CINCTURE, in Architecture, is
3. Unequal Chords in the ſame the ſame with Apophygee.
Circle, are not proportional to their CIRCLE, is a plain Figure, com-
Arches.
prehended under one Line only, to
CHOROGRAPHY, is a particular which Bounding Line all Right
Deſcription of ſome Country; as of Lines, that are drawn from a Point
England, France, or any part of in the middle of it, are equal to one
them, &c.
another. And is may be suppoſed
CHROMATIC, a Term in Muſic, to be generated thus :
being the ſecond of the three Kinds,
IF
1
1
1
5
1
CIR
CIR
If the Line AB be faſtened at then AC ⓇED +AEX CD =
one End to the Point A, and the ADX CE.
other Point or End B thereof be
B В
mov'd round in a Plane till it is re-
turn'd to the Place from whence it
went, that Line, in thus moving,
will deſcribe a Circle ; and the Point
D
B
4
A
!
1
E
А.
6. In a Circle the Sine of any
Arch is equal to half the Chord of
twice that Arch. The Square of
or. End B, the Circumference there- the Chord of any Arch is equal to
of: And the Point A will be the the Rectangle under the verſed Sine
Centre.
of that Arch, and the Diameter of
1. The Area of any Circle is e- the Circle.
the Circle. - The Sine of an Arch
qual to a Rectangle under the Dia- is to the Co-line of that Arch, as the
meter, and one Quarter of the Cir. Radius is to the Tangent of that
cumference.
Arch.-- The Radius is a mean Pro-
2. The Diameter of a Circle is portional between the Sine of an
proportional to the Circumference. Arch and the Co-fecant of thatArch.
3. If two Right Lines, AC, DE, - The Radius is, a mệan Propor-
terminating in the Periphery of a tional between the Tangent of an
Circle, do interſect each other in the Arch, and its Co-tangent.-- As the
Point B, either within the Circle, Radius is to a mean Proportional
or (being continued) without it, as between the Aggregate of the Ra-
in the fecond Figure, then A BxBC dius and Sine of an Arch, and the
BEXBD.
Difference between the Radius and
that Sine ; fo is twice that Şine to the
I
B
Sine of double that Archi,
D с C
D
7. In a Semi-circle, if AB be
C
the Chord of an Arch, and FD the
Chord of the Complement of that
В.
Arch to a Semi-circle; then will
the Difference between the Diame-
ter A D and the Chord A B, the
A
А
B
1
1
4. The Angle BAC made by the
F
Tangent A B, and the Chord AC
is equal to any Angle AEC, or
ADC, in the Alternate Segment
AEC of the Circle.
С.
A
5. Let ACDE be a Quadrila-
E D
teral Figure in the Circle, and the Chord FD, and the Radius AC,
Lines ĀD, EC, the Diagonals, be continual Proportionals.
8.
CIR
CIR
7. In a Semi-circle, if A B be the be continual Proportionals. Confe-
Chord of any Arch, and its Com- quently if the Radius be = 1; we
plement BD to a Semi-circle be ſhall have
in
be drawn, and the Diameter AD V2-V2+ Vztrat
be continued out to E; fo that DE
3.
be =AB: then will the Line D E for the Side of a regular Polygon
the Chord AF and the Radius AC of 96 Sides.
B
F
!
A
C G D
E
8. The verſed Sine of an Arch upon the Circumference of a Circle
drawn into the Radius, is equal be biffected by the right Line A D,
to the Square of the Sine of that and AC be drawn to E, fo that
Arch.--The Sine of an Arch drawn DE the Continuation of BD be
into the Radius, is equal to twice equal to AD, then will CE =
the Sine of that Arch drawn into A B.
its Co-line.--- The Square of the Ra 10. Thrice the Square of the
dius, the Square of the Chord of Radius drawn into the Chord of
any Arch, the Difference of the the third Part of an Arch, lefſen'd
Squares of the Diameter, and of by the Cube of this Chord, is equal
that Chord and the Square of the to the Square of the Radius drawn
Chord of twice that Arch, will be into the Chord of three times that
proportional.--- As the Difference Arch.- If three Arches of a Circle
between the Square of the Radius be equi-different, viz. Arithmetical
and the Square of the Tangent of Progreſſionals, the Radius, twice
an Arch leſs than 45 Deg, is to
the Co-fine of the middle Arch,
twice the Square of the Radius ; ſo the Sine of the common Difference
is the ſame Tangent to the Tangent of the Arches and the Difference of
of twice that Arch.
the Sines of the extreme Arches
9. If an Angle CAB ftanding are Proportionals.
11. In a Circle, if AB, BC, be
A
two Arches, and AE the Sine of
the Arch AB be continued out to F,
and B D be made equal to CB, and
B
the Diameters BM, DL, be drawn,
as alſo the Lines CHL, DI, perpen-
dicular to F A. Moreover, if the
D
Chords FD, DA, AL, LF, be
drawn, and the Lines HG, GA,
GI, IK, KF, KH. I ſay 1. EI
will be the Sine of CB, (BD)
AH (FI) the Sine of the Arche
AB to the Sine of the Arch BC. :
3. FH (AI) the Sine of the Arch
E
AB
:
2.
CI" R
CIR
AB the Sine of the Arch B C. 7. L G the co-verſed Sine of the Arch
4. DK the verſed. Sine of the Arch AB the Arch B C. 8. FD the
AB + the Arch BC. 5. D G the Chord of the Arch AB + the Arch
verſeà. Sine of the Arch AB - the CB, being equal to twice the Sine of
Arch BC. 6. LK 'the co-verſed the Arch AB + the Arch C B.
Sine of the Arch AB+ the Arch B. 9. AD the Chord of the Arcb AB
B
с
D
HO
E
F
А.
N
K
L
101
M
- the Arch CB equal to twice the laſt, concerning the Similarity of
Sine of the Arch AB - the Arch the Trapeziums, it follows, becauſe
CB. 10. F N the Sine of the Arch each of them have two Angles at
AB + the Arch BC. 11. AG the Circumference, viz. the one
the Sine of the Arch AB -- the Arch LAB + B C, the other equal to 1
BC. 12. LH the Co-line of the the Complement of AB + BC;
Arch AB+-the Co-fine of the Arch as alſo two right-angled Triangles
BC. 13. DI the Co-ſine of the form'd by the reſpective Diagonals
Arch AB the Co-fine of the Arch and Sides, which are reſpectively
BC, equal to the verſed Sine fimilar : Therefore the sides about
of the Arch AB - the verjed Sine the equal Angles will be propor-
of the Arch BC. 14. Lf the Co- tional; confequently the Trapeziums
Jine of the Arch AB + the Arch ALHG, FDIK, L FHK, and
BC. 15. Į LA the Co-fine of AIG D, are ſimilar.
AB the Arch BC. And, 16. 12. If the Arches AB, BC, CD,
Laſtly, the Trapeziums ALHG, DE, EF, &c. be equal, and the
FDIK, LFHK, and AIGD,
will be fimilar.
All the Articles, except the laſt, evi-
C
E
dently enough follow from the Con-
ſtruction and the Definition of Sines,
Co-fines, verſed Sines, and Chords í B
G
and from 1. 6. of this, that the
Chord of any Arch is twice the Sine A
of double that Arch. As to the
Chords
}
CIR
CIR
1
Chords AB, AC, AD, AE, &c. perpendicular to the Diameter A B;
be drawn; then it will be A B: then will ADX DB=CE XEF
AC :: AC:AB+AD:: AD:
AC+ AE :: AE: AD+AF ::
+ DE
AF: AE + AG.
F
17. If from a Point D in a Chord
c
AE, the Line D B be drawn, mak-
E
B
ing the Angle ADB = ACE in
AD
B
C
E
ED
A
B
D
0
1
B
17. If one Side AD of a Trape.
zium inſcribed in a Circle be con-
tinued out, the external Angle
EDC will be equal to the Angle
DE
A
E
D
A
С
B
>
the Segment ACE, of which A E B, oppoſite to the Angle ADC
is the Baſe ; and if the right Line its Complement to a Semi-circle.
ACB be drawn; then will 'A'BX 18. If the two Diagonals of a
AC be AE.
Trapezium, inſcribed in a Circle,
=
cut one another at right Angles,
15. If the Chords A B, CD, be the Sums of the Squares of each
at right Angles; the oppoſite Arches Pair of oppoſite Sides will be equal.
AD +CB=AC+DB.
and the Aggregate of the four
D-
Squares of the Segments of thoſe
Diagonals will be equal to the
А
Square of the Diamecer of the
B
E
Circle - If any Tangent meets the
Diameter continued out, and from
the Point of Contact a Perpendicu-
lar be let fall upon the Diameter,
C
the Rectangle under the Diſtance of
this Perpendicular from the Centre,
16. If the Point E be taken in and the point where the Tangent cuts
the Chord CF, and ED be drawn the Continuation of the Diameter,
will
D
3
2
2
CIR
CIR
will be equal to the Square of the C in the Periphery, cutting B D in
Radius:
F,G; then will BG +FD be
19. If the Point B be taken in
the Diameter CG of a Semicircle, BD.
and the Point A in that Diameter 22. If ABC be an equilateral
continued out ſuch, that making Triangle infcribed in a Circle, and
AD, the Difference of A C, CB, any right Line be drawn from the
it ſhall be AD:DC::CB:BE. Point C, cutting the Arch AB in
the Point D, and the Chords AD,
DB be drawn ; then will CD be
F
AD + D B.
Continue out BD to F, ſo that
DF be equal to AD, and join the
Points A, F, then will F B be equal
ADCB E G
AD + DB. Now becauſe the
Then any two Lines AF, FB, Angle A DB is į of two right
drawn from A and B to the Circum- Angles, or 120 Degrees, fince
ference, will be in a conſtant Ratio, DÅB + DBA = of one right
viz. that of A C to CB.
Angle or 60 ; for DAB= DCB,
20. If the Points A, C, be taken and DBA = DCA. Therefore
in the Diameter DE of a Circle,
equally diſtant from the Centre F,
F D
and any two right Lines A B, BC,
B
A
B
E
DA F CE
be drawn from them to the Cir-
cumference of the Circle ; the Sum
of the Squares of theſe Lines will be the Angle F D A= of one right
equal to A E + CE.
Angle or 60 Degrees. Conſequent-
21. In a Semicircle, if from AE ly the Triangle AFD will be equi-
the Ends of the right Lines AB, lateral; and becauſe the Angles
ED, each perpendicular to the ACD, A BD, ſtanding upon the
Diameter BD, and equal to the Arch A D are equal, and the sides
Side of an inſcribed Square or the AC, A B, as alſo AD, A F; there-
Chord of 90 Degrees, be drawn the fore the Triangles C'AD, A FB,
right Lines A C, EC, to any Point will be equal and ſimilar. Conſe
quently the Side F B will be equal
C
to the Side DC, that is, the Lines
AD, BD equal to the Line CD. :
D
23. If two Circles touch one an-
B
FG
other, any right Line drawn from
the Point of Contact will cut off
ſimilar Segments from the Circles,
and it will be divided at the Point
E
of Contact in the Ratio of the Dia-
L
meters.
A
CIR
CIR
A.
meters. If a right Line joining Join the given Points A, B, and
the Centres of two Circles be di-. continue out the Lines A B, CD,
vided in the Ratio of their Semi- to meet each other, as at D; make
diameters, any right Line drawn
thro' the Point of Diviſion will cut
off fimilar Segments from thoſe
B
Circles.
24. If a right Line MA be drawn
F
D
through the Centres L, K, of two
Circles, and the Point M be ſuch,
that any right Line MB drawn Df a mean Proportional between
from it, and cutting the ſaid Circles A, D, BD; then a Circle deſcribed
through the Points A,B,F, will touch
A
the right Line CD in the Point F,
B В
for FD=AD* B D.
Prob. 2. To draw a Circle thro'
K
a given Point A, to touch two right
Lines GH, IH, given in Poſition.
Continue out the right Lines GH,
JH, to meet in the Point H, and bil-
Ject the Angle GHI by the right
I
Line KH from A draw the right
F
Line AKL perpendicular to KH.
L
G
L
I
G
H
H
K
A
I
Then if a Circle be deſcribed by
Prob. 1.) thro' the Points A, L, to
M
touch either of the Lines GH, IH,
it will be the Circle required.
does cut off ſimilar Segments GF, Prob. 3. To draw a Circle CBD,
CB, from them; I ſay MAX to touch a given Circle K B, and
MÁ = MG X MB. Note, the two right Lines EC, FD, given
Point M may be taken between the in Poſition.
Circles.
Draw the right Lines IG, KH,
25. From the three laft Theorems parallel to E C, FD, and at a Di-
may be ſolved the following elegant ftance from them equal to the Semi-
and uſeful Theorems of Vieta's, re diameter AB of the given Circle ;
lating to the Deſcription of a Circle. and (by the laſt Problem) thro? the
The Problems are theſe :
Point A, draw a Circle AGH,
Prob. 1. To deſcribe a Circle to touch the two right Lines IG, KH,
through two given Points A, B, that 'in G, H. This done, if a Perpendi-
fhall touch a right Line CD given cular LC be let fall from L, the
in Poſition.
Centre of the Circle AGH to the
Line
CIR
CIR
Line E C, and a Circle BCD be
drawn about the Centre L, with the
E
B
D
A А
Amosas
E
I
B
H
I
K
G
the Line BC: I ſay, this will touch
H Н the Circle DEF in E too.
For draw the Line D B cutting
the Circle DEF in the Point E, and
join FE ; now becauſe the Angles
DFE=FE B, and FCB are each
Semi-diameter LC; this Circle will right Angles; a Circle will paſs thro'
touch the given one KB in the Peint the four Points F, E, B, C. Confe-
B, and the right Lines EC, FD, in quently DB DE=DFX DC=DA
the Points C,D, for AL=LG=LH,
X DH, whence A;H,E,B, are in a
and AB=GC=DH, and to their Circle; but E is in a Circle : where-
Differences BL, CL, LD, will be fore the Circle DEF either cuts or
equal.
touches the Circle A HEB, in E.
Prob.
4.
To draw a Circle AEB Draw the Diameter BI, then fince
thro' a given Point A, to touch a
this Circle is touch'd by B C in B, the
given Circle DEF, and a right Angle CBI will be a right Angle ;
Line BC given in Poſition.
and ſo I В will be parallel to DC,
Circle draw a right Line DC per Te, e'r, coincide ; and fo DÉF,
Thro' the Centre G of the given and joining IE, the Angles I E B,
DEF, are right Angles : Wherefore
pendicular to the Line BC, meeting BEI, are two fimilar Triangles, and
the given Circle in F, D, and draw
the right Line D A, which divide in the Circles circumſcribing them touch
H, ſo that DAX DH be=DCx DF, in the Point E.
and thro' the Points A, H. deſcribe
Prob. š. To deſcribe a Circle
(by Prob. 1.) a Circle AEB to touch LCM to touch two given Circles
DL, MP, and a right Line CZ
given in Pofition.
Let A,B be the Centres of the given
Circles, draw BZ perpendicular 10
CZ, make ZX AL, and draw
B
HT
HX parallel to CZ; then about
the Centre B, with a Radius, equat
to the Aggregate or Difference of the
Radius's of the firſt and ſecond Circles
D
F.
deſcribe a Circle MG, and thro' the
Centre A draw a Circle (by Prob. 4.)
to touch the Circle MG in G, and
the Right Line HX in Hj. and ved
1
I a
fi
€ IR
CIR
E be its Centre. Lally, if EC be Prob. 6. To deſcribe a:: Circ
drawx perpendicular to cz, and a DGB thro' two given Points B, D
to touch a given Circle EGF.
D
Let A be the Centre: of the given
Circle, join the Points B, D, and di-
vide B D in H, ſo that BD x BH
be equal to the Difference of the Squares
H
of AB, and AF, and from H draw
the Tangent HF to the Circle EGF,
and jein BF cutting the Circle in
G and F. Alſo join D G cutting the
A
I.
C
E
X
M
P
BR
G
W
D
E
A
I
D
$
D
A
Н.
E MC
B
G
NI
Н
PB
.
B
A
F
L
E
i
D
HT
0
F
THA
A
F 표
​X
B
Z
G
M
(B
D
H
Circle LCM be deſcribed about the
Gentre E, with the Radius E C, this
Circle will touch the given ones DL,
ME, and the Right Line C 2. The
Reaſon is ſufficiently evident from the
Conſtruction.
Circle
reme
Yo****
H
X MB
CIR
CIR
given Circles; and in KL take the
Point M fuch, that any Right Line
A
BM being drawn, fall cut off fimi.
lar Segments GF, CB, from the Cir.
F
cles; then draw the Right Line MI,
E
and divide it fo in N, that MIX
MN
MHX MA; and throʻ I, N;
draw a Circle B G I touching the
B
Circle ABCD, and let B be the
D
Point of contact. Join BM cutting
the Circles in B, C, F, G; then MG
- MH XMA MNx
MI; wherefore the Points N,I,B,G
are in the ſame Circle. But Gis
alſo in the Circle EFGH; where-
Circle in E, then if a Circle GBD fore the Circles. EFGH, IBGN,
be deſcribed, about the three points de mutually cut one another, or touch
G, B, D; the fame will be that in G. But BNI, BCD, touch in
B; wherefore the Segments BG, BC,
For becaufe BD XBH = AB
are ſimilar. But FG, BC, are
fimilar; wherefore F G, BG, are
AF", it eaſily follows (37.3.) that fimilar: and ſo the Circles E F G,
the Points D, G, F, H, are in a Cir- IBG, do touch in the Point G.
cle: and to the Angle DGB; of the Prob. 8. To deſcribe a Circle MLH
Quadrilateral Figure DGFH= to touch three given Circles whoſe
Angle FHB; and the Angle G, DB Centres are A, B, D.
or G-EF = (32. 3.) Angle HF B; About the Centre D defcribe a Cir-
therefore EF, BB are parallel: and cle with a Radius equal to the Diffe-
ſo the Triangles @ DB, GEF will be
rence or Sun of the Radii of the firſt
fimilar. Conſequently the Circle BGD and third Circles, and about the
touches the given Circle EGF in G, Centre B with the Radius B G equak
and paſſes through the given Points B,
D.
Prob. 7. To deſcribe a Circle GBI H
thro' a given Point I, to touch two
given Circles B A, FE.
Join the Centres K, L, of the A
quired.
2
E
$320133
G
T
B
A
K
D
E
L
H
G
B
E
M
M
N
M
to the Sum or Difference of the Radii
of the firſt and ſecond Circles; and
through A deſcribe a Circle AG F,
In 3 touching
CIR
CIR
22
G
touching the Circles B, D, in the or any right-lined Figure be made
Points G, F, then a Circle deſcribed equal to a given Circle. But it is
about E, with a Radius equal to AE not ſo eaſy to fhew from Arithmeti-
+AH, will touch the three given cal or Geometrical Principles, why
Circles in the points HM,E. There this cannot be done. I believe our
are more Caſes of thiş Problem, Want of knowing enough of the na-
which it is eaſy to ſupply.
ture of Incommenſurability is the
26. If it be required to divide à cauſe.Dr. Barrow, towards the
given Circle into two Segments, end of his fifteenth Mathematical
having a given Ratio of Ř to S; Lecture, ſays, that he is of opinion,
ſuppoſe BM D. to be a Semi-cycloid, that the Circumference of a Circle,
whoſe Baſe is AD, and Altitude and its Radius, are Lines of ſuch a
nature, as to be not only incom-
B
menſurable in Length and Square,
but alſo in Length, Square, Cube,
M
N
Biquadrate, &c. ad infinitum: for
H
(continues he) the side of the in-
fcribed Square is incommenſurable
to the Radius, and the
Square of the
Side of the inſcribed Octagon is in-
commenſurable to the Square of the
D P
А Radius; and conſequently the Square
of the octagonal Perimeter is incom-
AB. Divide AD in P in ſuch man- menſurable to the Square of the
ner, that AP:AD :: S-R:S Radiųs: and thus the Ambits of all
+R; and from P draw PM per- regular Polygons, inſcribed in a
pendicular to AD, and M N paral- Circle, may have their ſuperior
Jel to it, and draw the Right Line Powers incommenſurate with the
AO; then will the Segments AGO, co-ordinate Powers of the Radius;
AHO, be to one another as R to S. from whence the laſt Polygon, that
27.
If two Semicircles be deſcribed is, the Circle itſelf, does ſeem to have
upon the Diameter of a Semicircle, its Periphery incommenfurate with
ſo as to touch one another; the trili- the Radius. Which, if true,"will
nealSpace contain'd under thoſe three put a final ſtop to the Quadrature
Semicircles, is equal to a Circle of the Circle, ſince the Ratio of the
whoſe Diameter is a mean Propor- Circumference to the Radius, is al-
tional between the Diameters of the together inexplicable from the na-
leſſer Semicircles.
ture of the thing, and conſequently
28. In a Semicircle the Ratio of the Problem requiring the Explica-
a greater Arch to a leſs, is greater ţion of ſuch a Ratio is impoffible' to
than that of the Chord of the greater he folved, or rather it requires that
Arch to the Chord of the leſs, as is for its Solution which is impoſſible
elegantly demonſtrated by Ptolemy, to be apprehended. But concludes
in his Almageft.
he, this great Myſtery cannot be
29 A'very indifferent Mathema- explain'd in a few Words: But
tician does now know, that the Ra- if Time and Opportunity had per-
tio of the Diameter of a Circle to its mitted, I would have endeavoured
Circumference cannot be expreſſed to produce many things for the Ex-
in Numbers exactly'; 'nor can two plication and Confirmation of this
Right Lines be drawn expreſſing Conjecture.—Sir Iſaac Newton, in
that Ratic: neither can a Square, Ļib.1. of his Principia, has attempted
at
.
.
1
I
1
X 1
1
I
CI-R
CIR
at a Demonftration,' to ſhow the Im- the neareſt to the Truth of any that
poſſibility of the general Quadrature' have ever been publiſhed, as may
of oval Figures, by the Deſcription be ſeen in Mr. Jones's Synopſis, being
of a Spiral, and the Impoffibility of to 100 Places of Decimals.
determining the Interſections of that
31. The Ratio of the Diameter
Oval and Spiral (which muſt be the of a Circle to the Circumference
caſe, if the Oval is ſquareable) by will be nearly as 7 to 22, or 106
a finite Equation. But I muſt con to 333, or 113 to 355, or 1702 to
feſs this Method is not ſo clear and 5347, or i815 to 5702, or 100000
evident as might be wiſh'd. to 314159. The ſecond being more
30. There have been many Per- exact than the firſt, and the next
ſons, even many Ages ago, as well foregoing one ftill more fo than the
as in theſe later Times, who have next following. The Inveſtigation
given themſelves much pain, and at of theſe Ratio's chiefly depends upon
the ſame time greatly expoſed their the Theorem laid down under the
own Ignorance, by publiſhing pre-. word Ratio
tended Quadratures of the Circle; 32. If the Radius of a Circle be
and among the Moderns, no one 1. the Length of an Arch of 309
has, been more eminent than our
own Countryman Mr. Hobbs, who will be
+
notwithſtanding his Skill in ſome
3
3* 3
things, yet has ſhewn a moft obfti-
I
nate Ignorance in this. The Great
+
&*C.
Archimedes, in Libro de Dimenſione
5*9 7 X 27
9*812
Circuli, has firſt given a near Ratio and twelve times this will be equal
of the Diameter of a Circle to its to the whole Circumference. --- -If
Circumference in ſmall Numbers, to fix times the Radius be added in
being that of 7 to 22: but his De part of the fide of the inſcribed
monſtration is long and tedious. Square, the Sum will be nearly
Many Ages after him, Van Ceulen, the Circumference of the Circle; but
a Dutchman, in Libro de Circulo & leſs, that is, the Diameter being 2,
Adfcriptis, gave us a nearer Ratio the Semi-circumference will be 3 +
in larger Numbers, expreffed by 36 v 2 nearly,
Places of Figures; and was ſo fond 33. If to fix times the Radius be
thereof, that he order'd it to be put added 1 part of the fide of the in-
upon his Grave-Stone. After him, ſcribed Square, the Sum will be al-
Willebrord Snell, another Dutchman, moſt equal to the whole Circumfe-
in his Cyclometricus, gives the ſame rence of the Circle ; that is, the
Ratio to 36 Places of Decimals,being Diameter being 2, the whole Cir-
that of :1 to 3. 14. 15926, 53589, cumference will be 6+ to
✓
2.
79323, 84626, 43383, 27950, 28958 nearly.
nearly: which they effected by the
continual Biffe&tion of an Arch of a Circle, and the Half Circumference
34. If BC be the Diameter of
the Circle, after a manner moft ex-
ceedingly troubleſome and laborious.
D
After theſe came the indefatigable
Mr. Abraham Sharp, who gave that
H HT
Ratio to 72 Places of Decimals, in a B
ſheet of Paper, publiſhed about the
year 1706. But the very ingenious
E F
Mr. Macbin has carried this Ratio
BC
L 4
$
CIR
CIR
1
1
939
-
53
3
16
be
I
13
I
IS
BC be divided into three equal Parts
BE, EF, FC, and from E F be to
&c. or as 1 is to
drawn the Right Lines ED,FD
to the Point D, biſſecting the other
16
4.
16 4
Semi-periphery BDC; then will 5 239
t
2393
GD+ GH be nearly equal to a
Quadrant of that Periphery : but
.4
& c. ſo is the Dia.
leſs, though the Difference does not
55 2395?
exceed the Toso Part of the Diame: meter of a Circle to its Circumfe-
ter; that is, if B C be 2, the whole rence.
Circumference will be nearly =
38. If a be the Chord of an
Arch of a Circle, and b the Chord
8 Vz+Vī
8bwa
+8
of that Arch, then will
2 + V3
3
35. If the Chord of a Quadrant nearly equal to the Length of the
be 1. the Length of the Arch of a
Arch of the Circle,
Quadrant will be=1+ -- 39 Let r be the Radius of a Cir-
++
c. This cle, a any Arch, c the Chord, s the
Series was firſt taken notice of by Sine, v the verſed Sine, and t the
Dr. Gregory and Mr. Leibnitz; but Tangent of that Arch; then (1.)
Sir Iſaac Newton, in his Obſerva- will a be nearly =
tions upon the ſame, ſays, it is of
4
2 rv 72
no uſe, by reaſon of the very ſmall
Convergency thereof. For he ſays,
3
in order to get the Length of a qua-
drantal Arch to 20 Places of Deci- (2.) a =
41
It
mals, there will be occaſion for
rtv
nearly 5000000000 Terms of that
Series : to compute which it would
(3.) a = 2+
3
require a thouſand Years.
22
36. If the Diameter of a Circle
3 rs
and
:, then will the * Ž. a
(4.) a =
whole Circumference be the Sum
2**V7?
of theſe three Series's,
2r + V2
a3
(5.) a =
ar
+
rut av 2
+
3 5 7
9
14 rs to N 72
(6.) a
+
3 5
7
+ &
(7.) a =
3 m2 +
9
2 go2
هی
2
$
X
2
5
be 1,
a
2
X s.
a
225
a9
a 을
​co
EC.
wl
1
8
a
25
all
+
+
wels
1
I
9r+6V72
al4
al
t3
15
98
ato
t
&c. t
1
II
I
3
57
a²2
alo
2.8
مر
29
776 +
2 +
+
+
E c.
5
7
9
37. If the Radius of a Circle be
1, and a be = 2V3. then will I
the Circumference be a
3
1 / 3
9
3a
+
3a
tos
9
92
93
Ec. (8.) a =s
7
9 go
I X I X 53
IXIX 33 xsS
+
2 * 3 X 1 2 X 3 X 4x574
I XIX 3* 3 * 5 * 5
s7
+
2 X 3 X 4* 5 x 6x7 go
I as
(9.) and S = a-
g. 2
+
120 74
a
EQ C.
23
Bonustan
IT
1
CIR
I a9
362880 78
Esc, nitz's.
8 9
5040 g +
xs S
03
- 3.0²
40 X 4d &
I
аа
20
720 rs
a8
40320 q , &c.
CIR
i a7
Newton's, and the faţter Mr. Leibe.
If the Radius AE of a
Circle ber, and AB be a ; then
will the Area ACDB be = **
V z grau x
(10.) a= 270 x Ito
6x2r
*3
*7
509 a
бр. 4рға
5
112 1152r
+
fo
I 12 x 8 d3
&c. &c. Or, if AE be, and BĒ,x;
a4
the Area BD E will be = x x
(11.) v=
+
2453
*-*
*-76435,
& C. If the Cherd ED be drawn,
twice the Segment BDĘ will be
40. As 14 is to u, fo is the nearly equal to ED + BD x4 BE.
Square of the Diameter of a Circle or biſſecting RE in F,' and drawing
to the Area thereof nearly.
D.F; twice that Segment will be
In
Dr. Wallis's Arithmetic of Infinites, 4-DF +ED:
X 4 BE nearly.
we firſt find infinite Series's expreſ $95
ſing the Ratio of a Circle to the
D
Square of its Diameter ; there are
C
two of them. (1.) As,
3X3 X5 X 5.7. &c.
24* 4*6*6*8,
is to i; fo
Egc.
is the Circle to the Square of its
Diameter : This was found oật by
A
Him. 12 ) As
i to je
!
41. If x be the Radius of a Cir.
+
+ 28
cle and x the Distance of any given
+ 48
to stay
D
is to i, ſo is the Square of the Dia-
meter' of a Circle to the Circle B
E
itſelf ; or, as
It
F
A
P
2+49-4. &c.
K
G
is to i; fo is the Square of the
Diameter of a Circle, to the Circle
I H
itſelf: this is the Lord Bronker's, as
Wallis himſelf ſays.
If the Dia.
D
meter of a Circle be i, the Area
C
E
3
e
81. &C.
I
!
2.tzt
25
1
I
I
1
F
B
I 12
0
G
A
will be = 1
40
5
7
& C. - If the
1152
2816
Diameter of a Circle be 1, the A-
rea of the whole Circle will be
1- 3. to š
1 ts-t, &C.
The former of theſe is Sir Iſaac
H
M
T
L
K
Pun
CIR
CIR
OF
THE
HIGHER
Point P, in the Diameter from the ColleEtiones Mathemat. (6.) Gregory
Centre O, and m be any given Num- St. Vincent, in his Quadratura Cir.
ber; and if the Circumference beculi. (7.) Vincent Leotaudus, in his
divided into as many equal Parts Amenior Contemplatio Curvilineo-
AB, BC, CD, &c. as there are (8.) Van Grafen von Herbera
Units in 2 m, and from the Point P to ftein, in Diatome Circulorum. (9.)
all the Points of Diviſion be drawn All Treatiſes of Conic Sections, (for
the right Lines AP, BP, CP, DP, a Circle is a Conic Section.) (10.)
+&c. then will ÁP.x¡CPXEP, Vieta, in his works. (11.) Mr.
&c. be , *, according as Huygens, in his Inventa de Circuli
P falls within or without the Cir. Magnitudine.
cle; and the Product of BP DP
CIRCLB
x FP, &c. will be = guth + xm,
KIND, an idle Word of Wolfius,
This famous Theorem firſt appeared and ſome others, fignifying gene-
in Mr. Cotes's Harmonia Menfura- rally a Curve expreſied by the E-
rum; but without a Demonftra-
tion. Dr. Pemberton, in a little quation pour
quation gut = ax mull; which
Piece entitled Epiftola ad Amicum, indeed will be an Oval when m is
and Mr. De Moivre in his Miſcellan. an even Number ; but when m is
Analytica, have each demonſtrated an odd Number, the Curve will
it.
have two infinite Legs; as ſuppoſe
42. Thus have I given a few of m= 3, then the Curve FAMG
the moſt uſeful and elegant Proper- expreſſed by the Equation y3 =
ties of the Circle, extracted out of, ax? *3, where AP, x, PM, 9,
various Authors. Some of the Wri- and AB, a, will be one of Sir Iſaac
ters upon the Circle expreſsly or Newton's defective Hyperbola's, be-
occaſionally, are (1.) Euclid, in his ing according to him the 37th
Elements, lib. 3. (2.) Apollonius, in Species, whoſe Afymptote is the
his Conic Sections, and Tractatus de right Line DE at half right An-
Locis planis. (4.) Archimedes, in gles with the Abſciſs HI; and to
Libelle de Dimenfione Circuli, and his call ſuch a Curve a Circle, is mak,
Liber Affumptorum. (5.) Pappus, in ing a wrong uſe of Words.
D
H
1
B
I
AC
E
2
CIRCLES
1
C:IR
CIS
CIRCLES of Altitude. See Almi CIRCUM- POLAR STARS, are
canters.
fuch Stars, that being pretty near
CIRCLES of Declination on the to our North-Pole, do move round
Globe, by ſome Writers, are the it;, and in our Latitude never ſet,
Meridians on which the Declina- or go below the Horizon.
tion, or Diſtance from the Equator CIRCUMSCRIBED. A Figure, in
of any Planet or Star is accounted. Geometry, is ſaid to be circumſcrib-
CIRCLE EQUANT, in the old ed, when either the Angles, Sides,
Aftronomy, is a Circle deſcribed on or Planes of the circumſcribed Fi.
the Centre of the Equant; and the gure touch all the Angles of the
principal Ufe thereof is to find the Figure that is inſcribed.
Variation of the firft Inequality. CIRCUMSCRIBED HYPERBOLA,
CIROLES of Longitude on the is one of Sir Iſaac Newton's Hy-
Globe, are great Circles, pafling perbolas of the ſecond Order that
thro' a Star, and the Poles of the cuts its Aſymptotes, and contains
Ecliptic, where they determine the the Parts cut off within its own
Star's Longitude, reckon'd from the Space.
Beginning of Aries; and upon them CIRCUMVALLATIOə,'or the Line
the Latitudes of the Stars are ac- of Circumvallation, in Fortification,
counted.
is a Trench, borderd with a Para-
CIRCLES of Poſition, are Circles pet round about the Befieger's
paſſing thro' the common Interſec- Camp, within Cannon-lhot of the
tions of the Horizon and Meridian, Place, to hinder the Relief of the
and thro' any Degree of the Eclip- Beſieged, and to ſtop Deſerters. At
tic, or the Centre of any Star or the Diſtance of a Musket-fhot it is
other Point in the Heavens ; and commonly flank'd with Redoubts,
are uſed for finding out the Situa- and other ſmall Works, or with
tion, or Poſition of anyStar, &c. Field. Forts raiſed upon the moſt
CIRCULAR NUMBERS. Theſe, eminent Poſts. A Line of Circum-
by fome, are ſuch, whoſe Powersvallation muſt never be drawn at
terminate in their Roots themſelves ; the foot of a riſing Ground, for
as 5 and 6, whoſe Powers do end fear left the Enemy, having ſeized
in 5 and 6; the Square of 5 being on the Station, ſhould plant Cannon
25, and of 6, 36, &c.
there, and ſo command the Line.
CIRCULARVELOCity, a Term This Line is uſually about ſeven
in Aftronomy; and fignifies, that Foot deep, and twelve broad.
Velocity of any Planet or revolving CISSOID, is a Curve of the few
Body, which is meaſured by the cond Order, as AM, Am, conſiſting
Arch of a Circle.
CIRCUMAMBIENT. See Ambient.
CIRCUMFERENCE, is the outer-
moft bounding Line, or Lines of
M
any plain Figure.
CIRCUMFERENTOR, an Inftru-
ment uſed in Surveying, being a
Р P.
B
large Box and Needle, faſtend on
to the middle of a Braſs Index,
with Sights at each end of the
Index.
CIRCUMGYRATION, is the Mo-
7n2
ţion of any Body about a Centre.
of
CIS
CIS
of two infinite Hyperbolic Legs and at the ſame time ſhews how to
AM, Am, having a right Line AB find two mean Proportionals,and the
for a Diameter, and a right Line Roots of a Cubic Equation, with-
CC its Afymptote, and of ſuch a out any previous Reduction by
nature that calling AB, a, the Ab- means thereof. Let AG be the
ſciſs AP, x, and the Correſpon- Diameter, and P the Centre of the
dent Semi-Ordinate PM, or Pm, y, Circle belonging to the Cimaid;
it will be yj xa-*=* This and from F draw FD, FP, at right
Name was given to the Curve by Angles to each other, and let.FP be
Diocles an ancient Greek Geometri = AG; then if the Square P.ED
cian, being principally deviled for be ſo moved, that one side' EP
finding two mean Proportionals be thereof always paſſes through the
tween two given right Lines, but Point P, and the End D of the. o-
Sir Ifaac Newton in his Enumera- ther Side E, D, ſlides along the right
tio Linearum tertii Ordinis, neckons Line FD, the middle Point of the
it amongſt one of the defective Hy- Side ED, will deſcribe one Leg
perbola's, being according to him GC of the Ciſloid, and by capti
the 42d Species. In his Appendix nuing out F D on the other. Side F.
de Æquationum Confituftione Line- and turning the Square about by a
ari, at the End of his Arithmetica like Operation, the other Leg may
Univerfalis
, he gives the following be deſcribed.
elegant Defeription of this. Curve,
A
C
K.
D
G
There is another way, which I right Angles to the Afymptote FB.
thought upon to deſcribe this Curve Take two Squares, a ſingle one
by a continued Motion ; and it is NAM, and a double one or Tee
thus :
NPOPM, and faften the Angle
Let ANF be the generating of the ſingle Square in the Point A,
Circle, and AF the Diameter at ſo as to be moveable about the
1
no
B
M
N
Р
fame.
-
CIS
fame. Thus if the Leg PO of the how to find a right Line equat
double Square bë moved along AF, one of the Legs of this Curve, by
and the Interſection N of the Leg means of the Hyperbola ; bat fup-
AN of the ſingle Square, with the preffed the Inveſtigation, which how-
Leg NP of the double Square be ever may be ſeen in his Fluxións.
moved along the Circumference 4. The Ciffoidal Space contained
ANF of the Circle ; the Interſec under the Diameter A B, the A.
tion M of the other Leg AM of. fymptote BC, and the Curve AOZ,
the ſingle Square, with the other of tħe Ciffoid, is the Triple of the
Leg PM of the double Square, will generating Circle AN B.
deſcribe the Leg AM of the Cif Dr. Wallis treats of this Line
ſoid ; and after the fame manner in his Mathematical Works, Vol. I.
the other Leg may be deſcribed. pag: 545. and following.
This Curve may be deſcribed by Civil Day. See Day.
Points after the following manner : CIVIL YEAR, is the legal Year,
Join the indefinite right Line BC or annual Account of Time, which
at right Angles to AB, the Dia- every Government appoints to be
meter of the Semi-circle AOB, uſed within its own Dominions ;
and draw the right Lines AH, AF, and begins with us the 25th Day
AC, &c. then if you take A M= of March.
IH, AO=OF, ZCAN, &C. CLEPSYDRA, an Inſtrument of
the Points M, 0, 2, &c. will{form. the Ancients, particularly the Egyp-
the Curve A MOZ of the Cifroid. tians, to meaſure Time with, by
р G
K B
the running of Water out of one
Veffel into another.
M
There were many kinds of them:
But in all, the Water ran gently
thro' a narrow Paſſage from one
Vefſel into another; and in the
TH
lower was a Piece of Cork, or light
Wood, which, as the Veſſel fill'd,
roſe up by Degrees, and ſo fhew'd
the Hour.
F
But in theſe Inſtruments there
were two Inconveniencies: The
firſt whereof was that the Air, ac-
cording to its different Tempera-
ture, as to Heat, Cold, Denfity, c.
had an influence
upon
the Running
Z
of the Water, ſo as to make it mea-
fure Time unequally. And the fe-
cond, which was yet greater, that
the Water always ran flower out,
1. Draw the right Lines PM, according as its Quantity and Preſ-
KI, perpendicular to AB, then fure in the Vefſel abated.
AK PB and PNIK.
Mr. Varignon, in the Memoirs de
2. The Lines AK, PN, AP, PM, l'Academie Royale des Sciences, for
as alſo AP, PN, AK, KL, are con the Year 1699, lays down a gene-
tinual Proportionals.
ral geometrical Method of making
3. Sir Iſaac Newton, in his laft Clepſydra's, or Water-Clocks, with
Letter to Mr. Leibnitz, has ſhewn any kind of Veſſels, and with any
5
gives
CLI
CLO
1
given Orifices for the Water to run probable, that Pythagoras had it
out of.
from them ; who uſed to talk very
Vitruvius, in lib.9. of his Archi- much of the Efficacy of the Num-
te&ure, treats of theſe Inſtruments s ber Seven, being a Number he was
and Pliny, in chap. 60. lib. 7. ſays, extremely in love with.
that Scipio Nafica was the firſt who CLIMATE, is a part of the Su-
meaſured Time at Rome by Clepſy- perficies of the Earth, bounded by
dra's, or Water-Clocks.- Geſnerus, two Circles, parallel to the Equator,
in his PANDECTES, pag. 91. gives ſo that the longeſt Day in that Pa-
ſeveral Contrivances of theſe Inſtru- rallels neareſt to the Pole, exceeds
ments. There is Solomon de Caus, the longeſt Day in that Parallel
who treats of this Subject in his Rea- neareſt to the Equator, ſome cer-
fons of moving Forces, &c. So alſo tain definite Part of Time, viz. half
does Mr. Ozanam, in his Mathemati an Hour, till you come to Places fi-
cal Recreations, wherein is a Treatiſe tuate nearly under the Arctic Circle ;
of Elementary Clocks, tranſlated and a whole Hour, or even ſeveral
from the Italian of Dominique Mar- Days, when you go beyond it.
Linelli. You have alſo a Treatiſe - The ancient Greek Geographers
of Hour-Glaſſes, by Arcangelo Ma- reckoned only ſeven Climates from
ria Radi, callid Nova Scienza di the Equator, towards the North
Horologi Polvere.. See more, in Pole; and denominated them from
the Technica Curiofa of Gaſper Schot- ſome noted Place, thro' which the
tus; and Mr. Amonton's Remarques middle Parallel of the Climate pas-
& Experiences Phyſiques ſur la Con- fed. But the Moderns reckon up
ſtruction d'une nouvelle Clepſydre, twenty-four, as may be ſeen in Va-
exempte des défauts des autres. renius, page 319, prop. 13. chap.
CLIFF, or Cleff, a Term in Mu- 25 lib. 2.
fic, fignifying a certain Mark, from Clock, a well-known Inftru-
the Poſition whereof the proper ment, wherewith to meaſure Time,
Places of all other Notes, in a Piece conſiſting of ſeveral Wheels of va-
of Muſic, are known. And there rious Sizes moving one another, by
are four of them.
Teeth fitting into each other, which
The firſt of theſe Cliffs is called Wheels are continued in Motion by
Faut-Cliff, and belongs to the Baſs; the Force of a Weight, or Spring,
the Cefaut-Cliff, or Tenor-Clif; the and fhewing the Hour by the Sound
Counter-Tenor, or Beni-Clif : and of a Bell, and an Index moving a-
and the Treble or Gamut-Cliff. bout a circular Plate. Some Clocks
CLIMACTERICAL YEARS, are go but 24 Hours before they muſt
certain obſervable Years, being ſup- be wound up: Others eight Days :
poſed to be attended with ſome Others again, 32 Days; and ſome
great Mutation of Life, or Fortune. have been made to go a whole
Theſe are the ſeventh Year ; the Year, or longer.
twenty-firſt, made up of three times In the Diſquiſitiones Monafticæ of
ſeven the forty-ninth, made up of Benedictus Haeften, publiſhed in the
ſeven times ſeven ; the ſixty-third. Year 1644, he ſays, that Clocks.
being nine times ſeven ; and the were invented by Silveſter the IVth.
eighty-firſt, which is nine times a Monk of his Order, about the
nine ; which two laſt are called che Year 998, as Ditmarus and Bozius
Grand Clymacterical Years. Aulus have ſhewn; for before that time,
Gellius ſays, this Piece of Stuff came they had nothing but Sun-Dials, and
from the Chaldeans first. And it is Clepſydra's to tell the Hour.---Con-
rarde
1
Year 1372.
CLO
CO E
rarde Gefner, in his Epitome, pag. .CLOSE, in Mufic. See Cadence:
604, ſays, that Richard Walling CLOUDs, are a Congeries of Wa-
ford, an Engliſh Abbot of St. Albans, ters, drawn up from the Sea and
who flouriſhed in the Year 1326, Land into Vapours; which when
made a wonderful Clock by a moſt they are very nearly placed to one
excellent Art, the like of which another, appear denſe and thick
could not be produced by all Eu- but when they are more remote,
rope.- Moreri under the Word H. are clear and bright, and ſometimes
rologe du Palais, ſays, that Charles almoſt tranſparent.
the Fifth, call'd the wife King of Clouds ſwim in the Air at but a
France, order'd at Paris the firſt ſmall diſtance from the Surface of the
great Clock to be made by Henry Earth: For thoſe, who have taken
de Vic, which he ſent for from Ger. their Altitudes, do affirm, that they
many, and ſet it up upon the Tower do not exceed one Mile in Height,
of his Palace; and this was in the and many of them not above half a
John Froiſſart, in Mile.
chap. 28. vol. 2. of his Hiſtoire & CLOUTS, are thin Plates of Iron,
Chronique, ſays, the Duke of Bour- nail'd on that part of the Axle-tree
gogne had a Clock which founded of a Gun-Carriage which comes thro'
the Hour, taken away from the the Nave, through which the Lins-
City of Courtray, in the Year 1382. Pin goes.
And William Paradin, in his An COACERVATE VACUUM. See
nales de Bourgogne, ſays the ſame Vacuum.
thing
COALITION, is the gathering
There are ſeveral Treatiſes upon together, and uniting into fenfible
Clocks; the principal of which are, Maffes, the minute Corpuſcles that
Hieronymi Cardani de Varietate Re- compoſe any concrete or natural
rum Libri XVII.- Conrandi Dafy- Body; and a Coaleſcency is com-
podii Defcriptio Horologii Aftronomici monly taken for the ſame.
Argentinenfis in fummo Templi eretti. COASTING, is that Part of Na.
-Guidonis Pancirolli antiqua deper- vigation where the places aſſigned
dita & nova reperta. L'Uſage du are not far diftant, ſo that a Ship
Cardan, ou de l'Horologe Phyſique uni- may fail in fight of Land, or with
verſelle, parGalilée Mathematicien du in Soundings, between them.
Duc de Florence. Mr. Oughtred's Co-EFFICIENT of any generating
Opufcula Mathematica.- Mr. Huy- Term in Fluxions, is the Quantity a-
gens's Horologium Oſcillatorium. riſing by the Diviſion of that Term
Pendule Perpetuelle, par l'Abbe de by the generated Quantity.
Hautefeuille.- 7. J. Becheri Theo CO-EFFICIENT'S, in Algebra,
ria & Experientia de nova Tempo- are ſuch Numbers, or given Quan-
ris dimentiendi Ratione & Horologio tities, that are put before Letters,
rum Conſtructione. Clark's Ough. or unknown Quantities, into which
tredus explicatus, ubi de Conſtructione Letters they are ſuppoſed to be mul-
Horologiorum.--Horological Diſqui- tiplied, and ſo do make a Rectangle
fitions.--Mr. Huygens's poſthumous or Product with the Letters ; as
Works. Mr. Sully's Regle Artifici- here, 3 a, or bx, or Cxx ; where
elle du Temps, &c.- Mr Serviere's 3 is the Co-Efficient of 3a; b, of
Recueil d'Ouvrages Curieux.- Mr.
Mr. bx, and C of Cx.x.
Durham's Artificial Clock-Maker. In a Quadratic Equation the Co-
Mr. Camus's Traite des Forces Efficient is, according to its Sign,
Mouvantes -Mr. Alexandre's Traité either the Sum or Difference of its
Général des Horologies.
two Roots,
In
COL
COL
In any Equation the Co-Efficient ſtriking upon the Organ of Sight,
of the ſecond Term is always equal ſo as to produce that Senſation we
to the Sum of all the Roots, keep- call Colour.
ing their proper Signs.
Sir Iſaac Newton was the firſt that,
The Co-Efficient of the third from Éxperiments on Priſms, found
Term, is the Sum of all the Rect- there was a great Deformity in the
angles ariſing by the Multiplication Rays of Light; and from thence
of every two of the Roots, how found, that Colours are not Quali-
many ways ſoever thoſe Combina- ficátions of Light, derived from Re-
tions of two's can be had; as three fractions or Reflections of natural
times in a Cubic, fix in a Biqua- Bodies, but original and connate
dratic Equation, &c.
Properties, which in divers Rays
The Co-Efficient of the fourth are different ; fome Rays being diſ-
Term, is the Aggregate of all the poſed to exhibit a red Colour, and
Solids made by the continual Mul- no other ; fome a green, and no
tiplication of every three of the other ; and ſo of the reft. Nor
Roots, how often foever ſuch a are there only Rays proper and
Ternary can be had ; and ſo on, ad particular to the more eminent Co.
infinitum.
lours, but even to all their inter-
Coffer, in Fortification, is a mediate Gradations.
hollow Lodgment a-croſs a dry The leaſt refrangible Rays are
Moat, from fix to ſeven Foot deep, all diſpoſed to exhibit a red Colour;
and from fixteen to eighteen broad, and the moſt refrangible ones, are
the upper Part being made of Pie- thoſe that expreſs a Violet Co-
ces of Timber, raiſed two Foot a lour.
bove the Level of that Moat ; There are two ſorts of Colours
which little Elevation has Hurdles the one original and fimple, and
laden with Earth for its Covering, the other compounded of theſe.
and ſerves as a Parapet with Em- The original and primary Colours
braſures.
are red, yellow, green, blue, and
The Beſieged generally make uſe a violet purple, together with o-
of thefe Coffers to repulſe the Be- range, indigo, and an indefinitive
ſiegers, when they endeavour to Number of intermediate Grada-
paſs the Ditch. And they differ tions.
only in Length from the Caren The fame Colours in Specie, with
niers, which are alſo ſomething leſs theſe primary ones, may be alſo
in Breadth.
produced by Compoſition ; for a
COLD, is one of the primary Mixture of yellow and blue makes
Qualities of Body, and is no more green ; of red and yellow makes
than the arriving of the minute and orange; of orange and yellowiſh
inſenſible Parts of any Body at ſuch green makes yellow. And general-
a State, as that they are more ſlow- ly, if any two Colours be mixed,
ly or faintly agitated than thoſe of which, in the Series of thoſe gene-
our Fingers, or other Organs of rated by the Priſm, are not too far
Feeling; for from this Effect we diſtant from one another, they, by
ſay a Body is cold.
their mutual Alloy, compound that
COLLISION, is the ſtriking of Colour which in the ſaid Series ap-
one 'hard Body againſt another. pears in the Midway between them :
COLOUR, is that Quality of a But thoſe that are ſituated at too
natural Body, whereby it is diſpoſed great a diſtance, do not do ſo. O-
to modify Light falling upon it, and range and Indigo produce not the
inters
3
COL
COM
intermediate green, nor ſcarlet and and moſt fimple, according to ſome,
green the intermediate yellow. is feven Models long, comprehend-
Whiteneſs is the uſual Colour of ing its Baſe and Capital, and di-
Light, Light being a confuſed Ag- minifh'd a fourth Part of its Dia-
gregate of Rays, endued with all merer.
forts of Colours, as they are pro-
The Dorick, ſeven and a half, or
miſcuouſly darted from the various eight Diameters long, and its Baſe
Parts of luminous Bodies ; and of and Capital are ſomewhat more
ſuch a confuſed Aggregate is gene- beautified with Mouldings.
rated Whiteneſs, if there be a due The lonick Column, nine Diame,
Proportion of the Ingredients. ters long, and has its Capital ſet off
The Colours of all natural Bodies with Völuta's, or curled Scrolls, dif-
have no other Origin than this, viz. fering in that reſpect from others,
That they are variouſly qualified as well as its Baſe, which is pecu-
to reflect one fort of Light in greater liar to it.
plenty than another ; as Sir Iſaac The Corinthian, the richeſt of all,
Newton has ſhewn in the Philoſophi- being ten Diameters in Length, has
cal Tranſactions.
two Rows of Leaves for the Qrna-
The Senſations of different Co ment of its Capitals,
' with Stalks, or
lours ſeem to ariſe from hence, That Stems, from whence ſhoot forth
ſeveral ſorts of Rays do make Vi- ſmall Voluta's.
brations of ſeveral Bigneſſes, which, The Compoſite Column, is alfo
according to their Magnitudes, do ten Diameters long, and its Capi-
excite Senſations of different Co- tal is made like that of the Corin
lours; much after the fame manner thian.
that the Vibrations of the Air, ac COLURES, are two great Circles,
cording to their ſeveral Bigneſſes, imagin'd to paſs through the Poles
do excite Senſations of different of the World, one of them through
Sounds.
the Equinoctial Points Aries and
And it is probable that the Har- Libra, and the other through the
mony and Diſcord of Colours (for Solftitial Points, Cancer and Capri-
fome Colours, as of Gold, Yellow, corn; they being called the Equi-
and' Indigo, are agreeable to the noctial and Solftitial Colures.
Eyes, and others not) ariſe from COMA-BERENICES, a Northern
the Proportions of theſe Vibrations Conſtellation of fix'd Stars.
propagated through the Fibres of COMBINATION of Quantities, is
the Optic Nerves into the Brain, the manner of finding how many
juſt as the Harmony and Diſcords different ways they may be varied,
of Sounds ariſe from the Vibrations or taken one and one, two and
of the Air.
two, three and three, &C. as
COLUMN, is a kind of a round the Number of Combinations of
Pillar, compoſed of a Baſe, a Fuft, three Quantities abc, two and two
or Shaft, and a Capital, and ſerves are three, viz. ab, ac, bc. If
to ſupport the Entablement.
three Quantities are to be combin'd,
Columns are different, according and their Number is only three, as
to the different Orders, being ca- abc, then the Number of Combi-
pable of a great Number of Varia- nations will be only one, yiz. abc;
tions, with regard to Matter, Con. and if there are four Quantities
ſtruction, form, Difpofition, and abcd, and three to be taken, then
Ule.
the Combinations will be four, viz.
The Tuſcane, being the ſhorteſt abs, abd, bod, acd; and if the
M Number
i
COM
COM
X
Х
1
2
x
X
Х
I
2
were.
Number of Quantities to be com- poſed that Comets were only Me-
bin'd be called q, and u be the Num teors or Exhalations, ſet on fire in
ber of them to be taken, then the the higheſt Region of the Air, be-
Number of Combinations will be low the Moon. And this Opinion
9-
-uti
gemut2
had ſo far prevailed, that no body
quu+3
* thought it worth while to write con-
3
cerning the uncertain Motions of a
q-*+4x9-u+s, &c. For Vapour or Exhalation ; and ſo no-
4
S
thing certain about the Motions of
Example: Let the Number of the Comets can be found tranſmitted
Quantities to be combin'd be 6, and from them to us.
let 4 be the Number of them ta But Seneca, the Philoſopher, from
ken; then the Number of the the Confideration of the Phænome-
Combinations will be
na of two remarkable Comets of
6-4+1 6+2 6-4+3 his Time, made no fcruple to place
them among the Celeſtial Bodies,
3
and believed them to be Stars of
6~4++
= xxx=15.
equal Duration with the World,
4
tho he could not tell the Laws of
The Number of all the poſſible their Motion ; but propheſied that
Combinations beginning from the After-Ages would find out in what
Combinations of every two will be Parts of the Heavens the Comets
wander'd, what and how great they
29-ml; as when the Number of
Quantities be 5, then the Number
of the poſſible Combinations will firiť obſerved a Comet, that then ap-
Tycho Brahe, in the Year 1577,
be 25.6=26.
If u repreſents any Number of peared to have no Diurnal Paral.
lax, and conſequently was not only
no Aerial Vapour, but alſo much
Quantities, then will
higher than the Moon. And after-
wards Kepler found that the Comets
expreſs the poſlible Number of all moved freely thro' the Orbits of the
the Variations ; as if u = 4, then Planets, with Motions very little
45_-4
different from right-lin'd ones. And
Hevelius embracing the ſame right-
3
lin'd Motion of the Comets, ob-
COMBUST, a Term in Aſtrono- ferv'd many of them ; but com-
my. When a Planet is not above plain'd, that his Calculations did
eight Degrees and thirty Minutes
not agree to the Matters of Fact
diftant from the Sun, either before in the Heavens; and found that the
or after him, he is ſaid then to be Path of a Cornet was bent into a
combuſt, or in Combuſtion.
Curve-Line towards the Sun.
COMETS, are Stars, moſt of
But from the accurate Obſerva-
which have Tails, ſuddenly ariſing tions of the great Comet of the
in the Heavens, and appearing for Year 1680, Sir Iſaac Newton ſhews, ,
ſome time, do afterwards again dif- in his Principia, that Comets move
appear; and all the time that they in Conic Sections, having their Foci
are ſeen, they, like the Planets, in the Centre of the Sun, and by
every Day ſome certain Rays drawn to the Sun, do deſcribe
Length in their proper Orbits. Area's proportional to the Times;
Ariſtotle, and his Followers, ſup- and ſo, if Comets return in their
Orbits,
4+1
u
41
HI
1020
= 340.
I
move
6
?
}
COM
COM
Orbits, the Orbits are Ellipſes, and lib. 3. Dr. Halley, his Synopſis
the periodic Times are to the perio- Cometica, in the Philoſophical Tranſ-
dic Times of the Planets in the ſer actions, n. 218.
quiplicate Ratio of the principal COMMA, a Term in Muſic, bem
Axes. But the Orbits of Comets ing the ninth Part of a Tone, or the
are ſo near to Parabola's, that Pa- Interval whereby a Semi-Tone, or a
rabola's may be taken inſtead of perfect one exceeds the imperfect.
them, without any ſenſible Error. This is uſed only in the Theory of
The Planes of the Orbits of Co- Muſic, to ſhew the exact Proportion
mets are always inclined to the between Concords.
Plane of the Ecliptic; and ſome COMMANDING GROUND, in
move from Eaſt to Weft, ſome from Fortification, is ſuch as overlooks
Weſt to Eaſt, ſome from North to any Poft, or ſtrong Place, and is
South, and ſome from South to of three ſorts : Firſt, a Front com-
North.
manding Ground, which is an Height
The Bodies of Comets, accord- oppofite to the Face of the Poſt,
ing to Sir Iſaac Newton, are folid, which plays upon its Front. Second-
compact, fíx'd, and durable, liké ly, a reverſe commanding Ground,
the Planets, and ſhine by the Light which is an Eminence that can play
of the Sun-Beams reflected from upon the back of any Place, or Poſt.
them ; And the Tail of a Comet is Thirdly, an Enfilade Commanding
only a long and very thin Smoak, Ground, which is an high Place,
or Train of Vapours, which the that can, with its Shot, fcour all the
Head of the Comet emits from it, Length of a ſtraight Line.
by being vaſtly heated by the Sun ; COMMENSURABLE
MAGNI-
and always appears on that ſide of TUDEs, are ſuch as are meaſur'd by
the Comet oppoſite to the Sun. one and the ſame common Mea.
John Regiomontanus was the firſt fure; as, if the Magnitudes A, B,
who has ſhewn how to find the Mag-
nitude of Comets, their Diſtance
Ar
from the Earth, and their true Place
Blo
in the Heavens ; his 16 Problems
de Cometæ Magnitudine, Longitudine,
ac Loco, are to be found in an an the one 5, and the other
33
cient Book publiſhed in the Year ſur'd exactly by the Magnitude C,
1544, with the Title of Scripta ſuppoſed to be is then the Magni-
Joannis Regiomontani.
tudes A and B are called Commen-
Writings about Comets, are Tycho Surable.
Brahe, his Progymnaſmata Aftrono-
COMMENSURABLE NUMBERS,
miæ Inftauratæ.-- Kepler, of the whether Integers or Fractions, are
Comet (in High-Dutch) in the Year ſuch as have ſome other Number
1607, and de Cometis Libelli tres, which will meaſure or divide them
Hevelius's Prodromus Cometicus, without any Remainder: Thus, 4
containing an Hiſtory of the Comet and 6, or and are commenſu-
of the Year 1664. Alſo his Come rable.
tographia.- Dr. Hook, his poſthu COMMENSURABLE in Power.
mous Works.-- Mr. Caſſini's little Right Lines, by Euclid, are ſaid to
Tract of Comets Mr. Sturmius's be commenſurable in Power, when
Diſertatio de Cometarum Natura.- their Squares are meaſured by
Sir Iſaac Newton, his Principia one and the fame Space or Super-
Philoſophie Naturalis Mathemat, ficies.
Coma
be mea.
8
IS
M 2
COM
COM
COMMENSURABLE SURDS, are that during the Motion of the Ship,
ſuch Surds, that being reduced to the Chards may be nearly Horizon-
their leaſt Terms, become true fi- tal, and the Flower-de-Luce of the
gurative Quantities of their Kind; upper Chard will always point to-
and are therefore as a rational wards the North.
Quantity to a rational one.
This Inſtrument, tho' it be ſub-
COMMON Axis, in Optics. Seeject to Accidents, is of great uſe in
Axis.
Navigation; and all the confidera-
COMMON Divisor, is that
that ble Diſcoveries of Countries are ow-
Number that exactly divides any ing to the ſame.
two other Numbers, without a Re The Invention of it, by ſome, is
mainder.
attributed to one John Goia, of A-
COMMON MEASURE, is ſuch a malphi, in Campania, in the King-
Number that exactly meaſures two dom of Naples; who made the
or more Numbers without a Re- Chard thereof to conſiſt only of
mainder,
eight Points, viz. the four Cardinal,
COMMON MEASURE (greateft,) and four Collateral ones. Others
of two or more Numbers, is the fay it was the Invention of the Peo-
greateſt Number that can meaſure ple of China. And Gilbert, in Li-
them; as, 4 is the greateſt com bro de Magnete, affirms, That Pau-
mon Meaſure of 8 and 12.
lues Venetus brought it firſt into Italy
COMMON Ray, in Optics, is a in the Year 1260, having learned
Right Line drawn from the Point it from the Chineſe. And Ludi Ver-
of Concurrence of the two optical tomanus affirms, That when he was
Axes, thro' the Middle of the in the Eaſt-Indies, about the Year
Right Line, paſſing thro' the Cen- 1500, he ſaw a Pilot of a Ship direct
tre of the Pupil of the Eye.
his Courſe by a Compaſs, faften'd
COMPARTITION, in Architec- and framed as thoſe that now are
ture, is the uſeful and graceful Di- commonly uſed.
ftribution of the whole Ground-plat And Mr. Barlow, in his Navi-
of an Edifice into Rooms of Office, gator's Supply, Anno 1597, ſays, That
Reception, or Entertainment, &c. in a perlonal Conference with two
COMPARTMENT, in Architec- Eaſt-Indians, they affirmed, that
ture, is a peculiar Square or other inſtead of our Compaſs, they uſe a
figur'd Space, (for an Infcription, Magnetical Needle of fix Inches,
&c.) mark'd out in ſome orna and longer, upon a Pin in a Diſh
mental Part of a Building.
of white China Earth, filled with
COMPASS, in Navigation, is a
Water
in the bottom whereof
Circle, or Chard of Paſtboard, di- they have two Croſs-Lines for the
vided into thirty-two equal Parts, principal Winds, the reſt of their
called Rhumbs, or Points, repreſent- Diviſions being left to the Skill of
ing the thirty-two Winds, with the their Pilots. Alſo, in the ſame
initial Letters of their Names ſet to Book, he ſays, That the Portugueſe,
them, having a touched Needle or in their firſt Diſcovery of the Eaſte
Wire fix'd to it underneath, and in Indies, got a Pilot of Mahinde, thar
its Centre a Braſs Cell, or Conical brought them from thence in thirty-
Cavity, by means of which it three Days, within ſight of Ca.
hangs on an erect Pin, ſet up in licut.
the Centre of another ſuch Chard, COMPASS DIALs, are ſmall Ho-
fitted in a Wooden or Braſs Box, rizontal Dials, fitted in Braſs or
with Jambols, or Braſs Hoops-; fo Silver Boxes for the Pocket, and
j
are
NUMBERS,
are
at
COM
COM
are ſet North and South, by means grams AGE, FCE, made by
of a Compaſs, or touched Needle drawing two right Lines GE, FÉ,
belonging to them.
through the Point E, in the Diago-
COMPASS ES of Proportion, or Pro- nal; parallel to the Sides AB, BC,
portional Compaſſes, are ſuch that of any Parallelogram ABCD.
have two Legs, but four Points, In' every Parallelogram thefe
which, when opened, are like a Complements are equal.
Croſs, not having the Joint at the
COMPOSITE
End of the Legs,
as common Com- ſuch, that ſome Number beſides
paſſes : And ſome of theſe have Unity can meaſure; 'as 12, which
fixed Joints, others moveable ones ; is meaſur'd by 2, 3, 4, and 6.
upon the Legs of the latter of which
COMPOSITE NUMBERS,
be
are drawn the Lines of Chords, tween themſelves, are ſuch that
Sines, Tangents, & c. as on the have ſome common Meaſure beſides
Sector.
Unity; as 12 and 15, which may
Their Uſe is to divide Right be both meaſur'd by 3.
Lines, and Circles into equal Parts,
COMPOSITE Order, is the fifth
or to perform other Operations of Order of Architecture; and is ſo
the Sector one opening of called, becauſe its Capital is com-
them.
poſed of two Rows of Leaves proper
COMPLEMENT of any Arch or to the Corinthian Order, and the
Angle, to any other Arch or Angle, Voluta's of the Ionic. This Or-
(as of ninety Degrees, an hundred der is ſometimes called the Italic
and eighty Degrees, &c.) is the or Roman, as having been firſt in-
Arch or Angle, which, together vented by that people. Its Column
with that Arch or Angle, makes up is ten Diameters in Height, and
ninety Degrees, or a hundred and there are always Dentiles or ſimple
eighty Degrees, &c.
Modillions to its Cornice.
COMPLEMENT of the Courſe in COMPOSITION, is the reverſe of
Navigation, is the Number of Points the Analytic Method, or of Refo-
the Courſe wants of ninety Degrees, lution. It proceeds upon Principles
or eight Points, that is, of one ſelf-evident, on Definitions, Poftu-
fourth of the Compaſs.
latums, and Axioms, and a previ-
COMPLEMENT of the Courtain, ouſly demonſtrated Series of Propo-
in Fortification, is that Part of the fitions, ſtep by ſtep, till it gives a
Courtain which (being wanted) is clear Knowledge of the Thing de-
the Demi-Gorge.
monſtrated, This is what they
Complement of the Line of call the Synthetical Method, and is
Defence, is the Remainder of the uſed by Euclid, Apollonius, and moſt
Line of Defence after the Angle of of the Ancients.
the Flank is taken away.
COMPOSITION of Proportion. If
COMPLEMENTS in a Parallelo- there be two Ratio's, and it ſhall be
gram, are the two ſmall Parallelo is the Antecedent of the firſt Ra.
tio to its Conſequent, ſo is the An-
A
B tecedent of another to its Conſe-
G
quent. Then, by Compofition
of Proportion, as the Sum of
the Antecedent and Conſequent
of the firſt Ratio, to the Ante-
cedent or Conſequent of the firſt,
D
ſo is the Sum of the Antece-
dent
E
F C
M 3
COM
CON
)
or
z
.4
dent and Conſequent of the ſecond, to the Product of their Confequents,
to the Antecedent or Conſequent of is called a Compound Ratio : So 6 to
the ſecond : As, if A:B :: C:D, 72 is in a Ratio compounded of
then, by Compoſition, A+B: A 2 to 6, and 3 to 12.
(B) :: C+D: C (D)
The Exponent of a compound
COMPOUND Intereſt, is that Part Ratio is equal to the Product that
of it that treats of the Money pro- the Exponents of fimple Ratio's
duced from any Principal, and its produce.
Intereſt put together, as the Intereſt As if m be the Exponent of the
of that Principal becomes due.
A
C
That is, finding the new Principal Ratio
B
and n of j; then will
that is ſtill created by the Increaſe
of the growing Money at every Pay-
AC
mn be the Exponent of
ment, or rather at the Times when
BD
the Payments become due, is called
А
Compound Intereſt, or Intereſ upon of the Ratio compounded of
B
Intereft.
If R be the Amount for one
С
and
D:
Pound of one Year, then R will be
If there are never
fo
the Amount for two Years, R for tities, A, B, C, D, E, F, Bc. the
many Quan-
three Years, R for four Years, &c. Ratio of the firſt A to the laſt F,
As il. is to its Amount for any is compounded of the Ratio's of the
given time, fo is any propoſed Prin- Quantities being between the Ex-
cipal or Sum to its Amount for the
А B с D
ſame time.
tremes, viz.
B: T: D' E'
COMPOUND MOTION, is that
E
which is produced by feyeral For-
Egc.
F .
ces conſpiring together; and For-
ces are ſaid to conſpire, when the COMPRESSION, is the ſqueezing
Direction of the one is not contrary of a Maſs of Matter into a leffer
to the Direction of the other; as Bulk.
when the Radius of a Circle moves CONCAVE, or Concavity. This
about the Centre, and at the ſame ſignifies the hollowneſs of any
time a Point be conceived to go thing.
forwards along it.
CONCAVE-GLASS, or Lens, is
Whence every curv'd-lin'd Mo one that is flat on one ſide, and
tion is a Compound Motion. ground hollow on the other ; but
COMPOUND QUANTITIES, in uſually ſpherical. This, by fome,
Algebra, are ſuch as are connected is called a Plano-Concave, and if
together by the Signs + and , the Glaſs be Concave on both
and are expreſſed by the ſame Let- fides, it is called a Double - Con-
ters more than once, or elſe by the
ſame Letters unequally repeated; as, The Object AB, ſeen through
atb-c, and bb-b, are Compound a Concave-Glaſs, will appear in
Quantities.
an erect Poſture, but diminish'd in
COMPOUND RATIO. The Ra.
a compounded Ratio of FL X
tio that the Product of the Antece-, GM to GL X FM, ſuppofing
dents of two or more Ratio's has F to be the Point to which the
Ray
cave.
A
CON
A M
B
1
CON
Fig. 1.
A
M
P
M
LC
E
QR
N
F
F
the Eye.
1
c
Draw the right Line QQ, and
AC perpendicular to it in the Point
E, and from the Point C draw ma-
ny right Lines, CM cutting the
G
right Line Re in Q, and make
QN, AE = EF, viz.
Ray BC tends unrefracted, and G equal to an invariable Line: Then
the Curve, wherein are the Points
The Rays of the Sun, in their M, is called the firſt Conchoid ; and
Paſſage through a Concave-Glaſs, the other, wherein are the Points
are weakened after the Refraction; N, the ſecond ; the right Line Qe
and ſo the Effect of Concave. Glaffes being the Directrix, and the Point C
is contrary to that of Convex ones. the Pole. And from hence it will
The confuſed Appearance of a be very eaſy to make an Inſtrument
Point through any Concave-Glaſs, to deſcribe the Conchoid.
proceeds from the too great Diver-
The Line QQ is an Afymptote
gency of thoſe Rays that fall on to both the Curves, which have
the Eye ; and ſo becauſe the more Points of contrary Flexion.
remote the Eye is from the Glaſs, If OM= AE=a EC = b,
the leſs will the Rays diverge í MR=EP=x, ER=PM=y;
therefore, the further the Eye is then will a? b2 -- 2a² bx + a **
from a Concave-Glaſs, the more
= 62 42 — 26x3 + + + *?y?
diſtinct will the Appearance of any expreſs the Nature of the ſecond
Object through it be, thọ it will Conchoid ; and x4 + 26x3 +92 m2
be more faint.
4.62*2=a? b2 + 2a2 bx tax,
The apparent Place of Objects, the Nature of the firſt; and ſo both
ſeen through Concave-Glaſſes, is al- theſe Curves are of the third kind.
ways brought nearer to the Eye ;
The firſt and ſecond Conchoid
and this is the Reaſon why they do in reality make but one Curve
help ſhort-fighted Perſons, or ſuch of the third Order, having four in-
as can ſee nigh Objects only di. finite Legs and but onę Aſymptote
ſtinctly.
between them; and of theſe there
CONCENTRIC FIGURES, arę
are three different Species exprefled
ſuch as have the ſame common by the Equation, x xyy = -@**.
Centre.
6 x3. cx² . dx .
where a, b,
CONCHOID, is the Name of a c, d, e, are invariable Quantities,
Curve given to it by its Inventor the Abſciſs AP is x, and the Cor-
Nicomedes, and is thus generated ; refpondent Semi-ordinate PM, or
M 4
Phi,
!
CON
CON
जी
PU
In a
Fig. 3•
AF
E P
т.
fourth has a contrary Sign, there
Fig. 2.
will be another Species expreſſed by
that Equation, confifting of two
M
M
Conchoids and an Oval next to the
M
Convex Sides of one of them; and
P
when two Roots of that Equation
PE
be equal (but not the middle ones)
and the other two real and un-
m
equal ; there will be another Spe-
cies expreſſed by that Equation,
having a double Point next to the
Q
Convex Side of one of the Conchoids.
And, laſtly, when that Equation
has all its Roots real, unequal, and
with the ſame Sign, what is expref-
ſed by the Equation will be two
M
Ovals, ſo that the Equation xxyy
M
ax4. bx3, cx?. dx. e. ex-
P
preſſes fix different Species of Curves.
The first three of which will be de-
ſcribed by what has been ſaid a-
im
bove ; for if in the firſt Fig. the
Line EF be taken greater than
EC; the Conchoid of Fig. 2. will
Q
be had. If EF be
If EF be = EC, that of
Fig. 3. will be had ; and when EF
is leſs than EC, that of Fig. 1. or Fig.
Fig. 4.
4. will be had.
M
M
Sir Iſaac Newton, in the latter
Part of his Algebra, tells us, That
PHY
this Curve was uſed by Archimedes
and other Ancients in the Conſtruc-
E
tion of ſolid Problems ; and he
himſelf prefers it before other
Curves, or even the Conic Sections
in the Conſtruction of Cubic and
Biquadratic Equations, on account
of its Simplicity and eaſy Deſcrip-
Pm, y! For when the Equation tion, ſhewing therein the manner of
ax4. bx3.682. dx.e. has their Conſtruction by help of it.
four real Roots, and the two middle CONCRETE NUMBERS, are thoſe
ones be equal, the Curve will have that are applied to expreſs or de-
a Node, as at Fig. 2. when three note any particular Subject; as 3
Roots of that Equation be equal, Men, 2 Pounds, &c. Whereas, if
the Curve will have a triple Point, nothing be connected with the Nom-
as F, in Fig. 3. and when two of ber, it is taken abſtractly or uni-
the Roots are imaginary, the Curve verſally ; as 4 fignifies only an Ag-
at Fig. 4. will have only four infi- gregate of four Units, he they Men,
nite Legs. Moreover, when that Pounds, or what you pleaſe.
Equation has three real unequal CONCURRING, or CONGRUENT
Roots with the ſame Sign, and the FIGURES, in Geometry, are ſuch,
M
in
as
1
N
Z X
CON
CON
as being exactly laid upon one ano Area of its Baſe, multiplied into one
ther, will exactly meet, and cover third Part of its Altitude.
one another; and therefore it is a 2. All Cones ftanding upon the
received Axiom, that plane Figurès, fame Baſe, and being between the
exactly covering one another, are fame Parallels, are equal to one an-
equal among themſelves.
other.
CONDENSATION, is when any 3. The Superficies of a right
Mafs of Matter is thruſt into a lefs Cone, not taking in the Baſe, is
Bulk than it was before, by means equal to a Triangle, whoſe Baſe is
of Cold.
the Periphery, and Altitude the
Cone. If the immoveable Point Side of the Cone.
S be taken without the Plane, in 4. Of all Cones ftanding upon
which the Circle VXY is deſcrib'd; the fame Bafe, and being between
the fame Parallels, (that is, having
the ſame Altitude,) the Superficies
S
S of that which is the moſt oblique,
is the greateit, and ſo the Superfi-
cies of the right Cone is the leaſt
but the Proportion of the Superficies
V
of an oblique Cone to that of a
DI
D
V
Y
right one, or which is all one, the
Compariſon thereof to a Circle, or
X
the Conic Sections, has not yet been
determined.
and if the indefinite right Line SZ,
Dr. Barrow, in his Geometrical
drawn through that Point, moves Lectures, was the firſt who has
quite round the Circumference of ſhewn how to find a plane Curve
that Circle, then that Line will Superficies equal to the Surface of
generate a Superficies, and
the Solid an oblique Cone, which plane Su-
contain'd under the Baſe, or Circle perficies will be bounded by a Curve
VXY; and that part of the Sus of the third Order ; ſo that the Sur-
perficies between the Baſe and the face of an oblique Cone cannot be
Vertex, or Point S, is called a Cone; found, but by the Quadrature of a
and if the Line SD, or Axis be at Space contained under a Curve of
right Angles to the Plane of the the third Order, and right Lines :
Baſe, the Cone is called a right one ;
for if the Altitude of the Cone bec,
but if it be oblique, as in the ſecond the Diſtance from the Centre of the
Figure, the Cone is called an ob- Baſe to the Point in its Plane, upon
lique or ſcalene one.
which the Perpendicular falls be b,
Euclid, in his Eleventh Book, and any Abſciſs of the Baſe begin
gives a Definition of a Cone that is ning at the Centre be callid 6.
and
not general, it being only of a right at taacc be= d; the Fluxion
angled Cone; for he ſays a Cone of the Part of the Surface of the
is produced by the Revolution of
about the perpendicular Leg remain- VObxx = znabx+d+
the Plane of a right-angled Triangle, oblique Cone will be =
ing at reft.
And
1. Every Cone is one third Part
of the Cylinder, having the fame it is impoſſible to compare the Fluent
Baſe and Altitude; and to the Solic of this with any of the Conic Sec-
dity of any Cone is cqual to the tions. It may indecd be compar'd
܀
20
аа
I
to
CON
CON
1
aa,
Aa
ag
to Part of the Superficies of a right dorgius de Sectionibus Conicis.- Gre-
Cylinder, (whoſe Baſe is the Baſe of gory St. Vincent's Quadratura Circuli
the Cone) made by cutting the Cy- & Sectionum Coni 10 Libris compre.
linder thro' by the Periphery of an henfa.--De la Hire's Opus de Sec-
Hyperbola moving parallel to itſelf, tionibus Conicis. De Witt's Ele-
and at a given Diſtance from the menta Curvarum Dr. Wallis's
Baſe of the Cylinder ; the Semi-tranſ- Conic Sections. De l'Hospital's
verſe Axis of which Hyperbola is = Analytical Treatiſe of Conic Sec-
tions, and their Ure: Milnes's
✓24
and the Semi-conju- Methodo demonftrata.--Mr. Simpſon's
Elementa Sectionum Conicarum nova
b
Conic Sections. Mr. Muller's Conic
gate = Va++ d. Sections; and many others ſcarcely
worth while to mention.
5. The Centre of Gravity of a
Cone is three fourths of the Axis is the ſhorteſt of the two Axes ; and
CONJUGATE Axis of an Ellipſis,
diftant from the Vertex.
in the Hyperbola it is a mean Pro-
CONE of Rays, in Optics, are portional between the tranſverſe
all the Rays that fall from any Axis and the Parameter.
Point of an Object upon the Surface
CONJUGATE DIAMETERS of an
of any Glaſs, having its Vertex in Ellipſis, or Hyperbola, are two Dia-
that Point, and the Glaſs for its
meters ſo drawn, that one of them
Baſe.
is parallel to the Ordinates of the
CONFUSED VISION. See Viſion. Other.
Conge, a Term in Architecture.
Co NJU G ATE HYPE R BO L, A 's.
See Apophygee.
If there betwo oppoſite Hyperbola's,
CONGRUITY of Geometrical Fi- AM, am, whoſe principal Axis is
gures. See Concurring:
CONIC SECTIONS, are Curves
la
made by cutting a Cone by a Plane,
and leaving out the Circle and Tri-
angle ; are three in Number, viz.
B 7
the Ellipſis, Hyperbola, and Para-
bola.
Theſe Curves being all thoſe of
a
the ſecond Kind, or Order, are of
M.
M
vaft uſe in Mathematics. See more
of them under the words Ellipſis, the Line Aa, and Conjugate Axis
Hyperbola, and Parabola.
the Line Bb; and if there be two
The moſt ancient Treatiſe upon other Hyperbola’s, whoſe principal
Conic Sestions, 'is, that of Apollonius Axis is the Line Bb, and conjugate
Pergæus, containing, eight Books ; one the Line Aa, then theſe four
the four firſt of which have been Hyperbola's are called Conjugate
oftentimes publiſh’d. But Dr. Hal- Hyperbola's, the two former oppo-
ley's Edition has the whole cight. fite ones, being Conjugates to the
Pappus, in lib. 7. Collcet. Mathemat. latter.
ſays Euclid wrote four Books of CONJUNCTION, in Aftronomy,
Conics, which Apollonius afterwards is the Meeting of the Stars and
ſtole and publiſhed as his own, with Planets in the ſame Degree of the
four more Books added to them.- Zodiac, and is either apparent or
Amongſt the Moderns, there is My- true.
CON
1
as
CON
CON
CONJUNCTION apparent, is when other Repetitions of the former.
a right Line ſuppoſed to be drawn But there can be only ſeven or eight
through the Centres of the two fimple Conſonances, the perfect ones
Planets, does not paſs through the being the Uniſon, Eighth, and Fifth,
Centre of the Earth, but through with their Compounds.
the Eye.
CONSTANT QUANTITIES, are
CONJUNCTION true, is when ſuch that remain the ſame, while
that right Line being produced, others increaſe, or decreaſe. So the
paſſes through the Centre of the Semi-diameter of a Circle is a con-
Earth.
ſtant Quantity ; for while the Ab-
CONOID, is a Solid produced by ſciſs and Semi-Ordinates increaſe, it
the Circumvolution of a Section of remains the ſame.
the Cone about its Axis, and may CONSTELLATION, or Aſteriſin,
be either a
is a Company of fixed Stars, ima-
CONOID Elliptical. See Spheroid. gined (by the Ancients) to repre-
CONOID Hyperbolical. See Hy- lent the Name of ſomething, and
perbolical Conoid.
commonly called by the Name of
CONDID Parabolical. See Para- that thing. Of theſe there are
bolical Conoid.
forty-eight, twenty-three being
The Sections of all Conoids, Northern, and twenty-five Southern
made by Planes cutting them, will be ones.
the ſame as the Sections of a Cone. Some Zealots have been ſo vain,
CONSCRIBED, the ſame with to attempt the changing the
Circumſcribed. Which ſee.
Names of the Conſtellations, in
CONSÈCTARY, is a Deduction, giving them Appellations taken
or Conſequence, drawn from a pre- from the Scriptures, as venerable
ceding Propofition; and is the Bede, and Julius Schillerius, who
fame with Corollary
called, for Example, Aries, Peter ;
CONSEQUENT, in Mathematics, Taurus, Andrew; Andromeda, the
is the latter of the two Terms of a Sepulchre of Chriſt; Hercules, the
Ratio: As ſuppoſe the Ratio be of wiſe Men coming from the Eaſt; the
A to B, then B is ſaid to be the great Dog, David, &c.
Confequent.
And Weigelius, a quondam Pro-
Console, in Architecture, is an feſſor of Mathematics at Geneva,
Ornament cut upon the Key of an in his Cælum Heraldicum, has tranſ-
· Arch, which has a Projecture or ferred the chief Princes of Europe
Jetting, and upon occaſion, ſerves into the Heavens; as the Great
to ſupport little Cornices, Bufts, and Bear is changed into the Elephant
Baſes.
of the Kingdom of Denmark, &c.
CONSONANCE, in Muſic, is the But this Boldneſs ought not to
Agreement of two Sounds, the one be approved of; which, inſtead of
grave, and the other acute, being being uſeful, will beget Confuſion
compounded together by ſuch a in Aſtronomy: For the Names and
Proportion of each, as proves a- Signs of the Ancients are to be re-
greeable to the Ear.
tained, not only becauſe there can.
An Uniſon is the firſt Confonance, not be better ones put for then,
an Eighth the ſecond, a Fifth the but that the Writings of Aftrono.
third ; and then follows the fourth, mers, that have been as yet publiſid,
and the Thirds and Sixths, Major may be underſtood, and the Obſer-
and Minor. There are other Con- vations of the Ancients compared
fonances, being the Doublets, or with thoſe of the Moderns.
Cox
CON
CON
1
CONSTIPATION, is when the the fourth, will conſtruct the Equa-
Parts of any Body acquire a cloſer tion. Moreover, to find the two maſt
Texture than what they had be- fimple Loci, that will conſtruct an
fore.
Equation of 37. Dimenſions, having
CONSTRICTION, is the crouding extracted the ſquare Root of 37,
the Parts of any Body cloſe to- which is 6, the Remainder will be
gether, in order to Condenſation. 1, being leſs than 6; therefore one
CONSTRUCTION of Equations, of the Loci muſt be of the 6th, and
in Algebra, is the finding the un- the other of the 7th Degree. And
known Quantities or Roots of an theſe Loci will do for Equations of
Equation, either by ſtraight Lines, 38, 39, 40, 41, and 42 Dimenſions.
or Curves.
Francis Vieta, in his Canonica
1. All fimple Equations, or thoſe Recenſione Effectionum Geometricarum,
of one Dimenſion, may be conſtruct- and Marinus Ghetaldus, in his Opus
ed, by reſolving the Fractions that pofthumum de Reſolutione & Compoſa-
the unknown Quantity is equal to, tione Mathematica, as alſo Deſcartes,
into proportional Terms.
in his Geometria, have ſhewn how
2. All Quadratics may be con to conſtruct ſimple and quadratic
ftructed by means of a right Line, Equations. Deſcartes too, has ſhewn
and a Circle.
how to construct cubic and biqua-
3. All cubic or biquadratic Equa- dratic Equations, by the Interſec-
tions may be conſtructed by means tion of a Circle and a Parabola. So
of a Circle, or a given Parabola, or alſo has Mr. Baker, in his Clavis
Hyperbola.
Geometrica. But the genuine Foun-
4. All Equations may be con dation of all theſe Conſtructions was
ſtructed by the Interſection of two firſt laid and explained by Renatus
Loci. And the moſt fimple Loci Sluſius, in his Mefolabium, part 2.
that will conſtruct an Equation, inay This Doctrine is alſo pretty well
be found thus : Extract the ſquare handled by De la Hire, in a little
Root of the higheſt Power of the Treatiſe, entitled, La Conſtruction
unknown Quantity, and if there be des Equations Analytiques, joined to
no Reinainder, then each of the two his Conic Sections. Sir Iſaac New-
Loci muſt be of the fame Number ton, at the End of his Algebra, has
of Degrees as there are Units given the Conſtruction of cubic and
contained in that ſquare Root. biquadratic Equations mechanically ;
But if there be a Remainder, the and by the Conchoid and Cifroid,
fame is equal, leſs, or greater than as well as the Conic Sections, See
the ſquare Root: If it be equal, or alſo, Dr. Halley's Conſtruction of
leſs, the Degree for one of the cubic and biquadratic Equations ;
Loci will be the Root itſelf; and as alfo Mr. Colfon's, in the Philofo-
for the other, that Root plus Unity. phical Tranſactions; and the Mar-
If the Remainder be greater than quis De l'Hoſpitals Traite Analytique
the Root, then the Degree of both des Sections Coniques.
the Loci ſhall be the Root plus CONSTRUCTION, in Geometry,
Unity.
is the drawing ſuch Lines as are
As, if it were required to find the previouſly neceſſary for the making
moft fimple Loci that will conſtruct any Demonſtration appear more
an Equation of 12 Dimenſions, the plain and undeniable.
ſquare Root thereof is 3, and the CONTACT, is when one Line,
Remainder is 3 ; whence, a Locus Plane, or Body, touches another
of the third Degree, and another of and the Parts that do thus touch,
;
are
1
-
A
D
CON
CON
are called the Points, or Places of CONTINUED QUANTITY, is
Contact.
that whoſe Parts are inſeparably
CONTIGUITY, is only the Sur- joined and united together, ſo that
face of one Body's touching that of you cannot diſtinguiſh where one
another. But Continuity is the im- begins and another ends.
mediate Union of the Parts which CONTRA-Mure, in Fortification,
compoſe any natural Body s ſo that is a little Wall built before another
one cannot tell where one begins, Partition-Wall, to ſtrengthen it, ſo
and another ends.
that it may receive ng Damage from
CONTIGUOUS ANGLES, in Geo- the adjacent Buildings.
metry, are ſuch as have one Leg CONTRATE-WHEEL, is that
common to each Angle; and are Wheel in Watches, which is next to
ſometimes called adjoining Angles: the Crown, whoſe Teeth and Hoop
As the Angles ABC, CB Di lie contrary to thoſe of the other
CBD, DBĒ; DBE, EBA, are Wheels ; from whence it takes its
contiguous Angles.
Name.
CONTRAVALLATION, or the
Line of Contravallation, in Fortifi-
cation, is a Trench guarded with a
B
Parapet, and uſually cut round a-
bout a Place by the Beſiegers, to ſe-
cure themſelves on that lide, and to
ſtop the Sallies of the Gariſon. It
E
is without Muſket - ſhot of the
The Sum of any two contiguous Town; ſo that the Army forming
Angles is always equal to two right a Siege, lies between the Lines of
Angles. ,
Circumvallation, and Contraval-
CONTINENT, in Geography, is lation.
a great Extent of Land, compre CONTRE-QUEUE D'YRONDE, a
hending ſeveral Regions and King- Term in Fortification, the ſame as
doms; and which is not interrupted Counter-Swallow's-Tail.
or ſeparated by Seas. Of theſe
CONVERGING, (or Convergent)
there are reckoned four, viz. Europe, Rays, in Optics, are thoſe Rays
Afia, Africa, and America.
that, iſſuing from divers Points of
CONTINGENT LINE, the ſame
an Object, incline towards one ano-
with Tangent Line. This Line, in ther, till at laſt they meet, and
Dialling, is ſuppoſed to ariſe from crofs, and then become diverging
the Interſection of the Plane of the Rays; as the Rays AB, CB, do
Dial and Equinoctial; and is ſo converge till they come to the Point
called, becauſe it is a Tangent to a B, and then they diverge, and run
Circle, drawn upon the Plane of the
Dial, and is at right Angles to the
ſubſtilar Line.
A
D
CONTINUAL PROPORTIONALS.
If there be ſuch a Series of Quanti-
B
ties, that the firſt is in the fame
Proportion to the ſecond, as the
C
ſecond to the third, and the third
E
to the fourth, and the fourth to the
fifth, and ſo on, they are called con off from each other in the Lines
tinual Proportionals.
BD, BE.
Con-
CON
CON
CONVERSE, in Mathematics. One the former caſe is greater, or leſs,
Propoſition is called the Converſe of according to the greater or leſs Di-
another, when, after a Concluſion ſtance of the radiating Point.
is drawn from ſomething ſuppoſed in 4. If an Object be in the Focus
the converſe Propoſition, that con- of a convex Glaſs, and the Eye on
clufion is ſuppoſed; and then that, the other ſide of the Glaſs, the
which in the other was ſuppoſed, is Object will appear erect and di-
now drawn as a Concluſion from it. As ftinct.
thus; when two Sides of a Triangle 5. The Images of Objects, op-
are equal, the Angles under their 'pofite to a Lens, any how convex,
Sides are equal ; and on the con are diſtinctly painted and inverted
verſe, if thoſe Angles are equal, in the Focus thereof.
the two Sides are equal.
6. The Image ba of an Object
CONVEX-GLASS, or Lens, is a AB, delineated in the Focus d, of
Glaſs that has one of its Superficies a convex Glaſs, is to the Object it-
plain, and the other ſpherically ſelf, as to Diameter, in the Ratio
convex. This, by fome, is called of the Diſtance of the Image Cd,
a Plano-Convex.
to the Diſtance of the Object CD.
1. If AGB be a Convex Glaſs, 7. If the Eye O be in the Axis of
and F the Focus of Parallel Rays, a convex Lens, but between the
Focus d and the Lens, the Object
will appear in an erect Pofition, but
augmented, as to Diameter, in a
Ratio compounded of the Diſtance
E
of the Point F, to which the Ray
F
to
K
F
D
A
A
B
1549
ct
F
A.
D
B
G
1
T
I
CI
E
0
.....
d
and C the Centre of the Glaſs, then
will FD= 2CG -GD. And
fo if two thirds of the Thickneſs
GD be ſo ſmall, as to be neglected,
as often happens, then will Parallel
Rays unite at the Diſtance of the
Glaſs's Diameter, whether the flat
or convex Side of the Glaſs be
turned towards the luminous Body.
2. If KE be a Glaſs Convex both
ways, or a double Convex, and
C,0, be the Centres of the Con-
vexities, and F the Focus of Paral-
lel Rays falling upon the Glaſs, .
then will KO + CE : 2 OE ::
KO: FK.
3. The Focus of diverging Rays
is farther diftant from the Glaſs
than the focus of Parallel Rays;
and the Diſtance of the Focus in
F
TY
BE
>
20 min.
COR
COR
BE tends unrefracted from the Part of this End ſticking out is fome,
Lens EL, to the Diſtance of the times cut into the Figure of a Boul-
Eye OL, from the fame ; and of tin, Ogee, and ſometimes of a Face,
OD, the Diſtance of the Object &c. the upper Side being fiat.
AB, from the Eye to the Diſtance The Corbets are uſually placed,
FD of the ſame Object, from the for Strength's fake, juſt under the
Point to which the Rays tend unre. Semi-Girder of a Platform, and
fracted, that is, FL:OD :;OL ſometimes under the Ends of Cam-
:FD.
ber-Beams.
8. And if the Eye O be beyond Cor CAROL1, an Extra-Conſtel-
the focus, the Point F will fall be lated Star in the Northern Hemi-
yond the Object; and then FL: ſphere, ſituated between Coma Be-
FD :: OD: O L.
renices and Urſa Major, ſo called in
9. If the Object AB be ſo far Honour of King Charles II.
diftant from the Glaſs, that the COR HYDRÆ, a Fixed Star of
Ray B E, refracted to the Eye O, the firſt Magnitude in the Conſtel-
diverges from the Point F in the lation Hydra. Its Longitude is 142
Axis, between the Glaſs and the deg. 49 min. Latitude 22 deg. 23
Object, then it will appear inverted, min. and Right Aſcenſion 133 deg.
and the apparent Magnitude will
be to the true Magnitude, in the Cor Leonis. See Regulus, or
Ratio compounded of FL to FD, Bafilicus.
and of OD to OL.
CORDON, in Fortification, is a
COPERNICAN SYSTEM of the Row of Stones, made round on the
World, is the ancient Pythagorean Outſide, and ſet between the Wall
Syftem, which Nicholas Copernicus, a and the Fortreſs, which lies allope,
German, in a Treatiſe publiſh'd in and the Parapet, which ſtands per-
Latin about the Year 1566, revived, pendicular, after ſuch a manner,
after it had been for many Years that this Difference may not be of-
thrown out of doors ; and it ſup- fenſive to the Eye ; whence thoſe
poſes, that the Earth and the Pla- Corders ſerve only as Ornaments,
nets revolve about the Sun, which ranging round about the Place, being
ſtands ſtill, as their Centre ; and only uſed in Fortifications of Stone-
that the diurnal Motion of the Sun Work : For in thoſe made with
and fixed Stars is not real, but Earth, the void Space is filled up
imaginary, ariſing from the Motion with pointed Stakes.
of the Earth about its Axis.
CORDS, in Muſic, are the Sounds
CORBEILS, in Fortification, are produced by an Inſtrument or Voice.
little Baſkets about a Foot and an CORINTHIAN ORDER, of Ar-
half high, eight Inches broad at chitexture, being the fourth Order,
the bottom, and twelve at the top; is the richeſt and the moſt delicate
which, being filled up with Earth, of them all, and was invented by
are commonly ſet one againſt ano an Architect of Athens. Its Capital
ther upon the Parapet, or elſewhere, is adorned with Rows of Leaves,
leaving certain Port-Holes, from and of eight Voluta's, which ſup-
whence to fire upon the Enemy port the Abacus. The Height of its
under Covert.
Column is ten Diameters, and its
CORBET, in Architecture, is a Cornice is ſupported by Modillions.
ſhort Piece of Timber, placed in a CORNEA, is the hinder external
Wall, with its End ſticking out fix Tunic of the Eye, being like a pel-
or eight Inches ; and the under lucid Horn, very firm, of a ſpheri-
cal,
COR
COV
ca
cal, or rather ſpheroidical Figure, CORPUSCLES, in Natural Phin
ſtanding out behind the remaining loſophy, fignify the minute or ſmall
Part of the Ball of the Eye, and Parts of a Body. And
conſolidating the Eye and Scleroti CORPUSCULAR PHILOSOPHY,
is the Explanation of Things, and
CORNICHE, or Cornice, is the giving an Account of the Phæno-
third and higheſt Part of the Enta- mena of Nature by the Motions and
blature, and commonly fignifies the Affections of the minute Parts of
uppermoſt Ornament of any Wain- Matter.
ſcot, &c. in regard to the Pillar; CORIDOR, in Fortification, is
and is different, according to the the Covert-Way lying round about
different Orders of Architecture, the whole Compaſs of the Fortifica-
In the Tuſcan it is without Orna- tions of a Place, between the Out-
ment; and this Pillar, of all others, side of the Moat and the Palliſadoes.
has the leaft Mouldings. The Do Corvus, a Southern Conſtella-
ric is adorn'd with Dentils, like the tion, conſiſting of ſeven Stars.
Ionic, and which ſometimes has its Co-SECANT, is the Secant of an
Mouldings cut into it. The Corin. Areh, which is the Complement of
thian Pillar, of all others, has the another, to go Degrees,
moſt Mouldings, and thoſe very often CO-SINE, is the Right Line of
cut with Modillions, and ſometimes an Arch, which is the Complement
Dextils. The Compoſite has its of another, to 90 Degrees.
Dentils and Mouldings cut, with its COSMOGRAPHY, is a Deſcription
Channels or Chamferings under its of all the ſeveral Parts of the viſible
Platfond.
World, according to their Numbers,
CORNISH-RING of a Gun, is the Poſitions, Motions, Magnitudes, and
next from the Muzzle-Ring back their other Properties.
wards.
CO-TANGENT, is the Tangent of
COROLLARY, or Confe&ary, is a an Arch, which is the Complement
Conſequence drawn from ſomething of another, to 90 Degrees.
that has been already demonftrated; Co-VERSED SINE, is the re-
as, when it is demonſtrated, That maining Part of the Diameter of a
two Semi-circles can cut each other Circle, after the Verſed Sine is taken
but in one Point, therefore it follows from it.
from thence, That two whole Cir COVERT-WAY, in Fortification,
cles can cut one another but in two is a Space of Ground level with the
Points.
Field, on the Edge of the Ditch,
CORONA, in Architecture, is about twenty Foot broad, ranging
properly the flat and moſt advanced quite round the Half-Moons, and
Part of the Cornice, called by us the other Works, towards the Country.
Drip, becauſe it defends the reſt of This is otherwiſe called Coridor,
the Work from Wind and Water and has a Parapet raiſed on a Level,
But by Vitruvius it is often taken together with its Banquets and Gla-
for the whole Cornice.
cis, which from the Height of the
CORONA BOREALIS, the Parapet muſt follow the Parapet of
Northern Garland, a Conſtellation in the Place, till it is inſenſibly loſt in
the Northern Hemiſphere, conſiſting the field. It has alſo a Foot-Bank.
of about twenty Stars.
One of the greateſt Difficulties in
CORONA MERIDIONALIS, a Siege, is to make a Lodgment on
Southern Conſtellation, of thirteen the Covert-Way, becauſe the Be-
Stars.
fieged uſually palliſado it along the
Middle,
or
2
:
COU
COU
Middle, and undermine it on all ſides. COUNTER-Mine, is a wbterrar
This is called the Counterſcarp, be- neous Paſſage, made by the Belieged,
cauſe it is on the Edge of it.
in ſearch of the Enemy's Mine, to
COVING-CORNICE, is ſuch a give air to it, to take away the
Cornice, that has a great Caſemate, Powder; or by any other means to
or Hollow in it, which is commonly fruſtrate the Effect of it.
lathed and plaiſter'd upon Compaſs
COUNTER · PART, a Term in
Sprockets, or Brackets.
Mufic, only denoting one Part to be
COUNT-WHEEL, is a Wheel in oppoſite to another: As, the Baſe is
the ſtriking Part of a Clock, moving ſaid to be the Counter-part to the
round once in twelve or twenty-four Treble.
Hours. This by ſome is called the COUNTER-POINT, is the old
Locking-Wheel, becauſe it has com manner of compoſing Pieces of Mu-
monly eleven Notches in it at un-' fic, before Notes of different Mea-
equal Diſtances from one another, ſures were invented'; which was, to
in order to make the Clock ſtrike, ſet Pricks or Points one againſt an-
and it is driven round by the Pinion other, to denote the ſeveral Con:
of Report.
cords. The Length or Meaſure of
COUNTER-APPROACHES, are which Points was ſung according to
Works made by the Beſieged, to the Quantity of Words or Syllables
hinder the Approach of the Enemy; whereto they were applied.
and when they deſign to attack them COUNTERSCARP, is that Side of
in Form.
the Ditch that is next to the Coun-
COUNTER - BATTERY, is one try; or properly the Talus that
raiſed to play againſt another. ſupports the Earth of the Covert-
: Counter-BREAST-Work, the Way; tho' by this Word is under-
ſame with Falſe Bray.
ſtood often the whole Covert-Way,
COUNTER FORTS, are certain with its Parapet and Glacis. And
Pillars and Parts of the Walls of a ſo it muſt be underſtood, when it is
Place, diſtant from fifteen to twenty faid, The Enemy loaged themſelves on
Foot one from another, which are the Counterſcarp.
advanced as much as poſſible in the - COUNTER-SWALLOWS-TA11.,
Ground, and joined to the Height is an Outwork in Fortificacion, iis
of the Cordon by Vaults, to ſupport the figure of a ſingle Tenaille,
the Way of the Rounds, and part of wider towards the Place, that is, ac
the Rampart; as alſo to fortify the the Gorge, than at the Head, or
Wall, and ſtrengthen the Ground; next to the Country.
but are not now of much Uſe, unleſs COUNTER-TENOR, one of the
in large Fortifications.
mean or middle Parts of Muſic, he-
COUNTER-FUGUE, in Muſic, is ing called ſo, becauſe it is oppolite
when the Fugues proceed contrary to the Tenor.
to one another.
COURSE, in Navigation, is that
COUNTER-GUARDS, in Fortific Point of the Compais, or Coast of
cation, are large Heaps of Earth, the Horizon, on which the Ship is
in figure of a Parapet, raiſed above to be ſtcered from Place to Place; or
the Moat, before the Faces, and it is more properly che Angle chat
the Foint of the Baſtion, to preſerve is made by a Tangent to the Meri-
them; and then they conſiſt of two dian, and an inhnitely ſmall Part of
Faces, making an Angle-Saliant, a Rhumb-Line at the Point of Con-
and are parallel to the Faces of the tact.
Baſtion.
Cour
N
CRO
A
CRY
1
COURTINE, or Courtain, in For- its Breadth by the Length of the
tification, is the Front of the Wall Middle Periphery.
between the Flanks of two Baſtions ; CROWNED Horn-WORK, is a
or the longeſt Straight Line that Horn-Work with a Crown-Work be-
runs round the Rampart, drawn fore it.
from one Flank to the other, being Crown-Post, is a Poft which,
border'd with a Parapet five Foot in ſome Buildings, ſtands upright in
high, behind which the Soldiers the Middle, þetween two principal
ſtand, to fire upon the Covert-Way, Rafters, and there goes Struts or
and into the Moat.
Braces from it to the Middle of each
CRONICAL. See Acronical. Rafter.
Cross-MULTIPLICATION, is CROWN-WHEEL of a Watch, is
a Method; uſed by Workmen, of the upper Wheel next to the Bal-
caſting up ſuperficial Dimenſions of lance, which by its Motion drives
Feet, Inches, and Parts, by firſt it, and in Royal Pendulums is called
ſetting down a Length taken in Feet the Swing-Wheel.
and Inches, and ſetting the Feet CROWN-WORKS, in Fortification,
and Inches of another Length, by are certain Bulwarks advanced to-
which the former Length is to be wards the field to gain ſome Emi-
multiplied, directly under the Feet nence, confifting of a large Gorge,
and Inches of that Length ; and and two Wings that fall on the
then multiplying the Feet by the Counterſcarp near the Faces of the
Feet, and (croſs-wiſe) the Inches Baſtion; ſo that they are defended
of one Length by the Inches of the by them, and next to the Field ſhew
other, and dividing the Sum of the an entire Baſtion, being between two
Product by 12, and multiplying the Demi-Baſtions, the Faces whereof
Inches by the Inches, and dividing look towards one another.
them by 144.
CRYSTALLINE Humour of the
Cross-STAFF, or Fore-Staff, is Eye. This Humour lies immediately
& Mathematical Inſtrument of Box, ' next to the Aqueous within the
or Pear-Tree, confiſting of a ſquare Opening of the Tunica Uvea, and,
Starming of about three Foot long, like a Glaſs put over a Hole, col-
having each of the Faces thereof lects and refracts the Rays of Light
divided like a Line of Tangents, falling upon it, being very pellucid,
and four Croſs-Pieces of unequal in figure of a Lens unequally Con-
Lengths to fit on to the Staff, the
Halves of which are as the Radius's Kepler, in Paralip. in Vitellionem,
to the Tangent Lines on the Faces cap. 5. pag. 167. thinks, that the
of the Staff. This Inſtrument is foremoſt side of the cryitalline Hu-
uſed in taking the Altitudes of the mour is the Segment of a Spheroid,
Celeſtial Bodies at Sea.
generated by the Revolution of an
CROSSIERS, are four Stars in fi- Elliptis about its Axis ; and the
gure of a Croſs, ſerving thoſe that hinder Side, the Segment of an
fail in the Southern Hemiſphere, to hyperbolic Conoid, made from the
find the South Pole.
Revolution of an Hyperbola about
CROTCHET, a Term in · Muſic, its Axis.
being the fifth Note of 'Time.. But Schottus, in Libro de Univers.
Crown, in Geometry, is a plain Nat. Eg Art. part 1. lib. 2.
Ring, included between two con- ſays, That the cryſtalline Humour
centric Peripheries, and the Arca is not of the fame Figure in all
thereof will be had by multiplying Men, and even in the ſame Perſon,
it
vex.
pag. 68.
1
3
ſince u
3 u
CU B
CUB
it varies according to his Age; for All cubic Equations have three
it is more round in ſome chan others, Roots, either all real, or one real,
and in a Perſon of full Age it is and two imaginary.
turgid, but in old Age it is almoſt All cubic Equations may be re-
flat.
duced to this Form, x3 + 2x ter
CUBATUR E of a Solid, is the = 0; wherein the fecond Term is
Meaſuring the Space contained in wanting; and they may be extracted
it, or the finding the folid Content if q be affirmative, or even negative;
of it.
93
CUBE, is a ſolid Body, confifting provided that
be not greater
27
of fix equal Sides, being all Squares, than I rr.
The Solidity of any Cube is found
If x3 + x
by multiplying any one of its Sides, Equation, which has always two
4 = a be a cubic
or Faces by the Height.
Cubes are to one another, in the and the real Root be wanted ſup-
imaginary Roots, ſince q is negative,
triplicate Ratio of their Diagonals, poſe x = 4+ ż; then will u be =
or of the Sides of their Faces.
CUBE-Root of any Number or
Quantity, is ſuch a Number or 19+
199+
P3
And
Quantity, which, if multiplied into
27
itſelf, and then again the Product
thence ariſing by that Number or
x; therefore if
Quantity, (being the cube Root,) the fäid known Value of u be put
this laft Product ſhall be equal to
the Number or Quantity whereof it for the ſame in this Equation, we
is the cube Root; as 2 is the cube shall have the value of x in known
Root of 8, becauſe two times 2 is affirmative in the given cubic Equa-
Terms. In like manner; when 9 19
4, and two times 4 is. 8; and a fb
is the cube Root of a3 + 3abb + Roots, which is when 99 is greater
tion, and it has two imaginary
3 baa 463.
Every cube Number has three p3
than ; the Value of u will be
Roots; one real Root, and two 27
imaginary ones: as the cube Num-
ber 8 has one real Root 2, and two V
1 g+V499
and
imaginary Roots, viz. V
27
and V-341. And, generally, fo there will be a real Value of *.
if a be the real Root of any cube But when all the Roots of the given
Number, one of the imaginary Roots cubic Equation are real, they cannot
of that Number will be
be found by this means; becauſe in
and the other
this Cafe 99–
P3
will be a ne-
27
✓
gative Quantity; and fo its ſquare
Root is an impoſible Quantity. But
CUBIC EQUATion, in Algebra, Tables of Sines or the Trifection of
theſe Roots may be found by the
is ſuch an one wherein the unknown
an Arch of a Circle.
Quantities ariſe to three Dimen-
Firſt find the Sine which is to the
fions; as x3 = a3 -63, or x3 +
***=p, or *+fxx-abx Radius;- as ?? to and
mm n to per, &c.
P
having
3
3
P3
3-I
a tv
3 a a
2
3 аа
2
N 2
CU B
CUB
V 999
Faving found the Degrees of the 3 multiplying 3, produces 9; and
Arch anſwerable thereto, take again, 3 multiplying 9, produces.
part of thoſe Degrees, and double 27.
the Sine of them; then, if a fourth The Difference of two cube Num-
Proportional be found to this double bers, whoſe Roots differ by Unity,
Sine,
VP, and the Radius ; that is equal to the Aggregate of the
fourth Proportional will be one Va. Square of the Root of the greater,
lue of x in the cubic Equation x3 double the Square of the leſs, and
+ Pax +q=0.
the leſs Root.
The real Root of a cubic Equa-
CUBIC PARABOLA, a Curve as
tion x3.px.q=0, whoſe two others BCD of the ſecond Order, having
are imaginary, may be otherwiſe
found thus : let the Sine of the third
D.
Term px bet, then the Difference
between two mean Proportionals
M
39
between +
+ PP
2p
4P
А.
39
999
P
and
to
+ PD
2
4 PP
will be the value of x. And if the
Sine of px be --, the Sum of thoſe
B
mean Proportionals will be the Va-
lue of x. Or ſuppofing a = Vit, two infinite Legs CD, CB, tending
39
and b =
contrary ways. And if the Abſciſs
the Difference or Sum AP, x, touches the Curve in C, the
2 P
of two mean Proportionals between relation between AP (x) and PM (5)
is expreſſed by the Equation y
b + Vāatbb and b + a x3. 6x2. X. d. or when A falls
Vaatbb will be the value of x. in C, by the Equation y = a x3.
Cubic Foot of any Subſtance, is which is the moſt ſimple Equation
ſo much of it as is contained in a of the Curve.
Cube, whoſe side is one Foot.
If it be required to deſcribe the
CUBIC Hyperbola, is a Figure cubical Parabola by a continued
expreffed by the Equation xy? = a, Motion, you may do it thus, by
having two Afymptotes, and con- means of a Square and the equilate-
fiſting
of two Hypercola's, lying in ral Hyperbola : Thro' a given Point
the adjoining Angles of the Aſymp- A, draw the Right Line CAB, and
totes, and rot in the oppoſite An- DAE at right Angles to it, and
gles, like the Apolionian Hyperbola; draw FAG at half right Angles to
being otherwiſe called by Sir Iſaac CAE or D AB, and let DE, BC
Newton in bis Enumeratió Lineárum be Afymptotes to the equilateral
Tertii Ordinis, an Hyperboliſmus of Hyperbola's HFI, KGL; then
a Parabola; and is the 65th Species take a ſingle Square MAN, and
of his Lines, according to him. a double one DMPN. Faften the
CUBIC NUMBER, is that Num. · Angle of the ſingle Square MAN in
ber which is produced by multiply- the Centre A, lo as to be moveable
ing any Number by itſelf
, and then about the ſame. Then if the Leg
again the Product by that Number; DP of the double Square be moved or
as, 27 is a Cubic Number, ſince lid along the Afymptote DAE, and
at
5
1
CU B
C U. B
at the fame time the Interſection of and draw the right Line AM; from
the Leg AŅ of the fingle Square, M draw the Perpendiculars MN ad
MP to B D and A C: Draw NO
D
parallel to AM. Then if P Q be
made equal to NO, the Point &
H
m
K will be one point thro' which the
cubical Parabola muſt paſs. And
after the ſame manner may any
number of Points be found. There
G
are ſeveral other ways of finding
F
A
Points of the cubical Parabola; as,
by means of two Squares; by means
N
of the common Parabola, &c. But,
let this be ſufficient.
In the c bical Parabola, if A
I
M
L be the Axis, and QN the Baſe, and
E
nmrin
B
A
PR
A
and the Leg PN of the double
Square moves along the Curve KL
of the equilateral Hyperbola ; the
M
Interſection M of the other Leg
AM of the ſingle Square, vith the
Leg PM of the double Square, will
N
deſcribe the Part AM of the cubical
Parabola. And if the Interſection
of the sides of the fingle and RM be parallel to AQ; then will
double Squares be moved along the
other oppoſite Hyperbola HFI, RM be always as Qin -QR.
the Interſection m of the other Sides
Allo in the ſecond Figure, if the
will deſcribe the other Part Am of Right Line A P cuts the cubical
the cubic Parabola.
Otherwiſe, by means of Points. Let
ABC be an Ifoſceles Triangle, and
M
Fig2
B
A А
B
P
M
N
Parabola ABCM in three Points
Q
A, B, C, and from any Point P be
A
C
drawn the Right Line or 0. dinate
P
PM, cutting the Curve in on Point
M only : then will PM be always
as the solid AP X B? x CP; which
BD perpendicular to the Rife AC. is an effential Property of this
Take awy Point M in the Side B CCurve.
And
N 3
CUL
CUR
And hence it is eaſy to conſtruct
The Curve of this Parabola cage
a cubic Equation x3 ta ax = 63 not be rectified, not even by means
by the Interſection of this Curve, of the Conic Sections.
But a
and a right Line. See the Con- Circle may be found equal to the
ftr: &tion of a cubic Equation by Curve Surface generated by the
werns of the cubic Parabola, and Rotation of the Curve AM about
a right Line by Dr. Wallis, in his the Tangent AP to the principal
Algebra : As alſo the Conſtruction Vertex A.
of Equations of ſix Dimenſions, by Let MN be an Ordinate, and
means thereof and a Circle by Dr. MT a Tangent, at the Poine Mi
Halley, in a Lecture formerly read and let PM be parallel to AN.
at Oxford.
M
P
1
1
A
I
'N
A
Divide MN in the Point , in fand eight hundred Pounds.
Its
fuca manner that MO be to ON Load is about twelve Pounds, and
as TM is to MN. Then a mean it carries a Shot of five Inches and
Proportional between TM +ON a half in Diameter, weighing twenty .
and of AN will be the Semi- Pounds.
Diameter of a Circle equal to the CULVERING Ordinary, weighs,
Superficies deſcribed by that' Ro- four thouſand five hundred Pounds,
tation.
and twelve Foot long : The Weighd
The Area of a cubic Parabola, of the Ball ſeventeen Pounds five
is three fourths of its circumſcribing. Ounces.
Parallelogram.
CULVERING of the leaſt fize,
CYBO-CUBE, the fixth Power. is five Inches in Bore, eleven Foot
Ců.so-Cubo-CUBE, the ninth long, weighing about four thouſand
Power:
Pounds, carries a Shot three Inches
CULMINATION of a Star, in and a half in Diameter, weighing
Aſtronomy, is the Paſſage thereof fourteen Pounds niñé Ounces.
over the Meridian: And ſo a Star CUNEUS. See Wedge. .
is ſaid to culminate when it paſſes CURRENTS, are certain progreſo
over the Meridian.
ſive Motions of the Waters of the
CULVERING, a Species of Ord- Sea in ſevera). Places, either quite
nance; of which there are three forts, down to the bottora, or to a certain
viz. the Extraordinary, the Ordi- determinate Depth; and theſe carry
nary, and the leaft-fized Culvering: the Ships faſter, or elſe retard their
CULVERING Extraordinary, is Motion, according as the Current
five Inches and a half in Bore, thir- fpts with or againſt the Ship's
ccc Foot long, weighs four thou- Motioa.
CUR-
1
3
CYC
CUR
CURSOR, in Mathematical In- to folve Problems by their Inter-
ſtruments, is any ſmall piece that ſections, and to conſtruct Equations :
ſlides ; as, the Piece in an Equinoc- As if the Problem of Ward, in his
tial Ring-Dial that ſlides to the Young Mathematician's Guide, about
Day of the Month. Likewiſe the the May-pole upon a Hill, was to
little Ruler or Label of Braſs, being be conſtructed geometrically ; the
divided like a Line of Sines, and eaſieſt and moſt natural way of
ſliding in a Groove along the mid- doing it, would either be by an
dle of another Label, repreſenting Ellipfis, whoſe focal Diſtance is
the Horizon in the Analemma, is the given Baſe, and tranſverſe Axis
called a Curfor.
the Sum of the sides of the Tric
Curtated DistANCE, is the angle, and a Square whoſe angular
Diſtance of the Place of a Planet Point is moveable about one Focus,
from the Sun reduced to the E- and Ruler moveable about the other
cliptic.
Focus. Or elſe by deſcribing a Curve
ÇURTATION, is the Difference form’d (by moving a Square about
between the Diſtance of a Planet a given Point upon a Plane, and a
from the Sun, and the curtated Di- Ruler about another given Point
ſtance.
upon that Plane, in fuch manner
CURVATURE. This fignifies that the Ruler always paſſes through
Crookedneſs.
a given Point in one ſide of the
CURVE, the ſame as Crooked. Square) with the interſection of the
Curves, in Geometry, are ſuch Ruler and the other ſide of that
Lines, which running on continually Square, and then taking a Tbread
in all Directions, may be cut by one of the given Length, doubling it,
right Line in more Points than one. and putting it about the given Points
Or which include a Space with one upon the Plane, and moving it
right Line, either returning into titely about till the Point ſtretch-
themſelves or making infinite Ex- ing it falls in the faid Curve.
curſions.
Dr. Wallis, in chap. 70. of his
Curves are divided into Algebrai- Hiſtory of Algebra, fays, that Equa-
cal, or Geometrical, and Tranſcen- tions of 5 or 6 Diameters, may be
dent. And Geometrical ones into conſtructed by two Conic Sections.
thoſe of the firſt, ſecond, third, &c. And if higher Equations are to be
Order : See the Word Geometrical conitructed, there muſt be more Co-
Curve. Expreſs Writings upon nic Sections uſed to the Performance.
Curve Lines, beſides the Conic Sec- But here the Doctor is miſtaken, as
tions, are Archimedes's De Spiralibuse is now well known by a Geometri-
Dr. Barrory's Lectiones Geometri- cian even of the ſecond Claſs ;
Sir Iſaac Newton's 'Enumera- whence it is plain, the Doctor did
tio Linearum tertii Ordinis. Ster- not well underland this Doctrine,
ling's Illuſtratio tractatus Domini CUT-BASTION. See Baqiqsa,
Newtoni de Lineis tertii Ordinis. Cuverte, in Fortification, is a
Mr. Mac-Laurin's Geometria Orga- deep Trench about: four Batlom
nica.-Mr.Brakonridge's little Trea- broad, which is commonly funk in
tiſe of Curves.-- There are beſides, the middle of the great dry Ditch
ſeveral ſmall Diſcourſes upon Curves, till you. come to Water, and ſerves
in the Asta Eruditorum, the Me- both to, preysnt the Beſiegers Min-
moires de l'Academie Reyale des ing, and alſo the better to keep off:
Sciences, &c.
the Enemy
Two of theUſes of Curve Linęs, are CYCLE, is a perpetual Reroiu-
t.cn
N 4
сүс
C Y C
tion of certain Numbers, which ſuc Cycle of the Moon, is a Revolu-
cefſively go on from the firſt to the tion of nineteen Years, which be-.
laft, and then return again to the .gan one Year before Chriſt, in which
firſt, and ſo circulate perpetually. fpace of time the new and full
There are three principal Cycles, Moons return to the ſame Days of
viz. the Cycle of Indiction, the Cy- the Julian Year they were on be-
cle of the Moon, and the Cycle of the fore, and the begins again her Courſe
Sun.
with the Sun,
Cycle of Indi&tion, is a Revo The Cycle of the Moon, after
lution of fifteen Years, which firſt three hundred and twelve Years, will
began the third Year before Chriſt. not reſtore the new and full Moons
Chronologers diſagree about the to the fame Day of the Julian
Time that the Cycle of Indiction Year, but there will be an Error of
began; and alſo concerning the Uſe one whole Day.
that the Romans invented it for : Cycle of the Sun, is a Reyolu-
But, according to vulgar Computa- tion of twenty eight Years, in which
tion, the Year of Chriſt's Nativity time the fame Dominical Letter
was the third of this Cycle; and comes about again in the fame Or-
thus we are certain, that it was e- der, and Leap-Years expire, and
ſtabliſhed by Conſtantine in the Year the 29th Year the Cycle begins a-
312
gain.
If you ſubtract 312 from the The Uſe of this Cycle is to find
Year given, and divide the Remain- the Dominical Letter, which may
der by 15, and what remains, omit- be had from the following Table,
ting the Quotient, is the Year of when the Cycle of the Sun for a
the Roman Indiction; or if 3 be ad- given Year is known ; but this is
ded to the given Year, and the found by adding 9 to the given
Sum be divided by 15, the Remain- Year, and dividing the Sum by 28 ;
der, omitting the Quotient, will be for the Remainder is the Cycle
the Year of the Indiction.
fought.
1
A Table of the Cycle of the Sun, with the Dominical Letter an-
fwering to it.
9DC)
9DC/ '13 FE
IO B
17 AG
21 CB
22 A
25 ED
18 F
IGP
2 E ;
3
1)
4.
с
BA
G
7 F
8 E
A
II
14
D
15
C
16 B
19 E
G
23
26 C
27 B
28 A
12 G
20
D
24 F
Cycicip, or Trochoid, is a Curve; Point a in the Periphery of a Cir-
as Ą B C deſcribed by the given cle, while the Circle rolls along a
right Line, as AC from the Point
B
A, where the Curve begins, to the
Point C, where it ends.
L
1. The Cycloid is a Curve of the
mechanical kind; for the Relation
C of its Ordinates, (they being ſup.
poſed
62
1)
1
1
terms.
CY C
CYC
poſed, ſtraight Lines,) and Ab- fcribed Cylinder ; but does not give
fciffi's cannot be expreſſed in finite it, no more than the Demonftration
of the Ratio's aforeſaid, except that
2. If PL be drawn parallel to of the firſt. Honoratus Fabry, in
AD, the Semi-Baſe of the Cycloid, Synopſis Geom. gives us a ſhort Trea-
then will PM be equal to BM, tiſe of the Cycloid, wherein you
the Arch of the generating Circle; have four ways of demonſtrating
and ſo if the Arch BM be taken the firſt of the Theorems above;
for an Abſciſs, and the right Line as alſo the Demonſtrations of all the
PM for a Semi-Ordinate, and BM reft, with ſeveral other Theorems
=x, PM=y, the Nature of the about the Centres of Gravity of
Cycloid will be expreſſed by this the cycloidal Space, &c. which he
Equation, x=y .
himſelf ſays, he found out before the
3. The Cycloidal Space, or the Year 1658.
Space ABCD contain'd under the We learn from the Preface of Dr.
Curve of the Cycloid and the Baſe, Wallis's Treatiſe of the Cycloid,
is the Triple of the generating that Mr. Paſcal, in the Year 1658,
Circle.
propoſed publickly at Paris, altho
4. The Length of any Arch AP, without any Name, the two fol-
of a Cycloid, is equal to four times lowing Problems as a Challenge, to
the verſed Sine of half the Arch be ſolved by the Mathematicians
a H, of the generating Circle be- of Europe, with a Reward of twenty
tween the deſcribing Point a and Piſtoles for fo doing ; which were
the Bife of the Cycloid; whence to find the Dimenſion of any Seg-
the Length of the whole Cycloid is ment of the Cycloid cut off by a
equal to four times the Diameter of right Line parallel to the Baſe, and
the generating Circle.
the Solid generated by the Rotation
Some of the French (amongſt whom of the fame about the Axis, and
is Mr. Paſcal) will have this Curve about the Baſe of that Segment.
to be firſt taken notice of, and pro- Which ſet the Doctor upon writing
poſed to the Confideration of the the faíd Treatiſe upon that Curve,
Geometricans of trofe times by being a much better and compleat
Father Merſennus in the Year 1615. piece than any Authors who wrote up-
But Torricellius, (in Lib de Motu on the Cycloid before him : fop he
Gravium, publiſh'd Ann. 1644.) ſays gives the Surfaces of the Solids gene-
Galilæo mention'd it 45 Years be- rated by the Rotation of the cycloidal
fore, viz. Anno 1599. -Torricellius Space about its Axis, and about its
firſt ſhew'd the cycloidal Spice to Biſe, and other Determinations of
be three times the generating Cir- the Centres of Gravity, &c. Here
cle (tho' Mr. Paſcal will have Mr. he ſays too, that Sir Chriſtopher
Roberval to be the firſt) The Wren, Anno 1658, was the firſt who
Solid generated by the Rotation of found out a right Line equal to the
that Space about its Baſe to the Curve of the Cycloid ; and Ms.
circumſcribing Cylinder to be as 5 Huygens in his Horolog. Oſcillar,
to 8.- About the Tangent parallel mentions himſelf as the firft Inventor
to the Baſe, as 7 to 8. --- About the of the Segment of a Cycloidal
Tangent parallel to the Axis, as 3 Space, made by drawing a right
to 4.- He alſo ſays, that he could line parallel to the Bale at the
tell the Ratio of the Solid generated Diſtance of the Axis of the Curve
by the Rotation of the cycloidal from the Centre, being equal to a
Space about its Axis to the circum- right-lind Space, viz. to a regular.
Hexagon
C.Y L
C Y L
1
6
Hexagon inſcribed in the generating Cylinder is called a right one ; but
Circle, whoſe Demonſtration is to if not, an oblique or ſcalene one.
be ſeen in Wallis's ſaid Treatiſe. 1. The Section of every Cylinder
There are ſeveral other Authors by a Plane oblique to its Baſe, is an
who ſpeak of the Cycloid, as Mr. Ellipfis.
Farnat, Mr. Bernoulli, here and 2. The Superficies of a right Cy-
there in the Asta Eruditorum, Mr. linder is equal to the Periphery of
de la Hire, &c. too many to men- the Baſe, multiplied into the Length
tion; and in the Memoirs of the of its side.
Royal Academy of Sciences at 3. The Solidity of a Cylinder is
Paris, Ann. 1706, you have the equal to the Area of its Baſe, mul-
Doctrine of Cycloids, or rather tiplied into its Altitude
Epicycloids, generated by Curves 4. Cylinders of the ſame Baſe,
revolving upon themſelves.- This and ſtanding between the ſame Pa-
is the Curve that the Centre of Of- rallels, are equal.
cillation of a Pendulum moving in, 5. Every Cylinder is to a Sphe-
will deſcribe any Arches of it all roid inſcrib'd in it, as 3 to 2.
in the ſame time, and a Body falling 6. If the Altitudes of two right
in it from any given Point above to Cylinders be equal to the Diame.
another (not exactly) under it, will ters of their Baſes, thoſe Cylinders
come to this point, in a leſs time are to one another as the Cubes of
than in any other Curve, paſſing the Diameters of their Baſes.
thro? thoſe two Points.
CYLINDRICAL SPECULUM, is
CYGNUS, the Swan, a Conſtel- a Cylinder of poliſh'd Metal ; be-
lation in the Northern Hemiſphere. ing either convex or concave.
CYLINDER. . If any indefinite The Images of formous Objects,
right Line Sz, being without the ſeen by the Reflexion of the Sur-
Plane of the Circle V XY, moves face of a convex cylindrick Specu.
about the Circumference of that lum, are render'd deformed ; and
Circle always parallel to itſelf, until vice verſa the Images of deformed
Objects appear formous ; ſo that a
S
SI
Figure altogether confuſed, ſeeming
to be drawn without any manner of
x
Intent, being placed horizontally
near one of theſe Cylinders, will
appear in the Surface of the Cylin-
der the Face of a Man, or any other
formous Figure. But then the con-
VOY
Y fuſed Figure muſt be firſt drawn ac-
ZX
cording to Art.
If parallel Rays fall after ſuch a
it be returned to the ſame Place manner in the Superficies of a con-
from whence it went, then the inde- cave Cylinder, as to cut its Axis at
finite Solid contain'd under the right Angles, and their Inclination
Bafe or Circle V XY, and the Sun to the Speculum be leſs than fixty
perficies generated after this man- Degrees; after the Reflection, they
ner by the right Line SZ, is cal- will be united in a right Line, pa-
led a Cylinder, and the ſaid Super- rallel to the Axis, being at a Di-
ficies is called the Superficies of it ; ftance lefs than one fourth Part of
and if the Line SZ be perpendicu- the Diameter.
lar to the Plane of the Baſe, the The Rays AB, AD, which, from
the
tu
2l
x
1
vby
Zix
DAC
DAY
the fame Point A of the Axis, fall DADO, a Term in Architecture,
in the ſame Periphery H I of a con- uſed by ſome Writers for a Dye,
being the part in the middle of the
Pedeſtal of a Column, between its
F
Baſe and the Cornice.
DAILY MOTION of a Planet.
H
I
See Diurnal Motion.
DARKENBD Room. This is the
fame as Camera Obſcura ; being a
Room darkened all but in one licele
Hole, having a Convex-glaſs in it
А
to tranſmit the Rays of outward
Objects to a Piece of paper, or white
Cloth in the Room.
DARK TENT, by ſome Writers,
is the Name of a ſmall portable Ca-
cave Cylinder, after the Reflexion, mera Obſcura.
are united in the Point F, fo far di-
DATA, is the Term, in Mathe-
ſtant from C, the Centre of the matics for ſuch Things or Quanti-
Çircle, in the Periphery whereof the ties as are given or known, in order
Reflection is made, as the radiating to find out other things thereby,
Point A is diſtant from it.
which are unknown.
CYMATIUM, a Member of Ar-
Davis's QUADRANT, the com-
chitecture ; whereof there are two
mon Sea-Quadrant, or Back-ftaf.
forts, viz. the Doric and the Lesbic.
Day, is either natural or artis;
The Doric is a Member that has a ficial.
Concavity leſs than a Semi-circular
DAY (NATURAL,) is the Space
one, and a Projecture equal to half of Time determind by the Motion
the Altitude. The Lesbic is both of the Sun round the Earth in
concave and convex, having the twenty-four Hours, and begins at
Projecture equal to half the Alti- twelve at Night.
DAY (ARTIFICIAL,) is the
CYNOSURA, a Conſtellation con-
Time between the Sun's Riſing and
Gfting of ſeven Stars, being other. Setting. The Length of this varies
wiſe called Urſa Minor.
in different Places of the Earth ;
Cypher, or nought, noted thus, for under the Equinoctial the Arti-
(0); is that which being put beficial Days are but twelve Hours
fore a Figure, fignifies nothing, (un- long, and under the Poles they are
lefs in Decimals, where it augments,
half a Year.
þeing put before, in the ſame pro Civil, becauſe it is by divers Na-
The Natural Day is alſo called
portion, as when put after Integers.)
But after a Figure, it increaſes it by
tions reckon'd divers ways. The
tens; and ſo on, ad infinitum.
Babylonians began to account their
Day from the Sun-riſing : The
Jews and Athenians from the Sun-
fetcing, whom the Italians now fol-
D.
low, beginning their firſt Hour at
Sun-Set, The Egyptians began at
ACTYLONOMY, the Midnight, as we account the Aſtro-
Art of numbering on the nomiçal Day ; but the Umbri began
Fingers.
tude.
D
at Noon.
DECAGON,
D E C
D E C
1
1
DECAGON, in Geometry, is a Value of them decimally, as 2, 20,,,
plane Figure of ten Sides, and ten '30, & C. ſo when ſet on the left
Angles ; and if all the Sides are Hand of Decimal Fractions, they
equal, and all the Angles, it is cal-. decreaſe the Value decimally, as : 5
led a regular Decagon; and it may .05 .005 &c. But ſet on the left
be inſcrib'd in a Circle.
Hand of Integers, or on the right
Hand of Decimal Fractions, they
fignify nothing, but only to fill up
void Places. Thus, . 5000 or 2005.
is buts Units.
B
Arithmetical Operations may be
perform'd vafly ſooner by Decimal
А
Fractions than by Vulgar Fractions,
becauſe the Denominators being o-
mitted, the Rules of Addition, Sub-
traction, Multiplication, and Divi-
of
as
Decagon inſcribd in a Circle, and Numbers, regård being had to the
it be continued out to C, fo that Pointing, which is eaſy: Yet, by
BC=AD, then will AB:BC :: theſe, Operations will not always
come out exactly true ; but you
BC:AC.
If y be the Radius of a Circle, may, come as near the Truth as
poſſible, by bringing out niore Fi-
then will vag2 1r, cr
gures.
DECIMAL SCALES, are, in ge-
is to be the side of a neral, any Scales upon a ſquare
Rule, that are divided decimally,
Decagon infcrib'd in that Cirele.
being Scales of Money, Weights,
If the Side of a regular Decagon Meaſures, made from Tables bear-
be 1, the Area thereof will be nearly ing thoſe Names, and ſerve readily:
8 69 ; whence as 1 to 8.69, fo is by Inſpection only, to fhew you the
pearly the Square of the side of any Decimal Fraction that proporly
given Decagon to the Area of that belongs to any part of Money,
Decagon.
Weight, or Meaſure, &c.
DECIMAL FRACTIONS,
DECLINATION (APPARENT,)
ſuch that have 19, 100, 1000, is the Diſtance of the apparent
10000,&c. for their Denominator ; Place of a Planet from the Equi-
as, 146, 783, 8c, and noctial.
the Numerators, for Brevíty and DECLINATION of the S117or
Conveniency fake, are commonly any Star, or Point of the Heavens,
expreſs'd by a Point, or Comma, is its Diftance from the Equator,
fet on the left Hand thereof, thus, meaſar'd in the Arch of a great
is is on - 34 is and · 346 is Circle, perpendicular to the Eqtza-
So, the Denominators being tor. R:SO's Place :: S. greateſt
amicred.
Declination. S. of his preſent De-
Regiomontanus was the first that clination.
aſed Decimal Fractions in the Con The greateſt Declination of the
firection of the Tables of Sines, it- Sun, or of the Feliptic, was brf, as
bout A. D. 3464.
we know of, obſerved by Pyrhsas, at
As Cyphers' fet on the right Mafilia, about three hundred and
Hand of Integers do increaſe the twnty-fow Years before Chrift;who
obferring
2
are
DEC
D E F
obſerving that the Height of a Line, like a fiducial Edge, to cut
Gnomon was to the Shadow of it, the Degrees of the Limb: For'at
when the Sun was in the Meridian, any time when the Sun ſhines, by
ás, 3195!] to 90000, from thence having the Hour of the Day, you
concluded the Sun's greateſt Decli- may, get the . Declination of any
nation to be 23 deg. 52 min. 41 Wall or Plane by this Inſtrument.
fec. And Gajendus found the Sol DECLINING Erect-DIALS,
ftitial Shadow of the ſame Length, are thoſe whoſe Planes do ſtand
as it had been obſerved by Pytheas, perpendicular to the Horizon, and
near two thouſand Years before : decline, that is, do not face directly
And ſo he concluded, that the Sun's the four Cardinal Points. See E-
greateſt Declination, or that of the RECT Declining DIALS.
Ecliptic, is conſtant. But from a DECLINING ERECT - PLANES.
compariſon of the ſeveral Obſerva- See Ereet Declining Planes,
tions concerning this matter, the
1. Becauſe the Diſtance of the
Sun's greateſt Declination is com- Sun from the Centre of the Earth
monly accounted 23 deg. 30 min. is ſo vaſtly remote, that all Points
DECLINATION of the Sea-Com- of the Superficies of the Earth may
paſs, or of the Needle, is its Varia- be taken as if they were in the Cen-
tion from the true Meridian of any tre, the Styles of all Dials may
be
Place. See concerning this in Mr. conceived as Parts of the Axis of
Lowthorp's Abridgment of the Phin the Earth paſſing thro’ the Centre
loſophical Tranſaktions, Vol. 2. chap. of the Earth.
4. pag. 607. & feq. And in Fa. . 2. The Extremity of the Style of
ther Noel's Obſervationes Mathem. all Dials may be taken for the Cen-
& Phyfic. cap. 8. p. 108. & feqq. tre of the Earth.
DECLINATION (TRUE,) is the 3. The Hour-Lines drawn upon
Diſtance of the true Place of a all Dial-Planes, are the common
Planet from the Equator.
Sections of Hour-Circles of the
DECLINATION of a Wall, or Sphere with the Dial-Planes.
Plane for Dials, is an Arch of the The Equinoctial Circle upon all
Horizon, contained either between Dial-Planes, will be a ſtraight Line,
the Plane and the prime vertical and the Parallels of Declination will
Circle, if you reckon it from che be the Conic Sections.
Eaſt or Weſt; or elſe between the DecussATION, a Term in Op-
Meridian and the Plane, if you ac tics, fignifying the croſſing of any
count it from the North or South. two Lines, Rays, &c. when they
DECLINATORIES, are Inſtru- meet in a Point, and then go on
ments contriv'd for taking the De- ſeparately from one another.
clinations, Inclinations, and Recli DEFENCES, in Fortification, are
nations of Planes ; and are of ſeveral all ſorts of Works that cover and
kinds. The best whereof, for tak- defend the oppoſite Poſts, as Flanks,
ing the Declination, conſiſts of a Parapets, Caſemates, &c. No Mi-:
{quare Piece of Braſs, or Wood, ner can be fixed to the Face of a
with a Limb accurately divided into Baltion before the oppofite one be
Degrees, and every fifth Minute, if ruin'd, or till the Parapet of its
poſſible, having a horizontal Dial Flank be beaten down, and the
moving on the Centre, made for the Cannon in all Parts that can fire
Latitude of the Place it is to ſerve upon that place which is attack'd,
in, and which has a ſmall bit of are diſmounted.
fine Brass fixed on its Meridian DEFERENT, in the old Prole-
2
mais
D E F
D E G
1
A
maic Syſtem, is an imaginary Circle, likewife 16, whoſe Parts 1, 2, 4, 8,
which, as it were, carries about the make but 15.
Body of a Planet, and is the ſame Defile, in Fortification, is a
with the Excentric.
ſtraight narrow Line, or Paſſage,
Deficient HYPERBOLA, is a thro' which a Company of Horſe or
Curve having but one Aſymptote, Foot can paſs only in File, by mak--
and two Hyperbolic Legs running ing a ſmall Front; ſo that the Ene-
out infinitely next to the Afymptote my may take an opportunity to ſtop
contrary ways.
their March, and to charge them
This Name is given to the Curves with ſo much the more Advantage,
by Sir Iſaac Newton, in his Enu- in regard that thoſe in the Front
meratio Linearum tertii Ordinis : and Rear cannot reciprocally come
There are fix different Species of to the Relief of one another.
them which have no Diameters, ex DEFINITIONS, are our firſt Con-
prefled by the Equation xyy tey ceptions of things, by means where-
a x3 + 6x2 +0x +d. a t3 of, they are diſtinguiſhed among
being negative. When the Equa-themſelves, and from whence, what-
tion ax+ = bx3 tcx? + dx tee foever things being conceived by
has all its Roots real and unequal, them; the reſt are deduced. There
the Curve will have an Oval joined, are two kinds of Definitions, viz.
to it. If the two middle Roots are Nominal and Real.
equal, the Oval will join to the DEFINITION (NOMINAL,) is
Legs, and they will cut one another an Enumeration of ſuch known
in ſhape of a Nooſe. If theſe Roots Things that are ſufficient for the
are equal, the Nodus will be chang- diftinguiſhing of any propoſed Thing
ed into a very acute Cuſp or Point. from others; as is that of a Square,
If of three Roots, with the ſame if it be ſaid to be a Quadrilateral,
Sign the two greateſt are equal, the Equilateral, and Rectangular Fi.
Oval will vaniſh into a Point. If gure.
any two Roots are imaginary, there DEFINITION (REAL,) is a dis
will be only a pure Serpentine Hy- ftinct Notion of the Geneſis of a
perbola, without any oval Decuf- Thing, that is, which expreſſes the
ſation, Cuſp or conjugate Point ; manner how the thing can be done,
and when the Terms 6 and d are or made; as is this Definition of a
wanting, there will be the fixth Circle, viz. That it is deſcribed by
Species.
the Motion of a right Line about a
There are alſo ſeven different fixed Point.
Species of theſe Curves, each having DEFLECTION, is the Tendency
one Diameter, expreſſed by the E of a Ship from her true Courſe, by
quation aforeſaid when the Term ey reaſon of Currents, & c. which turn
is wancing. According to the va- her out of her right way. But this
rious Conditions of the Roots of the Word, by Dr. Hook is applied to
Equation à x3 = 6*2 + 6x +d, as the Rays of Light; that is, Deflec-
to their Reality, Equality, their tion of the Rays of Light is diffe-
having the ſame Signs, or two of rent both from Reflexion and Re-
them being imaginary.
fraction, and is made towards the
DEFICIENT NUMBERS, are Surface of the opacous Body per-
fuch, whoſe Parts, added together, pendicularly; and this is the fame
make leſs than the Integer whereof Property that Şir Iſaac Newton calls
they are the Parts ; as 8, whoſe Inflection.
Parts being 1, 2, 4, make but 7; DECREE, is the three hundred
and
A
1
D E M
D EN
and fixtieth Part of the Circumfe Ordnance. The common fort of
rence of a Circle. It is ſubdivided them are four Inches and a quarter
into fixty Parts, called Minutes, and Bore, two thouſand ſeven hundred
each of them again into fixty more, Pounds Weight, ten Foot long, car-
called Seconds, &c.
ries a Shot of ten Pounds eleven
DeLPHINUS, the Dolphin , a Ounces, is charged with ſeven
Conſtellation in the Northern He- Pounds four Ounces of Powder, and
miſphere, containing ten Stars. ſhoots point-blank an hundred and
Dem 1-BASTION, is a Fortifica- ſeventy-five Paces.
tion, having only one Face, and one Demi-CULVERING of the leaf
Flank.
fize, is four Inches and a quarter
Demi-CANNON, Lowell, the Bore, ten Foot long, two thouſand
Name of a great Gun. (The ordi- Pounds Weight. Its Charge is fix
nary ones are about fix Inches Bore, Pounds four Ounces of Powder, it
five thouſand four hundred Pound carries a Ball of four Inches Dia-
Weight ; ſome ten; ſome eleven meter, and of nine Pounds Weight,
Foot long; and carry a Shot of a- and its Level-range is an hundred
bout thirty Pound Weight.) It and ſeventy-four Paces.
carries point-blank an hundred and Demi-CULVERING, of the largeſt
fifty-fix Paces. Its Charge of fort, is four Inches and three quar-
Powder is fourteen Pound Weight. ters Bore, ten Foot and one third
There are alſo two ſizes of Demi- long, three thouſand Pounds Weight.
Cannon above this, which are ſome- Its Charge of Powder is eight
thing larger : As the
Pounds and eight Ounces, the Ball
Demi-CANNON Ordinary, which is four Inches and a half Diameter,
is fix Inches and a half Bore, twelve weighs twelve Pounds eleven Ounces,
Foot long, weighs five thouſand fix and it ſhoots point-blank an hun-
hundred Pound. Its Charge of dred and ſeventy-eight Paces.
Powder is ſeventeen Pounds, eight DBMIDITION, a Note in Muſic,
Ounces, carries a Shot of fix Inches being the ſame with Tierce Minor,
one eighth in Diameter, whoſe See Monochord,
Weight is thirty-two Pounds, and Demi-Gorge, in Fortification,
the Piece ſhoots point-blank an hun- is half the Gorge or Entrance into
dred and fixty-two Paces.
the Baſtion, not taken directly from
Demi-CANNON, of the longeſt Angle to Angle, where the Baſtion
fize, is fix Inches three fourths Bore, joins to the Curtain, but from the
twelve Foot long, fix thouſand Angle of the Flank to the Centre
Pounds Weight Its Charge is of the Baſtion, or Angle, the two
eighteen Pounds of Powder, and the Curtains would make, were they
Piece ſhoots point-blank an hundred protracted to meet in the Baftion.
and eighty Paces.
Demi-QUAVER, the laſt Note
Demi-Cross, is an Inſtrument of Time in Muſic.
uſed by the Dutch to take the Al DEMONSTRATION, is the Rea.
titudes of the Celeſtial Bodies at fons that are laid down for making
Sea, and conſiſts of a Siaff divided the Mind aſſent to the Truth or
into a Line of Tangents, and a Fallhood of a thing propoſed.
Croſs-piece, or Tranſom, and has Denes, the fame with Cauda
three Vanes. But we do not uſe Lucida, or Lion's Tail, a Star ſo cal-
this Inſtrument, our Sea Quadrant led. Which ſee.
being better.
DENOMINATOR of a Fraction, is
DEMI-CULVERING, A Piece of the Number or Letter below the
Line:
DE P
DES
rizon,
Line. Thus 4 and b, are the De- repreſented by the Hypotheneuſe and
nominators of the Fractions, and Perpendicular of a right-angled
plain Triangle, the Departure will
not be the Baſe of that Triangle.
DENOMINATOR of any Ratio,
DEPRESSION of the Pole. So
is the Quotient ariſing from the Di- many Degrees as you fail or
travel
viſion of the Antecedent by the Con- from the Poles towards the Zenith,
fequent, as, 6 is the Denominator you are ſaid to depreſs the Pole, be-
of the Ratio of 30 to 5, fince 5) cauſe it comes the fame Number of
30 (6; and this is alſo called the Degrees lower, or nearer to the Ho.
Exponent of the Ratio.
DENSITIES of Bodies, is their DESCANT, in Muſic, ſignifies the
Thickneſs; and a Body is ſaid to
Art of compoſing in ſeveral Parts,
be denſer, when it contains more and is threefold, viz. Plain, Figu-
Matter under the ſame Bulk than rative, and Double.
another Body.
DESCANT (DOUBLE) is when
The Denifities of any two Bodies the Parts are ſo contrivd, that the
are in a Ratio. compounded of the Treble may be made the Baſs; and,
direct Ratio of their Quantities of on the contrary, the Baſs the Treble.
Matter, and the reciprocal Ratio of
DESCANT (FIGURATIVE, or
their Bulks.
FLORID,) is 'that wherein Diſcords
DENTICLES, are Ornaments in are concerned as well (tho' not fo
a Cornice, cut after the manner of much) as Concords, and having all
Teeth.' Theſe are particularly af. the Variety of Points, Figures, Syn-
fected in the Doric Order : and the copes, Diverſities of Meaſures, and
'fquare Member whereon they are whatſoever elſe is capable of adorn-
cut, is called the Denticule.
ing the Compoſition.
DEPARTURE, in Navigation, is
DESCANT (PLAIN,) is the
the Eaſting, or Weſting of a Ship, Ground-work or Foundation of the
with regard to the Meridian it de- Mufical Compoſition, and wholly
parted or failed from; or it is the conſiſts in the ordinary placing of
Difference of Longitude between the many Chords.
preſent Meridian the Ship is under, DESCENSION OBLIQUE. See
and that where the laſt Reckoning Oblique Defcenfion.
or Obſervation was made; and, in
DESCENSION RIGHT. See
all Places, except under the Equator, Right Defcenſion.
it muſt be accounted according to
DESCENTS, in Fortification, are
the Number of Miles in a Degree the Holes, Vaults, and hollow Pla-
of the Parallel the Ship is in. ces, made by undermining the
The Departure, in Plain and Mer- Ground;
as the Counterſcarp, or
cator's Sailing, is always repreſented Covert-way; ſo that a Deſcent into
by the Baſe of a Right-Angle Tri- the Moat or Ditch, is a deep dig-
angle, where the Courſe is the An- ging into the Earth of the Covert-
gle oppoſite to it, and the Diſtance way, in Figure of a Trench, of
the Hypotheneufe. In the Plain which the upper Part is cover'd
and Mercator's Chart, as Radius to
with Madriers or Clays, againſt
the Diſtance, fo is the Sine of the Fires, to ſecure the Paſſage into the
Courſe to the Departure.
Moat.
But this is erroneous, except in
Descent of heavy Bodies. 1. If
very ſmall Diſtances; for if the Di. two Podies deſcend perpendicularly
ftance and Difference of Latitude be from any unequal Heights near the
Surface
DES
D ES
Surface of the Earth, the Lengths is, as the very ſmall Arches ad,
of the Lines that they deſcribe, are ae, which are equal to them. But
in the duplicate Ratio of the Times theſe very ſmall Arches are in the
or Velocities; and fo the Velocities ſubduplicate Ratio of their verled
are as the Times.
But if Bodies defcend pependi-
ad
cularly from any Heights whatſo-
ever, then this Proportion will not
hold.
If AEF be a Semicircle, and F
the Centre of the Earth, and a
A
А
D
B
D
E
T
!
t
Sines ab, ac, that is, the Lines
ab, ac, deſcribed by a deſcending
Body, are in the duplicate Ratio of
the Times, which is the Theorem
F
firſt laid down.
2. All Bodies near the Surface
Body falls from any Height A above of the Earth do deſcend perpendicu-
the Surface of the Earth to the larly at ſuch a rate, as that at the
Places B, C, and the Lines BD, end of the firſt Second of Time
CE, are drawn; as alſo the Lines they have deſcribed fixteen Fect one
FD, FE, then the Times of its Inch.
falling the Lengths AB, AC, will 3. The Velocity of a heavy Body
be expreſs'd by the trilineal Spaces deſcending in an inclin'd Plane at
FAD, FAE.
the end of any given time, is to the
The Lengths that a Body near Velocity that it would acquire by
the Surface of the Earth deſcends in deſcending perpendicularly in the
equal times, do increaſe according to ſame time, 'as the Altitude of the
the odd Numbers, 1, 3, 5, 7, 9, inclin'd Plane is to its Length.
&c.
4. The laſt Velocity acquired by
Hence, by way of Corollary, if the direct Deſcent, is to the lait
a Body falls from the Point a, the Velocity acquired in the ſame time
ſmall Diſtances ab, a c, compar'd by the oblique Deſcent, as the ab-
with a f the Semidiameter of the folute Gravity is to the relative
Earth, the trilineal Figures Fad, Gravity of the deſcending Body.
Fae, may be taken for right-angled 5. The Line deſcrib'd by the di-
Triangles, whoſe Areas will bc, to rect Deſcent is to the Line defcribed
one another, as the Lines bd, ce, in the ſame time by the oblique
ſince the Baſe a F is common, that Deſcent, as the Length of the
O
Plane
DES
D EW
ܠ
Plane to the perpendicular Height In all theſe Theorems concerning
of the Plane.
the Deſcent of Bodies on inclined
6. If the Line deſcribed by the Planes, the Lengths of the Planes
direct Deſcent be to the Line de muſt be inconſiderable, with regard.
fcribed by the oblique Deſcent, as to the Semi-diameter of the Earth;
the Height of the Plane to the for otherwiſe they are not true.
Length of the Plane, then the 12. The Time of the Defcent
Times of Deſcent ſhall alſo be in of a Body, through the Arch BC
that Proportion.
of a Semi-cycloid, is equal to the
7. If the Line deſcribed by the
direct Deſcent be to the Line de-
fcribed by the oblique Deſcent, as
the Height of the Plane to the
Length of the Plane, the laſt Velo BH
cities ſhall be equal
8. The laſt Velocities acquir'd
c
upon ſeveral inclined Planes of the
fame Heights, and however differ-
ing in Length, are equal.
Time of its Deſcent through any
9. The Time of an oblique De- other Arch A C.
ſcent through any Chord of a Circle, 13. Alſo a Body will deſcend
drawn from the loweſt Point of the from a given Point, as B, to a given
Circle, is equal to the Time of a Point C, ſooner along the Arch BC
direct Deſcent through the Diame- of a Cycloid, than along any other
ter of that Circle.
Curve, drawn through the Points
10. If a Body deſcends from the B,C.
Point A through any Number of 14. If Water runs out through a
ſmall Hole, made in the bottom of
AN E
a parabolic Conoid, the Surface of
the Water will deſcend equal Spaces
in equal Times.
15. If a Body be thrown down-
B
wards in a refifting Medium, with
ſuch a Velocity as fhall make the
Refiftance of the Medium equal to
the Acceleration of Gravity, it will
afterwards move on, or deſcend with
an uniform Motion.
16. The Velocity of a Body de-
fcending by its own Weight, in a
inclin'd Planes, AB, BC, CD, it refifting Medium, is always leſs
will acquire the ſame Velocity at than that Velocity that produces
the Point D, in the End of its Fall, the uniform Motion ; but conti-
as though it fell from the Point E nually approaches to it.
of equal · Height with A, in one Dew, are little Globules of Wa-
continued Plane ED.
ter, raiſed up from the Earth by
11. The laſt acquir'd Velocities Heat, which, for a while, ſwim up
of a Body, deſcending to the loweſt and down in the Air; and when
Point of a given Circle, through ſeveral of them convene into Drops,
different Chords, fhall be as thoſe by means of Cold, they then fall
Chords.
down again to the Earth.
LIO
1
D
DE-
3
DIA
DIA
DESCRIBENT, a Term in Geo DIA DROME. This is the ſame
metry, fignifying a Line or Super- with Vibration, or the Swing of a
ficies, that by means of the Motion Pendulum.
of it, a Superficies or Solid is de DIAGONAL, is a ſtraight Line
ſcribd.
drawn a-croſs a Figure, from one
DIACOUSTICS, or DIAPHO-
DIAPHO. Angle to another, and is called a
NICS, is the Confideration of the Diameter by fome.
Theſe are
Properties of refracted Sound, as it chiefly in quadrilateral Figures.
paſſes through different Mediums. As the Lines AC, BD, are the
But the
Diagonals of the Parallelogram
DIACOUSTIC Curve, or the ABCD.
Couſtic by Refraction, is generated Every Diagonal, as A C, divides
thus : If you imagine an infinite a Parallelogram into two equal Parts,
Number of Rays, B A, BM, BD, or Triangles, ABC, ADC.
I
HI
В
A
DIE
A
F
E
D
N
с
Two Diagonals AC, BD, of e.
very Parallelogram, do mutually bif-
fect each other, as in the Point E.
DIAGONAL SCALE, See Scales.
DIAGRAM, is a Scheme for the
M
Deſignation, or Demonſtration of
any Figure.
t
DIAL, or Sun-Dial, is the De-
ſcription of Lines upon a given
Plane, or on the given Superficies
of any Body, after ſuch a manner,
that the Shadow of a Gnomon, or
the Rays of the Sun, tranſınitted
through fome Hole, or reflected
from a very little reflecting Sub-
B
ſtance, ſhall touch given Lines at a
given Hour. And the manner of
&c. iſſuing from the ſame luminous this Deſcription is called Dialing.
Point B, to be refracted to or from The firſt Sun-Dial that was ſet up
the Perpendicular MC, by the gi- at Rome, was by Papyrius Curſus, ,
ven Curve AMD; and ſo, that about the 447th Year of the City,
CE, the Sines of the Angles of In on the Temple of Quirinus ; but it
cidence CM E be always to OG, went not right.
went not right. And about thirty
the Sines of the refracted Angles Years afterwards, M. Valerius Meja
OMG in a given Ratio, the Curve fala brought another out of Sicily,
HFN, which touches all the re and ſet it up upon a Pillar near the
fracted Rays AH, MF, DN, &c. Rofiram. But this went not righe
is called the Diacouſtic, or Couftic neither, becauſe not made for the
by Refraction.
Latitude of Rome. But about eleven
02 Years
DIA
DIA
Years after there was one ſet up, that DIALLING GLOBE, is an In-
went more exact.
ſtrument of Braſs, or Wood, with a
The Invention of Sun-Dials are Plane fitted to the Horizon, and an
by ſome attributed to Anaximenes ; Index particularly contrived, to
and by fome to Thales. And Vi- give a clear Demonſtration of that
truviuss among the various kinds of Art.
Dials he mentions, ſays, That Be DIALLING LINEs, or Scales,
rolus the Chaldean invented one are ſuch divided Lines; as being put
upon a reclining Plane, nearly paral- on Rulers, or the Edges of Qua-
lel to the Equinoctial.
drants, and other ſuch like Inftru-
There are a great many Authors ments, ſerve to ſhorten the Buſineſs
who have wrote upon Dialling. of Dialling.
Some of which are,-Vitruvias, in DIALLING SPHERE, is an In-
his Architecture, cap. 4. 6° 7. lib.9. ſtrument made of Braſs, with feveral
Sebaſtian Munfter, his Horologra- Semi-circles ſliding over one an-
phia.- John Dryander de Horologio- other, upon a moveable Horizon, to
rum varia Compoſitione.- Conrade demonſtrate the Nature of ſpherical
Gefner's Pande&ta. Andrew Scho. Triangles, and to give the true Idea
ner's Gnomonicæ. Fred. Comman- of drawing Dials on all ſorts of
dine de Horologiorum Deſcriptione. - Planes.
Joan. Bapt. Benedi&tus de Gnomonum Diameter of a Circle, is a right
Umbrarumque Solarium Ufu.-- Cla- Line that paſſes through the Centre
vius's Gnomonices de Horologiis-a of the Circle, and is both ways ter-
Joannes Georgius Schomberg, Exegeſis minated by the Circumference, and
Fundamentorum Gnomonicorum. does divide the Circle into two equal
Traité des HorologesSolaires, by Solomon Parts.
de Caus.-Joan.Bapt.Trolta's Praxis DIAMETER of a Curve, is a
Horologiorum. Defargues's Maniere right Line, as AC, that biffects
Univerſelle pour poſer l'Elieu & pla- the right Lines DE, DE, drawn
cer les Heures & autres choſes aux
Cadrans Solaires. Ath. Kircher's
А
Ars magna Lucis & Umbræ,- Ley-
bourn's Art of Dialing,--Ozanam's
B
Dialing. Hallum's Explicatio Ho D
E
rologii in Horto Regio Londini.-Trac-
tatus Horologiorum Joannis Mark.-
D
E
La Gnomonique ou l'Art de Tracer les
Cadrans, avec les Demonſtrations,
by Mr. de la Hire.- Wells's Art of
Shadows.
parallel to one another ; and are
DIAL (CYLINDRICAL,) is a either of a finite or infinite Length.
Dial upon the Convex Superfi Altho' a right Line biffecting all
cies of a Cylinder, where the parallel Lines drawn from one Point
Hour-Lines are Curves, drawn by of a Curve to another, is taken in a
means of the Sun's ſeveral Altitudes ſtrict ſenſe only for the Diameter of
every Day that he enters into the a Curve Line, yet it may not be a-
Beginnings of the Signs ; and the miſs more generally to define a Di-
Hour of the Day is ſhewn by the ameter, in ſaying, that it is that
Extremity of the Shadow of a Stile, Line, whether Right or Curve,
ſtanding at right Angles to the Sur- which biſſects all the Parallets drawn
face of theCylinder at the top thereof. from one Point to another of a
Curve
of
D JA
DI G
Curve ; fo that according to this, are allow'd for the Intercolumna-
every Curve will have a Diameter. tion.
And thence Sir Iſaac Newton's DIATESSARON, a Term in Mu-
Curves of the ſecond Order, have all fic, being otherwiſe called a perfeet
either a right-lin'd Diameter, or Fourth, and fignifies an_Interval,
elſe the Curves of ſome one of the conſiſting of one greater Tone, one
Conic Sections for Diameters. And leſſer, and one greater Semi-Tone,
many Geometrical Curves of the If the Tenfion of two Strings of
higher Orders, may alſo have for equal bigneſs be as 3 to 4, they
Diameters Curves of more inferiour will found a Diatefaron.
ones, and that ad infinitum.
DIATONIC, a Term ſignifying
DIAMETER CONJUGATE in the the ordinary ſort of Muſic, which
Ellipfis. See Conjugate Diameter. proceeds by different Tones, either
DIAMETER of Gravity, in any in aſcending or deſcending. It con-
Surface or Solid, is that Line in tains only the two greater and leſ-
which the Centre of Gravity is fer Tones, and the greater Semi-
placed.
Tone.
DIAMETER PRINCIPAL. See Diesis in Muſic, is the Diviſion
Principal Diameter,
of a Tone below a Semi-Tone, or
DIAMETEĘ TRANSVERSE. See an Interval compoſed of a leſſer and
Tranſverſe Diameter.
imperfect Semi-Tone. So that when
DIAMETRICALLY OPPOSITE, Semi-Tones are placed where there
is when two things are the moſt ought to be Tones, or when a
oppoqte to one another that they Tone is ſet where there ſhould
can be ; as one End of the Diame be only a Semi-Tone, this is called
ter of a Circle is to the other. Diefs.
DIAPASON, a Term in Muſic, Diesis (ENHARMONICAL,) is
being a Chord including all Tones, the difference between the greater
and is the ſame with what we call and lefler Semi-Tones.
an Eighth, or an Oitave, becauſe DIFFERENCE, is the Exceſs
there are but feven Tones, or Notes, whereby one Magnitude exceeds an-
and then the eighth is the fame other.
again as the firſt.
DifferENCE of Aſcenſion. See
If the Tenfion of two equal Aſcenſional Difference.
Strings be to each other, as, i to 2, DIFFERENCE of Longitude of two
their Tones will produce an Oétave. Places of the Earth, is an Arch of
DIAPENTĘ, or perfect Fifth, is the Equator contained betwcen the
the ſecond of the Concords making Meridians of thoſe two Places.
an Oétave with the Diate/aron. DIFFERENTIAL of any Quantity
If the Tenſion of two equal amounts to the ſame as the Fluxion
Strings be as 3 to 2, then they will of that Quantity. This Word is
found a Diapente.
not uſed by us.
DIAPHANOUS BODY, or Medi DIFFUSION, commonly ſignifies
um, is that through which the Rays the diſperſing of the ſubtle Exuvia
of Light freely paſs; as is Glais, of Bodies into a kind of Atino-
Air, Water, the Humours of the ſphere all round them.
DIGIT, in Aſtronomv, is the
D,1A STYLE, is a ſort of Edifice, twelfth Part of the Diameter of the
where the Pillars ftand at ſuch a Sun or Moon, and is uſed to expreſs
diſtance from one another, that the Quantity of an Eclipſe.
three Diameters of their thickneſs Dicits, or Monades, a Term in
Arth-
Eye, &c.
O 3
DIO
DIO
1
/
to-
Arithmetic, which fignifies any In- (Curve or right-lin'd, Concave or
teger under 10; as 1, 2, 3, 4, 5, 6, Convex, Spherical, or otherwiſe,
7, 8, 9.
and theſe greater or leſſer) of the
DILATATION, ſignifies a thing Glaſs or Water ; by which means
taking up more Space than it did the Objects ſeen thro' them, do, in
before.
appearance, alter their Magnitude,
DIMENSION, in Geometry, is Diſtance and Situation.
either Length, Breadth, or Thick The Ancients have treated of di-
neſs ; as, a Line hath one Dimen- rect and reflected Viſion ; but what
fion, viz. Length; a Superficies two, we have of reflected Viſion, is very
viz. Length and Breadth ; and a lame and imperfect. Joannes Bap-
Body or Solid has three, viz Length, tifla Porta, in a Treatiſe of Refrac-
Breadth, and Thickneſs. This Word tion, in nine Books, has endeavoured
is alſo uſed with regard to the Powers at rendring this Doctrine more per-
of the Roots of an Equation, which fect; but without any tolerable
are called the Dimenſions of that Root: Succeſs. The firſt who wrote
As in a cubic Equation the higheſt lerably well upon Dioptrics, was
Power has three Dimenſions. Kepler, who has demonſtrated the
DIMETIENT. The ſame with Properties of ſpherical Lens's very
Diameter.
accurately, in a Treatiſe firſt pub-
DIMINISHED ANGLE, a Term liſhed anno 1611.----After Kepler,
in Fortification. See Angle. Gallilæo has given ſomething of this
DIMINUTION, in Muſic, is no Doctrine in his Letters; as alſo the
thing elle but the abating ſomething Examination of the Preface of Jo-
of the full Value or Quantity of any hannes Pena upon Euclid's Optics,
concerning the Uſe of Optics in
DIOPTER, the ſame with the Aſtronomy. Deſcartes alſo pub-
Index or Alhidada of an Aſtrolabe, liſhed a Treatiſe of Dioptrics, com-
or ſuch-like inſtrument,
monly annexed to his Principles of
DIOPTRICS, is the Science of Philoſophy, wherein is the true Law
refracted Viſion; or it is that Part of Refraction found out by Snell;
of Cptics, which treats of the dif- but the Name of the Inventor ſup-
ferent Refiactions of Light, in its preſs'd, and the true Manner of
Paſage through different Nicdiums, Viſion more diſtinctly explain'd than
3.5 Air, Watci, Glaſs, E c.
by any before him. Herein is laid
Dioptrics is one of the moſt uſe- down the Properties of elliplical and
ful and pleaſant Sciences thac Man hyperbolical Glaffes, and the Praxis
ever had to do with, reſtoring of grinding Glaſſes.--Dr. Barrow
the biind to Sight with very little has treated of Dioptrics in a moft e-
eaſe, and at a very ſmall expence, legant manner, altho' ſomewhat too
bringing vallly remote Objects, as briefly, in his Optical Lectures, read
well as very ſmall oncs, witảin the formerly at Cambridge. There is
reach of the Eye, affording, both Mr Huygens's Dioptrics, a perfect
Pleaſure and Amazement, which on Work of its kind.. Molyneux's
therwiſe would never have been ſo Dioptrics, a heavy dull Piece, altho'
much as thought of; and all this by it may be uſeful to ſome. Mr.
means of the wonderful attractive Hartſoeker's French Eſſay of Diop-
Power in Glaſs and Water, cauſing trics.-Father Cherubin's Dioptrique
'the Rays of Light in their Paffage Oculaire, and La Vifion parfaite. -
thro' them to alter their Courle, Dr. David Gregory's Elements of
accordirg to the different Surfaces Dioptrics.- Traber's Nes vus Opti-
note,
er
CILS
!
DIR
DIR,
cus.-Zahr's Oculus Artificialis Tele- upon Planes, that directly face the
dioptricus. Dr. Smith's Optics, a Eaſt and Weſt, or are parallel to
compleat Work, of its kind. the Meridian of the Place.
Wolfius's Dioptrics, contain'd in his Theſe Dials Mew the Hour but
Elementa Matheſeos Univerſalis. from Sun-riſing to Noon, or from
DIPPING NEEDLE. If a mag- Noon to the Sun-ſetting; and the
netical Needle be duly poiſed about Hour-Lines are all parallel to one
an horizontal Axis, it will have a another, and at Diſtances from the
Direction of Altitude above the Ho- Hour-Line of fix, that are equal
rizon, beſides its Direction towards to the natural Tangents of the De-
the North, in an horizontal Pofi- grees in the ſeveral Hours.
tion, always pointing to a determi In theſe Dials the Style is paral.
nate Degree of Altitude or Eleva- lel to the Plane, ftands upon the
tion, above the Horizon, in this or Hour-Line of Six, and its Height or
that Place reſpectively. It is now Diſtance from the Plane is equal to
called a Dipping Needle. And Mr. the Diſtance of the Hour-Line of
Whiſton of late has endeavour'd to Nine, from the Hour-Line of Six, or
diſcover the Longitude by it.
to the Radius of the ſaid Line of
DIPTERON, in Architecture, a Tangents, being the Diſtances of the
Name which the Ancients attribu- Hour-Lines from the Hour-Line of
ted to thoſe Temples, which were Six.
encompaſſed with a double Row of It is very eaſy to draw one of
Pillars, making two Porticos, which theſe Dials for a given Latitude: For
they called Wings; but we com- having drawn the horizontal Line
monly call them iples.
AB, and the right Line AK from
DIRÉCT, in Aftronomy. A any Point A thereof,, making the
Planet is faid to be direct when it Angle BAK equal to the Comple-
goes forward by its proper Motion ment of the Latitude, with the
in the Zodiac, according to the Suc- Radius D E deſcribe a Circle, and
ceſſion of the Signs ; or when it ap- thro' the Centre D draw EC per-
pears ſo to do to an Obſerver ſtand- pendicular to AK; ſo that the
ing upon the Earth.
Circle may be divided into Quan
DIRECT Erect EAST and drants, and divide each of the
West DIALs, are Dials drawn Quadrants into fix equal Parts, and
altet
D
riin!
F
8
Hostel
E
I
$
B
Gi
of
from
DIR
DIR
from the Centre D to the Points of 4, 4. 5;5. 6,6. &c. to E C; and
Diviſion draw the right Lines D 4, theſe Parallels will be the Hour-
D
52
D6, D7, D8, D 9, D 10, Lines. A Weſt Dial is drawn after
Dú, and thro' the Points 4, 5, 6, the fame manner as appears in Fig.
7, 8, 9, 10, 11; draw Parallels
2. repreſenting a Weſt Dial.
L
OOTD
o
K
I
1
G
A
B
DIRECT ERECT South, or makes an Angle with the ſame, e-
NORTH DIALs, are Dials drawn qual to the Complement of the Ele-
upon Planes that direetly face the vation of the Pole; that of the
South or North, or are parallel to South Dial facing downwards, and
the prime vertical Circle, or to the that of the North upwards.
vertical Circle cutting the Horizon North Dials are but of little uſe ;
in the Eaſt and Weſt Points.
for from the time of the Autumnal
The Sun ſhines upon the South Equinox to the Vernal one, the
Dial of this kind, at the time of Sun does not ſhine upon them ; but
the Equinox, juft twelve Hours, or at the Vernal Equinox it begins to
from its Riſing to its Setting. For ſhine upon them, and as the Days
which reaſon there are twelve Hour- increaſe, it ſhines longer and longer.
Lines drawn upon it : But as the Some few Hours from its Riſing in
Days increaſe in Summer, the Sun the Morning, to a certain time be-
ſhines á leſs time upon them; that fore Noon, and from a certain time
is, he comes on the Dial after Six after Noon to its Setting, and the
in the Moming, and goes off it be- time after Six in the Morning of its
fore Six in the Evening; and the going off, will be equal to the
Proportion for finding the Time of time of its coming on after Six in
its coming on after Six, or going off the South-Dial, or any given Day
before Six, will be as Radius to the in Summer; and the time of its
Tangent of the Latitude, ſo is the coming on again in the Afternoon
Tangent of the Sun's Declination will be equal to the time of its go-
to the Sine of a certain Number of ing off on the South-Dial.
Degrees, which reduced into Time As the Radius is to thc Co-fine
will be that fought.
of the Latitude, ſo is the Tangent
The Style of theſe Dials ftands of the Angle, that any Hour-Line
upon the Hour-Line of twelve, and makes with the Hour-Line of
twelve
DIR
DIR
twelve, to the Tangent of the plane cal, AZPD the Meridian, PR
Angle, that that Hour-Line makes the Hour-Line of Six, and Axis of the
with the Hour-Line of twelve. World in a given Latitude BP, and
If A B be the Horizon, EF the RSP be any Hour-Circle ; then in
Equinoctial, DZ the prime Verti- the ſpherical Triangle QZ P right-
angled at 2, the Side Ze will re-
Z
preſent the right-lin'd Angle made
by that Hour-Line, with the up-
E
P right Meridian upon the Plane of a
South or North Dial ; ſo that to
find the ſeveral Hour-Angles, you
have given in that ſpherical Tri-
A
Bangle, the Angle ZPV, and the
V
Side ZP, the Complement of the
Latitude, to find the Side ZQ.
South or North Dials may be
drawn geometrically, thus : Draw
R
the upright Line A B for the Me-
ridian or Hour-Line of 12, and
D
S
F
A
high
d
H
YA
VITI
8
B
F
1/0 11 12
1
2
3
H
d
h
Ġ
C
B
16
A
taking
DIR
DIS,
taking any convenient Diſtance AC, If the Height of the Pole be
raiſe the indefinite Perpendicular greater than the Angle of Inclina-
CD, and make the Angle CAD tion, then the North Pole is ele-
equal to the Complement of the vated, and the Centre is below.
Latitude of the Place the Dial is If the Height of the Pole be lef-
made for; and at D make the An- ſer than the Angle of Inclination,
gle EDC equal to CAD, and then the South Pole is elevated, and
thro' E draw the right Line GH the Centre is above.
cutting the Line A 12 at right An In direct North Incliners the
gles. Make EB=ED, and with Sum of the Angles of Inclination
this as a Radius deſcribe a Quadrant and Elevation of the Pole, is the
of a Circle, and divide the ſame Height of the Style above the Plane,
into fix equal Parts, and thro' the or Angle that the Style makes with
Points of Diviſion draw the right the Plane.
Lines Ba, Bb, Bc, Bd, &c. to cut Inclining and Reclining Dials are
the Line GH; then right Lines not of much uſe, being only made
drawn from A thro' a, b, c, d, &c. for compleating a Body of Dials :
will be the Hour-Lines of 1, 2, 3, And after the Styles are rightly
4, 5. And if Ee, Ef, Eg, Eh, fixed, the beſt way of drawing the
be taken reſpectively equal to Ea, Hour-Lines upon them, if the Body
Eb, Ec, &c. and from A right be moveable, will be to get a good
Lines be drawn thro' e, f, g, h, &c. regular Dial firſt drawn upon the
theſe will be the Hour-Lines of 11, Body, and when the Sun ſhines
10, 9, 8, 7, and the Hour-Line of move it ſo, that the Shadow of the
6 will be perpendicular to A 12. Style ſucceſſively falls upon the
A North Dial is drawn exactly after Hour-Lines; for then if Lines are
the ſame manner, it being in reality drawn upon the Inclining and Re-
only a South Dial inverted, as ap- clining Planes of the Body, along
pears in the ad Figure.
the Shadows of their reſpective
Direct SOUTHWEST,North, Styles, they will be the ſame Hour-
or East RECLINERS, are thoſe Lines that the Shadow of the Style
Dials drawn upon Planes, which of the regular Dial fell upon. But
face any of the Cardinal Points of if the Body be not moveable, the
the Horizon.
Bufineſs muſt be done, by waiting
DIRECT SOUTH or NORTH till the Shadow of the Style of the
INCLINING DIALs,are ſuch whoſe Dial has gone over all the Hour-
Planes incline to the Horizon, and lie Lines,which may be done in one Day.
directly open to the South or North. DIRECT Ray, in Optics, is the
DIRECT SOUTH or NORTH Re- Ray proceeding from a Point of a
CLINING DI Als, are ſuch whoſe viſible Object, directly to the Eye,
Planes recline from the Zenith, and through one and the fame Medium.
lię directly open to the South or DIRECTION, a Term in Me-
Norih.
chanics, wherein, by the Line of
Theſe Dials are deſcribed after Direction,
Direction, is always meant the
the ſame manner as direct South Line of Motion, that any Body
Dials, the following Rule in placing goes in, according to the Force im-
the Style being only obſerved : In preſſed upon it.
South Incliners the Difference of DIRECTRIX, or Dirigent, a Term
the Angle of Inclination, and the in Geometry, fignifying the Line
Height of the Pole is the Height of Motion, along which the deſcri-
of the Style above the Plane. bent Line, or Surface, is carried in
the
1
DIS
DIV
the Geneſis of any Plane or ſolid nance, or thereabouts, fo that a
Figure.
Sight-line taken upon the top of
DiscoNTINUAL PROPORTION. the Baſe-Rings, againft the Touch-
See Diſcrete Proportion.
hole, by the Mark ſet on or near
DISCORDS, in Muſic, are certain the Muzzle, may be parallel to the
Intervals of Sounds, which being Axis of the Concavity of the Piece.
heard at the ſame time are unplea This is commonly done, by tak-
fant to the Ear; and theſe are the ing the two Diameters of the Bafe-
ſecond, fourth, and feventh, with Ring, and of the place where the
their Otaves, that is, all Intervals, Diſpart is to ſtand, and dividing the
but thoſe few that exactly terminate Difference between them into two
the Concords, are Diſcords. equal Parts, one of which will be
Notwithſtanding Diſcords found the Length of the Diſpart, which
unpleaſant, when heard by them - is ſet on the Gun with Wax or Pitch,
ſelves, yet being artfully mixed with DISSEMINATE VACUUM. See
Concords, they make the beſt Mu- Vacuum.
fic: And of all the Diſcords a fe DISSONANCE, in Muſic, is a dif-
cond is the moſt unpleaſant. agreeable Interval between two
Discrete (or Disjunct) PRO- Tones, which, being continued to-
PORTION, is when the Ratio of gether, offend the Ear.
two or more Pairs of Numbers or DISTANCE, in Navigation, is
Quantities is the ſame, but not con the Number of Degrees or Leagues,
tinual, that is, when the Ratio of &c. that a Ship has failed from any
the Confequent of one Pair of Num- given Place or Point.
bers, or Quantities, to the Antece DISTANCE of the Eye, in Per-
dent of the next Pair, is not the ſpective, is a Line drawn from the
fame, as of the Antecedent of one Foot of the Altitude of the Eye to
Pair to its Conſequent ; as 3:6 :: the Point, where a Line drawn at
8:16. are diſcrete Proportionals; right Angles to it will interſect the
becauſe the Ratio of 3 to 6 is equal Object.
to the Ratio of 8 to 16. But the DISTANCE of the Baſtions, in
Ratio of 3 to 6, or 8 to 16, is not Fortification, is the side of the exte-
the ſame as of 6 to 8.
rior Polygon.
DISCRETE QUANTITY, is ſuch DISTINCT Base, in Optics, is
as is not continuous, and joined to- that Diſtance from the Pole of a
gether; as Numbers, whoſe Parts Convex Glaſs, in which Objects be-
being diftinct Units cannot be unit- held through it appear diftinétly,
ed into one Continuum ; for in a and well defined, and is what is 0-
Continuum there are no actual de- therwiſe called the Focus.
terminate Parts before Diviſion ; DistincT VISION. See Vifion.
but they are potentially infinite. Dirone, a double Tone, or the
DISDIAPASON, a Term in Muſic, greater Third, is an Interval in Mu-
being a double eighth or fifteenth. fic, which comprehends two Tones.
Disk of the Moon, or any Planet, If the Tenfion of two equal
is the Circle made by cutting it Strings be as 4 to 5, or as 5 to 6,
thro' the Centre by a Plane perpen. they will found a Ditone,' or a Se-
dicular to a Line drawn from the mi-ditone.
Earth or Sun.
Divergent POINT. See Ver-
Dispart, a Term in Gunnery, tical Focus.
ſignifying the ſetting a Mark upon DIVERGENT (or Diverging)
the Muzzle Ring of a Piece of Ord- RAYS, in Optics, are thoſe Rays,
that,
DI V
DI V
that, iſſuing from a Point of a vi- in a long time, and yet continually
fible object, are diſperſed, and con- fill a very large Space with qdoria
ţinually depart from one another, ferous Particles.
according as they are removed from
Dr. Keil in his Vera Phyfica,
the Object.
Ļect. 5. has been at the pains to
DIVERGING PARABOLA. See calculate the Magnitude of a Par-
Parabola Diverging:
ticle of Afa Fætida, which will be
Dividend, in Arithmetic, is the rooftog of a Cubic
Number that is to be divided into Inch. And in the fame Lecture he
equal Parts by another Number. ſhews, that the Particles of the
DIVISIBILITY, is that Diſpo- Blood in the Animalculæ, that are
ſition of a Body, whereby it is con- obſerved in Fluids by means of Mi-
ceived to have Parts, into which it croſcopes, muſt be leſs than that
may actually or mentally be di- Part of a Cubic Inch which is ex-
vided.
preſſed by a Fraction, whoſe Nume-
Body is diviſible in infinitum ; rator is 8, and Denominator Unity
that is, you cannot conceive any with thirty Cyphers after it.
Part of its Extenſion, ever ſo ſmall, DIVISION, one of the four Rules
but that ſtill there may be a ſmal- of Arithmetic, is the finding of a
ler.
Number or Quantity ſuch, from
There are no ſuch Things as two given Numbers or Quantities,
Parts infinitely ſmall; but yet the that it ſhall be to one of the Num-
Subtility of the Parts of ſeveral Bo-bers or Quantities, as Unity is to
dies is ſuch, that they very much the other.
furpaſs our Conception. And there Division of Numbers, is only a
are innumerable Inſtances in Nature compendious Subtraction; for ſince
of ſuch Parts, that are actually ſe- the Diviſor is ſo many times con-
parated one from another.
tained in the Dividend as there are
1. Mr. Boyle mentions a filken Units in the Quotient, therefore con-
Thread, that was three hundred tinually ſubtracting the Diviſor
Yards long, which
weighed but two from the Dividend, and accounting
Grains and a half.
an Unit for each Time, the Sum
2. He alſo ſaid, that fifty ſquare of theſe Units is the Quotient.
Inches of Leaf Gold weighed but 1. One whole Number may be
one Grain. Now, if an Inch in divided by another, by the follow-
Length be divided into two hundred ing Rule: 1. Set a Point under the
Parts, the Eye may diſtinguiſh them lait of the Left-hand Places in the
all. Therefore, in one ſquare Inch Dividend, out of which the Diviſor
there are forty thouſand viſible Parts; may be taken, and the Number of
and in one Grain of Gold there are Places in the Dividend to the right
two Millions of ſuch Parts; which of that point incluſive gives the
may be yet further divided.
Number of Places of the Quotient;
3. A whole Ounce of Silver may as if 1096 825 were to be di-
of ,
which is afterwards drawn out into vided by 365. I ſet a Point under
6, and not 9; becauſe I cannot get
a Wire of 1300 Foot long.
365 in 109
But in 1096, 1
4. In odoriferous Bodies we can
still perceive a greater Subtility of may: And ſo the Quotient will con-
Parts, which are ſeparated from one
fiſt of four Figures. Hence there
another, for ſeveral Bodies ſcarce loſe are three Figures 825 to the right
any ſenſible Part of their Weight take the Diviſor 365 out of the
2. Try how often you can
firſt
D I V
DI V
firſt Part ( 1096 ) of the Dividend, which is always perform'd without
which will be always leſs than ten the Cyphers prefix'd to the Diviſor
times; and ſet the Number of times and Dividend. If the Dividend has
3 in the Quotient, then multiply the not fignificant Figures enough for
Diviſor thereby and ſubtract the to be divided by the Diviſor, or if
Product 1095 out of the ſaid Part after the Diviſion there be a Re-
1096
of the Dividend,
and ſet down mainder, you may proceed to what
Remainder. 3. To the right Degree of Exactneſs you pleaſe, by
of the Remainder ſet down the annexing Cyphers to the Right-hand.
next Figure of the Dividend, from - The Value of the Quotient after
which take the Diviſor as often as the Diviſion is ended may be found
you can, ſetting down the Number by this Rule, as well as that before
of times in the Quotient, multiply laid down,
Conſider how many
the Diviſor thereby and ſubduct the Decimal Places there are in the Di-
Product as before; and in this man- vidend, for ſo many muſt there be
ner the Operation muſt be repeated in the Quotient as the Dividend
to the end. 4. If the Divifor has has more than the Diviſor, and to
Cyphers towards the Right-hand, cauſe this, a Cypher muſt oftentimes
cut off ſo many of the Right-hand be prefix'd.
Places of the Dividend as there are 3. Vulgar Fractions are divided
Cyphers in the Diviſor, which an- by the following Rule. Multiply
nex to the Remainder when the O. the Numerator of the Dividend by
peration is finish'd.
the Denominator of the Divifor,
2. Diviſion of Decimal Fractions and the Product is the Numerator
is the ſame, as in whole Numbers ; of the fractional Quotient; and
but in finding out the true value of then inultiply the Denominator of
the Quotient, it is to be obſerved, the Dividend by the Numerator of
that the Diviſor being placed under the Diviſor, and the Product is the
the Dividual, the Figure anſwering Denominator of the fractional Quoc
it in the Quotient, mut always be tient.
in a like place with that Figure in To divide one Fraction by ano-
the Dividend, which is over the ther, is by the Nature of Diviſion
Unit's Place of the Diviſor ; as if to find how often the Diviſor, that
.0006528 were to be divided by is, how often ſuch a part of its
.032. If .032 be placed under the Numerator as is exprefled by the
firit Dividual :00065, it appears Denominator, is contain'd in the
.0006528
Dividend. In dividing any proper
thus,
And the ſecond
.032
Fractions by one another, the Di-
Decimal Place in the Dividend, ſtànds vidend being really the Product of
over the Place of Units in the Di. the Diviſor, and Quotient multi-
viſor; wherefore the firſt Figure 2 plied together, will be leſs than ei-
in the Quotient, muſt be in the fe- ther of them, when the Quotient
cond Decimal Place, and ſo the is a proper Fraction ; or when any
firit Place is to be ſupply'd with Fraction or whole Number is divid-
a Cypher. See the Operation ed by a proper Fraction, the Quo-
.032) .0006528 1.0204 tient will always be greater than the
Dividend.
64
4. Algebraic Diviſion is perform-
ed by taking to pieces what has
I 28
been compounded by Multiplica-
O
tion; as ab divided by a gives b
for
I 28
DIV
DI V
a
for the Quotient; bab divided by
at 4*
But when the Divifor
3b, gives 2a for the Quotient ;
16a8c3 divided by 2 ac, gives alſo conſiſts of ſeveral Terms, the
8bcc for the Quotient. But if the Divifion is performed as in Num-
Quantity to be divided cannot be bers
, in order to rightly
perform
thus reſolved by Diviſion, it is er which, the Terms of the Quantity
nough, when both the Quantities to be divided, as well as of the Di-
are not Fractions, to ſet down the vidend, ought to be orderly diſpoſed
Divifor underneath, with a fhórt
Line between them; thus ab di- . Letter, which is thought moſt con-
according to the Dimenfions of ſome.
venient for this purpoſe; fo that'
But when
vided by c, will be an
thoſe ſtand in the firſt Place in
the Quantities are Fractions, they which that Letter is of the moſt
are divided like vulgar Fractions, as Dimenfions; and thoſe in the fe-
cond, in which the Dimenſions of
e divided by , will be 60. it are neareſt to the greateſt ; and
ſo on to thoſe Terms which are not
If a Quantity to be divided be, at all multiplied by that Letter,
compounded of ſeveral Terms, its and ſo are to be laſt of all; as if
Divifion is performed by applying a3 to 2 aac - aab 3 abc to bbc,
each of its Terms to the Diviſor;
were to be divided by qob; it
as aat-4axmxx divided by a gives would ſtand thus :
t-zaac
ab
-6) aab --- 3 abc+-bbc ( au +2ac — be
aab
ad
1
al
a3
O + zaac-3abc
2aac2abc
O-abc-t-bbc
--abc+bbc
O
O
Or thus - 6+a) (66 = 2.20
3ac ta3
obtzar
(-1
tomaac
taa
c bb ach
2acc
+
- aa , te
b
+zaac
zac , tozaac
6
ta3
aa
1
Some begin Algebraic Diviſion Quotient will be an affirmative one :
from the laſt Terms ; but it comes ſo alſo, when a negative Quantity
to the ſame thing, if the Diviſion is divided by a negative one, the
be performed ſucceſſively. Note Quotient will be an affirmative
alſo, when an affirmative Quantity Quantity. And when an affirma-
is divided by an affirmative one, the tive Quantity is divided by a nega-
tive
1
1
DO D
DOM
tive one, or a negative Quantity by by J, of the Diſtance of that Face
an affirmative Quantity, the Quo- from the Centre of the Dodecahedron,
tient will be a negative Quantity.
which is the ſame as the Centre of
Division of Proportion. If four the circumſcribing Sphere.
Quantities be proportional, as a : The Side of a Dodecahedron, in-
b::c:d. then the Affumption of ſcribed in a Sphere, is the
greater
the Difference between the Antece- Part of the side of a Cube, inſcrib:
dents (amb, or b–a) to either the ed in that Sphere, cut into extream
Antecedent (an) or Conſequent (b) and mean Proportion.
of the firſt Ratio (a to b;) and the If the Diameter of the Sphere be
Difference between the Antecedents 10000, the side of a Dodecahedron, in
(c-d, or dac) to either the Ante- ſcribed in it, will be .35682 nearly,
cedent (6) or Conſequent (d) of the All Dodecahedrons are ſimilar, and
ſecond Ratio c to d, is called Divi are to one another as the Cubes of
fion of Proportion.
their Sides; and their Surfaces are
Divisor, in Arithmetic, is the alſo fimilar, and therefore they are
Number that divides another, or as the Squares of their Sides ;
that which ſhews into how many whence, as .509282 is to 10.51462,
Parts the Dividend is to be di- ſo is the Square of the Side of any
vided.
Dodecahedron to the Superficies
DIURNAL ARCH, is that Arch thereof; and as .3637 to 2.78516,
that the Sun, Moon, or Stars de- fo is the Cube of the side of any
ſcribe between their Rifing and Dodecahedron to the Solidity of it.
Setting.
DODECATEMORY. The twelve
DIURNAL MOTION of a Planet, Signs of the Zodiac, Aries, Taurus,
is ſo many Degrees and Minutes, &c. are ſo called, becauſe each of
&c. as any Planet moves in twenty- them is the twelfth Part of the
four Hours. And the Motion of Zodiac.
the Earth about its Axis is called its Dome, is a round, vaulted, or
Diurnal Motion.
arched Roof of a Church, or any
DIURNAL PARALLAX. See Pa- great Building.
rallax.
DOMINICAL LETTER, one of
DODECAGON, a regular Polygon, the firſt ſeven Letters of the Alpha-
confifting of twelve equal Sides and bet; wherewith the Sundays are
Angles; and in Fortification it is mark'd through the Year in the
a Place, with twelve Baftions. Almanack.
If the Radius of a Circle, in which If any given Year be added to
the Dodecagon is inſcribed, be=1, one fourth Part of it, omitting
then the side of the Dodecagon will Fractions, and you add 4 to the
be nearly .654. And as I is to the Sum, and divide the whole by 7,
Square of the side of any given and then ſubtract 7 from the Re-
Dodecagon, fo is 2.51956 to the A- mainder, this laſt Remainder ſhews
rea of it nearly.
the Order of the Dominical Letter
DODECAHEDRON, is one of the for chaç Year in the Alphabet : For
Platonic Bodies, or five regular So- Example į
lids, and is contained under twelve
In the Year
1725
equal and regular Pentagons,
The fourth Part is omit-
The Solidity of a Dodecahedron is ting Fractions,
431
found by multiplying the Area of To both which add
4
one of the Pentagonal Faces of it
by 12; and then this latter Product The Sum is
A
*}
2106
Which
DOU
D UP
1
cant.
Which divided by 7, leaves 4, and DOUCINE, in Architecture, is an
4 taken from 7, leaves 3 ;. where- Ornament of the higheſt Part of the
fore the Dominical Letter is C for Cornice, or a Moulding cut in fi-
that Year.
gure of a Wave, half Convex, and
DONJON, in Fortification, com- half Concave.
monly ſignifies a large Tower, or DOVETAILING, in Architec-
Redoubt of a Fortreſs ; from whence ture, is the way of faſtening of
the Garriſon may retreat in caſe of Boards or Timber together, by lec-
Neceflity, and capitulate with good ting of one Piece into another in-
Advantage.
dently, with a Dove-Tail Joint, or
DORIC ORDER of Architecture, with a Joint in figure of a Dove's
is the ſecond Order, and the moft Tail.
agreeable to Nature, having no Or Draco, a Conſtellation in the
naments on its Baſe, nor its Capi- Northern Hemiſphere; conſiſting
tal. Its Column is eight Diameters of thirty-three Stars.
high, and its Freeze is divided be DRAGON'S HEAD and TAIL,
tween Triglyphs and Metopes. are the Nodes of the Moon. See
This Order, which repreſents So- Nodes.
lidity, ought not to be uſed but in DRAGON-BEAMS, in Architec-
great and maſſy Buildings, as the ture, are two ſtrong Braces or Struts,
Outſides of Churches and public which ſtand under a Breaſt-Summer,
Places,
and meet in an Angle on the Shoul-
DOUBLE DESCANT. See Def- der of the Key-piece.
DRAUGHT COMPASSES , are
DOUBLÉ HORIZONTAL DIAL, Compaſſes with ſeveral moveable
is a horizontal Dial of Mr. Ough. Points, to draw fine Draughts in
tred's, with a double Gnomon; one Architecture, &c.
to ſhew the Hour on the outward DRAUGHT Hooks, are large
Circle, and the other to ſhew the Hooks fix'd on the Cheeks of a com-
Hoor on the Stereographic Projec- mon Carriage, two on each ſide,
tion drawn upon it. This finds the one near the Trunion-Hole, and the
Meridian, Hour, the Sun's Place, other at the Train.
Riſing, Setting, &c. and many other DRAW-BRIDGE, is a Bridge
Propofitions of the Globe.
made to draw up, or let down, as
DOUBLING the Cape, or a Point occaſion ſeryes, before the Gate of
of Land, in Navigation, is to come a Town or Caſtle: And they are
up with it, paſs by it, and ſo to made after ſeveral Faſhions ; but
leave it behind the Ship.
the moſt common are made with
DOUBLE, or FLANK'DTENAIL- Plyers, twice the Length of the
See Tenaille.
Gate, and a Foot in Diameter. The
Double Point, in Geometry, is inner Square is travers'd with a
one Point conſider'd as two infinitely Croſs, which ſerves for a Counter-
near ones, belonging to Geometri- Poiſe; and the Chains that hang
cal Curve Lines; or it is an infinite- from the other Extremities of the
ly ſmall Oval, whoſe bounding Line Plyers, to lift up, or let down the
is become ſo extremely ſmall, as to Bridge, are of Braſs or Iron.
be taken for two Points, diſtant DRIP, in Architecture. See
from each other every way by an Larmier.
infinitely ſmall Space; and in the Dry Moat. See Moat.
Ellipfis the following Equation will DUPLICATE PROPORTION, or
expreſs a double Point, viz. yy= RATio, is a Ratio compounded of
**+ 2ax
two
LE.
ga.
D UP
E A R
two Ratio's; as, the duplicate Ra DURABLE FORTIFICATION.
tio of a to b is the Ratio of aa to See Fortification.
bb, or of the Square of a to the DURATION, is the Idea we have
Square of b
of the Continuation of the Exiſtence
If three Quantities are in conti- of any thing.
nual Proportion, the firſt is to the
DIALLING.
See Dial.
third in the duplicate Ratio of the Dye, or Die, in Architecture,
firſt to the ſecond; or as the Square is any ſquare Body, as the Trunk,
of the firſt to the Square of the ſe- or notch'd Part of a Pedeſtal, being
cond.
that Part included between the Baſe
DUPLICATION; is the doubling and the Cornice.
of any thing
DYPTERE, or Diptere, in the
DUPLICATION of a Cubic, is to antient Architecture, was a kind of
find the Side of a Cube that ſhall be Temple, encompaſſed round with a
double in Solidity to a given Cube, double Row of Columns; and the
Several have attempted to do this Pſeudo-Diptere, or falſe Diptere,
geometrically; but it is in vain to was the ſame, only this was encom-
pretend to it, for it cannot be done paſſed with a ſingle Row of Columns,
without the Solution of a cubic E. inftead of a double Row.
quation; and fro a conic Section, or
ſome higher Curve, muſt be uſed
for determining the Problem.
The Solution of this problem de-
E.
pends upon finding two mean Pro-
portionals between two given Lines. ARTH. This Body of Land
For if the side of a given Cube be
and Water, whereon we dwell.
a, and the Side of a double Cube be Various have been, and now are
= y, then will 2 a3 =y3; or put the Opinions concerning the Shape
= 2 a, it will be a ab=33; of the Earth, by ſuch who are igno-
therefore it will be aa : yyo: y:b; rant of Geography. That of the
common People is, that it is a vaſtly
y y
or making z=
it will be a : % extended Plane, having a bottomleſs
Foundation. And of this Opinion
y:b. So that theſe four Quan- were Laftantius (in Lib. 3. C, 24)
tities will be continual Proportionals: and St. Auguſtine (in Lib. 16. De
conſequently ys the side of the Cube Civitate Dei,) and ſeveral other of
fought, is the ſecond of two mean the antient Fathers, and leſs-know-
Proportionals hetween a and b.
ing Philoſophers. Concerning the
This Problem of doubling the latter of which, ſee Ariſtocle's Book
Cube, formerly was propoſed by De cælo, Lib. 2. cap. 13. It is nog
the Oracle at Delphos, to the Inha- known who was the firſt that al-
bitants of that Ifand, who went to ferted, that the figure of the Earth
aſk what was to be done, to cauſe was ſpherical: but this we may be
the Plague then raging amongſt them ſure, that the Doctrine is very an-
to ceaſe? The Oracle made anſwer, tient, becauſe it the taking of Baby-
that before this could happen they lon by Alexander the Great, Ecliples
mult double the Altar, which was a were ſet down and computed for
Cube. Sce Valerius Maximus
, Lib. many Years bufore the Nativity of
8. alſo Eurocius's Commentary on Chrift, which without the Know-
Lib. 2. Archimedes De Sphara & Cy. ledge of the ſpherical Figure of the
lindre.
Earth could not have been done: it
P
being
ting b
É AR
E A R
being evident that Thales the Gre- ſetting of the Sun and Stars in every
cian was ſufficiently acquainted with Latitude, are agreeable to the Suppo-
this, becauſe he predicted an Eclipſe fition of the Earth's being ſpherical:
of the Sun.
All which could not be ſo, if the
1. That the Figure of the Earth Earth were of any other Figure.
is nearly ſpherical, is ſufficiently Moreover, when one ſtands upon the
confirmed from Eclipſes; eſpecially Shore, and fees a Ship afar off under
thoſe of the Moon, which are cauſed fail, making towards the Land; at
by the Shadow of the Earth falling firſt we ſee only the Topfails or
upon the Moon. And ſince this higheſt Parts, and at the ſame time
Shadow always appears circular, do manifeſtly behold the Convex
whether it falls to the Eaſt, Weft, Surface of the Sea interpoſed be-
or South, and its Diameter greater tween our Sight and the Hull or
or leſs, according as the Moon is lower Parts of the Ship, till ſhe
more or leſs diſtant from the Earth; approaches nearer, and this uni-
it is evident from Optics, that the formly every way alike, and pro-
Figure of the Earth is, in Appear- portionably to the ſeveral Diſtances ;
ance at leaſt, ſpherical. Alſo E- which is an evident Proof of the
clipſes of the Sun, which are cauſed Roundneſs of the Sea.--Laſtly, the
by the Interpoſition of the Moon Roundneſs of the Earth moſt mani-
between the Sun and thoſe Places feſtly appears from the Voyages of
where it appears eclipſed : I ſay it.. ſeveral Perſons of theſe latter Ages,
could not be determind when, and who have fail'd quite about the
in what Places ſuch Eclipſes ſhould ſame. For firſt of all Ferdinand
appear, and where not, if the Earth's Magellan, anno 1519, in 1124 Days;
Figure where unknown. And be- Francis Drake, an Engliſhman, anno
cauſe the Places where ſuch Eclipſes 1577, in 1056 Days; Thomas Can-
happen, and where not, are deter- diſh, another Engliſhman, anno 1586,
mind upon the Suppoſition of the in 777 Days; Simon Cordes, a Dutch-
Earth's Surface being ſpherical ; it man, anno 1590; Oliver Noort, an-
is evident that the fanie is ſpherical. other Dutchman, anno 1598, in 1077
-The ſpherical Figure of the Earth Days; William-Cornelius Schouteen,
is evinced alſo from the riſing and a third Dutchman, anno 1615, in
ſetting of the Sun, Moon, and Stars; Days; James Heremetes and john
which happen ſooner to thoſe who Huygens, anno 1623, in 802 Days,
live to the Eaſt, and later to thoſe conſtantly continuing their Courſe
living Weſtwardly: and that more or Wefterly, return'd again to Europe
leſs fo, according to the Roundneſs Eaſterly, obſerving all the way every
of the Earth.--So alſo going or fail. Phænomenon conſequent from the
ing to the Northward, the North Roundneſs of the Earth. -Altho*
Pole and northern Stars become the Surface of the Earth or Sea is
more elevated, and the South Pole said to make but one continued
and fouthern Stars more depreſs’d; Round, yet this, in reality, is not
the Elevation Northerly increaſing to be ſo itrictly taken, as to have no
equally with the Depreſſion Sou- Inequality in it; but as a Ball, tho*
therly; and either of them propor- it has ſome Duft or ſmall Grains of
tionably to the Diſtances.gone. The Sand upon it, may ſtill be ſaid to be
ſame thing happens in going to the round; fo tho' the Land, Hills, and
Southward. Beſides, the oblique Mountains be fomewhat raiſed above
Afcenfions, Defcenfions, Emerſions, the ſpherical Surface of the Sea, and
and Amplitudes of the riſing and fome Valleys depreſs'd below it, yet
becaufe
:
and
1
É À Ř
É A Ř
becauſe the greateſt of theſe Inequa- De Caufa Gravitatis, p. 154
lities has ſcarcely any ſenſible Ratio foll. wherein he makes the Ratio of
to the whole, the whole may well the polar Diameter to that of the
be affirm'd to be round.
Equator, as 571 to 835$ and Sit
It is not many years ſince the Ifaac Newton's Princip. Phil. Nat.
true Figure of the Earth has been Mathem. Lib. 3. where that Ratio
diſcovered; for ever before it was in the firſt Edition is as 689 to 692.
taken by Mathematicians and Geo-- See alſo a late Treatiſe, entitled
graphers as perfectly Spherical, ex- The Meaſure of the Earth, by ſeveral
cepting the ſmall Inequalities in its Frenchmen feñit to the North to mea-
Surface of Mountains, Valleys, &c. ſure the Earth, by order of the
But now it is evident, that the Fi- King of France, chiefly occaſioned
gure of the Earth is an oblate Sphe- by the Opinion of Mr. Caffini, who
roid, form'd by the Rotation of an would have the Figure of the Earth
Ellipſis about its leffer Axis. So to be a prolate or egg-form Sphe-
that thoſe Diameters are longeſt of roid, the Axis being longër than a
all belonging to the Circle between Diameter of the Equator.
the Middle of the Poles, or the E 2. On Suppoſition that the Sun's
quator; and thoſe more remote from Parallax be thirty-two Seconds, the
it, are ſhorter, till you come to the Earth's mean Diſtance from the Sun
Axis, joining the Poles of the Earth, will be $4,000,000 Miles. But Sir
which is the ſhorteſt of all. What Ifaac Newton takes the apparent
gave
the firſt Occafion to the Know- Diameter of the Earth from thë
ledge of this, was the Obſervations Sun to be twenty-four Seconds; and
of ſeveral Frenchmen in the Eaſt- In- ſo the Sun's Parallax twelve Seconds;
dies, about 70 Years ago, (ſee the and if ſo, the Sun's Diſtance will be
Hiſtory of the Royal Academy of Sci- much greater.
ences, by Mr. Du Hamel, p. 110, 3. Since the Earth is of a prolate
156, 206. and L'Hiftoire de l'Acad. ſpheroidal Figure, ſwelling out to
Roy. 1700, 1701.) who found that wards the Equator, and flatted or con-
Pendulums, the nearer they came to tracted towards the Poles ; ſo as the
the Equator, perform'd their Vibra- Diameter of it, at the Equator, is
tions Tlower. From whence it fol- longer than the Axis by about thirty-
lows, that the Velocity of the De- four Miles; upon this Account, there
ſcent of Bodies, or Gravity, is leſs ariſes a ſmall Inequality in the Mag-
in the Countries near the Equator nitude of a Degree of Latitude; for
than thoſe near the Poles. And this they increaſe from the Equator to
ſet Sir Iſaac Newton and Mr. Huy- the Poles by nearly the eight hun-
gens to work, to find out the Cauſe; dredth Part. But this Difference
which, they ſay, is the Revolution of Increaſe is ſo very ſmall, that in
of the Earth about its Axis: for meaſuring Degrees by Inſtruments,
ſince it moves much ſwifter at the it cannot be diſcover'd. Hence it
Equator than at the Poles, the Di- alſo follows, that heavy Bodies do
minution of the Weight of Bodies not tend directly to the Earth's
there, muſt be found greater than Centre, unleſs at the Poles and E-
near the Poles; and fo thoſe Parts quator, but every where perpendi-
of the Sea, ſituate near the Equator, cularly to the Surface of the Sphe..
being by this Cauſe made lighter, roid.
are chrown up to a greater height.
4. Diogenes Laertius fays, that
See this curious Subject fully handled Anaximander a Scholar of Thales,
by Mr. Huygens, in his Diſcourſe who lived about 550 Years before
the
P 2
E A R
E AR
+
.
the Birth of Chriſt, was the firſt who found out by thoſe who went before
gave an account of the Circumference him.-Snell relates from the Ara-
of the Sea and Land.--And his Mea- bian Geographer Abelfedea, who
fure thereof ſeems to be uſed by the lived about the gooth Year of
fucceeding Mathematicians, till the Chriſt, that about the 800th Year.
time of Eratoſthenes. Ariſtotle, at the of Chriſt, Maimon, an Arabian King,
end of Lib. 2. De cælo, ſays the having got together ſome ſkilful
Mathematicians, who have attempted Mathematicians, conimanded them
to meaſure the Circuit of the Earth, to find out the Circumference of the
make it 40000 Stadiums; and this Earth. And theſe accordingly made
is thought to be that of Anaximan- choiſe of the Fields of Meſopotamia,
der.--The next after Anaximander, wherein they meaſured under the
who undertook this Buſineſs, was fame Meridian from North to South,
Eratoſthenes, who lived about 200 until the Pole became one Degree
Years before Chriſt
. He makes the depreſs’d. And that Meaſure they
Circuitof the Earth to be 250000 found to be 56 or 56 Miles and a
(ſome ſay 252000) Stadiums, which half: and ſo according to them the
Pliny makes to be 31500 Roman Circumference of the Earth is 20160
Miles, each of which is reckon'd to or 20340 Miles.-It was a long time
be 10.o Paffes. He perform’d the after before any body elſe try'd to
thing by taking the Sun's Zenith perform this Buſineſs: but at length,
Diſtance, and meaſuring the Diſtance Snell, a Profeſſor of Mathematics
between two Places under the ſame at Leyden, in Holland, about 120
Meridian, as Cleomedes relates. But Years ago began again to ſet about
This Dimenſion was taken by many this Work, who with a great deal
of the ancient Mathematicians to be of Skill and Labour, by meaſuring
falſe; and chiefly Hipparchus, who large Diſtances under the ſame Pa-
lived 100 Years afterwards, and rallel, ſound one Degree to be 28500
added 25000 Stadiums to Eratoſthe. Perches, each of which is 12 Rhind-
mes's Circuit: but for what reaſon is land Feet, or 19 Dutch Miles, and
not known.--The next who mea- the whole Periphery 6840 Miles ; a
ſured the Earth was Poſidonius, who Mile being, according to him, 1500
lived in the time of Cicero and Perches or 18000 Rbindland Feet.
Pompey the Great; he makes the See more in his Treatiſe, called E-
Circumference to be 240000, (ac- ratoſthenes Batavus. -The next
cording to Cleomedes,) but 18coco Modern, who undertook this Mea-
Stadiums (according to Strabo.) He ſurement, was our Countryman Ri-
did it by the Altitudes of a Star, ckard Norwood, who in the Year
and meaſuring a Diſtance under the 1635, by meaſuring the Diſtance
ſame Meridian. Ptoleniy, in, his from London to York with a Chain,
Geogr. {ays, that Marinus, a cele- and taking the Sun's Meridian Alti-
brated Geographer, attempted ſome- tude, the 11th of June, with a Sex-
thing of this kind ; and likewiſe in tant of above five Feet Radius, found
.Lib. I. cap. 3. mentions himſelf as a Degree to contain 367205 Feet,
having try'd to do the thing after a or 69 Miles and a half and 14 Poles ;
different way from any body before and thence the Circumference of a
him, which was from Places under great Circle of the Earth is a little
different Meridians; but does not above 25036 Miles, and the Diame-
ſay how much he found it to be: for ter a little more than 7966 Miles.
hé till made uſe of the Number of And this Meaſure is allow'd by every
180000 Stadiums, which had been one to be as exact as any whatever.
See
E A R
ECL
See the Particulars of the whole Perihelium is in the Month of Dec
Affair in his Seaman's Practice. cember, viz. about the third or
The Meaſurement of the Earth fourth Day.
by Snell, tho' very troubleſome and EARTH-BAGS, in Fortification
ingenious, and much more accurate are the ſame with Canvaſs-Bags.
than any of the Ancients, being Which fee.
thought by ſome of the French, in EAVES-LATH, in Architecture,
the Reign of Lewis XIV. to be ſub- is a thick feather-edged Board, nail'd
ject to ſome ſmall Errors, the Affair round the Eaves of a Houſe for the
was renew'd, after Snell's way, by lowermoſt Tiles, Slates, &c. to reit
Mr. Picart and other Mathema- upon.
ticians, by the French King's Com EBBING and FLOWING of the
mand; they uſing for that Purpoſe Sea. See Tides.
a Quadrant of 3 French Feet Ra ECHO, is a Repetition of Sound,
dius, and found a Degree to contain cauſed by Reflection.
342360 French Feet. See Mr. Pi ECHINUS, from the Greek Echi
cart's Treatiſe, entitled La Meſure nos, the Shell of a Cheſnut, com-
de la Terre.-Mr. Caffini the younger, monly fignifies that Part of the
in the year 1700, by the French Quarter-Round which includes the
King's Command too, went about Ovum, or Egg, and ſometimes the
this Buſineſs, with a Quadrant of Quarter-Round itſelf.
io French Feet Radius for taking the ECLIPSE, is a Deprivation of the
Latitude, and another of 3. Feet Light of the Sun, or ſome Heavenly
for taking the Angles of the Trian- Body, by the Interpoſition of an-
gle: And found a Degree, from his other Heavenly Body between our
Calculation, to contain 2836. Toiſes, Sight and it. As an Eclipſe of the
or 69172.o Engliſh Miles. And this Sun is the Deprivation of its Light,
Meaſurement being perform'd with cauſed by the Interpoſition of the
all the Care and Exactneſs poſſible, Body of the Moon, between our
muſt be look'd upon as very near Sight and the Sun. An Eclipſe of
the Truth; and differs from our the Moon is the Deprivation of her
Norwood's only 8 Toiſes. See the Light, cauſed by the diametrical.
Hiſt, de l'Acad. Roy. an. 1702.
Interpofition of the Earth between
5. The Earth's Excentricity is a the Sun and Moon.
hundred and fixty-nine of ſuch Parts A total Eclipſe of the Sun or
as the Sun's Diſtance is a thouſand. Moon, is when their whole Bodies
The periodic Time of the Earth, in are obſcured: And a central Eclipſe
her Orbit, is three hundred and of the Moon, is when it is not only
fixty-five Days, five Hours, fifty- total, but alſo the Centre of the
one Minutes ; the Motion about its Moon paſſes through the Centre of
Axis is performed in twenty-three that Circle which is made by a
Hours, fifty-fix Minutes, four Se- Plane, cutting the Conę of the
conds; and its Axis makes an Angle Earth's Shadow at Right Angles,
with the Plane of the Ecliptic of with the Line joining the Centres
fixty-fix. Degrees, thirty-one Mis of the Sun and Earth. A partial
nutes.
Eclipſe, is when Part of the Body
6. The Earth's Horizontal Paral- of the Sun and Moon are only dar-
lax to an Eye at the Sun's Surface ken'd.
will be fixteen Minutes ; and it is 1. The Moon can never be e-
nearer the Sun in December than it clipſed, but when ſhe is in Oppo-
is in June, and conſequently its fition to the Sun, or at: Ful! ; and
likewiſe
P 3
ECL
ECL
likewiſe in or near the Nodés: And ſays is two hundred and twenty-
the Sun, but when he is in Con- three Synodical Months, or eighteen
junction with the Moon, and the Julian Years, ten Days, (when the
Moon is in or near the Nodes. Cycle, or Period contains five Leap
2. The Limit for Eclipſes of the Days,) and eleven Days (when four
Moon is about 11 deg. 40 min. on Leap Days) feven Hours, forty-
each ſide of the Node: And the three Minutes one Fourth; in which
Limit for thoſe of the Sun about time all Correſpondent New Moons,
16 deg. 40 min. on each ſide it. Full Moons, and Eclipſes return
Alſo the atmoft Latitude of the again. This Cycle is, by him, call-
Moon, that can permịt any Eclipſe ed the Saros, and is mentioned by
of the Moon, is about 1 deg. 2 min. Pliny in lib. 2. of his Natural Hi-
And the ſame utmoſt Latitude that ſtory.
can permit any Solar Eclipſe is 7. The principal Alteration of
about dez. 32 min.
the Time of the Day in all Eclipſes,
3. If you multiply the Number of depends upon the 'Exceſs of this
Lunar Months, accompliſhed from Period above an even Nurnber of
that which began the 8th of Ja. Days, which is ſeven Hours, and
nyary, N. S. in 1701. to that Month forty-three Minutes one Fourth; fa
in which any New Moon falls out, that the Cycle puts every Corre-
and add to the Product 33890, and ſpondent Eclipſe later than the fore-
divide the Sum by 43200; then if going almoft eight Hours: And fa
The Remainder or the Difference be- if three of thole Cycles are joined
tween the Diviſor and Remainder together, thoſe odd Hours and Mi-
be leſs than 4960, there will be nutes will amount nearly to one Day,
an Eclipſe of the Sun ţhat New and they will nearly bring the
Moon.
middle Point of the Correſpondent
4. Likewiſe if you multiply the Eclipſes to the fame Time in the
Number of Lụnar Months, accom- fame Place, which a ſingle Cycle
pliſhed from that which began the cannot do; and theſe three Cycleş
8th of January, N. S. 1701. to the together will be fifty-four Years, and
New Moon preceding any Full thirty-two or thirty-three Days.
Moon, and to the Product add 8. There will be elapfed nine
37326, and then divide the Sụm by hundred Years in the time that the
43200, if the Remainder or Diffe- Moon begins to enter the Ecliptic
rence between the Diviſor and the Limit for Eclipſes of the Moon on
Quocient be leſs than 2800, there one ſide, till it goes out of it on the
will be an Eclipſe of the Moon at other; in all which time there will
the faid Full.
be fifty Periods, and Eclipſes of the
5. All Eclipſes of the Moon are Moon each Period: And there will
of the ſame Magnitude all over the be elapſed twelve hundred and fixty
Earth, and begin and end at the Years from the time that the Moon
fame Times to all thoſe inhabiting begins to enter the Ecliptic Limit-
under the ſame Meridian. But E. for Eclipſes of the Sun on one ſide
clipſes of the Sun on various Parts the Node, till it goes out of it on
of the Earth, are different: They the other: During which long time
always begin on the Weſt Side the there will be ſeventy Periods, and
Sun, and end on the Eaſt.
fomewhere Eclipſes of the Sun each
6. Dr. Halley, in his Tables not Period. After which long Spaces
3.et publithed, takes notice of a Cy- of Time there will be no ſuch E-
cle, or Period, which Mr. Whillon clipſes for a much longer time.
9. The
1
ECL
E CL
9. The Motion of the Centre of equal; not only on account of the
the Shadow of the Moon, in E. Difference of the Moon's Motion at
clipſes of the Sun, is nearly right the beginning and ending of the
lined.
entire Eclipſe; which indeed is very
10. The Dimenſions of the Pen- inconſiderable, but chiefly by reaſon
umbra, or entire Eclipſe, and the of the Difference of the Obliquity
Extent of the total Shadow on the of the Horizon all the way of its
Earth, are continually different, ac- Paſſage.
cording to the different Elevations 15. The Duration of Solar E-
of the Sun and Moon above any clipſes is different, according as their
particular Horizon.
Middle happens about Six in the
II. The Figure of the entire Pen- Morning or Evening, or about
umbra, or general Eclipſe, and of Noon, or about any intermediate
the Umbra, or total Darkneſs, as Time. If that happens about Six
they appear upon every Country, on o'Clock, Morning or Evening, the
account of the different Obliquity diurnal Motion then neither much
of every Horizon, is different, and conſpires with, nor oppoſes the pro-
will make Ovals, or Ellipſes of per Motion of the Centre of the
different Species perpetually; and Shadow: and the Duration is almoſt
in the vaſt Penumbra it will be an the ſame as it would be if the Earch
Oval, being the Interſection of a had no diurnal Motion at all. If
conical and Tpherical Surface; but in that happens about Noon, the diur-
the ſmaller Umbra, or total Dark- nal Motion, moſt of all, conſpires
neſs, which is confind to a much with that proper Motion of the
narrower Compaſs, it very nearly Centre, and makes the Duration of
approaches to the Interſection of a the Eclipſe the longeſt poffible. If
Conic Surface with a Plane, which it happens in the intermediate
is a true Ellipſis.
Times, the diurnal Motion, in a
12. The Species of that Ellipfis leſs degree, confpires with the other
depends on the fame Altitude above Motion, and makes the Duration of
the Horizon at the time of total a mean Quantity, between that of
Darkneſs, as does the Poſition of other Caſes : But if it happens con-
its longer Axis on the Azimuth of fiderably before Six o'Clock in the
the Sun at the fame time. This Morning, or after Six in the Even:
Oval, when the Sun is of a con- ing, the diurnal Motion is back-
ſiderable Altitude, is almoſt an exact ward, and ſhortens that Duration
one; but when the Sun is near the proportionably
Horizon, it will be very long, and 16. The Computation or Calcu-
so leſs exact, becauſe the ſpherical lation of Eclipſes of the Sun, is at
Surface of the Earth is at a Diſtance beſt but a troubleſome Buſineſs ;
more remote from a Plane.
that of the Moon being eaſier than
13. The perpendicular Breadth of that of the Sun. The Moon's con-
the Shadow is neither that of the fifts in having the following Data :
longer, nor that of the ſhorter Axis 1. Her true Diſtance from the
of the Cone of Shadow; but that Node, at the mean Conjunction.
of the two longeſt Perpendiculars, 2. The true Time of the Oppo-
drawn from the Tangents, parallel ſition, together with the true Place
to the Diameter; along which the of the Sun and Moon, reduced to
Direction of the Motion is.
the Ecliptic. 3. The Moon's true
14 The Velocity of the Motion Latitude at the time of the true
of the Centre of the Shadow is un- Conjuncțion, and the Diſtance of
P
4
each
ECL
+
ECL
each of the Luminaries from the let CD be the Ecliptic, and A the
Earth ; as alſo their horizontal Centre of the Shadow ; thro' which
Parallaxes' and apparent Semi-dia- draw. Le perpendicular to DC.
meters. 4. The true horary Mo Let D, H, CQ, be Weſt, North,
tions of the Moon and of the Eaſt, and South. From A with the
Sun; and the apparent Semi-dia- Diſtance equal to the Sum AN of
meter of the Earth's Shadow. From the Semi-diameters of the Shadow
theſe being given, the Duration, AP, and that of the Moon PN
Beginning, Middle, End, and Quan- deſcribe a Circle DOC, and
tity of the Eclipſe, may be obtain'd with AP the Semi-diameter of the
from Addition, Subtraction, the Rule Shadow deſcribe another concen-
of Proportion, and Trigonometry: tric Circle ELF, which will ex,
A Type of an Eclipſe of the hibit the Section of the Earth's
Moon may be deſcribed in plano, Shadow during the Moon's Paffage.
when the Semi diameter of the Make AL equal to the Moon's
Moon and Earth's Shadow, as alſo Latitude at the Beginning of the
the Latitude at the Beginning and Eclipſe, and at L raiſe the Perpen-
End of the Eclipſe, are given: For dicular LN meeting the greater
H
S.
T
N
V
INOX
K
.
C
DE
A
Q
Periphery' towards the Weſt in the tion. From O and N, with the
Point N; then will the Centre Diſtance of the Moon's Semi-dia-
of the Moon be at N at the Begin meter, deſcribe the Circles PV,TX,
ning of the Eclipſe. In like man which will expreſs the Moon at the
ner make AS equal to the Moon's Beginning and End of the Eclipſe.
Latitude at the End of the Eclipſe, Laitly, from
and at S raiſe the Perpendicular os, cular to ON, then will the Centre
which being parallel to DC, is at of the Moon in the middle of the
the fame Diſtance from it, then Obfçuration be at I ; and ſo if a
xvill the Centre of the Moon be in Circle HK be deſcribed from 1,
O, at the End of the Eclipſe. Join with the Diſtance of the Moon's Se-
the Points 0 and N by a right mi-diameter ; it will repreſent the
Line ; then will ON be an Arch of Moon in her greateſt Obſcuration,
tne Orbit, which the Moon's Cen- and will define the Quantity of the
tre nroves thro' durįng her Obſcura- Eclipſe,
The
1
ECL
E CL
The Calculation of an Eclipſe of the Moon, when fining with a full
the Sun depends upon the following Face, to inſtantly loſe her Light
Data: 1. The mean Conjunction, and Colour; and interpreted this to
and from thence the true Conjunc- . be no other than a Token ſent by
tion, together with the Place of the God of ſome impending Calamity
Luminaries at the apparent Time that would happen to them. In
of the true Conjunction. 2. The the ſame place Plutarch takes no.
apparent Time of the viſible new tice, that Anaxagoras, who flouriſh'd
Moon at the apparent Time of but a little before Nicias, was the
the true Conjunction. 3. The ap- firſt who had the Boldneſs to com-
parent Latitude at the apparent municate in Writing the Cauſe of
Time of the viſible Conjunction. the lunar Light and Shadow. But
When theſe are once had, the his Opinion was yet conceald from
other Quæfita. may be obtained by the Public, who would not eaſily ad-
Trigonometry, and other Helps. mit or approve of any Writings
But to get the Data, the greateſt concerning the Cauſes of natural
Part of the Trouble conſiſts in the Appearances ; but looked upon all
Parallaxes of Longitude and Lati- ſuch who employed their Times
tude, which if there were no ſuch this way, as Men buſying themſelves
thing, it would make the Calcula- in vain Purſuits, and guilty of Im-
tion of folar Eclipſes the ſame as piety to bound and limit the Deity.
that of lunar ones.
with certain Laws; and for this was
17. M. De la Hire, has given the Protagoras baniſh'd from Athens, and
Deſcription of an Initrument to find Anaxagoras, when carried to Priſon,
out Eclipſes by, as may be ſeen in was releaſed by Pericleswith much ado.
Bion's Book of Mathematical In 19. Thales was the firſt who pre-
ftruments: You have alſo a geome- dicted an Eclipſe of the Sun ; and
trical way of projecting an Eclipſe Ptolemy in Lib. 6. of his Almageſt,
of the Sun, by a Pair of Compaſſes has fewn how to find an Eclipſe of
and a Sector, which may be ſeen the Sun by means of Parallaxes,
in Vol. II. of Sir Jonas Moor's which Regiomontanus in his Epitome
Mathematics: In Dr. Keil's Afro- Almageſti, Lib.6. has fully explain'd.
nomical Lectures : At the End of See alſo, concerning Eclipſes
my Tranſlation of Bion's Book of Hevelius's Machin. Cæleft. tom. 1.
Mathematical Inſtruments; and in c. 18. f. 372. et ſeq.- De la Hire's
a little Tract of the Uſe of the Sektor, Tabula Aſtronomicæ,-
.-- W'ing's A-
printed for Mr. Wright, a Mathe- ftronomia Britannica.-- Wideburg's
matical Inſtrument-maker.
Tractat, de Eclips. totali Solis et
18. Plutarch relates in his Life of Terre, anno 1715. d. 3. Maij.
Nịcias, that when Soldiers were comi Gregory's Element. Aſtronom. Phyſ
. &
manded to embark upon an Expe. Geom.-Wolfius's Elem. Aſtron. § 841.
dition, there happend that Nightan & $.913.- Leadbetter's Doétrine of
Eclipſe of the Moon, which very Eclipſes. See alſo the Tranſactions
much ſurprized their General, and of the Learned, publiſhed at Peterſ-
all the Soldiers, and by reaſon of burgh, wherein is a Method of com,
their Ignorance of the Cauſe there. puting Eclipſes by Series's. One of
of, it poffefſed them with great Ap- the principal Uſes of Eclipſes is to
prehenſions of ill Luck : For, ſays find the Longitude of Places. See
he, many knew the Cauſe of Eclipſes under the Word Longitude.
of the Sun, but they had not the ECLIPTIC, is a great Circle of
leaf ſuſpicion what ſhould make the Sphere, ſuppoſed to be drain
thro?
1
E L A
E L A
thro' the middle of the Zodiac, and rebounds back again in, will be e-
making an Angle with the Equinoc- qual to the Angle AFC.
tial (in the Points of Aries and Li-
bra) of 23 deg. 30 min. which is
А
E
the Sun's greateſt Declination. But
in the new Aftronomy, it is that
Path or Way among the fixed Stars
that the Earth appears to deſcribe
to an Eye placed in the Sun.
This is, by fome, called Via Solis, с
D
F
or the Way of the Sun, becauſe the
Sun, in his annual Motion, never If a String be trained like thoſe
deviates from this Line, as all other of a muſical Inſtrument, it ſhall be-
Planets do, more or leſs; from come elaſtic ; for the ſmalleſt Force
whence the Zodiac hath its Breadth. ſhall be ſufficient to bend it, tho
EFFECTION, is a Word uſed by it be ſtrained never fo hard; and
Geometers, in the ſame ſenſe with when that Force ceaſes, the Force
the Geometrical Conſtruction of Pro- that ſtrains it, ſhall bring it back
pofitions, and often of Problems and to its firſt Situation, and the String
Practices; which, when they are being once mov?d, fhall oſcillate
deducible from, or founded upon like a Pendulum, and perform them
fome general Propoſition, are called all, both great and ſmall, in the
the Geometrical Effections thereunto ſame time.
belonging
Moſt elaſtic Bodies, when ftruck,
EFFLUVIUMS, are the very ſmall give a muſical Sound, and the Rea-
Particles, or Corpuſcles that are ſon why fome do not, ſeems to be
continually emitted from Bodies,
either becauſe the Spring is too
ELASTICITY, is the ſame as weak, and the Motion too ſlow, or
Springineſs: And an elaſtic Body is becauſe the Elaſticity is too ſtrong,
that which gives way for a time (or and the vibrating Parts ſo ſhort,
leffens its Figure) to another Body, and the Sound ſo acute, and ſo foon
Striking or preſing it, but preſently over, that it cannot be perceived
recovers its former Figure by its by the Ear.
own natural Power : And a Body If the Magnitudes and Motions of
perfectly elaſtic, is one that recovers ſpherical Bodies perfectly elaſtic,
its Figure with the ſame Force it moving in the ſame right Line, and
loft it by.
meeting one another, are given, their
All Bodies in Nature, that we Motion after Reflection may be de-
know of, are in fome degree or
termin'd thus : Let the Velocities
other, elaſtic, but none of them are
of the Bodies A and B be called a
perfectly elaſtic; and from this Ela- and b reſpectively, and if the Bo-
fticity of Bodies proceeds that noted
Ijaw of Nature, viz. That Action А
B
and Re-action are always equal and
çontrary: For if there was no Ela-
fticity, this Law would not hold
good.
If the elaſtic Ball A ſtrikes
againſt the firm Bottom CD ob- dies tend the ſame way, and A
liquely in the Direction A F, the moving ſwifter than B, follows it,
Angle EFD, whoſe Side Fe it then the Velocity of the Body A
after
B
S
E L E
E L E
after the Reflection will be
they attract, or repel all kinds of
a AaB+26B
very light Bodies at a ſenſible Dia
and that of the Body ftance, when the attracting Body is
A/B
2aA-A-+6B
heated by being rubb'd. And this
But if the Electrical Attraction is nothing elſe
A+B
but the Attraction of Coheſion, ex-
Bodies meet, then changing the Sign cited by a ſtrong Attrition to act
of b, the Velocites after Reflection with leſs Force in a larger Sphere.
aAaB-26B
It is evident from ſeveral Experi-
will be
and
A+B
ments, that in electrical Attraction,
2a4-46A-6B
the Particles of Light and Æther
; either of which, if are forcibly repelled or driven away
AB
from the electrical Body, and that
they happen to come out negative, this Force reaches to a conſiderable
it follows that the Motion after Re- Diſtance, but is ſtrongeſt near the
flection tends the contrary way to electrical Body.
which A tended before the Reflection. If a Glaſs Tube fifteen or eighteen
And this is alſo to be underſtood of Inches long, and one Inch in Dia-
the Motion of the Body A in the ameter, he rubbed with a Cloth, it
former Caſe.
has a very ſenſible Electricity; for
The Cauſe of Elaſticity, in moſt if light Bodies, ſuch as Pieces of
Bodies, ſeems to be the repulſive Leaf-Gold and Soot be laid upon a
Force of their Particles ; for when Plane, and the Tube be brought
the elaſtic Body is compreſſed, its near them, they will be put in mo-
Pores are thereby contracted, and tion, attracted, repelled, and driven
made ſmaller ; ſo that many Par- ſeveral ways by the Tube. The
ticles, which were at ſome diſtance Tube acts at different Diſtances, ac-
before, are now brought nearer to- cording to the different State of the
gether, within the Sphere of each Air ; ſometimes at the Diſtance of
other's Repulſion; which Repulfion one Foot; but when the Air is full
grows ſtronger as the Compreſſion of Vapours, the Effect is diminiſhed;
increaſes, and the Particles are and the Tube muſt be rubbed all
forced cloſer to each other: Where- one way from the End that your
fore, if the Pores of a Body are Hand does not hold it with.
very large, it may admit of Com ELEMENTS, by Geometricians
preffion without much Elaſticity. and Natural Philoſophers, are uſual-
And hence alſo, we ſee the. Reaſon ly taken for the ſame as Principles ;
why the Elaſticity of Metals is in- and when they ſay the elementary
creaſed by hammering,
Principles of natural or mix'd Bodies,
Sir Iſaac Newton, in Prop: 23. they mean the ſimple Particles out
lib. 2. Princip. demonſtrates, That of which the mix'd Body is com-
Particles which mutually avoid, or poſed, and into which it is ultimate-
fly from one another by ſuch Forces iy reſolvable. The Word is alſo
as are reciprocally proportional to uſed for the firſt Principles or Rudi-
the Diſtances of their Centres, will ments of any Science ; as the Ele-
compoſe an elaſtic Fluid, whoſe ments of Euclid.
Denſity ſhall be proportional to its ELEVATION of a Mortar-piece,
Compreſſion.
fignifies the Angle which the Chace
ELECTRICITY, is that Property of the Piece, or the Axis of the Ca-
of ſome Bodies, as Amber, Jet, vity of the ſame makes with the
Sealing wax, Glaſs, &c. whereby Horizon.
ELE-
ELL
E L-L
or
ELEVATION of the Pole, is the PN will be equal to the Square of
Number of Degrees that the Pole the Ordinate PM; and ſince (draw-
is raiſed above the Horizon of any ing NO.parallel to AB) this Rect-
Latitude.
angle is leſs than that under AP,
ELLIPSIS, in Geometry, is a and the Latus rectum AF, by the
Curve Line as DE Freturning into Rectangle under AP and of,
NO and O F, being ſimilar to that
C С
under A B and AF ; the ſaid De-
ficiency made him call the Curve
by the Name of an Ellipſis.
The eaſieſt way of deſcribing this
Curve by a continued Motion when
the tranſverſe and conjugate. Axes
D
AB, ED, are given, is thus : Firſt
F
D
1
А.
B
G
B
A
itſelf, and is the common Section of
F C
the Surface of a Cone ACB, gene-
rated by a Plane, ſo cutting it as
E
when continued, it falls above the
Baſe AGB of the Cone.
The Reaſon of this Name which verfe Axis, ſuch that the Diſtances
take the Points F, f, in the tranſ-
Apollonius firſt gave to this Curve, CF,Cf, from the Centre C be each
is this: Let BA, ED, be any two
conjugate Diameters of an Ellipfis equal to V AC-CD, or ſuch that
(they are the Axes in this Figure) FDfD be each equal to AC;
and at the End A of the Diameter and having affix'd two Pins in the
BA, raiſe the perpendicular Af Points F, f, (which are calld the
Foci of the Ellipſis) take a Thread
equal in Length to the tranſverſe
F
Axis A B, and put about them, and
E
N
faften the two Ends of the Thread
together at M; then if this Thread
M
be drawn tight by means of a Pin M,
and the faid Pin be moved round
А
B
till it returns to the Place from
whence it firſt moved, and the
Thread at the ſame time being al-
ways kept tight, ſo as to form a
D
right-lin’d Triangle F Mf; the ſaid
Pin M. will defcribe an Ellipſis,
equal to the Latus rectum, being whoſe Axes are A B, D E. And by
a third Proportional to AB, ED, this means may Points, thro' which
and draw the right Line B F ; then the Curve is to paſs, be found ;
if any Point P be taken in BA, and for if with any Diſtance leſs than
an Ordinate PM be drawn, cutting the Axis A B you deſcribe an Arch
BF in the Point N, the Rectangle of a Circle about the Centre F, and
under the Abſciſs AP, and the Line with another Diſtance equal to what
the
an 의
​.
D
E
ELL
E L L
the ſaid Diſtance wants of being along a right Line, drawn in a
equal to A B, you deſcribe another
certain Poſition in that Plane ; for
Arch about fi interſecting the for- then the Interſection of the other
mer one; the ſaid Point of In- two of theſe Squares will deſcribe
terſection will be one Point of the an Ellipfis ;
or inſtead of two
Ellipfis.
Squares you may have only one, and
If it be requir'd to find Points a Ruler, and an Ellipfis will ſtill be
thro' which an Ellipſis of given con- deſcribed.
jugate Diameters AB, ED, is to
An excellent general way of find-
pals, it may be done thus : Conti- ing Points, thro' which this Curve
nue out CD to H, ſo that DH be and the other two conic Sections
DC, thro' H draw FG paral.. paſs, may be ſeen under the Word
lel to AB, and AF, BG parallel Geometrical Curve. See various ways
of deſcribing an Ellipſis in Gregory
H o G St. Vincent's Quadratura Circuli.
1. The Area of the Elliptic Space
is a mean Proportional between the
two Circles, having the tranſverſe
M
and conjugate Axes for their Dia-
Р
meters.
c
B
2. The Periphery of the Ellipfis
may be obtained by the following
Series.
to CH, and draw the Diagonal For if CB, half of one of the
AG. Take any Point P in AB, Axes of an Ellipſis be=r, and
and draw PO parallel to AF, cut- CD, the half of the other, = 1,
ting AG in N; then if P M be a and there be let fall a Perpendicular
mean Proportional between PN,
PO, the Point M will be a Point
D
G
of the Ellipſis. And thus may any
Number of Points be found for one
half the Ellipſis; and to find them
for the other half, it is but conti-
A
B
F
nuing out OP below AB, and
making Pm equal to PM, then
will m be a Point in the other half
of the Ellipſis.
GF to AB, which calla; then the
There are many other ways of Length of the Curve of the El-
deſcribing an Ellipfis by a continued
ge2 a3
Motion, and by means of Points, lipfis GB will be at
604
it
-As by moving the Angle of a
4 72 f? as -
Square along a right Line, and at
+
the famie time letting the End of
one ſide of the Square paſs along a 8c4f2 97 +26 a? --- 462 ,5 a?
given Point without that Line ; for
&c.
then the Extremity of the other
ſide of the Square will deſcribe an
And if the Species of the Ellipfis
Ellipſis.- By faſtening the Angles be determined, this Series will be
of two Squares in two Points upon
more ſimple; and if c = 2 r, then
a Plane, and cauſing the Interſection
Will BG=a+
+
of two sides of the Squares to move
G
r as
+
400$
II 2012
3
zas
96,3
2048 +
+
E L L
E L L
113 a7
A
1
хаа
N
2.2
3.3.5 d3
2.3.5-5-7 24
3419 29
thro' by the Curve of an Hyper-
+
+
& C.
458752
7549747278
C. bola (whoſe ſemi-tranſverſe Axis is
And if the ſaid Curve was an Hy-
ab
a,
in ſuch
perbola, the ſaid Series would ferve
for it, by making the even Parts of manner that the Semi-tranſverſe
all the Terms affirmative, and mak- moves in a Plane paſſing thro' the
ing every third, fifth, and feventh Axis of the Cylinder, the Plane of
Term negative.
the Hyperbolic Space moving always
3. In the Ellipfis, (ſee Fig. of n.2.) parallel to itſelf, and the Centre of
if á Semi-diameter C B be called a, the Hyperbola running along a Di-
the Semi-conjugate CD, b, the Ab- ameter of the Baſe of the Cylinder.
Iciſs CF, *, and the correſpondent In Mr. Simpſon's Book of Fluxions,
Ordinate FG, y; then will the E- you have the following Series for the
Rectification of the Curve of the
quation yy 2-— **, be Ellipſis, which ſeems to be the moſt
the moſt ſimple poffible, expreſſing elegant the nature of the thing will
the Curve of the Ellipſis, m and n
admit.
being invariable Quantities.
The Periphery of the Ellipfis is
The Rectification of the Curve of to that of a Circle, whoſe Diameter
the Ellipfis cannot be had from the is equal to the tranſverſe Axis of
Quadrature of any Space belonging
d 30
to the Conic Sections;
for if DC, cĚ the Ellipſis, as 1—
2.2.4.4
be the Semi-Axes, and dd=aa-bb,
viz, equal to the focal Diſtance,
Esc.
and cidd be = 64, then will 2.2.4.4.6.6. 2.2.4.4.6.6.8.8.
d
is to 1, where d is equal to the
j
V«c + 99, be the Fluxion Difference of the Squares of the
6
Axes apply'd to the Square of the
of the Arch DM of the Ellipſis, and tranſverſe Axis.
the Fluent of this Fluxion is not 4. If any two parallel right Lines,
to be exactly had but from the Qua- CD, HG, be drawn, terminating
drature of a Space contain'd under in an Ellipfis in the Points C,H,D,G,
a Curve of the third Order, whoſe and a third Line AB, terminating
Equation ſuppoſing the Abſciſs to in the fame in the Points A,B;
be and Ordinate y, will be uu yy
then will СEXED:HFXFG::
+dd yy=bb uu
But if z be a correſpondent Arch
A
of a Circle deſcribed about the
Centre A with the Radius A C, the
E
Fluxion of the Arch of the Ellipſis
H
1
D
will be canvaabb+ddyy. So that
B
G
the Rectification of the Curve of the
Ellipſis alſo may be had from the AEX EB: A F x FB. And fo,
Quadrature of the outward Curve when A B and C D happen to be
Surface of a Cylinder (whoſe Baſe conjugate Diameters, HG will be
is the Circle 'deſcribed upon the an Ordinate; and in this caſe A E
tranſverſe Axis of the Ellipfis) re-
E B, CE = ED, HF - F G.
maining after the Cylinder is cut whence CE® : HF :: ĀĒ” :
AF
66-73
Va
dd cc.
E L L
E L Ľ
AF X FB, which is a very noted drawn from the Foci, are equal to
Property of the Ellipfis.
one another
5.
If any two right Lines, touch-
ing an Ellipfis in the Points G, D,
A c
meet in the Point A, and from A
E
be drawn the right Line ALI,
meeting the Curve in the Points
L, I, and the Line GD joining
F D G
.PH
non
#
iza
G
f
C
I
9. If the Line L K be the tranſ.
A
K
verſe Axis of an Ellipfis, and Points
H,1, the two Foci, and the Rulers,
D
HG, IF, be in Length equal to
L K, and the Rule F G to HI;
the Points of Contact in the Point and if the Ends of the Rules, HG,
K; then will AL:AI :: KL:
KI. And ſo fince, when the right
G
Line LI paſſes thro' the Centre C
of the Ellipfis, it is biſſected ; there-
fore CK, CI, CA, are continual T
Proportionals. See more under Hy-
perbola.
6. In every Ellipfis a Parallelo-
gram, as EFGH, that circum-
ſcribes it, ſo that its Sides be paral-
lel to the two conjugate Diameters
K2, MI, is equal to the Rectangle L
H
I.
B
А
N
M
G. IF, be moveable about the Foci,
E
H, I, and the Rule F G be faften'd
to them, fo as to be moveable about
D
the Points F, G; then will the In-
terſection of the Rules HG, IF,
deſcribe ao Ellipfis.
A BCD, whoſe Sides are equal to That this Curve will be an Ellip-
the two Axes NO, PQ: See more fis, will appear thus. Join FH;
under Hyperbola.
for becauſe the Triangles FGH,
7. In every Ellipfis the Sum of FIH, have two Sides, FG,
the Squares of any two conjugate each equal to the two Sides HI, IF,
Diameters is equal to the Sum of and the Baſe FH common, the
the Squares of the two Axes. Angles FHG, HFI, will be equal ;
8. In every Ellipfis the Angles and ſo the Sides, FE, EH, are
ACF, GCE, made by the Tan- equal : Whence FI = HE + EI;
gent A E, and the Lines F C, CG, but FI is equal to LK; whence HE
12
:
FI
GH,
+
E L L
E L O
1
of EI
2
= to the Axis ; and conſe. videndo) cc: zaat-ubb :: yy : zaak
quently the Point E is in the Ellip 2bb-tyyxx. And again (di-
fis, whoſe Foci are H, I, and Axis videndo) cc : yy :: zaa-t-2bbcc :
LK; becauſe the Sum of the Lines zaat2bb-cc-xx. But ſince zaat
HE, I E, in the Ellipfis, are always 2bb-cc is = 466+46c-4cc. for a=
equal to L K.
btc. Therefore cc : yy :: 466+
10. If one End A of any two e- 4bctc : 466+4bcticxx. But
qual Rulers A B, BD, which are
moveable about the point B. like a 466+4bc+c=a+b. Conſequent-
Carpenter's Joint-Rule, be falten'd ly cc : yy :: atb ; atb - xx.
to the Rule LK, ſo as to be made which is a noted Property of the
moveable about the Point A, and Ellipfis.
the End D of the Rule DB 'be ELLIPTICAL COMPASS, is an
drawn along the side of the Rule Inſtrument for drawing of Ellipfes
at one Revolution of the Index, .
B
and confifts of a Croſs A BGH,
with Grooves in it, and an Index
CE, which is faften'd to the Croſs
I
K by means of Dove-tails at the Places
.2
2
ultra
D
E
G
G
P
B
AL
==B
KS
D
A
[
H
1
E
G
C, D, that ſlide in the Grooves ; ſo
L K, any Point E, taken in the that when the Index is turned about,
Side DB of the Rule, will deſcribe the End E thereof will deſcribe an
an Ellipfis, whoſe Centre is A, con- Ellipfis.
jugate Axis = 2 DE, and trans ELLIPTICAL DIAL, is ary In-
verfe = 2AB + 2 BE.
ftrument inade commonly of Braſs,
The following Demonſtration of with a Joint, to fold together, and
this Property being new, at leaſt to the Gnomons to fali flat, commo-
me, is the reaſon I put it down. diouſly contrived to take a little
Let us call BD, a; BE, b; DE,c; room in the Pocket. By it may be
A e, x
X; e E, y; he, u; h D, q; and found the true Meridian, Hour of
e D, z. Then xx+'-y=299 + the Day, Rifing and Setting of the
2uu (by 9. 2. Euclid). But ſince aa Sun, with ſeveral other Propoſitions
:99:: 66: uu. And (by compound of the Globe.
ing) aa:99 :: 2aatabb : 299+2uu. ELONGATION of a Planet, or
or cc:c--yy:: zaa+2bb: 2y+z1u; Angle of Elongation, in Aftronomy,
becauſe aa : yy :: cc : zz. Therefore is the Difference between the Sun's
.cc : 6cmyy :: zaart 2bbxxafcc-
yy. true Place, and the Geocentric Place
ſince xx7cc--yy=299+zuu. and (die of that Planet!
The
E M P
EN G
The utmoft Elongation of Venus ENCEINTE, a French Term, in
can be but forty-five Degrees, and Fortification, fignifying the whole
that of Mercury but thirty Degrees, Incloſure, Circumference, or Com-
which is the reaſon this Planet is ſo paſs of a fortified Place, conſiſting
rarely ſeen.
either of Baltions, or not.
EMBOLUS, is the Sucker of a ENDECAGON, a plain Figure, of
Pump, or Syringe; which when the eleven Sides and Angles.
Pipe of the Syringe is cloſe ſtopped, ENFILADE, in Fortification, fig-
cannot be drawn up but with the nifies a Situation of Ground, which
greateſt Difficulty ; and when for- diſcovers a Poſt according to the
ced up by main Strength, will, on whole Length of a right Line, fo
being let go, return again with great that it can be ſcoured with the Can-
Violence.
non, and render'd almoſt defenceleſs.
EMBRASURE, in Architecture, is Whence, to
the Enlargement made in the Walls, Enfile che Curtain or Rampart, is
to give more Light and greater Con- to ſweep the whole Length of it
venience to the Windows and Doors with the Cannon.
of a Building
ENGINe, in general, is any me-
EMBRASUres, in Fortification, chanic Inſtrument, compoſed of
are the Holes in a Parapet, through Wheels, Screws, Pullies, &c. by
which the Cannons are pointed to the Help of which a Body is either
fire into the Moat or Field. They moved or hinder'd from moving
1. When the Quantities of Mo-
from one another, every one of them tion, in the Weight and Power, are
being from fix to ſeven Foot wide equal, the Engine ſhall fand in E-
without, and about three within. quilibrio ; but when they are un-
Their Height above the Platform equal, the greater Quantity of Mo-
is three Foot on that fide toward tion ſhall overcome and work the
the Town, and a foot and a half Engine.
on the other ſide toward the Field; 2. Of Forces in themſelves equal,
that ſo the Muzzle may be funk on that which is nearelt to that Point
occaſion, and the Piece brought to of the Engine, about which the
Thoot low.
Weight and Power move, or upon
EMERs10n, in Aftronomy, is the which they ſuſtain each other, is
Time when any Planet, that is e- relatively the weakeſt upun the En-
clipſed, begins to emerge, or get gine ; for as the Engine works, the
out of the Shadow of the eclipfing neareſt Force moves the floweſt, and
Body. When any Body alſo, therefore has the leaſt Quantity of
lighter in Specie than Water, being Motion.
thruſt violently down into it, riſes 3. The Effect of any Force upon
again, 'tis ſaid to emerge out of the the Engine will not be changed; if,
Water.
without changing the Line of Direc-
EMINENTIAL EQUATION, a tion, it is only placed in ſome other
Word of no great uſe, is an artificial Point of che faine Line. The Na-
Equation, containing another Equa- ture of any Engine is explained,
tion eminently, and is uſed in the when it is known in what Circum-
Inveſtigation of the Area's of curv'd ſtances the Weight and Power will
Spaces.
be in Equilibrio upon that Engine.
EMPATTEMENT, by ſome is
4 In all Engines whatſoever, the
the ſame with Talus in Fortification. Weight and Power will be in Equi-
Which fee.
hbriy, when their Quantities are in
Q
the
E PA
EPI
the reciprocal Proportion of the Ve If the Golden Number be given,
locities, which the working of the and it be divided by 3, and the Re-
Engine will give them.
mainder be multiplied by 10, and
5. If an Engine be compoſed of added to the Golden Number, and
ſeveral fimple Engines, the Power is from the Sum, 30 be taken away,
to the Reſiſtance when it counter- the Remainder will be the Epact.
ballances it, in a Ratio compounded EPAULE, in Fortification, is the
of all the Ratio's, which the Powers Shoulder of the Baſtion, or the An-
in each ſimple Engine would have gle of the Face and Flank; whence
to the Reſiſtance, if they were ſe. that Angle is often called the Angle
parately applied.
of the Epaule.
ENGONASIS HERCULES, the EPAULEMENT, in Fortification,
Name given by Aſtronomers to one is a Side-work, made either of Earth
of the Northern Conftellations, con thrown up, of Bags of Earth, Ga-
taining about forty-eight Stars: - bions, or of Faſcines and Earth ; of
ENGYSCOPE, the fame with a which latter they make the Epaule-
Microſcope. Which ſee.
ments of the Places of Arms for
ENHARMONICAL, or ENHAR the Cavalry behind the Trenches.
MONIC, in Muſic, is uſually applied Epaulement, is uſed for a Demi-
to the laſt of the three Kinds, a- Baſtion, and ſometimes it fignifies a
bounding in Dieſes, which are the ſquare Orillon, which is a Maſs of
leaſt fenfible Diviſions of a Tone. Earth almoſt ſquare, faced and lined
See Dieſis.
with a Wall, and deſigned to cover
ENNE A DECATERIDES, the ſame the Cannon of a Caſemate.
with the Golden Number. Which EPICYCLE, is a ſmall Circle,
fee ; or the Cycle of the Moon. whoſe Centre is in the Circumfe-
ENNEAGON, is a Polygon of nine rence of a greater, or a ſmall Orb,
equal Sides.
which being fixed in the Deferent
ENTABLATURE, or ENTABLE- of a Planet, is carried along with
MENT, in Architecture, fignifies its Motion, and yet with its pecu-
the Architrave, the Freeze, and the liar Motion, carries the Body of the
Cornice together, and is different in Planet faſtend to it round about
the different Orders.
its proper. Centre; which ancient
ENVELOPE, in Fortification, is a Aftronomers attribute to all the Pla-
Mount of the Earth, ſometimes nets, for ſolving their Appearances,
raiſed in the Ditch of a Place, and except the Sun.
ſometimes beyond it, being either EPICYCLOID, is a Curve gene-
in form of a ſimple Parapet, or of rated by a Point taken in the Peri-
a ſmall Rampart, borderd with a phery of a Circle that rolls or re-
Parapet. Theſe Envelopes are volves upon the Periphery of ano-
made when one would only cover ther Circle, either within or with-
the weak Places with ſingle Lines, out it.
without any Deſign of advancing 'The Length of any part of the
toward the Field, which cannot be Curve, that any given Point in the
done but by Works that require a revolving Circle has deſcrib'd from
great deal of Breadth, ſuch as Horn- the Time it touched the Circle it
Works, Half-Moons, &c.
revoly'd upon, ſhall be to double
EPACT, is a Number expreſſing the versid Sine of half the Arch,
the Exceſs of a Solar Year above a which all that time touched the
Lunar one, and is only of uſe in Circle at reſt, as the Sum of the
finding the Age of the Moon. Diameters of the Circles, to the
Seni-
EPI
E QU
$emi-Diameter of the reſting Cir- Centre in the other Focus, and the
cle, if the revolving Circle moves Radius fhall be the principal Axis
of the Hyperbola, and any other
Point of the Hyperbola fall de-
ſcribe a Line of the fourth Order.
See concerning theſe Lines in
Lib. 1. of Sir Iſaac Newton's Prin-
cip. Mathem. Alſo Mr. De la Hire,
in his Memoires de Mathematique &
de Phyſique, wherein he ſhews the Na-
ture of this Line, and its Uſe in Me-
chanics. See alſo Mr. Mac-Laurin's
Geometria Organica.
EPISTYLE, in Architecture, is a
Maſs of Stone, or Piece of Timber,
laid upon the Capital of a Pillar,
Epocha, or ÉPOCHE, in Chro-
nology, ſignifies ſome remarkable
Occurrence, from whence ſome Na-
tions date and meaſure their Com-
upon the Convex-fide of the reſting putation of Time.
Circle: But if upon the Concave The Julian Epocha takes its
fide, as the Difference of the Dia- Name from Julius Cæſar's Refor:
meters, to the Semi-Diameter.
mation of the Roman Calendar,
If a Parabola moves upon ano which was done forty-five years be-
ther equal to it, the focus of it fore Chriſt, in the ſeven hundred
will deſcribe a right Line perpendi- and eighth Year from the Building
cular to the Axis of the Parabola at of Rome, and in the ſeven hundred
reſt, and at a Diſtance from it equal and thirty-firft Olympiad.
to the Diſtance of the Vertex from The Ethiopic, Abyfinian, or as
the Focus, and the Vertex of the ſome call it, the Diocletian Epocha,
Parabola will deſcribe the Ciffoid or others, the Æra of the Martyrs,
of Diocles, and any other Point becauſe it bore a Date with a very
thereof will deſcribe fome one of ſevere Perſecution ; this Epocha be-
the defective Hyperbola's of Sir gan Auguſt 29, A. D. 284. and in
Iſaac Newton, having a double Point the third Year of the Emperor Dio-
in the like Point of the Parabola at cletian. 'Tis uſed by the Egyptians
reft.
and Abiliyns.
If in like manner an Ellipſis re The Turkiſh, or Arabic Epocha,
volves upon another, equal and fi- which they call the Hegira, bears a
milar to it, the Focus will deſcribe' Date from Mahomet's Flight from
a Circle, whoſe Centre is in the Mecca, A. D. 622, July 16.
other Focus, and the Radius ſhall The Perfic, or Jefdegerdic Epocha,
be equal to the Axis of the Ellipſis; takes its Date either from the Coro-
and any other Point of the Plane of nation of the laſt Perſian King Jef-
the Ellipfis ſhall deſcribe a Line of degerdic, or Jeſdagerdis, as ſome ſay,
the fourth Order. The ſame may or from his being conquer'd rather
be ſaid alſo of an Hyperbola, re- by Ottoman the Saracen, which was
volving upon another, equal and ſia June 16. A. D. 632.
milar to it; for one of the foci EQUABLE MOTIONS, are fuch
will deſcribe a Circle, having its as always continue the ſame Degree
e2
of
E QU
EQU
, x, v
2
01
c
3
of Velocity,and are neither accelerat- ther, accompanied with the proper
ed nor retarded; but if there be an Signs +,
& c. and
Acceleration or Recardation of the known by this Mark =, amongſt
Velocity of two or more Bödies, them, fignifying that Number or
and it be exactly and uniformly literal Quantity, or all the Num-
the ſame in them both, or all, they bers, or literal Quantities, or both,
ſay, ſuch Bodies are
before it, to be equal to the Number
EQUABLY accelerated or retarded.
or literal Quantity, or all the
EQUALITY, is the exact Agree- Numbers or literal Quantities, or
ment of two Things, in reſpect of both, which are after it. Thus 2
Quantity
= 2, 5+ 3
28
4 + 2,
A plainer Definition of Equality, 7=2 I+3+3, asa,b=1,
is this; thoſe Things are equal to dd =gg + h), *x + ax = bb,
one another, which poſſeſs the ſame
bx3 ex
Place, or may be conceived to pof- *3 t-ax+bx=13,
+
feſs the ſame Place by the Flexion
f
and Tranſpoſition of their Parts. Vao, &c. are all Equations, figni-
See a learned Diſcourſe about this, fying reſpectively that 2 is equal to
by Dr. Barrow, in his rith and
2, that 5 + 3
- 2 (viz. 6.) is e-
12th Mathematical Lectures
qual to 8
4+2 (viz. 6.), y e-
EQUATION, or the total Proſta- qual to 2 I + 3 + 3, (viz. 7.) ;
phærefis, in the Ptolemaic Theory of a equal to a, b equal to c, dd equal
the Planets, is the Difference be-
to gg +bh, *x + ax equal to bb,
tween the Planets mean and true
6x3
Motion, and the Angle made by *3 to axa-t-bx equal to c3,
the Lines of the true and mean Mo.
tion of the Centre. But the
EQUATION, OF PHYSICAL PRO- equal to vão. You will alſo
STAPHÆRESIS, is the Difference fee frequently Numbers or literal
between the Motions of the Centre Expreſſions of Quantities, or a Mix-
of the Epicycle in the Equant, and ture of both, before the Sign
in the Eccentric. And the
Equality, and a Cypher o after it,
EQUATION, or Optical Pro-
or elſe a Cypher before and thoſe
STAPHÆRESIS, is the Angle made after, which by many is call'd an
by two Lines drawn from the Cen- Equation. But I think very wrong-
tre of the Epicycle to the Centre ly, for all that is really meant by
of the World, and of the Eccen- ſuch an Expreſſion is, that the Quan-
tric.
tities before or after ſuch a Sign
EQUATION of the Orbit, is the mutually, deſtroy each other: Or,
fame with the Total Profiaphæreſis, wh:n ſome of them be taken from
or liquation total.
the others, there will be no Diffe-
EQUATION, in Algebra, is an rece remaining.
Equality between one Number or Perhaps the calling ſuch an Ex-
Quantity, and one ; feveral, and priflion an Equation, might have
one ; ſeveral and ſeveral, or between given occafion for the Author of the
their Sums, Diffcrences, Products, Minute Philoſoplier, in his Diſcourſe
Quotients, Powers and Rocts, either called Tire Anal;i, pag 86. qu.APIP 40.
all expreffed particularly by the not only to talk Nonſenſe himſelf,
common Numerical Characters, or but charge theMathematicians of the
univerſally by the Letters of the preſint Āge to do ſo too. For, ſays
Alphabet, or by both theſe toge- he, is it 1.0t a general Cafe, or Rule,
that
+
с
ex
2
of
ܝܐ
1
1
E QU
E QU
that one and the ſame Co-efficient di- = 0, and x = b, or x-
-b=0
viding equal Produkts, gives
. equal then will traxx b=0, be a.
Quotients? And yet, whether ſuch quadratic Equation, having two af-
Co-efficient can be interpreted by o, or firmative Roots + a, and + b.
nothing, or whether any one will ſay In like nianner, when xsa, x=b,
that if the Equation 5*0 = 2 XO
be divided by , the Quotient on both *=, or x-a=0, x-b=0, x-6
Sides will be equal? Whether there-
=0; then will x---axa-bxare
fore a Caſe may not be general, o be a cubic Equation, having
with reſpect to all Quantities, and three affirmative Roots. See more
yet not extend to Nothings, or include of this in Harriot's Praxis Artis
the Caſe of nothing; and whether the Analyticæ, (who was the firſt that
bringing nothing under the Notion of explained the Nature of Equations
Quantity, may not have betray'd Mén after this way,) Deſcartes's Geometry,
into falſe reaſoning? Now herein and other Writers of Algebra.
he talks both ignorantly and unin- Vieta has explained their Nature
telligibly, and falſely ; 'for in the from the Analogy of their Terms ;
firſt place, a Co-efficient does not di- and Dr. Barrow, at the End of his
vide Products, but multiplies them, Geometrical Lectures, has given a
any one that is acquainted with Specimen of doing the ſame by
its Definition very well knows. In curve Lines.
the next place, whoever.calls o, or
Every Equation has as many
nothing, a Co-eficient? This would Roots as the unknown Quantity of
be talking ſtark nonſenſe, ſaying the firſt Term has Dimenſions, or
nothing is ſomething. Thirdly, as the Exponent thereof contains
what Mathematician (except this Units.
pretended one) ever called 2 X0 =
All Equations have as many af-
5.*.0, an Equation ? Or, would ſay, firmative Roots as there are Permu-
if it were divided by o, the Quo- tations of Signs ; and as many ne-
tient on both ſides will be equal. gative Roots, as there are Succef-
Fourthly, Does his aſking, whether ſons of them; as in the quadratic
a Cafe may not be general and ex- Equation x2 +x-60, there is one
tend to all Quantities, and yet not ex-
Succeſſion of Signs tot, and one
tend to Nothings, or include the Caſe Permutation + But the Equa-
of nothing, ſignify any more than tion has two Roots; one being the
ſaying a Caſe may be general, and affirmative one + A, and the other
extend to all Quantities; but it is the negative one - 3 Alſo in the
no caſe at all, when there is no-
cubic Equation 33-3*-10x+24
thing to make it one. Laſtly, who-
=o there are two Permutations of
ever brought nothing under the No. Signs + - and
and at, and one
tion of Quantity ; this would be a
Succeſſion
But it has three
Contradiction in Terms : What muſt Roots; two affirmative ones + 2,
'one take a Man to be, who aſſerts + 4, and one negative one
-3.
that nothing is ſomething ? For EQUATION (ANNUAL) of the
Quantity is allowed by all to be mean Motion of the Sun, and
ſomething; and of all people, I am Moon's Apogee and Nodes.
very ſure, no Mathematician will The Annual Equation of the
ever ſay this is nothing.
mean Motion of the Sun, depends
The Nature of Equations are
upon the Eccentricity of the Earth's
very well explained from their Ge- Orbit round him, and is fixieen
neration ; as if x be =4, or x-a' ſuch Parts, of which the inean 13-
a3
sta ce
1
I
EQU
E QU
1
or
Atance between the Sun and Earth is EQUILATERAL HYPERBOLA,
a thouſand; from whence, by ſome, is ſuch an one whoſe tranſverſe Dia-
'cis called the Equation of th: Centre ; ineter is equal to its Parameter; and
and this, when greateſt, is i deg. ſo all the other Diameters equal to
56 min. 20 ſec. the greateſt An- their Parameters, and the Aſymp-
nual.
totes of it do cut one another at.
EQUATION of the Moon's mean right Angles in the Centre.
Motion, is 11 min. 40 ſec. of its
Its molt fimple Equation, with re-
Apogee 20 min. and of its Node 9 gard to the tranſverſe Axis
, being
min. 30
fec. and theſe four Annual yy=xx aa; and with regard
Equations are always mutually pro- to the Conjugate yy. = x x + aa,
portionable to one another ; ſo that when a is the Semi-tranſverſe,
when any of them is at the greateſt, Semi-conjugate Axis. The Length of
the three others alſo will be greateſt; the Curve cannot be found by means
and when any one leſs, the reſt di. of the Quadrature of any. Space, of
miniſh in the ſame Ratio: Where- which a Conic Section is any Part
fore, the Annual Equation of the of the Perimeter ; altho' Mr. Leib-
Centre (of the Sun) being given, the nitz, in one of his Letters to Sir
other three correſponding Equa- Iſaac Newton, publiſhed in the Com-
tions will be given; ſo that one mercium Epiftolicum, is of opinion it
Table (i.e. of the Central Equa. could. See concerning the Deſcrip-
tion) may ſerve all.
tion of this Curve under the word
EQUATION of a Curve, is an E- Hiperbola.
quation Thewing the Nature of a
EQUILATERAL TRIANGLE. See
Curve by Expreſſion, the Relation Triangle.
between an Abſciſs, and a Cor-
EQUILIBRIUM, in Mechanics,
reſpondent Ordinate, (which was is when the two Ends of a Ballance
first done by Deſcartes in his Geo- hang ſo exactly even and level, that
metry) or elſe expreſſing the Rela- neither-doth aſcend or deſcend, but
tions of their Fluxions, &c. See Şir do both keep in a Poſition parallel
Iſaac Newton's Fluxions, & c.
to the Horizon, which is occafioned
EQUATION of Time, is a Space by their being both charged with an
of Time to be added to, or ſub- equal Weight.
tracted from the Time ſhown by the
EQUIMUL'TIPLES, are Num-
Sun, that thereby it may become bers or Quantities multiplied by one
equable, and is the Difference be- and the ſame Number or Quantity.
tween the Sun's mean Motion, and See Proportion.
its right Aſcenſion; and is greateſt EQUINOCTIAL, (in the Heavens)
about the latter End of January or Equator on the Earth, is a great
and October, it being then near Circle, whole Poles are the Poles
fifteen Minutes; and about the Be- of the World. It divides the Globe
ginning of April, June, and towards into two equal Parts, that is, the
the latter End of Auguſt, it is leaft, Northern and Southern Hemi-
being then leſs than a Minute. See ſpheres. It paſſes through the Eaſt
the Xitronomical Writers upon this and Weſt Points of the Horizon ;
Subject
and at the Meridian is raiſed as much
EQUATOR. See Equinoctial.
above the Horizon as the Comple-
EQUICRURAL. See I/oceles.
ment of the Latitude of the Place.
EQUICULUS,Or EQUUS MINOR, 1. Whenever the Sun cometh to
a Conſtellation in the Northern He this Circle, it maketh equal Days
miſphere, conhiting of four Scars,
and Nights all sound the Globe, be-
caure
E Q_U
É QU
Cauſe he then always riſes due Eaſt, whoſe Plane is
, parallel to the Equi-
and fets due Weſt, which he doth noctial.
at no other time of the Year : 1. The Hour-Lines on this Dial
whence it hath it's Name. All are all equally diſtant from one an.
Stars alſo which are under this other round the Periphery of a Cir-
Circle, or which have no Declina- cle, and the Style thereof is a ſtraight
tion do always riſe due Eaſt, and ſet Pin, or Wire, ſet up in the Centre
full Weſt, &C.
of the Circle, perpendicular to the
2. All People living under this Plane of the Dial.
Circle (which, in Geography, is 2. The Sun ſhines
upon
the
upper
called the Line,) have their Days Part of this, Dial-Plane from the
and Nights equal. At Noon the roth of March to the 12th of Sep-
Sun is in the Zenith, or directly tember, and upon the under Part'the
over their Heads, and cafts no Sha- other half of the Year.
dow.
3. There are ſome of theſe Dials
3. From this Circle (on the made of Braſs, & c. and ſet up in
Globe) is the Declination, or La- a Frame, to be elevated to any gi-
titude accounted on the Meridian. ven Latitude.
4. And the Circles which run EQUINOCTIAL ORIENT. See
through each Degree of Latitude or Orient.
Declination, are called Parallels of EQUINOCTIAL OCCIDENT.
Latitude, or Declination.
See Occident.
5. Through this Equinoctial all EQUINOXES, are the preciſe
the Hour-Circles are drawn'at right Times in which the Sun enters into
Angles to it ; and through the Poles the firſt Points of Aries and Libra;
of the World, at every fifteenth for the Sun moving exactly under
Degree on the Celeſtial Globe. the Equinoctial, he makes our Days
6. And the Equator on the Ter- and Nights equal. This he doth
reſtrial Globe is divided by the Me- twice a year, about the roth of
ridians into thirty-fix equal Parts.
March and 12th of September;
7. The natural Day is meaſured which therefore are called the Verg
by the Revolution of the Equator, nal and Autumnal Equinoxes.
and is ended when the fame Point 1. It is found by Aſtronomical
of the Equator comes again to the Obſervation, that the Equinoctial
ſame Meridian, which is in twenty- Points (which are the firſt Points of
four Hours.
the Signs Aries and Libra) go back-
8. Wherefore, ſince the Equator wards
every Year 5 fec.
(as all great Circles are) is divided 2. And our admirable Sir Iſaac
into three hundred and ſixty De. Newton, taking the Matter into
grees, each Hour muſt be i'w of that Conſideration, according to his Prin-
Number, or fifteen Degrees; there- ciples, found, by Calculation, that
fore one Degree of the Equator will they muſt recede 49 min. 58 ſec.
contain four Minutes of an Hour, which is furpriſingly near the
and fifteen Minutes of a Degree will Truth.
make a Minute of an Hour, or fixiy 3. The Space from the Vernal
Seconds; and conſequently four Se- to the Autumnal Equinox, is eight
conds anſwer to one Minute of a or nine Days longer than from che
Degree.
Autumnal to the Vernal, by reaſon
EQUINOCTIAL COLURE. See of the Poſition of the Perilielion of
Colure.
the Earth's Orbit near the Winter
EQUINOCTIAL DIAL, is one Solſtice.
EQUI
Q4
E RE
ER E
EQUINUS BARBATUS, a kind of ing Dial is to be drawn. Let RSP.
Comet. See Hippeus.
be an. Hour-Circle, and RXP
Erect DECLINING DIALS. another at right Angles to the A-
See Declining Erect Dials.
zimuth Circle ZGD. Then in
In Dials of this kind, as the Ra- the right-angled ſpherical Triangle
dius is to the Co-fine of the Plane's ZXP, ZX will be the Subſtyle's
Declination, ſo is the Co-fine of the Diſtance from the Meridian, which
Elevation of the Pole, to the Sine may be had by having given the
of the Style's Height. And as the Hypotheneuſe ZP, being the Com-
Radius is to the Sine of the Plane's plement of the Latitude, and the
Declination, ſo is the Co-Tangent Angle XZP, being the Comple-
of the Elevation of the Pole, to the ment of che Plane’s Declination,
Tangent of the Subſtyle's Diſtance the Side PX will be the Style's
from the Meridian; and as the Ra. Height, and the Angle ZPX the
dius is to the Co-Tangent of the Inclination of the Meridians.
Declination, ſo is the Sine of the Moreover, in the ſpherical Tri-
Elevation of the Pole to the Co- angle ZP Q, the Side ZQ, will
Tangent of the Inclination of the be the Angle that the given Hour-.
Meridians ; and as the Radius ; is Line RSQP makes with the
to the Sine of the Style's Height :: Meridian at the Centre of the Dial;
ſo is the Tangent of any Hour An- and this may be had from the gi-
gle : to the 'I'angent of the Hour-
ven Angles Z, P, and the Side ŽP
Arch.
between them.
All the Proportions above may Theſe Sort of Dials may be
be obtain’d from the Doctrine of drawn geometrically too, the Height
Spherical Triangles, and that after of the Style being firſt given. Sup-
the following manner: Let A B pofe ABC to be an horizontal
be the Horizon, EF the Equator, Line, and the Line BD at right
making an Angle with the ſame e- Angles to it, to be the Meridian or
qual to the Complement of the La- Hour-Line of 12. Make the An-
gle EBD equal to the Comple-
z
ment of the Latitude, and the An-
gle FBG equal to the Declina-
E
P
tion of the Plane, and draw ED
perpendicular to the Meridian.
V
Make FB equal to ED, and from
F let fall FG perpendicularſ to the
A
B Meridian BD, and make DH
G
equal to FG; and thro' B draw
BH, which will be the ſubſtylar
Line. This done, draw the Line
R
F
I K thro' H perpendicular to BH,
and this will be the Tangent or
D
Contingent Line, as it is calied, and
make the Angle HBL equal to
titude. DZ the prime Vertical, the Height of the Style, and from
AZPD the Meridian. PR the H let fall the right Line HM per-
Axis of the World and Hour-Circle pendicular to the Style B L. Lait-
of Six, in a given Latitude BP. Îy, make HN equal to HM, and
Z&D är Azimuth Circle, upon about the Centre N deſcribe a Circle
The Plane of which an eiect declin- HROP; which will be the Equi-
noctial.
S
ESP
E V E
noctial. Continue down the Meri- N thro' them to cut the Tangent
dian DB 'to cut the Tangent Line IK, in the Points 11, 1, 2, 3, 4, 5,
IK in the Point Q; and draw the &c. and if thro' theſe laſt Points
right Line QN, cutting the Equi- be drawn the right Lines B 11, B1,
noctial in R. Then if the Circum- B2, B3, B4, B5, &c.
&c. there
ference of this Circle be divided will be the Hour-Lines of 11, 1,
into 24 equal Parts, beginning at 2, 3, 4, 5, &c.
R, and right Lines be drawn from
À
13
M
K K
L
כ
;
12
12 쫭
​11
R
N
P
I
ERECT DIRECT PLANES, ortadel, and the firſt Houſes of the
DIALS, are thoſe that ſtand up- Town.
right, and face the four Cardinal ESTIVAL Occident. See Oc-
Points.
cident.
ERECT DIRECT, EAST, 'WEST, ESTIVAL ORIENT. See Orient.
South, or NORTH DIALs. See ESTIVAL SOLSTICE. See Sol
Ereft Dire? Planes.
fice.
ERIDANUS, or RADUS, a Sou Evection, or (being the ſame
thern Conſtellation, conſiſting of as) LIBRATION of the Moon, is an
twenty-eight Stars.
Inequality in her Motion, by which,
ESCALADE, or SCALADE, is a at or near the Quadratures, ſhe is
furious Attack upon a Wall, or not in a Line drawn through the
Rampart, carried on with Ladders Centre of the Earth to the Sun, as
to mount up upon it, without going ſhe is at the Syzygies, or Conjunc-
on in Form, breaking Ground, or tion and Oppoſition, but makes an
carrying on of Works to ſecure the Angle with that Line of about two
Men.
Degrees fifty-one Minutes.
ES PAULE, or EPAULB. See The Motion of the Moon about
Epaule.
its Axis is only equable, it perform-
ES PAULEMENT! See Epaule- ing its Revolution exactly in the
fame time as it rolls round the
ESPLENADE, a Term in Forti. Earth ; and thence it is that it nearly
fication, the ſame with the Glacis always turns the ſame Face towards
of the Counterſcarp originally ; but But this Equality, and the un-
now 'tis uſually taken for the empty equal Motion of the Moon in her
Space between the Glacis of a Ci- Ellipfis, is the cauſe why the
Moon,
ment.
US.
E V O
E X H
one 5
Moon, ſeen from the Earth, ap 1. When the Point B falls in A,
pears to librate a little upon its the Radius MC of the Evolute is
Axis, ſometimes from Eaſt to Weſt, equal to the Arch BC; but if not,
and ſometimes from Weſt to Eaſt ; to AB + the Arch BC.
and ſome Parts in the Eaſtern Limb 2. The Radius of the Evolute
of the Moon go backwards and CM is perpendicular to the Curve
forwards a ſmall Space, and ſome · AM.
that were conſpicuous, are hid, and 3. Becauſe the Radius MC of
then again appear.
the Evolute continually touches it,
Even NUMBER, is that which it is evident, from the Generation
can be divided into two equal Parts, of the Curve deſcribed from the E-
as 4, 6, 8, &C.
volution, that it may be deſcribed
EVENLY EVEN NUMBER, is through innumerable Points, if the
that which an even Number mea. Tangents in the Parts of the Evo-
fures by an even one, as 16 is an lute are produc'd until they be-
evenly even Number, becauſe 8, an come equal to their anſwerable
even Number, meaſures it by two, Arches.
an even Number.
4. The Evolute of the common
EVENLY ODD NUMBER, is that Parabola, is a Parabola of the fe-
which an even Number meaſures by cond kind, whoſe Parameter is 47
an odd one, as 20, which the even of the cominon Parabola.
Number 4 meaſures by the odd 5. The Evolutè of a Cycloid, is
another Cycloid equal and ſimilar
EVOLUTE CURVES, Ifa Thread to it.
FCM be wrapped, or winded 6. All the Arches of Evolute
about the Curve BCF, and then Curves are rectifiable, if the Radii
unwinded again, the Point M there- of the Evolute can be expreſſed
of will deſcribe the Curve AMM, geometrically.
This Doctrine of Evolute Curves
M
is very well explained and handled
M
by Mr. Huygens, in his Horologium
Oſcillatorium. See alſo what Dr.
James Gregory, Mr. Mac-Laurin,
and Sir Iſaac Newton in his Flu-
xions, have wrote upon this Sub-
B
ject.
EVOLUTION, in Algebra, figni-
fies the Extractions of the Roots of
any Powers,
EURI'T HMY, in Architecture, is
the exact Proportion between all the
Parts of any Building
EUS'TYLE, is the beſt manner
I of placing Columns, with regard
to their Diſtance, which Vitruvius
which Mr. Huygens, the Inventor, will have to be two Diameters and
calls a Curve deſcribd from Evolis- a Quarter, or four Modules.
tion; and the Curve BCF is the E EXAGON, the fame with Hexagon.
volute, the Part MC of the Thread Which ſee.
being called the Radius of the E EXHALATION, is any thing that
volute.
is raiſed up from the Earth by
3
.
C
Heat ;
EXP
E X T
Heat; as Vapours, Miſts, Fogs, ties be proportional, the Ratio of the
&c.
firſt to the third is ſaid to be the
EXHAUSTED RECEIVER, is the Duplicate of the Ratio of the firſt
Veſſel of Glaſs, &c. that ſtands to the ſecond, and of the ſecond to
upon the Body of the Air-Pump, the third ; therefore, according to
in order to have the Air pump'd out this, ; muſt be the double of ,
of it.
which is very falſe. But every one
EXHAUSTIONS, or the Method of knows the Logarithm of the Ratio
Exhauſtions, is the ancient Method of 1 to 9; that is, the Logarithm
of Euclid, Archimedes, &c. that of 9, is the double of the Ratio of
proves the Equality of two Magni- 1 to 3, or 3 to 9; that is, the
tudes by a Deduction ad Abſurdum, Logarithm of 3. From whence it
in ſuppoſing, that if one be greater appears, that Logarithms are more
or leſs than the other, there would properly the Exponents of Ratios,
follow an Abſurdity; and it is than numerical Quotients; and of
founded upon the firſt Propoſition this opinion ſeem Dr. Halley, Mr.
of the roth Book of Euclid. See Cotes, and others.
more of this Method in Prop. 2, EXPONENTIAL CALCULUS. See
10, 8c. lib. 12. Euclid.
Calculus Exponentialis.
EXPONENT of a Ratio, is the EXPONENTIAL CURVE, is that
Quotient ariſing from the Diviſion whoſe Nature is expreſſed by an
of the Antecedent by the Conſe. exponential Equation.
quent; as the Exponent of the Ra The Area of any exponential
tio of 3 to 2 is i }, and of the Curve, whoſe Nature is expreſſed
Ratio of 2 to 3 is Ž. And a Row by this exponential Equation, 2*=
of Numbers in an Arithmetical Pro- yi (making I tv=x,) will be
greſſion, beginning from o, being
placed over a Rank of geometrical
²+
U3
0.1.2.3.
Progreſſionals are called Exponents.
1. If the Conſequent be Unity,
the Antecedent itſelf is the Expo- 0.1.2.3.4. 0.1.2.3.4.5.
nent of the Ratio.
26 &c.
2. The Exponent of a Ratio is to
0.1.2.3.4.5.6.
Unity, as the Antecedent is to the
EXPONENTIAL EQUATION, is
Conſequent.
Altho' the Quotient of the Divi. that wherein there is an exponen-
fion of the Antecedent by the Con-' tial Quantity; as x*=y.
ſequent, is uſually taken for the
EXPONENTIAL QUANTITY, is.
Exponent of a Ratio ; yet in rea a Quantity whoſe Power is a vari.
lity, the Exponent of a Ratio ought able Quantity; as **, a*.
to be a Logarithm. And this ſeems EXTERIOR POLYGON. See Por
to be more agreeable to Euclid's De- bygon Exterior.
finition of Duplicate and Triplicate EXTERIOR TALUs, See Talus.
Ratio's in his sth Book,'than Quo EXTERMINATION Of the
tients. For 1, 2, and 9, are conti- known Quantity from an Equation, is
nual Proportionals ; now if be the the taking it away, or getting it
Exponent of the Ratio of 1 to 3, out of the Equation.
and or ş, the Exponent of the If there be two Equations, and an
Ratio of 3 to 9, and ; the Expo- unknown Quantity in each of them;
nent of the Ratio of 1 to 9; and has but one Dimenſion, it may be
Since Euclid fays, If three Quanti- exterminated by making an Equa-
I
I
O.1.2.
I
I
rott
سب کرد
1
Une
lity
E X T
EX T
ro byy-
c
c
C
lity between its different Values
axy-ds
Where-
found in each of them; as if a tox
be = bty, and cx + dy = 48. in x is reduced to one Dimenſion,
then in the firſt Equation *=6+ 'and ſo may be taken away, from
4g-dy
what has been already ſaid. In like
y-a, and x =
And
manner, if 33 xyt-abx, and
yu
48- dy
xx-xyt-e; in order to take out
then will bty-- a=
the laſt Equation muſt be multi-
9
plied by y; then will y3 = y**
wherein * is exterminated.
If the Quantity to be extermina- fion in both ; and fo xyy + abx=
xyyticy have the ſame Dimen-
ted be of one Dimenſion in one of
the Equations, and in the other it down to two Dimenſions. Then by
y is
has more, ſubſtitute its value in the
means of this and the moſt fimple
other Equation; as if x yy=as, of the given Equations yy=xx--XY
and x3 + 43 = bby -- aax: Then tcc, we may get out y entirely by
a3
what has been already ſaid.
in the firſt Equation x =
and
If there be ſeveral Equations, and
this Value being put for x in the
as many unknown Quantities, the
Buſineſs of exterminating an un-
ſecond, and it will be jo + 93 =
known Quantity muſt be performed
gradualiy; as if ax=yz, x+y=,
and
bby ; wherein x is gotten
5x=y+3%. If the Quantity »
уу
be inade choice of, the Value
aa
as
out.
y z
a
j z
5.72
a
When in neither of the two Equa-
of one of the other Quantities
tions the unknown Quantity to be
exterminated does conſiſt of one Di.
x or 2, ſuppoſe % (found by the
menfion, the Value of the greateſt for it in the ſecond or third Equa-
firit Equation) muſt be ſubſtituted
Power of muſt be found in each
Equation ; and if thoſe Powers be tion ; by which means we ſhall ob-
not the fame, the Equation having tain
+y=%, and
the leait Power, muſt be multiplied
-y +
by the Quantity to be exterminated, 3%. From whence at laſt z may
or by its Square, or Cube, &c. till be taken away, as above.
it has the ſame Power with that in When the unknown Quantity, is
the other Equation ; then an Equa- of ſeveral Dimenſions, it is ſome-
lity muſt be made between the Va- times very troubleſome to get it out,
lués of thoſe Powers, by which and the Labour will be very much
means a new Equation will arife, ſhorten’d by the following Exam-
wherein the greateſt Power of the ples, being as ſo many Rules.
Quantity to be exterminated will 1. From a x x +bx+c= 0,
be diminiſhed, and by a Repetition and f x x to g*th = 0, being
of the Operation will at length be exterminated, there
exterminated; as if xx taxi
ab-bg20fx ab-tobog
and axg (xxd3. And x be to
be exterminated, in the firſt Equa- x bf tagg
x bft aggoffxc=0.
tion xx will be
2. From a x3 + 6 x x + cx for
a x ge--13 doo, and f * x + gx + b
in the latter xx
0, x being exterminated, there
ccmes
out
:byg
b уу?
a a, and
; and
c.
comes
83
3 X X
45 * *
xbdfh
EXT
E X T
comes out ab-bg-20fx a bht Quantities] 1,-4, 6,
3 x ġ I, *
XX, + 3, and y reſpectively,
bh-og-2df x bf htch-dg
and there comes out 3 -- ** * * *
x aggtcft +3 agh+688 + dff
x df=o.
*9-6*** *4-3x+x3 + 6x x
3. From a x+ + 6*3 +4**+ -3x+x3: +3xx xxx +9x-3*3
dx tezo, and f x'x +g**
h=0, * being exterminated ,
3 % o. Then
there comes out ab-bg-2cf * tities, and multiplying, you have
blotting out the ſuperfluous Quan-
a b3 + bb-cg-2df x bf hht 27–18 xx+3**, -9xx+x0,+
agstoff schb-dgb tegs- 3** -18**+2x+=0. And order-
ing (duly) 46 + 18 x4
ze fb + 3 agh +688 - dff x + 27 =0.
dfhtzabb +3bg bandfg teff
Theſe Rules, to be found in Sir
xeff-bg? 2abxef g8=o.
Iſaac Newton's Algebra, may be
4. Froma x3 + 6** **+ carried higher at pleaſure ; but
d=0, and fx3 + g *2+bx+ then their Inveſtigation becomes
k
o, * being exterminated, very troubleſome. However there
have been ſome Perſons, who have
there comes out ah -bg
bg - 2cf been at the pains to compute a ge-
x aabh-achktakt-bh-g2df neral Rule for the Extermination
of the unknown Quantities from
toak +66 +228 + 3d
f Equations, wherein they have any
x a a kk : +odb-ddg-cckt Dimenſions whatever. But the Ap-
plication of the Rule to particular
2 bdk x agg + cff: + 3@gh Caſes, is oftentimes more tedious
+688 + dff-3a fk xdd f
-
than their Inveſtigation by the com-
mon way.
3 akabbt ogt dfx b c fkt bk Sir Iſaac has not ſhewn how he
-2 dg xbbfk:-bbk-3adh-cdf found them out; becauſe that fol.
xbbfk :-bbk – zadbcdf lows fo eaſily from what has
been
a g k =0,
For Example, to exterminate x
faid: For Example, in the firſt Rule,
out of the Equations x x + 5 x
we have xx +
3 y 30, and 3 * * - 2xy +40:
= 0,
i reſpectively ſubfticute in the firſt.
g *
b
Rule for abs, fg, and h, [theſe and xx+
There-
Quantities, viz] 1, 5,
f
3 y y i
bx
3, – 2y and 4; and duly obſerv-
gx
b
fore
ing the Signs + and --,
-, there
+
f f f
ariſes 4 + Toy + 18 yy x 4+
fo x =
And if this
by3 x 15 + 4yy
bf
27yy
ag
3yy=0, or 16+ 40 y +72yy Value of x be put in the Equation
+ 300 – 9033 + 69 94 = 0. a xx+b x ti=0; we ſhall have
În like minner that y may be a3 bh
2 aacf, accff
gotten out of the Equations ;3
+
myy--3x=o, and yy + xy-*x+
bf - agx x bfas
3=0, ſubſtitute in the ſecond ahh
Rule for a,b,c,d,fg,h, and x, [theſe bf-ag
+=; which E-
quation
Х
1
bx
C
+
a
+
O.
C
+
; and
U
a
u
1
ah-cf
20
X
2
bcf
E X T
E X T
XCO.
sb
а a
quation being clear'd of Fractions, quation, and the numerical Quo-
and then contracted as much as pof- tients be reſpectively affix'd to them ;
and if the Powers to be reſolv'd be
fible will become abbġ – 20f fubtracted from the Sum of the gi-
xah+bb-cgxbf tagstoff ven Parts, and the Difference be
After the ſame way, al- callid +b; and if, in the next
tho' with increaſing Trouble, the Place, the Sum of all the Co-effi-
other three Rules may be invefti- cients in the fecond Column be
gated. If I remember right
, Rho- made equal to s; and laſtly, if in the
nane in his Algebra has done this.
third Column there be put down
EXTERNAL ANGLES. See An- the Sum of all the Co-efficients,
gles External.
which call t; then will the Root z
EXTRA-MUNDANE SPACE, is
the infinite void Space, which, by be nearly = at
ſome, is ſuppoſed to be extended be
sstth
yond the Bounds of the Univerſe ;
is tv Iss tb2.
and conſequently, in which there is nearly a +
really nothing at all.
EXTRACTION of Roots, is the If azt bz² to 23 + d z4 t
Method of finding the Root of a exs+fzº, &c. Egy + y2 +
Number or the value of an un- i y3 + k + + lys + mjo, &C.
known Quantity of an Equation. then will the Root of this infinite
In moſt Books of common Arith-
—БАА
metic, you have the manner of ex- Equation be x = y +
tracting the Square and Cube Root
of a Number. The Analytical Wri - 2b AB - CA3
ters who fhew how to do this in 32+
y3 +
Species or Algebra, as well for
k-bB2- 26AC-3CAB? —d44
pure Powers, as adfected Equa-
tions, are Oughtred, in his Key
to Mathematics. Vieta, in his y?,&c. Where it muſt be obſerv'd,
Tractatus de numerofa Poteßatum that every Capital Letter is equal to
purarum atque affectarum Refolun the Co-efficient of each preceding
tione. Sir Iſaac Newton, in the Term; as the Letter B is equal to
Commercium Epiflolicum ; in his
bm-b A2
Fragmenta Epiſtolarum, publiſhed by the Co-efficient of
Mr. Jones. And in Dr. Wállis's
Algebra. Ozanam's Nouveau E-
1. The Denominator of every
lemens d’Algebre, lib. 2. p. 267.-
Co-efficient is always a.
Ralphſon, in his Analyſis Æquationum 2. The firſt Member of each Nu-
Univerſalis. Monſieur de Lagney. merator is always a Co-efficient of
Dr. Halley, in the Philoſophical the Series 8.3 + hy2 to i y3, &c.
Tranſactions. Mr. Colfon, in his viz. the firſt Numerator begins with
Commentary upon Sir Iſaac New. the Co-efficient g, the ſecond Nu-
ton's Fluxions; and many others. merator with the ſecond Co-effi-
If z, the Root of any adfected cient h, ETC.
Equation, be ſuppoſed to be com 3. In every Member after the
poſed of the Parts + a, or €, firſt, the Sum of the Exponents of
and if from the Quantity ate, or the Capital Letters is always equal
e, there he forni'd all the to the Index of the Power to which
Powers of x found in the given E- this Member belongs: Thus, if
you
a
F
E X T
F A C
you conſider the Capital Letter, EXTREMES' (DISJUNCT,) are
km B?—26 AC-36 A²B A4, the two circular Parts remote from
the aſſumed middle Part. See more
of this under Spherical Trigonometry.
which belongs to the Power y4, in
Eve, an Organ of the Body, re-
every Member you will ſee that
b Ba, 26 AC, 3¢ A? B, d A4 ; confifts of five Tunics, viz. the
preſenting whatever is viſible, and
the Sum of the Exponents of the Cornea, Sclerotica, Uvea, Choroide,
Capital Letters is 4.
Retina : And three Humours, the
4. The Exponents of the fame
Letters which are written before the Aqueous, Cryſtalline, and Vitreous.
Capitals, expreſs how many Capi-
tals there are in each Member.
5. The Numerical Figures that
F.
happen in theſe Members, expreſs
the Number of the Permutations,
which the Capital Letters of each
ACE, or FACADE, in Archi-
Member are capable of.
tecture, , is a flat Member
EXTREME and MEAN PROPOR-
which hath a great Breadth, and
divided in C, that the Rectangle outward Part of a great Building,
TION, is when a Line AB is to ſmall Projecture; as in Architraves,
or
which immediately preſents itſelf to
A
B view.
FACE of a Baftion, or, of the Bul-
under the whole Line A B and the wark, is the moſt advanced Part of
lefſer Segment C B is equal to the a Baſtion toward the Field, or the
Square of the greater Segment A C. Diſtance comprehended between the
How to divide a right Line after Angle of the Shoulder, and the
this manner is taught by Euclid, in flanked Angle.
Lib. 2. of his Elements of Geometry, Face of a Place, is the Curtain,
But no. Number can be fo divided together with the two Flanks raiſed
into two parts; as is well demon- above it, and the two Faces of the
ſtrated by Clavius, in his Commen- Baſtion that look towards one ano-
tary upon Lib.9. of Euclid. This ther, and flank the Angle of the
is alſo evident enough thus : Let a Tenail.
be the Number, and x the greater FACE prolonged, in Fortification,
Part, then the leſſer Part will be a is that part of the Line of Defence-
*; and ſo a a - ax = xx; and Rafant, which lies between the
a tavs
Angle of the Shoulder and the Cur-
thence x =
And tain; or, 'tis the Line of Defence-
Rafant diminiſhed by the Length of
fince the ſquare Root of 5 cannot be a Face.
had in Numbers exactly, it is plain FACIA, or FASCIA, ſignifies any
that the Value of x, partly conſilting flat Member, as the Band of an
of that ſquare Root, multiplied by Architrave, &c. There are ſome
a, cannot be had exactly in Num- who write Faſcie, grounded upon
bers neither.
the Latin Word Faſcia, a large T'ur-
Extremes (CONJUNCT,) in ban, which Vitruvius makes uſe of
right-angled ſpherical Trigonome- on the like Occaſion.
try, are the two circular Parts that FACTORS, in Multiplication, the
are next to the middle Part.
Part. And Multiplicand and Multiplicator are
called
2
FAS
F E L
called Factors, becauſe they do make make up the Parapets of Trenches,
or conſtitute the Product.
&c. Some of them are dipped in
Faint Vision. .See Vifion. melted Pitch or Tar, and being ſec
FALCATED. The Moon, or any on fire, ſerve to burn the Enemy's
Planet, is ſaid to appear falcated, Lodgments, or other Works.
when the enlightened Part appears FAUCON, a ſort of a Cannon,
in the Form of a Sickle, or Reap- whoſe Diameter at the Bore is five
ing-Hook, which is when he is Inches and a quarter, Weight ſeven
moving from the Conjunction to the hundred and fifty Pound, Length
Oppoſition, or from New Moon to ſeven Foot, Load two Pound and a
the Full; but from Full to a New half, Shot two Inches and a half
again, the enlightened Part appears Diameter, and two Pounds and a
gibbous, and the dark falcated.
quarter Weight.
FALCON. See Faucon.
FAUCONET, a ſort of Ordnance,
FALCONET. See Fauconet. whoſe Diameter at the Bore is four
FALSE ATTACK. See Attack. Inches and a half, Weight four
FALSE BRAYE, in Fortification, hundred Pounds, Length fix Feet,
is a ſmall Mount of Earth four Fa- Load one Pound and a quarter, Shot
thom wide, erected on the Level ſomething more than two Inches
round the Foot of the Rampart, Diameter, and one Pound and a
on the side of the Field, and ſepa- quarter Weight.
rated by its Parapet from the Berme, FAUSSE BRAY E. See Falſe Braye.
and the Side of the Moat. 'Tis FEATHER-EDGED, is a Term
made uſe of to fire upon the Enemy, uſed by Workmen, for ſuch Boards
when he is already ſo far advanced, as are thicker on one Edge, or Side,
that you cannot force him back than on the other.
from of the Parapet of the Body of Fellows, in Fortification, are
the Place; and alſo to receive the fix Pieces of Wood, each of which
Ruins which the Cannons make in form an Arch of a Circle ; and theſe
the Body of the Place.
joined all together by Duledges,
FALSE POSITION. See Poft make the Wheel of a Gun-Carriage.
tion.
Their Thickneſs is uſually the Dia-
FASCIA. See Facia.
meter of the Bore of the Gun they
FASCIÆ, from Bands, or Sevathes, ferve for, and their Breadth ſome-
are certain Places in the Diſks of the thing more.
Planets Mars and Jupiter, that ap FELLOWSHIP, or the Rule of
pear lighter, or inore obſcure than Fellowſhip, in Arithmetic, is a Rule
the reſt of their Bodies, being ter that teaches how, by having given
minated by parallel Lines, and ſeem the ſeveral Stocks of Perſons that
ſometimes broader, and ſometimes are Partners together in Trade ;
narrower, and do not always poſſeſs to proportion to every one of them
the fame Place of the Diſk.
his due Share of Lors or Gain.
A very broad, but di fkith Fafcia The Rule of Three, ſeveral ways
was obſerved in the middle of the repeated, will fully anſwer any
Planet Mars by Mr. Huygens, in the Quettion in chis Rule.
For as the whole Stock (or ge-
FASCINES, or FAGGOTS, in For- neral Antecedent) : is to the Total
tification, are ſmall Branches of thereby gained or loft, (which is
Trees, or B:vins, bound up in the general Conſequenc) :: ſo each
Bundles, which being mixed with Man's particular Share : is to his
Earth, ferve to fill up Ditches, to projer Share of Loſs or Gain.
FIBRES,
year 16;6.
1
FIG
1
FIB
Fieres, are the ſmall Threads, pounded of the inverſe Ratio of the
or Filaments, of which elaſtic Bo- ſquare Roots of the Weights, by
dies are, or may be ſuppoſed to be which the Chords are ſtretched, of
made.
the Ratio of the Lengths of the
1. The Elaſticity of Fibres con- Chords, and of the Ratio of the
fiſts in this, that they can, bé ex- Diameters.
tended, and taking away the Force 9. Every Particle of a ſtretched
by which they are lengthened, they String or Wire, any how fet in no-
will return to the Length which they tion, and cauſing Sound, uniformly
had at firſt.
vibrating backwards and forwards,
2. Fibres have no Elaſticity, un- with a very ſmall Motion, is always
leſs they are extended with à certain accelerated and retarded according
Force.
to the Law of the Vibration of a
3. When a Fibre is extended with Pendulum. The periodical Time of
ſo inuch Force, it loges its Elaſticity. One Vibration, being to the Time of
4. The Weight by which a Fibre the Deſcent thro' half the Length of
is increaſed a certaiu Length by its the String by the Force of Gravity,
ſtretching, is, in the different De. in the fubduplicate Ratio of the
grees of Tenſion, as the Tenſion Weight of the String to the Force
itſelf.
ſtretching it.. And from hence is
5. The leaſt Lengthenings of the is computed, that a Mufical String,
ſame Fibres are, to one another, ſounding. De la Solre, performs 250
nearly as the Forces by which the Vibrations in a Second of Time.
Fibres are lengthened. Therefore, FICHANT FLANK. See Flank.
in all the leaſt Inflections of a FICHANT LINE of Defence. See
Chord, Muſical String; or Wire, the Fixed Line of Defence.
Sagitta is encreaſed and diminiſhed Field-FORt. See Fortine.
in the fame Ratio as che Force with FIELD-Pieces, are ſmall Can-
which thë Chord is inflected. non, which are uſually carried along
6. In Chords of the ſame Kind, with an Army in the Field; ſuch
Thickneſs, and which are equally as Three Pounders, Minions, Sakers,
ftretched, but of different Lengths, Six Pounders, Demi-Culverins, and
the Lengthenings, which are pro- Twelve Pounders; and theſe being
duced by ſuperadding equal Weighes, finall and light, are eaſily carried.
are' to one another, as the Lengths FIELD-STAFF, is a Staff carried
of the Chords. If the Forces by by the Gunners, being about the
which the Fibres are ſtretched be Length of a Halbert, with a Spear
equal, and they are infected by equal at the End, which to cach Side has
Forccs, even in that Caſe allo che Ears ſcrew'd on like the Cock of a
Sagittă will be equal, however dif- Matchlock; and the Gunners fcrew
ferent the Thickncís be.
lighted Matches in theſe when they
7. If there be two equal and fimi are on Duty, this being called Arm-
tar Chords, but unequally ſtretched, ing the Field Staff
the Squares of the Times of the Firth, a Term in Muſic, being
Vibrations' are to one another in the ſame as Diaperte. Which fee.
veifely as the Weights by which the FIGURA L' (or FIGURATE),
Chords are' ſtretched.
NUMBERS, are ſuch as do, or may
3. Any Chords of the ſame kind repreſent fome Geometrical Figure,
being given, the Darations of the in relation to which they are always
Vibrations may be compared toge confider'd; as triangular Numbers,
ther; for they arë in a Ratio com- Pentagonal Numbers, Pyramidal
R Numbers,
1
FIG
F I G
Numbers, &c. Of which ſee more and Tranſverſum in the Hyperbola
under the reſpective Words.
and Ellipfis.
FIGURATIVE DISCANT. See FIGURE, in Geometry, is a
Diſcant.
Space encompaſſed round on all sides,
FIGURE, in Phyſics, or Natural and is either Rectilineal, Curvili-
Philoſophy, is the Surface or ter- neal, or Mixed.
minating Extremes of any Body. FIGURE of the Secants, is a me-
FIGURE, in Conic Sections, ac chanical Curve thus generated : Let
cording to Apollonius, is the Rectan- PO be a Tangent to the Circle
gle made under the Latus Rectum Q50, and let an infinite right Line
E
F
L
H
P
S
d
D
1
a G
Ko
І) А В
d
ch
PL
е e
POR revolve about the Centre 0, Radius QO of the Circle. The
cutting the Circle in S, and the reaſon of this is, becauſe the infinite
Tangent in P: then if upon the in- Secant POR revolving perpetually
finite Baſe, or abſciſſal Line A K, be about the Centre, round and round
taken the Point A, and afterwards again, will be affirmative, and ne-
the Abſciſs A B be taken upon the gative by turns, paſſing from the
fame, always equal to the circular one to the other as often as
goes
Arch Qs, and the correſpondent through Infinity (ſpeaking in the
Ordinate BC at right Angles to it, modern Style:) where it is to be
be equal to the Secant O P of that obſerved, that ſo much of the Curve
Arch, and moves along AK: By as appears in the Figure, is de-
this Motion the Extremity C of that ſcribed during the Motion of the
Ordinate will deſcribe the Curve Secant, from the Situation Qo, till
EDC, called the Figure of the Se- it has moved once and a half about.
The Quadrature of the Space
This Curve, in reality, conſiſts of ADCB will give the meridional
an infinite Number of ſuch Parts; Parts for a given Latitude in Mer.
of which EDC is one, having an cator's Chart. And this may be ob-
infinite Number of parallel Alym- tained by the Quadrature of an hy-
ptotes FG, HI, LK, drawn at perbolic Space, or, which is the
Diſtances from one another, each e- ſame thing, by the Logarithms : For
qual to half the Circumference of the if the Circle ol$ be a Meridian,
Circle QSO, which Parts do alter Ra Point of the Equator, and S a
nately fall above and below the abs Point whoſe Latitude is es, it is
ſciſſal Line AK: the leaſt Ordinates well known, that its meridional
being a d or AD, each equal to the Parts, or Latitude, is to its true
Magnitude
cants.
و ۶ نو
3
>
Nrr
و ور
-- و و 1
rr
F I G
FIG
Magnitude, as the Sum of the Se- ſeveral Sines; but in a given Ratid
cants ſtanding upon this, is to the to them.
Sum of ſo many times the Radius; Any Space ABC of this Curve
that is, as the curvilinear Area A B is ſquareable: For ſuppoſing r to be
CD, to the Rectangle D A B in. the Radius of the generating Circle,
ſcribed in it. Now if ol or AD and the Sine or Ordinate BC to té
be called r; A B or QS, X; and y, the Fluxion of that Area will be
BC or OP, y; we ſhall Have x =
and the Fluent of this
rr
and the Fluxion of
✓
rr - II
у
✓
уу - rr
Flužion will be rem
,
the Area ABCD will be
that is, the Space ABC is equal to
rry
the Rectangle under the Radius, and
: and the Fluent of this the verſed Sine of the Arch of thë
generating Circle to which the Ab-
Fluxion may be had from the Tables ſciſs A B is equal; fo that the Area
of Mr. Cotes's Harmonia, viz. the of the whole Space ACG is equal
6th Form of Fluxions, it being the to twice the Square of the Radius.
Logarithm of the Ratio of oto The Curve cannot be rectified
OP - PR, or of Radius to the even by means of any Space belong-
Tangent of half the Complementing to the conic Sections; for its
of the Latitude, the Radius being
the Module (as Mr. Cotes calls it)
V
Fluxion is j
of the Canon of Logarithms; that
уу
is, the Number 0.434294481903,
&c. in Brigg's or Ulacque's Loga- Subtangent will be
✓
rithms.
Уу
Figure of the Sines, is a me-
This Curve (as generally defined
chanical Curve ACG, generated above) is that, into which a ſtretch'd
much after the ſame way as the Fi- String or Wire perpetually conforms
gure of the Secants, the Difference itſelf, when it is fet a vibrating by
being only, that here every Ordi- a Quill or other ſmall Force; as
nate BC, anſwerable to the Abſciſs eaſily follows from what Dr. Taylor
A B, is the Sine of the correſpon- has ſaid, concerning the Motion of
dent Arch QS of the Circle," (ſee a ſtretch'd String, in the Philoſophia,
the Figure of the Secants,) inſtead of cal Tranſaktions, Nº 337-
being its Secant as OP: This Curve
The firſt who I can find took noë
confiiting of an infinite Number of tice of this Curve, was Father Fa-
Parts, fuch as ACG, alternately bri, in his Synopſis Geometrica, pub-
riſing above, and falling below the lithed about the year 1669, wherein
abfciffal Line Al; which, in reality, he gives a Diſcourſe concerning the
malse but one continued infinite fer-
ſame; and this makes me wonder;
pentine Line.
why Wolfius, in his Elementa Ma-
Note, Some define this Curve more thef. Univers, ſhould attribute the
Invention to Mr. Leibnitz ?
C
A/
G
FIGURE of the Tangent, is a me-
I
chanical Curve E A CD, generated
B
HT
like the figure of the Secants, (ſee
above under that Word) with this
enerally, by making the ſeveral Difference, that the Ordinate BC
Ordinates BC, not only equal to the is here equal to the Tangent QP
jo y.
r r
म
N
R2
of
FIR
FIX
1
to
of the Arch QS, to which the fixed Stars, or the Height of Hea
Abſciſs A B is equal ; the Curve ven. But more properly 'tis that
Space which is expanded or appears
D
arched over us above in the Hea-
vens.
С
FIRST MOVER. See Primum
Mobile.
Fissures, are certain Interrup-
А.
tions, that horizontally or parallelly
divide the ſeveral Strata, of which
the Body of our terreſtrial Globe is
compoſed.
Fixed Line of Defence, in For-
E
tification, is a Line drawn along the
Face of the Baſtion, and terminates
confifting of an infinite Number of in the Courtin.
ſuch Parts, of which EAD is one, Fixed Signs of the Zodiac, are,
and having a like Number of paral- by fome, Taurus, Leo, Scorpio, and
lel Aſymptotes at equal Diſtances Aquarius, being ſo called, becauſe
from each other.
the Sun paſſes them reſpectively in
FIGURES, in Arithmetic, are the the middle of each Quarter, when
nine Digits, or numerical Charac- that particular Seaſon is more ſettled
ters, 1, 2, 3, 4, 5, 6, 7, 8, 9, and o. and fixed than under the Sign that
FIGURES CURVILINEAL, are begins and ends in it.
ſuch as have their Extremities crook FIXED STARS, are ſuch that
ed; as Circles, Ellipfes, &c. conſtantly keep at the ſame Dif-
FIGURES MIXED, are ſuch as tance, with reſpect to each other.
are bounded partly by right Lines, 1. The firſt who compoſed a Ca-
and partly by crooked ones; as a talogue of the fixed Stars was Hip-
Semi-circle, Segment of a Circle, parchus of Rhodes, about a hundred
&c.
and twenty Years before Chriſt, who,
FIGURES PLANE, (or Plane Sure from his own, and the Obſervations
faces,) are ſuch as are terminated of ſome before him, collected a
and bounded by right Lines only. thouſand and twenty. two Stars, ac-
FIGURES RECTILINE AL., are cording to their proper Latitudes
thoſe that have their Extremities all and Longitudes : And ſo, in Pliny's
right Lines, as Triangles, Quadri- Judgment, dared to do a thing which
laterals, &c. Polygons regular, ir- God himſelf did not approve of, in
telling the Number of the Stars for
Filler, is any little ſquare Poſterity, and reducing them to a
Moulding, which accompanies or Standard.
crowns a larger.
2. Ptolemy augmented Hipparchus's
Finite, is what hath fixed and Catalogue with four Stars more.
determined Bounds or Limits ſet to And Ulegbeigh, the Grandſon of
its Power, Extent, or Duration. Tamerlane the Great, placed a thou-
FINITOR, the ſame with the ſand and ſeventeen in his Catalogue,
Horizon; and 'tis ſo called, becauſe who ſays in his Preface, That he
the Horizon finiſhes or terminates obſerved all that could be obſerved,
your Sight, View, or Proſpect. beſides twenty-ſeven in the South.
FIRMAMENT, by ſome Aftronos 3. The next who made a Cata-
mers, is taken for the Orb of the logue was Tycho Brahe, of ſeven
1
hundred
tegular, &C.
1
ments.
Year 1677
1
FIX
FIX
hundred and ſeventy-ſeven fixed Teleſcopes he ſtill found out more.
Stars, from his own Obſervations ; 'And Anthony Maria de Rheita affirms,
and would admit 'no Star into his that in the ſingle Conſtellation of
Catalogue, but what he had found Orion, he number'd above two thou-
out, and inveſtigated by his Inftru- fand Stars, by help of a Teleſcope.
9. Several fixed Stors, obſerved
4. Dr. Halley was the firſt who by the Ancients, vanith, or cannot
obſerved rightly the ſouthern fixed now be ſeen ; and new ones appear
Stárs at St. Helena, being three hun- for a time, and then vaniſh. The
dred and ſeventy-three in Number; Light of ſome Stars alſo diſappear,
and computed their Places for the and after a ſtated Period they ſhine
again : Among which is that emi-
5. Hevelius of Dantzic likewiſe nent one in the Neck of the Whale,
made a Catalogue of the fixed Stars, which for eight or nine Months is
containing one thouſand eight hun- not ſeen; and the other four or
dred and eighty-eight in all, viz. three Months it appears, varying its
nine hundred and fifty known by Magnitude.
the Antients, and fix hundred and 10. The fixed Stars, like the reſt
three, which he calls his own, and of the Planets, appear every Day
three hundred and thirty-five of Dr. to riſe and ſet, and to move with
Halley's, which could not be ſeen in a circular Motion from Eaſt to
the Horizon of Dantzic.
Weſt in twenty four Hours, in Cir-
6. But Mr. Flamſtead's Catalogue cles whoſe Planes are parallel to
of Stars, contain'd in his Hiftoria the Equator.
Cæleſtis, is far more numerous and II. The fixed Stars, beſides their
exact than any of the others; for it former apparent Motion round the
contains three thouſand Stars ; but Earth, ſeem to have another quite
inany of them cannot be obſerved contrary to that. By this they ap-
without a Teleſcope; ſo that you pear to change their Longitude, or
cannot obſerve above a thouſand Diſtance, from the Beginning of
by the naked Eye in the viſible Aries forward, according to the
Hemiſphere: And this ſeems won- Order of the Signs, or to move in
derful to many, that in a ſerene conſequentia, by a ſlow Motion of
Night, when the Moon does not about one Degree in ſeventy Years.
ſhine, at firſt ſight the Stars appear So that thoſe Stars, that in Hippar-
to be innumerable: But this pro- chus's time were in Aries, are now
ceeds from the Fallaciouſneſs of in Taurus, &c. And the Proceſſion
Viſion, proceeding from the ve- of the Terreſtrial Equinoxes is the
hement Twinkling of the Stars, Cauſe of this apparent Motion.
while the Eye obſerves them all to. 12. The Light of the fixed Stars
gether confuſedly, and without is much more Itrong and vivid than
Order.
that of the Planets, altho' their ap-
7. Yet the Number of the fixed parent Diameters are much leſs ;
Stars, obſervable by a Teleſcope, is becauſe the Stars, like the Sun, ſhine
vaſtly great ; for direct a good Te- by their own Light, and the Planets
leſcope to the Heavens, and there only by the Reflection of the Sun
will appear great Multitudes, eſpe 13. The fixed Stars twinkle much
cially in the Via Lactea.
more than the Planets; becauſe
8. Dr. Hook, with a Teleſcope of their apparent Diameters being very
twelve Feet, obſerved ſeventy eight ſmall, the leaſt Atom, or Particle
Stars in the Pleiades; and with longer of Mattér, floating in our Atmo-
ſphere,
R 3
F L A
F L O
1
are
{phere, will hinder, for a Moment, ought to be made, to flank the q.
the Stars being entirely viſible; as ther. Hence the Courtin is always
the thick Smoke of a Chimney the ſtrongeſt Part of any Place, be-
will do the Planets themſelves, cauſe 'tis
flanked at each end.
which will winkle in ſuch a FLANK (FICIANT,) is that from
caſe.
whence a Cannon playing, fireth its
14. The Diſtance of the fixed Bullets directly in the face of the
Stars from us is vaſtly great; be- oppoſite Baſtion.
cauſe they have no ſenſible Paral FLANK (RASANT,) is the Point
lax ariſing from the annual Motion from whence the Linė of Defence
of the Earth. Tho? Mr. Flamſtead begins, from the Conjunction of
ſays, that the annual Parallax of the which, with the Courtin, the Shot
Pole-Star is forty Seconda ; and Mr. only razeth the Face of the next
Huygens tells us, that with Teleſcopes, Baſtion, which happens when the
which would magnify the apparent Face cannot be diſcovered, but from
Diameter above a hundred times, the Flank alone.
he could never diſcover any ſenſible FLANK (RETIRED,) or the lower
Magnitude in the fixed Stars. or covert Flank, is that exterior
FLANK, in Fortification, is that Part thereof, whoſe advanced Part
Part of the Baſtion which reaches if it be rounded, is called the Oril-
from the Courtin to the Face, and lon; ſo that this Flank Retiré, as the
defends the oppoſite Face, the Flank, French call it, is only the Píatform
and the Courtin.
of the Caſemate, which lies hid in
There is alſo the oblique or fe- the Baſtion.
cond Flank, which is that part of FLANKS SIMPLE, Lines
the Courtin, where they can ſee to which go from the Angle of the
fcour the Face of the oppoſite Ba- Shoulder to the Courtin, and whoſe
fion; and is the Diſtance between principal Function is the Defence
the Lines Raſant and Fichani. of the Moat and Place,
The low, covered or retired Flank, FLANKED (or DOUBLE TE-
is the Platform of the Caſemate, NAILLE.) See Tenaille.
which lies hid in the Baſtion.
FLANKED LINE of Defence. See
FLANK, is alſo a Term of War, Rafant Line of Defence.
fignifying one ſide of a Battalion FLANKING ANGLE. See Angle.
of an Army; as to attack the Ene FLANKED ANGLE, is the An-
my in Flank, is to diſcover and fire gle formed by the two Faces of the
upon them on one ſide.
Baſtion, and ſo forms the point of
FLANK of the Courtin, or ſecond the Baltion.
Flank, is that part of the Courtin FL'A'r BasȚION. See Baſtion.
between the Flank and the Point, FĻAT-BOTTOM'D MOAT. See
where the Fichant Line of Defence Moat.
terminates.
FLAT CROWN. See Corona,
FLANK 4 Place, is to diſpoſe a Flie. That Part of the Ma.
Baſtion, or other like Work, in ſuch riner's Compaſs on which the thirty
manner, that there ſhall be no part two W.inds are drawn, and to which
of it, but what is defended ; ſo as the Needle is faften'd underneath,
you may from thence play upon they call the Flie.
Front and Rear. For any Fortifi FLOATING BRIDGE, is a Bridge
cation, that hath no Defence but made in form of a Redoubt, con-
juſt right forwards, is faulty ; and fiſting of two Boats, covered with
to render it compleąt, one Part Planks, which ought to be ſo ſolidly
framed
1
1
goz
Х
S
S2
е с
Х
Х
gic ત
sam 4 ^zni
-
e A
х
fza:
F LU
FL U
framed, as to bear both Horſe and =Q, and on-n=p:) will be
Cannon.
«А
FLUENT, or flowing Quantity of
ex
+
a Fluxion, is that Quantity (whether
fun
Line, Surface, Solid, &c.) of which е В
9-4
eD
it is the Fluxion; as the Fluent of X
fan
+
fan
* is x, and the Fluent of xy + yx
is xy; and fo of others.
the Letters A, B, C, D, &c. expref-
It is eaſy to find the Fluxions, in fing the neareſt antecedent Terms,
all caſes of given Fluents, and that
P
exactly ; but on the contrary, it is viz. A the Term ; B the Term
very difficult to find the Fluents of
given Fuxions. Indeed there are
infinite Caſes where there cannot be
c. This Series
exactly had, unleſs by the Quadra- where r is a Fraction or negative
ture of curve-lin'd Spaces ; and ſince
the Areas of Curves in order above
Number runs on infinitely; but when
the Conic Sections cannot be ac-
g is a whole affirmative Number, it
curately expreſſed in Numbers, A. becomes finite, conſiſting of ſo many
in r.
nalyſts have been obliged to be
The Fluents of the following
content with Fluents, expreſſed near
Forms of Fluxions may be had in
the Truth by Series's. This Doc-
trine was firſt invented by Sir Ifaac finite Terms, viz. of dż z
0 12 I
Newton, and notwithſtanding the
many Authors upon the Subject after Vetfx", when 6 is a poſitive
him (except Mr. Cotes) he has whole Number, and n an whole
handled the Buſineſs more profound-
on + 1
ly, and carried it much farther than Number. Of dż z
any of them, whether Englik of ve+fz", when e is a negative
Foreigners. See his Fluxions, and
Of
Quadrature of Curves. See alſo Mr. whole Number,
Cotes's Harmonia Menſurai um
On
11. The Fluent of the Fluxion
when e is ą poſia
+mt
vit f 2
ż will be
tive whole Number. OF
n ton - I
2. The Fluent of az
when 8 is a
& t f an
+onton
be
#mt
3. The Fluent negative whole Number. But when
o is a whole Number, and the Sign
of d zºž x et f2mm (where d, e, cannot be had but by the Quadra-
thereof is otherwiſe, the Fluents
f, expreſs any given Quantities, and ture of the Hyperbola or Ellipſis,
0, n, and m, the Indexes of the or by the Logarithms or Tables of
Powers of the Quantities to which Sines, or elfe by infinite Series's :
0+ It being an Hyperbola when the
they are affix'd, by making
Sign of f 21. is t, and an Ellipfis
d
when it is Fluents of the fol-
muti
lowing Terms of Fluxions may be
R 4 obtained
2
dzz
tim
az
#mti
0
d ž z
niż, will
an
%
n
N
Fremtr=un xe +fxen)*
F LU
F L U
on I
I. dizon
dizonthn
3.
dżiz
d
1
72
ent
2
I
dbtained by the Hyperbola and The Expreſſions of the Fluents of
moft of theſe forms of Fluxions
Ellipfis
.
may be ſeen in Sir Iſaac Newton's
ët if
Quadrature of Curves, and of all
-1
of them in Mr. Cotes's Harmonia
Menfurarum. From which Treatifę
et fa?
we learn, that Mr. Cutes could find
the Fluent of any Fluxion whoſe
d z zo?
0nI
Vet fan Form might be compared with
8+b?
or+r-1
dzzon + { ™-IV'et frem
{where d, e, fi
et fan
& + bz?
are any given Quantities, and
d zz
” any Index of the variable
$-
Quantity; any affirmative or
§ + bz? Vetfz"
pegative whole Number, and I
di
onth ni
6.
any Fraction,) by means of the Hy:
gth an Vetfan
perbola and Elliptis. Moreover, he
affirms, that he could find the
Fluents' of Fluxions of this Form
17. dż0 1 - 1
Ve+f2"
d
gth
+
1
or even
1-1 Vetf "
8.
et fat g 222
ktlah
of this diz
I
etfz+gz?n + h 3*
$+$z* *6 247
without any Exception or Limita-
tion, when is a poſitive or nega-
10.
tive whole Number, and l the De-
kt la et fa + gz²
nominator of the Fraction
11. dż z
e+fz" tozas" Number of this Series 2, 4, 8, 16,
32, &c. and ſeems to be of opinion
12.
that the Fluent of any rational
Vetf zu + 8 221
Fluxion depends upon the Meaſure
of Ratio's and Angles, or upon Lo-
¿zz V.etfxn +9227 garithms and the Tables of Sines i
thoſe being excepted, that may be
kt la
had otherwiſe in finite Terms. And
Dr. Smith, the ingenious Editor of
84:
this Work, ſays, he could have pute
k – 16", Ve+f2" + gz2n down the Fluents of
In all theſe Forms 8 is a whole
Number, poſitive or negative,
*7.12+*" +hx*+kz***!%$*
and
d zz
dizon
8 + hz?
8 n +
+
n
T
di An- !
9.
on I
d zz.
22
d
ܐ
is any
I
d zz
· I
-
an
1
džezoni
dż zno
pa
5
1
dž z
An
. 1
FLU
FLU
pound radical or fractional Quan
and
x tity, will be had, (tho':not exactly
et fan + g 224 th 232 in general) by first throwing thoſe
d
Exproſſions into infinite Series's of
Ř * Iza
ī, by means of the fimple Terms, and then finding the
Fluents of thoſe fimple Terms.
Meaſure of Ratio's and Angles ; Parts .yield to any Impreſſion ; and
FLUID BODY, is that whole
bụt that he began to be tired with by yielding are eaſily moved one
the Calculation
among another : And ſo it follows,
When an Equation for a Fluent is that Fluidity ariſes from hence, viz.
found, it is very often neceſſary to that the Parts do not ſtrongly co-
add to or ſubtract from it ſome in- here, and that the Motion is not
variable Quantity, in order to get hinder'd by any Inequality in the
the trųe Fluent, which Quantity is Surface of the Parts.
eaſily found by making the variable
1. Fluids agree. in this with ſolid
Quantity in the Expreſſion to vaniſh, Bodies, that they confitt of heavy
and putting what remains, with its Particles, and have their Gravity
Sign changed, to the ſaid Expreffion, proportionable to their Quantity of
as the Expreſſion b+*Vabfiax, Matter, in any Poſition of the
which the common Rules give for Parts.
the Fluent of x Vabtax, will tained in a Veffel, to keep it from
2. The Surface of a Fluid con-
not be the true Fluent of this Flu. Powing out, if it be not preſſed from
xion; it being too much by the above, or if it be equally preffed,
Quantity tab obtained from will become plain, and parallel to
the Horizon.
*bti Vabtax by, making x
3:
The lower parts of Fluids are
0; fo that the true Fluent will preſſed by the upper: This Preſ-
be 6 +x V'ab tax ſure is in proportion to the incum-
3b Vab
bent Matter, that is, to the Heigh
The Fluent of a ſimple Fluxion of the Liquid above the Particle
that is preſſed.
is found by ſtriking out the fluxio-
nary Letter, increafing the Index of Parts, which ariſes from the Gravity
4. The Preſſure upon the lower
the variable Quantity by 1, and di-
viding the last Expreſion by the In- of the Super-incumbent Liquid,
exerts itſelf every way, and every
dex thus increaſed; as the Fluent af
way equally.
mt!
5. In Tubes, whether equal or
a * * mwill be
and that
mFI
unequal, whether ſtraight or oblique,
a Fluid rifes to the fame Height.
-mtal 6. When Liquids of different
will be
AndGravities are contained in the ſame
-- m to
Vefſel, the heavieſt lies at the loweſt
the Fluent of a Fluxion conſiſting Place, and is preſſed by the lighter,
of any Number of fimple Terms and that in proportion to the Height
joined together by the Signs to and of the lighter,
will be equal to the ſeveral 7. The Bottom and Sides of a
Fluents of thoſe fimple Terms Veſſel, which contains a Liquid, are
joined in like manner by the Signs preſſed by the Parts of the Liquid,
tand; and the Fluent of a Com- which immediately touch them.
ex
I
am
ai
of
mm
This
FLU
F L U
This Action increaſes in proportion moves through different Liquids
to the Height of the Liquid.
with the fame Velocity, is as the
8. When a Solid is immerſed in Denſity of the Liquid.
a Liquid, it is preſſed by the Liquid 16. When the ſame Body moves
on all ſides ; and that Preſſure in- through the fame Liquid with dif-
creaſes in proportion to the Height ferent Velocities, this Reſiſtance in-
of the Liquid above the Solid. Bo- creaſes as the Square of the Ve-
dies very deeply immerſed are e- locity.
qually preſſed on all ſides.
17. The Refiftance from the Co-
9. A Body ſpecifically heavier heſion of Parts in Liquids, except
than a Liquid, being immerſed in a glutinous ones, is not very ſenſible.
Liquid, will defcend.
In ſwifter Motions the Reſiſtance
10. A Solid fpecifically lighter alone is to be confider'd, which is
than a Liquid, aſcends to the high- as the Square of the Velocity.
eft Surface of the Liquid. But ſup 18. When a Body is moved in
poſe a Solid of the fame ſpecific any Liquid, the more blunt the
Gravity with the Liquid at any Body is, by that means it is more
Height; the Liquid will ſuſtain the retarded. If the Body be not im-
whole Body.
merſed deep, the Reſiſtance is to be
11. All equal Solids, but of diffe. diſtinguiſhed from the Retardation.
rent ſpecific Gravities, when they are 19. When we ſpeak of the ſame.
immerſed into the fame Liquid, they Body, the one may be taken for the
lofe equal Parts of their Weight. other. From the Reſiſtance ariſes a
12. However the Denſities of e Motion contrary to the Motion of
qual Bodies differ among themſelves, the Body ; the Retardation is the
if they be immerſed in the fame Celerity, and the Reſiſtance itſelf
Liquids, the Weights which they is the Quantity of Motion.
loſe are in the Ratio of their Bulks. 20. The Retardations of any Mo-
13. The immerſed Parts of the . tions are, Firſt, as the Squares of
Bodies ſwimming on the Surface of the Velocities : Secondly, as the
the ſame Liquor, are to one ano-
Denfities of the Liquids, through
ther as the Weights of the Bodies. which the Bodies are moved
And the Parts which deſcend into Thirdly, inverſly, as the Diameters
the Liquid, by laying on of different of thoſe Bodies : Laſtly, inverſly, as
Weights, are to one another as thoſe the Denſities of the Bodies them-
Weights.
ſelves.
14.
If
any
Veſſel be filled with 21. The Reſiſtance of a Cylin-
a Liquid, and that Liquid be weigh- der, which moves in the Direction
ed, and if you make the ſame Ex- of its Axis, is equal to the Weight
periment with other Liquids, their of a Cylinder made of that Liquid,
Weights will be as their Denſities. through which the Body is moved,
15.
All Bodies moved in Fluids having its Baſe equal to the Body's
ſuffer a Reſiſtance, which ariſes Baſe, and its Height equal to half
from two Cauſes : The firſt is the the Height, from which a Body
Coheſion of the Parts of the Liquid: falling in vacuo, may require the
The ſecond is the Inertia, or Inac- Velocity with which the ſaid Cylin-
tivity of Matter ; the Retardation der is moved through the Liquid.
from the Coheſion of Parts is as the 22. When a Body, ſpecifically
Velocity itſelf. The Reſiſtance a heavier than a Liquid, is thrown up
riſing from the Inertia, or Inactivity in it, a Body riſes to a leſs Height
of Matter, when the fame Body than it would riſe in vacuo with the
famo
2
i
F L U
FLU
fame Celerity. But the Defects of even in Liquors that are not glas
the Height in a Liquid from the tinous.
Heights to which a Body would FLUTINGS, by the French called
riſe in vacuo with the ſame Celeri- Cannelures, are certain perpendicu,
ties, are nearly as the Squares of the lar Cavities cut length-ways around
Heights in vacuo.
the Shaft of a Column, and rounded
23 The Velocity of a Liquid, at at the two Extremes. Their Num-
any Depth, is the ſame as that which ber was at firſt limited to twenty-
a Body, falling from a Height equal four in the Ionic, and twenty in the
to that Depth, would acquire. Doric Order ; but that Limitation,
24. A Liquid riſes higher, if its ſome of our modern Architects
Direction be a little inclined, than have taken the liberty to diſpenſe
if it ſpouts vertically.
with.
25. The Reſiſtance of the Air has Flux and Reflux of the Sea.
a ſenſible Effect upon the Motions See Tide.
of Liquids; and in ſmall Heights, FLUXIONS, are the very ſmall,
the Defects of the Heights from the or rather indefinitely ſmall Particles
Heights in vacuo; are in the Ratio of Quantities, being called by this
of the Square of the Height of the Name by Sir, Iſaac Newton, who
Liquid above the Hole.
conſiders them as the momentane-
26. In the greateſt Heights of ous Increments of Quantities. For
{pouting Liquids, greater Holes are Example: Of a Line by the Flux
required. In all Heights there is a of a Point, and of a Superficies, by
certain Meaſure of the Hole, through the Flux of a Line, and of a Solid
which the Liquid will riſe to the by the Flux of a Superficies, and
greateſt Height poſſible.
the Doctrine of theſe infinitely ſmall
27. Liquids which ſpout obliquely, Parts, is likewiſe called Fluxions.
are not retarded from ſo many Fluxions are of valt uſe in the In-
Cauſes, nor ſo much as thoſe that veſtigation of the Nature of Curves,
ſpout vertically.
and in the Diſcovery of the Qua-
28. A Liquid ſpouting from a dratures of curvi-lind Spaces, and
Hole in the Centre, will go to the their Rectifications, and in perform-
greateſt Diſtance poſſible.
ing many other admirable Effections,
29. The Squares of the Quantities that can be done ſcarcely any other
Aowing out, are in the Ratio of the way.
Heights of the Liquids above the The Fluxion of any generated
Holes.
Quantity is equal to the Fluxions of
30. If through equal Holes a Lin all the ſeveral generating Terms,
quid runs out of a Cylinder, and multiplied into the Indexes of their
out of another Veffel of the fame Powers, and into their Co-Efficients
Height, and in which the Liquid continually.
is always ſupplied, ſo as to be kept If each Term of an Equation
at the fame Height,) in the time in whoſe Fluxion is required, be mul-
which the Cylinder is emptied, tiplied ſeparately by the ſeveral
there runs out twice as much Wa- Indexes of the Powers of all the
ter from the other Veffel as from Aowing Quantities contained in that
the Cylinder.
Term, and in every ſuch Multipli-
31. Beſides the Irregularities from cation, if one Root or Letter of the
Friction, and the Reſiſtance of the Power be changed into its proper
Air, there are ſeveral others arif. Fluxion, fo fhall the Aggregate of
įng from the Coheſion of the Parts, all the Products, connected together
by
1
F L U
FOC
in-32
XÀÄx
22
in
!
.
1
2
12
by their proper Signs, be the Flu-
the third Fluxion of
xion of the Equation deſired.
If the Fluxion of the Numerator
of any Fraction be multiplied by
the Denominator, and after it be * xon ; and ſo on. This was diſcover-
placed with the Sign-, the Flu- ed by Sir Iſaac Newton, in the Year
xion of the Denominator multiplied 1665.
by the Numerator; then will this be
Fly. See Flie.
the Numerator, and the Square of the
FLYING PINION, is a part of a
Denominator will be the Denomina- Clock, having a Fly, or Fan, there-
tor of the Fraction expreſſing the by to gather Air, and ſo to bridle
Fluxion of the given Fraction. the Rapidity of the Clock's Mocion,
FLUXIONS (SECOND, THIRD, when the Weight deſcends in the
&c.) are the Fluxions of Fluxions, ſtriking Part.
whichare conſidered as flowing Quan Focus of an Ellipſis, is a Point
tities themſelves : The ſecond Fluxi- in the longeſt Axis on each ſide the
ons being marked by two Points over Centre; from each of which if any
them : Thus, j; the third by three'; two right Lines are drawn, meeting
one another in the Periphery of the
thus x; and fo on.
Ellipfis ; their Sum will be always
If i be the Fluxion of the Quan- equal to the longeſt Axis ; and to
when an Ellipfis and its two Axes
tity x, and
be the Index of the are given, and the Foci are required,
Power of the fame, and if for x be you need only take half the longeſt
taken ***, and the Quantities Axis in your
Compaſſes, and ſetting
one Foot in the End of the ſhorter,
the other Foot will cut the longer
stil”, be expanded into a Series, in the Focus required.
Focus of an Hyperbola, is a Point
in the principal Axis within the
we fhall have atxl" =
oppoſite Hyperbola's; from whence,
if any two right Lines are drawn
meeting in either of the oppoſite
m² - mn
Hyperbola's, their Difference will
À ² x :
be equal to the principal Axis.
Focus of a Parabola, is a Point
m3n
in the Axis within the Figure, di-
1922—3m’ntamn?
+
ftant from the Vertex one fourth
623
Part of the Latus Rectum.
&c. wherein the ſecond Term Here I cannot help taking no-
tice of what is ſaid by the Editors
of the Aeta Eruditorum at Leipfic,
x n is the firſt Fluxion of x
for January 1705, who, upon the
m2_mn M-coming out of Sir Iſaac Newton's
the third Term
*ňx n Curves of the ſecond Order, ſpeak
thus concerning them, in the Style
of Mr. Leibnitz : (Cæterum Autor
is the ſecond Fluxion of x
and
non attingit Focos vel Umbilicos Cur-
m?3mənt2mn?
varum fecundi generis, & multa
the fourth Term
minus Generum altiorum, Cum ergo
6 73
e a
mu
an
n
n +
m
MM
12
22
cica
+
* * *
2 n2
1
xxkix m
-n2
<
2 n²
22
)
6.
1
FOC
FOCO
ea. Res abftrufioris fit Indaginis, & therwiſe, their Meaning ſhould have
maximi tamen in hoc genere Ufus been explain'd.
tum ad Deſcriptiones, tum ad alias Focus, in Optics, is, the Point
Propertates Curvarum, & Doctrina of Convergence, or Concurrence of
bæc Focorum ab illuſtriſſimo D. D.T. the Rays of Light made by the Re-
(Tſchurnhaus) profundius fit verfata; fraction, or the Reflection of a re-
Supplementum ejus pro his Curvis fracting or refle&ting Subſtance.
expectamus. In Engliſh thus: “But 1. In a Plano-Convex Glaſs, pa-
* ſince the Author has not meddled rallel Rays are united with the Axis,
with the Foci of the Curves of that is, the Focus is diſtant from the
• the ſecond Order, and much leſs Pole of the Glaſs a Diameter of
with thoſe of the Curves of high- the Convexity, if the Segment be
er Orders : Therefore, as theſe but thirty Degrees.
are of a more abſtruſe Enquiry, 2. In double Convex-glaſſes of
. and at the ſame time of the the fame Sphere, the focus is diſtant
greateſt Uſe, as well in the De- from the Pole of the Glaſs about the
• ſcription, as the Diſcovery of o- Radius of the Convexity, if the Seg-
"ther Properties of the Curves; and ment be but thirty Degrees.
whereas the moſt illuftrious 3. The Rays that fall nearer the.
• D. D. T. (Tſchurnhaus) is very Axis of any Glafs, are not united
deeply verſed in the Doctrine of with it ſo ſoon as thoſe that are
6. the Foci, we expect from him a farther off ; and the focal Diſtance
• Supplement to thoſe Curves. in a Plano-Convex Glaſs will not be.
Now, the Perſon who makes Ob, fo great when the Convex-fide is to
ſervations upon this Paſſage of the wards the Object, as on the contrary.
Compilers of the Leipfic Acts (in the 4. In viewing any Object or Body.
Commercium Epiftolicum, publiſhed by a Plano-Convex Glaſs, the Con-
by Order of the Royal Society at vex-lide muſt be turned outwards.
London), and which, as I have been Focus VIRTUAL, See Virtual
informed, was Sir Iſaac himſelf, Focus.
ſays, Compilatores Aftorum in fcri 1. In Concave Glaffes, when a
bendis Librorum breviariis a Cenfu- Ray falls from Air parallel to the
ris temerariis abflinere debent. Ex Axis, the Virtual Focus, by its firſt
hac Cenſura patet Animus Scripto- Refraction, is at the diltance of a Di-
ris in D. Newtonium. In Engliſh, ameter and a half of the Concavity.
* The Compilers of the Leipfic Acts 2. In Plano - Concave Glaſles,
• in their Abſtracts of Books, ſhould when the Rays fall parallel to the
* abſtain from raſh Cenſures ; but Axis, the. Virtual Focus is diſtant
• here the opinion of the Writer, from the Glaſs by the Diameter of
concerning Sir Iſaac Newton, the Concavity,
. fully appears.
And this is 3. In Plano-Concave Glaſſes, as
very juftly ſaid; for it is well 107: 193 :: ſo is the Radius of the
known, that the Curves of the fe- Concavity to the Diſtance of the
cond Order have no Foci. If by Virtual Focus.
Foci are meant ſuch Points that the 4. In double Concaves of the
Sum of any Number of right Lines fame Sphere, parallel Rays have
drawn from them to any point of their Virtua! Focus at the diſtance
one of theſe Curves ſhall be of a of the Radius of the Concavity.
given Length, which one muſt fup 5. But whether the Concavities
poſe they mean, if they mean any be equal or unequal, the. Virtual
thing by that Word, at leaf ; if o- Focus, or Point of Divergency of the
parallel
FOC
FOR
parallel Rays is determined by this · Heat at that time 9216 times; and
Rule: As the Sum of the Radii of this will have an effect as great as
both Concavities : is to the Radius the direct Rays of the Sun would
of either Concavity :: ſo is the have on a Body placed at one nine-
double Radius of t'other Concavity ty-ſixth Part of the Diſtance of the
: to the Diſtance of the Virtual Earth from the Sun, or on a Planet
Focus.
that ſhould move round the Sun at
6. In Concave Glaſſes, if the but a very little more than a Dia-
Point to which the incident Ray meter of the Sun's Diſtance from
converges, be diſtant from the Glaſs him, or that would never appear
farther than the Virtual Focus of farther from him than about thirty-
parallel Kays, the Rule for finding fix Minutes.
the Virtual Focus of this Ray, is Dr. Halley, in the Philoſophical
this : As the Difference between Tranſactions, N° 205. ſhews a ge-
the Diſtance of this point from the neral way of finding the Foci of
Glaſs, and the Diſtance of the Vir- ſpherical' Glaffes by Computation.
tual Focus from the Glaſs : is to So does Mr. Ditton, in his Fluxions.
the Diſtance of the Virtual Focus :: See alſo Dr. Gregory's Elements of
fo is the Diſtance of this point of Dioptrics.--Mr. Carré and Guiſnée
Convergence from the Glaſs : to the in the Memoires de l'Acad. Royale
Diſtance of the Virtual Focus of this des Sciences. And beſides theſe, ſe-
converging Ray.
veral others who have wrote upon
7.
In Concave Glaſſes, if the this Subject : Amongſt which, Dr.
Point to which the incident Ray Barrow's and Sir Iſaac Newton's
converges be nigher to the Glaſs Ways of finding geometrically the
than the Virtual Focus of parallel Foci of ſpherical Glaſſes, (to be
Rays, the Rule to find where it ſeen in Dr. Barrow's Optical Leca
croſſes the Axis, is this : As the tures) appear to me to be far more
Exceſs of the Virtual Focus more neat and elegant than any I have
than this point of Convergency elſewhere ſeen.
from the Glaſs : is to the Virtual FOLIATE, a Name given by
Focus :: fo is the Diſtance of this ſome (as the ingenious Mr. De
Point of Convergency from the Moivre in the Phil
. Tranfaét.) to
Glaſs: to the Diſtance of the Point Curve Line of the ſecond Order, ex-
where this Ray croſſes the Axis. preſſed by the Equation x3 + 3 =
To find the Focus of a Meniſcus a x y, being one Species of defec-
Glafs; ſee under the Word Me- tive Hyperbola's, with one Aſyinp-
nifcus.
tote, and conſiſting of two infinite
If there be a Burning-Glaſs of a Legs croſſing one another, and
Foot in Diameter, this will conſti- forming a ſort of Leaf. (See
pate or croud together all the Rays Species 42. of Sir Iſaac Newton's
of the Sun which fell before on the Lines of the third Order.)
Area of a Circle twelve Inches in FOMAHANT, a Star of the firſt
Diameter, into the Compaſs of one Magnitude in Aquarius, whoſe Lon-
eighth Part of an Inch, the Area's gitude is 329 deg. 17 min. Lati-
then of the two Circles will be as tude 21 deg. 3 min.
9216 to 1; and conſequently the FOOT-BRANK, or BANQUETTE,
Heat of the leſſer to the Heat of in Fortification, is a ſmall Step of
the greater, will be reciprocally as Earth, on which the Soldiers ftand
9216 to 1 : that is, the Heat in the to fire over the Parapet.
Focus will exceed the Sun's common FOR E-STAFF. See Croſs-Staf
FORT:
FOR
FOR
FORT, is a Caſtle or Place of the other Parts ; ſo that there may
ſmall Extent, fortified either by Art be no Place in which an Enemy can
or Nature.
lodge himſelf undiſcovered by thoſe
Fort-ROYAL, is that which that are within, and that both from
hath twenty-fix Fathoms for the the Front, the Sides, even from be-
Line of Defence.
hind, if poſſible.
FORT-STAR, is a Redoubt, con 2. The Fortreſs ought to com-
ftituted by re-entring and faliant mand all Places round about, and
Angles, which commonly have therefore all the Out-Works muſt
from five to eight Points. See more be lower than the Body of the
under the Word Sconces.
Place,
FORTIFICATION, or MILITA 3. The Works that are moſt re-
RY ARCHITECTURE, is the Art mote from the Centre of the Place,
fhewing how to fortify a place with ought always to be open to thoſe
Ramparts, Parapets, Moats, and that are more near.
other Bulwarks ; to the end, that 4. The Angle-Flanquant, or the
a ſmall Number of Men within, may Point of the Baſtion, ought to be,
be able to defend themſelves for a at leaſt, of ſeventy Degrees, or as
conſiderable time againſt the Af- fome ſay, (Mr. Vauban,) not more
faults of a numerous Army without; than a hundred, or leſs than ſixty.
ſo that the Enemy, in attacking 5. The Angle of the Courtin
them, muft of neceſſity ſuffer great ought never to be leſs than ninety,
Loſs.
or greater than a hundred Degrees ;
Fortification is either regular, or becauſe if it be larger, 'tis too much
irregular, and with reſpect to time, ſubject to the View of the Enemy,
may be diſtinguiſhed into durable 6. The greater the Flank and De-
and temporary
migorge is, in proportion to other
FortiFICATION (DURABLE) Things, the better, becauſe there is
is that which is raiſed to continue à both more room to retrench in, and
long while,
alſo becauſe there may be made re-
FORTIFICATION (IRREGU - tiring Flanks, which add very much
LAR,) is that where the sides and to the Strength of a Place.
Angles are not all uniform, equi 7. The Line of Defence ought
diſtant, nor equal one to another. never to exceed point-blank Muf-
ForTIFICATION (REGULAR,) ket-ſhot, which is about an hundred
is that which is built on a regular and twenty, or a hundred and twen.
Polygon, the Sides and Angles ty-five Fathoms.
whereof are all equal ; being com-
8 The Baltions that are not too
monly about a Muſket-ſhot one little, nor yet too exceſſively big,
from another.
are to be preferred before others;
FORTIFICATION (TEMPOR A- and the Angle of a Baſtion ſhould
RY,) is that which is erected upon not exceed a hundred, nor be leſs
an emergent occaſion for a little than fixty Degrees.
time. Such are all ſorts of Works 9: The greater the Angle that is
caft up for the ſeizing or maintain- made by the outward Polygon and
ing of a Poſt or Paſſage; as alſo the Face ſhall be, the greater is the
Circumvallations, Contravallations, Defence of the Face.
Redoubts, Trenches, Batteries, &c. 10. Whatſoever incloſes a dura.
1. Every Place within the Forti- ble Fortification, muſt be either
fication ought to be flanked, that Flank, Face, or Courtin, built fo
is, ſeen lide-ways, or defenfible from well, that the firſt Diſcharge of the
Cannon
FOR
FRA
Cannon may not be able to pierce diſtant one from another 120 Fa-
through it:
tiom'; but their Extent'and' Figure
11.°"Tis impoffible to fortify a are different, according to the si-
Triangle after the common way, tuation or Nature of the Ground,
becauſe the Angle of the Gorge is ſome of them having whole Baftions,
always leſs than ninety Degrees. and others only Dem'i-Battions.
12. The acuter the Angle at the They are made uſe of only for a
Centre is, the Place is by fo much timme, either to defend the Lines of
the ſtronger, becaufe it will' have Circumvallation, or to guard' ſome
the more Sides.
Paffage, or dangerous Poft:
13. In a regular Fortification the FRACTION, is a broken Num..
Face muld never be leſs than half ber, fignifying one or more Parts,
the Courtin; and the Faces of the proportionally. of any Thing di-
Baſtion ought to be defended by the vided : It confifts of two Numbers
ſmall Shot of the oppoſite Flank. ſet 'one over another, with a Line
14. Any Trenches are preferable between them, ask. In all Fractions,
to thoſe filled with Water, efpe- as the Numerator : is to the Deno-
cially in great Places, where Sallies, minator :: fo is the Fraction itſelf ;
Retreats, and Succours are frequently to that whole of which it is a Frac-
neceſſary; but in ſmall Fortreſſes, tion. Hence there may be infinite
Water-Trenches that cannot be Fractions of the fame Value' one
drained are beſt, becauſe there is no with another ; for there may be
need of Sallies, Succours, &c. infinite Numbers found, which ſhall
There are many Writings upon have the ſame Proportion one to
Fortification: Some of which are another.
Melder's Praxis Fortificatoria.-- Les 1. When the Numerator is lefs
Fortifications de Compte de Pagan.- than the Denominator, the Fraction
L’Ingenieur parfait du Sieur de Ville. is leſs than the whole, and conſe-
Sturmy's Architettura Militaris quently is what they call a proper
Hypothetica.--Blondel's Nouvelle Ma- Fraction.
niere de Fortifier les Places. The 2. But when the Numerator is
Abbé de Fay's Veritable Maniere de either equal to, or greater than the
bien Fortifier de M. Vauban.- L'In- Denominator, the Fraction is called
genieur François. Coborn's Nouvelle improper, becauſe 'tis equal to, or
Fortification tant pour un terrain bas greater than the whole. Thus is
& humide, que fec & elevé. equal to 1, and is equal to 1, and
Alexander de Grotte's & Donatus 3. Fractions are fingle or
Roſelli's Fortification.-Medrano's pound.
Ingenieur Francoiſe. The Chevalier 4. Single Fractions are ſuch as
de Saint Julien's Architecture Mili- have but one Numerator,, and one
taire. - Landsberg's Nouvelle Ma- Denominator, as, 1.
niere de Fortifier les Places. An 5. Compound Fractions, or Frac-
anonymous Treatiſe in French, called tions of Fractions, are ſuch as con-
Nouvelle Maniere de Fortifier les fift of more than one Numerator,
Places, tirée des Methodes du Cheva- and one Denominator, as of šof
lier de Ville, &c Ozanam's Traité , and are always connected by the
de Fortification.-- Memoires de l'Ar- Word of.
tillerie de Surirey de St. Remy.
6. All Fractions, whoſe Numera-
FORTINES, or FIELD-FORTS, tors and Denominators are proportio-
are Sconces, or little Fortreſſes, nal, are equal to one another As the
whoſe flanked Angles are generally Fractions , 38, ii, are all equal.
Every
1
com-
5
4 4 IZ
1
3
nator.
FIR
FRU
Every Fraction, ſuch as
pity that this has not hitherto been
Ec.
put in practice.
may
etfz+8z + b 23
FRIGID Zones. See Zones.
be reduced into as many ſingle ones, FRONT, in Perſpective, is the
as there are Roots in its Denomi- Orthographical Projection of an
Object upon a parallel Plane.
FRAISES, in Fortification, are Front, in Fortification, is what
pointed Stakes fixed in Bulwarks the French call Tenaille de Place,
made of Earth, on the one fide of and the face of a Place. It is that
the Rampart, a little below the Pa- which is comprehended between the
rapet. Theſe Stakes, being from Points of any two neighbouring Ba-
feven to eight foot long, are driven ftions, viz. the Courtin, and two
in almoſt half way into the Earth, Flanks, which are raiſed upon
the
and preſent their Points ſomewhat Courtin, and the two Faces of the
floping toward the field. They Baſtion, which look towards one
ſerve to prevent Scalades and Deſer- another.
tion.
FRONT-LINE, in Perſpective. See
FRAME, is the Out-Work of a Line of the Front.
Clock or Watch, confifting of the FRONTISPIECE. See Portale.
Plates and Pillars, and which con FRONTON, is a Part or Member
tains in it the Wheels, and the reſt in Architecture,
in Architecture, which ſerves to
of the Work.
compoſe an Ornament raiſed over
Freese, a Term in Architec- Doors, Croſs-Works, Niches, &c.
ture. See Freeze.
ſometimes making Triangles, and
FREEZE, a largé Flat-Member, ſometimes Parts of a Circle. It is
which feparates the Architrave from alſo called Faſtigium by Vitruvius,
the Cornice. The Word comes and Pediſment by the French.
from Latin, Phrygio, an Embroi FROZEN ZONES. See Zones.
derer ; the Freezes being frequent FRUSTUM, in Geometry, figni-
ly adorned with Figures in Baſs- fies a Piece cut off, or ſeparated
Relief, ſomewhat in imitation of from any Body ; as the Fruſtum of
Embroidery. The Freeze is fome a Pyramid or Cone, is a part or
times alſo expreſſed by the Word Piece of them cut off (uſually) by à
Zophoros, from the Greek, Zoophoros ; Plane parallel to the Baſe.
it being uſual for Animals to be re The Solidity of the Fruſtum of
preſented upon it.
a Pyramid with a ſquare Baſe will
Fresco, in Architecture, is a be had, by adding the Area's of the
Sort of Painting, which is made upper and under Baſes to a mean
upon the Plaiſtring of an Edifice be: Proportional between them, and
fore it be dry.
multiplying that Sum by one Third
Friction, is the Reſiſtance ari- Part of the Height of the Fruftum;
fing from the Motion of one Super- and as 14 to it nearly, ſo is the
ficies upon another, and is cauſed Solidity of the Frultum of a ſquare
by their Defect of Slipperineſs. Pyramid, to the Solidity of the
Mr. Romer and De la Hire have Fruftum of a Cone, whoſe Diame-
ſhewn in the French Memoirs, that ters at Top and Bottom, are equal
the Figures of the Teech of Wheels to the sides of the upper and lower
ought to be Epicycloids, that ſo Baſe, and Height equal.
their Reſiſtance may be the leaſt The following Demonſtration of
poſlible. And it is a great deal of the Theorem above, being not to be
S
found"
3
FRU
F US
I
х
X EF.
2
2
bix.
and Altitude Ef; conds ; ſo thaž che Fuſe must be
found every where, may not be dif- whoſe common Altitude is BC, and
pleaſing to fome. Let AD the Baſe Baſes equal right-angled 'Triangles,
G
T AD-BC
each equal to
FUGUE, in Muſick, is fome Part
confilling of four, five, fix, or any
B
c
Number of Notes begun by ſome
F
one fingle Part, and then ſeconded
by a third, fourth, fifth, and fixth
Part; if the Compoſition conſiſts of
A E D ſo many, repeating the ſame, or
of the Fruftum ABCD of a ſquare Parts follow, or come in one after
ſuch-like Notes ; ſo that ſeveral
Pyramid be called a, the
upper
Baſe
another in the ſame manner, the
B C, b; the Height EF, c; the
Height EG of the whole Pyramid leading Parts Hill flying before thoſe
y; and the Height FG of the Py-
ramid BGC, X; then a : ý::
FUGU E-DOUBLE, is when two
or more different Parts move toge-
and a3 : a ay::
:a ay::b: x; alſo a : y ::
b3 : bbx; therefore a3 : aay;;63 ly interchanged by ſeveral Parts.
ther in a Fugue, and are alternate-
:bbx; and a : 63 :: a ay:bb x;
FULIGINOUS VAPOURS, by
and (dividendo) a3-63: aay-bbc fome, are thick, impure, and footy
::a3 : a ay; and fo a3 ---b3:a ay Vapours.
bbc:; a: y; that is, as ambic;
FURNITURE of a Diál, are ſuch
that is, ambxa atóbtab: a ay Lines as are drawn thereon for
-6bx::a-b:c; that is, ab Ornament ; as the Parallels of De-
xaa tbbt ab: aap-bbx :
clination, Length of the Day, Azi-
coxă=6:03 ; and amb xaat byloniſh and Jewiſh Hours, &c.
muths, Points of Compaſs
ob Fab:ccxa-::a ay
FUSAROLE, is a ſmall round
that is, a atbb-tab:cc::
Member in Architecture, cut in
a a y-bó x : 03 ; that is, c xaat form of a Collar, with ſomewhat
bb tab.c3 :: aav - 660:63; long Beads, under the Echinus, or
Quarter-hound of Pillars of the Do-
wherefore cxaa +66 + ab
аар •
bbx; and of the one will rick, Ionick, and Compoſite Orders.
Fuse, or Fusil, of a Bomb or
be equal to į of the other.
But { Granado-Shell, is that which makes
of a ay-bbx is equal to the Fru- the whole Powder, or Compoſition
ftum ; therefore of cx aatbbtab in the Shell, take fire, to do the
will be equal to the Fruftum. defigned Execution. 'Tis uſually a
This may be demonſtrated other- wooden Pipe or Tube filled with
wife, by fuppofing the Fiuftum of Wild-Fire, or lome fuch Compofi-
the Pyramid to conſiſt of one right- tion, and is deſigned to burn to long,
angled Parallelip pedon, whoſe Al- and no longer, as is the lime of the
títude i. E F, and side of the ſquare Motion of the bomb from the
Bire BC: of four Pyramids; the Mouth of the Mortar to the Place
Sides of each of whoſe square Baſes where it is to fall, which Time Mr.
AO-BC
Anderſon makes to be about
is
Se-
27
end of four cqual criangular Priſms, contrived either from the Nature of
che
6 bx
:53
3
1
2
G
G A L
G A U
the Wild-Fire, or the Length of the Firë of the Beſieged. Theſe Gal-
Pipe which contains it, to burn juftleries are frequently made uſe of in
that time.
the Moat, already filled with Fag-
Fust, in Architecture,' ſignifieś gots and Bavins, to the end that
the Trunk or Shaft of a Column; the Miner may approach ſafe to
being that Part comprehended be the face of the Baftion, when the
tween the Baſe and the Capital. Vi- 'Artillery of the oppoſite Flank is
truvius calls it Scapus.-
diſmounted.
Fusy, is that Part of a Watch GARDECAUT, or GAR D-DU.
about which the Chain or String Cord, is that which ſtops the Fuſy
is wrapped, and is that which the of a Watch, when wound up, and
Spring draweth, being in form com for that end is driven up by the
monly taper.
In larger Works, String. Some call it Guard-Cock,
going with Weights, it is cylindri- others Guard-du-Gut.
cal, and is called the Barrel.
GAUGE-Point of a ſolid Mea-
fure, is the Diameter of a Circle,
whoſe Area is equal to the ſolid
G.
Content of the fame Meaſure , as
the Solidity of a Wine-Gallon be-
ABIONS, a Term in Forti- ing 231 Cubic Inches, (according to
fication, ſignifying Bakets Wincheſter Meaſure; ) if you con-
made of Ofier-Twigs, equally wide ceive a Circle to contain ſo many
at the top and bottom, about four Inches, the Diameter of it will be
Foot in Diameter, and from five 17.15; and that will be the Gauge-
to fix high; which being filled Point of Wine Meaſure: and an
with Earth, are ſometimes uſed as Ale. Gallon containing 288 Cubic
Merlons for the Batteries, and ſome- Inches, by the ſame Rule;, the
times as a Parapet for the Lines of Gauge Point for Ale-Meafure will
Approach, when it is requiſite to be 19.15.
carry on the Attacks through a GAUGING, is finding the Capa-
ftony or rocky Ground, and to ad- cities or Contents of all Sorts of
vance them with extraordinary Vi. Veſſels which hold Liquids, Powders,
gour. They ſerve alſo to make Meal, Corn, &c.
Lodgments in fome. Pofts, and to The common Rule for finding the
ſecure other Places from the Shot Contents of all Ale and Wine Caſks,
of the Enemies, who nevertheleſs is to take the Diameters at the
endeavour to ſet the Gabions on Bung, and at the Head ; by which
fire with pitched Faggots, to render you may find the Areas of the Cir-
them uſeleſs.
cle there ; then you muſt take two
GABLE-End of a Houſe, is the thirds of the Area of the Circle at
upright Triangular-End from the the Bung, and one third of the Area
Cornice, or Eaves, to the top of of the Circle at the Head, and add
its Roof.
them together into one Sum, which
GAGE.Point. See Gauge-Point. Sum multiply'd by the internal
GALLERY, in Fortification, is Length of the Caſk, gives the Content
a cover'd Walk, the Sides whereof in ſolid Inches, which you may
are Muſket-Proof, conſiſting of a into Gallons, by dividing by 282
double Row of Planks lined with for Ale, and 231 for Wine Gallons.
Plates of Iron ; the top being ſome The Writers upon Gauging are,
times covered with Earth or Turf, Hunt, Everard, Douharty, Shet-
to hinder the Effect of the artificial tleworth, &c.
GAUGH
+
turn
S2
>
G E O
G AZ
one End
GAUGING - Rod. This Rod, GEMINI, one of the twelve Signs
whoſe Uſe is to find the Quantities of the Zodiac, being the third in
of Liquors contained in any kind order ; alſo a Conſtellation of that
of Veſſels, is uſually made of Box- Name.
Wood, and confifts of four Rules, GENERATING LINE, or Fi-
each a Foot long, and about three GURE, in Geometry, is that which
Eighths of an Inch ſquare, joined by its Motion or Revolution pro-
together by three Braſs Joints ; by duces any other Plane or Solid Fi-
which means the Rod is render'd gure. Thuş a Right Line moved
four Foot long; when the four Rules any way parallel to itfelf, generates
are quite open'd, and about one Foot a Parallelogram; round a Point in
when they are folded together. the fame. Plane, with
i. On the firſt Face of this Rod faften'd in that Point, it generates a
are placed twe. Diagonal Lines, one Circle'; one entire Revolution of a
for Beer, and the other for Wine; Circle in the ſame Plane, generates
by means of which, the Content of the Cyeloid ; the Revolution of a
any common Veſſel in Beer or Wine- Semicircle round its Diameter, gea
Gallons may be readily found, in nerates à Sphere, &c. Sir Iſaac
putting the Rod in at the Bung-Hole Newton uſes the word.
of the Vefiel, until it meets the In GENERATED, or
GENITED
terſection of the Head of the Veſſel, QUANTITY, in a very large Senſe,
with the oppoſite Staves to the is taken for whatever is produced
Bung-hole.
either in Arithmetic, by Multipli-
2. On the ſecond Face of this cation, Diviſion, or Extraction of
Rod are a Line of Inches, and the Roots ; or in Geometry, by the In-
Gauge-Line, which is a Line ex- vention of the Contents, Areas, and
preſſing the Areas of Circles, whoſe sides of Figures.
Diameters are the correſpondent GENESIS, in Geometry, is the
Inches in Ale-Gallons.
Formation of any Plane or Solid
3. On the third Face are three Figure by the Motion of ſome Line
Scales of Lines. The firſt is for or Surface, which Line or Surface
finding how many Gallons there is is always called the Deſcribent ; and
in a Hogſhead, when it is not full, that Line, according to which the
lying with its .Axis parallel to the Motion is made, is called the Di-
Horizon. The ſecond Line is for rigent.
the fame Uſe as that for the Hogf GEOCENTRIC, ſignifies any Pla-
head. The third Line is to find
to find net or Orb that has the Earth for
how much Liquor is wanting to fill its Centre, or the fame Centre with
up a Butt when it is ſtanding. the Earth.
4. Half way the fourth Face of Geocentric LATITUDE of a
Gauging-Rod are three Scales of Planet, is the Angle, which a Line
Lines, to find the Liquors wanting joining the Planet and the Earth,
in a Firkin, Kilderkin, and Barrel, makes with the Line drawn perpen-
lying with their Axes parallel to the dicular to the Plane of the E.
Horizon.
cliptic.
GAZONS, in Fortification, are GeoceníRIC PLACE of a Pla-
Pieces of freſh Earth cover' with net, is a Point of the Ecliptic, to
Graſs, cut in form of a Wedge, which the Planet ſeen from the
about a Foot long, and half a Foot Earth is referred.
thick, toline Parapets, and the GEOD ÆSIA, Surveying, or the
Tranſverſes of Galleries.
Art of meaſuring Land.
2
GEODE-
i
4
1
or
>
GEO
GEO
GeodeTICAL Numbers, are, tko. A Line of the fourth
ſuch as are conſidered according to Order, is that whoſe Equation
thoſe vulgar Names or Denomina- has four Dimenſions, or which
tions, by which Money, Weights, may be cut in four Points by a
Meaſures, &c. are generally known, right Line whoſe moſt general
or particularly
divided by the Laws Equation is g4 + ax + xy3+
and Cuſtoms of ſeveral Nations.
GEOGRAPHICAL MILE, is the cx2 +-dxte x y2 +-f3x3 +8x? +-hxtk
Sea-Mile, ör Minute; being one xy +1x4+m*3 + uxzit patq=o.
fixtieth Part of a Degree of a great And ſo on ad infinitum.
Circle on the Earth's Surface.
And a Curve of the firft kind
GEOGRAPHY, is the Science that (for a right Line is not, to be rec-
teaches and explains the Properties kon'd amongſt Curves) is the ſame
of the Earth, and the Parts thereof with a Line of the ſecond Order ;
that depend upon Quantity. and a Curve of the ſecond Order
Some of the Geographical Wri- the ſame as a Line of the third;
ters amongſt the Ancients were Pto- and a Line of an infinite Order is
lemy, Pliny, Strabo, and Jolin de Sa- that which a right Line can cut in
croboſco. Amongſt the Moderns we an infinite Number of Points, ſuch
have Cluverius, Heslin, Ricciolus, as a Spiral, Quadratrix, Cycloid, the
Varenius, Morden, Boboun, Echard, Figures of the Sines, Tangents, Se.
Gordon, &c.
cants, and every Line which is ge-
GEOMETRICAL ALGE- nerated by the infinite Revolutions
BRAIC CURVES, are thoſe whoſe of a Circle or Wheel.
Ordinates and Abſciſſes being right In each of the ſaid Equations x is
Lines, the Nature thereof can be the Abſciſs, and y a correſpondent
expreſſed by a finite Equation, hav- Ordinate, making any given Angle
ing thoſe Ordinates and Abſciſſes with it; a, b,c,d, &c. given Quan-
in it.
tities, affected with their Signs +
Geometrical Lines or Curves are and whereof one or more may
divided into Orders, according to be wanting, provided by ſuch De-
the Number of Diinenſions of the fect the Line does not become one
Equation, expreſſing the Relation of an inferior Order.
between the Ordinates and Abfciffa's, 1. The moſt complicated or ge-
or according to the Number of neral Equation of geometrical Lines
Points, by which they may be cut
by a right Line. So that á Line of of all Orders is tax+b xy91
the firſt Order will be only a right f cx? + dx te x , hem
Line expreſſed by the Equation
x 3.4-ma2 +
ytaxtb=0. A Line of the ſe- 883.thx? tkx+1 x Mootor-3 +- &c.
cond or quadratic Order, will be
+ m *+ r *
xn-
the Conic Sections, and Circle,
Its at t
whoſe moſt gi neral Equation is px3 + &c. t q=o. where n
22 taxfb xy tex?+dxte=o. expreſſes the Order of the Line, and
A Line of the third Order, is that a, b, c, d, e, g, h, k, l, &c. m, 1,'s,
whofe Equation has three Dimen- P; &c. q conſtant Quantities, va
fions, or may be cut by a right riouſly affected with the Signs +
the
Line in three points, whoſe most
being the Sum of the natural Num-
general Equation is y3 + axtb xy2 bers decreaſing from ntito o,
+ cxtdate xut fxstex?+hx and the Number of the Co-efficients
S 3
L
or
GEO
GEO
}
*
2
х уу
2
2
F
2
or invariable Quantities will be finitely diftant, that Ordinate in the
2² +3n
Equation defining the Curve, will
not afcend to ſo many Dimenſions
2. The general Equation ya + fes a right Line where the Ordinate
as the Curve; ſo that x=a expref-
ax+6 x y tex? + d x to
* ta x
of all Curves of the firſt kind may y is of no Dimenſion.
be tranfmuted into a more ſimple of the Tecond Order which run on
= bx? tcxtd, expreſſes all Lines
one, ſtill expreſſing them all, viz.
z-= f*+ gx+, where z is ad infinitum. xt a
the
ordinate
, & the Abſciſs, and bx? tcxtd xy+ ex3 +fx?+gx+h
f, g, h, conſtant
Quantities. For by all Lines of the third Order that runs
extracting the Root, y will be =
out to Infinity, and generally
ax+6
++ và x + a + xếa x y = b**+-+-+da*-2,
2 abx+4dx+66+4€; that is, &c. * yote+
ſuppoſing, P=4c+a?, q=2ab+4d, exs+fx2 + gx3, &c.
and r=b6 +4€, it will be y=
ax+6
xos, &c. + b gry + k x1
+ 1 px +qxtr; and +1x42, &c. +'q, expreſſes all
Curves that run out infinitely.
if again we ſuppoſe x = y
4. The general Equation of all
axtb
P
fo
Curves of the ſecond kind may be
, f ig
and
4
tranſmuted to the four following
particular Equations ſtill expreſſing
h
we ſhall have %= them all, viz, xy? --- yax3 +6x?
4
txtd. xy=ax3 +6x² + xtd.
Vfx2 +gx+h, and ſo z? = fx? yy=a x3 + bx? tcx + d. and
*g*+h.
y = ax+6*? +0x + d. The firſt
Hence when the Term fx2 is af- of which Equations repreſents a Fi-
firmative, the Curve expreſſed by gure, having ſix hyperbolical Legs
the Equation z? = f*2 +8x+ with three Afymptotes, forming an
will be an Hyperbola. When the Iſoſceles Triangle, if the Term ax3
fame is negative, an Ellipfis ; and be affirmative. "But if the Term ey
when the fame is abſent, a Parabola ; be abſent, the three Afymptotes
ſo that there are but three different meet in a Point, in the Abſciſs; and
Species of Curves of the firſt kind.
of theſe Curves, which Sir Iſaac
When the Root of the Quantity Newton calls redundaint Hyperbo.
in the Vinculum being Part of the la’s, there are nine different Species
Value of y can be extracted, the without Diameters; twelve with but
Locus of the given Equation will one Diameter ; two with three Dia-
be a right Line.- When the Terms meters ; nine with three Afymptotes,
j2 and cm? are wanting, the Curve converging to a common Point. But
es preſſed by the Equation above when axi is negative, the Figure
will be an Hyperbola, when the expreſſed by that Equation will be
Abſciſs is either an Afymptote, or a defective Ilyperbola ; of which
parallel to it, and the Ordinates are there are fix different Species, hav-
parallel to the other Aiymptote. ing but one Aſymptote, and only
3. If the Ordinate of a Curve be two hypei bolical Legs, junning out
parallel to a Tangent at a Point in-- contrary ways ad infinitum. the
Afymptote
1
GEO
G E O
Afymptote being the firſt and prin- oppoſite Angles of the Aſymptotes,
cipal Ordinate ; and when the Term but in the adjacent Angles; there
ey is not abſent, the Figure will being two different Species of theſe
have no Diameter ; but if abſent, it Curves, called by Sir Iſaac Newton,
will have one Diameter. And of the Hiperbolifmæ of a Parabola.
theſe latter, there are ſeven different The ſecond Caſe of Equations,
Species. If the Termax3 be viz. xy=ax3 + bx? + cx tod,
abfent, but bx? not, the Figure ex- expreſſes a Figure having two hy-
preffed by the Equation remaining perbolical Legs to one Afymptote,
will be a parabolical Hyperbola, being the principal Ordinate, and
having two hyperbolical Legs to two parabolical Legs.
one Afymptote, and two paraboli The third Caſe of Equations ye
cal Legs converging one and the ax3 + 6*2 + cxt-d, expreffes a
ſame way. And when the Term ey Figure having two parabolical Legs
is abſent, the Figure will have but running out contrary ways; and of
one Diameter ; but when not, it theſe there are five different Specics.
will have no Diameter. And of this Sir Iſaac Newton calls them Di-
latter there are four different Spe- verging or Bell-form Parabola's.
cies, according to Sir Iſaac Newton. See more concerning them under the
In the firſt Caſe of the Equation, Word Parabola Diverging.
when the Terms a x3, bx2, are
The fourth Caſe of Equations
wanting, that is, when the Equa- yax3 +6*" text de expreſſes
tion becomes xyz tey=(x+d, it
a Parabola with contrary Legs, viz.
expreſſes a Figure conſiſting of three the Cubical Parabola.
Hyperbola's oppoſite to one ano 4. Thus, according to Sir Iſaac
ther, one lying between the parallel Newton, there are but 72 Specie: of
Aſymptotes, and the other two Lines of the third Order. But Mr.
without, having three Afymptotes, Sterling afterwards found out four
one of which is the firſt and princi- more Species of redundant Hyper-
pal Ordinate, and the other two bola's ; and I myſelf two more of
are parallel to the Abſciſs, and e the deficient Hyperbolas expreſſed
qually diſtant from it; or elſe two by the Equation xyy=bx? * <*
oppoſite Hyperbola's without the +d. When bx?+extodo has
Afymptotes, and a Serpentine Hyn two unequ.il negative Roots, and
perbola between them ; there being two equal negative Roots i ſo that
four different Species of theſe Curves in reality there are 78 different Spe-
called by Sir Iſaac Newton, the cies of Lines of the third Order.
Hyperbolifmæ of an Hyperbola.. 5. How the ſeveral Equijors for
When the Term c*? is negative, all Lines of the third Order when
the Figure expreſſed by the Equa- the Ordinates are parallel to an t-
tion xy2 tey - cx? +d, is a ſymptote, may be tranſmuted into
Serpentine Hyperbola, having only the four particular Equations above
one Aſymptote, being the principal mentioned, is elegantly enough
Ordinate, or elſe a conchoidal Fi. ſhewn by Mr. Sterling in his lilu-
gure ; there being three different pratio Tractatus D. Nentoni de E-
Species of theſe Curves, called by numeratione Linearum tertii Oidinis.
Sir Iſaac Newton the Hyperboliſmæ The ſame is done by Mr. Nichol
of an Ellipfis.
too, in the Memoires de l'Academie
When the Term c *2 is abſent, Royale de Sciences, Anna 1728. Lut
the Equation ey? tey=d expreſies trilingly long and tedions. Alho '
two Hyperbolas, not lying in the I have ſaid that the Lives of the
S 4 third,
G E O
G E O
y, or
third Order confift either of hyper- ries's.' And this, no doubt, made
bolical or parabolical Parts, yet Mr. Nichol, in the Memoires above
ſome of them have beſides, Ovals related, give a Specimen of perform-
belonging to them, either ſeparate ing the Buſineſs by finite Equations i
from the infinite Legs, or joining and ſince him, I myſelf have wrote
to them; they have alſo double a little Treatiſe, ſewing almoſt by
Points, which make a part of the Inſpection not only how the ſeveral
Curve, and other notable Diftinc- Species of 'thoſe Curves ariſe from
tions, as may be ſeen in Sir Ifaac the previous Deſcription of other
Newton's Enumeration of theſe Curves (whoſe Abſciſs is x, and
Lines, where you have their Fi- Ordinate the whole Value of
gures as well as the Qualifications Part of that Value) but alſo the
of the ſeveral Equations expreſſing manner of finding any Number of
each different Species, chiefly ariſing Points through which they muſt
from the Equation, expreſſing the paſs, and that after a way the moſt
Value of the Ordinate y in the fimple and natural the thing ſeems
Terms of the Abſciſs x ; giving no
to admit.
Ordinate, as often as that Value is 6. Sir Iſaac Newton tells us, that
the Square Root of a negative Curves may be generated by Sha-
Quantity, or Part of thac Value ; an dows. He ſays, if upon an infinite
infinitely ſmall Ordinate ; a finite Plane illuminated from a lucid Point
one ; or an infinitely great one: for the Shadows of Figures be project-
Example, in the firt Caſe of Equa- ed ; the Shadows of the Conic Sec-
tions xy2 -ey=ax3 +6x? tcx tions will be always Conic Sections ;
+d, it will be found by extract- thoſe of the Curves of the ſecond
kind will be always Curves of the
ing the Root that y =
+
ſecond kind; thoſe of the Curves
of the third kind will be always
Vaxt +bx3 toxi+dx+fee; Curves of the third kind; and ſo
on ad infinitum,
ſo that y will be poſſible as often as
And, like as a Circle by projecting
ax4 +683 +6**+ dx tee is its Shadow generates all the Conic
affirmative, and impoſſible when the sections, ſo the five diverging Pa-
fame is negative: And the Number rabola's by their Shadows will ge-
of Times that this can happen will nerate, and exhibit all the reſt of
appear from the Deſcription of a
the Curves of the ſecond kind : and
biquadratical Parabola, whoſe Ab ſo ſome of the moſt ſimple Curves
ſciſs is x, and Ordinate ax446x3+
of the other kinds may be found,
c*?+ dx+ce.
which will form, by their Shadows
Mr. Sterling in the Treatiſe afore- upon a Plane, projected from a lucid
ſaid has ſhewn how to find the Fi- Point, all the reit of the Curves of
gures and ſeveral Species of theſe that' fame kind. But as Sir Iſaac
Curves by throwing the Value of Newton has neither demonſtrated
the Ordinate y into an infinite Series, what he here fays, nor has parti-
which certainly is a very ſhort and cularly ſhewn how his Curves of the
general way of doing the thing ;
ſecond Order may be derived from
but at the ſame time is both difti- the Shadows of the diverging Para-
cult, unnatural, and obſcure ; and bola's, you have in the French Me-
more eſpecially 10 ſuch who are not
moires à Demonſtration of ihir, and
well verſed in the Doctrine of Se- of the ſecond Order, which may be
a Specimen of a few of the Curves
generated
1
e
2%
G E O
GEO
generated by a Plane's cutting a Order, of which I imagine there
Cone or Solid formed from the are ſome thouſands, not ſo much by
Motion of an infinite right Line reaſon of the Difficulty of the
along a diyerging Parabola (having thing, as the want of Inclination to
an Oval) always paſſing thro' purſue it. Four or five Years ago
given or fixed Point above the Plane I was very fond of this Buſineſs, and
of that Parabola.
have now by me ſome hundreds of
Mr. Mac-Laurin, in his Orga- the Curves of the third Order ;
nica Geometria, ſhews how to de- but finding the Number behind ſtill
ſcribe ſeveral of the Species of very great, my Inclination began
Curves of the fecond Order, eſpe- to abate, till at length I grew quite
cially choſe having a double Point, tired of the Work, and laid it a-
by the Motion of right Lines and fide. The Abbé Bragelonge in the
Angles ; but a good commodious French Memoires of the Royal Aca-
Deſcription by a continued Motion demy, has given a Diſcourſe upon
of thoſe Curves which have no dou- ſome of the Curves of this Order,
ble Point is (by Sir Iſaac Newton) which is both long and tedious, and
ranked amongſt the moſt difficult very far ſhort of a compleat Trea-
Problems.
tiſe on this Subject; and at the End
As nobody before Sir Iſaac New- he promiſes an Enumeration of the
ton ever did, or I believe could, give ſeveral Species of theſe Curves.
the Figures, various Species, and But ſince I have not yet feen any
principal Properties of the Curves ſuch thing, he may perhaps have
above the Conic Sections, (altho' in fallen into my Condition.
the Preface to De Witts's Elementa The General Equation of all
Linearum Curvarum a Treatiſe Curves of the 'third kind may be
upon the Curves of the ſecond Or- reduced to the following ten parti-
der was promiſed); ſo it is my firm cular Equations, which were com-
belief, that no one after him will municated to me by my ingenious
be able to enumerate the ſeveral Friend Mr. Duncomb Smith, who is
different Species, and exhibit the very well ſkill'd in theſe things.
Figures of the Curves of the third
1. y4+f*?y?. +gxy3 +hxpy+iy2 +kxytly
2. y4 +fxy3 +8x2; +hxyz +ixytky
3. x2y3 +fy3 78*?ythy3 +-ky
4. *?y? +fy toys fbxtiy
sax+ +6x3 + x2 +dxte
5. 93 +fxyz +gx2y+hy
6. 33 +fxyz +gxythy
7. y+texy+xy3+gay* by°Fixy+ky
8. *3y +exy3-4-f*y+gy2 +bxy tiy
9. x3ytey3 +fxyz +gxy+hy
=ax3 +63? tcxtd.
10. *3ytey3 +fy2 +gxy+hy
If it be ſo difficult to underſtand again, what an infinite Increaſe of
the Nature, Properties, and Num- Difficulty will ariſe in apprehending
ber of the Curves of the ſecond and the Nature of the infinite-infinite
third Kinds, how much more ſo Number of Curves which do not
muſt it be to attain to a glimpſe lie in the ſame Plane? When one
of that infinite Number and Variety duly conſiders this, it muſt be con-
expreſſed by the Equations of the feffed that the moll ſkilful and pene-
ſucceeding higher Dimenſions ? Apd trating Mathematician poſſible, may
really
GEO
GEO
1
really be ſaid to know little or no- the Art of Meaſuring the Earth,
thing at all concerning the Nature but it is now the Science of what-
of Curve Lines, however he may ever is extended, ſo far as it is ſuch ;
otherwiſe think. Thoſe who have that is, of Lines, Superficies, and
a mind to ſee how far this Doctrine Solids.
has been advanced, with regard to GEOMETRY, as related by
Curves of the higher Orders, as Proclus, had its firſt riſe in Egypt,
well as thoſe of the firſt and ſecond where the Nile annually overflow-
Orders, may conſult Mr. Mac- ing the Country, and covering it
Laurin's Organica Geometria, and with Mud, obliged Men to diſtin-
Mr. Braikonridge's Exercitatio Geo- guiſh their Lands one from another
metrica de Curvarum Deſcriptione. by the Confideration of their Fi-
All geometrical Lines of the odd gure; and to be able alſo to mea-
Order, viz. the third, fifth, feventh, fure the Quantity of them, and to
&c. have at leaſt one Leg running know how to plot it, and lay them
on infinitely ; becauſe all Equations out again in their juft Dimenſions,
of the odd Dimenſions have at leaſt Figure, and Proportion; after which,
one real Root. But vaft Numbers 'tis likely a 'farther Contemplation
of the Lines of the even Orders are of thoſe Draughts and Figures
only Ovals ; amongſt which there helped them to diſcover many ex-
are ſeveral having very pretty Fl. cellent and wonderful t roperties be-
gures, fome being like fingle Hearts, longing to them, which Speculation
others double ones, others in figure continually was improving, and is
of Fiddles, others again ſingle Knots, ſtill to this very day. But the Geo-
double Knots, & C.
metry of the Ancients was contain'd
Two geometrical Lines of any within narrow Bounds, as well as
Order will cut one another in as their other Mathematical Specula-
many Points as the Number ex- tions, for it only extended to right
preſſes, which is produced by the Lines and Curves of the firſt kind,
Multiplication of the two Numbers or Order ; whereas now Lines of
expreſſing thoſe Orders. And Mr. infinite Orders are received in Geo-
Braikonridge, in the Preface to his metry.
Treatiſe aforeſaid, fays, Mr. George Geometry is divided into Specu-
Campbell, now Clerk of the Stores lative and Practical : The former
at Woolwich, has got a neat Demon- treating of the Properties of Lines
tration of the ſame, which he hopes and Figures ; ſuch as Euclid's Ele-
he will publiſh.
ments, Apollonius's Conics, &c. And
GEOMETRICAL PLANE. See the latter ſhews how to apply theſe
Plane.
Speculations to Uſe in Life.
GEOMETRICAL PROGRES Plato thought the word Geometry
SION, or PROPORTION. See Pro a very ridiculous Name for this
greſion.
Science, and ſubſtituted in its place
GEOMETRICAL SOLUTION of the more extenſive Name of Men-
a Problem, is when the Thing is ſuration ; and after him, others
folved according to the Rules of gave it the Title of Pantometry.
Geometry, and by ſuch Lines as But this is too ſcanty; for it not
are truly geometrical, and agreeable only enquires into, and demonſtrates
to the Nature of the Problem. the Quantities of Magnitudes, but
GEOMETRIC PLACE, or Locus. alſo their Qualities, ziz Species,
See Locus.
Figures, Ratio's, Pofitions, Tranſ-
GEOMETRY, originally ſignifies formations, Deicriptions, Divifions,
how
GEO
GI V
how to find their Centres, Diame- Torricellius, in his Opera Geometrica.
ters, Tangents, Aſymptotes, Curva- -Viviani, in his Divinationes Geo-
tures, & c. ſome ſay it is the Science metricæ in quintum Librum Apol-
of enquiring, inventing, and demon- lonii Pergei adhuc defideratum. -
ſtrating all the Affections of Mag- Theodoſius, in his Spherics.--Serenius,
nitude. And Proclus calls it the in his Section of the Cone and Cyline
Knowledge of Magnitudes and fi- der. Gregory St. Vincent, in his
gures, and their Limitations ; alſo Quadratura Circuli ; and many ou
of their Ratio's, Affections, Pofitions, thers. Add to theſe Dr. Barrow's
and Motions of every kind.
Geometrical Lectures.-- Bullialdus's,
The Writings upon Geometry Schooten's, and Dr. Gregory's Exer-
are very numerous ; ſome ſpecula- citationes Geometricæ.- De Billy's
tive, and others practical. A- Treatiſe de Proportione harmonica.-
mongſt the former are the well- La Lovera's Geometria veterum pro-
known Elements of Euclid, firſt mota. Viviani's Exercitatio Ma.
wrote by him in Greek more than thematica.-- Herberftein's Diotome
2000 Years ago ; but in theſe later Circulorum.
but in theſe later Circulorum.- Palma's Exercitatio-
Ages tranſlated into various Lan nes in Geometriam. Apollonius de
guages. Orontius Finæus, Anno Sestione Rationis. The Writers upon
1530, publiſhed a Commentary up- Practical Geometry, are Clavius,
on the firſt fix Books ; and ſo did Mallet, de la Hire, Taquet, Ozanam,
James Peletarius, Anno 1557. Ni. Wolfius, and many others, which I
colas Tartaglia alſo publiſhed about hall omit to mention.
the ſame time a Commentary upon
GIBBOUS, is a Term uſed in re-
all the
15
Books. After which ference to the enlighten'd Parts of
Clavius did the like.
the Moon, while ſhe is moving from
There is a Greek Commentary Full to the firſt Quarter, and from
upon Euclid's firſt Book by Proclus : the laſt Quarter to the Full again ;
As alſo thoſe of Campanus and for all that time the dark Part ap-
Theon, upon the whole Books. There pears horned and falcated, and the
are alſo Commandine's, Dee's, Schu- light one bunched out, convex or
belius's, Herlinus's, Daſypodius's, gibbous.
Ramus's, Herigon's, Barrow's, Ta GIRDERS, in Architecture, are
quet's, Dechales's, Furnier's, and the largeſt Pieces of Timber in a
Scarborough's Euclid, with many o- Floor. Their Ends are uſually fa-
thers too many to mention here. ſten'd into the Summer or Breaſt-
There are many modern Writers Summers, and the Joiſts are framed
of the Elements of Geometry, as in at one end to the Girders. No
well as Euclid; ſuch as Borellus, Girder Mould lie leſs than ten Inches
Pardies, Arnald, Sturmy, Lamy, Po into the Wall, and their Ends ſhould
lynier, Marchetti, Wolfus, &c. A- be laid in Lome, &c.
mongſt thoſe who have exceeded
Given, is a Word often uſed in
Euclid in the Elementary Geome- Mathematics, and fignifies fome-
try, we have Archimedes in his thing which is ſuppoſed to be
Treatiſes of the Sphere and Cylinder, known. Thus, if a Magnitude,
of the Dimenſion of the Circle, of be known, or that we can find an-
Conoids and Spheroids, of Spirals other equal to it, they ſay 'tis a
and the Quadrature of the Parabola. given Magnitude If the Poſition
Kepler, in his Nova Stereometria of any thing be ſuppoſed as known,
Doliorum Vinariorum.- Cavalerius, they ſay, given in Pofition. Thus
in his Geometria Indivifibilium. if a Circle be actually deſcribed up-
on
E L O
GNO
on any Plane, they fay, its Centré account, visi that the Diſtances
is given in Poſition ; its Circumfe between Places upon the Rhumb
rence is given in Magnitude ; and are all meaſured by the ſame
the Circle both in Poſition and Mag- Scale of equal Parts, and the Di-
nitude. But a Circle may be given ſtance of any two Places in the
in Magnitude only; as, when only Arch of a great Circle, is nearly
its Diametėr is given, but the Cira repreſented in this Chart by a
cle not actually deſcribed. If the ſtraight Line ; and ſo, if Land-Maps
Kind or Species of any Figure be were inade according to this Pro-
given, they ſay, given in Species : jection, they would, in my opinion, be
if the Ratio between any two better than thoſe that are made any
Quantities is known, they are ſaid other ways whatſoever. But this
to be given in Ratio.
Chart will never be of fo excellent
GLACIS, a floping Bank in For- Uſe to Seamen, as Mercator's ; be-
tification. It ſignifies a very gentle cauſe the Meridians, Parallels, and
Steepneſs; but is more eſpecially particularly the Rhumb-Lines, being
taken for that which rangeth from all Curves in the Globular Chart, but
the Parapet of the cover'd Way, to ſtraight Lines in that of Mercator ;
the Level on the ſide of the Field. fraight Lines are vaſtly more eaſy
GLOBE, the ſame as Sphere. to draw and manage than Curves,
Which ſee.
eſpecially ſuch as the Rhumb-Lines
When a Globe has all the Parts on the Globular Chart are.
of the Earth and Sea drawn or de This Projection is not new, but
lineated on its Surface, like as on a on the contrary very ancient ; . for
Map, and placed in their natural it is mentioned by Ptolemy in his
Order and Situation, it is called an Geography; as alſo by Blundevill, in
artificial terreftrial Globe.
his Exerciſes.
: But if upon the Superficies there GNOMON, in a Parallelogram, is
of, be painted the Images of the a Figure made of the two Comple-
Conſtellations, and the fixed Stars, menis, together with either of the
with the Circles of the Sphere, it Parallelograms about the Diagonal ;
is called an artificial cælefiial as in the Parallelogram Aổ, the
Globe.
Gnomon is M +xfoz+N, or
Both theſe Globes, in order to M+N+*+%.
fhew the Nature of the Sphere, and
reſolve Aſtronomical and Geogra-
B
phical Problems, are fitted and
M
moveable in Braſs Meridians, and
X
theſe Meridians are ſet in Notches
N
made in broad wooden Circles re-
Z
preſenting the Horizon.
GLOCULAR CHART, is a Name А
given to a Repreſentation of the GNOMON, in Dialling, is the
Surface, or fome Part of the Sur- Style, Pin or Cock of any Dial,
face of the terraqueous Globe upon whoſe Shadow ſhews the Hour.
a Plane, wherein the Parallels of La- The Gnomon of every Dial repre-
titude are Circles nearly cor centric; ſents the Axis of the World.
the Meridian's Curves bei ding to GNOMONIC PROJECTION of the
wards the Poles, and the Raumb- Sphere, is the Repreſentation of the
Lines alſo Curves.
Circles of the Sphere, upon a Plane
This Chart is valuable upon this that touches the Sphere, or elſe on
2
!
one
GOR
GRA
on one that does not cut it, the Ėye elſe but the prolonging of the Cour-
being ſuppoſed in the Centre of the tines from their Angle with Flanks,
Sphere.
to the Centre of the Baſtion whete
In this Projection, (which all they meet; but when the Baſtion is
Plane Sun-Dials may be faid to be flat, its Gorge is a right Line,
of, from whence it derives its Name, which terminates the Diſtance com-
viz. from Gnomonics, or Dialling,) prehended between two Flanks.
all the great Circles of the Sphere GORGE of the Ravelin, or of 4
are repreſented by ſtraight Lines, of Half-Moon, is the Space contained
an indeterminate Length. All leſſer between the Extremities of the two
Circles, parallel to the Plane of Faces on the side of the Place.
Projection, will be Circles ; and all Gothic (or MODERN). ARCHI-
leſfer Circles, oblique to the Projec- TEGTURE, is that which is far re-
tion-Plane, will be either Parabola's moved from the Manner and Pro-
Ellipſes, or Hyperbola's, according portions of the Antique, having its
to their different Obliquity.
Ornaments wild and chimerical, and
GNOMONICS.
The ſame with its Profiles incorrect However, it
Dialling.
is oftentimes found very ſtrong, and
Golden NUMBER. See Cycla appears very rich and pompous, as
of the Moon.
particularly in ſeveral Engliſ Ca-
If i be added to the Year, and thedrals. This manner of Building
the Sum be divided by 19, the Re- came originally from the North,
mainder, after Diviſion, is the Gol whence it was brought by the Goths
olen Number.
into Germany, and has ſince been
GOLDEN RULE. See Rule of introduced into other Countries.
Three
GRANADO, is a little hollow
GORGE, GULLA, or Neck, in Globe, or Ball of Iron, or other
Architecture, is the narroweſt Part Metal, about two Inches and a half
of the Tuſcan or Doric Capitals,
Doric Capitals, in Diameter, which being filled with
lying between the Aſtragal, above fine Powder, is, fet on fire by the
the Shaft of the Pillar, and the An-, means of a ſmall Fuſee, faſtened to
nulets. It is alſo a kind of concave the Touch-Hole: As ſoon as it is
Moulding, larger, but not ſo deep kindled, the Caſe flies into many
as a Scotia, which ſerves for Com- Shatters, much to the Damage of
partments, & c.
all that ſtand near. Theſe Grana-
GORGE, in Fortification, is the does ſerve to fire cloſe and narrow
Entrance of the Platform of any Paſſages, and are often thrown with
Work.
the Hand among the Soldiers, to
Gorge, in ail other Outworks, diſorder their Ranks; more eſpe-
is the Interyal betwixt the Wings cially in thoſe Potts where they ſtand
on the side of the great Ditch. But thickeſt, as in Trenches, Redoubts,
it ought to be obſerved, that all the Lodgments, &c.
Gorges are deftitute of Parapets ; GRAVITY, is that force by
becauſe, if there were any, the Be- which Bodies are carried, or tend
fiegers, having taken poffeffion of a towards the Centre of the Earth.
Work, might make uſe thereof, to GRAVITY (4BSOLU'TE,) is the
defend theinſelves from the Shot of whole Force by which any Body,
the Place; ſo that they are only tends towards the Centre of the
fortified with Palliſadoes, to prevent Earth.
a Surprize.
GRAVITY (ACCELERATE,) is
GORGE of a Bajlion, is nothing the Force of Gravity conſider'd, as
growing
GRA
GRA
growing greater, the nearer it is to 7. In all Places equi-diſtant from
the attracting Body or Point. the Centre of the Earth, the Force
GRAVITY (RELATIVE,) is the of Gravity is nearly equal.
Exceſs of the Gravity in any Body, 8. Gravity equally affects all Bo-
above the ſpecific Gravity of a Fluid dies, without regard to their Bulk,
it is in.
Figure, or Matter; ſo that abftract-
GRAVITATIon, is a Preſſure that ing from the Reſiſtance of the Me-
a Body, by the Force of its Gravity, dium, the moſt compact and looſe,
exerts on another Body under it. the greateſt and ſmalleſt Bodies
1. All Bodies are mutually heavy, would deſcend equal Spaces in equal
or gravitate mutually towards each Times, as appears from the quick
other; and this Gravity is propor- Deſcent of very light Bodies in the
tional to the Quantity of Matter; exhauſted Receiver.
and at unequal Diſtances it is in 9. There are various Opinions of
verfly, as the Square of the Diſtance. Philoſophers concerning the Cauſe
And ſo the Sun and Planets mutually of Gravity ; but the moſt probable
gravitate towards each other; the is, that of a very ſubtle Fluid,
Satellites of Jupiter and Jupiter; which encompaſſes the Earth and
the Satellites of Saturn and Saturn; Air, that freely pervades the Pores
and the Moon and the Earth. of all Bodies: For the Endeavours
2. On the Surfaces of Bodies that of ſuch a Fluid to detrude all earthly
are Spherical and Homogeneous, Bodies from it, together with ſome
the Gravities will be in the Ratio other Properties, may make all Bo-
compounded of the Denſities and dies move towards the Centre of the
the Diameters.
Earth : And that there is ſuch a
3. If a Body be placed in a Fluid, is ſewn by Experiments.
Sphere that is Homogeneous, Hol-
10. Sir Iſaac Newton, in his Op-
low, and every where of the ſame tics, the laſt Edition, propoſes the
Thickneſs, it has no Gravity, let it following Queries concerning that
be placed where it will.
ſubtle Medium, which is the Cauſe
4. In an homogeneous Sphere, of the Gravity and Attraction of
Gravity decreaſes in coming towards Bodies.
the Centre, in the direct Ratio of 1. If in two large tall Cylindrical
the Diſtance from the Centre. Veſſels of Glaſs inverted, two little
5. By Gravity all Bodies deſcend Thermometers be ſuſpended, ſo as
towards a Point, which either is, not to touch the Veſſels, and the
or is very near to the Centre of Air be drawn out of one of theſe
Magnitude of the Earth and Sea, Veſſels, and theſe Veſſels thus pre-
about which the Sea forms itſelf into pared be carried out of a cold
a ſpherical Surface; and the Pro. Place into a warm one, the Ther-
minences of the Land, conſidering moineter in vacuo will grow warm
the Bulk of the Whole, differ but as much, and almoſt as ſoon as the
inſenſibly therefrom.
Thermometer which is not in va-
6. This point or Centre is fixed cuo; and
cuo ; and when the Veſſels are
within the Earth, or at leaſt hath carried back into a cold Place, the
been ſo ever ſince we have any au- Thermometer in vacuo will grow
thentic History. For a Conſequence cold almoſt as ſoon as the other
of its Shifting, tho' never ſo little, Thermometer. Is not the Heat of
would be overflowing of the low the warm Room conveyed thrcugh
Land on that Side of the Globe to the Vacuum by the Vibrations of
wards which it approached. a much ſubtler Medium than Air,
which,
GR A
GRA.
which, after the Air was drawn Liness And doth not the gradual
out, remained in the Vacuum ? Condenſation of this Medium ex-
And is not this Medium the ſame tend to ſome Diſtance from the Bo-
with that Medium by which Light dies, and thereby cauſe the Infle-
is refracted and reflected ? and by xions of the Rays of Light, which
whoſe Vibrations Light communi- paſs by the Edges of denſe Bodies, at
cates Heat to Bodies, and is put into ſome diſtance from the Bodies.
Fits of eaſy Reflexion and eaſy 4.
Is not this Medium much rarer
Tranſmiſſion ? And do not the Vi- within the denſe Bodies of the Sun,
brations of this Medium in hot Bo- Stars, Planets, and Comets, than in
dies contribute to the Intenſeneſs the empty Celeſtial Spaces between
and Duration of their Heat? And them ? And in paſſing from them to
do not hot Bodies communicate their great Diſtances, doth it not grow
Heat to contiguous cold ones, by denſer and denſer perpetually, and
the Vibrations of this Medium, pro- thereby cauſe the Gravity of thoſe
pagated from them into cold ones great Bodies towards one another,
And is not this Medium exceedingly and of their Parts towards the Bo-
more rare and ſubtle than the Air, dies ; every Body endeavouring to
and exceedingly more elaſtic and go from the denſer Parts of the
active? And doth it not readily per- Medium towards the rarer? For if
vade all Bodies ? And is it not (by this Medium be rarer within the
its elaſtic Force) expanded through Sun's Body than at its Surface, and
all the Heavens ?
rarer there than at the hundredth Part
2. Doth not the Refraction of of an Inch from its Body, and
Light proceed from the different rarer there than at the fiftieth Part
Denſity of this Ætherial Medium of an Inch from its Body, and rarer
in différent Places, the Light' re there than at the Orb of Saturn; I
ceding always from the denſer fee no reaſon why the Increaſe of
Parts of the Medium ? And is not Denſity ſhould ſtop any where, and
the Denſity thereof greater in free not rather be continu'd through all
and open Spaces, void of Air, and Diſtances from the Sun to Saturn,
other groſſer Bodies, than within and beyond. And though this In-
the Pores of Water, Glaſs, Cryſtal, creaſe of Denſity may at great Di-
Gems, and other compact Bodies ? ſtances be exceeding flow, yet, if the
For when Light paſſes through elaſtic Force of this Medium be ex-
Glaſs, or Cryital, and falling very ceeding great, it may ſuffice to im-
obliquely upon the farther Surface pel Bodies from the denler Parts of
thereof, is totally reflected, the the Medium towards the rarer, with
total Reflection ought to proceed all that Power which we call Gra-
rather from the Denſity and Vigour vity. And that the elaſtic Force of
of the Medium without, and be- that Medium is exceeding great,
yond the Glaſs, than from the Ra- may be gathered from the Swiftneſs
rity and Weakneſs thereof.
of its Vibrations. Sounds move
3.
Doth not this @cherial Me- about 1140 Engliſh Feet in a Second
dium in paſſing thro' Water, Glaſs, of Time, and in ſeven or eight
Cryttal, and other compact and Minutes of Time they move
denſe Bodies inco empty Spaces, bout one hundred linoliſh Miles.
grow deníer and denſer by degrees, Light moves from the Sun to
and by that means refract the Rays about ſeven or eight Minutes of
of Light nor in a Point, but by Tine, which Diltance is about
bending them gradually in Curve- 70000coo Engliſh Miles ; fuppofing
the
us in
}
GR A
GR A
gether.
the horizontal Parallax of the Sun and all groſs Bodies, perform their
80 be about 12 ſec. And the Vi- Mocions more freely, and with leſs
bracions or Pulſes of this Mediam, Reſiſtance in this Æthereal Medi-
that they may cauſe the alternate um, than in any Fluid, which fills
Fits of eaſy Tranſmiſſion and eafy all Space adequately, without leav-
Reflexion, muft ble ſwifter than ing any Pores, and by conſequence
Light, and by conſequence above is much denſer than Quickſilver or
70oooo Times ſwifter than Sounds. Gold? And may not its Refiftance
And therefore the elastic Force of be ſo ſmall, as to be inconſiderable ?
this Medium, in proportion to its Den- For inſtance, if this Æther (for ſo I
fity, muſt be above 700000 x 700000 will call it) ſhould be ſuppoſed
(that is above 490000000000) Times 700000 Times more elaſtic than our
greater than the elaſtic Force of the Air, and above 700000 Times more
Air, is in proportion to its Denſity. rare, its Reſiſtance would be above
For the Velocities of the Pulſes of 600000000 Times leſs than Water :
elaſtic Mediums are in a ſubdupli- And ſo fmalla Reſiſtance would
cate Ratio of the Elafticities and the ſcarce make any ſenſible Alteration
Rarities of the Mediums taken to in the Motions of the Planets in ten
thouſand Years. If any one would
5. As Attraction is ſtronger in aſk how a Medium can be ſo rare,
ſmall Magnets than in great ones, in let him tell me how the Air, in the
proportion to their Bulk; and upper Parts of the Atmoſphere, can be
Gravity is greater in the Surfaces above an hundred thouſand thcuſand
of ſmall Planets than in thoſe of Times rarer than Gold ? Let him
great ones, in proportion to their alſo tell me how an electric Body
Bulk; and fmall Bodies are agitated can, by Friction, emit an Exhala-
much more by electric Attraction tion ſo rare and ſubtile, and yet ſo
than great ones; ſo the Small- potent, as by its Emiſſion to cauſe
neſs of the Rays of Light may con no ſenſible Diminution of the Weight
tribute very much to the Power of of the ele&tric Body, and to be ex-
the Agent, by which they are re- panded through a Sphere, whoſe
fracted. · And ſo, if any one ſhould Diameter is above two Feet, and
fuppoſe that Æther (like our Air) yet to be able to agitate and carry
may contain Particles, which en- up Leaf Copper, or Leaf- Gold, at
deavour to recede from one another, the Diſtance of above a Foot from
(for I do not know what this A. the electric Body? And how the
ther is,) and that its Particles are Efluvia of a Magnet can be ſo rare
exceedingly ſmaller than thoſe of and fubtile, as to paſs through a
Air, or even thoſe of Light: The Plate of Glaſs, without any Reſif-
exceeding Smallneſs of its Particles tance, or Diminution of their Force,
may contribute to the Greatneſs of and yet ſo potent, as to turn a
the Force, by which thoſe Particles magnetic Needle beyond the
may recede from one another, and Glaſs ?
thereby make that Medium ex 7. Is not Viſion performed chief-
ceedingly more rare and elaſtic than ly by the Vibrations of this Me-
Air, and by conſequence exceedingly dium, excited in the bottom of the
leſs able to reſiſt the Motions of Pro- Eye, by the Rays of Light, and
jectiles, and exceedingly more able propagated through the folid, pel-
to preſs upon groſs Bodies, by endea- lucid, and uniform Capillamenta of
vcuring to expand itſelf.
the optic Nerves into the Place of
6. May not Planets and Comets Senſation? And is not Hearing per-
formed
GRA
GRA
formed by the Vibrations either of ſmall Diſtances performs the Chy-
this or ſome other Medium, excited mical Operations of Fermentation,
in the auditory, Nerves by the Tre- & c. and reaches not far from
mors of the Air, and propagated the Particles with any ſenſible
through the folid, pellucid, and uni- Effect.
form Capillamenta of thoſe Nerves 10. All Bodies feem to be com-
into the Places of Senfátion; and ſo poſed of hard Particles ; for og
of the other Senſes.
therwiſe Fluids would not con-
8. Is not animal Motion perform- geal.
ed by the Vibrations of this Me 11. Even the Rays of Light ſeem
dium, excited in the Brain by the to be hard Bodies.
Power of the Will, and propagated 1 2. Now if compound Bodies are
from thence through the folid, pel fo very hard, as we find ſome of
lucid, and uniform Capillamenta of them to be, and yet are very porous,
the Nerves into the Muſcles, for. and conſiſt of Parts, which are on-
contracting and dilating them! I ly laid together, the fimple Para
fuppoſe that the Capillamenta of the ticles which are void of Pores, and
Nerves are each of them folid and were never yet divided, muft be
uniform, that the vibrating Motion harder ; for ſuch hard Particles bo-
of the Æthereal Medium may be ing heaped up together, can fcarce
propagated along them from one touch one another in more than a
End to the other uniformly, and few Points, and therefore muſt be
without Interruption; for Obſtruc- feparable by a much leſs Force than
tions in the Nerves create Palfies. is requifite to break a folid Particle,
And that they may be ſufficiently whoſe Parts touch in all the Space
uniform, I ſuppoſe them to be pel- between them, without any Pores
lucid, when viewed ſingle, tho' the or Interſtices to weaken their Co-
Reflections in their Cylindrical Sur- heſion. And how ſuch very hard
faces may make the whole Nerve Particles, which are only laid to-
(compoſed of many Capillamenta) gether, and touch only in a few
appear opake and white; for Opa- Points, can ſtick together, and that
city ariſes from reflecting Surfaces, fo firmly as they do, without the
ſuch as may diſturb and interrupt Affiſtance of ſomething which cauſes
the Motions of this Medium.
them to be attracted or preſs'd to-
9. The Parts of all homogeneal wards one another, is very difficult
hard Bodies, which fully touch one to conceive. ,
another, ſtick together very ſtrong 13. The ſame thing I infer alſo
ly: And for explaining how this from the cohering of two poliſhed
may be, ſome have invented hooked Marbles in vacuo, and from the
Atoms, which is begging the Que- ftanding of Quickſilver in the Baro
ſtion; and others tell us, that Bo- meter at the Height of fifty, fixty,
dies are glued together by reft, that or feventy Inches, or above, when-
is, by an occult Quality, or rather ever it is well purged of Air, and
by nothing; and others, that they carefully poured in, ſo that its Parts
{tick together by conſpiring
Motions, be every where contiguous, both to
that is, by relative Reſt amongſt one another, and to the Glaſs. The
themſelves. I had rather infer Atmoſphere by its Weight Breſles
from their Coheſion, that their the Quickflver into the Glaſs, to
Particles attract one another by the Height of twenty-nine or thirty
fome Force, which, in immediate Inches : And ſome other Agent
Contact, is exceeding ſtrong, and at raiſes it higher, not by preſſing is
T T
into
GRA
GRA
into the Glaſs, but by making its open Air, (as hath been tried be-
Parts ſtick to the Glaſs, and to one fore the Royal Society,) and there-
another; for upon any Diſcontinua- fore are not influenced by the Weight
tion of Parts, made either by Bubbles, or Preſſure of the Atmoſphere. .
or by ſhaking the Glafs, the whole
15. If two plain poliſhed Plates
Mercury falls down to the Height of of Glaſs, three or four Inches broad,
twenty-nine or thirty Inches. and cwenty or twenty-five long, be
14. Moreover, if two plain po- laid, one of them parallel to the
liſhed Plates of Glaſs (ſuppoſe two Horizon, the other upon the firſt,
Pieces of a poliſhed Looking-Glaſs} ſo as at one of their Ends to touch
be laid together, ſo that their Sides one another, and contain an Angle
be parallel, and at a very ſmall of about ten or fifteen Minutes, and
Diſtance from one another, and the fame be firſt moiſten'd on their
then their lower Edges be dipped inward Sides, with a clean Cloth,
into Water, the Water will riſe up dipped into Oil of Oranges, or Spi-
between them ; and the leſs the Din sit of Turpentine, and a Drop or
ſtance of the Glaſſes is, the greater two of the Oil or Spirit be ler
will be the Height to which the fall upon the lower Glaſs at the o-
Water will riſe. If the Diſtance be ther End; ſo ſoon as the upper
about the hundredth Part of an Glaſs is laid down upon the lower,
Inch, the Water will riſe to the ſo as to touch it at one end as a-
Height of about an Inch; and if bove, and to touch the Drop at the
the Diſtance be greater or leſs in other end, making with the lower
any Proportion, the Height will
be Glaſs an Angle of about ten or fif-
reciprocally proportional to the Di- teen Minutes, the Drop will begin
ftance, very nearly : For the attrac- to move toward the Concourſe of
tive Force of the Glaſſes is the ſame the Glaſles, and will continue to
whether the Diſtance between them move with an accelerated Motion
be greater or leſs, and the Weight of till it arrives at that Concourſe of
the Water drawn up is the ſame, if the Glaſſes ; for the two Glaſſes at-
the Height of it be reciprocally pro- tract the Drop, and make it run
portional to the Height of the that way towards which the Ato
Glaſſes. And, in like manner, Wa traction inclines. And if, when the
ter aſcends between two Marbles, Drop is in motion, you lift
up
that
polithed plain, when their poliſhed End of the Glaſſes where they meet,
Sides are parallel, and at a very and towards which the Drop moves,
little Diſtance from one another : the Drop will aſcend between the
And if ſlender Pipes of Glaſs be Glaſſes, and therefore is attracted.
dipped at one End into ſtagnating And as you lift up the Glaffes more
*Water, the Water will riſe up with- and more, the Drop will aſcend ſlow-
in the Pipes, and the Height to which er and flower, and at length reft, be-
it ariſes will be reciprocally propor- ing then carried downward by its
tional to the Diameter of the Ca Weight, as much as upwards by the
vity of the Pipe, and will be equal Attraction. And by this means you
to the Height to which it riſes be- may know the Force by which the
tween two Planes of Glaſs, if the Drop is attracted at all Diſtances
Semi-Diameter of the Cavity of the from the Concourſe of the Glaſſes.
Pipe be equal to the Diſtance be 16. There are therefore Agents in
tween the Planes, or thereabouts. Nature able to make the Particles of
And theſe Experiments ſucceed after Bodies ſtick together by very Atrong
: the fame manner in vacuo, as in the Attractions. And it is the Buſineſs of
expe-
1
out.
GUL
GUN
experimentalPhiloſophy to find them Gulf, in Geography, is ſuch a
Part of the Ocean, as runs up into
Great Bear. See Urſa Major. the Land, thro' narrow Paſſages, or
GREAT CIRCULAR SAILING, Streights ; as the Gulf of Florida,
is the manner of conducting a Ship in America; the Arabian Gulf, or
in, or rather pretty near the Arch Red-Sea in Africa; the Perſian Gulf
of a great Circle, that paſſes through in Afia; the Gulf of Venice, or the
the Zenith of the two Places froin Adriatic Sea in Europe.
whence, and to which ſhe is bound. GUNTER'S-LINE, or the Line of
GREAT Circles of the Globe or Numbers, is the common Line of
Sphere, are thoſe whoſe Planes paffing Numbers, invented by Mr. Gunter,
through the Centre of the Sphere, di- a Profeſſor of Geometry at Greſham-
vide ic into two equal Parts or He- College. It is only the Logarithms
niſpheres : of which there are ſix laid off upon ſtraight Lines ; and its
drawn on the Globe, viz. the Me- Uſe is for performing Operations of
ridian, Horizon, Equator, Ecliptic, Arithmetic, by means of a pair of
and the two Colures. Which fee. Compaſſes, or even without, by
GREGORIAN YEAR. The new fliding two of theſe Lines of Num.
Account, or new Style, inſtituted bers by each other.
upon the Reformation of the Kalen GUNTER'S QUADRANT, is a
dar, by Pope Gregory XIII. (from Quadrant of Wood, Braſs, &c. being
whom it takes the Name) in the partly a Stereographical Projection
Year 1582. Whereby ten Days be- upon the Plane of the Equinoctial,
ing taken out of the Month of Oc- the Eye being in one of the Poles,
tober, the Days of their Months go where the Tropic, Ecliptic, and
always ten Days before ours : As Horizon, are Arches of Circles;
for inſtance, their eleventh is our
but the Hour-Circles are all Curves
Which« new Style or
drawn by means of the ſeveral Al-
Account, is uſed in moſt Parts be- titudes of the Sun for ſome particu-
yond the Seas; and is called from lar Latitude every Day in the Year.
Pope Gregory, the Gregorian Account. The Uſe of this Inftrument, is to
GRENADO. See Grenado. Sheil. find the Hour of the Day, the Sun's
GROUND-PLATES, in Archi- Azimuths, &c. and the other.com-
tecture, are the outernioſt Pieces of mon Problems of the Globe; as
Timber, lying on near the alſo to take the Altitude of an Ob
Ground, and framed into one ano. ject in Degrees; Put theſe Quadrants,
ther with Morteſſes, and Tennons as commonly fold by Initrument-
of the Joiſts, the Summer and Gir- Makers, are but of very little uſe,
ders; and ſometimes the Trimmers on account of their Inaccuracy, and
for the Stair Care and Chimney-way, the ſmall Radius they are made to.
and the Binding-Joifts.
They may indeed ſerve Country-
GUERITE, in Fortification, is a Fellows to tell what is a clock to
ſmall Tower of Wood or Stone, half an Hour, or a Quarter perhaps ;
placed uſually on the Point of a as likewiſe to amuſe their ignorant
Baſtion, or on the Angles of the Neighbours.
Shoulder, to hold a Centinel, who Note, This Quadrant is by no
is to take care of the Ditch and to Means to good as Collins's, in find-
watch out againſt a Surprize. ing the Hour of the Duy.
GULA, or GULLET. See OEfa-
GUNTER'S-SCALE, uſually called
phagus.
by Seamen the Gunter, is a large
GULBE, in Architecture, the fame plain Scale, with the Lines of ar-
as Gorge.
tificial
firſt Day.
or
T 2
HAL
HE A
tificial Sines, Tangents, and verſed tain Meteor, in figure of a bright
Sines, laid off upon ſtraight Lines Circle, encompaſſing the Sun, Moon,
on it, fo contrived to a Line of or a Star, eſpecially the Moon.
Numbers upon it, that by means Theſe Halo's do ſometimes ap-
of this Scale, and a pair of Com- pear colourd, like the Rainbow :
paffes, all the Caſes of plain and And Sir Iſaac Newton, in his Op-
ſpherical Trigonometry may be ticks, gives a Hint at their Solution ;
ſolv'd tolerably exact, and conſe- where he ſhews that they ariſe
quently, all Queſtions in Naviga- from the Sun, or Moon's ſhining
tion, Dialling, &c. may be work'd through a thin Cloud, conſiſting of
by it.
Globules of Hail or Water, all of
The Name of this Scale is from the ſame ſize.
the firſt Inventer Mr. Gunter. It HARMONICAL,
or MUSICAL
is now commonly put upon Sec: PROPORTION.
Three or
four
tors, being there call'd Artificial Quantities are ſaid to be in an
Lines.
Harmonical Proportion ; when in
Gutte, or Drops, in Archi- the former Caſe, the Difference of
tecture, are certain Parts in figure the firſt and ſecond ſhall be to the
of little Bells, which being fix in Diffurence of the ſecond and third,
Number, are placed below the Tri- as the firſt is to the third ; and in
glyphs, in the Architrave of the the latter, the Difference of the firſt
Doric Order. Theſe are thus named and ſecond to the Difference of the
from their Shape, reſembling the third and fourth, as the firſt is to the
Drops of Water, that having run fourth : For Example, 2, 3, and 6,
along the Triglyphs, ftill hang un are harmonically proportional: For
der the Cloſure between the Pil 1:3::2:6. If proportional Terms
Jars.
in the former Cafe are continu'd,
there will ariſe an harmonical Pro-
greffion.
If there be three Quantities in an
H.
harmonical Progreſſion, the Diffe-
rence between the ſecond and twice
AL F-MOON, in Fortifica- the firſt, is to the firſt, as the fe-
tion, is an Out-Work having cond is to the third. Alſo the Sum
only two Faces, forming together a of the firſt and laſt is to twice the
Saliant-Angle, which is flank'd by firſt, as the laſt is to the middle
ſome part of the Place, and of the one.
other Baſtions.
If there be four Quantities in
HALP-Moons are ſometimes an harmonical Proportion, the Dif-
raiſed before the Courtains, when ference between the ſecond and
the Ditch is a little wider than twice the firſt, is to the firſt, as the
'it ſhould be ; and they are much third to the fourth.
the ſame as Ravelins, only the HARMONY, is an agreeable or
Gorge of the Half-Moon is made pleaſant Union between two
bending in, like a Bow, and moft more Sounds, continuing together
commonly covers the Point of a at the ſame Time.
Baſtion ; whereas Ravelins are placed HEAD-ANGLES. See Angles.
before the Courtain ; but they are HEAT, in a hot Body, is the A-
defective, as being not well flank'd. gitation of the Parts of the Body,
HALF-TANgent. See Scale. and the Fire contained in it; by
HALO, or HALLO, is a cer, which Agitation a Motion is pro-
duced
HA
or
HEA
HE A
duced in our Bodies, exciting the the Declinations are contrary, is e
Idea of Heat in our Minds; and qual to a Circle into the Sine of
Heat, in reſpect of us, is only that the Altitude at Six, in the Summer.
Idea ; and in the hot Body is no- Parallel, and conſequently thoſe
thing but Motion.
Differences are as the Sines of La-
Heat, in all Bodies, is a Motion titude into, or multiplied by thè,
that may be infinitely diminiſh'd, Sines of the Declination.
and there may be ſuch a Motion, 10. The Tropical Sun under the
tho' it be not fenfible to us, becauſe Equinoctial has of all others the
often we cannot diſcover any thing leaſt Force under the Pole : It is
of Heat.
greater than any other Day's Healt
1. No Heat is ſenſible to us, un- whatſoever, being to that of the E-
leſs the Body that acts upon our quinoctial, as 5 to 4.
Organs of Senſe has a greater De 11. The Heat of the Sun for any
gree of Heat than that of our Or- ſmall Portion of Time, is always
gans.
as a Rectangle, contain'd under the
2. The Heat of a Body is not in Sine of the Angle of Incidence of
proportion to the Quantity of Fire. the Ray, producing Heat at that
3. Several heated Bodies will be Time.
come lucid, if their Heat be in 12. From the following Table,
creaſed.
and theſe Properties of the Sun's
4. Heat may be ſo increaſed, that Heat, we may have a general Idea,
in ſome Bodies the Attracting Force of that part of Heat that ariſes fim-
is overcome by the Repelling Force; ply from the Preſence of the Sun.
and in this Caſe the Particles fly The Table foewing the Quantity of
from each other, and acquire an E- The Table foewing the Quantity of
laftick Force, ſuch as the Particles
Heat to every oth Degree of La-
titude,
of Air have.
5. The Equinoctial Heat of the
Sun in Sun in Sun in
Sun, when he becomes Vertical, is
Lat.
ne
yo.
as twice the Square of the Radius.
6. Under the Equinoctial, the
18341 18341
Heat of the Sun is as the Sine of
19696
20290 15834
the Sun's Declination.
18797
21737 13166
30
7. In the Frigid Zones, when the
17321 22651
10124
40
Sun fets not, the Heat is as the Cir-
15321 230486944
cumference of a Circle into the 50 12855 22991 3798
Sine of the Altitude: Theſe Aggre 60
22773 1075
gates of Warmth are as the Sines of
70
6840 23543
the Sun's Declination; and at the 80 3473 24.675
fame Declination of the Sun, he 90
25055
are as the Sines of the Latitudes;
and generally they are as the Sines But the different Degrees of Heat
of the Latitudes into the Sines of and Cold in differing Places, de-
the Declination.
pend in a great meaſure upon the
8. The Equinoctial Day's Heat Accidents of the Neighbourhood of
is every where as the Co-line of the high Mountains, whoſe Height ex-
Latitude.
ceedingly chills the Air brought by
9. In all Places where the Sun the Winds over them ; and of the
ſets, the Difference between the Nature of the Soil, which variouſly
Sanımer and Winter-Heats, when retains the Heat, particularly the
Sands,
O
20000
IO
20
inio
10000
000
000
0000
000
T 3
H EL
HEL
متك
A
Sand, which in Africa, Arabia, and
в с
generally where ſuch fandy Defarts
are found, do make the Heat of the
D
Summer incredible to thoſe that
have not felt it.
HEGIRA, a Term in Chronolo-
G
gy, fignifying fthe Epocha, or AC-
M
count of Time uſed by the Arabians
and Turks, who begin their Com-
putation from the Day that Maho-
N
met was forced to make his Eſcape
from the City of Mecca, which hap-
pend on Friday July 16. A. D. 622.
under the Reign of the Emperor towards the Centre A of the Circle,
Heraclius.
is what is call’d the Helicoid, or spi-
HEIGHT of a Figure. See Alti- ral Parabola.
tude of a Figure.
2. If the Arch BC, as an Ab.
Height of the Pole. Şee Alți- ſciſſe,be called x; and the Part CÉ
tude of the Pole.
cf the Radius, as an Ordinate to it
HELIACAL RISING, is when a be called y; then the Nature of
Star, having 'been under the Sun- this Curve will be expreſs’d by ls
Beams, gets out ſo as to be ſeen a- Fyy; ſuppoſing. I equal to thệ
gain.
Latus Reétam of the Parabola.
HELIACAL-SETTING of a Star, AELICOSOPHY, is the Art of
is when it, by the near Approach delineating all sorts of Spiral Lines
of the Sun, firſt becomes inconſpi- in Plano. .
cuous. This is reckon'd in the
HELIOCENTRIC PLACE of a
Moon, but at ſeventeen Degrees di- Planet, is that Point of the Ecliptic
ſtance, or thereabouts ; but in other to which the Planet, ſeen from the
Stars, 'tis as ſoon as they get di- Sun, is referred, and is the fame as
ftant, or come near the Sun by the the Longitude of the Planet ſeen
space of a whole Sign.
from the Sun.
HELICE MAJOR and Minor ; HELIOSCOPES, are
a fort of
the ſame with Urſa Major and Mi- Teloſcopes fitted ſo, as to look on
the Body of the Sun without of.
HELICOID PARABOLA, or the fence to the Eyes.
PARABOLIC SPIRAL, is a Curve 1. Becauſe the Sun may be ſeen
which ariſes from the Suppoſition through colour'd Glaſſes without
of the Axis of the common Apollo- Hurt to the Eye i therefore, if the
rian Parabola being bent round in- Object and Eye-Glaſſes of a Tele-
to the Periphery of a Circle, and ſcope be made with colour'd Glaſs,
is a Line then paſſing through the as Red and Green, and equally co-
Extremnities of the Ordinates, which lour'd and pellucid, that Teleſcope
do now converge towards the Cen- will become a Helioſcope.
tre of the ſaid Circle.
2. But Mr. Huygens only uſed a
1. Suppoſe the Axis of the com- plain Glaſs blacked at the Flame of
mon Parabola to be bent into the a Lamp or Candle on one side, and
Periphery of the Circle BDM, then placed between the Eye-Glaſs and
he Curve BFGNA which paſſes the Eye, and that will anſwer the
through the Extremities of the Or. Deſign of an Helioſcope very well.
dinates CF, DG, which converge HELISPHERICAL Line, is
Rhumb
nor.
HÉP
Η Ε ΤΑ
Rhumb-Line in Navigation ; and is taken for a Place that hath ſever
ſo called, becauſe on the Globe it Baſtions for its Defence.
winds roạnd the Pole ſpirally, and HEPTANGULAR FIGURE, in
ftill comes nearer and nearer to it. Geometry, is that which confifeth
Şee more of this under Rhumb Line. of ſeven Angles.
Helix, in Geometry, is the HER ISSON, in Fortification, is a
ſame as Spiral. Which fee. Beam armed with a great Quanti-
HEMISPHERE, is the Half of ty of ſmall Iron Spikes or Nails,
the Globe or Sphere, when 'tis fup- having their Points outward, and is
poſed to be cut through the Centre ſupported by a Pivot, upon which
in the Plane of one of its greateſt it turns, and ſerves inſtead of a Bar-
Circles. Thus the Equator divides rier to block up any Paflage. They
the Terreſtrial Globe into the Nor- are frequently placed before the
thern and Southern Hemiſphere; Gates, and more especially the Wic-
and the Equinoctial, the Heavens af, ket-Doors of a Town or Fortreſs,
ter the fame Manner.
to ſecure thoſe Paſſages which muſt
1. The Centre of Gravity of a of neceſſity be often opened and
Hemiſphere, is five Eighths of the ſhut.
Radius diſtant from the Vertex. HERMETICAL SEALING, or
2. The Horizon alſo divides the Hermes's SeaL, or to seal or flop
Earth into two Hemiſpheres, the up any Glaſs hermetically, is to head
one light, and the other dark, ac the Neck of the Glaſs till it be juſt
çording as the Sun is above or be ready to melt, and then with a
low that Circle.
Pair of hot Pinchers to pinch or
3. Alſo Maps or Prints of the cloſe it together.
Heavens, Conſtellations, &c. paſted HERMITAN, is the Name of a
on Boards, are ſometimes called dry North and North-Eaſterly Wind,
Hemiſpheres, but uſually Plani- which uſually blows on the Coaſts
Spheres.
of Guinea in Africa ; but ſometimes
4. The Writers of Optics prove, it blows alſo from other Foints.
That a Glaſs - Hemiſphere unites HERSE, in Fortification, is a
the Parallel Rays at the Diſtance Lettice, or Portcullice, made in the
of a Diameter and one Third of form of a Harrow, and beſet with
a Diameter from the Pole of a many Iron Spikes. It is uſually
Glafs.
hung by a Cord faſten'd to a Mou.
HEMITONE, in Muſic, was what linet, which is cut in caſe of a Sur-
we now call an Half-Note.
prize ; or when the firſt Gate is bro-
HENDECAGON, in Geometry, is ken with a Petard, to the End that
& Figure that hath eleven Sides, and the Herſe may fall, and top up
as many Angles.
the Paſſage of the Gate, or other
HENDECAGON, in Fortification, Entrance of a Fortreſs. Theſe Her-
is taken for a Place defended by ſes are alſo often laid in the Roads
eleven Baſtions.
to incommode the March, as well of
HenIOCHUS, one of the Nor- the Horſe as of the Infantry.
thern Conſtellations. See Auriga. HBRSILLON, in Forcification,
HEPTAGON, in Geometry, is a is a Plank ſtuck with Iron Spikes,
Figure of ſeveral Sides and Angles; for the ſame Uſe as the Herſe.
and is called a Regular Hepta HETERODROMUS Vectis, or
gon, if thoſe Sides and Angles be e- LEAVER, in Mechanics, is that
qual.
where the Hypomachlion is placed
HEPTAGON, in Fortification, is between the Power and the Weight;
and
'
T 4
HET
H I T
and where the Weight is elevated Way; as thoſe who live between
by the Deſcent of the Power, and the Tropicks and Polar Circles,
contrariwiſe.
whoſe Shadows at Noon in North
HETEROGENEAL NUMBERS are Latitude, are always to the North-
mix'd Numbers, confiſting of whole ward, and in South Latitude to the
ones, (or Integers,) and of Frac- Southward.
tions.
HEXACHORD, a certain Inter-
HETEROGENEAL SURDS, are val of Concord of Muſic, common-
ſuch that haye different Radically called a Sixth; and is twofold,
Signs ; as baa, bb, 19, 118, viz. the Greater and Leffer.
&c.
The greater Hexachord is com-
If the Indexes of the Powers of poſed of two greater Tones, two
the Heterogeneous Surds be divided lefſer Tones, and one greater Semi-
Tone, which are five Intervals ; but
by their greateſt common Diviſor,
the leſſer Hexachord confifts only of
and the Quotients be ſet under the
Dividends, and thoſe Indexes be
two greater Tones, one lefſer Tone,
and two greater Semi-Tones.
multiplied croſswiſe by each other's
The Proportion of the former, in
Quotients; and before the Products
be fet, the common Radical Sign Numbers, is as 3 to 5; and that of
the latter, as 5 to 8.
✓, with its proper Index; and if
HEXAGON, în Geometry, is a Fi-
the Powers of the given Roots be
involved alternately according to
gure of fix Sides and Angles ; and if
the Index of each other's Quotient, called a Regular Hexagon.
thoſe Sides and Angles be equal, 'tis
and the common Radical Sign be
The Side of every Regular Hexa,
prefix'd before thoſe Products, then
gon inſcribed in a Circle, is equal
will thoſe two Surds be reduced to
others, having but one common Ra-
in Length to the Radius of that
Circle.
dical Sign: Ås to reduce
As I is to 4.672, fo is the Square
Vaa and 70.
of the side of any Regular Hexa-
gon to the Area thereof nearly.
2) vaa (2vbbe
HexAHEDRON, one of the Pla.
tonic Bodies, is the ſame as the
Cube, being a regular Solid of fix
equal Sides or Faces.
HEXASTYLE, an'antient Build-
bb
ing, which had fix Columns in the
HETEROGENEAL Light, by Face before, and fix alſo behind,
Sir Iſaac Newton, is ſaid to be that and is the ſame with the Pſeudodip .
which conſiſts of Rays of different teron.
Degrees of Refrangibility : Thus, Hip-Roof, in Architecture, is
the common Light of the Sun or ſuch a Roof as hath neither Gable-
Clouds is heterogeneal, being a Mix- Heads, Shred-Heads, nor Jerkin-
ture of all ſorts of Rays.
Heads. Theſe Hip-Roofs, by fome,
HeTEROGENEOUS PARTICLES, are called Italian Roofs.
are ſuch as are of different Kinds, Hippeus, or EQUINUS, a Co-
Natures, and Qualities, of which mer which ſome will needs have to
generally all Bodies confift.. reſemble a Horſe. But the Shape
Hete Roschi, in Geography, of this kind of Comet is not always
are ſuch Inhabitants of the Earth as alike, as being ſometimes Oval, and
have their Shadows falling but one ſometimes imitating a Rhomboides.
Irs
4
I
X
2
4
Waaaa,
1
1
)
H OM
HOR
Its Train, in like manner, is fome HOMOGENEAL NUMBERS, ard
times . ſpread from the Front or thoſe of the ſame Nature and Kind.
Fore-Part; and at other Times from HOMOGENEAL SUR Ds, are ſuch
the Hinder-Part : Therefore they as have one common Radical Sign;
are diſtinguiſhed into Equinus Bar-
as ,
ja, tb, or 163, or , O.
batus, Equinus Quadrangularis, and
Equinus Ellipticus.
HOMOGENEOUS PARTICLES,are
Hips, in Architecture, are thoſe ſuch as are all of the fame Kind,
Pieces of Timber which are at the Nature, and Properties ; as the Parts
Corners of a Roof. They are a
of pure Water, of meer Earth with-
good deal longer than the Rafters, finer Metals; ſuch as Gold, Silver,
out Salt in it, or the parts of the
becauſe of their oblique Pofition, for &c. 'Tis uſed in oppoſition to He-
they are level at every Angle.
Hircus, a fixed Star, the fame terogeneous; which fee.
with Capella.
HOMOGENEAL Lighr, is that
Hircus, a Name given by ſome
whoſe Rays are all of one Colour,
to a sort of a Comet encompafied and Degree of Refrangibility, with-
out any Mixture of others. See
by a kind of Mane, ſeeming to be
Colours.
rough and hairy, by reaſon of its
HOMOGENEUM COMPARATIO-
Rays appearing like Hair. It is al-
ſo ſometimes without any Train or
NIS, by Vieta, is the abſolute Num-
Buſh.
ber in a Quadratic, or Cubic, &c.
HoBits, are a ſort of ſmall Mor- Equation ; and this Number always
tars from ſix to eight Inches dia. poffefſeth one side of the Equation,
meter : Their Carriages are like and is the Product of the Roots
thoſe of Guns, only much ſhorter. multiplied into one another.
HOMOLOGOUS Sides or An-
They are very good for annoying
the Enemy at a
diſtance with
GLES of two Figures, are thoſe that
ſmall Bombs, which they will throw keep the fame Order from the Be-
two or three Miles; or in keeping ginning in each Figure; as in the
of a Paſs, being loaded with Car-
two fimilar Triangles ABC, DEF,
touches.
E
HOLLOW-Tower, in Fortifi-
B
cation, is a Rounding made of the
Remainder of two Brizures, to join
the Curtain to the Orillon, where
the ſmall Shot are play'd, that they
C
may not be ſo much expoſed to the
A
View of the Enemy.
D
F
HOMOCENTRIC. The ſame with the Sides AC, DF; AB, DE;
Concentric.
BC, EF; as alſo the Angles A, D;
HOMODROMUS Vec't Isor LEA- B, E; C, F, are Homologous.
VER, is one where the Weight is in Hoop-WHEEL. See Detenta
the middle between the Power and Wheet.
the Fulcrum, or the Power in the Hokezon, is that great Circle
middle between the Weight and the which divides the Heavens and the
Fulcrum.
Earth into two parts, or Hemi-
HOMOGENEAL, fignifies of the ſpheres, diſtinguiſhing the Upper
ſame Kind or Sort, or that which from the Lower. It is either Sena
differs not in Nature, &c. The ſame fible or Apparent, or the Racional
with. Homogeneou.
of True Horizon.
ܗ ܐ
1
are
H OR
HOR
1. The Senſible or True Horizon firſt Letters of their Names annex'd
is that Circle which limits our The Uſes of this Circle on the
Light, and may be conceived to be Globe are,
made by ſome great Plane, or the 1. To determine the Riſing and
Surface of the Sea.
Setting of the Sun, Moon, or Stars ;
2. It divides the Heavens and and to ſhew the Time thereof by the
Earth into two parts; the one light, Hour-Circle and the Index.
and the other dark; which
2. To limit the Increaſe and De-
ſometimes greater or leſſer, accor- creaſe of Day and Night : For when
ding to the Condition of the Place, the Sun riſes due Eaſt, and ſets
&c.
Weſt, the Days and Nights are e-
3. It determines the Riſing and qual; but when he riſes and ſets to
Setting of the Sun, Moon, or Stars the North of the Eaſt and Weſt, the
in any particular Latitude ; for Days are longer than the Nigḥts ;
when any one of theſe appears jult but the Nights are longer than the
at the Eaſtern Part of the Horizon, Days, when the Sun riſes and fets
we ſay it riſes ; and when it doth to the Southward of the Eaſt and
fo at the Weſtern Part, we ſay it Weſt Points of the Horizon.
fets. And from hence alſo the Al-
3. To ſhew the Amplitude and
titude of the Sun or Stars is ac- Point of the Compaſs the Sun riſes
counted, which is their Height above and ſets upon.
the Horizon.
HORIZONTAL LINE, or BASE
HORIZON Rational, Real, or of a Hill, is the Line AB drawn
True, is a Circle which encompaſſes
the Earth exactly in the Middle,
and whoſe Poles are the Zenith and
Nadir; that is, the two Points, one
exactly over our Heads, and the
other under our Feet.
HORIZON on the Globe,or Sphere,
B
is a broad Wooden Circle encom-
paffing it round, and repreſenting
the rational Horizon, having two upon a Plane parallel to the Hori-
Notches on the North and South zon whereon the Hill is ſuppoſed to
Parts of it for the Brazen Meridian ſtand.
to ſtand in. On this broad wooden
HORIZONTAL Dial, is one
Horizon ſeveral Circles are drawn, whoſe Plane is parallel to the Ho- ,
the innermoſt of which is the Num- rizon of any Place.
ber of Degrees of the twelve Signs In all Horizontal Dials the Style
of the Zodiac, viz. thirty Degrees makes an Angle equal to the Lati-
to each Sign.
tude of the Place, and the Angles :
Next to this you have the Names that the Hour-Lines make with the
of thoſe Signs : then the Däys of Meridian, may be found by this
the Month, according to the Ju- Proportion : As the Radius. is to the
lian Account, or Old Style, with the Sine of the Latitude, ſo is the Tan-
Kalendar according to the Foreign gent of any Hour's Diſtance from
Account, called New Style ; and 12 to the Tangent of the Angle
without theſe is a Circle divided in that the Hour-Line of that Hour
to thirty-two equal Parts, which makes with the Hour-Line of 12.
make thirty-two Rhumbs, or Points The Reaſon of this proportion
of the Mariner's Compaſs, with the for finding the ſeveral Hour-Anglés,
will
1
А.
1
HOU
upon a Plane.
HOR
will appear from what is ſaid under rallel one to another, are terminated
the Word Direet Ereet South or at the Gorge of the Work, and ſo
North Dials, for in the Figure there, preſent themſelves to the Enemy.
in the right-angled Spherical Tri HOROLOGIOGRAPHY, is the Art
angle AVR, we have given the of making Dials, Clocks, or other
Angle ARV for the Hour, and the Inſtruments to ſhew the Time of the
Side A R for the Latitude ; to find Day.
the Side A V, being the Angle that HOROMETRY, is the Art of
the Hour-Line of the given Hour meaſuring or dividing the Hours,
makes with the Meridian upon the and keeping account of Time.
Plane of the Dial.
HOROPTER, in Optics, is a right
Horizontal Dials may be drawn Line drawn through the Point of
Gcometrically, aſter the very fame Concurrence, parallel to that which
manner as direct or erect South or joins the Centre of the Eye.
North Dials. See the Figures for HOROSCOPE, in Aftrology, fig-
this purpoſe under theſe Words. _nifies the firſt Houſe, or Aſcendant,
Only in this Caſe the Angle ADC and is that Part of the Zodiac which
muſt be made equal to the Lati- is riſing at the time of the Calcula-
tude, and not the Complement. tion of a Scheme.
HORIZONTAL LINE, is any HORSE-SHOE, in Fortification,
Line drawn parallel to the Horizon is a work of a round, and ſome-
times oval Figure, raiſed in the
HORIZONTAL LINE of a Dial, Ditch of a marſhy Place, or in low
is a right Line drawn through the Ground, and bordered with a Para-
Foot of the Style parallel to the pet. It is made to ſecure a Gate,
Horizon,
or to ſerve as a Lodgment for Sol-
HORIZONTAL PARALLAX. See diers to prevent Surprizes, or to
Parallax.
relieve an over-tedious Defence.
HORIZONTAL PROJECTION. HOUR, is the twenty-fourth Part
Se Projection.
of a natural Day, containing fixty
HORIZONTAL Range, or LE. Minutes, and each Minute fixty Se-
VEL RANGE of a piece of Ord- conds, & c. Theſe are aſtronomical
nance, is the Line that a Ball de- Hours, which always begin at the
ſcribes parallel to the Horizon or Meridian, and are reckoned from
Horizontal Line when the Piece is one Noon to the next Noon.
level
1. But fome Hours are begun to
1. The Horizontal Ranges are be accounted from the Horizon;
the ſhorteſt
. And ſome Pieces of which, when the Account begins at
Cannon will make them fix hun- the Sun's Riſing, are called Babylo-
dred Paces, and fome but a hun- niſh Hours, which begin with the
dred and fifty ; and the Ball, with Sun's Riſing, and reckon on twenty-
the Range of fix hundred Paces, four Hours to his Riſing again the
will
go from nine to thirteen Foot nexč Day.
into the Earth.
2. Others are reckon'd after the
HORN-WORK, in Fortification, fame manner, only they begin at
is an Outwork, which advanceth the Sun's Setting inſtead of his Ri-
toward the Field, carrying in the fing; and theſe are called Italian
Forepart, or its Head, two Demi- Hours, becauſe the Italians account
Baſtions, in Form of Horns: Theſe their Time after this faſhion.
Horns, Epaulments, or Shoulder 3. There is yet another kind of
ings being joined by a Curtain, ſhut Hours,' which are caMed prvih
up on the side by two Wings, pa. Hours; becauſe of old the Jet
accounted
PWS
H Y A.
H Y D
on both
accounted their time this way. Humour of the 'Eye contain'd bea
They are one twelfth Part of the twixt the Tunica-Retina and the
Day or Night, reckoned from the Uvea.
Sun-riſing to the Sun-ſetting, (if HY BERNAL OCCIDENT. Sec
the Days or Nights belong or
Occident.
ſhort;) and theſe were called, as we HYBERNAL ORIENT. See O-
find in the Holy Scripture, the rient.
Firft, Second, and Third, &c. Hours HYDATOIDES, is the watery
of the Day or Night.
Humour of the Eye contained be-
HOUR-CIRCLES, the ſame with twixt the Tunica-Retina and the
Meridians, are great Circles, meeting Uvea.
in the Poles of the World, and croſ HYDRA, a Southern Conftella-
ſing the Equinoctial at right Angles. tion, confifting of twenty-fix Stars,
They are drawn through every and imagined to repreſent a Water-
fifteenth Degree of the Equinoc Serpent.
tial and Equator, and
HYDRAULICS, is the Science of
Globes are ſupplied by the Meri- the Motion of Fluids, eſpecially
dian, Hour-Circle, and Index. Water, under which is contain'd the
The Planes of the Hour-Circles Structure of all Fountains, Engines
are perpendicular to the Plane of the to carry or raiſe Water, or which
Equinoctial, which they divide into are mov'd by Water, and ſome for
twepty-four equal Parts
other Uſes.
HOUR-Lines on a Plane Dial, Some of the Writings upon Hy-
are the Interſections of the Plane draulics and Hydroſtatics, are År-
of the Dial, with the Planes of the chimedes, in his Libris de Infidentibus
Hour-Circles of the Sphere. humido.--Marinus Ghetaldus, in his
HOUR-SCALE, is a divided Line Archimedes promotus. Thoſe of
on the Edge of Collins's Quadrant, Mr. Oughtred. Mr. Mariotte, in his
being only two Lines of Tangents Treatiſe of the Motion of Water, and
of forty-five Degrees each, fet to other Fluids.-- Mr. Boyle, in his Hy-
gether in the middle ; and the Uſe droftatical Paradoxes. Franciſcus
of it, together with the Lines of Tertius de Lanis, in his Magiſterium
Latitudes, is to draw the Hour- Naturæ & Artis. Mr. Lamy, in
Lines of Dials that have Centres, his Traité de l'Equilibre des Liqueurs.
by means of an equilateral Triangle, -Thoſe of Mr. Rohault.-Dr. Wal-
drawn on the Dial-Planes.
lis, in his Mechanics- Thoſe of
HURDLES, or CLAYS, in Forti- Mr. Dechales.- Sir Iſaac Newton,
fication, are made of thick and in lib. 2. of his Princip. Philof. Nat.
ſmall Twigs of Willow, or Ofiers, Johannes Ceva, in his Geometria
being five or fix Foot high, and Motus.--Thoſe of Johannes Baptiſta
from three to four Foot broad. Balianus. Mr. Gulielmeni, in his
They are interwoven very cloſe Menſura Aquarum Fluentium.
together, and uſually laden with Thoſe of Mr. Herman. Thoſe of
Earth, that they may ſerve to ren- Mr. Wolfius. Mr. s'Grave fande.
der Batteries firm, or to conſolidate Mr. Muſchenbroek.--Mr. Leopold,
the Paſſage over muddy Ditches, or - Hero of Alexandria, his Liber
to cover Traverſes and Lodgments Spiritalium, tranfated by Comman-
for the Defence of the Workmen dine into Latin Salomon de Caus,
againſt the artificial Fires or Stones in his French Book of Machines.
that may
be caſt
upon
them.
Caſper Schottus, his Mechanica Hy-
HYALOIDES, is the virtreous draulico-Pneumatica.- George An-
drea
H Y G
H Y P
drea Bockler in his Arthițe&tura Curi- bottom hang a Weight of about a
ofaGermanica.
Auguſtine Rammil- Pound ; let thereon, or unto the bot-
leis.Lucas Antonius Portius.--Stur tom of the Weight, be faftened an
my
in his German Treatiſe of the Index of about a Foot long, and un-
Conſtruction of Mills. --Switzer, &c. der it, on a Table, or on a piece of
HYDRAULICO - PNEUMATICAL Board, place a Circle, divided into
ENGINES, are thoſe that raiſe Water what Number of Degrees you pleaſe,
by means of the Spring, or natural and fit it ſo that the Centre of the
Force of the Air.
Index may hang juſt over the Cen-
HYDROGRAPHICA È CHARTS, are tre of the Circle. After it has hung
certain Sea-Maps, delineated for the thus two or three Days, to ſtretch
Uſe of Pilots and other Mariners ; the Cord, you may begin to mea-
wherein are marked all the Rhumbs fure by it the Degrees of Moiſture
or Points of the Compaſs, and Meri or Drought in the Air ; for the Cord
dians parallel to one another, with will twiſt one way, and contract it-
Shelves, Shallows, Rocks, Capes, & c. ſelf for wet, and untwiſt itſelf again
HYDROGRAPHY, is an Art which on the contrary way for dry.
teacheth how to deſcribe and mea-
HYPERBOLA, is a Curve made
fure the Sea; giving an Account of by cutting a Cone by a Plane that
its Tides, Counter-Tides, Soundings, falls within the Circular Baſe of the
Bays, Creeks, &c. as alſo Rocks, Cone, being neither parallel to the
Shelves, Sands, Shallows Promonto- Side of the Cone, nor cuts it thro’the
ries, Harbours, Diftance from one Vertex,and which Plane, if continued,
place to another, and other Things will cut the oppoſite Cone. As the
remarkable on the Coaſts.
Curve CAG is an Hyperbola, if the
HYDROSTATICS, is the Science of
the Gravitation of Fluids, and of
their Adion, when demerſed in So-
lids.
This is a part of Philofophy which
F
ought to be looked upon as the moſt
ingenious of any, the Theorems and
A
Problems of this Art being handſome
Productions of Reaſon, and afford-
ing Diſcoveries not only pleafing, but B
G G
Н.
alſo ſurpriſingly wonderful and uſeful.
HYDROSTATICAL BALLANCE.See Plane AG, continu'd out, cuts the
Ballance.
oppoſite Cone in D, and is not pa-
HYEMAL Solstrce. See Solſtice. rallel to the Side FH, nor does pafs
HYGROMETER, is a Philoſophical through the Vertex E.
Inſtrument, which meaſures the Dry 1. If one End of a long Rule
neſs and Moiſture of the Air.
fM O be faftened in the Point f,
Hygroscope, is an Inſtrument taken on a Pláne, in fuch a manner,
ſhewing the Increaſe and Decreaſe of that it may turn freely about that
the Dryneſs of the Air.
fixed Pointf, as a Centre ; and one
The Hygroſcope of Mr. Moly- End of the Thread FMO (being in
neux, being a very ſimple and good Length leſs than the ſaid Rule) be
one, is made thus :
fixed to O, the other End of the
Faften a Piece of Whipcord, of a- Rule, and the other End of the
bout four Foot long, to a Hook or Thread be fixed in the Point F, ta-
Staple, in ſome convenient Place of ken on the Plane ; then if the Rule
the Ceiling of a Room, and at the FMO be turned about the fixed
Point
:
C
?
1
HY P
H Y P
Point f ; and at the ſame time you
keep the Thread OMF always in
an equal Tenſion, and its Part MO
K
C
H
E
B
ID
Z
X
ů
1
E
B
A
OF
V
G
А
L
D
1
the other Side EC of the given Angle
cloſe to the Side of the Rule, by CEB, will deſcribe an Hyperbola.
means of the Pin M; the Curve
Otherwiſe by means of Points. Let
Line AX deſcrib'd by the Motion AB, DE, (Fig. 1.) be the Axes in-
of the Pin M, is one Part of an Hy-
perbola.
Fig. 1.
And if the Rule be turned about,
and moves on the other Side of the
M
fixed Point F, the other Part AZ of
D
the ſame Hyperbola may be deſcrib'd
after the fame Manner.
A
PB
But if the End of the Rule be
faftened in F, and that of the Thread
in f, (the Rule and Thread keeping
D M
the ſame Lengths,) you may deſcribe
another Curve Line za x after the
fame Manner, which will be oppo-
Fig. 2.
ſite to X AZ, and is likewiſe an Hy-
perbola.
A
The following Deſcription of an
Hyperbola by a continued Motion,
F
being that of Mr. De Witt’s, in his
G
'Elementa Linearum Curvarum, is
pretty enough. Let KL, GH be
D
the Afymptotes, take the Point A be-
tween them, and having faſtend a
Ruler AB to the Point B in the
С E
lo
Side E B of a given Angle CEB, terſecting one another in C; take a
move the Side EB of that given ny Point P in AB, and from D or
Angle along the Line KL, always draw the right Line DP, or EP;
co-inciding with it ; then if the Rule then thro' p draw the right Line
AB be at the ſame time carried about M
m parallel to DE, and make PM,
the fix'd Point A upon the Plane ; and Pm each equal to DP or EP;
the. Interſection C of that Rule with then will the Points M, m be two
Points
-
H Y P
H Y P
Points of the oppoſite Hyperbola's : fcribe the Portion Ee of an Hyper-
and thus may an infinite Number of bola,
Points be found. Or (Fig. 2.) let 3. Any Parallelogram defcrib'd aa
AC, CB be the Afymptotes, and bout an Elliphs, or between the Con-
D'a given Point. Draw any right Line jugate Hyperbola's, ſo that the four
EF thro’ the Point D, terminating Points of Contact may be join'd by two
in the Afymptotes, and make FG, Diameters GH, IF only, which
equal to DE, then will the Point G therefore will be Conjugates, is equal to
be one point thro' which an Hyper- the Parallelogram deſcribd about the
bola is to paſs ; and thus may any two Axes Aa, Bb; and conſequently
Number of Points be found ; and of all ſuch Parallelograms are equal to
all Ways to deſcribe an Hyperbola by one another.
means of Points, this is the eaſieſt.
From F, the Extremity of one
2. If there be given the two Foci Diameter, draw the Line ÉD paral-.
C, F, of an Hyperbola, and the Ver- lel to the other Diameter GH, (con-
tex É, and it is requir’d to deſcribe tinued out in the oppoſite Hyperbo-
an Hyperbola to theſe Foci and Ver- la's,) meeting the Axis (produced in
the Ellipfis) in the Point T and from
Let KF=CE, ſo that EK be G the Extremity of the Diameter
the tranſverſe Axis, and take three GH draw the Line GD parallel to
Rules CD, DG, and GF, ſo that the Diameter IF, meeting DF in
CD=GFEK, and DG=CF.
Let the Rules CD, GF, be of an
D
indefinite Length beyond C, D, and
G
D
A
tex,
E
F
€
T
PA
a
H
A
F
I
С
K к
B
G
T:
A А
I
1
more
H
1
1
have Slits in them the Breadth of
B
the Pin that is to deſcribe the Hyper-
bola. Moreover, let theſe Rules have
Holes made in them at C, F, in or D: And from the Point Flet fall
der to faften them to the Foci C and the Perpendicular FP to the Axis
F, by means of Points, and at the Aa; then GD, DF, will touch the
Places. DG, they are to be joined Ellipfis, and the Hyperbola's bG, af
by the Rule DG. This being done, in Ĝ, F, the Extremities of the Con-
if a Pin be put in the Slits, viz. the jugate Diameters ; and ſo the Paral-
-common Interſection of the Rules lelogram CGDF will be one Fourth
CD, GF, and mov'd along, cauſing of that deſcribed about the Ellipſis,
the two Rules GF, CD, to turn a or between the Conjugate Hyperbo-
bout the Foci C, F, that Pin will de- la's, having the Condition mentioned
1
in
.
+
CCXX
+
7
tt
1
2
CCXX
tt
•c cx x
tt
ccx x
tt
ccx x
HYP
H Y P
in the Theorem: Therefore, if
CE be drawn perpendicular'to DF, Ellsp. or xx+ tt in Hyp.
(produced in the Conjugate Hyperbo-
la's,) we are to prove that CG (= and CE’ xDF (=DGʻ) is =
DF) CE is =C6xCa=of the 71862 +44 c4 x2 770c? **
Rectangle under the two Axes.
which
4. Call Ca, t, Cb, c, and Cp, %,
+ +*?*c?+++++
then ca” (tt) : APxPa (tt-**in is = Ca x C = 7 ttc; as it
the Ellipfis
, or xx~tt in the Hyper- evidently appears by multiplying the
bola) :: Cb (cc): FP =CC Denominator by tttcc: And there-
fore CaxC6=CGXCE, and ſo
In the Ellipſis, or
cc in Hyp. 4CEXEG=4CbxCa.W.W.D.
5. If from any Point Min an Equi-
And CF=xxtosa
in El- lateral Hyperbola there be drawn the
Right Line MG parallel to the A-
fymptote CN, and the Right Line OM
lip.or xx_c1 + in Hyp. becauſe parallel to the other Afymptote CS; 1
FPCis a right-angled Triangle. Again lay the, Rectangle under OMⓇOC
CP(x) : Ca (1): : Ca (4): CT = will always be equal to the Square of
the Line RQ, which is drawn from
And PT'=x
2ttt the Point A, wherein the Axis CP
cuts the Curve, parallel to the Afymp-
and FT* =*x+66-2717 + to CS, and terminating in tbe other
Afymptote CN.
in Ellip. or xx (
62tt
For draw the Ordinate MP, and
raiſe the Perpendicular AQ, and let
+
in Hyp, becauſe FPT
fall A R perpendicular to CN; now
CR,RR,RA, are equal to one
is a right-angled Triangle. Now, the another, becauſe the Axis CP does
Triangles FPT, CET, are ſimilar,
N
becauſe the Angles at E and P are
AM
right ones, and the Angle ETC in
the Ellip, common, (but in the Hy-
Q
perbola the Angle ETC=PTF,)
74
whence FT" (xxFre— 2+1+
R
t
64
***
4
XX
CCXX
#4
CCXX
+
1
Р
CCXX
or
G
7***): Ep* («c
-cc): :: CT" (09): CE
2
CCXX
t4 c²
S
Farther, the
24 - 2² x² + c²x²
biſe& the Angle form'd by the A-
Square of the Semi-Conjugate, viz. fymptotes, and AQ is perpendicular to
CG (=Ca+C6 - TF in the Axis, and AR parallel to CS:
Then call CN, s, CQ,m, the Se-
Ellip. or = CF +Cb -Cain mi-Axis CA, a ; and the Line AQ,
(which is equal to the Semi-conju-
Hyp. ) is tt-**+
gate Axis) b." The Triangles
CAQ,
CPN,
CCXX
in
}
i
22
m
2
aa
.):
:PM
mm
MN
M
mm
v boss
b):OM or
mm.
H Y P
H Y P
CPN, are fimilar : Therefore CQ, Point of contact to the Diameter Aa :
(m) : 'CN (s):: C A (a) : CP= I ſay, CP, CA, C T are in conti-
nual Proportion.
s. And CQ (m) : CN ():: QA Suppoſe the Arch MN to be in-
finitely ſmall, and draw NQ_parallel
(6): NP=bs. But by the Nature to PM, and M R parallel to A a;
now the ſmall Triangle MNR will
of the Hyperbola CA' (a a): All be ſimilar to the Triangle TPM, be-
(66) :: CP+CAⓇ AP
cauſe the very finall Arch M N may
bbss 5
be looked upon as being the Pro-
Canon
longation of the Tangent TM.
Now call A C, a, the Semi-Conju-
bb.
gate, CB, b, the Subtangent, TP, s,
Whence,
the Part, A P, x, the Semi-Ordináte,
bs v boss
bb.
MP, y, and the ſmall Right Line
PQMR, e: Then, becauſe the
| Again, the Triangles RAQ, OMN, Triangles T PM,MRN, are fimi-
are ſimilar, and both Iſoſceles, there lar TP (s): PM () :: MR (e) :
fore, A Q(): AR
R (2) :: M N RN=Then, if QN (7) +
2yye
-66 : OM or ON + yen) be put for yy in
z abbx-bbxx
Vs5-mm
the Equation yy=
And fo-CO
4
expreſſing the Nature of the Ellipfis ;
or GM-SH
Vrs— mm and A Q(x te) for x, we ſhall
And
2 y yie
have jyt
=
+
4 4 4 zabbx -+-2abbe-bbxx2bbex-bbee,
which is equal to RA or
and if the former Equation be ſub-
W. W. D.
tracted from this, then will2
yye
+
4
The Demonſtration of this property yyee 2 abbezbbex-bbee
being eaſy and new, (at leaſt to me,)
was the Cauſe of my laying it down and dividing by e; and afterwards
here.
ſtriking out all the Terms adfected
6. If A a be any Diameter of an
Ellipſis, or Hyperbola, C the Centre, leſs than the others, and then will
with c, becauſe they are infinitely
and if the Right Line TM touches
z abb- 2 bbx
уу
this Ellipfis in the Point M, and the lyy
Ordinate MP be drawn from the
a bbbbx
and again ſubſtitus
R
A
2abbx-bbx*
ting for yy, its Value
and bb will be gotten out; and
U
then
2
aa
2
4
+
Уyee
ОСхом
$s
ss
In m
$
SS
1
M m
2
a a
2
ce
SS
a a
=
=>
$
Aa
MN
1
а а
E
IQ
aa
La
* Η Υ Ρ
H Y P
then s will be
2'a -*.
** gci
Now,
a
TCxPC, (2.4.*
- x.
ta
a
*
that is, amit
xa-
-*)
I
I
x, that is,
is =A * (a a) which was the thing
to be demonſtrated.
8
Esc. The ambiguous Sign
7
being to in the Hyperbola, and in
the Circle and Ellipſis. Whence if
the Square circumſcribing the Circle
be zi, the following Series will be
had, viz.
+
+ + It
3
15 24
+ tat
63 80
99
Eʻc.
8. In theſe Series ***
3 35999
&c. expreſſes the Area of the Circle
+
8
48
&c. the Area of the Equilateral
1
I
1
I
.
+
+
35
N
48
I 20
1
M
IB
a
T
o
I
Q
PA
+
I20
After the fame manner we prové
this in the Hyperbola, only obſerving
here that the Equation, expreſſing
the Nature of the ſame, is yy =
2 abbx+bb xx
А A
D
a a
܀
1
:
7. If QF A be a Sector, contain'd
under two Right Lines, meeting in
B
the Centre Q, and the Conic Curve
T
A
F
Hyperbola BCEF, when B C is the
double of EF, and the inſcribed
Square is=_. The Numbers 3,
4
8, 15, 24, being ſquare ones lef-
fen'd by Unity.
9.
All the Properties of Diame-
F A, the Point A being the Extre- ters
, Tangents, and Foci
, &c. in the
mity of the Axis : And if a Tangent Hyperbola, are the ſame as thoſe in
in É meets the Tangent in A in the the Ellipfis, only uſing Differences
Point T, and AT be call’d t ; and for Sums. As for Example, as the
the Rectangle under half the Latus Square of the Semi-Conjugate, or ſe-
Restum, and half Latus cond Axis B C, is to the Square of
verſum, be fuppoſed = 1, then thall the Ordinate KI; fo is the Square
the Sector of the Hyperbola, Circle of CP, half the principal Axis, to
or Ellipſis, divided by the Semi-Latüs the Rectangle under Ñ I and PI.
Again, the Difference of the two
Tranſverſum be=
the Foci to
++ + Lines drawn from
3
the Curve, is always equal to the
prin
+3
ES S
!
2
H Y P
H
N
rea or Space contained between the
Curve of the Hyperbola and the
Blue
whole Ordinate. If CL=b, and Le
C
x, and CF=a, and QP=y,
P
then a? =by+xy; and if a=b
1, the Space between the Afymp,
totes will be expreſſed by x - Lx
К.
I
+4x3+4x++LxS,&c.
3
4
5
Any Hyperbolic Space GEHG,
principal Axis. Alſo the Difference of the fame Height g E bg; (whoſe
is to any other Hyperbolick Figure
of the Squares of any two conjugate Latus Roelum and Tranſverſum, as in
Diameters are always equal to the the Circle, are equal, and alſo both
Difference of the Squares of the equal to DE, the Latus Tranfver-
conjugate Axes.
1o. Any two Lines drawn in the ſum of the former Space) :: as the
Hyperbola, parallel to each other. Conjugate Axis AB: is to the La-
and cut by a third, have the fame
tus Tranfverfum DE.
Property as is mention'd of two pa-
D
rallel Lines drawn in the Ellipſis.
See Ellipfis.
А B
HYPERBOLICAL CYLINDROID,
is a ſolid Figure, whoſe Generation
is given by Sir Chriſtopher Wren, in
Philof. Tranſact. Nº 48. There are
two oppoſite Hyperbole, joined by
the Tranſverſe Axis, and thro' the
Centre there is a Right Line drawn
G
II
at Right Angles to that Axis, then
the Hyperbola are ſuppoſed to re-
The Area of any aſymptotical hy.
volve; by which Revolution a Body perbolic Space D E VB, may be
will be generated, which he calls an thus found by means of the Loga-
Hyperbolic Cylindroid; and whoſe rithms. Take the Differences of the
Baſes, and all Sections parallel to
them, will be Circles. And in Nº
53. of the Tranſactions, he applies it
to the Grinding of hyperbolical Glaſ-
ſes ; and he ſays, they muſt be ei-
S
D
ther formed this way, or not at all.
B
HYPERBOLIC Space, is the A-
c
V
Logarithms of the Numbers expreſſing
the Ratio of D E to B V, and find
the Logarithm of that Difference ;
to which add the conſtant Loga-
rithm 0.3622156887; and the Sum
will be the Logarithm of a Num-
ber expreſſing the Space E DBV,
E
th
1
11
:)
3
P
U 2
in
H Y P
H Y P
10000000000.
2
2
in ſuch Parts as the Oblong CD is Conoid thro' the Axis : If A H be
=
{AG, and you make HF: AF
HYPERBOLICUM ACUTUM, is a
:: AF : FK
Solid made hy the Revolution of the
and with the Ver-
infinite Area of the Space contained tex K, Centre F, and the fame Axis
,
between the Curve and the Afymp- another Hyperbola KLM be de-
tote, in the Apollonian Hyperbola, fcribed, ſuch that the Latus Rec-
tun's and tranſverſe Axes of the two
turning round that Afymptote. This
produces a Solid or Body infinitely Hyperbolas be reciprocally propor-
long ; and yet, as Torricellius plainly tional
, and let B C meet this Curve
demonſtrates, who gave it this Name) in M, and draw A L parallel to
it is equal to a finite Solid or Body.
MB, then it will be as the Space
HYPERBOLIC Conoid, is a Solid ALMB is to BC, fo is the Curve
generated by the entire Rotation of Surface of the Hyperbolic Conoid to
the hyperbolic Space FAG about its Circular Baſe.
A Cylinder equal to the Solid ge-
ст
rated by the Rotation of the hyper-
bolical Space AFGB about the Se-
mi-Conjugate Axis AC, may be found
thus : Let P be a third Proportional
A
.2
A
i
E
F
G
the Part A F of the tranſverſe Axis.
If AC be the ſemi-tranſverſe
Axis, it will be, as 2 AC + AF B В
is to 3 AC +AF, fo is the Cone
whoſe Baſe is the Circle deſcribed by
G D
FG, and Altitude A F, to the Co-
noid deſcribed as above.
If F be the Centre, A E the to AC and AF; then a Cylinder
Tranfverſe Axis, and A G the Latus the Radius of whoſe Baſe is FG,
and Altitude a fourth Proportional to
AC+P, AC+;P, and AF, will
F
be equal to the Solid deſcribed, as
above ; and the curve Superficies of
K
the ſaid Solid may be had by the
Quadrature of the hyperbolic Space.
I
HYPERBOLOIDEs, or HYPERBO-
A
LIFORM FIGURES, are Hyperbola's
of the higher Kind, whoſe Nature
is expreſſed by this Equation ay +
bom
MC
B
D
and if m be greater than
atn
Re&tum of the Hyperbola CAD n, the Hyperboliform Space is ſquara-
being the section of an Hyperbolic ble ; but otherwiſe not.
Hy-
1
H
I.
JA
H Y P
I D E
'HYPERTHYRON, in Architecture, or Dorick Capital, which lies be-
is a large Table, uſually placed over tween the Ethinus and the Aſtragal;
Gates or Doors of the Dorick Or- and is otherwiſe called the Collar,
der, above the Chambranle, in form Gorge, or Frize of the Chapiter.
of a Frize.
HypeThre, in ancient Architec-
ture, was two Ranks of Pillars all
about, and ten at each Face of any
Temple, &c. with a Periſtyle within
of fix Columns.
ACOB'S-STAFF, a Mathe-
HYPOMOCHLION, FULCRUM, or matical Inſtrument for taking
Prop, in Mechanics, fignifies the Heights and Diſtances. The ſame
Roller, which is uſually ſet under the with Croſs-Staff.
Leaver, or under Stones or Pieces ICHNOGRAPHY, in Perſpec-
of Timber, to the end that they tive, is the View of any thing cut
may be more eaſily lifted up, or re off by a Plane parallel to the Ho-
moved.
rizon, juſt at the Bafe or Bottom
HYPOTHENUSE in a right-angled of it. And in Architecture, is ta-
Triangle, is that Side which lub- ken for the Geometrical Plane or
tends the right Angle.
Platform of an Edifice, or the Ground-
In all right-angled Triangles, the Plot of a Houſe or Building deli-
Figure deſcribed upon the Hypo- neated" upon Paper, deſcribing the
thenuſe as a Side, is equal to the form of the ſeveral Apartments,
Sum of the two Figures deſcribed Rooms, Windows, Chimneys, &c.
upon the other two sides of that and this is properly the Work of
Triangle, being all three ſimilar. the Maſter-Architect or Surveyor,
HYPOTHESIS, is the ſame with being indeed the moſt abſtruſe and
Suppoſition; or it is a Suppoſition difficult of any.
of that which is not, for that which ICHNOGRAPHY, in Fortification,
may be ; and it matters not whe- is ,in like manner, the Plane or Repre-
ther what is ſuppoſed to be true, be ſentation of the Length and Breadth
fo' or not; but it muſt be poſſible, of a Fortreſs; the diſtinct Parts of
and ſhould always be probable. which are marked either upon the
Dr. Barrow ſays, Hypotheſes, or Ground itſelf, or upon Paper.
Poftulatums, are Propoſitions affu ICOSAHEDRON, is a regular Bo-
ming or affirming fome evidently dy, conſiſting of twenty Triangular
poffible Mode, Action, or Motion of Pyramids, whoſe Vertexes meet in
a Thing, and that there is the ſame the Centre of a Sphere ſuppoſed to
Affinity between Hypothefes and Pro- circumſcribe it, and ſo have their
blemas, as between Axioms and The Height and Baſes equal : Therefore,
orems. A Problem ſhewing the Man- the Solidity of one of thoſe Pyra-
ner, and demonſtrating the Poffibi- mids being multiplied by 20, the
lity of ſome Structure, and an Hy- Number of Baſes gives the folid
potheſis affuming fome Conſtruction Content of the Icojahedron.
which is manifeſtly poſſible.
Ides of a Month, among the Ro-
HYPOTRACHELION, in Architec- mans, were the Days after the Nones
ture, is the Top or Neck of a Pillar, were out. They commonly fell out
or the moſt ſender Part of it, which on the 13th of every Month, ex-
toucheth the Capital. It is taken cept in March, May, July, and Octo-
by ſome for that Part of the Tuſcan ber,( which they call the Full Months,
U 3
as
1
I M A
1 M A
.
a
2
2
às all the others were called Holm the Object, and as far behind the
low,) for then they were on the Speculum, as the Object is diſtant
15th, becauſe in thoſe Months the before it.
Nones were on the 7th.
In Convex Speculums, the Image
JET D'EAU, is the French Word is farther diſtant from the Centre
for a Pipe of a Fountain, which of the Convexity, than from the
caſts up the Water into the Air. Point of Reflexion, and the Image
M. Maristte, in his Treatiſe Du appears leſs than he Object.
Mouvement des Eaux, &c. faith, That İMAGINARY Root of an Equa-
a Jet d'Eau will never riſe fo high tion, are thoſe Roots or Values of
as its Reſervatory, but always falls the unknown Quantity in an E-
ſhort of it by a Space, which is in quation, which are wholly or part-
a ſubduplicate Ratio of that Height; ly expreſs’d by the Square Root of
and this he proves by ſeveral Expe- a negative Quantity, and of which
siments.
in every Equation their Number is
He faith alſo, That if a greater
As tvaa, and
branches out in ſmaller
always even.
ones,
diftri.
buted to different Jets, the Square -V-aa are the Roots of the E.
of the Diameter of the main Pipe quation xx'taa=0.
So alſo
muſt be proportioned' to the Sum of
all the Expences of its Branches. -V-aa, and Vaa
And particularly he ſaith, That if
the Refervatory be 52 Foot high, are the two Imaginary Roots of the
and the Adjutage half an Inch in Equation xx tax taa=
Diameter, the Pipe ought to be The Imaginary Roots of Equa-
three Inches in Diameter.
tions may be found by the follow-
IGNIS-FATUUS, is a certain Me- ing Rule : Conſtitute a Series of
teor that appears chiefly in the Fractions, whoſe Denominators are
Summer Nights, for the moſt part the Numbers in this Progreſſion 1,
frequenting Church - Yards, Mea- 2, 3, 4, 5, &e. going on ſo far aş
dows, and Bogs, as conſiſting of a the Number expreſſing the Dimen-
somewhat viſcous Subſtance, or a ſion of the Equation, and Numera-
fat Exhalation ; which being kin-
tors the fame Series of Numbers in
dled in the Air, reflects a kind of a contrary Order ; and divide each
thin Flame in the Dark, yet with- of theſe Fractions by that next be-
out any ſenſible Heat, often flying fore it, and place the Fractions a-
about Rivers, Hedges, &c. becaule riſing over the intermediate Terms
it meets with a Flux of Air in thoſe of the Equation ; then under each
Places. This Meteor is well known of the intermediate Terms, if its
among the common People under Square multiplied by the Fraction
the Name of Will-of-the-Whiff, or over it, be greater than the Product
Jack-with-a-Lanthorn.
of the Terms on each ſide it, place
ILLUMINATIVE MONTH, is the Sign +; but if not, the Sign ;
that Space of the Time that the and under the firſt and laſt Term
Moon is viſiblc, betwixt one Con- place the Sign + Then will that
junction and another.
Equation haye ſo many imaginary
I'M'AGE, in Optics, is the Ąp. Roots, as there are Mutations of
pearance of an Object, by Reflexion the under-written Signs from + to
or Refraction.
and “tot. And when there are
In all Plane Speculums the Image two or more Terms wanting at the
appears of the fame Magnitude as ſame time, the Sign
is to be
placed
}
1
I M A
IM A
placed under the firſt of the deficient
3 to
Terms, the Sign + under the le- duct of the 4th and 6th, as
4
cond, the Sign
under the third,
and ſo on, varying the Signs, ex-
m-4; and ſo on : fo that from
cepting that under the laft of the 5
deficient Terms, the Sign + muſt hence is gain'd the Fraction which
be always put when the two neareſt he directs you to put over the ſer
Terms on each ſide the deficient veral Terms of the Equation; and
Terms have contrary Signs.
the Reaſon of the following Part
Sir Iſaac Newton was the firſt who of his Rule chiefly follows from
gave a general Rule to find the ima. the Suppoſition that the Roots of
ginary Roots of an Equation, which any Equation, when real and une-
he has done in his Algebra, and in- qual, myft become equal before they
deed is the very fame with this here. can be imaginary : or contrariwiſé,
Jaid down. But as he himſelf ob- if imaginary, muſt become equal
ſerves, it will fometimes fail of diſco. before they can be real, upon the
vering all ſuch Roots, for fome Equa- augmenting the unknown Quantity.
tions may have more imaginary Roots The very ingenious Mr. Mac-Lau-
than can be found by this Rule, tho' rin in the Philofoph. Tranſactions,
this ſeldom happens. He has not fub- has given a Demonftration of this
joined the Demonſtration which very Rule of Sir Iſaac, together with
eaſily follows from his Rule for find one of his own, that will never
ing the Unciæ of the ſeveral Powers fail. So alſo has the learned
of a binomial Root ; for when the Mr. Campbell; both froin very la-
Roots of any Equation are all equal, borious and perplexing Compu-
and m be the Dimenſion, the Unciæ(or tations : I had almolt ſaid too long
Numbers prefix'd) to the firft Term and hard for one of a mode-
rate Patience and Capacity ever to
will be 1 ; to the ſecond
examine and be convinced of their
Truth,
the third
The real Roots of all Equations
(having imaginary ones) may be
fourth
eaſily found from common Alge-
i to bra ; that is, from the following
Theorem, viz. the Sum of the
3
the fifth
Squares of any Number n pf une-
3
4 qual Quantities, will be greater than
and ſo on ; and the Square of the the Sun of all the poſſibie Varieties
fecond Term will be to the Pro. of all the Products of the ſeveral
Quantities taken two and two, mul-
duct of the ift and 3d as
to
tiplied by ,,; whịch Theorem
; the Square of the third Term follows from this, that the Sum of
to the Product of the ſecond and the Square of two unequal Quan-
tities is greater than twice their Pro-
· fourth, as
i.
the duct. And this laſt from this, That
3
if four Quantities are proportional,
Square of the fourth to the Product the Sum of the greateſt and leaſt
of the 3d and 5th, as
is always greater than the Sum of
3
4; the two others.
the Square of the 5th to the Pro-
The
m
; to
I
ml
X
; to the
2
ma
MI m2
Х
2
3
1
1
m-O m2
X
1
x
m
O
1
MI
2
INI m2
to
2
ni
2
712
3
to
3
V 4
I MP
INC
are
The real and imaginary Roots of ftrument made of Braſs with a Box
Equations may be found alſo from and Needle and Staff, uſed to mea-
the Method of Fluxions, apply'd to ſure Land.
the Doctrine of Maximums and Mi-
IMPERVIOUS.
Bodies are ſaid
nimums; that is, to find ſuch a Va- to be impervious to others, when
lue of * in an Equation, expreſſing they will neither admit the Rays of
the Nature of a Curve, made equal Light, &c. nor the Effluvia of other
to y, an Ordinate which correſponds Bodies do paſs thro' them.
to the greateſt and leaſt Ordinate. IMPOST, in Architecture, is a
But when the Equation is above Plinth, or little Cornice, that crowns
three Dimenſions, the Computation a Piedroit or Pier, and ſupports the
will be intolerably laborious. See Couſſinet, which is the firſt Stone
Mr. Sterling's Treatiſe of the Lines that a Vault or Arch commences.
of the Third Order.
IMPROPER FRACTIONS,
The chief Uſe (that I know) of ſuch as have their Numerators equal
this Invention of imaginary Roots, to, or greater than their Denomi:
is to diſcover the various Figures and nators, as %, 11)
18. &c. Which are
Species of Curve-Lines.
not Fractions properly ſpeaking,
IMMENSE, is that whoſe Ampli- but either whole or mix'd Numbers ;
tude or Extenſion no finite Meaſure and are only in the form of Frac:
whatſoever, or how oft foever re- tions, in order to be added, ſub-
peated, can equal.
ſtracted, multiplied, or divided,
IMMERSION, is the plunging of &c.
any thing under Water.* 'Tis alſo INACCESSIBLE HEIGHT, or
uſed by Aſtronomers, to ſignify that DISTANCE, is that which cannot be
any Planet is beginning to come meaſured, by reaſon of ſome Impe-
within the Shadow of another; as diment in the way; as Water, &c.
in Eclipſes, whenever the Shadow INCEPTIVE of Magnitude, is a
of the eclipſed Body begins to fall word ufed by Dr. Wallis, expreſſing
on the Body eclipſed, we ſay, that ſuch Moments or firſt Principles, as
is the Time of Inmerſion; and tho' of no Magnitude themſelves,
when it goes out of the Shadow, is yet are capable of producing ſuch.
the Time of Emerfion.
Thus a Point hath no Magnitude
IMPENETRABILITY, is the Di- itſelf, but is inceptive of it. A Line
ftinction of one extended Subſtance conſider'd one way, hath no Magni,
from another, by which the Exten- tude as to Breadth, but is capable
fion of one Thing is different from by its Motion of producing a Sur-
that of another; ſo that two Things face which hath Breadth, G C.
extended, cannot be in the fame INCIDENCE POINT, in Optics,
Place, but muſt of neceſſity exclude is that Point in which a Ray of
each other.
Light is ſuppoſed to fall on a Piece
IMPERFECT CONCORD. See of Glaſs.
Concords.
INCIDENT RAY, in Catoptrics
IMPERFECT NUMBERS, are and Dioptrics.
are and Dioptrics. See Ray of Inci,
ſuch whoſe aliquot Parts taken all dence.
together, do either exceed, or fall INCLINATION, is a word fre.
ſhort of that whole Number of quently uſed by Mathematicians,
which they are Parts ; and theſe and fignifies the mutual Approach,
are two ſorts, either abundant or de- Tendence, or Leaning of two Lines,
ficient. Which ſee.
or two Planes, towards each other,
IMPERIAL TABLĘ, is an In. ſo as to make an Angle,
The
1
1
het en
j
INC
INC
The Inclination of two Planes is right Line drawn in the Plane
the acute Angle made by two Lines through the Point, where it is alſo
drawn one in each Plane, and per- cút by a Perpendicular drawn from
pendicular to their common Sec- any Point of the inclined Line. As
tion.
the Line CD inclines to the Plane
INCLINATION of the Axis of the AB, and the Inclination thereof is
Earth, is the Angle which it makes meaſur'd by the Angle EDC,
with the Plane of the Ecliptic, or made by the inclin'd Line CD, and
the Angle between the Planes of the
Equator and Ecliptic.
CV
INCLINATION of Meridians, in
Dialling, is the Angle that that
B
Hour-Line on the Globe, which is
perpendicular to the Dial-Plane,
E
D
makes with the Meridian,
INCLINATION of a Plane, in
Arm Toni FILTRATI GULTICUT
Dialling, is the Arch of a vertical
Circle, perpendicular to both the the Line ED drawn in the Plane
Plane and the Horizon, and inter- from the Point D, through the
cepted between them.
Point E, where a Perpendicular let
INCLINATION of the Planes of fall from any Point F, in the in-
the Orbits of the Planets to the Plane clined Line to the Plane, cuts it.
of the Ecliptic, are, thus : Saturn's INCLINING DECLINING DI.
Orbit makes an Angle of two De- ALS. See Declining Inclining Dials.
grees thirty Minutes, upiter's one INCLINING DIRECT SOUTH,or
Degree and one Third, Mars's is a North DIALs. See Direct South
little leſs than two Degrees, Venus's or North Inclining Dials.
is three Degrees and one Third, and INCLINED PLANE, is chat which
Mercury's is almoſt ſeven Degrees. makes an oblique Angle with the
The Inclination of the Orbit of a Horizon, Any Body, as A, laid
Planet may be found by having its upon an inclin'd Plane, loſes Part of
Latitude and Diſtance from the its weight, and the Weight B re-
Node'given ; for the Latitude is one quired to ſuſtain it is to the Weight
Side of a right-angled ſpherical
Triangle; the Diſtance from the
Node, the other Side ; and the An.
gle oppoſite to the Latitude, the In-
AO
B
clination of the Orbit.
INCLINATION of a Planet, is an
Arch of the Circle of Inclination,
comprehended between the Eclip D
E
çic and the Place of a Planet in his
Orbit.
of A, as the Height EC of the
INCLINATION of a Ray, in Diop- Plane to the Length DC of it,
trics, is the Angle which this Ray And from hence it follows that the
makes with the Axis of Incidence, Inclination of the Plane may be ſo
in the firſt Medium, at the Point little, that the greateſt Weight may
where it meets the ſecond Medium. be fuſtain'd on it by the leaſt
INCLINATION of a right Line Power.
ta a Plane, is the acute Angle which INCOMMENSURABLENUMBERS,
this right Line makes with another are fuch as have no cominon Divi.
for,
2
+
IN C
INC
2 u
Some of
for, that will divide them both e- tifice conſiſts in this, that the Quân-
qually, as 3 and 5.
tity to be made equal to a Square be
INCOMMENSURABLE QUANTI. algebraically expreſſed, and put es
TIES, are thoſe which have no a- qual to a Square ; and in ſuch
liquot Part, or any common Mea manner that the Equation thereby
ſure that may meaſure them; as may be reduced to one Dimenſion
the Diagonal, and Side of a Square : with reſpect to the unknown Quan:
for altho' that each of thoſe Lines tity of the Problein.
have infinite aliquot Parts, as the
Half, the Third, &c. yet not any be ſolved by an univerſal way of
Part of the one, be it ever ſo little, Solution, altho' different from that
can posſibly meaſure the other, as of determinate Problems; and o-
is demonſtrated in. Prop. 117. El. 10. thers again are to be come at par-
Euclid.
ticularly from a good Skill in the
INCOMPOSITE NUMBERS, are Properties of Numbers.
the ſame as Prime Numbers. See 1. If xx and yy be two Square
Prime Numbers.
Numbers to be found, whoſe Dif.
INCREMENT, or DECREMENT, ference a is given, it will be a=
is the Increaſe or Decreaſe of a
- yy; and a tyy=xx; fo
Quantity. There is a learned Latin that atyy muſt be a Square. Take
Treatiſe of the Doctrine of Incre-
2 (at pleaſure) = xty; then will
ments, publiſhed by Brooke Taylor, uuta
F.R.S. See more of this under
Series.
INCURVATION of the Rays of
2. If xx and yy are two Square
Liht. See Light and Refration.
Numbers to be found, whoſe Sum
INDETERMINED PROBLEM, is aa is given ; then muſt aa myy be
that which is capable of an infinite a ſquare Number, and taking u at
Number of Anſwers: As to find pleaſure, let xta be = uy; then
two Numbers, whoſe Sum, together
with their Product, ſhall be equal
uuti
to a given Number, or to make a
3. If aa,bb two given ſquare Num-
Rhomboides ſuch, that the Rect- bers be to be reſolved into two other
angle under the Sides be equal to ſquare Numbers xx and
it will
a given Square ; both of which bê aatbb = xx + yy ; and ſo
Problems will have infinite Solu-
a a tobbyy muſt be a Square
tions.
Number. Put y=2-b, and *
In ſome indeterminate Arithme-
2b+2 au
tical Problems there are more un 4%-a, then will
=%,
known Quantities than Equations,
uuti
and yet no one can be taken at 6 + 2au - buu
pleaſure ; and of theſe fort are all y
uuti
Problems, in which the Quantities
bu tauu — a
are to be equal to Squares or Cubes.
In theſe problems other Quantities
uu+1.
muſt be taken at pleaſure, and from 4. There are many other Pro-
theſe are determined the Quantities blems of this fort; ſuch as to find
of the indeterminate Problems; and two Numbers, whereof if to one you
when a Quantity, whoſe Conditions add the Square of the other, the
are determined, muſt be made equal Sum will be a ſquare Number,
to a Square or Cube, the whole Ar To find three Numbers whoſe Sum
a uz
will x
وزو
and X
2
IS
IND
4 y II
1
be = ng
i ſo
INC
is a Square, and two added together Remainder ; and ſince it muſt alſo
Thall make a Square. --To find three divide 2000- 21 y without a Re-
ſquare Numbers in an Arithmetical mainder, it muſt alſo divide the
Progreſſion, with an infinite Variety Difference of theſe two laſt Expreſ-
of others, which any one of him- fions, viz.
-17 y + 1989 and
ſelf may eafier propoſe than ſolve.
2000 -21 y, which Difference is
The following two Problems, viz.
to find a Cube Number which ad- 4y - 11. Let
ded to all its aliquot Parts ſhall
17
make a ſquare Number : And to
then will y be =
171+1
fo
find a ſquare Number, which added
4
to all its aliquot Parts ſhall make a
17 n tu
Cube, were formerly propoſed by that
muſt be a whole
Monſieur Fermat, as a Challenge to
4
all the Mathematicians of Europe ; Number. But 161 +12 is divi-
the former of which Dr. Wallis at ſible by 4; therefore the Difference
firit very oddly anſwered, viz. by n~I 17ntnu 16 +12
of
{aying that I was ſuch a Nuniberº;
and
4
4
4
for, ſays the Doctor, I is a CubeNum- mult be a whole Number : So that
ber, which added to all its aliquot n may be 1,5,9, 13, 17, &c. And
Parts being none, makes the ſquare
Number 1. But this was taken by accordingly y will be =
1X17+10
the French as a ſhuffling Anſwer un-
4
worthy ſuch a Man as the Doctor,
5*17+
and indeed I think fo too; for it is = 7, or
4
talking idlely, to ſpeak of adding a
Number to its aliquot Parts, when 9x17 + 11 13 X 17+1
4, or
it has no aliquot Parts. But after-
wards the Doctor gave many An-
17 *.17 til
ſyvers to thoſe Problems, as well as =58, or
=7s, or
ſome others of' a difficult Nature.
See the Letters that paſſed between 21x17x11
Dr.'Wallis, the Lord Bronker, Sir
=94; that is, if to
4
Kenelme Digby, &c. to be ſeen in 7 you add ſucceflively 17, you
the Dödtor's Works.
will have five of the Values of
If it be required to find what for ot17 is =24. 24+i7.is =41.
Number of Guineas and Piſtoles will 41717 is = 58. 58+17 is = 75.
make one hundred Pounds :
and 757-17 is =94; then the con
Put x for the Piſtoles, and y for reſpondent Number of Piltoles will
the Guineas ; then will 17x+219 .be 109, 88, 67, 46, 25, and 4.
be = 2000, and ſo x =
The Writers upon deterininate
2000–21 y
Now this muſt be a
Problems, are Diophantus, Kerſey,
Preylet, Ozanam, Kirkby, &c.
whole Number, in order to which INDEX, Charaltariſlic, or Exso-
find the neareſt Number to 2000. nent of a Logarithan, is that which
But leſs, that 17 will divide without News of how many Places the ab-
a Remainder, which is eaſily done ſolute Number belonging to the Low
by dividing 2000 by 17, and then garithm doth confift, and of ivhat
multiplying the Quotient by 17, the nature it is, whether Integer or a
Number will be 1989; then will Fraction. Thus, in this Logarithm
37 divide-1797-1989 without a -2.523421, the Number ſtanding on
the
24, or
4
4
4
17
I ND
IND
1
3
3
4
2
4
8.
6
3
3
S
S
the Left-hand of the Point is cal. I, becauſe as W x is a mean Propor-
led the Index; and becauſe it is 2, tional between 1 and x, fo į is an
Shews you that the abſolute Number Arithmetical Mean between ☺
anſwering to it, conſiſts of three and 1.
Places; for 'tis always one more
than the Index. If the abſolute
And the Exponent of ✓ x will
Number be a Fraction, then the be, becauſe as VF is the firſt of
Index of the Logarithm hath a ne-
the two mean Proportionals be-
gative Sign, and is marked thus,
tween i and x; ſo is the firſt of
2.523421.
the two Arithmetical Means between
INDEX of a Quantity, is that Quan- o and 1.
tity fhewing to what Power it is to
be involved ; as a3 fhews that a is
For ſince 1. x. xx. xxx, are con-
to be involved to the third Power ; tinually proportional, therefore their
where 3 is the Index, and a +6*+1, Cubes, or any other Roots, will be
alſo continually proportional ; that
Mhews that at is to be raiſed to
the Power nti, where nti is the is, v3:* (=1.) ŕ *.
✔ XXX
Index.
(=x)
If a Series of Geometrical Pro So alſo, I. X. XX. x3.x4.45. :
greſſionals be in this Order, 1. *. XX. Wherefore the Roots of the 5th
*3.44.45.20.x7, &c. Their In- Power of thoſe Quantities will be
dexes or Exponents will be in A-
rithmetical Progreffion, and ſtand = : That is, 11. 3x. V x2 * x3.
thus, 0, 1. 2. 3. 4. 5. 6. 7. But ♡ x4. V 85. (=x.)
if they are Fractions, as
Alſo for the fame Reaſon, the
then
x3 x4
x5 xo x7
Exponent of VX4, will be
their Exponents will be Negative, the Letter (or Power) over that of
N. B. Always place the Index of
and ſtand thus,
the Radical Sign.
1.- -2.-3.-4.-5.-6.--7.
Thus in Fractions, the Exponent
For if you ſuppoſe x=2, then
of
will be — 1, of 785
will
will be —-, of as will be
=, &c.
Or if you expreſs the Geometri- —- of a will be — , &c.
calSeries by means of the Exponents,
it will ſtand thus, x
N. B. V x, and xt, or Yxandxf,
And if it were expreſſed thus *°;
then it will be xI; becauſe 2 is or o4, and X are only two dif-
the Denominator of the Ratio, in ferent ways of Notation for one and
which Unity is not affected. Thus the ſame thing; the former in the
old, the latter in the new way.
alſo
and
So likewiſe
and 1=*°, x'=x, *2 = xx,
*2
&c.
Allo the Exponent of V * will be one; and most is *3,&c.
The
1
I
I
X
1
1
I
5
1*
2
;
I
I
I
I
I, and
U
1, and
XX
1
1
3
*
2
&c.
S
1
1
A
I
44
2:3
and a
2
are all
1
!
1
!
3
I.
NP
I
VJ
I
one
two
way, x
X 3
1
XX
1
x²,
11
3
and x
- }
3 into x
* Ñ 208,
1
wy
I
I
I
(===
*
IN D
INF
The way of reading or expreſling and negative Power, at equal Di-
Quantities ſo denoted, is thus, a ſtances from it : That is, NP.
is Unity divided by the Cube of x,
and if it were x
it muſt be read,
Wherefore 1=N x NP.
Unity or One divided by the Cube-
Root of the 7th Power of x.
Note alſo, That the Sum of the And dividing all by NP, No
Exponents of any two Numbers
or Quantities, in any Geometric Np. So that is all
NPP
Progreſſion, makes the Exponent
with N-2p.
of the Product of thoſe
Terms.
And to add ſome Examples of
Multiplication and Diviſion in this
Thus, x2+}, or té, is the way
3 / 3
way,
x
of expreſſing the Produ£t of xf into
√xs
1
$+
or *-*i is the *****
들
​Product of a
.
&c.
Alſo x * or * is the And Toys divided by will
Product of a
3 into itſelf, or the ſtand in this Notation ; thus,
Square of x
3
15
/x5
And the Difference between the
:-$=($=:,&c.
Exponent of the Quotient ariſing by *
Diviſion of the greater by the leſs.
INDICTION. See Cycle of In-
Thus x3, or x, is the Expo. di&tion.
INDIVISIBLES, in Geometry,
nent of the Quotient of *} by x3,
are ſuch Elements or Principles as
&c.
any Body or Figure may ultimately
Let p repreſent the Exponent of be reſolved into; and thefe Elé-
N, any Number at pleaſure ; and ments or Indiviſibles are in each pe-
let pr
culiar Figure ſuppoſed to be infi-
T'hen will NPN', Npty nitely ſmall.
N’, and No+2=N3, Np+ 3
This Method of Indiviſibles, is
N4, &C.
only the ancient Method of Ex-
hauſtions, a little diſguiſed and
Or if p=3; then will NpN3, contracted. It was firſt introduced
and Np+3 =NS, &c.
by Cavallerius, in his Geometria In-
And negatively, N= N3, divifibilium, Anno Dom 1635. Pur-
and NP+3 =Nº, &c.
ſued after by Torricellius in his
Works, printed 1644. And again,
Alfo, as o is an Arithmetical Mean by Cavallerius himſelf in another
between a poſitive and a negative Treatiſe, pabliſhed in 1647 And is
Quantity equally diſtant from it ;
now allowed to be of excellent uſe in
(i.e.) 6.0.6. are arithmetically the ſhortening of Mathematical In-
proportional : So is 1 a Geometrie veſtigations and Demonſtrations.
cal Mean between an affirmative
INFINIT e, or INFINITELY.
GREAT
!
X
llll
1
INF
INF
GREAT QUANTITY, is that which of the Rays of Light, and Mr. de la
has no Bounds, Ends, or Limits. Hire faith, he found that the Beams
INFINITELY SMALL QUANTI. of the Stars being obſerved in a
TY, is that which is ſo very ſmail, deep Valley, to paſs near the Brow
as to be incomparable to any finite of an Hill, are always more re-
Quantity, or which is leſs than any fracted than if there were no ſuch
aflignable Quantity.
Hill, or the Obſervations were made
1. No infinite Quantity can be on the top thereof, as if the Rays
augmented or leſſen'd, by adding or of Light were bent down into a
taking from it a finite Quantity: Curve, by paſſing near the Surface
Neither can a finite Quantity be of the Mountain.
augmented or leſſen'd, by adding INFLECTION - Point of any
or taking from it an infinitely {mali Curve, in Geometry, fignifies the
Quantity
Point or Place where the Curve be-
2. If there be four Proportionals, gins to bend back again a contrary
and the firſt is infinitely greater way,
When a Curve Line, as
than the ſecond ; then the third AF K, is partly Concave and partly
will be infinitely greater than the
fourth.
3. If a finite Quantity be divided
by an infinitely ſmall one, the Quo L
tient will be an infinitely great one;
and if a finite Quantity be multi-
plied by an infinitely ſmall one, the T
Product will be an infinitely ſmall
A
But if by an infinitely great one,
the Product will be a finite Quan-
P
M
tity.
If an infinitely ſmall Quantity be
F
E.
multiplied or drawn into an infi-
nitely great one, the Product will
be a finite one.
INFINITE SERIES. See Series.
INFLECTION, in Optics, is
B
multiplicate Refraction of the Rays
of Light, cauſed by the unequal
Denſity of any Medium, whereby Convex towards the right Line A B,
the Motion or Progreſs of the Ray or towards a fix'd Point, then the
is hinder'd from going on in a right Point F, that divides the Concave
Line, and is inflected or deflected from the Convex Part, and fo is at
by a Curve, faith the ingenious Dr. the Beginning of one, or the End of
Hook, pag. 217. who firſt took no- the other, is called the Inflektion
tice of this Property" in his Micro- Point, or Point of Inflection, as long
graphy. And this, he faith, differs as the Curve being continu'd to-
both from Reflection and Refraction, wards F, keeps its Courſe the fame.
which are both made at the Super- But the Point K is called the Point
ficies of the Body, but this in the of Retrogreſion, where it begins to
middle of it within.
reflect back again towards that
Sir Iſaac Newton diſcovered alſo. Part or Side where it took its 0
by plain Experiments, this Inflection riginal.
1. If
one.
..
INF
I N F
1. If thro' the Point F be drawn theſe Suppoſitions to find ſuch a
the Ordinate EF, as alſo the Tan- Value of AE, as that the Ordinate
gent FL, and from any Point, as M, EF fhall interſect the Curve A FK
on the ſame fide as AF, be drawn in F, the Point of Inflection or Re-
the Ordinate MP, as likewiſe the trogreſſion.
Tangent MT; then in thoſe Curves But to determine the Inflection
that have a Point of Inflection, the or Retrogreſſion in Curves, whoſe
Abſciſs AP continually increaſes, Semi - Ordinates CM, Cm, are
and the Part AT of the Diameter, drawn from the fixed Point C, draw
intercepted between the Vertex of CM infinitely near to Cm, and
the Diameter, and the Tangent make mH=Mm, let Tm touch
MT, increaſes until the Point P the Curve in M; now the Angles
falls' into E, after which it again CnT, CMm, are equal, and ſo
begins to diminiſh; whence the the Angle Cm H, while the Semi-
Line AT muſt become a Maximum Ordinates increaſe, does decreaſe, if
AL, when the Point P falls in the the Curve is Concave towards the
Point E.
Centre C, and increaſes if the Con-
2. In thoſe Curves that have a vexity turns towards it. Whence
Point of Retrogreſſion, the Part AT this Angle, or which is the ſame,
increaſes continually, and the Ab- its Meaſure will be a Minimum or
ſciſs increaſes ſo long, till the Point Maximum, if the Curve has a Point
T falls in L; after which it again of Inflection or Retrogreſſion ; and
diminiſhes. Whence AP muſt be- fo may be found, if the Arch TH,
come a Maximum, when the Point or Fluxion of it; be made equal to
T falls in L.
0, or Infinity. And in order to find
3. IF AE = x, EF=y, then will the Arch TH, draw mL, ſo that
the Angle Tm L be equal to mCL;
whoſe Fluxion then if C my, mr = x,
j
ta
which is
=i, we ſhall have y: 3 :: 3 :
ſup-
j?
I
Again draw the Arch HO to the
poſing a conſtant, being divided by Radius CH; then the ſmall right
s, the Fluxion of A L muſt become Lines mnr, OH, are parallel ; and
fo the Triangles o LH, mLr, are
nothing, that is, -
O;
ja
T
that multiplying by j”, and dividing
I
.by -- N, y=0; which is a gene M
ral Form for finding F the Point of
H
Inflexion or Retrogreſſion in thoſe
R
Curves, whoſe Ordinates are paral-
lel to one another. For the Na-
ture of the Curve AFK being
given, the Value of j may be found
in s; and taking the Fluxion of
this Value, and ſuppofing 'Å inva-
riable, the Value of y will be found
in , which being put equal to no-
thing, or Infinity, ſerves in either of
fimilar;
AL
مدل
mT
flip , توپ رو به نو
jy = 0; fo
m
IN S
I N T
INSTANT, is an infinitely ſmall
Part of Duration that takes up the
time of only one Idea in our Minds,
without the Succeſſion of another,
wherein we perceive no Succeſſion
M m
AT
IL O
R
at all.
No natural Effect can be produced
in an Inſtant.
From whence, follows the Reaſon
why a Burden ſeems lighter to the
Perſon carrying it in the Air, the
C
fafter he moves; and why the faſter
any one ſlides or ſcates upon Ice, the
fimilar; but becauſe HI is alſo per- leſs liable the Ice is to break, or
pendicular to mL, the Triangles even bend.
LHI, mLr, are alſo fimilar :
INTACTÆ, are right Lines to
which Curves do continually ap-
Whence :::::: : ; that is, proach, and yet never meet with
them. Theſe are uſually called
the Quantities mT, mL, are equal; Afymptotes: Which ſee.
But HL is the Fluxion of Hr, which
Integers, figniſies in Arithme.
is the Diſtance of Cmy: But tic, whole Numbers, in contradi-
HL is a negative Quantity, becauſe ſtinction to Fractions.
while the Ordinate CM increaſes, INTENSION, in Natural Philo-
their Difference rH decreaſes; fophy, ſignifies the Increaſe of the
whence it to yy--- yy=0, which Power, or Energy of any Quality,
is a general
Equation for finding the ſuch as Heat, Cold,&c. for of all the
Point of Inflection or Retrogreſſion. Qualities, they ſay, they are in-
INFORMED STARS, are ſuch of the tended and remitted, that is capable
• fixed Stars, as are not caſt into, or of Increaſe and Diminution.
ranged under any Form. See Sporades. The Intenſion of all Qualities in-
INGRESS, in Aftronomy, fignifies creaſes reciprocally, as the Squares
the Sun's entring the firſt Scruple of the Diſtances from the Centre
of one of the four Cardinal Signs, of the radiating Quality decreaſes.
eſpecially Aries.
INTERCALARY DAY, is the
INHARMONICAL RELATION, a odd Day put in or inſerted in the
Term in Muſic. See Relation In- Leap-Year.
harmonical.
INTERCEPTed Axis, a Term
INSCRIBED, in Geometry, a Fi- in Conic Sections, fignifying the
gure is ſaid to be inſcribed in another, fame with Abſciſa. Which ſee.
when all the Angles of the Figure in INTERCOLUMNATION, in Ar-
ſcribed touch either the Angles, Sides, chitecture, is the Space between two
or Planes of the other Figure. Columns, which, in the Doric Oru
INSCRIBED BODIES. See Re- der, is regulated according to the
gular Bodies,
Diſtribution of Ornaments in the
INORDINATE PROPORTION, is Frieze ; but in other Orders, ac-
where the Order of the Terms are cording to Vitruvius, is of five difu
diſturbed.
ferent kinds, viz. Picnojlyle, Syſtyle,
INSCRIBED HYPERBOLA, is Euſłyle, Diaſtyle, and Arcoſtyle.
ſuch an one as lies entirely within This the Latins expreſs by the
the Angle of its Afymptotes, as the Word Intercolumnium.
Conical Hyperbola doth.
IN-
I N T
INV
INTEREST, is the Sum reckoned ſeventh, with their Varieties. Com.
for the Lot or Forbearance of ſome pound ones are all thoſe that are
principal Sum lent for (or due at) a greater than an Octave, as the ninth,
certait time, according to ſome tenth, eleventh, c. with their Van
certain Rate; and therefore is cal- rieties.
led Principal, becauſe it is the Sum An Interval is alſo divided into
that procreates the Intereſt, or from Juſt or True, and into Falfe. All
which the Intereſt is reckoned, and the above-mentioned Intervals, with
is either Simple or Compound. their Varieties, whether Major or
INTEREST SIMPLE, is counted Minor, are Juſt; but the Diminu-
from the Principal only, and is eaſi- tive or Superfluous ones are all
ly computed by the ſimple or com- Falſe. An Interval is alſo divided
pound Golden Rule.
into a Conſonance and Diſſonance.
Interior POLYGON. See Po- Which fee.
lygon Interior.
INTERVAL of the Fits of eaſy
INTERIOR TALUS. See Talus. Reflection, and of eaſy Tranſmiſion of
INTERNAL ANGLES. See An- the Rays of Light, is the Spaces bea
gles Internal.
tween every Return of the Fit and
INTERSECTION, in Mathema- the next Return.
tics, ſignifies the cutting of one Line Theſe Intervals Sir Iſaac Newiort
or Plane by another ; thus we ſay, ſhews how to collect, and thence to
that the mutual Interfection of two determine whether the Rays ſhall-
Planes is a right Line.
be reflected or tranſmitted at their
INTERSTELLAR, a Word uſed ſubſequent Incidence on any pellua'
by ſome Authors to expreſs thoſe čid Medium.
Parts of the Univerſe that are with INTESTINE Motion of the Parts
out and beyond our ſolar Syſtem, Of Fluids. Where the attracting
and which are ſuppoſed as Planetary Corpuſcles of any Fluid are elaſtic,
Syſtems moving round each fixed they muſt neceſſarily produce an in-
Star as the Centre of their Motion, teſtine Motion; and this, greater or
as the Sun is of ours ; and if it be leſſer, according to the Degrees of
true, as 'tis not improbable, that their Elaſticity and attractive Forces.
each fixed Star may thus be a Sun For two elaſtic Particles, after
to fome habitable Orbs that may meeting, will fly from one another
move round it, the Interſtellar (abſtracting from the Reſiſtance of
World will be infinitely the greater the Medium) with the ſame Degree
Part of the Univerſe,
of Velocity that they met together
INTERTIES, in a Building, are with.
thoſe ſmall Pieces of Timber that But when, in leaping back from
lie horizontally between the Som- one another, they approach other
mers, or between them and the Cell Particles, their Velocity will be ina
or Reaſon.
creaſed.
INTERVAL, in Muſic, is the Di INVERSE Method of Fluxions, is
ſtance between any two Sounds, the Method of finding the flowing
whereof one is more grave, and the Quantity from the Fluxion given,
other more acute. They make fe- and is the ſame with what the foo
veral Diviſions of an Interval, as reign Mathematicians call the Cale
firſt into Simple and Compound: culus Integralis.
The Simple Intervals are the Octave, INVERSE Method of Tangents, is
and all that are within it, as the ſe- the manner of finding an Equation
cond, third, fourth, fifth, fixth, and of a Curve, or conſtructing a Curve,
X
by
ION
ISA
;
by means of a given Tangent, or rilis at Rome, now the Church of
any other Line, whoſe Determina- St. Mary the Ægyptian, are theſe :
tion depends upon a Tangent ; as to 2. The entire Order from the
find a Curve Line, whoſe Subtangent Superficies of the Area to the Cor-
nice, are twenty-two Modules, or
is 29), or whoſe Subtangent is a eleven Diameters.
2. The Column with its Baſe and
third Proportional to pay and y; Capital, contains eighteen Modules.
or whoſe Subnormal is a conſtant
4. The Entablature (i.e. the Ar-
Quantity; or whoſe Subtangent is chitrave, Friſe, and Cornice) contains
equal to the Semi-ordinate or to four Modules.
find a curve-lin'd Space, whoſe in 5. The Voluta of the Capital is
definite Area is expreſſed by Vx,
of an oval Form.
6. The Columns in this Order
or by a Vaat**, &c. And the are often hollowed, and furrowed.
Solution of moſt of theſe problems with twenty four Gutters ; and
depend upon the inverſe Method of ſometimes 'tis done only to the third
Fluxions.
Part of the Column, reckoning from
: INVERSE PROPORTION, or Pro- the bottom, and then that third
portion by Inverſion. See Proportion. Part hath its Gutters filled with
INVERSE RATIO, is the Aſſump- little Rods or Battoons, all the Parts
tion of the Conſequent to the Ante- of the hollow above being left empty.
cedent, like as the Antecedent to Iris, is that fibrous Circle next
the Conſequent; as if A: B: C: to the Pupil of the Eye, diſtinguiſh-
D; then by Inverſion of Ratio's B : ed with Variety of Colours. See
A:: D: C.
Uvea Membrana.
INVOLUTE FIGURES. The 'Tis ſo called from its Similitude
Curve AMM (ſee Evolute Curves) to à Rainbow, (in Latin, Iris.)
is what is called an Involuie Curved Alſo thoſe changeable Colours
Figure.
which ſometimes appear in the
INVOLUTION, in Algebra, is the Glaſſes of Teleſcopes, Microſcopes,
raiſing up any Quantity from its &c, are called Ires' for the ſame
Root to any other affigned ; as fup- reaſon; as is that coloured Spectrum,
poſe a +b were to be ſquared, or which a triangular priſmatic Glaſs
raiſed up to its ſecond Power, they will project on a Wall, when placed
ſay, involve a tob, that is, multiply (at a due Angle) in the Sun-Beams.
it into itſelf, and it will produce See · Rain-Bow.
aa+2 ab-tbb.
IRRATIONAL NUMBERS. See
INWARD FLANKING ANGLE, Surd Numbers.
in Fortification, is made by the IRRATIONAL QUANTITIES.
Courtin, and the Razant Flanking See Rational Quantities.
Line of Defence.
IRREGULAR BODIes, are So-
Ionic ORDER, in Architecture, lids, which are not terminated by
is the third Order, and is a kind equal and like Surfaces.
of Mean between the ſtrong and de IRREGULAR FORTIFICATION,
licate Orders. Its Capital is adorn- See Fortification.
ed with Volutes, and its Cornice IRREGULAR LINEs, or CURVES,
with Denticules.
See Regular.
1. The Proportions of this Pillar, ISAGON, in Geometry, is fome-
as they are taken from the famous times uſed for a Figure conſiſting of
one in the Temple of Fortuna Vi- equal Angles.
ISLES,
a
ISO
I SO
ISLES, in Architecture, are the and of the other unequal, that is
Sidës; or Wings of a Building. the greater, whoſe two sides are
ISOCHRONE Vibrations of a Pen- equal.
dulum, are ſuch as are made nearly 3. Of Iſoperimetrical Figures,
in the ſame Space of Time, as all the whole Sides are equal in Nümber,
Vibrations or Swings of the ſame that is the greatest; which is Equi-
Pendulum are; whether the Arks it lateral and Equiangular. From hence
deſcribes be longer or ſhorter : for follows that common Problem of
when it deſcribes a ſhorter Ark, it making the Hedging or Walling,
moves ſo much the ſlower; and that will wall in one Acre, or even
when a long one, proportionably any determinate Number of Acres,
fafter.
(which call a,) hedge or wall in any
IsOCHRONA LINE, is that in greater given Number of Acres, be
which a heavy Body is ſuppoſed to
it what it will. Which let bie b, as
deſcend without any 'Acceleration : likewiſe always a Square. In order
And Mr. Leibnitz, in the Act. Erud. to which, call x one side of an Oba
Lipl. for Feb. 1689. hath a Dif- long; (whoſe Area is the Number
courſe on this Subject : In which he
ſhews, That an heavy Body with a of Acres a) then will bè che os
Degree of Velocity acquired by the
Deſcent from any Height, may de- ther Side, and 2
+- 2 x, will be
fcend from the ſame Point by an in-
finite Number of Ifachronal Curves, the Ambit of the Oblong. Which
and which are all of the ſame Spe- muſt be equal to four times the
cies, differing from one another only
in the Magnitude of their Parame- Square Root of b; that is; 2 +
ters ; ſuch as are all the Quadrato-
Cubical Paraboloids, and conſe- 2x=4b. Whence the value of
quently ſimilar to one another. à will be eaſily had, and you may
He ſhews alſo there, how to find make infinite Numbers of Squares
a Line, in which a heavy Body de- and Oblongs that have the ſame
ſcending, ſhall recede uniformly from Ambit, and yet ſhall have different
ä given Point, or approach uniform: given Area's. See the Operation.
Let vbd.
ISOMERIA, in Algebra, is the ža +4.**
ſame with Converſion of Equations, Then
4
d
(ſee Equations, N°. 1.) or of clearing
And a +2**= 2 dx.
any Equation from Fractions.
Iso P Ë R I METRICAL FIGURES,
2 dås
in Geometry, are fuch as have equal
Perimeters, or Circumferences.
And * x di .
1. Of Iſoperimetrical Regular Fi-
gures, that is the greateft that con- Ånd xx+di+dd=c+Idda
or the morit Angles
, and conſequent. And x==1
ly a Cirele is the greatelt of all Fi-
to do
gures that have the fame Ambit as
it has.
Whence
2. Of two Iſoperimetrical Tri-
angles, having the ſame Baſe, x
whereof two sides of one are equal,
+ Add tudi
As
ly to it..
bg
Alo 2 * *
a.
2
v
2
1
Ź
1
i
P
K E Y
As if one side of the Square be all the Planets : It is diſtant from
10, and one Side of an Oblong be the Sun at a mean Rate 5201. If the
19, and the other 1, then will the Earth's mean Diſtance be 1000, its
Ambits of that Square and Oblong Excentricity is 250. The Inclina-
be equal, viz. each 40, and yet the tion of its Orbit is 1°. 20'. Its
Area of the Square will be 100, and Periodical Time is 43332 Days, 12
of the Oblong but 1.9.
Hours, and it revolves about its
ISOSCEL&S TRIANGLE. See Axis in nine Hours 56 Minutes.
Triangle.
The Magnitude of Jupiter is about
ISTHMUS, in Geography, is a 2460 Times greater than our Earth.
little Neck, or Part of Land joining 1. In the Year 1664, Campani,
a Peninſula to the Continent. by help of an excellent Teleſcope,
JULIAN PERIOD, is a Cycle of obſerv'd certain Protuberances, and
7980 Conſecutive Years, produced Inequalities in the Surface of this
by the continual Multiplication of Planet. As alſo the Shadow of his
the three Cycles, viz. That of the Satellites, and kept his Eye upon
Sun of 28 Years, that of the Moon them till they went off the Diſk.
of 19 Years, and that of the In 2. In the ſame Year, May 9, two
diction of 15 Years; ſo that this Hours, P. M. Mr. Hook, with a Te-
Epocha, although but artificial or leſcope of twelve Foot, obſerved a
feigned, (and which was the Inven- ſmall ſpot in the biggeſt of the
tion of the famous Julius Scaliger) three obſcurer Belts of Jupiter; and
is yet of very good uſe; in that within two Hours after, he found
every Year within the Period is di- that the ſaid Spot had moved from
ftinguiſhable by a certain peculiar Eaſt to Weft above half the Length
Character; for the Year of the Sun, of the Diameter of Jupiter.
Moon, and Indiction, will not be the 3. Mr. Caſini obſerved alſo, near
ſame again, till the whole 7980 the ſame time, a permanent Spot in
Years be revolved. Scaliger fixed the Dik of Jupiter ; by whoſe Help
the Beginning of this period 764 he not only found that Jupiter turns
Years before the Creation.
about upon his own Axis, but alſo
For the finding the Year of the the Time of ſuch Converſion, which
Julian Period, you have this Rule : he eſtimates to be nine Hours, and
Multiply the Solar Cycle by 4845, 56 Minutes : Which was alſo con-
the Lunar by 4200, and the Indice firm'd by better Obſervations of a
tion by 6916:
Spot in the Year 1691. The Equato-
Then divide the Sum of the Pro- rial Diameter of Jupiter to his Po-
ducts by 7980, and the Remainder
lar one, Sir Iſaac Newton computes
of the Diviſion (without having re- to be as 40 to 395 .
gard to the Quotient) ſhall be the
Year enquired after.
JULIAN YEAR, is the old Ac-
count of the Year, inſtituted by Ju-
K.
lius Cæfar, which to this day we
uſe in England, and call it the Old A L E NDA R. See Can
Style, in contra-diſtinction to the
lendar.
New Account, framed by Pope Gre-
KALENDS.
See Calends.
gory, which is eleven Days before Key, in Muſic, is a certain Tone,
ours, and is called the New Style. whereto every Compoſition, whether
JUPITER, the Name of one of it be long or fhort, ought to be fit-
the Planets. This is the biggeſt of ted or deſigned; and this Key is ſaid
to
!
L'A D
LAT
to be either flat or ſharp, not in re- therefore their Breadth muſt be two
ſpect of its own Nature, but with Diameters of the Shot, and their
relation to the flat or ſharp Third, Length for double-fortified Cannon
which is joined with it.
2 and of the Shot; før ordinary
Keys of an Organ, Harpſicord, or Cannon it muſt not exceed 2; but
Spinnet, are the horizontal Rows of for Culverins and Demi-Culverins,
ſmall Pieces of Wood, or Ivory, it may be three Diameters of the
or both; which the Fingers ſtrike Shot, and 3 and { for leffer Pieces,
upon to play, or cauſe the Inftru. in order to load at twice : If you
ment to found.
will load at once, this Length of
Knots. There are two Sorts of the Ladle muſt be double. And ob-
Knots uſed at Sea: One they call ſerve this, that a Ladle nine Balls
a Bowling-Knot, becauſe by this in Length, and two Balls in Breadth,
Knot the Bowling-Bridles are fa- will hold juſt the Weight of the
ftend to the Crenyles. This is very Iron Shot in Powder.
faſt, and will not ſlip.
LAMPADIAS, a kind of bearded
The other is a Wall-Knot ; which Comet, reſembling a burning Lamp,
is a round Knob, or Knot, made being of ſeveral Shapes; for ſome-
with three Strands of a Rope. This times its Flame or Blaze runs taper-
Knot ſerves for the Top-Sail, Sheet, ing upwards like 'unto a Sword,
and Stoppers.
and ſometimes it is double or treble
The Diviſions of the Log-Line pointed.
are thus called. Theſe are uſually LANGREL-Shot, is a ſort of
ſeven Fathom, or forty-two Feet a- Shot uſed at Sea. It is made of
funder, but they thould be fifty two Bars of Iron, with a Joint in
Feet ; and then as many Knots as the middle, by which means it can
the Log-Line runs out in half a be thorten'd, and ſo put the better
Minute, ſo many Miles doth the into the Gun; and at each end there
Ship fail in an Hour ; ſuppoſing her is an Half-Bullet, either of Lead or
to keep going at any equal Rate, Iron. When 'tis diſcharged, it flies
and allowing for Yaws, Lee-Way, out at length, and is of uſe to cut
&c.
the Enemy's Rigging, &c.
LARBOARD, the Left-hand Side
of a Ship, when you ſtand with your
Face to the Head,
L.
LARMIER, a flat ſquare Member
in Architecture, which is placed on
ABEL, is a long thin Braſs the Cornice below the Cimalium,
Ruler, with a ſmall Sight at and jets out fartheft; being ſo cal-
one end, and a Centre-Hole at the led from its Uſe, which is to diſperſe
other, commonly uſed with a Tan- the Water, and to cauſe it to fall aç
gent-Line on the Edge of a Circum- a diſtance from the Wall, Drop by
ferenter, to take Altitudes, &c. Drop, or, as it were, by Tears :
LACUNAR, in Architecture,' is For Larme, in French, ſignifies a
an arched Roof or Cieling, more e- Tear. See Corona.
ſpecially the Planking or Flooring LATERAL EQUATION, in Alge-
above the Porticoes.
bra, is the ſame with ſimple Equa-
LADLE, an Inſtrument to load tion, which has but one Root, and
great Guns with Powder. It ought may be conſtructed by ſtraight Lines
to be ſo proportioned, that two only,
Ladles-full may charge the Piece ; LATION, is the Tranſlation or
.
Motion
L
X 3
,
L AT
A
L ES
Motion of a Body from one place Cone, as the Line ED, in the fol-
to another in a right Line; and ſo lowing Figurë.
is much the ſame as Local Motion. LATUS PRIMARIUM, is a right
LATITUDE of a Place, is an . Line belonging to a Conic Section,
Arch of the Meridian of that Place, drawn through the Vertex of the
intercepted between its Zenith and Section of the Cone, and within it,
the Equator; or 'țis an Arch of the as the Line É È or D D in the pre-
Meridian intercepted between the ceding Figure.
Pole and the Horizon; and there LEAP YEAR, or BISSEXTUE,
· fore is called the Pole's Height. is every fourth Year; and is ſo calá
LATITUDE, in Navigation, is led from its leaping a Day more
the Diſtance of a Ship from the E- that Year ihan in a common Year.
quinoctial, either North or South, For in the common Year any fixed
and is counted on the Meridian; fo Day of a Month changeth ſucceſſive-
that if a Ship fails towards the E- ly the Day of the Week. If the
quino&ial, ſhe is ſaid to dépreſs the Year be divided by 4, and nothing
Pole; but if ſhe fails from the E- remains, 'tis Leap Year; but if 1,
quinoctial, ihe is ſaid to raiſe the 2, or "3, it is ſo many Years after
Pole; and if the ſails from the Equi- Leap-Year.
noctial, either North or South, her Leaver. See Lever.
Way gained thus is called her Dif Leaves, are the Nötches of the
ference af Latitude.
Pinion of a Watch. See Pinion.
LATITUDE of a Star, or Planeta Lee, a "Sea-Term, by which' is
is its Diitance from the Ecliptic, generally meant the Part oppoſite to
being an Arch of a Circle of Longi- the Wind.
tude, reckoned from the Ecliptic to Legs of a Triangle. When one
wards its Poles.
Side of
a Triangle is taken as a
LATITUDE HELIOCENTRIC of Baſe, the other two are called Legs.
a Plänet. See Heliocentric.
LeMMA, is a Term uſed chiefly
LATUS Recrum, a Term in by Mathematicians, and" fignifies a
Conics, being the ſame with the Pa- Propofition, which ferves previouſly
rameter.
Which ſee.
to prepare the way for the more
LATUS TRANSVERSUM" of the eaſy Apprehenſion of the Demon-
Hyperbola, is a right Line lying be- ftration of ſome Theorem; or for the
tween the Vertexes of the two oppo: Conſtruction of ſome Problem.
fite Sections ; or that part of the Lens, is a Term in Optics for a
common Axis, which is between the ſmall Convex, or Plano-Convex,
Vertexes of the upper and lower Concave, or Concavo-Convex Glaſs.
Leo, is the fifth of the twelve
Signs of the Zodiac, and is marked
thus 2
D
Lepus,the Hare, a Southern Con-
ſtellation, containing thirteen Stars.
LESSER Circles of the Sphere,
are thoſe whoſe Planes do not paſs
through the Centre of the Sphere;
and which do not divide the Globe
into two equal Parts, but are paral-
lel to the greater Circles; as the
Tropics and Polar Circles, and all
Parallels of Declination and Alti-
tudes;
E
4
L E V
'LE V
tude ; which latter being parallel to three Foot long, and about an Inch
che Horizon, are called Almicanters. in Diameter, bent up ſquare at both
LEVANT, in Geography, is pro- Ends to receive two Glaſs-Tubes of
perly the Eaſtern-ſide of any Con- three or four Inches, faſtend to
tinent or Country, or that on which them. In this Tube is pour'd com-
the Sun riſes ; but now with our mon or colour'd Water through ore
Seamen, it fignifies the Mediterra- of the Ends, until there is ſo much
nean Sea, and eſpecially the Èaftern
as to appear in the Glaſs-Tubes.
Part of it; and our Trade thither This Initrument being ſet upon a
is called the Levant Trade ; and a three-legged Staff, is fit for Uſe.
Wind that blows from thence out of There are many more nice and
the Streights-Mouth, is called a Le- compound Inſtruments of this kind;
vant Wind.
as may be ſeen in Mr. De la Hire's
Level, is an Inſtrument where- and Picard's Treatiſes of Levelling;
by we find an horizontal Line, and in Mr. Bion's Book of Mathematical
continue it out at pleaſure, and by Inſtruments; and in the Tranſactions
this means find the true Level for of the London and Paris Royal Societies.
conveying Water to ſupply Towns, LEVELLING, is the Art of find-
make Rivers navigable, drain Bogs, ing a true horizontal Line, or the
&c. Of theſe Inſtruments there Difference of Aſcent or Deſcent be-
are ſeveral kinds, of which a very , tween any two Places, in order to
good one for ſhort Diſtances, is this drain Moats, Marſhes, and Mioraſſes,
following; which conſiſts of a round &c. or to convey Water from Place
Tube of Braſs or other Matter about to Place.
If a Station be taken, more than fifty French Fathoms, it muſt be car-
rected from the following Table of Corrections,
Lines.
Parts.
Stations, Correétions.
Fathoms. Inches.
0
>
50
100
I
I
O
3
5
O
I
1
1
150
200
250
300
350.
400
450
500
550
600
650
700
750
4
9
3
2
O
2
O O
3
4
4
5
6.
7
7
8
To
II
4
3
I
800
0
850
I
Il
II
goo
950
O
1000
O
1
LEVER,
X 4
L E V
L I B
1
LEVER, the ſecond mechanical FR to FM, that is, FN to FP;
Power, is an inflexible right Line, and as FP to FM, that is, as FN
made uſe of to raiſe Weights, either * FP: FM XFP. And ſince FP
weighing nothing itſelf, or of ſuch is in both; therefore as FŅ:
Weight as may be balanced. The iM.
Lever is threefold.
The Action of a Power P, and
the Refiftance of the Weight M,
F F
increaſe in proportion to their Di.
MC
ftance from the Fulcrum; and
P therefore that a Power may be able
to ſuſtain a Weight, it is required,
T
that the Diftance of the Point in theç
Lever to which it is applied, be to
M
P
the Diſtance of the Weight, as the
Weight to the Intenſity of the
Power ; which, if it be ever folita
tle increaſed, will raiſe the weight.
Levity, is the Diminution or
PO
Want of Weight in any Body, when
M compared with another that is hea-
vier; and in this ſenſe is oppoſed tą
1. Sometimes the Fulcrum F is Gravity.
placed between the Weight P and
LIBRA, one of the twelve Signs
The Power M.
of the Zodiar, being exactly oppo-
2. Sometimes the Weight is be- fite to Aries.
tween the Fulcrum and the Power.
LIBRATION of the Moon, (ſec
3. And often alſo the Power acts Evection) is of three kinds.
between the Weight and the Ful-
1. Her Libration in Longitude ;
crum F,
which is a Motion ariſing from the
Plane of that Meridian of the Moon,
M (which is always, nearly, turned to-
wards us,) being directed not to the
Earth, but towards the other Focus
of the Moon's Elliptical Orbit ; and
fo to an Eye on the Earth ſhe ſeems
to librate to and again in Longi-
tude, or according to the Order of
the Signs in the Zodiac. This Li-
F
R bration is of no Quantity twice in
each Periodical Month, viz. when
If FM be a Lever, and the the Moon is in her Apogæum, and in
Weight P hangs any where there- her Perigæum; for the Plane of her
on, and FR be the Horizon, then Meridian above-mention'd, is di-
the Power M that will keep the rected alike to both the Foci.
Weight P at any Elevation M FR, 2. Her Libration in Latitude ;
acting in the Direction SM, per- which ariſes from hence, That her
pendicular to FM, in which Di- Axis not being perpendicular to the
rection the Action of the Power is Plane of her Orbit, but inclined to
a Maximum, will be to the Weight it, ſometimes one of her Poles, and
P, as FN to FM. For it is as ſometimes the other, will nod (as
they
P
ObmWv
괴
​Z
1
1
S
1
L I B
L. I G
they call it) or dip a little towards the former Libration in Latitude,
the Earth, (as is the Caſe of the depending upon the Light of the
Poles of the Earth towards the Sun,) Şun, will be compleated in her Sy-
and conſequently ſhe will appear nodical Month. Greg. Aftron. Lib.
to librate a little, and to ſhow fome- 4. Se&t. 10.
times more of her Spots, and ſome LIFTING Pieces, are Parts of
times leſs of them, towards each a Clock which do lift up and un-
Pole ; which Libration depending lock the Detents in the Clock-part.
on the Poſition of the Moon, in re Light, is Fire entring our Eyes
ſpect of the Nodes of her Orbit with in ſtraight Lines ; and by the Mo..
the Ecliptic, (and her Axis being tion thereof that it communicates
perpendicular nearly to the Plane to the Fibres in the bottom of the
of the Ecliptic) is very properly ſaid Eye, it excites the Idea of Light.
to be in Latitude.
I. A rectilinear Motion is the
3. And this is compleated in the Motion of Light, as it appears
Space of the 'Moon's Periodical from its being eaſily ſtopped by an
Month, or rather, while the Moon Obſtacle.
is returning again to the fame Po 2. And that an irregular Motion
fition, in reſpect of her Nodes. is more proper for it, may be proved,
4. There is alſo a third kind of becauſe the Rays that come directly
Libration ; by which it happens, from the Sun to the top of a Moun-
that though another part of her is tain, produce no Heat ; whilſt in
not really obverted to the Earth, the Valley, where the Rays are a.
as in the former Libration, yet ano- gitated with an irregular Motion by
ther is illuminated by the Sun: For ſeveral Reflexions, there is often
ſince her Axis is perpendicular near- produced a very intenſe Heat.
ly to the Plane of the Ecliptic, 3. That there is Light where
when the Moon is moſt Southerly, there is not fire, is beyond all
in reſpect of the Ecliptic North doubt ; for we daily fee hot Bodies
Pole, ſome Parts nearly adjacent to that do not ſhine.
it will be illuminated by the Sun ; 4. As to the Motion of Light, it
while, on the contrary, the South is plain, that it is performed in
Pole will be in Darkneſs. In this Right Lines; but whether it be
Caſe therefore, if it happens that Succeſſive or Inſtantaneous, is dir-
the Sun be in the ſame Line with puted ; that is, whether at the
the Moon's Southern Limit, then ſame Moment that a Body begins
will fhe, as the proceeds from Con- to ſhine, the Light is ſenſible at any
junction with the Sun towards her Diſtance ; or whether the Light
aſcending Node, appear to dip her goes on ſucceſſively to Places more
Northern Polar Parts a little into and more diftant.
he dark Hemiſphere, and to raiſe 5. It ſeems clearly to follow from
her Southern Polar Parts as much ſeveral Aſtronomical Obſervations,
into the Light. And the contrary that that Motion is ſucceſſive, and
to this will happen the next Fort Philoſophers did not long doubt of
night, while the New Moon is de- it; but by ſome later Obſervations,
ſcending from her Northern Limit; the Concluſions drawn from the
for then her Northern Polar Parts former are weakened, and we are
will appear to emerge out of Dark, obliged to confeſs that the Motion
neſs, and the Southern Polar Parts of Light has ſomething unknown
to dip into it. And this ſeeming to us.
Libration, or rather theſe Effects of
Mr.
>
LIG
L I G
1
Mr. Romer, from a great Num- Storm ; Quickſilver when ſhaken in
ber of Aſtronomical Obſervations for vacuo ; a Cai's Back or a Horſe's
the ſpace of 10 Years, inferr'd the Neck rubbed by the Hand in the
vaft Swiftneſs of the Motion of the dark; Wood, Fleſh and Fiſh when
Sun's Light, by means of the Eclip- putrefied.
ſes of Jupiter's Satellites. · From 10. Every viſible Point of any
whence Mr. Huygens in his Treatiſe Object emits Rays of Light into all
de Luni. P. 8,9. computes the Mo- Parts, from whence that Point is
tion of Light to be 1100000000 viſible.
Feet in one Second. Notwithſtand-
ing this, Mr Caffini and Miraldus, Sir Iſaac Newton, in his Optics,
from a great Number of Aftronomi-
cal Obſervations, will have Mr. Ro-
propoſes the following Queries.
mer and Huygens to be miſtaken.
See the Mem. de l'Academ. Royal de 1. Do not, great Bodies conſerve
Scien. anno 1707.
their Heat the longeſt, their Parts
6 The Preſence of the Air is of- heating one anothers and may not
ten neceffary for the Production of great denſe and fixed Bodies, when
Light.
heated beyond a certain Degree, e-
7. 'Tis probable, that the Rays mit Light ſp copiouſly, as by the
of Light which fall upon Bodies, Emiſſion and Reaction of its Light,
and by that means are reflected or
and the Reflexions and Refractions
refracted, begin to bend before they of its Rays within its Pores, to grow
arrive at the Bodies ; and that Light ſtill hotter, till it comes to a cer-
is reflected, refracted, and inflected tain Period of Heat, ſuch as is that
by one and the ſame Principle, act. of the Sun And are
not the
ing variouſly in various Circum- Sun and fixed Stars, great Earths
Itances.
vehemently hot, whoſe Heat is
8. 'Tis probable alſo, that Bo- conſerved by the Greatneſs of the
dies and Light act on each other : Bodies, and the mutual Action and
Bodies in emitting, reflecting, re- Reaction between them, and the
fracting, and inflecting it ; and Light, Light which they emit, and whoſe
by heating them, and putting their Parts are kept from fuming away,
Parts into a vibrating Motion, where- not only by their Fixity, but alſo
in Heat conſiſts.
by the vaſt Weight and Denſity of
9. All fixed Bodies, when heated the Atmoſpheres incumbent upon
beyond a certain Degree, do emit them, and very ſtrongly compref-
Light and ſhine ; and this Shining fing them, and condenſing the Va
and Emiſſion of Light is probably pours and Exhalations which ariſe
cauſed by the vibrating Motion of from them ? For if Water be made
the Parts; and all other Bodies a warm in any pellucid Veſſel emptied
bounding with earthy Particles, and of Air, that Water in the Vacuum
eſpecially when they are ſulphu- will bubble and boil as vehemently
reous, when their Parts are ſuffi- as it would in the open Air in a Vel-
ciently agitated, do emit Light; ſel ſet upon the Fire, till it receives
whether this Agitation be cauſed a much greater Heat. For the
by Attrition, by Percuſſion, by Pu- weight of the incumbent Atmo-
trefaction, or a vital Motion in an ſphere keeps down the Vapours,
Animal Body, c. or any other and hinders the Water from boils
way. Thus the Sea-water ſhines in a ing, until it grows much hotter than
LS
3
LIG
L IM
is requifite to make it boil in vacuo. to make Senſations of the ſeveral in-
Alſo a Mixture of Tin and Lead termediate Colours ?
being put upon a red-hot Iron in
3. May not the Harmony and
Vacuo, emits a Fume and Flame ; Diſcord of Colours ariſe from the
but the ſame Mixture in the open Proportions of the Vibrations pro-
Air, by reaſon of the incumbent pagated through the. Fibres of the
Atmoſphere, does not ſo much as Optick Nerves into the Brain, as
emit any Fome which can be per- the Harmony and Diſcord of Sounds
ceiyeç by Sight. In like manner ariſe from the Proportions of the
the great Weight of the Atmo- Vibrations of the Air ? For ſome,
ſphere, which lies upon the Globe Colours, if they be viewed together,
of the Sun, may hinder Bodies there are agreeable to one another, as
from riſing up, and going away thoſe of Gold and Indigo,and others
from the Sun in the form of Va- diſagree.
pours and Fumes, unleſs by means See more of the Nature of Light
of a far greater Heat than that in Dr. Boerhaave's Chemiſtry.
which on the Surface of our Earth LIKE QUANTITIEs, in Algebra,
would very eaſily turn them into are, ſuch as are expreſſed by the
Vapours and Fumes. And the ſame Letters equally repeated in
fame great Weight may condenſe each Quantity. Thus, 5 b and 4 b,
thoſe Vapours and Exhalations as and off and 2! f, are like Quan-
ſoon as they ſhall at any time he tities ; 5 b, and 4 b6, and 10 ff, and
gin to aſcend from the Sun, and afff, are unlike ones ; becauſe the
inake them preſently fall back 2- Quantities have not every where
gain into him, and by that Action tha fame Dimenſions, nor are the
increaſe his Heat much after the Letters equally repeated.
manner that in our Earth the Air
Like SIGNS, in Algebra, are
increaſes the Heat of a Culinary when both are Affirmative, or both
Fire. And the ſame Weight may Negative ; but if one be Affirma-
hinder the Globe of the Suo from tive, and the other Negative, they
being diminiſhed, unleſs by the E- are 'Unlike Signs. Thus, + 30 ,
miſſion of Light, and a very ſmall and + 2d, have Like Signs, but
quantity of Vapours and Exhala-
- 3ff, and +ff, have Unlike
Sigos.
2. Do not ſeveral ſorts of Rays
LIKI FIGURES, See Similar Fi-
make Vibrations of ſeveral Bigneſ-
gures.
ſes, which, according to their Big-
neſſes, excite Senſations of ſeveral
LIKE FIGURES," are in the du-
Colours, much after the ſame man-
plicate Ratio of their Homologous
Sides.
ner that the Vibrations of the Air,
according to their ſeveral Bigneſſes,
LIKE ARCHES of a Circle, are
excite Senſations of ſeveral Sounds ſuch as contain an equal Number
And particularly, do not the moſt of Degrees.
refrangible Rays excite the ſhorteſt Like SOLID FIGURES, are to
Vibrations for making a Senſation one another in the duplicate Ratio
of deep Violet the leaſt refrangible, of their Homologous Sides.
the largeſt for making a Senſation LIMB, ſignifies the uttermoft
of a deep Red, and the ſeveral in- Border or graduated Edge of an A-
termediate forts of Rays, Vibrati- ſtrolable, Quadrant, or the like Ma-
ons of feveral intermediate Bigneftes thematical Initrument; or the Cir-
cumference
tions.
LIN
L IN
cumference of the Primitive Arch Plane of a great Circle perpendicu.
in any Projection of the Sphere in lar to the Plane of the Projection,
Plano : Alſo the outermoſt Border and that oblique Circle which is
of the Sun's or Moon's Diſk, in an projected, interfects the Plane of the
Eclipſe of either Luminary Projection : Or it is the common
LIMBERS, in Gunnery, are a Section of a Plane paſſing thro' the
kind of Train joined to the Carriage Eye's Point, and thro' the Centre
of a Cannon upon a March ; it is of the Primitive, and at Right An-
compoſed of two Shafts, wide e gles to any oblique Circle which is
nough to receive a Horſe between to be projected, and in which the
them, (which Horſe is called the Centre and Pole of ſuch a Circle
Fillet-Horſe.) Theſe Shafts are
will be found.
joined by two Bars of Wood, and LINE of Direction of the Earth's
a Bolt of Iron at the End, and have Axis, in the Pythagorean Syſtem of
a Pair of ſmall Wheels. On the Aſtronomy, is the Line connecting
Axel-Tree riſes a ſtrong Iron Spike, the two Poles of the Ecliptic, and
on which the Train of the Carriage of the Equator, when they are
is put upon a March : But when a pro jected on the Plane of the fore
Gün is on Action, theſe Limbers mer.
are run out behind her.
Line of the Section, in Perſpec-
LIMIT of a Planet, is the greateſt tive, is the Interſectionor Contact of
Heliocentrick Latitude. Which fee. the plain to be projected with the
LIMITED PROBLEM, fignifies a Glafs or Diaphanous Plane.
Problem that hath but one, or a
Line of Lines, on the Sector, is
determined Number of Solutions ; a Scale of equal Parts on each Leg,
as to make a Circle paſs thro' three and running from the Centre. This
Points given, not lying in a Right is divided into 100 equal Parts, and
Line, to deſcribe an Equilateral ſometimes into more, when the In-
Triangle on a Líne given, &c. ſtrument is large.
LINCH-Pins, are thoſe Pins that Line of Numbers. See Gunter's
keep on the Carriage of a Piece of Line.
Ordnance.
LINE, in Fortification, is that
LINE, a Line in Geometry, is a which is drawn from one point to
Quantity extended in Length only, another, in delineating a Plane up-
and is ſuppoſed to have no Breadth on Paper : But in the field it is
or Thickneſs. It is made by the fometiines taken for a Ditch bounded
Motion of a Point.
with its Parapet, and ſometimes for
Line is alſo the 12th Part of an a Row of Gabions, or Sacks of
Inch.
Earth, extended in length on the
LINE of True 7 of a Planet, is a gainſt the Enemy's Fire
. Thus they
Ground, to ſerve as a Shelter a.
Place
Apparent
right
ſay, when the Trenches were car-
LINE The Earth's Centre ried on within
thro'
30 Paces of the
Glacis, we drew two Lines, one
the
from
on the Right Hand, and the o-
Planet, and continued as far as the ther on the Left, for a Place of
fixed Stars.
Arms.
LINE of Meaſures, in the Stereo Line CAPITAL, is that which
graphic Projection of the Sphere in is drawn from the Angle of the
Planı, is that Line in which the Gorge to the Angle of the Baſtion.
LINB
drawn Eye of the Spec-s the
1
t
LIN
LĨ N
Line CentrAL, is that which tinued Trench, with which a Cir-
is drawn from the Angle of the cumvallation or Contravallation is
Centre, to that of the Baſtion. ſurrounded, and which maintains
LINE of Defence, is that which a Communication with all its Forts,
repreſents the Courſe of the Buliet Redoubts, and Tenables.
of any ſort of Fire-Arms, more eſpe LINE of the Baſe, is a right
cially of a Muſquet-Ball, according Line which joins the Points of the
to the Situation which it ought to two neareſt Baſtions.
have to defend the Face of the Ba-. To line a Work, is to ſtrengthen
ftion.
a Rampart with a firm Wall, or to
Line of Defence Fixed or Fichant, encompaſs a Parapet or Moat with
is that which is drawn from the a good Turff, &c.
Angle of the Curtain, to the flank LINEA APSIDUM, or the Line of
ed Angles of the oppoſite Baſtion; the Apfes, in the old Aftronomy, is
nevertheleſs without touching the a Line paffing through the Center
Face of the Baſtion. This muſt ne of the World, and of the Excen-
ver exceed 800 Feet, which they tric; and whoſe two Ends are, one
reckon the Diſtance a Muſquet-Ball, the Apogæum, the other the Peri-
will do Execution.
gæum of the Planet That Part of
LINE of Defence Razant, is that this Line which lies between the
which being drawn from a certain Center of the World and that of
Point of its Curtain, razech the the Excentric, is called the Excen-
Face of the oppoſite Baſtion. This tricity.
is called alſo the Line of Defence Line of greateſt or leaſi Longitude
Stringent or Flanking.
of a Planet, is that part of the Linea
LINE of Approach, or of Attack, Aplidum reaching from the Center
fignifies the Work which the Ben of the World to the Apogæum, or
ſiegers carry on under Covert, to Perigaum of the Planet.
gain the Moat, and the Body of the LINE of mean Longitude, is one
Place.
drawn through the Centre of the
LINE of Circumvallation, is a World at Right Angles, to the Linea
Line or Trench cut by the Beſiegers Apſidum, and is there a
new Dia-
within Cannon-ſhot of the Place, meter of the Excentric, or Different;
which rangeth round their Camp, and its extreme Points are called
and ſecures its Quarters against the the mean Longitude.
Relief of the Beſieged.
Line of the mean Motion of the
LINE of Contravallation, is a Sun, in the old Aftronomy, is a
Ditch bordered with a Parapet, Right Line drawn from the Center
which ſerves to cover the Beſiegers of the World as far as to the Zodiac
on the Side of the Place, and to of the Primum Mobile ; and parallel
ſtop the Salleys of the Garriſon. to a Right Line drawn from the
LINES within fide, are the Moats Center of the Excentric, to the
towards the Place, to prevent the Center of the Sun ; which latter
like Salleys.
Line they call alſo the Line of the
LINES without fide, are the mean Motion of the Sun in the Ex-
Moats towards the field, to hinder centric, to diſtinguiſh it from the
Relief.
former; which is the Line of mean
Lines of Communication, are thoſe Motion in the Zodiac of the Pri-
that run from one Work to another.
mum Mobile.
But the Line of Communication, LINE of the Sun's true Motion, is
more eſpecially fo called, is a con a Line drawn from the Centre of
the
1
7
L IN
L IN
7
the World to the Center of the Sun, Excentric to the Center of the
and produced as far as the Zodiac Planet.
of the Primum Mobile,
LINE of the Apogæum of a Pla-
Line HORIZONTAL, is a right net, in the Old Aftronomy, is a right
Line parallel to the Horizon. Line drawn from the Centre of the
1. In Dialling, it is the common World, through the Point of the
Section of the Horizon and the Dial- Apogæum; as far as the Zodiac of
Plane.
the Primum Mobile,
2. In Perſpective, it is the com Line of the Nodes of a Planet,
mon Section of the Horizontal in the New Aſtronomy, is a Right
Plane, and that of the Draught.or
Line drawn from the Planet to che
Repreſentation, and which paffes Sun, being the common Interſec-
thro' the principal Point.
tion of the Plane of the Planei's Or-
LINE GEOMETRICAL, in Per- bit with that of the Ecliptic.
fpective, is a Right Line drawn any LINE EQUINOCTIAL, in Dial-
how on the Geometrical Plane. ling, is the common Interſection of
LINE TERRESTRIAL, in Per- the Equinoctial, and the Plane of
fpective, is a Right Line, wherein the Dial.
tbe Geometrical Plane and that of Lined Moaf, a Term in Forti-
the Picture or Draught interſect one fication. See Moat.
another.
LINĖ AR Numbers, are ſuch
LINE of the Front, in Perſpective, as have relation to Length only ;
is the common Section of the verti as (v.gr.) ſuch as repreſent onė
cal Plane and of the Draught. Side of a plain Figure ; and if the
Line of Station, in Perſpective, plain Figure be a Square, the Linear
according to fome Writers,' is the Number is called a Root.
common Section of the Vertical and LINEAR PROBLEM, in Mathe-
Geometrical Planes. Others, as matics, is ſuch an one as can be
Lumy, mean by it the perpendicular folved geometrically by the Inter-
Height of the Eye above the Geo- ſection of two Right Lines. This
metrical Plane. Others, a Line on is called a Simple Problem, and is
that Plane, and perpendicular to capable but of one Solution.
the Line exprefling the Height of LINE.SUBSTYLAR, is that Line
the Eye.
on which the Style or Cock of a
LINE OBJECTIVE, in Perſpec- Dial is erected, and is the Repre-
tive, is the Line of an Object, from fentation of ſuch an Hour-Circle as
whence the Appearance is ſought is perpendicular to the Plane of that
for in the Draught or Picture. Dial.
Line of Gravitation of any heavy
Line SYNODICAL, in reference
Body, is a Line drawn through its to fume Theories of the Moon, is a
Center of Gravity, and according Right Line ſuppoſed to be drawn
to which it tends downwards. through the Centres of the Earth
LINE of Direction of any Body and the Sun ; and if it be produced
in Motion, is that according to quite through the Orbits, 'tis called
which it moves, or which directs the
and determines its Motion.
Line of the true Syzygies : But a
Line of the ſwifteſt Deſcent of Right Line imagined to paſs through
a heavy Body, is the Cycloid. the Earth's Centre, and the mean
Line of the Anomaly of a Planet, Place of the Sun is called the
in the Ptolemaic Syſtem, is a Right Line of the mean Syzygies.
· Line drawn from the Center of the Lines of Chords, Sinés, Tangents,
Secants,
.
1
:
one of them, viz. AP (which may
LOC.
LOC
Secants, Verfed Sines, &c. See when the point fought is in the
Scale.
Circumference of a Circle ; 'Solid,
LINSTOCK, is a ſhort Staff of when the Point required is in the
Wood about three Foot long, hav- Circumference of a Conic Section ;
ing at one End a Piece, of Iron di or laſtly Surſolid, when the point is
vided into two Branches, .each of in the Perimeter of a Line of an
which hath a Notch to hold a Piece higher kind.
of Match, and a Screw to faften it LOCK-SPIT, a Term in Fortifica-
there. The other End of the Staff tion, ſignifying the ſmall Cut or
is thod alſo with Iron, and pointed Trench made with a Spade, to mark
to stick into the Ground ; 'tis uſed out the firſt Lines of any Work that
by the Gunners in firing Cannon. is to be made.
LIQUIDS, are ſuch Bodies as LOCKING-WHEL. See Count-
have all the Properties of Fluidity, Wheel, a Term in Watch-work.
(ſee that Word ;) and withal, have Locus. If there be two un-
their Particles ſo formed, figured, known and indeterminate Right
or diſpoſed, that they do adhere to Lines AP, PM, making any An-
the Surfaces of ſuch Bodies as are gle (APM) with each other at
immerſed in them, which we call pleaſure ; and if the Beginning of
Wetting ; and this Property of Li-
quid Bodies is ſometimes called H4 be called x) be fixed in the Point A,
midity or Moiſture.
and the faid AP indefinitely ex-
List, in Architecture, is a little tends itſelf along a right Line given
ſquare Moulding, ſerving to crown in Poſition; and the other PM
or accompany a larger, or on occa- which may be called y, continually
fion to ſeparate the Flutings of a
Columnn. It is ſometimes called
Fillet, and ſometimes Square.
M
Lister, a ſmall Band, or a kind
E
of a Rule in the Mouldings of Ar-
M
chitecture : Alſo the Space between
the Channellings of Pillars.
DP А, B
LITERAL Algebra. See All
gebra.
LIZIERE, a Term in Fortifica-
tion, being the ſame with Berm.
G
M
Which ſee.
LOCAL PROBLEM, in Mathe-
matics, is ſuch an one as is capable
M.
of an infinite Number of different
M
M
Solutions : So that the point which
is to reſolve the Problein, may be
indifferently taken within a certain
Extent. As ſuppoſe any where, in
DUP PAPP P
B
ſuch a Line, within ſuch a plain
Figure, &c. which is called a Geo-
metric Lucus, and the Problem is
ſaid to be a local or indetermined
one, and this local Problem may
be either fimple, when the Point
fought is in a Right Line ; Plane,
alters
+
e
:
M GM
Ż OC
LÔ C
-
alters its Pofition, and is always pa- cumference, then by the Nature of
rallel to itfelf: (that is, if all the the Circle, we Thall have always
PM's be parallel to one another.) PM (y) = DP X PB (aa-- *x,)
Then if there be an Equation, ſuppoſing BD-the Diameter of the
wherein are both the unknown Circle. Therefore the Locus of all
Quantities x and
,
mix'd with the Points M is the Circumference
known ones, which expreſſes the of a Circte.
Relation of every AP (x) to its 3. If all the PM's be ſuppoſed to
Correſpondent PM (y), the Curve tend from one fide of the Line A B,
pafling thro' the Extremities of all
as towards Q; and then they be
the Values of y, that is, through ſuppoſed to tend from the other ſide
all the Points M, is' called in ge- of the ſaid Line, as towards G;
neral a Geometric Locus ; and in then it muſt be obſerved, that their
particular the Locus of that Equa- Values from Poſitives (which they
jion.
are ſuppoſed to be when tending to-
1. For Example: Let us ſuppoſe, wards Q,) will become Negative,
6 x and fo ſhall we have PM
- ya
(Fig. 1.) that the Equation y =
Moreover, if the Point P be ſup-
expreſſes always the Relation of the poſed to fall from A towards B, and
Line AP (*) to PM (v), which afterwards the contrary way, as
make any Angle APM at pleafure from A towards D; then all the
with one another : In the Line A P A P's on this ſide A will become
affume AB=a, and from B draw negative, and conſequently we have
AP X.
BE=b, parallel to PM, and on
And a geometric Lo-
cus muſt paſs through the Extremi.
the ſame fide; then the indefinite
Line AM is called in general a
ties of all the Values (as well poſi-
Geométric Locus ; and in particular,
tive as negative) of one of the un-
known Quantities y, which anſwer
bx
the Locus of the Equation y=
to the Values both poſitive and ne-
gative of the other unknown Quan-
For if the Right Line MP be drawn tity x. Therefore, if the Right
from any one of its Points M pa- Line QAG be drawn parallel tó
rallel to BE, the ſimilar Triangles PM, a geometric Locus may be
ABE, A PM, will give always found in the four Angles B Ali
this Proportion, viz. A B (a) : BE BAG, GAD, DAQ; as in the
bx ſecond Example (Fig. 2.) or only in
(b) :: AP (x) : PM (y) = ſome of the Angles, as in the firſt
Cafe (Fig. 1.) For in the ſecond
And therefore the Right Line AE Example, ſuppoſe that AP be = x,
is the Locus of all the points M.
and PM = y, the Point M being
Moreover, (Fig. 2.) if yy= a a taken firſt in the Quadrant Q B;
x x expreſſes the relation of AP then if the Point M be taken
to PM, and the Angle APM be a afterwards in the Quadrant G B, we
Right Angle, then the Circumfe. ſhall have AP=x, and PM =
rence of a Circle, whoſe Radius is
--y; if M be taken on DG, we
the Right Line A B = a taken in ſhall have AP=x, and PM
AP, is called in general a Geometric
y : And finally, if M be taken
Locus, and, in particular, the Locus
on DQ, we ſhall have AP
- X,
of the Equation yy = aa
and PM y.
And in all theſe
For if the Perpendicular MP (y) be Cafes (by the Nature of the Circle)
drawn from any Point M of the Cir. there will come out the fame Equa-
tion
1
>
6 x
1
,لا
a
fore ya
2
BAQ
1
LOC
toc
tion yg=aa--** ; becauſe the when either of the unknown Quan-
Squares of ty, and to x, are the tities x apd y, or both of them to-
fame in all Caſes, viz. yy and **. gether, are found therein of diffe-
Moreover, in the firſt Example, if rent Dimenfions : So in the firſt
you maké AP=x, and PM=y, in Degree, if this Equation be pro-
the firſt Point M taken (on the ſame
fide as E) upon A E (produced to- poſed, y too, the Terms
wards A) in the Angle GAD, we
fhall have AP-
bx
*, and PM–
y
c will be different.
y; and ſince the Triangles ABE,
APM, are ſimilar, the following Moreover, in the ſecond Degree, if
Proportion will be formed, viz.
A B (a) : BE (6) :: AP (-x): you ſuppoſe yyt
2 bxy
- 2cy
bx
PM (y) =-
; and there-
fxx
+8*+hx -- hh +11
b.x
Which is the ſame
2b xy
S0; then the Terms yy,
Equation as was formed, by ſuppo.
fing the Point M to fall in the Angle
f x x
- 2 cy's
, g* to h *
4. The ancient Geometricians
did call Plain Loci, ſuch that are
hh + ll, ſhall be every one of
them different.
Right Lines or Circles; and Solid
Loci, thoſe that are Parabola's, El-
6. When the unknown Quantities
lipſes, or Hyperbola's ; and Surd-
X and y, have but one Dimenſion
Solid Loci, ſuch that are Curves of in a given Equation, and their Pro-
a fuperior Gender than Conic Sec- duct x y is not in the ſame, then the
tions. But the Moderns do difin-
Locus of that Equation will be al-
guiſh Geometric Loci into different ways a ſtraight Line ; and it may
Kinds or Degrees. For under the be reduced to ſome one of the four
firſt Degree are comprehended all following Formula's ; 1. y=
bx
**.
the Loci, wherein the unknown
Quantities x and y are found in E-
bx
6x
quations only of one Dimenſion; 2. y
+. 3. y
under the ſecond, all thoſe wherein
bx
thoſe unknown Quantities have two
4. C
Dimenſions, and ſo on; where you
may obſerve, that there muſt be no 6. When any Equation of two
Rectangle or Product of the un Dimenſions is given, and it is re-
known Quantities x and y in the quired to know which of the Conic
Equations for the Loci of the firſt Sections will be the Locus of it :
Kind or Degree ; and in the Equa Bring over all the Terms of the
tions for the ſecond, thoſe Quanti- Equation to one ſide ; ſo that one
ties muſt form a Product, as Xy of Member thereof be o, then there
no more than two Dimenſions; and may happen two Caſes :
in Equations for the third, a Pro Cafe 1. When the Plane xy is
duct * *y, or tyy, of three Di not in the given Equation. 1. If
menſions, &c.
there be but one of the Squares yy
5. The Terms of the Equation or x * therein, then the Locus
of a Locus are ſaid to be different, will be a Parabola. 2. If both the
Y
Squares
a
1
1-
a
1
LOC
L O C
A
meters.
LOCUS
Squares yy and x x are found therein the Loci of Equations of three Di-
with the ſame Signs, then the Lo- menſions. Euclid, Apollonius, Ari-
cus will be an Ellipfis or a Circle. fæus, Fermat, Viviani, have alſo
3. If the ſaid two Squares are found wrote of the Loci.
therein with different Signs, then LOCUS AD LINEAM, is when the
the Locus thereof will be an Hy- Point that ſatisfies the Problem, is
perbola, or the oppoſite Sections re- found in a Line, whether Right or
garding their Diameters.
Curve, and that by reaſon of the
Cafe 2. When the Plane xy hap- Want of one Condition, only to
pens to be in a given Equation. 1. render the Problem determinate al-
İf neither of the Squares yy and together.
xx, or but one of them, are found LOCUS AD SOLIDUM, is when
in the Equation, then the Locus three Conditions are wanting to the
will be an Hyperbola between its Determination of the Point fought,
Afymptotes. 2. If the Squares yy and ſo it will be found in a Solid;
and xx are found therein with dif- and this may be included either
ferent Signs, then the Locus ſhall under a Plane, Curve, or mix'd Su-
be an Hyperbola regarding its Dia- perficies, and thoſe either determi-
3. If the ſaid two Squares nate or indefinitely extended.
have the fame Signs, the Square yy
AD SÚPERFICIEM, is
muſt be freed from Fractions, and when there being two Conditions
then the Locus ſhall be a Parabola, wanting to determine any point that
when the Square of half the Frac- fatisfies any Problem, that Point
tion multiplying xo be equal to the may be taken throughout the Ex-
Fraction multiplying x*; an El- tenſion of ſome Superficies, whether
lipſis or Circle, when the ſame is Plane or Curve.
lels; and finally, an Hyperbola, or As the Locus of an Equation,
two oppoſite ones, regarding their wherein there are two, variable
Diameters, when the fame is Quantities, is a Right Line or a
greater.
Curve Line ; fo the Locus of an E-
The beſt way of finding the Loci quation containing three unknown
of Equations of two Dimenſions, is Quantities, will
Quantities, will be always
by extracting the Root after the Superficies. As the Equation **
nianner of Des Cartes. See his tyytzz=aa, repreſents the
Geometry; as alſo Sterling's Illuftra- Superficies of a Sphere, whoſe Ra-
tio Linearum tertii Ordinis. The dius is a, wherein x flows along a
Doctrine of theſe Loci is very well Diameter of a great Circle from the
handled too by De Witt, in his Centre,
Centre, y flows at right Angles upon
Elementa Curvarum ; Mr. Craige, the Plane of the Circle from the
in his Tractatus de Figurarum Cur Extremity of x, and z is a Perfen-
vilinearum Quadraturis Eg Locis dicular from the Extremity of
Geometricis; and the Marquis de the superficies of the Sphere. In
l'Hoſpital; in his Analytic Treatiſe
bbxx
like minner
of Conic Sections, have treated of this
= yy + z z,
Subject; and Bartholomæus Intieri, repreſents the Curve Superficies of
in his Aditus ad nova Arcana Geo- a right Cone, where x Rows along
metrica detegenda, has ſewn how the Axis from the Vertex, y
to find the Loci of Equations of the pendicular to x, z to y, a equal to
higher Orders : So alſo has Mr. the Axis, and b equal to the Radius
Sterling, in his Treatiſe aforeſaid, of the Baſe.
of the Baſe. See ſomething of this
given an infance or two of finding in a French Treatiſe entitled Re-
cherches
a
3
y, to
аа
is per-
:
to
L OG
LOG
cherches ſur les courbes a double broken) of a given Number, yet
Courbure.
ſuch Indexes or Exponents, that the
LOdgment of an Attack, is a ſeveral Powers or Roots they expreſs
Work caft up by the Beſiegers, dur- are the natural Numbers. 1, 2, 3, 4,
ing their Approaches in a dange-5, &c. 10 or 100000, &c.
rous Poſt, where it is abſolutely ne. (as if the given Number be 10, and
ceſſary to ſecure themſelves againſt its Index be aſſumed 1,0000000,
the Enemy's Fire ; as in a Cover'd- then the 0,0000000 Root of 10,
Way, in a Breach, in the bottom of which is i, will be the Logarithni
a Moat, or elſewhere. This Lodg- of 1; the 0,301036 Root of 10,
ment conſiſts of all the Materials which is 2, will be the Logarithm
that are capable to make Reſiſtance; of 2. the 0,477121 Root of 10,
viz. Barrels and Gabions of Earth, which is 3, will be the Logarithm
Paliſadoes, Woolpacks, Mantelets, of 3; the 0,61'2060 Root of 10,
Faggots, &c.
the Logarithm of 4. the 1,041393
LOG-Line, is one to which the Power of 10 the Logarithm of 11 ;
Log is faſten'd, which is wound the 1,079181 Power of 10 the Low
about a Reel for that purpoſe, fixed garithm of 12 ; and ſo on,) being
in the Gallery of the ship. This chiefly contrived for the Eafe and
Line, for about to Fathom from Expedition of performing Arithme-
the Log, hath, or ought to have tical Operations in large Numbers,
no Knots or Diviſions ; becauſe pointing out the Product of two
ſo much ſhould be allowed for the Numbers by the Addition of their
Log's being clear out of the Eddy Logarithms, the Quotient of their
of the Ship’s Wake before they Diviſion by the Subitraction of their
turn up the Glaſs; but then the Logarithms, and the Powers and
Knots or Diviſions begin, and Roots by the Doubling, Tripling,
ought to be at leaft 50 Foot from &c. Halving, Trifecting, &c. the
one another ; tho' the common er: Logarithms; and founded upon this
roneous Practice at Sea is to have conſideration, that if there be any
them but ſeven Fathom, or 42 Foot Row of geometrical proportional
diſtance.
Numbers, as 1, 2, 4, 8, 16, 32, 64,
Tho' this at beſt be but a preca- 128, 256, &c. or 1, 10, 100, 1000,
rious way, 'tis however the moſt 10000, &*c. and as many Arithme-
exact of any in uſe, and much bet- tical Progreſfional Numbers adapted
ter than that of the Spaniards and to them, or ſet over them, begin-
Portugueſe, who gueſſed at the Ship's
0,1, 2, 3, 4, 5,
Way by the running of the Froth ning with o, thus,
1,2,4,8, 16, 32,
or Water by the Ship’s fide; or
than that of the Dutch, who uſed to
6, 7, &c.
0, 1, 2, 32
1, 10, 100, 1000,
heave over a Chip into the Sea, and
ſo to number how many Paces they 4,
Eoc.
then will the Sum of
could walk on the Deck, while the 10000, &c.
Chip fwam or paſſed between any any two of thoſe Arithmetical Pro-
two Marks or Bolt-Heads on the greſſionals added together be that
fide.
Arithmetical Progreffional, which
LOGARITHMS, are the Indexes anſwers to, or ftuds over che Geo-
or Exponents, (moſtly whole Num- metrical Progreílnai, being the
bers and decimal Fractions, con- Product of the Multiplication of
fiſting of 7 Places of Figures at thoſe two Geometrical Prozreiliunals
leaft) of the Powers or Roots (chiefly under which the tivo aífum'd A-
Y 2' fithmecical
or
64, 128, c.
LOG
L OG
rithmetical Progreſſionals ſtand ; of the Nature of a Ratio, and denies
and if thoſe Arithinedical Progreſ- it to be any manner of Quantity.)
fionals be ſubtracted from each o- thoſe Gentlemen's Definitions muſt
ther, the Remainder will be the A- be either Nonſenſe, or very near it.
rithmetical Progreſſional ſtanding The firſt Makers of the Loga-
over that Geometrical Progreſſional rithms had a very laborious and
which is the Quotient of the Divi- difficult Talk to perform ; they firſt
fion of the two Geometrical Progreſ- made choice of their Scale or Sy-
fionals belonging to the two firſt ſtem of Logarithms, that is, what
aſſumed Arithmetical Progreſſionals, Sett of Arithmetical Progreſſionals
and the Double, Triple, &c. of any ſhould anſwer to ſuch a set of Geo-
one of the Arithmetical Progreſſio- metrical ones, for this is entirely
nals, will be the Arithmetical Pro- arbitrary; and for ſome Reaſons
greſſional ſtanding over the Square, the Decuple Geometrical Progreſ-
Cube, c. of that Geometrical fionals, 1, 10, 100, 1000, 10000,
Progreſſional, which the aſſum'd A. &C. and the Arithmetical one,
rithmetical Progreſional ſtands over ; 0, 1, 2, 3, 4, &c. or 0,000000;
as well as the Ž, j, &c. of that A. 1,000000; 2,000000 ; 3,000000;
rithmetical Progreſſional, will be the 4,000COO, &c. was thought moſt
Geometrical Progreſſional anſwering convenient. After this they were to
to the Square Root, Cube Root, get the Logarithms of all the inter-
&c. of the Arithmetical Progreſ- mediate Numbers between 1 and
fional over it ; and from hence a-
10, 10 and 100, 100 and 1000,
riſes the following common, tho' 1000 and 10900, &c. Hic Labor
lame and imperfect Definition of hoc Opus fuit. But firſt of all they
Logarithms, viz. that they are ſo ma were to get the Logarithms of the
ny
Arithmetical Progreſionals an- prime Numbers 3, 5, 7, 11, 13, 17,
Jwering to the ſame Number of Geo. 19, 23, &c. and when theſe were
metrical ones.
Whereas if any one once had, it was eaſy to get thoſe
looks into the Tables of Logarithms, of the Compound Numbers made
he will find that theſe do not at all up of the prime ones, by the Ad.
run on in an Arithmetical Progreſ- dition or Subtraction of their Lo-
fion, nor the Numbers they anſwer garithms.
to in a Geometrcial one. Theſe laft In order to this, they found a
being themſelves Arithmetical Pro- mean Proportional between 1 and
10, and its Logarithm will be { that
Dr. Wallis, in his Hiſtory of Al- of 10; and ſo given, then they a-
gebra, calls Logarithms the Indexes gain found a mean Proportional be
of the Ratio's of Numbers to one tween the Number firit found and
another.- Dr. Halley, in the Phi- Unity, which Mean will be nearer
loſophical Tranſactions, Nº 216. ſays, to i than that before, and its Loga-
they are the Exponents of the Ra- rithm will be of the former Loga-
tio's of Unity to Numbers.-- So rithm, or of that of 10 ; and
alſo Mr. Cotes, in his Harmonia having in this manner continually
Menſurarum, ſays, they are the found a mean Proportional between
Numerical Meaſures of Ratio's; but i and the laſt mean, and biffected
all theſe convey but a very confuſed the Logarithms, they at length,
Notion of Logarithms. Nay, if what after finding 54 ſuch means, came
the great Dr. Barrow ſays, in one to a Number 1,000000000000000
of his Mathematical Lectures, be 1278191493200323442 ſo near to
admitted for Truth, (where he treats i as not to differ from it ſo much
t
greſſionals.
as
samantha
1
LOG
LOG
as totoo. 0.000006 Part, and garithm of 3. So alſo having found
found its Logarithm to be 0,0000 the Logarithms of 13, 17, and 19,
0000000000005551115123125782 and alſo of 23 and 29, they did
702, and, 0000000000000001 278 eaſily get thoſe of all the Numbers
1914932003235 to be the Difference between 10 and 30, by Addition
whereby i exceeds the Number of and Subtraction only ; and ſo hav.
Roots or mean Proportionals found ing found the Logarithms of other
by Extraction, and then by means prime Numbers, they got thoſe of
of theſe Numbers they found the the Numbers compounded of them.
Logarithms of any other Numbers But ſince the way above hinted
whatſoever, and that after the fol- at, for finding the Logarithms of
lowing manner: Between a given the prime Numbers is ſo intolerably
Number whoſe Logarithm is want- laborious and troubleſome, the more
ed and 1, they found a mean Pro- ſkilful Mathematicians that came
portional as above, until at length after the firſt Inventors, employing
a Number (mix'd) be found, ſuch a their Thoughts about abbreviating
ſmall Matter above I, as to have i the thing, had a vaſtly more eaſy
and 15 Cyphers after it, which are and ſhort way offer'd to them from
followed by the ſame Number of the Contemplation and Menſura-
fignificant Figures ; then they faid, tion of hyperbolic Spaces contained
as the laſt Number mentioned above between the Portions of an A-
is to the mean Proportional thus fymptote, Right Lines perpendicu-
found, ſo is the Logarithm above, lar to it, and the Curve of the Hy-
viz. 0, 0000000000000000555111 perbola : For if ECN be an
5123125782702 to the Logarithm Hyperbola, and AD, AQ the A-
of the mean Proportional Number
ſuch a ſmall Matter exceeding 1, as
E
D
but now mentioned ; and this Lo-
garithm being as often doubled as
the Number of mean Proportionals
(form'd to get that Number) will
be the Logarithm of the given
Number.. And this was the me-
M
N
thod that Mr. Briggs took, to make
the Logarithms. But if they are A B P
to be made to only ſeven Places of
Figures, which are enough for com- fymptotes, and 'AB, AP, AQ, &c.
mon Uſe, they had only occaſion taken upon one of them be repre-
for to find 25 mean Proportionals, ſented by Numbers, and the Ordinates
or, which is the ſame thing, to ex BC, PM, QN, &c. be drawn from
th Root of io.
the ſeveral Points B, P, Q, ETC. to
Now having the Logarithms of 3, the Curve, then will the Quadri-
5, and 7, they eaſily got thoſe of line Spaces BCMP, PMNO, &c.
2, 4, 6, 8, and 9; for ſince 2, viz. Their Numerical Meaſures, be
the Logarithm of 2 will be the the Logarithms of the Quotients of
Difference of the Logarithms of 10 the Diviſion of AB by AP; AP by
and 5; the Logarithm of 4, will AQ,&c. Since when AB, AP,AQ,
be two times the Logarithm of 2 ; &care continual Proportionals, the
the Logarithm of 6, will be two ſaid Spaces are equal, as is demon-
times the Logarithm of 3 ; and the ftrated by ſeveral Writers concerning
Logarithm of 9, three times the Lo- Conic Sections. Amongit which,
fee
C
.
tract the 35557777
7
Y 3
2
L OG
L OG
ſee the Quadrature of the Circle, by N and C by a Right Line, which
Gregory St. Vincent; and the Mar- continue out Book-ways
quis de la Hoſpital's Conic Sections ; the Aſymptotes in R and S. Alſo
which likewiſe may be briefly de- join NM, MC, draw the Tangent
monſtrated thus : Join the Points TMV thro' M, and draw the
to meet
S
V
Z
*
M
N
R QT
P B A
the Right Line AMŽ cutting NS the Hyperbolic Segments NMN;
in Z. Now fince (by Suppofition) MCM equal. Wherefore at length
AP: AB :: AQ: AP; therefore the Trapezium Q NMP Seg-
(dividendo) AP-AB:AQ – AP:: ment NMN is = Trapezium
AB: AP. But ſince (by the Na- PMCB - Segment MCM; that
ture of the Curve) AB : AP :: is, the Hyperbolic Spaces QNMP,
PM : BC; therefore AP-AB: PMCB are equal.
AQ-AP :: PM : BC. Again This may be demonſtrated without
PM: ON :: BC :PM. and ſo conſidering any one Property of the
(componendo) PM +ON: BC+ Hyperbola, except that of the Rect-
P.M :: PM : BC. But lince (before) angles AB > BC, AP PM, AQ *
APAB : AQAP :: PM : BC; QN,&c. being equal to one another;
therefore AP-AB : AQ-AP :: for ſuppoſe BC to be a given Or-
PM+NQ : BC + PM. And ſo dinate, and let an infinite Number
PM+NQX QP= BCPM of Ordinates ab,. cd, PM, ef, gb,
x PB, and the half of the one e-
qual to the half of the other; that
is, the right - lin'd Trapeziums
QNMP, PMCB are equal.
Again, ſince (by Nat. Curve) TP-
AP and R Q = AB, and BC :
PM :: PM : ON (by Sup.) and
AP: AB :: BC PM; therefore
TP (AP): PM :: RQ (AB) : QN.
B
Wherefore the Triangles RN, A
ac Pry Q
TMP are ſimilar ; and ſo the
Right Line RS is parallel to the QN, &c. be drawn parallel to it
Tangent TV. Conſequently NC from the Points a, c, P, e, g, h, 0.
will be an Ordinate to the Diameter infinitely near to each other, and
AZ, and NZ=ZC, and the right- all deſcending in a continual geo-
lin'd Triangle NZM=ZCM, and metrical Progreflion; then will Ba,
duith
N
ас,
to 5
Den
2
X3
ita
1
2
+
3
1
TŐ
I
LOG
L OG
ac, cP, Pe, eg, &Q, & c. be con- +*5*, &c. and taking the Fluents,
tinual Geometrical Progreſſionals ; we ſhall have the Area AFDB
and ſo all the little Rectangles CB
# 3 8:4
x Ba, ab x ac, cd xcP, PM X
+
+
Pes efxeg, g h * g Q, &c. will &c. and the Area AFáb =*+
3 4 5
be equal to one another; and any
Number B M of them will be equal
+
to the ſame Number PN of them :
+
&c. and
3
4
that is, fince the hyperbolic Space
2x3
BN differs but by an infinitely ſmall the Sum bd DB = 2*+
Quantity from the Sum of all ſuch
little Rectangles, the hyperbolic ***+ **?+ 5*9, Sc. Now if
AB or Ab be
Space B CMP will be equal to the being = 0.9. and CB=1.1.
x Cb
hyperbolic Space PMNQ: Having putting this Value x the E-
by
thus ſhewn that theſe hyperbolic quations above, we ſhall have the
Spaces numerically expreſſed may Area bd DB=0.200670695462151
be taken for Logarithms, I think it for the Terms of the Series will
may not be amiſs to fhew a ſhort Spe- ftand as you ſee in this Table.
cimen from our great Sir Iſaac New-
ton, of the Method how to meaſure 0.2000000000000000ift
theſe Spaces, and conſequently how 6666666666666= 2d
the Logarithms may be conftructed. 40000000000= 3d
Term
Let CA=AF be = 1, and ABS
285714286=4th
Softhe
2222222=5th
Series
Ab=x; .then will tx be=BD,
181826th
1547th
and 5=bd; and putting theſe
I=8th j
0.2006706954621511.
Expreſſions into Series's, it will be
If the Parts Ad, and A D of this
Area be added ſeparately, and the
lefſer DA be taken from the greater
dA, we ſhall have Ad-AD=x?t
28
E c.
D
3
0. 0100503358535014. for the
SI
Terms reduced to Decimals will
ſtand thus :
С ВА В PGP
0.0100000000000000
5000000000CO
3333333333
itx=1-***-x3+x+~**,
25000000
Gc.and ==i+**********
1667
14
+*5, &c. and
x x
itx
0.0100503358535014
**?3-433-*5*, &c. and
Now if this Difference of the
Sit********** **** Areas be added to and ſubtracted
Y 4
from
74 4
go
+*
F
1
2
4
HQ
1
200000
܀
L OG
L OG
+
1
1.2
X 2
0.8
1.2
2X2
0.8
I
10 X 100
1000 :
= 13, and
from their Sum before found, half (that is, inſiſting upon the Parts of
theAggregateo.105360515.6578263 the Abſciſs 1.2, 0.8, and 1.27 0.9)
will be the greater Area Ad, and will be the Area A FHG when
half the Remainder 0.09531017980
43248 will be the leſſer Area AD.
CG is 2. Alſo ſince
By the fame Tables theſe Areas
AD and Ad will be obtain'd alſo =3. the Sum 1.0986122886681097
when AB= Ab are ſuppoſed to
be Too or CB=1.01, and cb= of the Areas belonging to and
0.8
0.99. if the Numbers are but duely
transferred to lower Places, as
2, will be the Area AFGH, when
CG=3. Again fince
0.0200000000000000
5.
6666666666
and 2x5=10; by a due Addi-
400000
tion of Areas will be obtain'd
28
1.6093379124341004 = AFHG,
Sum 0.0200006667066695=6D
when CG=S. and 2.032585092
9940457 = AFHG, when C'G
0.0001000000000000
10; and ſince 10x10=100;
and
and
50000000
3333
V 5*10*0.98 =7, and 10 X 1.1
1000 X 1.091
0.00010000 50003333=Ad-AD
ll, and
7 XII
Half the Aggregate o 0100503358 1000 X 0 998
535014 = Ad, and half the Re-
= 499; it is plain
mainder 0.0099503308531681 =
AD.
that the Area AFHG may be
And ſo putting AB=Abobo, found by the Compoſition of the
or CB =1.001, and C 6
Areas found before when CG=100,
there will be obtain'd Ad = 1000, or any other of the Numbers
0.00100050003335835, and AD= above mentioned ; and all theſe A-
0.00099950013330835.
reas are the Hyperbolic Logarithms
After the ſame manner, if AB= of thoſe ſeveral Numbers.
Ab be 0.2, or 0.02, or 0.002 ;
Having thus obtained the Hyper.
theſe Areas will ariſe,
bolic Logarithms of the Numbers
Ad= 0.2231435513142097, and 10,0,98, 0.99, 1.01, 1.02, which
AD = 0.1823215576939546, or may be done in about an Hour or
Ad = 0.0202027073175194, and two's time. If the Logarithms of
AD = 0 1098026272961797, or
the four laſt of them be divided
Ad= 0.002002, and AD0001. by the Hyperbolic Logarithm
From theſe Areas thus found, o-
2.3025850929940457 of 1o, and
the Index 2 be added, or which is
thers may be eaſily had from Addi- the ſame thing, if it be multiplied by
tion and Subtraction only. For its Reciprocal 0.4342944819032518,
we ſhall have the true Tabular Lo-
Gnce
08
= 2, the Sum of garithms of 89, 99, 100, 101, 102.
the Areas 0.6931471805599453 be- Intervals, and then we ſhall have
Theſe are to be interpolated by ten
longing to the Ratio's and
the Logarithms of all the Numbers
0.8
0.9 between 980 and 1020 ; and all be-
tween
2
and Cb0.999,
1.2
1.2
Х
09
I 2
1.2
1
+
I 2123,
TO
4-
+
2
100I
102
6-17;
= 23
29;
+
72
2n
+ +
3
x3 3
exs 5
= 43;
+
21
n
LOG
L OG
tween 980 and 1090, being again dx d x3
interpolated by ten Intervals, the 2
&c. to the Loga-
Table will be as it were conſtructed; rithm of the leffer Number ; for if
then from theſe we are to get
the
Logarithms of all the prime
Num- CG, CP, and the Ordinates psy
the Numbers are repreſented by Cp
bers and their Multiples leſs than
PQ be raiſed ; if n be wrote for
100, which may be done by Ad-
CG, and x for GP or Gp, the
dition and Subtraction only. For
Area ps QP or
** +
84 X 1020 V8 x 9963
= 2;
272
9945
x3
984
EG c. will be to the Area
=3;=5;
V 98
98=7; P=N;
=7;=uHG, as the Difference between
the Logarithms of the extreme
988
Numbers, or 2 d, is to the Diffe-
13 ;
7X1T
6 4x13 rence between the Logarithms of
9936
986 the lefſer, and of the middle onet
- 19;
;
which therefore will be
16 X 27
2x17
dx
dx²
dx3
992
999
c.
31;
37 ;
32
27
3n
984
989
987
EC.
24
41 ;
23
32
5%
9911
9991
dx da3
= 53;
=dx +
+ E°C. the
17
13X13
9882
9949
dx
59;
=61;
two firſt Terms d+ of this
2 X 81
3 * 49
994 9928 Series, being ſufficient for the Con-
- 67 ;
71 ;
14
= 73 ; ftruction of a Canon of Logarithms,
8x17
even to 14 Places of Figures, pro-
9954
= 79 ;
=83;
vided the Number whoſe Logarithm
is to be found be leſs than 1000,
9894
which cannot be very troubleſome,
98;
97 ;
and thus
6x17
becauſe x is either 1 or 2, yet it is
having the Logarithms of all the not neceſſary to interpolate all the
Numbers leſs than 100, you
Places by help of this Rule, ſince
have
nothing to do but interpolate them the Logarithms of Numbers which
ſeveral times thro' ten Intervals,
are produced by the Multiplication
Now the void Places may be filled or Diviſion of the Number laſt
up by the following Theorem. Let found, may be obtain'd by the
n be a Number, whoſe Logarithm had before by the Addition or Sub-
Numbers whoſe Logarithms were
is wanted ; let x be the Difference
between that and the two neareft traction of their Logarithms. More-
over, by the Difference of their Lo.
Numbers, equally diſtant on each
fide, whoſe Logarithms are already garithms and by their ſecond and
fide, whoſe Logarithms are already third Differences, if neceſſary, the
void Places may be ſupplied more
rence of their Logarithms; then
the requird Logarithm of the Num- expeditiouſly; the Rule aforegoing
ber n will be had by adding dt being to be applied only where the
Con.
= 47 i
2 12
I 2n
II X
21
996
9968
7 x 18
I 2
7x16
LOG
LOG
i
XX
2M
22
Zz
Continuation of ſome full Places is of the Number %. If the Number
wanted, in order to obtain theſe Dif. exceeds 1000, the firſt Term of
ferences.
7
By the ſame Method Rules may the Series, viz. is ſufficient to
4*
be found for the Intercalation of
Logarithms, when of three Num- get the Logarithm to 13 or 14 Pla-
bers the Logarithm of the leſſer and
ces of Figures, and the ſecond Term
of the middle Number are given, or
will give the logarithm to 20 Pla-
of the middle Number and the
ces; and if x be greater than 10000,
greater; and this altho' the Num- the firſt Term will exhibit the Los
bers ſhould not be in Arithmetical garithm to 18 Places of Figures.
Progreſſion. Alſo by purſuing the This Series is eaſily found out and
Steps of this Method, Rules may be deduced from the Confideration of
eaſily diſcovered for the Conſtruc- the Hyperbolic Spaces aforeſaid.
tion of the Tables of artificial Sines
Mr. Cotes, in his Harmon. Menſur.
and Tangents, without the Help of at the Beginning ſays, if the Sum of
the natural Tables. Thus far the
two Numbers be į and their Dif.
great Newton, who ſays, in one of ference x, and you ſuppoſe M=
bis Letters to Mr. Leibnitz, that 0,434294481903,&c. viz. theValue
he was ſo much delighted with the of the Subtangent of the Logarith-
Conſtruction of Logarithms, at
mic Curve, to which Briggs's Loga-
his firſt ſetting out in thoſe Studies, rithms are adapted, and you take
that be was aſhamed to tell to how
many Places of Figures he had car
-=A, A B, B
=C
ried them at that time; and this
was before the Year 1666, becauſe, C ED, &c. then will the Lo-
(ſays he) the Plague made him lay
aſide thoſe Studies, and think of o- garithm of the Quotient of the Di-
ther things.
viſion of the grealer by the leſs be
Dr. Keil, in his Little Traft of = A + B + C + D, &c. So
Logarithms, at the end of his Como that to find the Logarithms of 'the
mandine's Euclid, has given the fole prime Numbers 11, 13, 17, 19, 23,
lowing uſeful Series for finding the &c. you need but find the Product
Logarithms of great Numbers. Let of the two Numbers deficient from
x be an odd Number, whoſe Loga- either of them by 1, and exceeding
rithm is wanted ; the Numbers it by 1, which will always exceed
tai and xti will be even, and that Product by 1; then to the Low
ſo their Logarithms will be had, garithm of the Quotient of the Di.
and the Difference of theſe Loga- viſion of that Square by the ſaid
rithms which call y; alſo there is Product, found by the Rule but now
given the Logarithm of a Number, expreſſed, add the Logarithm of
which is a Geometrical Mean be that Product, which is always made
tween x~I and xtı, viz. equal up of the given Logarithms of the
to į the Sum of the Logarithms. prime Numbers, being leſs than the
given prime Number, and the Sum.
Then the Series y x
will be the Logarithm of the pro-
f
44 24x3 poſed given Number.
Mr. Mercators Logarit notechnia,
7
13
+ +
ſet forth An. 1668, was the first public
2520039
Treatiſe of the Conſruction of Lo-
&c. will be equal to the Logarithm garithms by the Hyper bola, that is,
by
2%
1
1
+360xs
181
15120x7
1
L OG
L OG
by help of infinite Series's, nearly Doctor in his Logarithmical Do-
expreſſing the Afymptotical Hyper- ctrine ; all which are entirely a-
bolic Spaces in Number. And after voided, and the whole feems clear
him Dr. Gregory, and others did to any Arithmetician and Geome-
the ſame thing ; but no one has trician of the leaſt Capacity from
ſhewn how to perform the Buſineſs the Confideration of the Hyperbola,
ſo perſpicuous and elegant as Sir as above-mentioned.
Ifaac, as will eaſily appear upon
Mr. Cotes too, at the Beginning
comparing his Way above mention- of his Harmon. Menfur. has done
ed with any other extant. Dr. this Buſineſs in imitation of Dr.
Halley too, (in Tranſ. Philos. Nº Halley, altho' more ſhort, yet with
216.) has given their Nature and the fame Obſcurity: for I appeal to
Conitruction (after a fort) without any one, even of his greateſt Ad-
any mention of the Hyperbola ; cho' mirers, if they know what he would
it is evident, that all the while he be at in his firſt Problem, viz. to
had the Hyperbola and the Menfu- find the Meaſure of a Ratio from
ration of the aſymptotical Spaces the Terms of the Problem itſelf,
under Conſideration; but rather than (which ſhould always be done) with-
exprefly mention them, becauſe he out having firſt known ſomething of
will not uſe Geometrical Figures in the matter from other Principles, as
an Affair purely Arithmetical (as Mr. the Hyperbola, &c.
Jones, in his Synopſis, ſays) he per The Lord Naper, a Scotch Baron,
plexes and ſtrains his Reader's Ima was the firſt who found out Loga-
gination with ſeveral almoſt unintel- rithms, having publiſh'a at Edin-
ligible Ways of Expreſſion ; ſuch as burgh, Anno 1614. Tables of Lo-
an infinite Number n of equal Ratio's garithmic Sines and Tangents for
or Ratiunculæ, in a continued Scale of the Uſe of Trigonometry, in a
Proportions between the two Terms. Treatiſe, entitled Canon Mirificum
Logarithmorum, computing them to
of any Ratio, as i and i+xor its". every Degree and Minute, and mak-
Then it will be the firſt Mean or ing the Logarithm of the Radius o ;
Root of the infinite Power it*"; fo that as the Logarithm of the
and let x (ſays he) be a Ratiuncula șines increaſe, the Sines themſelves
or Fluxion of the Ratio of 1 to itx. decreaſe, and thoſe of the Sines and
-We may value Ratio's by the Num- Tangents greater than the Radius,
ber of Ratiuncula contain'd in each.
are defective or leſs than o.
Ản infinite Number of Means may fally allow'd to be the firſt Inventor
Altho' the Lord Naper is univer-
be 'taken between the Terms of any of the Logarithms, yet Mr. Wolfe,
Ratio, provided the ſame Proportion in his Lexicon Mathem. fays, that
be every where obſervd. And thoſe
Ratiunculæ being hitherto confider's Kepler in his Rudolphin Tables (chap
as having the ſame Magnitude in all 3. p.!!.) mentions one Job Byrge,
Ratio's, the Logarithms of Ratio's Years before their Publication by
as having the Logarithms ſeveral
are as the Number of Ratiunculæ con-
tain'd between their Terms; and the Lord Naper, and complains of
therefore the Logarithm of any Num- him that he was Hominem cunctato-
ber is found by taking the Difference
rem & fecretorum fuorum Cuftodem,
between Unity and the infinite Root qui Fætum in Partu deßituit, non ad
of that Number, &c. There and
re-
U ſus publicos educavit. But to
ſeveral other are the unintelligible,
turn to the Lord Naper ; afterwards
or at least obſcure Expreſſions of the he thought of a more convenient
Form
LOG
L OG
Form of them; and having com- bers to which the Logarithms are
municated his Deſign to Mr. Henry fitted, only run from 1 to 1000,
Briggs, the Savilian Profeſſor of which may be ſufficient for many
Geometry at Oxford ; theſe two Caſes.-But amongſt thoſe,Sherwin's
jointly undertook the bringing of Tables of Logarithms, firſt publiſh-
Logarithms into a more conveniented at London, Anno 1705, are much
Form ; but the Lord Naper dying the beſt. In theſe you have the
before they had done, the whole Logarithms of all Numbers from 1
Burthen remaining was laid upon to 101000, conſiſting of ſeven Places
Mr. Briggs's Shoulders, who, with of Figures, with the Differences of
prodigious Labour, and great Skill, the Logarithms and the proportio-
made a Canon of Logarithms, ac- nal Parts ſet againſt them, by means
cording to that new Form, for the of which may be eaſily found the
Numbers from 1 to 20000, and Logarithm of any Number from I
from 90000 to 101000, to 14 to 10000000 ; ſo far, to wit, as
Places of Figures, which was pub- theſe Logarithms are expreſſed by
liſhed at London, Anno 1624. only feven Places of Figures. You
This Canon was again publiſhed have alſo the Logarithms of the
in Holland, by Adrian Vlaque, Anno Sines, Tangents, Secants, &c. to
1628. but filled up with the Loga- every Minute, and other uſeful
rithms of thoſe Numbers omitted Tables,
by Mr. Briggs; but theſe Logarithms As the Hyperbolic Logarithm of
are continued to but 10 Places of 10 is to Briggs's Logarithm of 10,
Figures. Mr. Briggs alſo computed ſo is the Hyperbolic Logarithm of
the Logarithms of the Sines and any Number to Briggs's Logarithm
Tangents to every Degree, and too of that ſame Number ; and if I be
Part of a Degree to 15 Places of the Hyperbolic Logarithm of any
Figures, to which he ſubjoined the Number greater than 1, then will it
natural 'Sines, Tangents, and Sccants,
12 13 14
'which he had before computed to +
6+24, &c. be that
15 Places of Figures. And theſe
Tables, together with a Treatiſe of Number ; but if leſs, it will be I-
their Conſtruction and Uſe, was pub-
IL 13 14
+
+
7
liſhed at London, Anno 1633. after
Egc. Theſe
24'
Mr. Briggs's Death by Henry Gele- Series's are Sir Iſaac Newton's, and
brand, under the Title of Trigono- may be ſeen in his laſt Letter to
metria Britannica.- Benjamin Ur- Mr. Leibnitz.-If an artificial Tan-
finus, in his Trigonometry, has given gent of any Arch a be t, and the
us a Canon of Logarithms to e- artificial Secant s, and the whole
very 10 Seconds. And Mr. Wolfe, Quadrant 9, and r the Radius ; then
in his Mathematical Lexicon, ſays,
а?
аб
that one Van Lofer had computed will s be =
them to every ſingle Second ; but
1273
his untimely Death prevented their
17a8
62a'o
Publication. Within this 60 Years
+
Eg c. and
2520r? 28350rº
there have been publiſh'd many com-
pendious Tables of Logarithms of (ſuppoſe 2a-q=et=et
6r
+
Numbers, Sines and Tangents, par-
es
61 e?
ticularly at the Ends of Books of
+
Navigation, conſiſting of only ſeven 24m+ 504070
Places of Figures, where the Num- & c. And if the artificial Secant of
459
I
2
I
2
st
at
+
+
4575
+
03
+
27709
72576 78,
413
322
724
3r3
14.15
374
452 26
45 rs ,
wo
13
67 +
2779
+
725768, c. But here it muſt
LOG
LOG
45° bes, and stl be any ar-
3. Whence there may be fuppo-
tificial Secant, then will its Arch be fed infinite Kinds of Logarithmic
12
Curves, if ** : 2 :: ly : Izl;
19ti-
+
Since the Ordinates pm continually
decreaſe, while the Ratio of A N to
&c. and 2a9 pm continually increaſes with the
Abfciffa, the Curve continually ac-
t5
6177
cedes to the Axis AX; but will
2474
5040 76 never meet it, and fo AX is an A.
fymptote to the Curve.
From the Definition of the Lo-
be obſerved that the artificial Radius to find Points thro’ which it is to
be obſerved that the artificial Radius garithmetical Curve, it appears how
is o, and when q is greater than 2a, paſs, which may be done too by
or the artificial Secant of 24° is
greater than the given Secant, the means of the Tables of Logarithms.
Signs are to be changed. Theſe an invariable Right Line. The in-
The Subtangent of the Curve is
Series's are Dr. James Gregory's, finite Space contain'd under the A-
fent Anno 1670 to Mr. Collins.
fymptote AX, the Curve NM in-
LOGARITHMIC CURVB. If the
finitely continued towards M, and
Right Line AX be divided into any the Ordinate AN is equal to the
Number of equal Parts, and if in Rectangle under A N and the Sub.
the Points of Diviſion A, P, R, SC tangent. Any Part NM of the
be joined the Right Lines AN,PM, Curve is rectifiable by means of the
pm, &c. continually proportional Subtangent ; for if PM be y, and
and parallel ; the Points N, M, m, the Subtangent a, the Fluxion of
Eg c. will be in the Curve called the
Logarithmic Curve.
an ithe Abfcilices A P, AJ, &c. the Part N M will be v staa,
are the Logarithms of the Ordi. And the Fluent of this may be had
nates PM, pm, &c.
by means of the Curve and Sub,
Whence if AP=x, Ap=v, tangent. See Mr. Cotes's Harmonia.
PM=y, pm=%, and the Loga Sir Iſaac Newton, in the ſecond
rithms of jy and z=ly, and l%; Book of his Princip. demonſtrates
that a Projectile deſcribes this Curve
A
N
when moving in a Medium, whoſe
Reſiſtance is as the Velocity of a
р
M
Body moving in it.- Concerning
this Curve, ſee Mr. Huygens's Dif-
cours ſur la cauſe de Pefanteur, pag.
P
176. and Guido Grando's Demonſtra-
tio Theorematum Huygenianorum circa
In
P Р
Logiſticam feu Logarithmicam Line-
am; as alſo Mr. Bernoulli's Diſcourſe
in the A&ta Eruditorum, Anno 1696,
then *ly and v=lzs, and ſo LOGARITHMIC SPIRAL. If the
*: 0 :: 1yilz, that is, the Deno- Quadrant of a Circle ANB be di-
minators of the Ratio's of AN to vided into any Number of equal
PM, and AN to pm, are to one Parts in the Points N, n, n, &c. and
other as the Abſciſſa's AP, AP, from the Radii CN, C1, Cn, &c.
be
2.
in
X
pag. 216.
A
1
LOG
L ON
772
be cut off CM, Cm, Cm, &c. con cation and Diviſion is ſaved, he
tinual Proportionals, the Points M, calls Logiſtical Arithmetic. Though
m, m, &c. will be in the Logarith- fome, by
mic Spiral. "Whence the Arches LOGISTICS, will underſtand the
AN, N n, nn, &c. are the Loga- firſt general Rules in Algebra, of
sithms of the Ordinates CM, Cm, Addition, Subtractions, &c.
&c. and there may be imagined an LOGISTIC SPIRAL.
See Loga-
rithmic Spiral.
А
N
LOGISTICAL Line, is that
which is otherwiſe called the Loo
M
P
n
garithmic Line, where the Ordinates
apply'd at equal Parts of the Axis
are in geometrical Proportion.
72
P
in
LONGIMETRY, the Art of mea-
furing Lengths or Diſtances, or to
p
take the Diltance of Trees, Steeples,
or Towers, &c. either one, or ma-
C
ny together; for which purpoſe the
B
Theodolite is reckoned to be the beſt
infinite Number of different Curves Inſtrument.
of this kind.
LONGITUDE of a Place, is an
Dr: Halley, in the Philoſophical Arch of the Equator intercepted
Tranſactions, has happily apply'd between the Meridian of that Place,
this Curve to the Diviſion of the and the firſt Meridian
and the firſt Meridian ; or 'tis more
Meridian Line in Mercator's Chart. truly the Difference, either Eaſt or
See alſo Mr. Cotes's Harmonia, Welt, between the Meridians of any
Guido Grando's Demonſtratio Theore two Places, counted on the E-
matum Huygenianorum ; the Asta E. quator.
ruditorum, An. 1691. p. 282, and LONGITUDE in the Heavens, is
foll.
an Arch of the Ecliptic, counted
LOGISTICAL ARITHMETIC, from the Beginning of Aries, to the
was formerly the Arithmetic of ſexa- Place where a Star's Circle of
geſimal Fractions, and uſed by A- Longitude croffes. the Ecliptic ; fo
Itronomers in their Calculations. I that "ris much the ſame as the Star's
ſuppoſe it was ſo called from a Place in the Ecliptic, reckoned from
Greek Treatiſe of one Barlaamus, a the Beginning of Aries.
Monk, who wrote about Sexage LONGITUDE of the Sun or Star
fimal Multiplication very accurate- from the next Equinoctial Point, is.
ly, and entitled his Book Logiſtice. the Number of Degrees and Mi-
This Author Volius, in his Books nutes they are from the Beginning
de Scientiis Mathematicis, places a of fries or Libra, either before or
bout the Year 1350, but miſtakes after them, which can never be
it for a Treatiſe of Algebra. more than 180 Degrees.
Thus alſo Shackerly, in his Ta LONGITUDE, in Dialling. The
bula Britannicæ, hath a Table of Arch of the Equinoctial intercepted
Logarithms adapted to Sexagefimal between the ſubſtilar Line of the
Fractions, which therefore he calls Dial and the true Meridian,' is called
Logiſtical Logarithms ; and the ex the Plane's Difference of Longitude.
peditious Arithmetic of them, which LONGITUDE, in Navigation, is
is by this means obtained, and by alſo the Diſtance of one Ship or
which all the Trouble of Multipli- Place, Eaſt or Weſt from another,
(counted
LON
LON
(counted in proper Degrees;) but 2. Others being fully ſatisfied of
of in Leagues or Miles, or Degrees the Impracticableneſs of the Method
of the Meridian, and not in thoſe of Eclipſes for finding the Longi-
proper to the Parallel of Latitude, tude at Sea, have thought of doing
it is commonly called Departure. it by a Clock or Watch: Which
1. Several ways have been thought indeed, if it could be made to go
of to find the Longitude at Sea; right all the time of a long Voyage,
the great Deſideratum of the Art of would give the Longitude at any
Navigation, for doing of which time, when the true Hour of the
ample Rewards have been promiſed Day or Night could be had under
by ſeveral Nations ; as by the E. any Meridian, or in any Place of
clipſe of the Moon, her Tranſit o the Earth: For the Clock going
ver, or Appulſe to any eminent true for the Meridian it was firſt fet
fixed Star; the Eclipſes of Jupiter's at, will ſhew the true Hour of the
Satellites, &c. which are all true Day or Night under any Meridian,
in Theory, and may be practiſed a or in any Place of the Earth; and
fhore with the greateſt exactneſs. then the true Hour being found by
For the time of any one of theſe the Sun or Stars in the place where
Phænomena being truly calculated the Ship is, the Difference between
for the Meridian of London (ſuppoſe, that and the Clock's Hour will be
or any other ;) and Tables may be the Difference of the Meridian in
eaſily made of all of them, which Time, or Longitude in Degrees.
the Navigator may carry to Sea 3. But it is not eaſy to make
with him. If then he could but ſuch a Movement, as will keep go-
obſerve the time of the Eclipſe or ing in, all Weathers, and all Cli-
Tranſit at Sea with accurate exact mates truely, eſpecially in ſome of
neſs, the Difference of Time of the the Southern ones, where the Dews
Eclipſe happening to him ſooner or are ſo great as to ruft the Parts of
later than at London, would give it ; and ſo retard, if not ſtop its
him the exact Longitude of the Motion.
Place of the Ship, either Eaſt or 4. Another Inconveniency is, that
Weſt from the Meridian of London : in different Latitudes the Hours
But the misfortune is, ſuch an Ob- ſhewn by the Clock, will be diffe-
ſervation of an Eclipſe, and the rent from thoſe ſhewn by it for che
exact Time of the Immerſion, or E- Latitude to which it is fitted
merfion of the deficient Body into, a Clock at London made to thew
or out of the Shadow, is not to be the Time there, when carried under
made without Teleſcopes of ſuch a the Equinoctial, will go too flow by
length, as the Motion of the Ship 2 or 3 Minutes, and the Law of
will not permit to be uſed at Sea : the Retardation as you go South-
Tho' by the by, if Ships were fent wards is not yet well known.
with good Inſtruments, and Men 5. Notwithſtanding this, Mr.
that know how to uſe them, to do Hluygens in his excellent Horolovium
this at all the Capes and Head- Oſcillatorium, mentions two Clocks
lands of the World, it would be a that were formerly made by his
thing of the greateſt uſe; and by Directions there laid down, being
fettling the Longitude of all thoſe carried to Sea in an Engliſh Ship,
Places, would cut all long Voyages in company with three other Ships,
into many ſhort ones, and afford which very much affifted the Cap-
means of continually rectifying the tain to judge of the true Place of
Dead reckoning at Sea. But to return the Ship For the Captain faid,
when
;
as
LON
'L U N
when they had failed from the has been approved of and recom-
Coaſt of Guinea to the Iſland of St. mended by Mr. George Graham, Dr.
Thomas, under the Equinoctial Cir- Smith, and Dr. Barker, as I have
cle, and there ſet the Clocks to the been informed.
Sun ; they failed Weſtwardly about LOWER FLANK, or RETIRED
70 Miles, and then directed their FLANK. See Flank, a Term in
Courſe towards the African Shore; Fortification.
and when they continued on upon LOXODROMIQUES, is the Art
a Courſe for about 2 or 300 Miles, or Way of oblique Sailing by the
the Captains of the reſt of the Ships Rhumb, which always makes an
fearing they ſhould want Water be- equal Angle with every Meridian,
fore they could arrive at the Coaſt je. when you fail neither directly
of Africa, would häve them go to under the Equator, nor under one
get Water at the American Iſlands, and the ſame Meridian, but oblique-
called the Caribbes ; and a Conſul- ly or a-croſs them. Hence the
tation being held thereupon, the Table of Rhumbs, or the Traverſe-
Journals and Reckonings of each Table of Miles, with the Difference
Ship were produced, all which dif- of Longitudes and Latitudes, by
fer'd from the Captain's, who had which the 'Sailor may practically
the Clocks aboard; one 120 Miles, find his Courſe, Diſtance, Latitude,
another roo, and the third ſtill more. or Longitude, is by ſome called by
But the Captain himſelf ſaid, he ga- this Name of Loxodromiques ; and
thered from his Clocks, that they ſuch Tables as ſerve truly and ex-
were not more than above 30 Miles peditiouſly to find the ſeveral Requi.
from one of the African Iſlands, lites, or to reſolve the Caſes of
call'd del Fuego, nigh to the Coaſt Sailing, are called Loxodromical
of Africa, and might arrive at the Tables.
fame the next day. And ordering LUCIDA CORONA, a fixed Star
them to direct their Courſe accor of the ſecond Magnitude, in the Nor-
dingly, they ſaw the faid Ifland the thern Garland, whofe Longitude is
next day at Noon, and in a few 217 Deg. 38 Min. Latitude 44 Deg.
Hours after arrived at the ſame. 23 Min. Right Aſcenſion 230 Deg.
Mr. Huygens in the fame Book,
fays, that afterwards by the Com LUCIDA HYDRA. See Cor Hy-
mand of Lewis the XIVth, the dræ.
French and Dutch made various Ex LUCIDA LYRA, a bright Star
periments with his Clocks; but of the firſt Magnitude, in the Con-
with various Events, which he attri- ftellation Lyra, whoſe Longitude is
buted often more to the Negligence 10 Deg. 43 Min. Latitude 6. Deg.
and Unfilfulneſs of the Perſons to 47 Min. Right Aſcenſion 276 Deg.
whoſe Care they were committed, 27 Min. And Declination 38 Deg.
than to the Faults in the Clocks
themſelves.
See more, pag. 17.
LUMINARIES, the Sun and Moon
Horol. Oſcillat.
are ſo called by way of Eminence ;
6. But the moſt ingenious and beſt for their extraordinary Luftre, and
Clock that ever was, or perhaps the great Quantity of Light that
ever will be made for this purpoſe, they afford us.
is that of Mr. Harriſon of "Leather LUNAR CYCLE. See Cycle of
Lane, London, as I have been in the Moon.
formed by Perſons, whom I take to LUNARY MONTHS, are either
be very good Judges; and which Periodical, Synodical, or Illumina-
3
tive.
12 Min.
30 Min.
L UN
MAG
tive. Which ſee in their proper Level of the Water, and hath a Pa-
Places.
rapet three Fathom thick.
LUNATION of the Moon, is the LUPUS, a Southern Conſtellation,
Time between one New Moon and conſiſting of two Stars.
· another ; and this is greater than the LYRĂ, the Harp, a Conſtellation
Periodical Month by two Days and in the Northern Hemiſphere, conſiſt-
five Hours ; and is called the Synodi- ing of 13 Stars.
cal Month, conſiſting of 29 Days, 12
Hours, and three Quarters of an
Hour,
LUNES, or LUNULÆ, in Geo-
M.
metry, are Spaces contain'd under a
Quadrant of a Circle, and a Semi ACHINA BOYLEIANA,
circle ; being called thus, becauſe Mr. Boyle's Air. Pump.
they repreſent the Figure of the MACHINE, or ENGINE, in Me-
Moon, when leſs than half full; as chanics, is whatſoever hath Force
the Space ABGC is the Lune. ſufficient either to raiſe or atop the
If the Line A B is drawn, as alſo Motion of a Body. Theſe Machines
the Line AE, at Right Angles to are either Simple or Compound.
BD; I ſay the Triangle ABE is Simple Machines are commonly
equal to the Part A BQ of the Lune, reckoned to be fix in Number, viz.
and ſo the whoſe Lune is equal to the Ballance; Leaver, Pulley, Wheel,
the Triangle ADC.
Wedge, and Screw. To theſe might
be added the inclined Plane ; ſince
'tis certain that the heavieſt Bodies
B
may be lifted up by the means there-
of, which otherwiſe could ſcarce be
moved.
Compound Machines or Engines
E
А.
are innumerable, in regard that they
C
may be made out of the Simple, al-
moſt after an infinite Manner.
MADRIER, in Fortification; is a
thick Plank, armed with Plates of
fron, and having a Concavity ſuffi-
cient to receive the Mouth of the
s
Petard when charged, with which is
is apply'd againit a Gate, or any
LUNETTES, in Fortification,are En- thing elſe that you deſign to break
velopes, Counter-guards, or Mounts down. This Term is alſo appropri-
of Earth caft up before the Curtain, ated to certain flat Beams, which are
about five Fathom in breadth, where- fixed to the Bottom of a Moat, to
of the Parapet takes up three. They ſupport a Wall. There are alſo Ma-
are uſually made in Ditches full of driers lined with Tin, which are co-
Water, and ſerve to the ſame pur- vered with Earth, to ſerve as a De-
poſe as Falſebrayes. Theſe Lunettes fence againit artificial Fires.
are compoſed of two Faces, which MAGIC SQUARE, is when Num-
form a re-cntring Angle ; and their bers in Arithmetic Progreſſion are diſ-
Platform being only twelve Foot poſed into ſuch parallel and equal
wide, is a little raiſed above the ranks, as that the Sums of each Row,
G
Z
29
M A G
M A G
as well diagonally as laterally, ſhall both pointing itſelf, and alſo ena-
be equal.
bling a Needle touched upon it, and
Thus theſe nine Numbers, 2, 3, then poiſed, to point towards the
4, 5, 6, 7, 8, 9, and 10, being dif- Poles of the World.
Sturmius, in his Epiſtola Invi-
511013
tatoria Dat. Altrof. 1682, obſerves,
4
68
that the attractive Quality of the
912 17
Magnet hath been taken Notice of
beyond all Hiſtory ; but that it
poſed into this ſquare Form, they do was our Countryman Roger Bacon,
every way directly, and diagonally who firſt diſcovered the Verticity of
make the fame Süm : As likewiſe it, or its Property of pointing towards
thoſe
the Pole ; and this about 400 Years
49
Numbers ;
fince. The Italians firſt diſcovered,
3013914811110119/28
that it would communicate this Vir-
38/47 7911827 29 tue to Steel or Iron. The various
45 61 8117 26 35 37
Declination of the Needle, under
different Meridians, was firſt diſco-
5 14 16 25134136/45
vered by Sebaſtian Cabott ; and its
13.15 24 3342 44 4
Inclination to the nearer Pole by
21123132141431 3 12
our Countryman Robert Noman. The
2213114014912 211/20
20
Variation of the Declination, ſo that
’tis not always the ſame in one and the
Magic LanThorn, a little Op- fame Place, he oblerves, was taken
tic Machine, by the means of which notice of but a few Years before,
are repreſented on a Wall, in the by Hevelius, Auzout, Petit, Volcka-
dark, many Phantaſms and terrible
mer, and others.
Apparitions, which are taken for the
The Properties or Phænomena of
Effect of Magic, by thoſe that are ig- this wonderful Stone, as they have
norant of the Secret.
been diſcovered by Gilbert, Kircher,
This Machine is compoſed of a Cabeus, Des Cartes, and others, are
concave Speculum from one Foot to
there :
four Inches Diameter, reflecting the 1. That in every Magnet there are
Light of a Candle, which paffcth two Peles, one pointing North, the
through a little Hole of a Tube, at other South ; and if a Stone be cut
whole End there is faften'd another or broken into never ſo many Pie-
double Convex-Glaſs of about three
ces, there are theſe two Poles in each
Inches Focus ; between theſe two are Piece.
ſucceſſively placed many ſmall Glaſ 2. That theſe Poles in divers Parts
fes, painted with different Figures, of the Globe, are diverſely inclined
of which the moſt formidable arc al- towards the Earth's Centre.
ways choſen, and ſuch as arc inoſt
3. That theſe Poles, tho' contra-
capable of terrifying the Spectators ; ry to one another, do help mutually
ſo that all theſe Figures may be re toward the Magnet's Attraction and
preſented at large on the oppoſite Suſpenſion of Iron.
Wall.
4. If two Magnets are ſpherical,
Magnet, or LOAD-STONE, is a one will turn or conform itſelf to
Foflile approaching to the Nature of the other, fo as cither of them would
Iron-Ore, and endowed with the do to the Earth ; and that after they
Property of attracting of Iron, and of have fo conformed or turned them-
ſelves,
1
MAG
M A G
felves, they endeavour to approach capped, than it can alone ; and that
to join each other ; but if placed in though an Iron Ring or Key be fuf-
a contrary Poſition, they avoid each pended by the Loadſtone, yet the
other.
Magnetical Particles do not hinder
5. If a Magnet be cut through the that Ring or Key froin tutning
Axis, the Parts or Segments of the round any way, either to the Right
Stone, which before were joined, will or Left.
now avoid and fly each other. - 14. That the Force of a Load-
6. If the Magnet he cut by a Sec- ſtone may be variouſly increaſed or
tion perpendicular to its Axis, the leffened by the various Application of
two Points which before were con Iron, or another Loadſtone to it.
joined, will become contrary Poles, 15. That a ſtrong Magnet, at the
one in one, the other in the other leaſt Diſtance from a leſſer or a wea-
Segment.
ker, cannot draw to it a piece of
7. Iron receives Virtue from the Iron adhering actually to ſuch leſſer
Magnet by Application to it, or bare or weaker Stone ; but if it comes to
ly from an Approach near it, though touch it, it can draw it from the
it doth not touch it; and the Iron re other : But a weaker Magnet, or
ceives this Virtue variouſly, accor even a little piece of Iron, can draw
ding to the Parts of the Stone 'tis away or ſeparate a Piece of Iron,
made to touch, or made to approach contiguous to a greater or ſtronger
to.
Loaditone.
8. If any oblong Piece of Iron 16. That in our North Parts of
be any how applied to the Stone, the World, the South Pole of a
it receives Virtue from it only as to Loadſtone will raiſe up more Iron
its Length.
than the North Pole.
9: The Magnet loſes none of its 17. That a Plate of Iron only, but
own Virtue by communicating any no other Body interpoſed, can im-
to the Iron, and this Virtue it can pede the Operation of the Loadſtone,
communicate to Iron very ſpeedily; either as to its attractive or direc-
though the longer the Iron touches tive Quality. Mr. Boyle found it true
or joins the Stone, the longer will in Glaſſes ſealed hermetically ; and
its communicated Virtue hold ; and Glaſs is a Body as impervious as
a better Magnet will communicate molt are, to any Efluvia.
more of it, and fooner than one not 18. That the Power or Virtue of
fo good.
a Loadſtoue may be impaired by
10. That Steel receives Virtue lying long in a wrong Pofture, as alſo
from the Magnet better than Iron. by Ruſt, Wet, &c. and may be quite
II. A Needle touched by a Mag- deſtroyed by Fire.
net, will turn its Ends the ſame way The Orb of the Activity of Mag-
towards the Poles of the World, as nets is larger or leſs at different
the Magnet will do.
times ; which is confirmed by what is
12. That neither Loadſtone nor found in fact to be true of our noa
Needles touched by it, do conform ble Loadſtone, which is kept in the
their Poles exactly to thoſe of the Repoſitory of the Royal Society ; for
World ; but have uſually ſome Va- that will keep a Key, or other Piece
riation from them; and this Varia- of Iron, ſuſpended to another, ſome-
tion is different in divers Places, and times at the Diſtance of eight or ten
at divers Times in the ſame Place. Foot from it ; but at other times,
13. That a Loadſtone will take not beyond the Diſtance of four
up much more Iron when armed or Foot.
MAG
2 2
Μ Α Ν
Μ Α Ρ
MAGNETICAL AMPLITUDE, is the Top, whereof the Miners make
an Arch of the Horizon, contained uſe, to approach the Walls of a
between the Sun at his riſing or ſet. Town or Caſtle.
ting, and the Eaſt and Weſt Point MAP, is a Deſcription of the
of the Compaſs ; or it is the diffe- Earth, or ſome particular Part there-
rent riſing or ſetting of the Sun of, projected upon a plain Superfi-
from the Eaſt and Weſt Points of cies; deſcribing the form of Coun-
the Compaſs; and is found by ob- tries, Rivers, Situations of Cities,
ſerving the Sun at his riſing or fet- Hills, Woods, and other Remarks.
ting, by an amplitude Compaſs. Anaximander the Scholar of Thales,
MAGNETISM, or MAGNETICAL about 400 Years before Chriſt, is ſaid
ATTRACTION, is the Virtue or to have been the firſt Inventor of
Power that the Loadſtone has of Geographical Tables, or Maps ; and
drawing Iron to it.
the Peutingerian Tables, publiſhed
MAGNETICAL AZIMUTH, is an by Cornelius Peutinger of Aufburgh, ,
Arch of the Horizon, contained be- contain an Itinerary of the whole
tween the Sun's Azimuth Circle, and Roman Empire ; all Places, except
the Magnetical Meridian ; or it is Seas, Woods and Deſerts, being put
the apparent Diſtance of the Sun down according to their mealured
from the North or South Point of Diſtances, but without any mention
the Compaſs ; and may be found by of Latitude, Longitude, or Bearing:
obſerving the Sun with an Azimuth Ptolemy of Alexandria, who lived
Compaſs, when he is about ten or about the 144th Year of Chriſt, in-
fifteen Degrecs high, either in the vented Meridians and Parallels, the
Forenoon or Afternoon.
better to define and determine the Si-
MAGNETICAL MERIDIAN. See tuations of Places, brought Maps to
Meridian.
a much greater Degree of Perfection
MAGNETICAL NEEDLE, is the than before. But Ptolemy himſelf
touched Needle of the Compaſs. owns, that thoſe Maps going by his
MAGNIFY, is a Word uſed chieây Name, were copied from others that
with regard to Microſcopes, being were made by Marinus Tyrus, &c.
only the bringing the Object nearer with ſome Improvements of his own
to the Eyes, and letting ſome Parts added. But from his Time till about
of it be ſeen, which before were not the 14th Century, whilft Geography
diſcoverable by the bare Eye. lay dead, no new Maps were pub-
MAGNITUDE. The ſame as Big- lithed. Mercator was the firſt of
neſs or Greatneſs.
Note, and next to him Ortelius, who
MANTELETS, in Fortification, are undertook to make a new Set of
a kind of moveable Penthouſes, and Maps, with the modern Diviſions of
are made of Pieces of Timber fawed Countries and Names of Places; for
into Planks ; which being about three want of which, Ptolemy's were grown
Inches thick, are nailed one over
almoſt useleſs. After him many o-
another to the height of almoſt fix thers publiſhed Maps, but for the moſt
Foot. They are generally caſed with part were mere Copies of his.-To-
Tin, and ſet upon little Wheels; wards the middle of the laſt Century,
ſo that in a Siege they may be Mr. Bleau in Holland, and Mr. San-
driven before the Pioneers, and ſerve ſon in France, publiſhed new Sets of
as Blinds to ſhelter them from the Maps, with many Improvements from
Enemy's Small-thot. There are alſo the Travellers of thoſe Times; which
other Sorts of Mantelets, covered on were afterwards copied, with very
little
$
1
MAR
M A T
little Variation, by the Engliſh, Tube, and ſaw its Body as large
French and Dutch; the beſt of theſe very near as the Moon at Pull; and
being thoſe of Mr. De Wit, and in it he obſerved ſeveral Spots, and
Viſcher.-Maps being by ſo many particularly a triangular one ; which
blind Copiers likely to fall into having a Motion, he concluded the
much Obſcurity and Error, Mr. Planet to have a turbinated Mo-
De Liſe, an ingenious French Geo- tion round its Centre.
grapher, made a compleat Set of 4. In the Year 1666, February
Maps, both of the old and new the 6th in the Morning, Mr. Caf.
Geography, corrected and improv'd fini, with a 16 Foot Teleſcope, ob.
from the Surveys ſeveral European ſerved two dark Spots in the firſt
Nations had made of their reſpec- Face of Mars, moving fiom Eleven
tive Countries, the Obſervations of at Night until Break of Day.
the beſt Travellers in all Languages, 5. February the 24th, in the
and the Journals of the Royal So- Evening, he ſaw two other Spots
cieties of London and Paris-Con- in the other face of this Planet,
cerning Maps, fee Varenius's Geogr. like thoſe of the firſt, but much
Gener. hb. 3. cap. 3 Prop. 4. p. m. bigger; and continuing the Obſer-
445. and foll. Fournier's Hydrogr. vations, he found the Spots of thoſe
lib. 4. c. 24. and foll. f. 667. Wol- two Faces to turn by a little and a
fius's Elem. Hydrogr. c. 9. John little from Eaſt to Weſt, and fo re-
Newton's Idea of Navigation, and turn at the Space of 24 Hours, 40
Mead's Conſtruction of Globes and Minutes to the ſame Situation,
Maps.
wherein they were ſeen at firſt.
MARINE BAROMETER. See
6. Whence he concluded, that
Barometer.
the Revolution of this Planet round
MARS, the Name of one of the its Axis, is performed in the Space
Planets which moves round the Sun of 24 Hours, 40 Minutes, or there-
in an Orbic between that of the abouts.
Earth and Jupiter.
7. The Magnitude of Mars to the
1. The mean Diſtance of Mars Magnitude of the Earth, is as 216
o from the Sun is 1524 ſuch to 343, and its apparent Diame-
Parts, of which the Earth's is 1000, ter, according to Mr. Flamſlead and
its Excentricity 141, the Inclinaticn Caſini, is 35.
of its Orbit i Deg. 52 Min. Its
8. That Mars hath an Atmo.
Periodical Time 686 Days, 23 ſphere,like ours, is argued from the
Hours. Its Revolution about its Phænomena of the fixed Stars ap-
Axis is performed in 24 Hours, 40 pearing obſcured, and, as it were,
Min.
extinct, when they are ſeen juſt by
2. This Planet ( as well as the the Body of Mars ; and if ſo, a
reſt) br:rows its Lighc from the Spectator in Mars will hardly ever
Sun; und has its Increaſe and De- ſee Mercury, unleſs it may be ſeen
creaſe of Light like the Moon ; and in the Sun, when that Planet palles
it may be ſeen almoſt biſfected when over his Dilk like a Spoi, as he
in his Quadratures with the Sun, or doth ſometimes to us.
in his Perigæon, but never cornicu. MATHEMATICS, originally fig.
lated or falcated, as the other In- nify any Diſcipline or Learning,
feriors.
(Mathefis :) But now, 'tis properly
3. March 10. 1665. Dr. Hook ob- that Science which teaches or con-
fcrved this Planet, with a 36 Foot templates whatever is capable of
being
2 3
M A T
M A T
being numbered or meaſured, as it Diviſion has been kept to by ma-
is computable or meaſurable.
ny of the more modern Mathema-
And the Part of Mathematics ticians, altho' ſome baniſh from
which relates to Number only, is hence Geodeſy and Logiſtics.
called Arithmetic ; that which relates That the Romans, eſpecially in
to Meaſure in general, whether the Times of their Emperors, were
Length, Breadth, Motion, Force, not Lovers, or even had a juft No-
E c. is called Geometry.
tion and valuable Opinion of Mathe.
MATHEMATICS may be rec- matics, abundantly appears from ſe-
koned either,
veral of their Writers, reckoning
1. Pure, fimple, or abſtracted, the Mathematicians amongſt Con-
which conſiders abftracted Quanti- jurers and Soothſayers. Tacitus (in
ty, without any relation to Matter, 1. Annal.) calls Mathematicians a
or ſenſible Objects. Or,
Brood of Fellows treacherous to
2. Mix'd Mathematics, which is thoſe above them, falſe to thoſe
interwoven every where with phyfi. who put their truſt in them, which
cal Conſiderations.
(ſays he) will be always prohibited,
MATHEMATICs alſo are divided and always retain'd in our City.
into,
Seneca (in his Play of Claudius) ſays,
1. Speculative, which propoſes it is manifeſt the Mathematicians
only the ſimple Knowledge of the ſometimes ſpeak Truth ; alluding
thing propoſed, and the bare Con- to the thouſand lying Predictions of
templation of Truth or Falſhood. the Death of Claudius, concerning
And,
which at laſt they ſpoke true. Ju-
2. Practical, which teaches how lius Paulus lib. 5. cap. 21. Senten-
to demonſtrate ſomething uſeful, or tiarum, ranks Mathematicians a.
to perform ſomething that ſhall be mongſt cunning Men and Aſtrolo-
propoſed for the Benefit and Ad- gers. Dio (in lib. 49. Hiftoriarum)
vantage of Mankind.
ſays, chat Agrippa cauſed the Aftro-
Ariſtotle (in 1. Met. 5.) fays, the logers and Magi to be removed
Pythagoreans were the firft amongſt from the City. Tacitus, that in the
the Greeks that meddled with Ma- Reigns of Tiberius and Claudian, the
thematics, and divided them into Senate paſs'd a Decree for baniſhing
four Parts, two Pure and Primary, the Mathematicians and Conjurer
viz, Arithmetic and Geometry ; out of Italy. In the Cod. Juſtinian.
and the other two Mix'd and Se- (lib. 9. titule 18 ) it is ſaid, that the
condary, as Muſic and Aſtronomy. Art of Geometry is neceſſary to be
Plato (in 7. de Rep.) divides them learn'd and uſeful to the Public ;
into five Parts, Arithmetic, Geo- but the mathematical Art is dam-
metry, Stereometry, Muſick and nable, and ought to be forbid en-
Aſtronomy; and Ariflotle himſelf tirely. Theſe and many other In-
added to them, Optics Mechanics, ſtances of the vile abuſe of the word
and Geodeſia ; and Proclus (in his Mathematics amongſt thoſe degene-
Comment upon Euclid's firſt Book) rating Ages of the Romans, are to
ſays, that Geminus, who liv'd about be found in Tacitus, Suetonius, & Co
the Time of Pompey the Great, di- But during the Time of the Roman
vided Mathematics into Arithmetic, Commonwealth,when learning flou-
Geometry, Geodeſy, Logiſtics, Op- riſhed more amongſt them, the Ma-
tics, Canonics, or Harmonics, Me- thematicians were in elteem, and di-
chanics, and Aftrology. And this ftinguiſhed from Fortunetellers 3 for
Cicero
6
М А Т
M A T
Cicero (in Lib. de Divin.) ſays, Do and produces wonderful Effects ;
you imagine that Conjurers can re which is the fruitful Parent of, I
folve whether the Sun be greater
I had almoſt ſaid, all Arts, the un-
than the Earth, or ſo great as it ap-
• fhaken Foundation of Sciences,
pears; or whether the Moon has
' and the plentiful Fountain of Ad-
an inherent Light, or borrows it vantage to human Affairs. In
from the Sun ? To tell theſe things, • which laſt reſpect, we may
be
ſays he, belongs to the Mathema + ſaid to receive from Mathematics,
ticians, and not the Conjurers. • the principal Delights of Life, Se-
No body (that I know has fo ' curities of Health, Increaſe of
elegantly ſet forth and deſcrib'd the ( Fortune, and Conveniences of La-
Uſes of Mathematics, as the great
• bour. That we dwell elegantly
Dr. Barrow, in his Prefatory Ora and commodiouſly, build decent
tion upon his Admittance into the Houſes for our ſelves, erect ſtately
Profefforſhip at Cambridge ; his Temples to God, and leave won-
Words (tranſlated) are, · The Ma derful Monuments to Pofterity :
• thematics (ſays he) effectually · That we are protected by thoſe
• exerciſes, not vainly deludes, nor Rampires from the Incurſions of
• vexaciouſly torinents ftudious the Enemy, rightly uſe Arms, and
• Minds with obſcure Subtilties, artfully manage War ; fkillully
perplex'd Difficulties, or conten range an Army, that we have fate
' tious Diſquiſitions ; which con Traffick through the deceitful
quers without Oppoſition, tri “ Billows, paſs in a direct Road
umphs without Pomp, compels 'thro' the trackleſs Ways of the
6 without Force, and rules abſo • Sea, and arrive at the deſign'd
lutely without the Loſs of Liber • Ports, by the uncertain Impulſe
ty ; which does not privately o-
of the Winds : That we rightly
ver-reach a weak Faith, but open • caft up our Accounts, do Buſineſs
ly aſſaults an armed Reaſon, ob expeditiouſly, diſpoſe, tabulate, and
tains a total Victory, and puts on
o calculate ſcatter'd Ranks of Num-
• inevitable Chains; whoſe Words bers, and eaſily compute them,
' are ſo many Oracles, and Works' tho'ugh expreſſive of huge Heaps
as many Miracles ; which blabs • of Sand, nay immenſe Hills of A-
• out nothing raſhly, nor deſigns toms : That we make pacifick Sepa-
any thing from the purpoſe. But rations of the Bounds of Lands, exa-
• plainly demonſtrates and readily • mine the Momentums of Weights
performs all things within its . in an equal Balance, and diitrie
compaſs; which obtrudes no falſe • bute every one his own by a juit
• Science, but the very Science itſelf, Meaſure ; that with a light Touch
• the Mind firmly adhering to it, as
we thruit forwards Budies which
• ſoon as poſſeſsid of it, and can way we will, and ſtop a huge Refil-
never after of its own accord de ance with a very ſmall Force; that
• ſert it, or be deprived of it by we accurately delineate the Face
any Foice of ochess : Laſtly, ſays r of this earthly Orb, and lubject
* he) the Mathematics which de " the Oeconomny of the Univerle to
. pend upon Principles clear to the our Sight : That we aptly digelt
• Mind, and agrceable to Expe- the flowing Series of Time, di-
' rience, which draws certain Con ſtinguiſh what is acted by due
• clufions, inſtructs by profitable • Intei vals, rightly account and
• Rules, unfolds pleaſant Queſtions, diſcern the various Returns of the
z
Seaſons,
6
6
6
6
4
1
M A T
Μ Α Τ
* Seaſons, the ſtated periods of the incredible Force and Sagacity of
• Years and Months, the alternate our own Minds by certain Expe-
* Increaſements of Days and Nights, 'riments, as to acknowledge the
o the doubtful Limits of Light and Bleſſings of Heaven with a pious
« Shadow, and the exact difference ( Affection.
• of Hours and Minutes ; that we • I omit the advantageous Spur
« derive the ſolar Virtue of the to our Reaſon, which accrues
• Sun's Rays to our Uſes, infinitely from this mathematical Exerciſe,
o extend the Sphere of Sight, en • both effcctually to turn aſide the
large the near Appearances of • Strokes of true Arguments, and
Things, bring remote Things near, warily decline the Blows of falſe
diſcover hidden Things, trace ones; to difpute ftrenuouſly,as well
· Nature out of her Concealments, as judge folidly, with a readineſs
• and unfold her dark' Myſte • of Invention, a justneſs of Me-
• sies: That we delight our Eyes “thod, and clearneſs of Expreſſion.
• with beautiful Images, cunningly - In like manner, there Diſci-
imitate the Devices and pourtray plines do inure and corroborate
the Works of Nature. Imitate, the Mind to a conſtant Diligence,
• did I ſay? nay excel; while we ' in Study, to undergo the Trouble
• form to ourielves things not in • of an attentive Meditation, and
• being, exhibit things abfent, and cheerfully contend with ſuch Dif-
• repreſent things pait ; that we re « ficulties as lie in the way ; they.
create our Minds, and delight our wholly deliver us from a credu-
· Ears with melodious Sounds, at . lous Simplicity, moſt ítrongly for-
temperate the unconftant Undula-
tify us againit the Vanity of
« tions of the Air to muſical Tunes, * Scepticiſm, effectually retrain us
• add a pleaſant Voice to a fapleſs • from a raih Prefumption, moſt
Log, and draw a ſweet Eloguence 'çafily incline us to a due Affent,
• from a rigid Metal ; celebrate our pertedly ſubject us to the go-
• Maker with a harmonious Praiſe, vernment of right Reaſon, and
• and not unaptly imitate the bler. inſpire us to wreſtle againſt the
6 fed Choirs of Heaven: That' we 'unjult Tyranny of falſe Prejudices.
" approach and examine the jnac . If the Fancy be unſtable and fluc-
• ceſſible Seats of the Clouds, diſtant • tuating, it is as it were poiſed by
• Tracts of Land, unfrequented Paths " this Ballaſt, and ſteadied by this
• of the Sea; lofty Tops of Moun • Anchor ; if the Wit be blunt, it
: tains, low Bottoms of Valleys, and is ſharpen'd upon this Whetſtone ;
deep Galphs of the Ocean that • if luxuriant, it 'is pared by this
we ſcale the etherjal Towers, Knife ; if headftrong, it is reftrain'd
freely range through the celeſtial by this Bridle; and if dull, it is
• Fields, meaſure the Magnitudes, rouſed by this Spur. The Steps
« and determine the Interſtices of are guided by no Lamp more
• the Stars, preſcribe inviolable clearly thro''the dark Mazes of
• Laws to the Heavens themſelves, Nature, by no Thread more
« and contain the wandering Cir- • furely thro the intricate Turn-
• cuit of the Stars within ffrict ings of the Labyrinths of Philo-
Bounds : Laſtly, that we compre- ' fophy; nor, laſtly, is the bottom
• hend the huge Fabric of the Uni of Truth founded more happily
• verſe, admiſe and contemplate the by any other Line. I will not
• wonderful Beauty of the Divine mention with how plentiful a Stock
• Workmanſhip, and ſo learn the of Knowledge the Mind is fur-
niſhed
C
6
6
6
6
6
Μ Α Τ
M A T
s niſhed from there, with what be twice as denſe aś another, and
< wholeſome Food it is nouriſhed, takes up twice the Space, 'twilt bar
6 and what fincere Pleaſure it en four times as great. This Quantity
joys. But if I ſpeak further, I of Matter is beſt diſcoverable by
• Thall neither be the only Perſon, Weight, to which 'tis always pros
nor the first who affirm it, that portionable ; as Sir Iſaac Newtonia
& while the Mind is abſtracted and by most accurate Obſervations on
• elevated from ſenſible Matter, Pendulums, found true by Expe-
diſtinctly views pure Forms, con rience.
ceives the Beauty of Ideas, and MAXIMIS and MINIMIS, The
inveſtigates the Harmony of Pro- Mathematicians call that Method
portions ; the Manners themſelves whereby a Problem is reſolved,
are ſenſibly corrected and im-
and im- which requires the greateſt or leaft
proved, the Affection compoſed Quantity attainable in that cate,
• and rectified, the Fancy calmed Methodus. de Maximis & Minimis.
and ſettled, and the Underſtanding 1 If any flowing Quantity in aa
• raiſed and excited to more Divine Equation propoſed be required to be
. Contemplations All which I determined to any extrenie Value :
* might defend by the Authority, 2. Having put the Equation into
• and confirm by the Suffrages of Fluxions, let the Fluxion of that
• the greatest Philoſophers, &c.' Quantity (whoſe Extreme Value is
Thus far the great Dr. Barrow. fought) be ſuppoſed = 0; by which
The firſt who publiſhed a Mathe means all thole Menibers of the I-
matical Curſus, was Peter Herigor, quation in which it is found, will
Anno 1644:- After him came out vaniſh, and the remaining ones will
Caſpar Schottus's, chen Sir Jonas give the Determination of che Max
Moore's New Syſtem of Mathematics. mum or Minimum desired.
-Dechales's Curſus, or Mundum MEAN ANOMALY. See Anomaly,
Mathematicum.-- Leybourn's Courſe MEAN CONJUNCTION, is when
of Mathematics.-De Graaf's Cur- the Mean Place of the Sun is the
jus, in Dutch.---Ozanam's Cours de fame with the Mean Place of the
Mathematique.-Taylor's Treaſure of Moon in the Ecliptic. And a
the Mathematics.-Wolfius's Elemen MEAN OPPOSITION, is when
fa Matheſeos Univerſal. -Sturmy's the former is in Oppoſition to the
Mathefis Juvenilis, ones's Synop. latter.
fis, &c.
MEAN MOTION, is that where-
MATTER, or Body, is an im- with a Planet, or any Point or Line
penetrable, diviſible, and paſſive is ſuppoſed to move equally in its
Subſtance, extending into Length, Orbit, and is always proportional
Breadth, and Thickneſs. This, when to the Time.
conſidered in general, remains the Sir Iſaac Newton, in his Theoryan
ſame in all the various Motions, the Mooil, fays, That the Sun and
Configurations, and Changes of Na- Moon's mean Motions from the
tural Bodies, being capable of put- vernal Equinox at the Meridian of
ting on all manner of Forms, and of Greenwich, are as follows, vis, the
moving according to all manner of lait Day of December 1680, Old
Directions and Degrees of Velocity; Style, at Noon, the Sun's mean M:-
the Quantity of Matter in any Body, tion 9 fig. 20 deg. 34 min. 45 fec.
is its Meaſure ariſing from the joint That of the Sun's Apogæum 3 fig.
Confideration of the Magnitude and 7. deg. 23 min. 30 ſec. The Moon's
Denſity of that Body; as if any Body mean Motion 6 lig. 1 deg. 45 min.
45
"S
1
A
1
M E A
M E A
TION.
4 : 6::
45
ſec. That of the Moon's Apo- drawn from the Sun $ to P, the Ex-
gæum 8 fig. 4 deg. 28 min. 5 ſec. tremity of the conjugate Axis of the
That of the aſcending Node of the Ellipfis the Planet moves in, and
Moon's Orbit 5 fig. 24 deg, 14 min. this is equal to the Semi-Tranſverſe
35
ſec. And December the laſt Day, Axis DC, and is ſo called, becauſe
1700, Old Style, at Noon, the Sun's it is a Mean between the Planet's
mean Motion was 9 fig. 20 deg. 43 greateſt and leaft Diſtance from the
min. 50 ſec. That of the Sun's A- Sun.
pogæum 3 fig. 7 deg. 44 min. 20
MEAN DIAMETER, in Gauging,
fec. The mean Motion of the Moon is a Geometrical Mean between the
10 fig. 15 deg. 19 min. 50 ſec. Of Diameters at Head and Bung in any
the Moon's Apogæum 11 fig. 8 deg. cloſe Caſk.
18 min. 20 ſec. And of the aſcend MEAN and ExTREAM PROPOR-
ing Node 4 fig. 27 deg. 24 min. 20
See Extream and Mean Pro-
ſec. For in twenty Julian Years, portion.
or in 7305 Days, the Sun goes thro ' MEAN or MIDDLE PROPOR-
20 rev. 9
min. 4 fec. The Motion TIon between any two Lines or Num-
of the Sun's Apogæum 21 min. The bers, is that which hath the ſame
Moon's Motion 247 rev. 4 fig. 13. Proportion to a third Term, that the
deg. 34 min. 5 ſec. The Motion of first bears to it.
the Moon's Apogæum 2 rev. 3 ſig.
1. Thus 6 is a mean Proportional
3 deg. 50 min. 15 ſec. Ofthe Node, between
4.
and
9,
becauſe
i rev. 26 deg. 50 min. 15 ſec. All 6:9.
the aforeſaid Motions are from the 2. The Square of a mean Pro.
Point of the vernal Equinox. And portional is equal to the Rectangle
if from them be ſubtracted the under the Extremes.
Proceffion, or Retrograde Motion of 3. Two mean Proportionals be-
the Equinoctial Point itſelf, which tween two Extreams cannot be
was moved in the mean Time in found by a ſtraight Line and a Cir-
Antecedentia, viz. 16 min. 40 ſec. cle; but it may be done by the
the Motions will remain in reſpect Conic Sections very eaſily, or by the
of the fixed Stars in 20 Julian Conchoid, or Cifroid.
Years ; the Motion of the Sun, 19 MEASURE, in Muſic, is a Quan-
rev. u 1 fig. 29 deg. 52 min. 24 ſec. tity of the Length and Shortneſs of
That of the Sun's Apogæum 4 min. Time, either with reſpect to natural
20 ſec. of the Moon, 247 rev. 4 fig. Sounds pronounced by the Voice,
13 deg. 17 min. 25 fec. Of the or artificial, drawn out of Muſical
Moon's Apogæum, 2 rev. 3 fig. 3 Inſtruments; which Meaſure is ad-
deg. 33 min. 35 ſec. Of the Moon's juſted in Variety of Notes, by a
Node, i rev. 27 deg. 6 min. 55 ſec. conſtant Motion of the Hand' or
MÉAN DISTANCE of a Planet Foot, down or up, ſucceſſively and
from the Sun, is the Right Line SP, equally divided ;
Down or Up, is called a Time or
Meaſure, whereby the Length of a
Semi-Breve is meaſured, which is
therefore termed the Meaſure-Note,
P
or Time-Note,
MEASURE of an Angle, is an Arch
5
of a Circle deſcribed about the An-
gular Point
Measure of a Number, is the
3
Number
ſo that every
M E A
Number that meaſures it; as 2 is whoſe Side is one Inch, Foot, -Yards
the Meaſure of 4.
or other determinate Length.
MEASURE of a Ratio, is a Loga MEASURE of a Superficies, or
rithm.
plain Figure, is a Square, whoſe Side
MEASURE of a Solid, is a Cube, is one Inch, Foot, I ard, &c.
Here follow feveral very uſeful TABLES of
different Meaſures.
A TABLE of the Foreign Meafures, carefully compared with
0
the ENGLISH.
1
008
0
1
2
OO
o
II
O
II
O
O
II
Suppoſe an Engliſh The Engliſh Foot di-
Foot divided into vided into Inches,
rooo equal Parts, and Decimal
thofe bere men Parts of an Inch.
tioned are in Pro-
portion to it, as
follows.
London
Foot
1.000
O I 2
Paris
the Royal Foot 1,068
Amſterdam
Foot
.942
3
Brill
Foot 1.103
ΟΙ 2
Antwerp
Foot .946
11. 3
Dort
Foot
1.184
@ 02
Rynland, or Leyden
Foot
1.033
I
4
Lorrain
Foot
.958
4
Mechlin
Foot
.919
Middleburgh
Foot
.991
II 9
Straſburgh
Foot
.9.20
Bremen
Foot
.964
II 6
Cologn
Foot
.954
II 4
Frankfort ad Mænam Foot
4
Spanish
1.001
Toledo
Foot
.899
7
Romàn
Foot
.967
6
On the Monu Ceſtucicus
O 11
ment of Stavilius,
.972
Bononia
Foot
1.204
OZ
4
Mantua
Foot 1.569
06 8
Venice
Foot
1.162
9
Dantzick
Foot
.944
3
Copenhagen
Foot
965
6
Prague
Foot
1.026
OO
3
Riga
Foot 1.831
099
Turin
1.062
1
7
The Greek
1.007
00
Paris Foot, according to Dr.
Bernard
2
.948
Foot
1
OO
0
10
O
11
{
}
7
I
I
I
OI
II
1
I
00
Foor
Foot
Y
I
1.056
Uni-
ΜΕ Α
3.976
2.076
3
2.260
Univerſal
Foot
1.089
Old Roman
Foot
970
Bononian Fool of M. Auzout
1.140
Lyons
Ell
Bologn
El
Amiterdam
ElL
2.269
„Antwerp
EIL
2.273
Rynland, or Leyden Ell
Frankfort
El
1.826
Hamburgh
EN
1.905
Leipfick
EN
2.260
Lubeck
Ell
1.908
Noremburgh
EN
2.227
Bavaria
Ell
.954
Vienna
Ell
1.053
Bononia
EIL
2.147
Dantzick
Ell
1.903
Florence
Brace, or Ell
1.913
Spaniſh, or Caftile
Palm
•751
Spaniſh Vare, or Rod, which is
four Palms
Liſbon
Vare
Gibraltar
Vare
Toledo
2.685
Palm
.861
Naples
Brace
2.100
Canna
6.880
Genoa
Palm
.830
Milan
Calamus
Parma
Cubit '1.866
China
Cubit
1.016
Cairo
Cubit 1.824
Babylonian
Old Greek Cubit
Roman
Turkiſh
Pike
2.200
Perlian
Aral
3.197
2 oo 8
2 03 2
2 00
2
2 03 I
1
09 9
1
10 8
2 03 1
1
09 8
2 03 3
II
4
1 00 6
2
OI
7
1 10 8
1
O
O 09 0
1
11
is}
3 001
I
00
O
2.750
2.760
Vare
)
09 6
2 09
2 09 I
2
08
2
009
2 01 2
6 10
O 09
o6
4
1
2
1 099
1
06
Teen
1
06
13
종
​05
2 02
4
· 3 02
3
Ал оли моля не
6.544
10
00
{
"}
{
Toto
A TABLE
MEA
A TABLE of Engliſh Long Meaſure.
Inches.
r
3 Palm
3 Span.
9
Foot.
12
i
4
6
18
Cubit.
2
1 2
2 |Yard.
21
1
60
Pace.
20
1779
OM
2
I
36
4 3
45
Ell.
5
3
6
5 35 1 2 3
72
24
8 6
4
198 66
16
5 12 / 417 2
Pole.
660
440
176 132 100
40
Furl.
6336012112017040 15 280 13520 11760 11408 1056 1880 1320 8 M.
Fatb.
22
II
2
220
79202640 880
.
A TABLE of Square Meaſure.
Inches Square.
144) Feet-17.
1296 9
Yards fq.
2 77 Paces so
39204 272.25 30.25 10.89 Poles fa.
1568162 10890
435.6 40 Rood Sq.
43560
4840 1742.4
4 Acres 19.
30976001115136102400 2560 640 Miles.
360
25
1210
1
160
A TABLE of Dry Meaſure.
}
Pints,
8 Gallons.
16
8
16
2-1 Pecks.
64
128
256
32
2
4. Bufbels.
8 2 Strikes.
16
Carnock,
4
or Coom
Seem, or
32
8
4
Quarter.
48 24
6
320
80
40
12 Laft.
512
64
2
102
I 2
3072
5120
384
640
20
TO
itt.
8tb.
16
| 101
64 128 | 256 | 512 3072 5120 llroy.
56 11c.
26. 4. C. 124 C. 40 C. Aver:
122
14. pz
7tb.
14
A
1
M E A
A TABL'E of Wine-Meaſure, Honey, Oil
, &c,
Pints.
8
Gall.
!
18
144
1
Rundl.
13
35
I
3}
43
2
Barrels.
Terces.
15
I ( 1 / 2
22
3 4
6 8
Hogſhead.
Punch. 1
Batt, or Pipe
1 2
Tun, 12 3 4
;
31 252
42 336
63
504
84 672
726 1008
252
2016
. 8
2
2
7
14
ATABLE for Beer-Meaſure.
Pints.
Gall. 8
Firk.
9 72
Kilderk.
22 18
144
Barrels.
2 4 / 36
288
Hogſhead. 2
4
8
72576
A TABLE for Ale-Meafure.
Pints.
Gall. 8
Firk.
9
64
Kilderk.)
128
Barrels.
2 4) 36256
Hoghead.) 2 8)
2
18
4
72
512
MECHANICS,
MEC
M E C
MECHANICS, is the Geometry As to the Deſcriptions of Ma-
of Motion, being that Science which chines, we have Strada, Zeifingius,
ſhews the Effect of Powers or mov- Beſſon, Auguſtine de Ramellis, Boetler,
ing Forces, ſo far as they are ap- Leopold, Sturmy, Perrault, Limbergh,
plied to Engines, and demonſtrates &c.
the Laws of Motion.
MECHANIC Powers, as they
Mechanics was very imperfect a are called) are fix, viz. the Ballance,
mongſt the Ancients. All that is the Leaver, the Wheel, the Pulley,
to be found of theirs upon this Sub- the Wedge, and the Screw; to ſome
ject, are Archimedes
, de Centro Gra or other of which, the Force of all
vitatis Figurarum Planarum, and mechanical Inventions muft necef-
Pappus in lib. 8. Colleet. Mathemat. ſarily be reduced. See thoſe Words.
of the five mechanical Powers: nor MECHANICAL PHILOSOPHY, is
have ſome of che more modern Au- the ſame with the Corpuſcular,
thors done much more ; ſuch as
ſuch as which endeavours to explicate the
Guido Ubaldus's Liber Mechanicorum. Phänomena of Nature from mecha-
- Rohault's Tractatus de Mechanica. nical Principles, i.e. from the Mo-
---Lamy's Mechanics in French. tion, Reſt, Figure, Poſition, Magni-
Oughtred's Mechanical Inſtitutions. tude, &c. of the minute Particles of
-Cafatus's Mechanica.
Matter. And theſe Principles are
Further Improvements are to be frequently called
found in Gallileo's Mechanical Dia. MECHANICAL CAUSES: And
logues.- Torricellius's Libri de Motu alſo the
Gravium naturaliter Deſcendentium MECHANICAL AFFECTIONS of
& Projectorum—- Balianus's Tracta- Matter.
tus de Motu naturali Gravium. MECHANICAL CURVE, is one
Huygens's Horologium Oſcillatorium. whoſe Nature cannot be expreſs'd
-Leibnitz's Reſijtentia Solidorum, in by an Algebraic Equation.
Acta Eruditor. An. 1684. p. 319. and MECHANICAL SOLUTION of a
Varignon's Papers in the Comment. Problem, in Mathematics, is either
Academ. Reg. Scienc. An. 1702. p.87. when the thing is done by repeated
-Borellus's Tractatus de Vi Percuſſion Trials, or when the Lines made ule
nis; de Motionibus Naturalibus a of to ſolve it, are not truly geome-
Gravitate pendentibus ; de Motu A- trical. Thus the Method of Nico-
nimalium - Huygens's Tractatus de medes, Eratoſthenes, Pappus, and
Motu Corporum ex Percuſione. Vieta, for finding two mean Pro-
Wallis's Tractatus de Mechanica, portionals ; and that of Nicodemus
seckon'd by ſome a very good Piece. and Dinoſtratus, for dividing an
- Keil's Introduction to true Philo- Angle into any Parts affigned, by
ſophy. - De la Hire's Mechanics. means of the Quadratrix, is mecha-
Mariotte's Traité du Choc des Corps nical ; becauſe the former is done
-Dechales's Treatiſe of Motion. by repeated Trials, and the latter
Pardies's Diſcourſe of Local Motion, by means of a Curve that is not
- Parent's French Elements of Me- truly geometrical.
chanics and Phyſics.---Sir Iſaac New Medium,in Natural Philoſophy,
ton's Principia.- Ditton's Laws of fignifies that peculiar Conftitution
Motion.-- Herman's Pbronomia. of any Space or Region through
s'Graveſande's Phyſics.-Euler's Tra- which Bodies move. Thus the Æ.
Etatus de Motu.- Defaguliers's Me- ther is fuppofed by ſome to be the
chanics. --Muſchenbroeck's Phyſics, Medium in which the Planets and
heavenly Bodies move; and by the
means
&c.
M E R
M ER
means of which it is, that all Ani- rately) the Diſtance of any Parallel
mals, as Ibſects; Birds, Beaſts, and of Latitude from the Equator is to
Mien, can breathe and live: But that Arch of Latitude (extended
Water is the Medium in which into a right Line) as the Curve-
Fiſhes live and move. Glaſs is alſo lin'd Space contain'd under the Ra-
called a Medium.
dius, ſo much of the Curve of the
MEMBRESTO, in Architecture, Figure of the Secants (ſee that Word)
is the Italian Term for a Pilarter, as is cut of by an Ordinate, raiſed
that bears up an Arch. Theſe are at the Extremnity of the right.lin'd
often Auted, but not with above Abſciſs or Arch of Latitude, that
feven or nine Channels. They are Abſciſs and that Urdinate, is to the
frequently uſed to adorn Door-Caſes, Rectangle under the Radius, and
Gallery - Fronts, and Chimney- the ſaid Abſciſs.
Pieces, and to bear up the Corniſhes 1. Though the plain Chart be
and Freszes in Wainſcot.
very eaſy and uſeful in ſhort Voy-
MENISCUS GLASSES, are thoſe ages, if you fail home in or near
which are Convex on one ſide, and the oppoſite Rhumb you went by,
Concave on the other..
as the Ancients, who being Coaſters,
As the Difference of the Seini- did before the Uſe of the Compaſs;
Diameters of the Convexity and yet foraſmuch as few Places, or in-
Concavity, to the Semi-Diameter of deed none, but ſuch as lie under
the Concavity, ſo is the Diameter the Equinoctial, can therein be ex-
of the Convexity to the Focal preſſed according to their true si-
Length
tuation and Diſtance one from ano-
MENSUR ABILITY, is an Apti- ther; but if they, be laid down
tude in a Body, whereby it may be true by the Courſe and Diſtance,
applied or conformed to a certain the Difference of Longitude will be
Meaſure
| falſe; if they be laid down by the
MENSURATION,
or Measu- Courſe and Difference of Longitude,
RING, is to find the ſuperficial then will che Diſtance and Difference
Area, or ſolid Content of Surfaces of Latitude be more than it ſhould
and Bodies.
be; and if they be laid down by
MERCATOR'S CHART, or PRO- the Diſtance and Difference of Lon-
JECTION, is a Projection of the gitude, (which in many Cafes' is im-
Face of the Earth in plano, wherein poffible,) then the Difference of La-
the Meridians, Parallels, and Rhumb- titude will always be too little, and
Lines, are all ſtraight Lines, and the Rhumb too wide from the Me-
the Degrees of Longitude are all ridian; and if they be laid down
equally diſtant from cne another; by their Latitude and Departure,
but the Degrees of Latitude increaſe then the Courſe will be wide, and
towards the Poles in the fame Pro- the Diſtance too much, &c
portion, that the Parallel-Circles on 2. It was the great Study of our
the Globe decreaſe, viz, in the Ra- Predeceſſors to contrive ſuch a Chart
tio of the Radius to che Sine. Com- in plano, with ſtraight Lines, on
plement of the Latitude ; or, the which all or any parts of the World
Diſtance of any Parallel of Latitude might be truly ſet down, according
from the Equator, is always as the their Longitudes, Latitudes,
Sum of all the Secants anſwerable Bearings, or Diſtances.
to every Point in that Arch of Lati-
3. A way was hinted for this near
tude to the ſame Sum of ſo many two thouſand Years ſince by Ptolemy',
times the Radius, or (more accu- and a general Map according there-
to,
to
1
1
/
or returns.
M ER
M E R
to, made in the preceding Age by ABC, the Line Ac repreſents the
one Mercator, but the thing de- Diſtance: A b the Difference of La-
monſtrated, and a ready way ſhewed titude, bc the Departure ; as in
of deſcribing it, was not till Mr.
Wright taught to enlarge the Meri-
A
dian-Line by the continual Addi-
tion of Secants ; fo that all Degrees
of Longitude might be proportional
to thoſe of Latitude, as on the Globe:
Which he has done after ſuch an
excellent manner, that in many re-
C
ſpects it is far inore convenient for
the Navigator's Uſe, than the Globe
itſelf, and will truly ſhew the Courſe
and Diſtance from Place to Place,
B
which way foever a Ship fails forth, ,
plain Sailing; AB the Meridional
4. The Meridian Line, in Merca. Difference of Latitude, according
tor's Chart, is a Scale of Logarith- to the true Chart, commonly called
mic Tangents of the Half-Comple- Mercator's Chart, and BC 'the Dif.
ments of the Latitude.
ference of Longitude.
The Differences of Longitude on
For the Departure as the Radius
'any Rhumb, are the Logarithms to the Diſtance Ac: ſo is the Sine
of the ſame Tangents, but of a dif- of the Courſe b Ac, to bc the De-
ferent Species; being proportioned
parture.
to one another, as are the Tangents For the Difference of Latitude :
of the Angles made with the Me. As the Radius to the Diſtance Ac,
ridian.
fo is the Sine of Acb, the Comple-
Hence any Scale of Logarithmic ment of the Courſe, to the Difference
Tangents is a Table of the Diffe- of Latitude Ab.
rences of Longitude, to ſeveral La-
For the Difference of Longitude,
titudes, upon ſome determinate
As the Radius, to AB the Meridio-
Rhumb or other ; and therefore, as nal Difference of Latitude, ſo is the
the Tangent of the Angle of ſuch a Tangent of the Courſe B AC, to
Rhumb, to the Tangent of any o BC the Difference of Longitude.
ther Rhumb; fo is the Difference
2. Both Latitudes and Courſe gi-
of the Logarithms of any two Tan venį to find the Diſtance, Departure,
gents, to the Difference of Longi- and Difference of Longitude.
tude on the propoſed Rhumb, inter For the Diſtance: As the Radius
cepted between the two Latitudes, to Ab the Difference of Latitude ;
of whoſe Half-Complements you ſo is the Secant of 6 A c the Courſe,
took the Logarithmic Tangents. to Ac the Diſtance
Here follow the ſeveral Cafes, and
For the Departure: As the Ra-
dius is to the Diſtance Ac; ſo is
their Proportions, in Mercator's
Sailing.
the Sine of b Ac the Courſe, to bo
the Departure.
1. One Latitude, Courſe, and For the Difference of Longitude :
Diſtance given : to find the other As the Radius is to AB, the Meri-
Latitude, Departure and Difference dional Difference of Latitude; ſo is
of Longitude.
the Tangent of BAC the Courſe, to
In the Right-angled Triangle BC the Difference of Longitude.
3.
Аа
MER
M ER
3. Both. Latitudes and Diſtance Sine of b Ac the Courſe, to the De-
given : to_find the Courſe, Depar- parture bc.
ture, and Difference of Longitude.
6. Both Latitudes and Departure
For the Courſe: As the Diſtance given: to find the Courſe, Diſtance,
Ac is to the Radius, ſo is Ab the and Difference of Longitude.
Difference of Latitude, to the Sine For the Courſe: As the Diffe-
of Acb the Complement of the rence of Latitude Ab, is to the Ra-
Courſe,
dius, ſo is the Departure bc, to the
For the Departure : As the Ra. Tangent of b A c the Courſe.
dius is to the Diſtance Ac, ſo is the For the Diſtance: As the Sine
Sine of b Ac the Courſe, to bc the of Ach the Complement of the
Departure.
Courſe, to the Difference of Latitude
For the Longitude: As the Ra- Ab; ſo is the Radius to the Diſtance
dius is to the Meridional Difference Ac.
of Latitude AB, fo is the Tangent For the Longitude: As the Ra-
of BAC the Courſe, to BC, the dius is to the Meridional Difference
Difference of Longitude.
of Latitude AB; ſo is the Tangent
4. Both Latitudes and Difference of BAC the Courſe, to the Dif-
of Longitude given: to find the ference of Longitude BC.
Courſe, Diſtance, and Departure. MERCATOR'S SAILING, is the
For the Courſe: As A B the Me Art of finding on a Plane the Mo-
ridional Difference of Latitude, is tion of a Ship upon any aſſigned
to the Radius ; fo is BC, the Diffe- Courſe, true in Longitude, Latitude,
rence of Longitude, to the Tangent and Diſtance; the Meridians being
of BAC, the Courſe.
* all parallel, and the Parallels of La-
For the Diſtance: As the Ra- titude ſtraight Lines.
dius is to A B the Difference of La MERCURY, is the Name of one
titude, fo is the Secant of BAC the of the Planets, revolving about the
Courſe, to Ac the Diſtance.
Sun.
For the Departure: As the Ra MERCURY, is the leaſt diſtant
dius is to Ac the Diſtance ; fo is from the Sun of any of the Planets;
the Sine of b Ac the Courſe, to bc its mean Diſtance from the Sun is
the Departure. Or, as the Radius 387 of ſuch Parts of which the
is to Ab the Difference of Latitude; Earth's is 1000, its Excentricity 80,
ſo is the Tangent of b Ac the Courſe, the Inclination of its Orbit is 6 deg.
to bc the Departure.
52 min. It performs its Revolution
5. One Latitude, Courſe, and round the Sun in 87 Days, 23 Hours.
Difference of Longitude given, to Its greatelt Elongation is about 28
find the other Latitude, Diſtance, Degrees. There has not yet been
and Departure.
obſerv’d any Spots in it; neither do
For the Latitude: As the Tan we krowy whether it revolves about
gent of BAC the Courſe, is to BC its Axis ; but it is probable it does.
the Difference of Longitude ; fo is Its Magnitude to that of the Earth
the Radius, to the Meridional Diffe- is as 216 to 3+3.
rence of Latitude A B.
In the Years 1736, 1743, 1756,
For the Diſtance: As the Radius 1769, 1776, 1782, 1789, in October,
is to Ab, the Difference of Latitude; this Planet will be ſeen in the Sun
ſo is the Secant of bAc the Courſe, near the aſcending Node ; and in
to Ac the Diſtance.
the Years 1753, 1786, 1799, it
For the Departure : As the Ra- will appear in the Sun, in the
dius is to Ab the Diſtance, ſo is the Month of April, near the other
Node.
Meri-
1
1
a
M E R
M E R
: MERIDIAN, is a great Circle And in the Centre C'E erect a Pin
paffing through the Poles of the of about a Foot long, perpendicu-
World, and both Zenith and Nadir,
crofling the Equinoctial at Right
Angles, and divides the Sphere into
two equal Parts, one Eaſt, the other
West, and hath its Poles in the Eaſt
and Weſt Points of the Horizon.
'Tis called Meridian, becauſe when
dl
E
the Sun comes to the South Part of
D
this, Circle, 'tis then Meridies, Mid-
B
Day, or High-Noon; and then the
Sun hath his greateſt Altitude for
that Day, which therefore is cal-
B.
led the Meridian Altitude.
Theſe Meridians are various, and
change according to the Longitude lar to the Plane. About the time
of Places ; ſo that they may be ſaid of the Tropics before Noon, from
to be infinite in Number : for that Nine to Eleven, and in the After-
all Places from Eaſt to Weſt have noon, from about One to Three,
their ſeveral Meridians ; but there mark the Points B, b, &c. A, a,
is (or ſhould be) one fixed, which is &c. wherein the Shadow of the
called the Firſt Meridian.
Pin terminates, and bifect the Arches
MERIDIAN on the Globe or Sphere, AB, ab, &c. in D, d, &c. Then
is repreſented by the Brazen Circle, if the ſame ſtraight Line D E does
in which the Globe hangs and biſect all the Arches AB, ab, &c.
turns. 'Tis divided into four go's, that will be the Meridian Line
or 360 Degrees, beginning at the fought.
Equinoctial on it. Each
This Method would be very exact,
way
the Equinoctial, on the Celeſtial if the Sun moved as the fixed Stars
Globes, is counted the South and do; but becauſe the Sun hath a
North Declination of the Sun proper Motion, as a Planet, there
or Stars ; and on the Terreſtrial will be ſome inconſiderable Error,
Globe, the Latitude of Places, North which yet may be corrected; for
or South.
ſince the Sun in one Minute of an
Upon the Terreſtrial Globes there Hour moveth as much by his daily
are uſually drawn 36 Meridians, Motion, as he loſeth in fix Hours
thro' every 10th Degree of Longi- by his proper Motion, you ſhall
tude.
add as much in the Way which the
MERIDIAN Line, is the com- Shadow goes in the laft Marks, as
mon Section of the Meridians, and that Shadow moveth in one Minute,
the Plane of the Horizon, and ſo which you may meaſure by a Pen-
runs on North and South.
dulum ; fo the laſt Points will not
1. To draw a Meridian Line, be taken juſt in the Circles, but a
there are ſeveral ways, and many little without them.
Inſtruments have been contriv'd for If A B, AC, and AD, be three
that purpoſe ; but the following Shadows, made in one Day, upon
Method is a very eaſy and good an Horizontal Plane, by the Pin
In an horizontal Plane, which A E, perpendicular to that Plane,
is eaſy to determine, deſcribe feve- the Meridian Line may be drawn
ral Concentric Circles BA, ba, &c. thus :
Аа 2
from
one.
If
M E R
M ER
.
If two of theſe Shadows are equal, MERIDIAN Line, on Gun.
then the Line drawn from the Point Ter's SCALE, is divided unequally
A, perpendicular to a Line joining towards 87 Degrees, (whereof 70
their Extremes, will be the Meri- Degrees are about one half) in ſuch
dian; but if not, let AC be the manner as the Meridian in Merca-
leaft
. In the Point A you muſt tor's Chart is divided and number'd.
raiſe the Lines AF, AG, and AH,
Its Uſes are many. For, i. It
perpendicular to AB, AC, and AD, ſerves to graduate a Sea-Chart ac-
and equal to AE, and join FB, cording to the true Projection. 2.
GC, HD. Now from FB, HD, Being joined with a Line of Chords,
take FI, HK, equal to GC; and it ſerves for the Protraction and
from the Points I and K draw the Reſolution of ſuch right-lined Tri-
Right Lines IL, KM, perpendicu- angles as are concerned in Latitude,
lar to AB, AD, and from the Longitude, Rhumb, and Diftance,
Points L, M, you muſt let fall two in the Practice of Sailing; as alſo in
pricking the Chart truly at Sea.
IP
MERIDIAN (MAGNETICAL) is
a great Circle paſſing through or by
the Magnetical Poles ; to which Me-
ridians, the Compafs (if not other-
wiſe hinder'd) hath reſpect.
MERIDIONAL DISTANCE, in
Navigation, is the ſame with the
Departure, Eaſting or Weſting, or
under which the Ship now is, and
I B
any other Meridian ſhe was before
N
under.
MERIDIONAL PARTS, Miles,
or MINUTES, in Navigation, are
E
the Parts by which the Meridians
Q ୧
in Mercator's Chart do increafe, as
the Parallels of Latitude decreaſe.
H
And the Co-fine of the Latitude
of any Place being equal to the Ra-
M
dius or Semi-Diameter of that Pa-
rallel, therefore in Mercator's Chart,
K
this Radius being the Radius of the
Equinoctial, or whole Sine of 90°,
D)
the Meridional Parts at each of the
Arches contained between that Lati-
more Perpendiculars LN, MO, to tude and the Equinoctial do decreaſe.
the Line joining L and M, which
The Tables therefore of Meridio-
let be equal to LI and M K. Now, nal Parts, which you have in Books
Jet P be the Interſection of the Lines of Navigation, are made by the
joining the Points M, L, and O.N.. continual Addition of Secants, and
Then if a Right Line be drawn calculated in ſome Books (as in Sir
thro P and C, a Perpendicular AQ Jonas Moore's Tables) for every
from A to the Line CP, will be the Degree and Minute of Latitude; and
Meridian. See the Demonſtration of there will ſerve either to make or
this in Van Schouten's Exercitationes graduate a Mercator's Chart, or to
Geometrica,
work Mercator's Sailing.
L
F
MERLON,
MIC
MIC
Merion, in Fortification, is that a half, that are contained between
Part of the Parapet which lies be- the two parallel Hairs of the Mi-
twixt two Embraſures, being from crometer in the focus of the Object
cight to nine Foot long on the fide , Glaſs of a Teleſcope, are proporcio-
of the Cannon, and fix on the ſide nal to the Revolutions of the Index
of the Field ; as alſo fix Foot high, required to ſeparate the Hairs, ſo
and eighteen thick.
as to catch thoſe Diameters or Di.
MESOL ABIUM, is the Name Itances.
of an Inſtrument for finding mean Concerning this Inſtrument, ſee
Proportionals.
what Mr. Auzout ſays in a little
METAL. The Outſide or Sur. Treatiſe of it contain'd in divers
face of a Piece of. Ordnance is called Ouvrages de Mathematique & de Phi-
the Superficies of her Metal: When fique, par Meſſieurs de l'Academie
the Mouth of a great Gun lies lower Royale des Sciences, Mr. de la Hire's
than her Breech, they ſay, the lies Aſtronomicæ Tabula ; Mr. Townley,
under Metal; but if ſhe lies truly in the Philof. Tranſact. N° 21.
level, point blank, or right with Wolfius, in his Elem. Affron. §. 508.
the Mark, they ſay, ſhe lies right Dr. Hook, in the Philofoph. Tranſait.
with her Metal..
N° 29. Mr. Hevelius, in the Aeta i
METOPs, is the ſquare Space be- Eruditorum, Anx. 1708. Mr. Bel-
tween the Triglyphs of the Doric Maſer, in his Micrometria. But the
Freeze, which among the Ancients Micrometers of the ingenious Mr.
uſed to be adorned with the Heads George Graham, are far better than
of Beaſts, Bafons, Vaſes, and other thoſe of any body elſe, both as to
Inſtruments uſed in facrificing. A Structure and Workmanſhip.
Demi-Metops is a Space ſomewhat MICROPHONES, are Inſtruments
leſs than half a Metops, at the Cor-. contrived to magnify ſmall Sounds,
ner of the Doric Freeze.
as Microſcopes do ſmall Objects.
MICROCOUSTICs, the ſame with Microscope, is a dioptric In-
Microphones.
ftrument, by which minute Objects
MICROMETER, is an Inſtrument are very much augmented, and ſeen
fitted to a large Teleſcope in the diſtinctly. Some of theſe are called
Focus of the Object-Glaſs, for mea- fingle ones, being ſuch that have but
furing the apparent Diameters of one ſmall Lens; others are com-
the Celeſtial Bodies, and ſmall Di- pound ones, conſiſting of ſeveral
ſtances that do not exceed a De. Lens's.
gree, or a Degree and an half.
1. We are uncertain where and
There are ſeveral ſorts of theſe by whom Microſcopes were invent-
Inſtruments, whereof ſome are ed; but this we know, that they
Movements conſiſting of a Plate or were unknown till the Year 1613,
Face divided like a Clock or Watch, becauſe, Hieronymus Surturus, 'who
with an Index or Hand, which being writ a Book that Year of the In-
turn'd, moves two fliding Plates of vention and Fabrick of the Te-
Braſs that carry two parallel Hairs, leſcope, makes no mention of them.
and counts on the Plate the Revolu 2. Mr. Huygens, in his Dioptrics,
tions of the Screws that move the will have one Drebbel, a Dutchman,
Plates, whoſe Threads are extreme to be the inventor, of the Double or
ly fine.
Compound Microſcope in the Year
The apparent Diameters for the , 1621; and Franciſcus Fontana, a
Diſtances of any Objects that are Neapolitan, in a Book of Obſerzia-
leſs than a Degree, or a Degree and tions, publiſhed by him in the Ver
16,6,
Аа 3
MIC
MIC
1
E
1646, fays, that he himſelf happen.' Lens's. If the Diameter of a Sphere
ed upon the Invention of the Com- be is of an Inch, it will magnify
pound Microſcope in the Year the Diameter of an Object in the
1621.
Ratio of 1 to 170 nearly; the Su-
3. If an Object be placed in perficies in the Ratio of i to 2890o,
the Focus of the Convex-Lens of a and the Solidity in the Ratio of i
ſingle Microſcope, and the Eye be to 4913000.
very near on the other ſide, the 6. The more an Object is ampli-
Object will appear diſtinct in an e- fied by a Microſcope, the leſs Part
rect Situation, and augmented in the thereof is comprehended' at one
Ratio of the Focal Diſtance of the view.
Lens, to ſuch a Diſtance, at which, 7. The Appearance of any given
if the Object was placed, the naked Object, formed by any given Glaſs
Eye would perceive it diſtinctly, or Combination of Glaſſes, becomes
which is about eight Inches for obſcure in ſuch proportion as its
good Eyes.
Magnitude increaſes.
4. If the Object AB be placed 8 Equal Appearances of the fame
in the Focus F, of a ſmall Glaſs Object, formed by different Combi-
Sphere, and the Eye be put in the nations, become obſcure in ſuch
Focus G, the Object will appear di- proportion, as the Number of Rays
conitituting each Pencil decreaſes,
F
that is, in proportion to the Small-
AH -IB
neſs of the Object-Glaſs.
9. Wherefore, if the Diameter of
the Object-Glaſs exceeds the Dia-
meter of the Pupil, as many times
as the Diameter of the Appearance
exceeds the Diameter of the Object;
the Appearance ſhall appear as clear
and bright as the Object itſelf.
10. The Diameter of the Object-
I
Glaſs cannot be ſo much increaſed,
without increaſing at the ſame time
G
the focal Diſtances of all the Glaſſes,
and conſequently the Length of the
Inſtrument: Otherwiſe the Rays
ftinet, and in an erect Poſture aug- would fall too obliquely upon the
mented, as to Diameter in the Ra- Eye-Glaſs, and the Appearance be-
tio of 1 of the Diameter E I to the come confuſed and irregular.
Diſtance of about eight Inches. If 11. Sir Iſaac Newton, in his Op-
the Diameter of the ſmall Sphere tics, Book II. Part III. ſays, That
be is of an Inch; then CE
if Microſcopes are or can be ſo far
and FEM to, and ſo FC =
to improved as with ſufficient Diſtinct-
Whence the true Diameter of the neſs to repreſent Objects five or fix
Object to the Apparent, is as I to hundred times bigger than at a Foot
103 nearly.
Diſtance they appear to the naked
3. Microſcopes made of ſmall Eye; he hoped that we might be
Glaſs Spheres will magnify Objects able to diſcover ſome of the greateſt
more than thoſe made of Lens's ; of the Corpuſcles of Bodies; and by
becauſe ſmall Glas Spheres may be one which would magnify three
made far more little than thoſe of or four thouſand times, perhaps, all
3,
thoſe
232
1
3
M.ID
M I L
hoſe that produce Blackneſs might IG be drawn; then the Angle
"be diſcovered. And if this could IGH is the Courſe, G I the Di.
be attained to, (viz. by Glaſſes to
diſcover the Conſtituent Particles of
G
Bodies) he fears it would be the ut-
moſt Improvement of this Senſe of
ſeeing; for it ſeems impoſſible to
1
H
ſee the moſt ſecret and noble Works
of Nature within the Corpuſcles,
becauſe of the Tranſparency of the
Corpuſcles.
12. The fame Gentleman in the
Philofoph. Tranſ. N° 88, from the Dif-
R
ference he had found between com-
pound and ſimple Colours, takes oc-
cafion to communicate a way for ſtance, and IH the Departure in
the Improvement of Microſcopes by middle Latitude Sailing. And
Refraction, viz. by illuminating the
As the Radius is to RI the Dif-
Object in a darken'd Room with ference of Longitude, ſo is the Sine
Light of any convenient Colour not of HRI the Complement of the
too much compounded ; by which middle Latitude, to HI the Depar-
means the Microſcopes will with ture ; and as GH the Difference
Diſtinctneſs bear a deeper Charge, of Latitude is 'to the Radius, fo is
and a larger Aperture.
HI the Departure to the Tangent
Some of the Writings about Mi- of HGI the Courſe.
croſcopical Obſervations, are Fran And as the Sine of HGI. the
cifcus Fontana's Obſervationes cæ- Courſe, to IH the Departure,
leftium terreſtriumque Rerum. Hook's ſo is the Radius to IG the Di-
Micrography
. Malpighius's Anato- ftance.
mia Plantarum; his Tractatus de MILKY-WAY, VIA LACTEA,
Qvo incubato, de Bombyce, de Viſcerum or GALAXY, is a broad white
Atructura. Leewenhoeck's Arcana Path or Track, encompaſſing the
Naturiæ deteta, Bonanni's Micro- whole Heavens, and extending it.
graphia curiofa.
ſelf in ſome Places with a double
MIDDLE LATITUDE, in Navi. Path ; but for the moſt part with
gation, is half the Sum of two La- a ſingle one. Some of the Ancients,
titudes. And
as Ariſtotle, &c. imagined that this
MIDDLE LATITUDE SAILING, Path confifted only of a certain
is the manner of ſolving the ſeveral Exhalation hanging in the Air ;
Cafes of Mercator's Sailing, without but by the Teleſcopical Obſervations
the Meridional Parts, by taking the ic hath been diſcovered to conſiſt of
middle Latitude ; and this nearly an innumerable Number of fixed
agrees with Mercator's Sailing. Stars, different in Situation and
If the Line GR be drawn, and Magnitude, from the confuſed Mix-
the Angle GRI be made at R, e ture of whoſe Light, its whice Co-
qual to the Coniplement of the mid- lour is ſuppoſed to be occafioned.
dle Latitude : And the Difference of It paſſes chrough the Conitellations
Longitude be ſet from R to I, and of Caffiopeia, Cygnus, Aquila, Perfens,
the Perpendicular IH be let fall, Andromeda, Part of Ophiucus and
and the Difference of Latitude be Gemini, in the Northern Hemi-
ſet off from H to G, and the Line ſpheres ; and in the Southera, it
takes
1
Aa 4
M IN
M I T
takes in Part of Scorpio, Sagittarius, 360 Degrees, with ſeveral Dials grá-
Centaurus, the Argonavis, and the duated thereon, generally made for
Ara.
the Uſe of Miners.
Metrodorus, and ſome Pythago MINIM, a Term in Muſic; being
reans, thought the Sun had once the fourth Note of Time, and is
gone in this Track inſtead of the mark'd thus q.
Ecliptic; and conſequently, that its MINION, à fort of a Cannon, is
Whiteneſs proceeds from the Re- either large or ordinary. The large
mains of his Light. As the Galaxy Minion is one of the longeſt Size,
is compoſed of an Infinity of ſmall and has its Bore three Inches and a
Stars, ſo it hath uſually been the quarter Diameter, and is a thouſand
Region in which new Stars'appear, Pound Weight. Its Load is three
as the Star in Caliopæia, which was Quarters of a Pound of Powder : Its
ſeen A.D. 1572, that in the Breaſt Shot three Inches Diameter, and
of the Swan, and another in the three Pound three Quarters Weight:
Knee of Serpentarius, and ſeveral Its Length eight Foot, and its Level-
others, which have appeared for a Range an hundred and twenty-five
while, and then become inviſible Paces.
again.
The ordinary Minion: Its Bore
MILITARY ARCHITECTURE, is three Inches in Diameter, and
the ſame with Fortification. weighs about eight hundred, or ſeven
MINE, in Fortification, is a Hole hundred and fifty Pounds Weight:
dug or made by a Pioneer under the It is ſeven Foot long : Its
. Load two
Rampart, or under the Face of the Pounds and a half of Powder: Its
Baſtion, whereto there are ſeveral Shot near three Inches Diameter,
oblique and winding Paſſages: When and weighs three Pounds and four
it is finiſhed, divers Barrels of Pow- Ounces ; and it ſhoots point-blank
der are placed therein, together with an hundred and twenty Paces. i
a Train or Saucidge; and the Quan MINUTE, is the both Part of a
tity of Powder is proportioned to Degree or Hour.
the Height and Weight of the Body MINUTE, in Architecture, is
which is to be blown up:
fometimes taken for a part of a
There are alſo Mines ſprung in Module.
the Field, which are called Fougades. MITRE, in Architecture, is the
The Alley or Paſſage of a Mire is Workmen's Term for an Angle,
ufually about four Foot ſquare ; at that is juſt forty-five Degrees, or
the End of which is the Chamber of half a Right one; and if it be a
the Mine, as they call it. The far- Quarter of a Right Angle, they call
ther it is carried on, the more it is it a Half Mitre. And they have an
ſubject to be diſcovered by the Ene- Inftrument made to this Angle,
my. Therefore, 'tis beſt not to aim which they call the Mitre Square;
at mining too far, and to make a with which they ſtrike Mitre Lines
new one where the former takes no on their Quarters or Battens; and
Effect. Concerning theſe, fee Lam. for Diſpatch they have a Mitre-Box,
bion, in his Praxis Architectonica; as they call it, which is made of two
Sz: ire de St. Femy, in his Memoires Pieces of Wood, each about an Inch
d'Artillerie, Tom. I. p.154, and thick, and one is nailed upright
foll. Wolfus, in his Eiement. Pyro- upon the Edge of the other; the
tech. § 147. & feq.
ujiper Piece hach the Mitre-Lines
MINE-DIAL, is a Box and Nee ftruck upon it on both sides, to
dle, with a brats Ring divided into direcł the Saw in cutting the Mitre-
Joints
: do
7
.
I
::
: 4
M OD
M OM
Joints readily, by only applying the Soffit or Bottom of the Drip, in
the Piece into this Box.
the Ionic, Compoſite, and Corinthian
Mix'D-LINED Figure, is one Cornices, and ought to correſpond
conſiſting of ſtraight and crooked to the Middle of Columns. Theſe
Lines.
are particularly affected in the Co-
MIXBD NUMBER, is one that is rinthian Order, where they are al-
part integer, or a whole Number, ways enrich'd with carved Works.
and part a Fraction; as 4%, 10, In the lonic and Compoſite they are
Egc.
'more ſimple, having ſeldom any
MIXED RATIO, or PROPOR- Ornaments, excepting ſometimes a
TION, is when the Sum of the An- ſingle Leaf underneath.
tecedent and Conſequent is com MODULE, in Architecture, is a
pared with the Difference between little Meaſure, by which we mean
Antecedent and Conſequent, as if any Bigneſs or Extent taken at plea-
4 : 3 :: 16:12
ſure, to meaſure the Parts of a Build-
Then
b
::cid
ing by, and is uſually determined
by the lower Diameter of the Co-
28
lumn and Pilaſters. Vignola's Mo-
a to b:a-b :: ctdic-dº
dule, which is equal to the Semi-
· Moat, in Fortification, is a hol- Diameter of the Column, is divided
low Space or Ditch dug round a into twelve Parts in the Tuſcan and
Town or Fortreſs which is to be de- Doric, and into eighteen in the reſt
fended; wherefore, the Length and of the Orders. The Module of Pal-
Breadth often depends upon the Na- ladio, Scammozzi, M. De Cambray,
ture of the Soil, according as it is and M. Deſgodetz, which is likewiſe
marſhy or rocky: But Moats in ge- equal to the Semi-Diameter, is di.
neral may be from fixteen to twenty- vided into thirty Parts.
two Fathom broad, and from fiften MOINEAU, is a Name the French,
to twenty-five Foot deep.
and fome Modern Writers of Forti-
Dry Moat, is that which is deſti. fication, give to a little Plat-Baſtion,
tute of Water, and ought to be which is raiſed before a Curtain that
deeper than one that is full of Water. is too long, and which hath two
Lined Moat, is that whoſe Scarp other Baltions at the Ends of it; for
and Counterſcarp are caſed with a they being out of Muſket-Shot, one
Wall of Maſons-Work lying in Ta or the other muſt be defended by
lus, or a Slope.
ſome ſuch thing as this Moincau or
Flat-bottom'd Moat, is that which Plat-Baſtion.
hath no floping, its Corners being Sometimes the Moineau joins to
ſomewhat rounded. All Moats muſt the Curtain, and ſometimes is dif-
be well flanked, and in general ſo joined from it by a Moat.
wide, as that no Ladder, Tree, &c. MOMENTS, are ſometimes taken
can reach a-croſs them. If the for the least and moſt inſenſible Parts
Ditch be dry, or has but little War of Time; as when we ſay, ſuch a
ter, there is uſually another ſmall thing was done in a Moment.
Trench cut quite along the Middle 1. In Mathematics, Moments are
of it.
ſuch indeterininate and inſtable Parts
MODEL, in Architecture. See of Quantity, as are ſuppoſed to be
Module.
in a perpecual Flux, i. e. either con-
Modes, in Muſic. See Mood. tinually decreaſing or increasing ;
MODILLIONS, in Architecture, which latter are taken for affirma-
are little inverted Conſoles undir tive and poſitive Moments, and the
former
MOM
.:
MOM
ones.
i
former for negative or ſubtractible Moments of the Parts of that Body
And theſe continually in- and therefore, where the Magni-
creaſing or decreaſing Particles are tudes and Number of any Particles
ſuppoſed to be infinitely ſmall; for are the ſame ; and where they are
as foon as ever they come to be of moved with the fame Celerity, there
any finite Magnitude, they ceaſe to will be the ſame Moments of the
be Moments. Moments therefore Wholes.
are to be looked upon not as the 6. M. - Leibnitz, Huygens, Ber-
generative Principles of finite Mag- noulli, Wolfe, and ſome other Fo-
nitude; but to be inceptive only of reigners, have all been drawn into
them..
an horrid Error concerning the Mo-
2. And becauſe 'tis the ſame thing, menta, or Force of falling Bodies:
if in the room of theſe moments, for they ſay, that the Forces of
the Velocities of their Increaſes or falling Bodies, at the Ends of the
Decreaſes be made uſe of, or the Fall, are not as the Velocities into
finite Quantities proportionable to the Quantities of Matter ; but as
ſuch Velocities; this Method of Pro- the Squares of the Velocities into
ceeding, which conſiders the Mo- the Quantities of Matter. And all
tions, Changings, or Fluxions of the Proof of this, by Experience,
Quantities, hath come to be called is a fallacious one, of, ſuſpending
Fluxions.
Balls by Threads to the Ceiling over
3. Moments, or Momenta, alſo in Veſſels of congealed Tallow, Clay,
a Phyſical Şenſe, as they are uſed Wax, or any other yielding Sub-
in reference to the Laws of Motion, ſtance; and then letting the Balls
fignify the Quantities of Motion in fall, and make Pits in the yielding
any moving Bodies; and fometimes, Subſtance: for when the Balls were
fimply the Motion itſelf : and they equal, and one weigh'd one Pound,
define it to be the Vis infta, or and the other two, and the lighter
Power by which any moving Bodies Ball hung twice the Height of the
do continually change their Places. other from the Surface of the Tal-'
4. And in comparing the Motions low; yet they made Pits in the Tal-
of Bodies, the Ratio of theſe Mo. low of the ſame Depth: And from
ments is always compounded of the this Experiment they would have
Quantity of Matter in, and the Ce- their Momenta to be equal, and con-
lerity of the moving Body ; ſo that ſequently their proper Weights are
the Moment of any moving Body in the reciprocal Ratio of the Spaces
may be conſider'd as a Rectangle which the ſaid Bodies deſcribe by
under the Quantity of Matter into their Fall; and becauſe theſe Spaces
the Celerity. And fince 'tis certain, are in the ſame Ratio as the Square
that all equal Rectangles have their of the Velocities; therefore, the
Sides reciprocally proportionable, Force of a falling Body is as the
(14 E. 6 Eucl.) therefore if the Body itſelf into the Square of the
Moments of any moveable Bodies Velocity at the End of the Fall.
are equal, the Quantity of Matter 7. M. s'Graveſande, in his Inſti-
in one, to that of the other, will tutiones Philofophiæ Newtoniana, con-
be reciprocally, as the Celerity of tradicts himſelf about this matter;
the latter to the Celerity of the for- for he ſays, pag. 75. Dum preſſione
mer, and vice verſa.
corpus acceleratur, manente equali
5. The Moment of any moving preſſione in corpus agenti, non augetur
Body may be conſidered alſo as celeritas æquabiliter. And therefore,
the Aggregate or Sum of all the according to this, if I take him
right,
res
MON
MON
right, the Motion of a Body that they called it a Demi-Diton, or a
falls freely ſhort Spaces, is not ac Tierce-Minor ; and laſtly, if the
celerated equally in equal Times : Terms were as 24 to 25, they called
And ſo the Celerity which is acqui- it à Demiton, or Dieze.
red in the Fall, is not as the Time The Monochord being thus divi-
in which the Body has fallen ; and ded, was properly that which they
conſequently the Spaces gone thro' called a Syſtem, of which there were
from the beginning of the Fall, will many kinds, according to the dif-
not be to one another, as the Squares ferent Diviſions of the Monochord.
of the Times or Velocities in which MONOTRIGLYPH,
a Term in
the Body fell ; and yet in the Expe- Architecture, fignifying the Space
riments, that he and Polenus has of one Triglyph between two Pila-
made to prove, that the Forces of ſters, or two Columns.
falling Bodies are as the Matter into Mood, in Muſick, ſignifies cer-
the Square of the Velocity, this tain Proportions of the Time, or
new Propoſition follows from the Meaſure of Notes. Theſe Moods or
Spaces gone thro' by the Fall of Modes, of ineaſuring Notes, were
Bodies, being as the Squares of the formerly four in Number, viz.
Times.
1. The Perfect of the More, in
8. See concerning this in the Aeta which a Large contained three
Eruditorum, An. 1686. p. 161. 'Hi- Longs, or a Long three Breves, a
foire des Ouvrages des Scavans, An. Breve three Semi-Breves, and a Se-
1690. P. 451. Journal Literaire, mi-Breve three Minims.
Tom. XII. p. 1, 190. Polenus, in 2. The Perfect of the Lefs, where-
Libro de Caſtellis, &c. But Dr. in a Large comprehended two Longs,
Defaguliers has ſhewn them all to a Long two Breves, a Breve three
be falſe in this Point, in the Philofo- Semi-Breves, and a Semi-Breve two
phịcal Tranſaktions, Nº 375, 376. Minims.
MONADES. See Digits.
3. The Imperfect of the More, in
MONOCHORD, a kind of Inftru- which a Large contained two Longs,
ment'anciently of ſingular Uſe for a Long two Breves, a Breve two Se-
the regulating of Sounds: But ſome mi-Breves, and a Semi-Breve three
appropriate the Name of Mono- Minims.
chord to an Inſtrument that hath 4. The Imperfe&t of the Lefs, is the
only one ſingle String, as the 'Trum- fame with that which we call the
Common Mood, the other three being
The Ancients made uſe of the now altogether out of uſe; altho®
Monochord to determine the Pro- the Meaſure of our coinmon Tri-
portion of Sounds to one another; ple-Time is the ſame with the
when the Chord was divided into Mood Imperfect of the Mare, except
two equal Parts, ſo that when the that we reckon but two Minims to
Terms were as i and 1, they call'd a Semi-Breve, which in that Mood
them Unifons ; but if they were as 2 to comprehended three. In our coin-
1, they call’d them Oet aves, or Dia mon Mood, two Longs make one
pafons; when they were as 3 to 2, Large, two Breves a Long, two Se-
they called them Fifths, or Diapenmi-Breves a Breve, &c. proceeding
tes; if they were as 4 to 3, they in the ſame Order to the laſt or
call'd them Fourths, or .Diatellarons : ; ſhorteſt Note : So that a Large con-
if the Terms were as 5 to 4, they tains two Longs, four Breves, eight
call'd it Diton,or Tierce Major ; but Semi-Breves, fixteen Minims, thirty
if the Terins were as 6 to 5, then two Crotchets, fixty-four Quavers,
១១
Be-
pet-Marine.
моо
Μ Ο Ο
1
Beſides theſe Moods of Time, five the Time in which the Moon runs
others relating to Tune, were in through the Zodiac, and therefore
uſe among the ancient Grecians, is accounted by the Motion of the
which were termed Tones or 7 dnes Moon : And ſo the Lunar Month
by the Latins; the Deſign of either is either Periodical, which is the
being to fhew in what Key a Song Time of the Moon's Motion from
was ſet, and how the different Keys any one Point of the Zodiac to the
had relation one to another. fame again, and is ſomething leſs
Theſe Sorts of Moods were diſtin- than 27 Days and eight Hours; or
guiſhed by the Names of the ſeveral elſe Synodical, which is the Time
Provinces of Greece, where they between New Moon and New Moon,
were firſt invented ; as the Doric, and is ſomething more than 29 Days
Lydian, Ionic, Phrygian, and Æolic. and a half.
Doric Mond confitted of ſlow-tuned 1. There is alſo a Solar Month,
Notes, and was proper for the ex which is the Time that the Sun
citing Perſons to Sobriety and Piety. takes up in running through one
Lydian Mood was likewiſe uſed in of the Signs of the Zodiac, and is
ſolemn grave Muſic ; and the De- almoſt 30 Days and a half.
ſcant or Compoſition was of flow 2. And both theſe Solar and Ly-
Time, adapted to ſacred Hymns or nar Months, are either Aſtronomi-
Anthems.
cal, like thoſe abovementioned; or
Ionic Mood was for more light and Civil, which are various, according
ſoft Muſick ; ſuch as pleaſant amo- to the Uſage of accounting in diffe-
Tous Songs, Sarabands, Courants, rent Places, Cities, and Nations.
Jigs, &c.
3. The Egyptians accounted by
Phrygian Mood was a warlike kind. Solar Months, each of 30 Days ;
of Mulick, fit for Trumpets, Haut- and to compleat their Year, after 12
boys, and other Inſtruments of the ſuch Months, they added five Days,
like Nature, whereby the Minds of which the odd Hours made’up.
Men were animated to undertake 4. But moſt of the ancient Na-
Military Atchievements, or Martial tions accounted by the Lunar Syno-
Exerciſes.
dical Month ; as the Jews, Greeks,
Æolic Mood, being of a more airy, and the Romans, till 7. Cæſar's
ſoft, and delightful Sound, ſuch as Time; and as the Mahometans do to
our Madrigals, ſerved to allay the this day. And becauſe theſe Months
Paſſions by the means of its grate- did not contain an exact Number of
ful Variety, and melodious Harmo- Days, to adapt them to Civil Com-
ny.
putation, they accounted alternately
Theſe Moods or Tones were di one Month to have 30, and the next
ſtinguiſhed into Authentic and Play- 31 Days ; and by this nieans they
al, with reſpect to the dividing of made two ſuch Civil Months' to be
the Oftave into its Fifth and Fourth. equal to two Lunar ones of 29 Days
The former was when the Fifth and a half : and they brought it to
poffeſſed the Lower Place, according paſs, that the New Month, for a
to che harmonical Diviſion of an Run of many Years, did not much
Ostave; and the other was when it deviate from the first Day of the
stood in the Upper Place, according Civil Month.
to the Arithmetical Diviſion of the Moon. The Periodical Revolu-
faine. Ostave.
tion of the Moon, in reference to
MONTH, properly ſpeaking, is the fixed Stars, is 27 Days, ſeven
Hours,
ir
0
Μ Ο Ο
M O O
Hours, 43 Minutes : And in the from the Action of the Sun, di-
ſame Space of Time, by a ſtrange ſturbing the Motion of the ſecon-
Correſpondence and Harmony of dary Planets) ſhe moves ſwifter, and
the two Motions, it revolves the deſcribes (by a Radius drawn from it
fame Way about its own Axis ; to the Earth) a greater Area in pro-
whereby (one Motion converting it portion to the Time, hath an Orbit
to, as the other turns it from the leſs curved, and by that means comes
Earth) the ſame side is always ex nearer to the Earth in her Syzy-
poſed to our Sight.
gies or Conjunctions, than in the
1. The Librations of the Moon's Quadratures, unleſs the Motion of
Body, which occaſion that the ſame her Eccentricity hinders it : Which
Hemiſphere exactly is not always Eccentricity is the greateſt, when
expoſed to our Sight, ariſe from the the Apogæum of the Moon happens
Eccentricity of the Moon's Orbit, in the Conjunction ; and is leaft,
from the Perturbations by the Sun's when the Apogæum happens at the
Attraction, and from the Obliquity Quadratures ; and her Motion is
of the Axis of the Diurnal Rotation ſwifter alſo in the Earth's Aphelion,
of the Moon's own Orbit, without than in its Peribelion. The Apogæum
the Knowledge of which Circum- alſo goes forward ſwifter in the
ſtances, her Phænomena would be in- Conjunction, and goes ſlower at the
explicable ; but by the Confideration Quadratures ; but her Nodes are at
of them are very demonftrable. reſt in the Conjunctions, and do
2. The mean horary Motion of recede moſt ſwiftly in the Quadra-
the Moon, in reſpect of the fixed tures.
Stars, is 32 Minutes, 56 Seconds, 8. The Moon alſo perpetually
27 Thirds, 12 Fourths and a half. changes the Figure of her Orbit,
3 The Moon is diſtant from the or the Species of the Ellipfis the
Earth, according to moſt Aſtrono- moves in.
mers, 59: According to Vindeline, 9. There are alſo ſome other In-
60; Copernicus 60}: Kircher, 601 equalities in the Motion of this pla-
And according to Tycho, 564 Semi- net, which can hardly be reduced
diameters of the Earth. Sir Ifaac to any certain Rule: As the Velo-
Newton thinks the Diſtance ought cities or Horary Motions of the A-
to be eſteem'd about 61. There- pogæum and Nodes, and their E-
fore the mean Diſtance may be rec- quations, and the Difference be-
kon'd 60.
tween the greateſt Eccentricity in
4. She is nearer the Earth at her the Conjunctions, and the leaſt in
Syzygy, than in the Quadrature by the Quadratures; and that Inequa-
ofth Part of the Diſtance.
lity which is called the Variation of
5. According to Mr. Callini, the the Moon: All theſe do increaſe and
Moon's greateſt Diſtance from the decreaſe annually, in a Triplicate
Earth is 61, the mean Diſtance 56, Ratio of the apparent Diameter of
and the leaſt Diſtance.52 Semi-dia, the Sun: And this Variation is in-
meters of che Earth.
creaſed and diminiſhed in a dupli-
6 The Power of the Moon's In- cate Ratio of the Time between the
fluence as to the Tides, is to that Quadratures ; as Sir Iſaac Newtox
of the Sun as 6 to one. Sir Iſaac proves in many places of his Prin
Newton.
cipia.
7. As to the Inequality of the 10. That curious Perſon found
Moon's Motion, (which proceeds the Apogæum in the Moon's Syzy-
gies
MOO
1
Μ Ο Ο
gies to go forward 23 min. each or Quantity of Matter in the Moon
Day, in reſpect of the fixed Stars ; to that of the Earth, is as I to 26
and to go backward 16 min. š nearly.
.
each Day in the Quadratures : And 16. The Plane of the Moon's Or.
therefore the middle annual Mo- bit is inclin'd to that of the Eclip,
tions he eſtimates at 40 Degrees. tic, and makes with it an Angle of
11. That the Cauſe of the ſecon about five Degrees ; and its Decli-
dary Light of the Moon, as they nation varies, and is greateſt when
call it, that is, the obſcure Part of the Moon is in the Quadratures, and
her appearing like kindled Aſhes,juſt leaſt when ſhe is in her Syzygies.
before and after the Change of the 17. By means of the Spots in the
new Moon, is the Sun's Rays re. Moon, the Lunar Ellipſes are more
flected from the bright Hemiſphere accurately obſerved than formerly,
of the Earth to thoſe dark Parts of
to the great Advancement of Geo-
the Moon ; and thence again re graphy and Navigation in ſettling
flected to the Earth deftitate of the the Longitudes of Places; for the
Sun's Light.
Immerſion and Emerſions of theſe
12. Sir Iſaac Newton makes it a Spots, from the Shadow of the
Propoſition to enquire into the Fi- Earth, are moſt nicely determined.
gure of the Moon ; and ſuppoſing 18. Although the Moon's Period
it, at its firſt Original to have been round the Earth be in 27 Days, 7
a Fluid, like to our Sea, he calcu- Hours, and three Quarters, (which
lates, that the Attraction of our is the Periodical Month) yet becauſe
Earth would raiſe the Water there to in the Space of a Periodical Month,
near go Foot high, as the Attraction the Earth alſo with its Satellite,
of the Moon raiſeth our Water the Moon, is moved forward ala.
to 12 Foot : Whence the Figure moſt an entire Sign ; therefore the
of the Moon muſt be a Spheroid Point of the Moon's Orbit, in the
whoſe greateft Diameter extended, laft Conjunction, or New Moon,
will paſs through the Centre of our will be gotten too far to the Weſt-
Earth ; and will be longer than the ward : and therefore the Moon can-
other Diameter perpendicular to it, not come yet to a new Conjunction
by 180 Foot ; and from hence it with the Sun, but wants of it two
comes to paſs, that we ſee always Days and five Hours; which muſt
the ſame Face of the Moon : For be paſs'd before the entire Lunation
the cannot reft in any other Poſition, will be over, and before the Moon
but will continually endeavour to hath exhibited all her Phaſes. Theſe
conform herſelf to this Situation, two Days, and five Hours therefore
Prop. 38. Lib. III.
being added to the Periodical Month,
13. Mr. Azout ſays, that this pla make the Synodical one, which con-
net's Diameter never appeard to fifts of 29 Days, 12 Hours, and
him above 33 min. and never leſs three Quarters.
than 24 min. 45 ſec.
19. The Moon diſturbs the Mo-
14. Sir Iſaac Newton reckons the tion of the Earth, and the common
mean Diameter of the Moon to be
Centre of Gravity of thoſe Bodies
32 min. 12 ſec. as the Sun's is 31 deſcribe that Orbit about the Sun,
min. 27 ſec.
which we have hitherto ſaid that
15. The Denſity of the Moon he the Earth deſcribed ; becauſe we
concludes to be to that of the Earth, overlook'd the Action of the Moon;
as 9 to 5 nearly ; and that the Maſs but the Earth really deſcribes an ir-
regular Curve.
20.
i
MOO
моо
20. The Gravity of the Moon tures, is to the Force which accele-
towards the Earth, is increaſed by rates or retards the Moon in its
the Action of the Sun, when the Orbit.
Moon is in the Quadratures ; and 28. And the Radius is to the Sum
it is an Augmentation or Addition or Difference of one and a half, the
to the Gravity of the Earth towards Co-Sine of double the Diſtance of
the Sun.
the Moon from the Syzygy, and half
21. The Earth's Diſtance from the Radius ; as the Addition of
the Sun remaining the ſame, the Gravity in the Quadratures, to the
abovemention’d Addition of Gravity Diminution or Increaſe of Gravity
increaſes and diminiſhes in the Rain that Situation of the Moon, con-
tio of the Diſtance of the Moon cerning which the Computation is
from the Earth.
made.
22. The Diſtance of the Earth
29. The Moon is leſs diſtant from
from the Sun remaining the ſame, the Earth at the Syzygies, and more
the Gravity of the Moon towards at the Quadratures.
the Earth decreaſes more ſlowly in 30. In the Quadratures and Syzy-
the Quadratures, than according to gies, the Moon deſcribes Area's by
the inverſe Ratio of the Square of Lines drawn to the Centre of the
the Diſtance from the Centre of the Earth, proportional to the Times.
Earth.
31. The Area's, by Lines drawn
23. The Force which diminiſhes to the Centre of the Earth, are not
the Gravity of the Moon in the Sy- exactly proportional to the Times
zygies, is double that which increaſes at all Times.
it in the Quadratures.
32. 'The Apſides of the Moon go
24. In the Syzygies, the diſturb- forward, when the Moon is in the
ing Force is directly as the Diſtance Syzygies : In the Quadratures, the
of the Moon from the Earth, and Apfides go backwards, that is, move
inverſly as the Cube of the Diſtance in Antecedentia.
of the Earth from the Sun.
33. The Progreſs, conſidering one
25. At the Syzygies the Gravity entire Revolution of the Moon, ex-
of the Moon towards the Earth, re ceeds the Regreſs, Cæteris Paribus.
ceding from its Centre, is more di 34. The Apfides go forward faſteſt
miniſhed, than according to the in- of all in a Revolution of the Moon,
verſe Ratio of the Square of the Di- ſuppoſing the Line of the Aplides
ſtance from that Centre.
in the Nodes ; and in that very
26. In the Motion of the Moon Caſe they go back the Noweft of all
from the Syzygies to the Quadrature, in the fame Revolution.
the Gravity of the Moon towards 35. Suppoſing the Line of the
the Earth is continually increaſed, Aplides to be in the Quadratures,
and the Moon is continually re- the Apfides are carried in Conſe-
tarded in its Motion : But in the quentia, the leaſt of all in the Syzy-
Motion from the Quadrature to the gies; but they return the ſwifteſt in
Syzygy, every Moment the Moon's the Quadratures ; and in this caſe,
Gravity is diminiſhed, and its Mo- in one entire Revolution of the
tion in its Orbit is accelerated. Moon, the Regreſs exceeds the Pro-
27. As the Radius is to the Sine, greſs.
and an half of double the Diſtance 36. The Excentricity of the Ora
of the Moon from the Syzygy; fo the bit, evefy Revolution undergoes va-
Addition of Gravity in the Quadra- rious Changes. It is the greateſt
of
M OO
моо
1
of all, when the Line of the Apſides tures, and in one whole Revolution
is in the Syzygies ; but the Orbit is of the Moon, the Force which in-
the leaſt Excentric of all, when the creaſes the Inclination exceeds that
Line of the Apfides is in the Qua- which diminiſhes it; therefore the
dratures.
Inclination is increaſed, and it is
37. The Ratio between the Addi- the greateſt of all, when the Nodes
tion of Gravity in the Quadratures, are in the Quadratures.
and the Force, which renoves the 44. All the Errors in the Moon's
oon out of its Orbit, is the Ratio Motion are ſomething greater in
of the Cube of the Radius to three the Conjunction than in the Op-
times the Product of the Sines of poſition.
the Diftances of the Moon from the
45. All the diſturbing Forces are
Quadrature, and of the Node from inverſly, as the Cube of the Diſtance
the Syzygy; as alſo of the Inclination of the Sun from the Earth, which
of the Plane,
when it remains the ſame, they are
38. This Force is increaſed as ás the Diſtance of the Moon from
the Moon advances towards the Sy- the Earth. Conſidering all the diſ-
zygy, and as the Nodes recede from turbing Forces together, the Dimi-
it.
nution
of Gravity prevails.
39. Conſidering one entire Revo-
46. The Motion of the Moon be-
lution of the Moon, Cæteris Pari- ing conſidered in general. The Gra;
bus, the Nodes move in Anteceden- vity of the Moon towards the Earth
tia ſwifteſt of all, when the Moon is diminiſhed coming near the Sun,
is in the Syzygies; then flower and and the Periodical Time is the grea-
flower, till they are at reſt, when teft ; as alſo the Diſtance of the .
the Moon is in the Quadratures. Moon (Cæteris Paribus) the greateſt,
40. The Line of Nodes does ſuc- when the Earth is in the Perihelion.
ceſſively acquire all poffible Situa MORTAR-PIECE, is a kind of
tions in reſpect of the Sun; and very ſhort Piece of Cannon, or Ord-
every Year goes twice thro' the nance, thick and wide, proper for,
Syzygies, and twice thro' the Qua- the diſcharging of Bombs, Carcaf-
dratures.
ſes, Stones, &c. It is uſually mount-
41. If we conſider ſeveral Revo- ed on a Carriage, the Wheels where-
lutions of the Moon, the Nodes in of are very low.
one whole Revolution go back very
faſt, the Nodes being in the Qua- Mr. Anderſon's TABLE of the re-
dratures ; then ſlower, till they come quifite Weight of Powder for all
to reſt, when the Line of Nodes is
Mortars, from 6 to 20 Inches dia-
in the Syzygies.
meter.
42. By the fame Force with
which the Nodes are moyed, the
Inch. Decin.! Pounds. Ounces.
Inclination of the Orbit is alſo 6.
13
changed ; it is increaſed as the
Moon recedes from the Node, and 7.
05
diminiſhed as it comes to the Node.
7
43. When the Nodes are come to
8
the Syzygies, the Inclination of the
8
ი6
Plane of the Orbit is the leaſt of
9.
14
all; for in the Motion of the Nodes
9. 5
3.
06
from the Syzygies to the Quadra 10,
3. 141
O,
OI
IO
ini Na
ол олол оло
00
2.
10,
10.
II.
II.
12.
tinnonoo i
10
9.
10.
I
II.
О, Ол ол оо олол Ол ол Ол ол
upon it.
M O T
MOT
Ounces.
Inch. Decem. | Pounds.
4. If two Bodieş, moving uni.
formly, go with unequal Velocities,
5
08
the Spaces which will be pafsd over
03 by them in unequal Times, will be
5
15, to one another in a Ratio com-
12
pounded of that of the Velocities,
I 2.
and that of the Times.
13.
09
5. The Motions of all Bodies are
13. 5
IO
as the Rectangles under the Velo-
14
111
cities, and the Quantities of Mat-
14 5
14 ter.
15
13. 03
6. The Motions of Bodies inclu-
15.. 5
14 09 ded in a given Space, among them-
16.
16. 16
ſelves, will not be changed by the
16.
17. 09 Motion of that Space uniformly for-
17. 0
19. 03 wards in a ſtraight Line.
17.
20. 15
7. Every Body will continue in
18.
22. 12
its State, either of Reſt or Mo-
18.
24 II
tion, uniformly forward in a Right
19.
26.
13 Line, unleſs it be made to change
19. 5
28. 14 that State by ſome Force impreſſed
20. o
31.
04
8. The Change of Motion is pro-
See the Deſcription of Mortars portionable to the moving Force
by Methins, in his Artiller. part 3. impreſſed, and is always according
c. 18, and foll. And Buckner's Are to the Direction of that Right Line,
till. part. 1. f. 78.& ſeq. As alſo Su- in which the Force is impreſſed. ,
rire de Saint Remy's Memoires d'Ar 9.' The Quantity of any Motion
tillerie, Part 2. P. 352. & feq. is diſcoverable by the Joint-Confi-
Motion, is a Continual and deration of the Quantity of Matter
Succeſſive Mutation of Place, and is in; and the Velocity of the moving
either Abſolute or Relative.
Body : For the Motion of any
1. Abſolute Motion, is the Change Whole, is the Sum of the Motions
of the Locus Abfolutus of any moving of all the Parts.
Body, and therefore, its Celerity 10. The Quantity of Motion,
will be meaſured by the Quantity which is found, by taking either
of the abſolute Space, which the the Sum of Motions made the fame
moveable Body has run through. Way, or the Difference of thoſe
But,
which are made contrary Ways, is
2. Relative Motion, is the Muta- not at all changed by the Action
tion of the Relative or Vulgar Place of Bodies one upon another.
of the moving Body, and ſo hath II. In all kind of Motions what-
its Celerity accounted or meaſured ever, rolling, ſliding, uniform, ac-
by the Quantity of relative Space, celerated, or retarded, in right Lines,
which the moveable Body runs over. or in Curves, & c. the Sum of the
3. All Motion is of itſelf Rectic Forces which produce the Motion
linear, or made according to of all Parts of its Duration, is al-
ſtraight Lines, with the ſame con- ways proportionable to the Sum of
ftant uniform Velocity ; if no exter- the Paths, or Lines, which all the
nal Cauſe makes any Alteration in Points of the moving Body deſcribe.
its Direction.
Bb
1 2,9
Μ Ο Τ.
MUL
12. The Product of the Duration Places, at the side of the Barriers,
of all uniform Motions, multiplied through which People paſs on Foot.
by the Force which began the Mo MOYENAU, (a French Term) in
tion, is always proportionable to Fortification, is a ſmall flat Baftion,
the Product made by the Path, or commonly placed in the middle of
Line of Motion multiplied by the an over-long Curtain, by which the
Maſs or Quantity of Matter in the Baſtions at the Extremities are not
moving Body.
well defended from the Small-Shot,,
MOTION of the Apogeum, in the by reaſon of their Diſtance ; ſo that
Ptolemaic Syſtem, is an Arch of the this work is proper for placing in
Zodiac of the Primum Mobile, con it a Body of Muſqueteers to fire
tained between the Line of the Apo- upon the Enemy from all Sides.
geum, and the beginning of Aries. MULTANGULAR FIGURE, is
MOTION COMPOUNDED, See one that has many Sides and Angles.
Compound Motion.
MULTILATERAL, in Geome-
MOULDINGS. Under this Name try, are thoſe Figures that have
are comprehended all thoſe Jettings more than four Sides.
or Projectures beyond the naked MULTINOMIAL Root. See Po-
Wall, à Column, &c. which only lynomial.
ſerve for Ornament ; whether they
MULTIPLE PROPORTION, is
be ſquare, round, ſtraight or crook- when the Antecedent being divided
ed. Of theſe there are ſeven kinds by the Confequent, the Quotient is
more confiderable than the reſt, more than Unity; and the Reaſon
viz. the Doucine, the Taton or Heel, of the Name is, becauſe the Con-
the Ovolo or Quarter-Round, the ſequent muft be multiplied by the
Plinth, the Afragal, the Denticle, Index, or Exponent of the Ratio,
and the Cavetto.
to make it equal to the Antecedent.
Movement; the ſame with Thus 12 is multiple in proportion
what many do call an Automaton, to 4, becauſe being divided by 4,
and with us fignifies all thoſe Parts the Quotient is 3, which is the Dee
of a Watch, Clock, or any ſuch cu- nominator of the Ratio; ar
and the
rious Engine, which are in Motion, Conſequent 4 being multiplied by
carry on the Deſign, or anſwer the 3, makes the Antecedent 12; where-
End of the Inſtrument.
fore 3 is fub-multiple of 12.
MOULINET, a French Term, ſig MULTIPLE SUPER-PARTICU-
nifying a Turn-Stile ; 'tis uſed in LAR PROPORTION, is when one
Mechanics, and fignifies a Roller, Number or Quantity contains ano-
which being croſſed with two Le ther more than once, and ſuch an
vers, is uſually applied to Cranes, aliquot Part.
Capitans, and other Sort of Engines MULTIPLE SUPER-PARTIENT
of the like Nature, to draw Cords, PROPORTION, is when one Num-
and heave up Stones, Timber, &c. ber or Quantity contains another
Alſo a kind of Turn-Stile, or woo- divers Times, and ſome Parts be-
den Croſs, which turns horizontally fides.
upon a Stake fixed in the Ground, MULTIPLICATION, is, in ge-
and is uſually placed in Paffages, neral, the taking or repeating of
to keep out Horſes, and to oblige one Number or Quantity as often ·
Paſſengers to go, or come one by as there are ſuppoſed Units in the
other Number : The Number mul-
Theſe Moulinets are often ſet up tiplied, is called the Multiplicand,
near the Out-Works of fortified the Number multiplying, the Muka
tiplicator;
one.
.
M U L
M U L
tiplicator or Multiplier ; and that Examples, 9764
which is found or produced, is cal-
3
led the Product,
MULTIPLICATION, is only a
31292
compendious Addition, effecting at
once, what in the ordinary Way of
5326 Multiplicand
Addition would require many Ope-
427 : Multiplier
rations : For the Multiplicandis
only added to itſelf, or repeated, as
37282
often as the Units of the Multi-
10652
plicator do expreſs it. Thus if 6
21304
were to be multiplied by 4, the Pro-
duct is 24, which is the Sum ari-
2274202
Product.
fing from the Addition of 6 four The Reaſon of theſe Rules depends
times to itſelf.
upon the following Propofition, viz.
In all Multiplication, as I is to that the Product of any two Num
the Multiplicator ; fo is the Multi-bers is equal to the ſeveral Products
plicand to the Product.
made by multiplying all the Parts .
1. Multiplication of whole Num of the one, by the other, or all the
bers is perform'd by the following Parts of the other.
Rules. "If the Multiplier be leſs 2. To multiply a Fraction by a
than 10, ſet it under the firſt Figure Fraction, is to take the Multiplicand
of the Multiplicand, and having ſo many ſuch Parts of a Time as is
drawn a Line underneath, let each fignified by the Maltiplier ; to do
Figure thereof, beginning at the which, multiply the Numerators of
place of Units, be multiplied by the two Fractions together, for the
the Multiplier, and fet each ſingle Numerator of the Fraction defired,
Product (if leſs than 10) under its and their Denominators for the De-
reſpective Figure of the Multipli- nominator of the Fraction, which
cand ; but if it be 10, or any Num- is the Product of the two given
ber of ro's with ſome Over-plus, Fractions ; as multiplied by is
fubſcribe that Over-plus ; but if
without, fet down a Cipher, and or 35; and the Product will
5 X7
always for every 10, reſerve i to
be added to the next Product, and Fractions multiplying each other.
the
the Number ſubſcribed will be the
3. The Multiplication of Decima)
Product of the whole.
When the Multiplier conſiſts of Fractions, is the ſame with that of
ſeveral Figures, let the Multipli- duct there muſt be always as many
whole Numbers, only in the Pro-
cand be multiplied by each Figure Decimal Places, as are both in
of the Multiplier, as before, begin- the Multiplier and Multiplicand.
ning with the firſt, and placing the
Examples :
ſeveral Products thereof underneath
Multiplicand 759.2
each other in ſuch order, that the
5.037
firſt Figure or Cipher of each Pro-
duct may be in the ſame place (of
53144
Units, Tens, &c.) with its reſpec-
22776
tive multiplying Figures ; then add
37960
theſe particular Products together,
and the Sum of them will be the
Product 3524.0904
Product of the whole Multiplica,
tion.
Bb 2
.0096
2 X 3
MUL
MUR
::
)
i
)
.0096
MULTIPLICATOR, in Arithme-
.072
tic, is the Number by which you
multiply, or the Number multiply-
192
ing:
672
MULTIPLIER, the fame with
Multiplicator.
.0006912
MURDERERS, are ſmall pieces
of Ordnance, either of Braſs or I-
-45
ron, having Chambers (that is,
.029
Charges made of Braſs or Iron) put
in at their Breeches : They are moſt-
.405
ly uſed at Sea, at the Bulk-Heads
90
of the Fore-caſtle, Half-Deck, or
Steerage, in order to clear the Decks,
,01305
when any Enemy boards the Ship;
4. Multiplication in Algebra, is they are faſtend and travers’d by a
performed when the Quantities are Pintle, which is put into a Stock.
fimple, by an immediate joining of Music, is one of the ſeven Sci-
the Letters, or if the fimple Quan- ences, commonly called Liberal, and
tities have Numbers before them, comprehended alſo among the Ma-
by ſetting the Product of the Mul- thematical, as having for its Object
tiplication of thoſe prefixed Num- diſcrete Quantity or Number ; but
bers, before the Letters thus joined; not conſidering it in the Abſtract
as a multiplied by b is ab, aa like Arithmetic ; but with relation
multiplied by cd, is aacd; 5€ to Time and Sound, in order to make
multiplied by 7gc, is 35 ego; and a delightful Harmony.
fo of others. But if the Quantities This Science is alſo Theoritical,
to be multiplied be Compound, then which examineth the Nature and
every ſimple Quantity in the Mul- Properties of Concords and Diſcords,
tiplication is to be multiplied by explaining the Proportions between
each, fimple Quantity of the Multi- them by Numbers : And Practical,
plier, and the Signs t- and -- muft which teacheth not only Compofi-
be ſet between the ſeveral Products, tion, that is, the manner of com-
always obſerving to prefix the Sign poſing all Sorts of Tunes, or Airs ;
fito that Product ariſing from the but alſo the Art of Singing with the
Multiplication of two ſimple Quan- Voice, or Playing upon Muſical In-
tities having both the Sign + pre- ftruments.
fix'd;, or both the Sign-; and to Some of the Ancients who have
prefix the Sign - to the Product, wrote of Harmony, are Ariſtoxenus,
when the signs of the fimple Quan- Euclid, Plutarch, Ptolemy, Pfellus,
tities are different; for Example, Porphyry, Briennius, Nichomachus,
afe multiplied hy b, will produce Alipius, Gaudentius, Bacchius, Quin-
abteb; a - e multiplied by b, tilian, Caffiodorus, Capella, Boetius,
will be ab eb; a tetoc mul- Proclus, 56. And ſome of the Mo-
tiplied by z will be azteztucz: derns are Mcibomius, Wallis, Del-
Alſo ac-bdfef multiplied by cartes, Merſennus, Faber, Holder,
8-tho-k will give acg bdg Sauveur (in the French Memoires,
+geftach-bdh i efh - An. 1901,1707, 1711,) Dechales (in
ack + bdk - efk.
his
4
tom. Mundi Mathematici,) M.
MultIPLICAND, in Arithmetic, Perault, Mr. Malcolme, Mr. Remeau,
is the Number to be multiplied. Mr. Euler, &c.
It
1
1
M U S
NA'T
It is very eaſy to conclude, from cellent Mr, Handel himſelf, deferved.
what we have upon Muſic from the ly named the Prince of Muſicians,
Ancients, that it was very imper- both for his Compoſition and Exe-
fect and deficient; and notwith- cution upon the Organ and Harpfi-
ſtanding the fabulous Wonders, it cord, has abundantly and wonderfully
is ſaid to produce upon Men's Paſ- performed his part.
fions in thoſe times, yet now-a-days MUSKET-BASKETS, in Fortifi-
I believe, the inoft ſkilful of their cation, are Bakets of about a Foot
Muſicians would little or ſcarcely and a half high, and eight or ten
move one at all: for it is moſtly Inches diameter at the bottom, and
agreed, that the ancient Greeks had a full Foot at the top: They are
not the Uſe of Concert Mufic, viz. filled with Earth, and are ſet on low
of different Parts founding at once, Parapets, or Breaſt-Works, or on
but only folitary, for one ſingle ſuch as are beaten down, that the
Voice or Inſtrument ; or elſe the Muſqueteers may fire between them
ſame Piece ſung or ſounded by fe at the Enemy, and yet be tolerably
veral Voices or Inſtruments to- well fecured againſt their Fire.
gether ; but ſome Octaves, or per MUTULE, in Architecture, is a
haps Fifths above the others. Gui- kind of ſquare Modilion, fet under
do Aretinus is ſaid to be the firſt the Cornice of the Doric Order, and
who invented and brought Sym- ſo called from the Word Mutilus,
phony or Concert into Mufic; but maimed or imperfect, becauſe they
what Progreſs he made, and what repreſent the Ends of the Rafters,
were his Compoſitions, we do not which are crooked or bent, in like
know. In a word, one may ven manner, as the Beams or Joints are
ture to affirm from the whole of repreſented by the Triglyphs in the
'what we find wrote on the Subject, Frize of the ſame Order.
that Muſic did not begin to arrive
at any tolerable Perfection, till to-
wards the End of the laſt Century,
when the great Purcel and prodi-
gious Corelli oblig'd the World with
their moſt agreeable and harmonical ADIR is that Point of the
Compoſitions ; then it was that Mu Heavens under the Earth,
fic began to advance apace, and re-
which is diametrically oppoſite to
ceive great Improvements from ma the Point directly over our Head,
ny other ingenious Compoſers and viz. the Zenith; ſo that they are
Performers of ſeveral European Na- both as it were the Poles of the Ho.
tions, eſpecially the Italians and rizon, and diſtant from it on each
Engliſh, and now ſeems to be brought fide ninety Degrees, and conſequent-
near its utmoſt Perfection ; fince all ly fall upon the Meridian, one a-
the agreeable Combinations of the bove the other under the Earth;
various Continuance, Riſing, Falling, and whatever Diſtance one of them
and Mixtures of Tones, mutt be has from the Equator, and one of
contain'd within certain Limits, the Poles of the World, the ſame,
whoſe Number may not be ſo great on the contrary, has the other from
as is generally imagined; and be the oppoſite Pole and adverſe Part
cauſe of the great Number of Per- of the Equator.
ſons who have for more than thirty
NAPIers, or NAPER'S-BONES,
Years faft paſt, applied themſelves or Rods, are a kind of larger Mul-
to this Art. Among whom the ex- tiplication-Table, contriv'd upon
four-
N.
N
Bb 3
NOC
NAU
four-ſquare Wooden or Ivory Rods · NAUTICAL PLANISPHERE, is
by the Lord Napier, for the more a Deſcription of the Terreſtrial
eaſy muịtiplying, dividing, and ex. Globe upon a Plane, for the Uſe of
tracting the Roots of great. Num- Mariners ; and is either the Plane
bers.
Chart, as they call it, where the
NATURAL DAY. See Day. Parallels of Latitude are all of the
NATURAL HORIZON; the ſame fame Length with the Meridians ;
with Senſible Horizon.
and which therefore is very erro-
NATURAL PHILOSOPHY, is the neous, except in fhort Voyages,
ſame with what is uſually called
uſually called and near the Equator : Or Merca,
Phyſics, viz. that Science which con- cator's Chart, where the Meridians
templates the Power of Nature, the are increaſed in proportion, as the
Properties of natural Bodies, and Parallels ſhorten, that is, as the
their mutual Actions one upon an- Secants of the Arch contained be-
other.
tween the Point of Latitude, and the
NAVIGATION, is the Art of Equator,
Sailing, whereby the Mariner is in NEBULOUS STARs, are certain
ſtructed how to guide a Ship from fixed Stars of a dull, pale, and ob-
one Port to another, the ſhorteſt and ſcurith Light. Theſe ſeen through
fafeſt way, and in the ſhorteſt time: good Teleſcopes, appear to be Clu-
And this is two-fold, either
Iters of ſmall Stars.
Improper, which is called Coaſting, Niedle. See Box and Needle.
in which the Places are at no great
NEGATIVE QUANTITIES, in
diſtance one from another, and the Algebra, are ſuch as have before
Ship fails uſually in fight of Land, them the Negative Sign, and which
and is within Soundings. Now, for are ſuppoſed to be leſs than no-
the Performance of this, there is re- thing.
quired a good Knowledge of the Neipe TIDES, written alfo
Lands, the Uſe of the Compaſs, Nepe or Neep, are thoſe Tides
the Lead, or Sounding Line, and (when the Moon is in the middle
fuch Books as Rutter's, &c.
of the ſecond and laſt Quarter)
Proper, is where the Voyage is which are oppoſite to the Spring-
performed in the vaſt Ocean, out Tides ; and as the higheſt of the
of fight of all Land; and here is Spring. Tides is three Days after the
neceffary not only the Knowledge Full or Change, ſo the loweſt of
of the Lead, Compaſs, &c. But the the Neep is four Days before the
Maſter muſt be a thorough Sailor Full or Change.
or Artiſt, and underſtand well Mer NEW.EL, in Architecture, is the
cator's Charts, Azimuth, and Ampli- upright Port that the Caſe of Wind-
tude Compaſs, Log-Line, and all good ing-Stairs turns round about.
Inſtruments for Celeſtial Obſerva Niche, in Architecture, is a Ca-
tions that can be uſed at Sea. vity left deſignedly in the Wall of a
Some of the Writers upon Navi- Building, to place a Statue in.
gation, are Varenius, Wright, Nor NOCTURNAL, is an Inſtrument
wond, Newhouſe, Seller, Ricciolus, made of Box, Ivory, or Braſs, to
Hodgſon, Jones, Atkinſon, Harris, take the Altitude or Depreſſion of
Patoun, &c.
the Pole-Star, in reſpect to the Pole
NAUTICAL CHART, the ſame itſelf, in order to find the Latitude,
as Sea-Chart.
and nearly the Hour of the Night.
NAUTICAL COMPAss, the ſame 1. There are ſeveral Sorts of Noc-
as Sea-Campaſs.
turnals, of which ſome may be Pro-
jections
NOC
N ON
jections of the Sphere ; ſuch as the to move about the Centre of the. In-
Hemiſpheres or Planiſpheres on the ſtrument.
Plane of the Equinoctial; but the NocTuRNAL ARCH, 'is that
Seamen uſe only two, and the man. Space in the Heavens which the
ner of uſing either is the ſame. One - Sun, Moon, or Stars, runs thro'
of them is fitted for the Pole-Star, parallel to the Equator, from their
and firſt of the Gardes of the Little Setting to their Rifing.
Bear; and the other for the Pole NOCTURNABLE, is an Inſtru-
Star, and the Gardes or Pointers ment uſed to find how much the
(as ſome call them) of the Great North Star is higher or lower than
Bear.
the Pole at all Hours of the Night.
2. The Inſtrument confifts of three NODATED HYPERBOLA. So
Parts .or Pieces; the largeſt of Sir Iſaac Newton calls a peculiar
which hath a Handle to hold it by, kind of Hyperbola, which by turning
when you would obſerve ; and op- round decuſſates, or croſſes itſelf.
poſite to the- Handle, there is a See Sir Iſaac Newton's Tractatus de
ſmall Tooth or Point, which (if it Enumeratione Linearum tertii Or-
be made for the Little Bear) ſtands dinis.
againſt the 25th of April; but if NODEs, in Aſtronomy, are the
for the Great Bear againſt the 17th Points of the Interſection of the Or-
of February, which are the Times bit of the Sun, or any Planet, with
of the Year when thoſe Stars come the Ecliptic; ſo that the Point where
to the Meridian at Twelve at Night. a Planet paſſes over the Ecliptic,
On this bigger Part or Piece there out of Southern into Northern La-
are two Circles deſcribed; the outer- titude, is called the North or Af
moſt hath the Months and their cending Node. And where it de-
Days, and the innermoſt hath the ſcends from North to South, 'tis the
Hours of a natural Day. On South or Deſcending Node.
the backſide of this Piece alſo are Nodus, or Node, in Dialling,
32 Points of the Compaſs deſigned is a certain point in the Axis or
and marked, and their intitial Let- Cock of a Dial, by the Shadow of
ters.
which, either the Hour of the Day
3. The ſecond Part of the Noc. in Dials without Furniture, or the
turnal hath two Circles deſcribed on Parallels of the Sun's Declination,
it ; of which the outermoſt is divided his place in the Ecliptic, the Ita-
into 297, equal Parts for the Days lian or Babyloniſh Hours, &c. are
of the Moon's Age, and the inner- ſhewn in ſuch Dials as have Furni-
moſt into 24 Hours ; and at the Be- ture.
ginning of the Days of the Moon's NONAGESIMAL DEGREE, is
Age, and at Twelve there is a Tooth the higheſt Point, or goth Degree
to be ſet to the Day of the Month of the Meridian.
in the upper Part.
Nones of a Month, are the next
4. The third Part is an Index Days after the Kalends, which is
with a fiducial Edge, iſſuing from the firſt Day in March, May, June,
the Centre ; and muſt be ſo long, and October ; the Romans accounted
that a good Part of it may extend fix Days of the Nones; but in all
beyond the outermoſt or biggeſt the reſt of the Months but four.
Piece. Theſe chree , Parts are ſo They had this Name probably, be-
order'd, that by means of a ſmall cauſe they were always nine Days
hollow Braſs Socket they are made incluſively, from the firſt of the Nones
24
+
Bb 4
to
។
f
, , o Lindsok, gundargitekturede
N U M
OBL.
to the Ides, i. e. reckoning inclu NUMERATION, in Arithmetic,
fively both thole Days.
is the true Diſtinction, Eſtimation,
NORMAL, the ſame with Per- and Pronunciation of Numbers, or
pendicular, or at Right Angles; and the Rule to read any Number, tho'
'tis uſually ſpoken of a Line, or a never ſo great, and to haỹe a diſtinct
Plane that interſects another perpen- Idea of each Place or Figure of it.
dicularly
NUMERATOR of a Fraction, is
NORTHERN SIGNS of the Eclip- that Part of it which ſhews or num-
tic or Zodiac, are thoſe fix which bers how many of thoſe parts which
conſtitute that Semi-circle of the E- any Integer is ſuppoſed to be divided
cliptic, which inclines to the North- into, are expreſſed by the Fraction.
ward from the Equator; as Aries, Thụs in , 6 is the Numerator,
Taurus, Gemini, Cancer, Leo, Vir: (which ſtands always above the
go.
Line) and ſhews you, that if any
Notes, in Muſic, are certain Whole be divided into 8 Parts, you
Terms invented to diſtinguiſh the number and enumerate, or take 6
Degrees of Sound, and the Propor- of them, i.e: three Quarters.
tion of Time belonging to it.
1. Theſe Notes relating to the
Diſtinctions of Sound, are ſeven in
O.
number, viz. Gamut, Aire, Bemi,
Cefaut, Geſolrate, Alamire, Befabe BELISKin Architecture, is
mi, Cefolfaut.
a kind of quadrangular Pyra-
2. And the Notes relating to mid, very tall and ſlender, raiſed in
Time, are nine in Number, viz. a a public Place, to ſhew the Large-
Large, Long, Breve, Semi-Breve, neſs of ſome enormous Stone, or to
Minim, Crotchet, Quaver, Semi- ſerve as a Monument of ſome me.
Quaver, and Demi-Semi-Quaver. morable Tranſaction.
3. But the Large and Long are OBJECT-GLASS, of a Teleſcope
how of little Uſe, as being too long or Microſcope, is that Glaſs which
for any Voice or Inſtrument (the is placed at that End of the Tube,
Organ only excepted) to hold out which is next the Object.
to their full Length; although their OBJECTIVE-LINE. See Line-
Reſts are ſtill very often uſed, more Obje&tive.
eſpecially in grave Muſic, and Songs OBLIQUE ANGLEs. See Angles
of many Parts.
Oblique.
NUCLEUS, is by Hevelius and OBLIQUE ASCENSION, is that
others uſed for the Head of a Co- Degree and Minute of the Equinoc-
met, and by others for the central tial which riſeth with the Centre of
Parts of any Planets.
the Sun or Star, or with any Point
NUCLEUS, in Architecture, is of the Heavens, in any oblique
the middle Part of the Flooring of Sphere.
the Antients, conſiſting of Cement, OBLIQUE CIRCLE,in the Stereo-
which they put betwixt a Lay, or graphical Projection of the Sphere,
Bed of Pebbles, cemented with Mor is any Circle that is Oblique to the
tar made of Lime and Sand.
Plane of Projection.
NUMBER, is whatever is referr'd OBLIQUE DESCENSION, is that
to Unity; or it is a Collection of Part of the Equinoctial which ſets
Units, and is that which teacheth with the Sun or Star, or with any
us to kno v how many any of the Point of the Heavens, in an oblique
Objects of cur Knowledge are. Sphere.
OBLIQUE
i
OBT
OC.T
OBLIQUE Force, is that whole Occidental, when it fets after the
Line of Direction is not at Right Sun.
Angles with the Body on which it OCCULTATION, in Aftronomy,
is impreſt. The Ratio,which ſuch is the Time that a Star or Planet is
an oblique Force, to move a Body, hid from our Sight, when eclipſed
bears to a direct or perpendicular by the Interpofition of the Body of
Force, will be as the Sing of the An- the Moon, or ſome other Planet
gle of Incidence is to the Radius. between it and us.
OBLIQUE PLAINS, in Dialling, OCEAN, is by Geographers taken
are ſuch as recline from the Zenith, for that great Collection of Waters,
or incline to the Horizon.'
or large Sea, which compaffes in the
OBLIQUE SAILING, is the Apó whole Earth, and into which the
plication of the Method of calcuother leffer Seas do uſually run.
lating the Parts of oblique Plane If, This great and univerſal 0-
Triangles, in order to find the Dif- cean, is ſometimes by Geographers
tance of a Ship from any Cape, divided into three Parts. As, 1. The
Head-Land, &c.
Atlantic and European Ocean, lying
OBLIQUE SPHERE, is where between Part of Europe, Africa, and
the Pole is elevated any Number of America. 2. The Indian Ocean,
Degrees leſs than 90 Degrees, and lying between Africa, the Eaſt-In-
conſequently the Axis of the World, dian Iſlands, and New-Holland. 3.
the Equator, and Parallels of De- The great South-Sea, or the Pacific
clination, will cut the Horizon ob- Ocean, which lies between the Phie
liquely.
lippine Iſlands, China, Japan, and
OBLONG, in Geometry, is the New-Holland on the Weſt, and the
ſame with a Rectangle - Parallelo- Coaſt of America on the Eaſt.
gram, whoſe Sides are unequal. adly, The Surface of the whole
OBSCURA CAMERA. See Ca- Ocean, or of all the Seas of the
mera Obſcura.
Globe, Mr. Keil computes, in his
OBSERVATION. T'he Seamen Examination of Dr Burnet's Theory
call an Obfervation the taking the of the Earth, to be 85490506 ſquare
Sun or any Star's Meridian Altitude, Miles ; and therefore ſuppoſing the
in order thereby to find their Latin Depth to be a Quarter of a Mile,
tude; and how they do this, you the Quantity of Water in the whole
will find under thật Word: And is 213726261 cubic Miles.
they call finding the Latitude, by OCTAGON, in Geometry, is a
the Name of Working an Obferva- Figure of eight Sides and Angles :
tion,
And this, when all the Sides and
OBTUSE ANGLEs. See Angles. Angles are equal, is called a Regu-
OBTUSE ANGULAR Section of a lar Octagon, or one which may be
Cone. So the ancient Geometers inſcribed in a Circle.
called that Conic Section, which If the Radius of a Circle circum-
fince, by Apollonius, is called the ſcribing a Regular Octagon be ar,
Hyperbola, becauſe they conſidered and the Side of the Octagon =y;
it only in ſuch a Cone, whoſe Sec-
tion through the Axis is a Triangle, then y= vi Lag V 27.
obtuſe-angled at the Vertex.
OCTAHEDRON, is one of the
OBTUS E-ANGLED TRIANGLE, regular Solids, conſiſting of eight
is one that has an obtuſe Angle. equal and equilateral Triangles.
OCCIDENTAL, (i. e. Wef ward) The Square of the Side of the
in Aftronomy, a Planet is ſaid to be Octahedron, is to the Square of the
Diameter
2
2r
OP.P
OPP
AD
Diameter of the circumſcribing OPPOSITE SECTIONS, are the
Sphere, as 1 to 2.
Hyperbola's D, C, made by cutting
If the Diameter of the Sphere be
2, the Solidity of the Ołtahedron
inſcribed in it, will be 1,33333,
nearly.
Octave, or Eighth,in Muſic,
is an Interval of eight Sounds.; eve-
Ty Eighth Note in the Scale of the
G
Gamut being the ſame, as far as the
Compaſs of Mufick requires.
OCTOSTYLE,, in Architecture,
is the face of an Edifice adorn'd
B
with eight Columns
OGEE. See Cima.
OPACOUS BODIES, are thoſe
thro' which the Rays of Light have the Oppoſite Cones A, B, by the
no Admiſſion.
ſame Plane. Theſe Hyperbola's are
Sir Iſaac Newton in his Optics, always, equal and ſimilar.
Book II. ſhews, That the Opacity If the oppoſite Superficies be cut
of all Bodies ariſeth from the Mule by a Plane making the oppoſite
titude of Reflexions cauſed in their Hyperbola's (or Sections) ÖES,
internal Parts : And he fhews alſo, o Ge: I ſay, both thoſe Hyperbo-
that between the Parts of the O- la's will be perfectly alike and
pake, and coloured Bodies, there equal.
are many Spaces, either empty or Let AF D be the Triangle paſ-
repleniſhed with Mediums of other fing thro' the Axis at Right An-
Denfities ; and he ſhews the true or gles to the Plane of the Hyperbola
principal Cauſe of Opacity to be oes, and ſuppoſe LFI 'to be a
This Diſcontinuity of their Parts ;
becauſe ſome Opake Bodies become
-L
КНИ
tranſparent by filling their Pores
with any Subitance of equal Den-
fity with their Parts.
Open FLANK, in Fortification,
is that Part of the Flank which is
covered by the Shoulder or Oril:
lion.
OPENING of the Trenches, is the
firſt breaking Ground of the Beſieg-
ers, in order to carry on their At-
tacks againſt the Town.
OPHIUCUS. One of the Northern
Conſtellations, containing thirty
E
Stars.
OPPOSITE ANGLES. See Angles.
OPPOSITE CONes, are two Si-
milar Canes, as A, B, having the
D
ſame common Vertex G, and alſo
B
the ſame Axis.
0
Tri-
A
אתם מעוניינינגפוטושוויווי• גריי
A
2
O PT
ORB
Triangle, in the fame Plane as the may comprehend the whole Doc-
Triangle AFD; this ſhall paſs trine of Light and Colours, and all
thro' the Axis of the oppoſite Cone, the Phænomena of viſible Objects.
and will cut the Hyperbola oG' e at Euclid long ago wrote of Optics,
Right Angles. Let A D, and L1, but with no great Skill. See Dr.
be parallel common Sections of thoſe Gregory's Euclid, and Herigon's Cur-
Triangles, and the Baſes of the op- jus Mathemat. ſo did Ptolemy in 10
poſite Cones. Draw the Right Line Books, but his work is loft. After
KF B thro' the Vertex F, in the theſe came out Alhazen the Aram
Plane of the Triangles, parallel to bian's Optics, (who wrote about the
the common Diameter GE of the Year 110o) a voluminous, tedious
Oppoſite Sections. Now, our Buſi- Piece: then Vitellio's about the Year
nes is to prove, that LHXHI 1270 ; and Peccam's, an Arch-
(= H): ACCD=0C) :: Biſhop of Canterbury, about the Year
H EXGH:GCX E C.
1279; alſo Roger Bacon, of Oxford,
Becauſe the Triangles A BF, began to write of Optics about the
ACG, and DBF, DCE, are fi ſame time. Amongthe moreModern,
milar. We have A'B:BF: : AC: you have Agulonius and Scheiner
CG, and BD:BF::CD:EC; the Jeſuit ; Taquet, Traber, Barrow,
Zahan, Kircher, Newton, and not
therefore A BxB D:BF :: AC long ago Dr. Smith, and Mr. Mar-
XCD:CGxEC, by multiply- tin.
ing the Antecedents and Conſe Optic PLACE of a Star or Pla-
quents of both Proportions by each net, is that Point or Part of its Or-
other.
bit, which is determined by our
Again, becauſe the Triangles Sight, when the Star is there ; and
ABF, IHG, and BDF, HLE, this is either true, when the Ob.
are ſimilar, therefore A B : BF :: ſerver's Eye is fuppoſed to be at the
HI : HG; BD:BF::LH: Centre of the Earth or Planet he
HE. And ſo multiplying the Ante- inhabits ; or apparent, when his
cedents and Conſequents of both the Eye is at the Circumference of the
Proportions by one another, and Earth.
you will have AB X BD:BD:BF
ORB, is only a hollow Sphere.
:: HIxLH:HG * H E. But it of the Earth in its Annual Revolu-
ORBIS MAGNUS, is the Orbit
was prov'd before thatABxBD: BF tion round the Sun.
A CXCD:CGXEC. There All the Ancients, and the Aftro-
fore HIxL H:HG ⓇHE:: AC nomers before the great Kepler ſup-
xCD:CGXE C, and fo H I x poſed this Orbit to be a perfect Cir-
LH:AC XCD::HG X HE: cle ; but he proves it to be an El-
C GⓇEC.
lipfis ; the remoteſt End of whoſe
Opposition, is that Poſition or longer or tranſverſe Diameter is
Aſpect of the Stars or Planets, when eight Signs, and eight Degrees di-
they are 6 Signs, or 180 Degrees ftant from the firſt Star in Aries,
diſtant from one another, and is and having the Sun in one of its
marked thus, bl.
Focal Points.
OPTICKS, taken properly and ORBIT of any Planet, is the Curve
fimply, is that Science which teaches that it deſcribes, about the Sun.
the Properties of a
a direct Vi The Orbits of all the Planets are
“fion ; but in a larger Senſe it Ellipſes, having the Sun in their
2
2
CO.n-
ORD
OR D
1
common Focus : But the Elliptic qual to the third Part terminated at
Orbit of the Earth, by the Action the Curve on the other Side: This
of the Moon, is ſenſibly disfigur'd; Line ſhall cut, after the fame man-
as alſo the Orbit of Saturn, by the ner, all others parallel to theſe, and
Action of Jupiter, when they are in . meeting the Curve in three Points ;
Conjunction.
that is, ſhall ſo cut them, that the
ORDER, in Architecture, is a Sum of the two parts on one side
particular Arrangement of Projec- of it, ſhall be equal to the third
tures; or 'tis a certain Rule for the Part on the other.
Proportions of Columns, and for
And therefore, theſe three Parts,
the Figures which ſome of the Parts
one of which is thus every where
ought to have on account of the equal to the Sum of the other two,
Proportions that are given themmay be called Ordinate Applicates
There are fix, viz the Tuſcan Order, alſo: And the interfccting Line, to
Doric Order, Ionic Order, Corin- which the Ordinates are applied,
thian Order, Compoſite Order, and the may be called the Diameter; the In-
Attic Order.
terfection of the Diameter and the
ORDER of Curve Lines. See Curve may be called the Vertex; and
Geometric Lines.
the Point of Concourſe of any two
1. The chief Properties of the Diameters, the Centre.
Conic Sections are everywhere And if the Diameter be Normal
treated of by Geometers ; and of to the Ordinates, it may be called
the ſame Nature are the Properties the Axis; and that point where all
of the Curves of the ſecond Gender, the Diameters terminate, the gene-
and of the reſt ; as from the follow- ral Centre.
ing Enumeration of their principal
Properties will appear.
Afymptotes and their Properties.
2. For, if any Right and Paral 3. The Hyperbola cf the firſt
lel Lines be drawn and terminated Gender has two Aſymptotes ; that
on both sides by one and the ſame of the ſecond, three; that of the
Conic Section ; a Right Line bif- third, four; and it can have no
fecting any two of them, ſhall biſ- niore, and ſo of the reft. And as
fect all the relt ; and therefore, the Parts of any right Line lying
ſuch a Line is called the Diameter of between the conical Hyperbola, and
the Figure ; and all the right Lines its two Aſymptotes are every where
ſo biſfected, are called Ordinate Ap- equal ; ſo in the Hyperbola's of the
plicates to that Diameter ; and the ſecond Gender, if any right Line be
Point of Concourſe to all the Dia- drawn, cutting both the Curve and
meters, is called the Centre of the its three Afymptotes, in three Points;
Figure ; as the Interſection of the the Sum of the two parts of that
Curve, and of the Diameter, is cal. Right Line being drawn the ſame
led the Vertex, and that Diameter way from any two Aſymptotes to
the Axis, to which the Ordinates are two Points of the Curve, will be
normally applied: And ſo in Curves equal to the third Part drawn a
of the ſecond Gender ; if any two contrary Way from the third Afymp-
right and parallel Lines are drawn tote, to a third Point of the Curvę.
meeting the Curve in three Points, a
Latera Tranfverfa & Recta.
right Line which ſhall cut thoſe Pa-
rallels, ſo that the Sum of two Parts 4. And as in Non Parabolic Co-
terminated at the Curve on one side nic Sections, the Square of the Or-
of the interſecting Line ſhall be e- dinate Applicate, that is, the Rect-
angle
1
ORD
ORD
angle under the Ordinates, drawn at cach Side by the Curve ; the firft
contrary Sides of the Diameter, is being cut by the third, and the fem
to the Rectangle of the Parts of the cond by the fourth ; as here the
Diameter, which are terminated at Rectangle under the Parts of the
the Vertexes of the Ellipfis or Hy: firſt, is to the Rectangle under
perbola, as a certain given Line, the Parts of the third, as the
which is called the Latus Rectum, Rectangle under the Parts of the le-
is to that part of the Diameter that cond, is to that under the Parts of
lies, between the Vertexes, and is the fourth : So when four ſuch right
called the Latus Tranfverſum: So Lines meet a Curve of the Second
in Non-Parabolic Curves of the Se- Gender, each one in three Points,
cond Gender, a Parallelepipedon, un- then ſhall the Parallelopipedon under
der the three Ordinate Applicates, is the Parts of the firſt right Line be
to a Parallelopipedon under the Parts to that under the Parts of the third
of a Diameter terminated at the as the Parallelopipedon under the
Ordinates, and the three Vertexes Parts of the ſecond Line is to that
of the Figure in a certain given under the Parts of the fourth.
Ratio : If you take three right Lines
to the three parts ofia Diameter fi Hyperbolic and Parabolic Legs.
tuated between the Vertexes of the
Figure, one anſwering to another; - All the Legs of Curves of the fe-
then theſe three right Lines may be cond and higher Genders, as well
called the Latera Recta, of the Fi as of the firſt, infinitely drawn out,
gure, and the Parts of the Diame- will be of the Hyperbolic or Para-,
ter between the Vertices, the Latera bolic Gender; and I call, that an
Tranfverfa. And as in the Conic Hyperbolic Leg, which infinitely ap-
Parabola, having to one and the ſame proaches to ſome Afymptote ; and
Diameter but one only Vertex, the that a Parabolic one, which hath no
Rectangle under the Ordinates is Aſymptote. And theſe Legs are
equal to that under the part of the belt known from the Tangents :
Diameter cut off between the Ordi- For, if the Point of Contact be at
nates and the Vertex, and a certain an infinite Diſtance, the Tangent of
Line called the Latus Rectum : So an Hyperbolic Leg will coincide
in the Curves of the Second Gender, with the Afymptote; and the Tan-
which have but two Vertexes to the gent of a parabolic Leg will recede
fame Diameter, the Parallelopipe in infinitum, will vaniſh, and no
don under the three Ordinates, is where be found. Wherefore, the
equal to the Parallelopipedon under Aſymptote of any Leg is found, by
the two Parts of the Diameter cut ſeeking the Tangent to that Leg at
off between the Ordinates and thoſe a Point infinitely diſtant : And the
two Vertexes, and a given Right Courfe, Place, or Way of an infi-
Line ; which therefore may be cal- nite Leg, is found by ſeeking the
led the. Latus Rectum.
Poſition of any right Line, which is
The Ratio of the Rettangles under parallel to the Tangent where the
the Segments of Parallels.
off infini,
tum: For this right Line is directed
Laſily, As in the Coric Sections, towards the ſame way with the in-
when two Parallels, terminated on
each Side at the Curve, are cut by
or ORDINATE
two other Parallels terminated on APPLICATES, are parallel Lines ·
finite Leg.
ORDINATES,
MM,
ORG
ORG
P
MM, terminating in a Curve, and that is, a Conic Section : And to
biffected by a Diameter, as AP. find which of the Conic Sections
will be deſcrib'd according to the
A
various Magnitude of the given
Angles FCQ, and KSH, and
Polition of the Line A E, deſcribe a
P
Segment of a Circle on the given
M
M
Line CS; containing an Angle e-
M
M
qual to the Complement of the gi-
ven Angles FCO, and KSH to
four Right Angles : If the given
Right Line A E meets that Circle
The half of which, as MP, is pro- twice, the Curve will be an Hyper-
perly the Semi-Ordinate, but it is bola : If it touches it, a Parabola:
uſually called the Ordinate. And if the Right Line A E falls
ORDNANCE,
are all ſorts of quite beſide the Circle, the Curve
great Guns uſed in War.
deſcribd will be an Ellipfis.
ORDONNANCE , fignifies the 2. While the Right Line A E
ſame thing in Architecture that it remains, and the Sum of the given
does in Painting; to wit, the Com- Angles FCO, and KSH, the
poſition of a Building, and the Dif- Species of the Curve will be the
poſition of all its Parts; it being fame; and in no caſe will a Circle
this that determines the Bigneſs of be deſcrib’d, but when the Right
the ſeveral Members, whereof a Line A E goes out to Infinity.
Building is compoſed.
3. If the given Angles above are
ORGANICAL DESCRIPTION of mutually the Supplements of each
Curves, is the Deſcription of them other to two Right ones, and the
upon a Plane, by means of In- Line A E meets CS continu'd out;
Itruments.
there will be an Hyperbola deſcrib'd:
1. If the given Angles FCO, If A E be parallel to CS, a Para-
and KSH move about two Points bola will be deſcrib'd.
S and C given in any Plane, and 4. If the infinite Right Lines
the Concurrence of the Legs CF, GH, DB cut one another at Right
SK, be moved along the Right Angles, and the Angle B of a Square
Line A E given in Poſition in that
Plane; then will the Concurrence
G
K
Pc
F
E
D
E
A S
B
P
Р
F
7
oc H
H
P, if the other Legs.CO, SH, de- ABC be faſten'd to the Point B ir
ſcribe a Curve of the firft kind, the Right Line DB, ſo as the Square
may
N
G
ORG
ORT
may be moveable about it; and This Term is alſo appropriated
if FG be a Ruler moveable about to certain long, and thick Pieces of
the Point E in the Right Line Timber, armed with Iron Plates at
DB, then if the Interſection G the Ends, and ſeparated one from
of the Ruler and one Side B B another. They are hung with
of the Square be carried along the Cords over the Gates of a Town
Right Line HG, the Interſection or Fortreſs, and in caſe of a Sur-
F of that Ruler, and the other Side prize, let fall perpendicularly ; by
CB (continued out upon occafion) which means the Paſſage is ſtopped,
will deſcribe one of the Conic Sec- fo that the Enemy cannot eaſily
tions; which will be an Ellipfis, remove or hoift up all the wooden
when the Point-E is taken between Bars with a Leaver, or any other
D and B; an Hyperbola, when D Machine fet under them : On which
is between E and B; and a Para- account, theſe Orgues are to be
bola, when E is at an infinite Di- preferred before Herſes or Port-
ftance, that is, when the Ruler ale cullices, becauſe the Pieces whereof
ways moves parallel to DB.
the latter conſiſt are joined together;
5. If ECA be a Right-angled ſo that when any Part iš hung or
Triangle, and the Sides AC, A E heaved up, the whole Machine is
likewiſe removed.
Theſe Orgues
IZ
therefore are much better than Port-
cullices.
ORIENTAL, in Aſtronomy: A
Planet is ſaid to be Oriental, when
E
it riſes in the Morning before the
Sun.
ORILLON, in Fortification, is a
ſmall Rounding of Earth lined with
X C
a Wall, which is raiſed on the
A Shoulder of thoſe Baftions that have
Caſemates to cover the Cannon in
be continued out, and if any point the retired Flank, and to prevent
X be taken in AC, and the Per- their being diſmounted by the E-
pendicular XG be drawn; then if nemy.
from the Point G be drawn the There are alſo other ſorts of O-
Right Line CG, and XZ
be made rillons, properly called Shoulderings,
equal to CG, the Point Z will be which are almoſt of a ſquare Fi-
in a Conic Section, which will be gure; they are called Epaulments.
an Ellipfis, when AC is greater Orion, a Southern Conſtellation,
than CĒ; an Hyperbola, when AC confifting of 39 Stars.
is leſs; and a Parabola, when AC ORLE, a Term in Architecture ;
is equal to CE.
the ſame with Plinth, which fee.
ORGUES, in' Fortification, are ORNAMENT, in Architecture, is
many Harque-Buffes, linked toge- any Piece of carved Work, ſerving
ther, or divers Muſket-Barrels laid as a Decoration in Architecture :
in a Row, within one wooden Stock, But the Word in Vitruvius and Vig-
ſo that they may be diſcharged nola, is uſed to ſignify the Entable-
either all at once, or ſeparately. ment.
They are made uſe of to defend Ortell, a Term in Fortifica-
Breaches, and other Poſts that are tion; the ſame with Berme, which
attacked.
fee.
ORTHO.
ÖRT
ORT
1
!
;
1
ORTHODROMIQUES, is the Art Line, 'as EF, or GH; and is al
of Sailing in the Arches of fome ways comprehended between the
great Circle : For the Arch of every
great Circle is Orthodromia, or the
Thorteft Diſtance between any two
с
Points on the Surface of the Globe,
ORTHOGRAPHY; in Mathema-
A
B.
tics, is the true Delineation of the
fore-right Plane of any Object.
D
I. In Architecture 'cis taken for
the Model, Platform and Delinea
tion of the Front of a Houſe that is
PG
HE
to be built and contrived according
to the Rules of Geometry ; accord- extreme Perpendiculars AP, and
ing to which Pattern, the whole BE.
Fabric is erected and finiſhed. 4. The Projection of the Right
2. In Perſpective, the Orthogra Line A B, is the greateſt when & B
phy of any Body or Building is is parallel to the Plane of the Pro-
the fore-right side of any Plane ; jection.
that is, the side or Plane that lies 5. From hence' it is evident, that
parallel to a ſtraight Line; that may a Line parallel to the Plain of the
Þe imagined to paſs thro' the out- Projection, is projected into a right
ward Convex-Points of the Eyes, Line equal to itſelf; but if it be ob-
continued to a convenient Length. lique to the Plane of the Projec-
The word Schenography is uſed tion, 'tis projected into one which
by. Lamy, and others in the ſame is leſs.
fenſe.
6. A plain Surface, as ABCD,
In Fortification, it is the Pro at right Angles to the Plane of the
file or Repreſentation of a Fortreſs, Projection, is projected into that
made after ſuch a manner, that the right Line (as AB) in which it cuts
Length, Breadth, and Height of its the Plane of the Projection. Hence
ſeveral Parts may be diſcovered. it is evident, that the Circle BCAD
ORTHOGRAPHICAL Projec- ftanding at right' Angles to the
TION of the Sphere, is the drawing
с
the Superficies of the Sphere on a
Plane which cutteth it in the mida
dle, the Eye being placed at an infi-
하
​nite Diſtance vertically to one of the
Hemiſphere's.
B
A
1. The Rays by which the Eye,
at an infinite diſtance, perceives any
Object, are parallel.
2. A Right Line perpendicular
to the Plane of the Projection, is
D
projected into a Point, where that
right Line cuts the Plane of the Pro- Plane of the Projection, which paſſes
jection.
thro' its Centre, is projected into
3. A right Line, a: AB, or CD, that Diameter A. B, in which it cuts»
not perpendicular, but either paral- the Plane of the Projection.
lel or oblique to the Plane of the 7. It is likewiſe evident, that
Projection, is projected into a right any Arch as cc is projected into o'o
3
equal
2
1
1
I.
It
OSC
O VA
equal to Ca, Cb, which is the 5. The Length of a Pendulum
right Sine of that Arch; and the that will perform its Oſcillations in
complemental Arch cA is projected a Second, is 39.125 Inches, or three
into o A, the verſed Sine of the Feet 3.125.
ſame Arch CC.
6. The ſhorter the Oſcillations in
8. A Circle parallel to the Plane the Arch of a Circle are, the truer
of the Projection, is projected into will the Pendulum meaſure Time,
a Circle equal to itſelf ; and a Cir or the more Iſochronal will the Of
cle oblique to the Plane of the Pro- cillations be.
jection, is projected into an Ellipfis. OSTENSIVE DEMONSTRA-
OSCILLATION, is the reciprocal Tions, are ſuch as plainly and di-
Aſcent and Deſcent of a Pendulum. rectly demonſtrate the Truth of any
1. If a ſingle Pendulum be fuſpend- Propofition ; in which they are di-
ed between two Semi-Cycloids B C, ftinguiſhed from A pogogical ones,
CD, that have the Diameter CF or Deductiones ad abfurdum, five ad
impoſſibile, which prove the Truth
propoſed, by demonſtrating the Ab-
С
ſurdity or Impoſſibility of aſſerting
the contrary
OSTENSIVE
DEMONSTRA-
B
D TIONS, are of two forts ; ſome of
which barely (but directly) prove
E
the Thing to be, which they call
on; and others demonſtrate the
Thing from its Cauſe, Nature, or
А.
eſſential Properties, and theſe are
called in the Schools doth.
of the generating Circle equal to OTACOUSTICS, are Inſtruments
half the Length of the String, ſo which help or improve the Senſe
that the String, as it'oſcillates, folds of Hearing.
about them; all the Oſcillations, OVAL, in Architecture, the fame
however unequal, will be Iſochro- with Echinus. Some write it Ova,
nal in a non-reſiſting Medium. becauſe of its Figure, being like an
2. The Time of an whole Oſcil- Egg; it is placed in the Mouldings
lation, thro' any Arch of a Cycloid, of the Cornices for Ornament; and
is to the Time of the perpendicular in a Pillar it is placed next to the
Deſcent thro' the Diameter of the Abacus.
generating Circle, as the Periphery OVAL FIGURE, in Geometry, is
of the Circle to the Diameter. a Figure bounded by a Curve Line
.3: If two Pendulums deſcribe fi- returning into itſelf.
milar Arches of Circles, che Times A Figure bounded by circular
of the Oſcillations are in the fub. Arches, ſo meeting as to coincide at
duplicate Ratio of their Lengths. the Points of meeting with the Tan-
4. The Number of Iſochronal gents to the Arches, and as to ap-
Oſcillations made in the ſame time pearance not differing from an El-
by two Pendulums, are reciprocally lipſis, is by Artificers call's an Oval,
as the times wherein each of the and may be thus deſcribed, to any
Oſcillations are made. The Times given Length and Breadth. Let
of the Oſcillations in different Cy- the given Length AB and Breadth
cloids, are in the ſubduplicate Ratio DE cut one another at right An-
of the Length of the Pendulums. gles, and in half at the Point C,
Cc aflume
O V A
O V A.
K
H
affume the right Lines AF, LB in I. This done, about the Centres
equal ; but leſs than the Breadth K I, with the Diſtance IN, deſcribe
D
two Arches NDO,ME A, and the
Oval required will be deſcribed.
Altho? it is uſual to call only a
N
G
Curve, reſembling an Ellipfis, an O-
val ; yet there are really in nature
A
B
F
an infinite Variety of Geometrical
M
A А
Ovals, of very different and plea-
fant Figures, expreſſed by Equa-
I
tions of all Dimenſions about the
ſecond, and more eſpecially thoſe
CD, and aboạt the Centres F, L, of the even Dimenſions ; as the E-
with the Diſtances AF, LB, de- quation a ayy= 34 + ax3 repre-
ſcribe two ſmall Arches MAN, ſents the Oval B, in ſhape of the
ABO; take DG equal to AF, Section of a Pear thro' the middle,
join FG, which divide into two and is eaſily deſcribed by means of
equal Parts at H, and fron H draw Points ; for if a Circle be deſcribed
the Right Line HI perpendicular whoſe Diameter AC=a, and AD
to FG, meeting the Diameter CE at Right Angles to AC be =AC,
D
N
B
M
M
Α Ρ
i
and any Point P be taken in AC, Ovals, amongſt which the following
and PM be drawn parallel to AD, twelve are, moſt obſervable; for if
and DP be drawn, and then NO; the Equation axt=bx3 + x² + x
and if PM be taken equal to NO, te has four real unequal Roots ;
the Point M will be one Point of the three Species, as appears in
the Oval fought.
Fig. 1, 2, 3, will be expreffed by
In like manner the Equation yt- the given Equation. When the two
py?=~2x4+ bx3 c3? + da leſſer Roots are equal, the three
*e, expreſſes ſeveral very pretty Species as appears in Fig. 4,5,6,
Fig. 1.
Fig. 2.
Fig. 3.
88 ☆ o
87 e
Fig. 4
Fig. 5:
Fig. 6.
will
7
}
OUT
PAL
will be expreſſed. When the two 10. when two Roots are equal, and
leffer Roots become imaginary ; two more ſo, the Species will be as
the three Species, as appear in Fig.7, appears in Fig. 11, and when the
8,9, will be expreſſed. And when two middle Roots become imaginary,
the two middle Roots are equal, the the Species will be as appears in
Species will be as appears in Fig. Fig. 12.
Fig. 7.
Fig. 8.
Fig. 9.
ਕੋਰ
Fig. 10.
Fig. 11.
Fig. 12.
88
∞
8 8
1
OUTWARD Flanking Angle, or Works muſt always be plain, and
the Angle of the Tenaille, is that com- without Parapets ; left, when taken,
prehended by the two Flanking- they ſhould ſerve to ſecure the Be-
Lines of Defence.
fiegers againſt the Fire of the re-
OUT-WORKS, in Fortification, tiring Beſieged; wherefore the
are all ſorts of Works, which are Gorges of Out-Works are only pal-
raiſed without the Incloſure of a liſadoed, to prevent a Surprize.
Place, and ſerve for its better De Ovolo, in Architecture ; fee
fence, and to cover it from the E- Quarter-round.
nemy, in the Plain without; as OXYGONE, the ſame with an ao
Ravelins, Half - Moons, Horn- cute-angled Triangle, and in ge-
Works, Crown - Works, Counter- neral
Guards, Tenailles, &c.
OXYGONIAL, is acute-angular.
1. It is a general Rule in all
Out-Works, that if there be ſeveral
of them, one before another, to
cover one and the ſelf-fame Tenaille
P.
of a Place, the nearer ones muſt gra-
dually, and one after another, com ALLET, is a Term belong-
mand thoſe which are fartheft ad ing the Ballance of a Watch,
vanced out into the Campagne ; or Movement.
that is, muſt have higher Ramparts,
PALLIFICATION, in Architec-
that ſo they may overlook and fire ture, is the piling the Ground-
upon the Beſiegers, when they are Work, or ſtrengthening it with
Maſters of the more Outward Piles or Timber driven into the
Works.
Ground, when they build upon a
2. The Gorges alſo of all Out- moiſt or marthy Soil.
Сс 2
PAL-
PAL
PAR
1
1
4
PALLISADES TURNING, are an PARABOL A, is a Curve, as EDF,
Invention of Mr. Coeborne's : For, in made by cutting a Cone by a Plane
order to preſerve the Paliſadoes of DG, parallel to one of its- Sides,
the Parapet from the Beſiegers Shot, as BC.
he orders them ſo, that many of
them ſtand in the Length of a Rod,
B
or in about ten Foot, and turn up
and down like a Trap; ſo that they
are not in fight of the Enemy, but
only juſt when they bring on their
Attack, and yet are always ready
to do the proper Service of Palli-
ſades.
A
C
PALLISADOES, or PALLISADES,
in Fortification, are ſtrong wooden
E
ſharp-pointed Stakes, fix or ſeven
Inches ſquare, eight Foot long, of
I. All Diameters D C of a Para-
which three Foot is in the Ground; bola, are parallel to the Axis B A,
ſet up half a Foot ſometimes one
and ſo are parallel to one another.
above another, with a croſs Piece
From A draw the Line AE,
of Timber that binds them together. which may be biſfected by the Dia-
Some of theſe are alſo ſometimes meter DC in the Point L;, and
arm'd with 'two or three Iron thro' any Point K in the Axis draw
HKM. Alſo from the Points
Spikes.
1: Theſe Palliſadoes are uſually H, E, C, draw the Semi-Ordinates
fixed in the void Spaces without the HB, EF, CO, to the Axis, which
Glacis near the Baſtions and Cur- will be all perpendicular to the
tains ; and in Avenues of all ſuch fame; then call the given Line CO,
Poſts as are liable to be ſurprized by
H
the Enemy, or carried by Aſſault.
Sometimes they are driven down-
right in the Ground, and ſometimes
G
ſtand at an acute Angle towards the D
I L
Enemy, that if they ſhould throw
Cords about them to pull them up,
A
they may ſlip off again.
B
IR
ź. Palliſadoes are always planted
on the Berme of Baſtions, and at the
M
Gorges of Half-Moons, and other
Out-Works: They alſo palliſade u-
ſually the Bottom of the Ditch ;
and to be ſure, the Parapet of the or GT, or DB, a; and BH, y; and
Cover'd-Way : And tho' ſometimes the Parameter to the Axis p. Now
they have placed theſe Palliſadoes
three Foot, from the ſaid Parapet OA= FA
and GL
outwards towards the Campagne ;
P
P
yet of late they have been planted
(becauſe the Triangles
in the very middle of the Cover'd-
way. All Palliſadoes ſhould fland EFA, EGL, being ſimilar, and
ſo cloſe, as to admit between them the Side EA biſfcčted in I, the
only the Muzzle of a Musset, or Side EF ſhall be biſfected in G,
*Pike.
and
1
C
НА
aa
даа
2 ad
II
m (=0 A) and EL" =aa
more
424
) (490
(5)
FLD
2 ay
F
F
?
2
2
2
2
PAR
PAR
and G L ſhall be = FA) and LC This being done, if you ſlide DG,
the side of the Square along the
Rule B C, and at the ſame time
keep the Thread continually tight
+ And becauſe the Tri- by means of the Pin M, with its
PP
Part MO cloſe to the ſide of the
angles. E FA, HBK, are ſimilar,
B
therefore EF (2 a) : FA
G
AM
:: HB (y): BK =
But
-ВА BK +OA = IC=
P
M
yy-zay+aa
ſince IL is =
K A and OA = LC. Therefore
XO
O
Z
y-zay+aa
L
P P
Square DO: The Curve AMX,
which the Pin deſcribes by this
:: EG': (a a) HD (yy-zayt-aa) Motion, is one Part of a Parabola.
:: E L : HI, becauſe the Trian, and moves on the other ſide of the
And if the Square be turn'd about,
gles DHI, GEL, are ſimilar, and fixed Point F, the other Part A MZ
confequently EC:IC :: EL : LA. of the fame Parabola may be de-
And drawing a Perpendicular from fcribed after the like manner ; to
the Point Mºto DC, and reaſoning that the Line XAZ will be one and
the ſame Curve.
after the ſame manner, you will have
3. To, draw a Tangent to the
CL : CD :: LA (=LE): IM. Parabola ; let AE be the Axis, DF,
Whence IM is = lH; and be- EG, two Ordinates infinitely near
cauſe the Point K is taken at plea-
fure in the Axis ; therefore all Right
F
Lines drawn parallel to E A, ſhall
H Η
be biſſected by the Line DC, and
ſo the ſame ſhall be a Diameter ac-
cording to the Definition, and the
Lines EA and HKM ſhall be Or-
A B DE
dinates to it.
2. If the Rule BC be placed
upon a Plane, together with the
Square GDO, in ſuch manner,
to each other, and FH parallel to
that DG, one of its Sides, lies a-
AE: Let p be the Parameter, AF
long the Edge of that Rule; and if the Tangent_(to be drawn) = a,
BD
you take the Thread FMO equal
=*, D'E=s, and DF -
=y.
in Length to DO, the other Side Then px=yy, and px + ps =
of the Square, and fix one End
thereof to the Extremity of the yy +2 GHxy+GH. But fince
Side DO, and the other in any GH is infinitely ſmall GH is in-
Point F, taken in the Plane on the finitely leſs than 2 GHxy, and ſo
fame fide of the Rule as the Square : may be rejected ; ſo that px +ps
2
2
G
1
2
2
Сс 3
PAR
PAR
1
2 ys
3a²
5a4
1
A
$
998; &c. will be the
=y+2 GHxyj and if from Ř Z is equal to RM into the Para-
this Equation be taken px -ýj, meter ; and ſo is a conſtant Quan-
we ſhall have ps 32 GĦ y; city.
and fo's : GH :: 2 j : p. But be 9. If a be the Parameter, and y
cauſe of the ſimilar Triangles ADF,
2 33
FHG, it will be s: GX :: a:vi =PM, then yt
therefore 2y:p :: a : , and ſo pa
2yy, or pa = 2 px, fince px =
437 10y9
yy; wherefore a=2x, or the Sub- yaº.
tangent AD= 2 BD.
Length of the Curve AM of the
4. The Right Line FM drawn Parabola.
from the Focus F, in the Axis to The Length of the Curve of the
the Extremity of the Semi-Ordinate Parabola may be obtained by means
PM, is equal to the Abſciſfæ AP of the Quadrature of the Hyperbo-
and AF, the Diſtance of the Fo- lic Space, which was firſt taken
cus from the Vertex.
notice of by Mr. Huygens, in the
Year 1657; for if there be two
oppoſite equilateral Hyperbola's,
A
whoſe tranſverſe Axis is equal to
the Parameter of the Axis of the
A
Parabola ; then the Space contain'd
under that Tranſverſe Axis, the
Curves of the oppoſite Hyperbola's,
N and a Right Line drawn parallel to
M
that Tranſverſe Axis will be equal
P
to the part of the Curve of the Pa-
rabola, whoſe Semi-Ordinate is
qual to the Diſtance of the ſaid Pa-
rallel from the Tranſverſe Axis of
LA
the Hyperbola drawn into the
R 2 Latis rectum of the Axis of the Pa-
rabola. Hence the Length of the
Curve of the Parabola may be had
5. The Square of the Semi-Ordi- by means of the Logarithms, and
nate PM is equal to the Rectangle that after the following manner.
under the Abſciſs AP, and the Let x be the Abfciſs and y the Se-
Parameter.
mi-Ordinate of the Parabola ; ſay,
6. The Rectangle under the Sum
as the conſtant Number 0,434294
of any two Semi-Ordinates, and is to the Logarithm of the Ratio of
their Difference, is equal to a Rect-
v
уу
angle under the Parameter, and the
****** to y, fo is of the
Difference of the Abfcifles.
4
In the Parabola, the Sub-Tangent Parameter of the Axis, to a fourth
PT is twice the Abſciſs AP, and
the Sab.Normal PQ=1 thé Pa. Number, which added to xx+
rameter, and ſo is a conſtant Quan-
4
tity.
will be the Length of the Curve
7. The Focus of the Parabola is of the Parabola, whoſe Abſciſs is x,
at ſuch a diſtance from the Vertex, and Ordinate zy.
that the Semi-Ordinate FN =1 PARABOLIC CONOID, is a Solid
the Parameter.
generated by the Rotation of a Pa-
8. The Rectangle under LR and rabola about its Axis.
This
Ky pro
e-
1
✓
1
$
PĄ R
PAR
This Solid is La Cylinder of the A Circle equal to the Curve Su-
fame Baſe, and Altitude ; for the perficies of a Parabolic Conoid, is
Circular Planes parallel to the Baſe, thus moft elegantly found by Mr.
are as the Numbers in Arithmeti- Huygens, in his Horolog. Draw the
cal Progreſſion, from the Nature of 'Tangent (Fig. 1.) MT, and divide
the Parabola ; that is, are as the the Ordinate MP in O, ſo that
Ordinates of a Triangle.
MO be to OP, as MT to MP;
Fig. 1.
T
A
B
OP
M
N
1
Fig. 2.
M
N
T
А
PBQ
then if between TM +OP and to the Curve Superficies of the Pa.
MN, you find a mean Proportional rabolic Conoid generated from the
BC; the Circle deſcribed with BC Parabola AMP.
for its Radius, will be equal to the PARABOLA Carteſian, is a Curve
Curve Surface of the Parabolic Co- of the ſecond Order expreſſed by
noid generated by the Rotation of the Equation xy-ax+bx2+x+d,
the Parabola MAN about its Axis containing four infinite Legs, viz.
AP. Mr. Huygens does not de two Hyperbolic ones, MM, B m,
monſtrate this ; but it is eafily e- A E being the Afymptote) tending
nough done from hence : Let (Fig.2.)
BNR be a Parabola deſcribed to
any Axis BQO, whoſe principal N
M
N
Vertex is B, with a Latus reftum to
the Axis, equal to four times the
Latus reftum, fuppoſe L, of the
M
Axis AP of the Parabola AM, and
let BQ be = L, and QO=AP,
and draw the Ordinates N, OR.
Then it will be as the Diameter of
m
'a Circle is to its Circumference, ſo
is the Parabolic Trapezium QNRO
1
1
A/B
Сс 4
icona
1
!
PAR
PAR
}
;
contrary ways, and two Parabolic, PARABOLA Diverging, is a Name
Legs. DN, MN joining them, be- given by Sir Iſaac Newton to five
ing the 66th Species of Lines of the different Lines of the third Or-
third Order, according to Sir Iſaac der, expreſſed by the Equation yy
Newton, call'd by him a Trident, ax3 +6x2+x+d. The firſt (Fig.1.)
and is made uſe of by Deſcartes, in being a Bell-Form Parabola, with
the third Book of his Geometry, for an Oval at its Head, which is the
finding the Roots of Equations of cafe when the Equation wax3 +
fix Dimenſions by its Interfections bx2+x+d, has three real and
with a Circle. Its moſt ſimple Equati- unequal Roots; fo that one of the
on is xy=x3 ta3, and Points through mof fimple Equations of a Curve
which it is to paſs may be eaſily of this kind is pyy=x3 tax? +-aax.
found by means of a common Pa- The ſecond a Bell-Form 'Para-
rabola whoſe Abſciſs is ax? 4-6x-4r, bola, with a conjugate Point or
and an Hyperbola whoſe Abſciſs is infinitely ſmall Oval at the Head,
d
being the caſe when the Equation
for y will be equal to the Sumax3 +bx? tcxtd has its two
or Difference of the correſpondent leſſer Roots equal , the moſt fimple
Ordinates of this Parabola and Hy. Equation of which is pyy=*3-axx.
perbola.
The third (Fig. 2.) a Parabola,
Deſcartes, in the aforeſaid Book,
Shews how to deſcribe this Curve
Fig. 1.
by a continued Motion, viz. by
taking a fixed Point upon a Plane,
and without that point ſuppoſing
an infinite Right Line to be drawn
upon that Plane in a given Poſition,
and then taking a Parabola drawn
upon a ſeparate Plane, and having
aſſumed a point in the Axis, and
faften'd a Ruler to the ſame, as alſo
Fig. 2.
to the point aſſumed upon the Plane,
he moves the Plane of the Parabola
along, ſo as its Axis always coin-
cides with the Line drawn upon
the Plane in a given Poſition, and
then the Interſections of the Curve
Fig. 3.
of the Parabola and the long Ruler
will deſcribe upon the Plane the
Carteſian Parabola. Mr. Mac-
Laurin, in his Organica Geometria,
deſcribes this Curve by carrying
the Interſection of one ſide of a
Square, whoſe angular Point is fa-
ftend in one Aſymptote, and a
parallel Line to the fame Afymptote
along the Curve of an Hyperbola ;
for then the Interſection of the o-
ther fide of that Square, and the
Parallel Line will deſcribe the Care
Beſian Parabola.
with
Fig. 4.
.
PAR
PAR
with two diterging Legs croſſing ſuch manner that the angular Point
one another like a Knot, which of the Square always coincides with
happens when the Equation o=ax3 the side of the Bevel firſt mentioned ;
+bx? tcxtd has its two greater then will this Point of the Square
Roots equal; the moſt fimple Equa- trace out upon the Plane a part of
tion being pyy=x3 +axx. The a diverging Parabola.
fourth (Fig. 3.) a pure Bell-form If a Solid generated by the Ro-
Parabola, being the caſe when o= tation of a ſemi-cubical' Parabola
ax: +6x2 tcxtd has two imagi- about its Axis be cut by a Plane,
nary Roots, and its moſt ſimple E- each of theſe five Parabolas will be
quation is pyj=3 ta3, or Pyg = exhibited by its Sections ; for when
*3-taax.--The fifth (Fig. 4.) a Pa- the cutting Plane is oblique to the
rabola with two diverging Legs, Axis, but falls below the Axis, the
forming at their meeting a Cuſpe Section will be a diverging Para-
or double Point, being the caſe bola, with an Oval at its head.-
when the Equation osax+6x2 + When oblique to the Axis, but
cxtd has three equal Roots ; lo paſſes thro' the Vertex, the Section
that pyy=x3 is the moſt ſimple will be a diverging Parabola, having
Equation of this Curve, which in an infinitely ſmall Oval at its head.
deed is the Semi-cubical or Nelian -When the cutting Plane is oblique
Parabola.
to the Axis, falls below it, and at
Points thro' which theſe five Pa- the ſame time touches the Curye-
rabolas muſt paſs may be very ex-
Surface of the Solid, as well as cuts
peditiouſly found by firſt taking a it, the Section will be a diverging
common Parabola, and a Right Parabola, with a Nodus or Knot ---
Line perpendicular to the Axis ſo When the cutting Plane falls above
ſituated, that a Line drawn from any the Vertex, either parallel or ob-
Point in this Line parallel to the lique to the Axis, the Section will
Axis, and terminating in the Curve be a pure diverging Parabola.-
ſhall be =axbxtc; for then a And when the cutting Plane paſſes
mean Proportional between this thro' the Axis, the Section will be
Line and any aſſumed Value of x a ſemi-cubical Parabola.
will be the Length of an Ordinate I might have been much more
of the five diverging Parabolas, full and particular about theſe
correſponding to the aſſumed Value Curves, as I am in my Treatiſe
All five of theſe Curves may of Curves that I have by me ; but
be deſcribed too by a continued it would ſwell this Book too much.
Motion, by means of a ſquare Be PARABOLIC PYRAMIDOID, is
vel and common Parabola ; for if a folid Figure, thus named by Dr.
the Angle of a Bevel (containing Wallis, from its Geneſis, or Forma-
a right Angle) be carried along the tion, which is thus :
Curve of a common Parabola, one Let all the Squares of the Ordi-
ſide thereof keeping parallel to the nates of a Parabola be imagined to
Axis of the Parabola ; and if at the be ſo placed, that the Axis ihall ſo
ſame time one side of a Square paſs thro' all their Centres at Right
paſſes through a given Point, not Angles ; and the Aggregate of theſe
within the Parabola, and the Inter- Planes will form the Parabolic Pro
ſection of the other side of the Be- ramidoid, whore Solidity is gain'd
vel and of the Square paſſes along by multiplying the Baſe by half the
a Right Line drawn from the ſaid Altitude.
Point perpendicular to the Axis, in PARABOLIC SPACT, is the Area
con-
of
PAR
PAR
contained between the Curve of the cedes farther from the Sun or Cen-
Parabola, and a whole Ordinate tre of Attraction. Thus if a Planet
AB.
in A moves to B, then is SB-SA
=bB, the Paracentric Motion of
C
D
that Planet.
B
1L
A
B
B
A А
This is of the circumſcribing
Parallelogram ACDB, in the com-
mon Parabola.
The Quadrature of the Parabola
was firſt found out by the great Ar-
rbimedes ; but his Demonſtration,
altho' very ingenious, is both long
and tedious. It is more elegantly
done by means of the Solidity of PARACENTRIC SOLLICITATI-
a ſquare Pyramid, which is of a ON of Gravity or Levity, (which is
Parallelepipedon, having the ſame all one with the Vis Centripeta,) is
Baſe and Altitude; for every Ordi- in Aſtronomy expreſſed by the Line
nate to a Tangent to the Vertex of AL drawn from the Point A, pa-
a Parabola, taken as an Abſciſs, will rallel to the Ray SB, (infinitely
be as a correſpondent ſquare Sec near SA,) until it interfects the Tan-
tion of the Pyramid, by a Plane pa- gent BL.
rallel to the Baſe; and the Sum of PARALLACTICAL ANGLE, is
all thoſe Ordinates, as the Sum of the Difference of the Angles C EA,
all thoſe Spaces, therefore, c. and BTA, under which the true and
PARABOLIC SPINDLE, is a So. apparent Diitances from the Zenith
lid made by the Rocation of a Semi-
are feen.
parabola about one of its Ordinates, PARALLAX, or PARALLAX of
and is equal to is of its circum- Altitude, is CB (or the Angle TSE,
fcribing Cylinder.
which may be taken for it) the
PARABOLIC SPIRAL. See He- Difference between the true Place B
licoid Parabola.
of the Planet S, and the apparent
PARABOLOIDES, or PARABO-
LIFORM CURVES, are Parabola's
А.
B
of the higher kind.
The Equation for all Curves of
this kind being a
"nx=ym, and
the Proportion of the Area of any
one to the Complement of it to the
circumſcribing Parallelogram will
be as m to n.
PARACENTRIC MOTION
ON of Im-
T
petus, is a Term in the New Aſtro-
nomy, for ſo much as the revolving Place C of the fame; this is equal
Planet approaches nearer to, or re 10 the Difference between AB, the
true
SD.
1
nearer
to
PAR
PAR
true Diſtance from the Zenith A, tal Parallax, is 19. 1! 25". and
and the apparent Diſtance A C.
the leaſt 54' 5":
PARALLAX of Afcenfion or De 6. The Horizontal Parallax of
ſtenſion, is an Arch of the Equinoctial, Mars, when greateſt is about 25",
whereby the Parallax of Altitude and that of the Sun is about 10%.
augments the Aſcenſion, and dimi-
PARALLEL-LINES, i: Geome-
niſhes the Deſcenſion of a Planet. try, are thoſe which iun always
PARALLAX of Declination, is an equi-diftant from each other ; lo
Arch of a Circle of Declination, that if they were infinitely pro-
whereby the Parallax of Altitude duced, they would neither go tar-
augments or diminiſhes the Declina- ther from, nor come
tion of a Planet.
each other; and their Distance is
PARALLAX of Latitude, is an always meaſured by a Perpendicu-
Arch of a Circle of Latitude, where- lar, which, wherever it be taken is
by the Parallax of Altitude aug- of the fame Length, or is always
ments or diminiſhes the Latitude.
equal to itſelf.
PARALLAX of Longitude, is an 1. Sir Iſaac Newton, in the 22d
Arch of the Ecliptic, whereby the, Lemma of the firſt Book of his Prinə
Parallax of Altitude augments or cipia, defines Parallels to be ſuch
diminiſhes the Longitude.
Lines that tend to a Point infinitely
1. The Parallax in the Zenith, diſtant.
is nothing, but in the Horizon the
2. Or Parallel Lines may be de-
greateſt.
fined thus : IF A be a Point with-
2. The Sines of the Parallactical out a given indefinite Right Line
Angles AMT, AST, at the ſame or CD; the ſhorteſt Line, as AR
equal Diſtances SZ, from the Ze-
nith are in the reciprocal Ratio of
E
А
M
Z
$
1
A
C
B
D
L
that can be drawn from A to it, is
perpendicular ; and the longeſt, as
R
T
E A, is parallel to CD.
3. A Right Line ZZ falling
the Diſtances TM, and TS, from on two parallel Lines P P and PP,
the Centre of the Earth.
makes the alternate Angles of
3. The Sines of the Parallactical
Angles of the Stars Mand S, equally
Z
diſtant from the Centre of the Earth
T, are as the Sines of the apparent
P -
P
Diſtances ZM, and Z S, from the
Zenith. The fixed Stars have no
BE
ſenſible Parallax.
-P
4. The Horizontal Parallax is the
ſame, whether a Star be in the true
Horizon, or the apparent Horizon.
Z
5. The Moon's greateſt Horizon-
and
do
F
ma
PAR
PAR
and é=b; alſo o=d, and a=g, is the Diſtance to the Co-fine of the
and the two internal Angles c+b, Latitude.
or etf = two right ones.
2. Given, the Difference of Lon-
PARALLEL-PLANEs, are thoſe gitude between two Places under
Planes which have all the Perpen- the ſame Parallel ; required their
diculars drawn betwixt them equal Diſtance.
to each other; that is, when they The Canon is, As the Radius is
are every, where equally diftant. to the Difference of Longitude : So
PARALLEL-RULER, is an In- is the Co-ſine of the Latitude to the
ftrument of Wood, Braſs, Silver, Diſtance.
&c. conſiſting of two Parallel-Rules 3. Having the Diſtance between
that open and ſhut parallel to one two Places in the ſame Latitude
;
another; and is of great uſe in all required, their Difference of Lon-
Parts of Mathematicks, where ma- gitude.
ny Parallel-Lines are to be drawn ; The Canon is, As the Co-fine of
and is particularly uſeful in redu- the Latitude is to the Diſtance : So
cing of any multangular Figure to a is the Radius to the Difference of
Triangle.
Longitude.
As ſuppoſe the Multangular Fi PARALLEL SPHERE, is where
gure A B C DE is to be reduced the Poles are in the Zenith and
into the Triangle GCB, by means Nadir, and the Equator in the Ho-
of the Parallel-Ruler ; firſt conti- rizon, which is the caſe of ſuch (if
nue out the Side A B, and laying any ſuch there be) who live direct-
one side of the Inſtrument to the ly under the North or South Poles.
Points A, D, open the other to the The Conſequences of this pofi-
Point E, and where it cuts the Line tion are, that the Parallels of the
AG, as in F, make a Mark ; this Sun's Declination will alſo be Pa- -
being done, lay one side of the rallels of his Altitude.
Ruler to the points F, C, and open The Inhabitants can ſee only
the other to the Point D, and it ſuch Stars as are on their Side of
the Equinoctial; and they muſt
D.
have fix Months Day and fix Months
continual Night every Year; and
the Sun can never be higher with
them, than 23 Degrees 30 Minutes,
which is not ſo high as he is with
us in February
G
FA .B PARALLELS of Altitude, or Al-
macanters, are Circles parallel to
will cut the Line BG in G; then the Horizon, imagined to paſs thro'
draw the Line CG, and the Tri- every Degree and Minute of the
angle GCB will be equal to the Meridian, between the Horizon and
Multangular Figure ABCDE. Zenith, having their Poles in the
PARALLEL SAILING; in Navi- Zenith. And on the Globes thefe
gation, is failing under a Parallel are deſcribed by the Diviſions on
of Latitude ; of this there are but the Quadrant of Altitude, in its
three Caſes.
Motion about the body of the
1. Given, the Departure and Di- Globe, when 'tis ſcrew'd to the
ftance ; required, the Latitude. Zenith of any Place.
The Canon is, As the Difference PARALLELS of Latitude on the
of Longitude is to the Radius : So Terreſtrial Globes, are the ſame with
Parallela
)
E
с
A
6
PAR
PAR
Parallels of Declination on the Ce to one another in the duplicate Ra-
leftial : But the Parallels of Lati- tio of their homologous Sides.
tude on the Celeſtial Globes are ſmall 4. The Area of any Parallelo-
Circles parallel to the Ecliptic, i- gram is had by multiplying one of
magined to paſs thro' every Degree its Sides by a Perpendicular let fall
and Minute of the Colures, and are from one of the oppoſite Angles.
repreſented there by the Diviſions 5. In any Parallelogram the Ag-
of the Quadrant of Altitude, in its gregate of the Squares of the Sides
Motion round the Globe, when it is equal to the Aggregate of the
is ſcrewed over the Poles of the E- Squares of the Diagonals.
cliptic:
PARALLELOGRAM,. is alſo an
PARALLELS of Declination, are Inſtrument made of five Rulers of
Circles parallel to the Equinoctial, Braſs or Wood, with Sockets to ſlide
imagined to paſs thro' every De- or ſet to any proportion, uſed to
gree and Minute of the Meridians enlarge or diminiſh any Map or
between the Equinoctial, and each Draught, either in Fortification,
Pole of the World.
Building, or Surveying & C.
PARALLEL RAYS, in Optics, PARALLELOGRAM PROTRAC-.
are thoſe that keep an equal Di-. Tor, is a Semi-Circle of Braſs with
ſtance from the viſible Obječt to the four Rulers, in form of a Paralle-
Eye, which is ſuppoſed to be infi- logram, made to move to any An-
nitely remote from the Object. gle : One of which Rulers is an In-
PARALLEL CIRCLES, on the dex, which ſhews on the Semi-Cir-
Globes; the ſame with the Leffer cle the Quantity of any inward or
Circles.
outward Angle.
PARALLELS alſo on the Terref-
PARALLELEPIPEDON, is a ſo-
trial Globe, are Circles drawn thro' lid Figure contained under fix Paral-
the middle of every Climate, di- lelograms, the Oppoſites of which
viding them into two halves, which are equal and parallel ; or 'tis a
are called Parallels.
Priſm, whoſe Baſe is a Parallelo-
PARALLEL I'S M of the Earth's gram. This is always triple to a
Axis, is the Earth's keeping its Pyramid of the fame Baſe and
Axis in its annual Revolution round Height.
the Sun, in a Poſition always pa-
PARALOGISM, is a pretended
rallel to itſelf, which it doth nearly, Demonſtration or Method of argu-
but not exactly ; for tho' the Dif- ing, but which is in reality falla-
ference be inſenſible in one Year, cious and falfe.
it becomes ſenſible enough in many · PARAMETER, by fome, as My-
Years.
dorgius, and others, called the La-
PARALLELOGRAM, in Geome tus Re&um of a Parabola, is a third
try, is a Right-lined Quadrilateral Proportional to the Abſciffa and
Figure, whole oppoſite Sides are pa- any Ordinate.
rallel and equal.
But in the Ellipfis and Hyper-
1. The oppoſite Angles of all bola, it is a third Proportional to
Parallelograms are equal to one an two conjugate Diameters.
other.
PARAPET, in Fortification, isan
2. All Parallelograms that are
Elevation of Earth and Stone upon
between the ſame Parallel-Lines, the Rampart, behind which the
and on the ſame Baſe, are equal. Soldiers ſtand ſecure from the Ene-
3. All ſimilar Parallelograms are my's great and ſmall Shot, and
where
:
1
PAT
P E D
where the Canon is planted for the Earth turns round its Axis. This
Defence of the Town or Fortreſs. Point is conſidered as vertical to the
Every Parapet having its Embra- Earth's Centre, and is the ſame
ſures and Merlons, is about fix Foot with what is called the Vertex, or
high on the fide of the Place, and the Zeniih in the Ptolemaic Pro-
from four to five in that towards jection.
the Country
So that this Diffe The Semi-Diameter of this path
rence of Height forms a kind of of the Vertex is always equal to the
Glacis above, from whence the Complement of the Latitude of the
Muſqueteers mounting the Banquet Point or Place that deſcribes it ;
of the Parapet, may eaſily fire into that is, to that Place's Diſtance from
the Moats, or at leaſt upon the
the the Pole of the World.
Counterſcarp. It ought alſo to Pause or Rest, in Muſic, is a
be from eighteen to twenty Foot Silence, or artificial Intermiſſion of
thick, if made of Earth ; and from the Voice or Sound, proportioned
fix to eight, if of Stone. The Earth to a certain Meaſure of Time, by
is much better than Stone, becauſe the Motion of the Hand or Foot.
Stone will fly to pieces when bat Theſe Pauſes or Reits are always
tered, and do miſchief,
equal to the Length or Quantity of
This word Parapet is alſo given the Notes whereto they are annexed,
to any Line that covers Men from and therefore are called by the
the Enemy's Fire: So there are Pa- fame Names, as a Long-Reſt, Breve-
rapets of Barrels, of Gabions, of Refl, Semi-Breve-Reft, &c.
Bags filled with Earth, & c.
PEDESTAL, in Architecture, is a
PARASTÆ, in Architecture, are ſquare Body, with a Baſe and Cor-
the ſame with Pilaſters ; the Italians nice, ſerving as a Foot for the Co-
call them Membretti.
lumns to ſtand upon ; it is different
PARHELII and PARHELIA are in the Orders.
fuch Phænomena, as we call Mock 1. The Tuſcan Pedeſtal, being the
Suns, being the Repreſentations of moſt ſimple of all, hath only a
the Face or Figure of the true Şun Plinth for its Baſe, and an Aſtragal
by way of Reflexion in the Clouds. crowned for its Cornice.
PARTICLES, are the very ſmall 2. The Doric Pedeſtal (according
Parts of which any natural Body is to Palladio) borrowing the Attic
ſuppoſed to be compounded ; and Baſe, ought to have for its Height
theſe are often called the conſtituent 2 of Diameters of the Column
or component Particles of any na- taken before : But no Pedeſtals to
tural Body.
this Order are ſeen among the an-
Pate, in Fortification, is a kind cient Buildings.
of Platform like what they call an 3. The Ionic Pedeſtal is two Dia-
Horfeſoe, not always regular, but meters, and about two thirds high.
generally oval, encompaſſed only 4. The Corinthian Pedeſtal hath
with a Parapet, and having nothing the fourth Part of the Column for
to flank it; and is afually erected in its Height, being divided into eight
marſhy Grounds, to.cover a Gate of Parts ; whereof one muſt be allowa
a Town.
ed for the Cimafium, two others
Path of the Vertex, is a Term for the Baſe, and the reſt for the
frequently uſed by Mr. Flamſtead, Dye or Square.
in his Doctrine of the Sphere, and 5. The Compoſite Pedeſtal ought
fignifies a Circle deſcribed by any to have the third Part of the Pillar
Point of the Earth's Surface, as the for its Height.
PEDI-
+
23 Stars.
ΡΕ Ν
Ρ Ε Ν
PEDIMENT, in Architecture, is Thus BGSC is a Pencil of Rays,
an Ornament that crowns the Or- and the Line BLC, is called the
donnance, finiſhes the Fronts of Axis of that Pencil.
Buildings, and ſerves as a Decora PENDULUM, is a Weight hang-
tion over Gates, Windows, Niches, ing at the End of a String, Chain,
&c. it is ordinarily of a triangular or Wire, by whole Vibrations or
Form, but ſometimes makes an Swings to and fro, the Parts or
Arch of a Circle.
Differences of Time are meaſured.
Peers, in Architecture, are a 1. The Velocities of Pendulums
kind of Pilaſters or Buttreſſes for in their loweſt Points, are as the
Support, Strength, and ſometimes Chords of the Arches they fall from
Ornament.
or deſcribe.
PEGASUS, a Conſtellation in the 2. The Lengths of Pendulums
Northern Hemiſphere ; containing (which are always accounted from
the Centre of Oſcillation, to the
Pelicoides, is the Name given Centre of the Ball or Bob) are to
by ſome to the Figure BCDA, each other in a duplicate Ratio of
contained under the two inverted the Times in which their Vibra-
Quadrantal Arches AB and AD, tions are reſpectively performed ;
and the Semi-Circle BCD, whoſe or are as the Squares of the Vibras
tions performed in one and the ſame
C
time; wherefore, the Times muſt
be in a ſubduplicate Ratio of the
Lengths. Sir Iſaac Newton demon-
ftrates, Cor. 2. Prop. 54. Princip.
that if the Force of the Movement
B
D of a Clock required to keep a
Pendulum fo adjuſted, that the.
whole Force or Tendency down-
wards ſhall be as the Line which
ariſes by dividing the Rectangle
F
under the Semi-Arch of the Vibra-
A
E
tion and the Radius, is to the Sine
of that Semi-Arch, then all the Of
Area to the Square AC, and that cillations ſhall ſtill be made in the
to the Rectangle EB.
ſame Space of Time.
PENCIL of Rays, in Optics, is a
3. 'Tis faid, that Ricciolus was
double Cone of Rays joined toge- the firſt that attempted to meaſure
ther at the Baſe ; one of which Time by the Pendulum, and there-
hath its Vertex in ſome other Point in he was followed, tho' nearly
of the Object, and the Glafs GLS about the ſame time, by Langrenus
for its Baſe ; and the other hath Vendelinus, Merſennus, Kircherus,&c.
its Baſe on the fame Glaſs, but its Some of which declare they knew
Vertex in the Point of Convergence, nothing of Ricciolus's Attempt ;
as at C.
but the firſt that applied it to a
G
Movement, Clock, or Watch, was
Mr. Chriſtian Huygens, and who
B
L
brought it alſo to a good Degree of
Perfection. See his Horologium of
cillatorium.
PENDULUMS-ROYAL, are thoſe
3
Clocks,
S
PER.
P E R
Clocks, whoſe Pendulum ſwings Se- tute-Law, of fixteen Foot and a half
conds, and goes eight Days, a in length.
Month, &*c. Thewing the Hour, Perfect CONCORDS, in Muſic;
Minutes, and Seconds.
ſee Concords,
PENINSULA, in Geography, is a PERFECT Fifth, the fame with
Portion of Land, which is almoſt Diapente : which fee.
ſurrounded with Water, and is join PERFECT NUMBERS, are ſuch
ed to the Continent only by an Títh- whoſe aliquot, or even parts joined
mus, or narrow Neck of Land ; as together, will exactly make that
Africa, the greateſt Peninſula in the whole Number, as 6 and 28, &c.
World, is joined to Afia, and that of For of fix, the half is three, the third
the Marea to Greece, &c
part two, and the fixth part one,
PentAGON, in Geometry, is a which added together, make fix;
Figure having five Sides, and five and it hath no more aliquot parts in
Angles : If all the Sides be equal, and whole Numbers ; fo twenty-eight,
alſo the Angles, it is called a Regu- which has theſe parts, viz. 14,794,2,
lar Pentagon.
and 1, exactly make 28 ; which
The ſide of a Regular Pentagon, or therefore is a Perfect Number,
one which can be inſcribed in a cir- whereof there are but Ten between
cle, is in power equal to the fide of One, and one Million of Millions.
an Hexagon and Decagon, inſcribed To find a Perfeff Number, that is,
in the ſame Circle.
a Number which ſhall be equal to all
PENTANGLE, a Figure having its aliquot parts taken together. Let
five Angles.
the Number fought be = 34*, be-
PenƯMBRA, in Aftronomy, is a ing ſuch as that it can be reſolved
faint kind of a Shadow, or the ut into its aliquot Parts or Factors :
moft Edge of the perfect Shadow, Now the aliquot Parts thereof will be
which happens at the Eclipſe of the ity+ya+y3,&c.until the Exponent
Moon; ſo that it is very difficult to becomes equal to n, and xtyx +32*
determine where the Shadow begins, t-y3x, &c. likewiſe until the Expo-
and where the Light ends.
nent be =n. Now from the Nature
PER AMBULATOR, the ſame as of a perfect Number 1 ty+ya+y3,
the Surveying-Wheel, is an Inſtru- &c. fox-tyxty2x+y3 x, &c. may
ment made of Wood or Iron, com- be =jMx; whence ity+ya+y3,
monly half a pole in Circumference, &c. =y"x-3-7x72x-3x, &c.
with a Movement, and a Face divi-
ded like a Clock, with a long Rod and
i tytys ty3, &C.
of Iron or Steel, that goes from the jh-1-3
---33,&c.
Centre of the Wheel to the Work: Now that y may be an Integer (the
There are alſo two Hands, which (as Number of aliquot parts in any par-
you drive the Wheel before you) ticular Caſe, if jy be expounded by
count the Revolutions ; and from a Number, will not be differens
the Compoſition of the Movement, from their Number in the general
and by the Diviſions on the Face, Form) it is neceſſary that yo-y-
fhew how many Yards, Poles, Fur- y2---y}, &c. be = 1; which cannot
longs, and Miles, you go. The Uſe happen in any other Caſe, but when
of this Inſtrument is to meaſure
y=2, then will be =i+242+
Roads, Rivers, and all level Lands, 23, E C. =1+2+4+8, &C.
with great expedition.
and the perfect Number 21 x :
Perch, a Meaſure, by our Sta- Therefore the Problem, though
propoſed
X.
PER
PER ..
propoſed as a kind of indeterminate a Place or Building encompaſſed with
one, is determinate : If n=1, then Pillars ſtanding round about within
will x=1+2=3, and conſequently the Couri : But this word Periſtyle
the perfect Number 2^x=6.
If
is ſometimes taken for a Row or
n=2, then will x=1+2+4=7;
Rank of Columns, as well without
as within any Edifice, as in Cloy-
whence 2" x=28: If n==3, then
ſters and Galleries. Sometimes this
will x=1+2+4+8=15; there-
was called Antiproſtyle.
fore 2"x=120.
Peritere, in Architecture, is a
PERIGÆON, or PerIGÆUM, is a place encompaſſed round with Co-
Point in the Heavens, wherein a lumns, and with a kind of Wings
Planet is ſaid to be in its neareſt Di- about it. Here the Pillars Stand
ſtance poſſible from the Earth. without, whereas in the Periſtyle
Perihelion, is that point of a they ſtand within.
planet's Orbit, wherein it' is neareſt PERITROCHIUM.
See Axis in
to the Sun.
Per itrochio.
PERIMETER, is the Bounds. of
PERIÆCI, are thoſe Inhabitants
any Figure.
of the Earth, who live under the
Period, in Chronology, figni- fame' parallels, but under oppoſite
fies a Revolution of a certain Num- Semi - Circles of the Meridian :
ber of Years ; as the Metonic Peri- Whence they have the ſame Seaſons
od, the Julian Period, and the Ca- of the Year, viz. Spring, Summer,
lippic Period: which ſee in their Autumn, and Winter, at the very
proper places.
fame Time ; as alſo the ſame length
PeriODICAL, is the Term for of Days and Nights ; for 'tis in the
whatſoever performs its Motion, fame Climate, and at an equal Die
Courſe, or Revolution regularly, fo ftance from the Æquator : But the
as to return again, and to diſpatch Changes of Noon and Midnight are
it always in the ſame period, or alternate one to the other.
ſpace of Time. Thus the periodi PERMUTATION of Quantities.
cal Motion of the Moon, is that See Variation and Combination.
whereby the finiſhes her Courſe PeRPENDICULAR, in Geome-
round about the Earth in a Month; try, is when a Right Line ftandeth
and this is in 27 Days, 7 Hours, 45 fo upon another, that the Angles on
Minutes, and is called the Moon's either ſide are equal ; then this
Periodical Month ; which is the ſpace Right Line, which fo ftandeth, is
of Time that the Moon finiſhes her perpendicular to that upon which it
Revolution in
ftandeth. A Right Line is ſaid to be
PERIPHERY, in Geometry, is the PERPENDICULAR to a Plane,
Circumference of a Circle, or of when 'tis perpendicular to more
any other Regular Curvilineal Fi- than two Lines drawn in that plane.
gure.
One Plane is perpendicular to ano.
PERISCII, are the Inhabitants of ther, when a Line in one plane is
the two frozen Zones, or thoſe that perpendicular to the other Plane,
live within the Compaſs of the Ar PERPETUAL Motion. By this
Elic and Antarktic. Circles ; for as Term ought to be meant an uniň-
the Sun never goes down to them terrupted Communication of the
after he is once up, but always round fame Degree of Motion from one
about, ſo do their Shadows. Whence part of Matter to another, in a Circle
the Name.
(or ſuch-like Curve 'returning into
PERISTYLE, in Architecture, is itſelf ) ſo that the fame Quantity of
Dd
Mattes
1
PER
PER
Matter ſhall return perpetually un the Deſcent of that preponderating
diminiſhed upon the firſt Mover : part, will be loſt in its Aſcent; and
And perhaps, if Men had rightly then the Wheel thus loaded, as ſoon
underſtood that this is the true as the Friction hath deſtroyed the
Meaning of a perpetual Motion, A- Motion given it, will for a while
bundance of Expence both of Mo- vibrate like other pendulous Bodies,
ney and Reputation might have been and then at laſt ſtand ſtill. Conſe-
ſaved, by the vain Pretenders to this quently no perpetual Motion by
piece of impoſſible Mechaniſm.
Wheel-Work.
1. When a Wheel, or other Ma PERSEUS, a Conſtellation in the
chine, once ſet in motion, will, Northern Hemiſphere, conſiſting of
without additional Actions on it, 38 Stars.
continue to move with the fame, or Persic Order of Architetture,
a greater Velocity, with which it is where the Bodies of Men ſerve
firſt moved, as long as the Matter inſtead of Columns to ſupport the
of which it conſiſts, remains the Entablature; or rather the Columns
ſame ; ſuch a Motion, by Mecha are in that Form.
nics, is called Perpetual.
The Riſe of it was this : Pauſa-
2. But ſince Bodies have not in nias having defeated the Perſians,
themſelves power to move thena- the Lacedemonians, as a Märk of
ſelves, and therefore have not pow• their Victory, erected Trophies of
er to increaſe or diminiſh a Motion the Arms of their Enemies, and then
given them; if they are not acted repreſented the Perſians under the
on by other Bodies, they will conti- Figures of Slaves, ſupporting their
nue fo to move, and with the ſame Porches, Arches, or Houſes.
Velocity : But all revolving Bodies PERSPECTIVE, is an Art that
ſuffer Friction with thoſe, by which teaches us the Manner of delineating
they are ſuſpended ; and the Velo- by mathematical Rules ; that is, it
cities of thoſe Bodies are therefore ſhews us how to draw geometrically
continually leſſen'd by the Action of upon a plane, the Repreſentations of
Friction. Therefore, a Wheel, or Objects according to their Dimen-
other Machine, ſet in motion with- fions, and different Situations ; in
out additional A étions on it, will not ſuch manner, that the ſaid Repre-
continue to move with the ſame Ve- ſentations produce the ſame Effects
locity, tho' the Matter of which it upon our Eyes, as the Objects
conſiſts remains the fame : But, on whereof they are the pictures.
the contrary, this Velocity will be Some of the Writers upon Per-
continually diminiſhed.
spective are Defargues, de Bolle, An-
3. Moreover, fince, by numberleſs drea Albertus, Lamy, Franciſcus Ni-
Experiments, the moſt polite or ceron, Pozzo, Ditton, Prick, Grave-
burniſh'd Bodies ſliding over one an- fande, Hamilton, &c.
other, loſe all the Motion which PERSPECTIVE AERIAL, is a
hath been given them, and in a proportional Diminution of the Li-
Thort Time : Therefore every Wheel, neaments and Colours of a Picture,
or any other ſuch Machine will, in when the Objects are ſuppoſed to be
a ſhort Time, loſe its Motion.
very remote.
4. Hence it appears,
that the
PERSPECTIVE LINEAL, is the
perpetual Motion is not to be ex Diminution of choſe Lines in the
pected by a ſingle Wheel.
plane of a Picture, which are the
5. And if any Contrivance cauſes Repreſentations of other Lines very
one part of a Wheel to preponderate remote.
another; whatſoever is gained by
PERSPEC
PHY
PHY
PeRSPECTINE MILITARY, is ledge of all Natural Bodies, and of
when the Eye is ſuppoſed to be in- their proper Natures, Conftitutions,
finitely remote from the Table or Powers, and Operations.
Plane.
PHYSIOLOGY, PHYSICS, or NA-
Pertica, a ſort of a Comet; the TURAL PHILOSOPHY, is the Sci-
ſame with Veru.
ence of natural Bodies, and their
PeTARD, in Fortification, is an various Affections, Motions, and O-
Engine of Metal. in the form of an perations. This is either,
high - crown'd Hat, with narrow 1. General, which relates to the
Brims, which being fill'd with very Properties and Affections of Matter
fine Powder, well primed, and then or Body in general. Or,
fixed with a Madrier or Plank, bound 2. Special and Particular, which
falt down, with Ropes running thro' confiders Matter as formed or diſtin-
Handles, which are round the Rim guiſhed into ſuch and ſuch Species,
of the Mouth of it, ſerves to break or determinate Combinations.
down Gates, Port.cullices, Draw 3. Dr. Keil, in his Introductio ad
bridges, Barriers, &c. This En- Phyficam, reckons four Claſſes or
gine is from 7 to 8 Inches deep, Sorts of Philoſophers, which have
and 5 broad at the Mouth; the treated of Phyſics or Natural Philo-,
Diameter at the Bottom or Breech fophy.
is an Inch and a half, and the Weight 4. Thoſe who delivered the Pro-
of the whole Maſs of Metal is from perties of natural Bodies under Geo-
55 to 60 Pounds, generally requi- metrical and Numeral Symbols,
ring about 5 Pounds of Powder for as the Pythagoreans and Platoniſts.
the Charge. They are alſo uſed in 5. The Peripatetics, who explain-
Countermines, to break through into ed the Natures of Things by Mat-
the Enemy's Galleries, and to diſap- ter, Form, and Privation ; by ele-
point their Mines.
mentary and occult Qualities ; by
PHÆNOMENON, in Natural Phi- Sympathies, Antipathies, Faculties,
loſophy, fignifies any Appearance, and Attraction, &c. and theſe did
Effect, or Operation of a Natural not fo much endeavour to find out
Body, which offers itſelf to the Con- the true Reaſons and Cauſes of
fideration and Solution of an Enqui- Things, as to give them proper
rer into Nature.
Names and Terms ; ſo that their
PHASES, fignifies the Appearance, Phyſics is a kind of Metaphyſics.
or the Manner of Things fhewing 6. The Experimental Philosophers,
themſelves ; and therefore in Aftro- who by frequent and well-made
nomy is uſed for the ſeveral Pofi- Trials and Experiments, as by Chy-
tions, in which the Planets (eſpeci- miſtry, &*c. fought into the Natures
ally the Moon) appear to our Sight; and Cauſes of Things: And to theſe
as obſcure, horned, half-illuminated, almoſt all our Diſcoveries and Im-
or full of Light, which, by the Help provements are due; and much more
of a Teleſcope, may likewiſe be ob. would they have done, if they had
ſerved in Venus and Mars.
not fallen into Theories and Hypothe-
PHROCYON, a fixed Star of the les, which they forced oftentimes
ſecond Magnitude, in the Conſtel. their Experiments to maintain, whea.
lation Canis Minor, whoſe Longi- ther they could or not.
tude is un Degrees, 23 Minutes,
7. The Mechanical Philoſophers,
Latitude 15 Degrees, 57 Minutes. who explicate all the Phænomena of
Physics, or NATURAL PHI- Nature by Matter and Motion, by
LOSOPlly, is the ſpeculative Know the Texture of Bodies, and the Fi-
Dd 2
gure
PIL
Ρ Ε Α
gure of their Parts; by Efluvia, from the Wall, but more uſually
and ather ſubtile Particles, &c. And contiguous to it, or let within it, ſa
in ſhort, would account for all EF as it does not ſhew above one fourth
fects and Phænomena by the known or fifth Part of its 1:hickneſs. The
and eſtabliſhed Laws of Motion Pilafter is different in ſeveral Or-
and Mechanics : And theſe are, in' ders, and borrows occaſionally the
conjunction with the laſt named, Name of each ; having the ſame
the only true Philoſophers.
Ornaments and the ſame Propor-
Picket, in Fortification, is fome- tions with the Columns.
times uſed for a Stake, ſharp at one PILLAR, in Architecture, is a
end, to mark out the Ground and kind of a round Column diſengaged
Angles of a Fortification, when the from any Wall, and made without
Engineer is laying down the Plane' any proportion ; being always either
of it; theſe are uſually pointed too maſſive or too Nender : Such
with Iron : There are alſo larger are the Pillars which ſupport the
Pickets, which are drove into the Vaults of Gothic Buildings.
Earth, to hold together Faſcines or PINION, in a Watch, is that
Faggots, in any Work caft up in lefſer Wheel which plays in the
hatte. And Pickets alſo are Stakes Teeth of another. Its Notches,
drove into the Ground by the Tents (which are commonly 4, 5, 6, 8,
of the Horſe in a Camp, to tie their &c.) are called Leaves, and not
Horſes to. And Pickets were alſo Teeth, as in other Wheels.
drove into the Ground before the The Quotient or Number of
Tents of the Foot, where they refted Turns to be laid upon the Pinion of
their Mukets or Pikes round about Report, is found by this Proportion:
them in a Ring. When a Horſe-
When a Horfe- As the Beats in one Turn of the
man hath committed ſome confi- great Wheel, to the Beats in an
derable Offence, he is often ſen- Hour : So are the Hours of the
tenced to ſtand on the Picket; which Face of the Clock, (viz. 12, or 24.).
is to have one Hand drawn up as to the Quotient of the Hour-Wheel,
high as it can be ftretch'd, and then or Dial-Wheel, divided by the Pi-
he is to itand on the Point of a Pic- nion of Report, i.e. the Number
ket or Stake only with the Toe of of Turns which the Pinion of Re-
his oppoſite Foot; ſo that he can port hath in one Turn of the Dial-
neither ſtand or hang well, nor eaſe Wheel.
himſelf by changing Feet.
P'IN-WHEEL. See Striking-
PieDOUCHE, in Architecture, is Wheel.
a little ſquare Baſe ſmoothed, and Pisces, is the twelfth and laſt
wrought with Mouldings, which Sign of the Zodiac, being a Con-
ferves to ſupport a Buſt or Statue ftellation confiſting of 35 Stars.
drawn half-way, or any ſmall Fi Piscis MERIDIANUS, a Sou-
thern Conſtellation containing twelve
PIED-DROIT, in Architecture, Stars.
is a ſquare Pillar differing from a Place, is that Part of Space
Pilaſter in this reſpect, that it hath which any Body takes up ; and
no Baſe or Capital: It is taken alſo with relation to Space is either ab-
for a part of the Jaumbs of a ſolute or relative; as Mr. Locke ob-
Door or Window.
ſerves.
PILASTER, in Architecture, is
2. PLACE, alſo is ſometimes ta.
a kind of a ſquare Column, fome- ken for that Portion of infinite Space,
times ſtanding free, and detach'd which is poſſeſſed by, and compre-
hended
gure in Relief.
-
PL A
PL A
hended within the material World, PLACE PLANE. See Locus
and which is thereby diſtinguiſhed Plane.
from the reſt of the Expanſion. PLACE SIMPLE. See Locus
3. PLACE is uſually diſtinguiſhed Simple.
into internal Place, which, pro PLACE SOLID. See Locus
perly ſpeaking, is that Part of Space Solid.
which any Body takes up and fills;
Placè SUR SOLID. See Locus
and External Place, which, accord- Surfolid.
ing to Ariſtotle, is determined by PLACE of the Sun, Star, or Planet,
the Surfaces or Confines of the ad. is the sign of the Zodiac, Degree,
joining or ambient Bodies : But it Minute, and Second of it, which the
is better divided into abſolute, which Planet is in; or it is that Degree of
is the former internal Place; and the Ecliptic reckoned from the Be-
into relative Place, which is the ap- ginning of Aries, which the Planet's
parent ſecondary or ſenſible Poſi- or Star's Circle of Longitude cut-
tion of any Body, according to the teth; and therefore is often called
Determination of our Senſes, with the Longitude of the Sun, Planet, or
reſpect to other contiguous or ad- Star.
joining Bodies.
PLAIN ANGLE. See Angle.
4. Place of Arms, when taken in 1. Sides of a plain Angle, are the
the general, is a ſtrong City which Lines forming it.
is pitched upon for the Magazine of 2. Vertex of any Angle, are the
any Army. But a Place in Fortifi- Points wherein the Lines forming it
cation uſually ſignifies the Body of meet.
a Fortreſs. And a
3. Meaſure of a plain right-lined
5. Place of Arms in a Gariſon, is Angle, is an Arch of a Circle dé-
a large open Spot of Ground in the ſcribed about the Vertex, contained
middle of the City, where the great between the sides of the Angle.
Streets meet, or elſe between the 4. Equal right-lined Angles, are
Ramparts and the Houſes, for the Ga- ſuch whereof the Area's of Circles
riſon to rendezvous in, upon any ſud- deſcribed from their Vertexes, and
den alarm, or other occaſion. And the intercepted between their Sides, are
6. Place of Arms of a Trench, or proportional to their Radii, or,
of an Attack, is a Poſt near it, ſhel- which is the ſame thing, do contain
tered by a Parapet .or Epaulement, the ſame Number of Degrees.
for Horſe ,or Foot to be at their 5. Eucl. in Prop. q. lib. 1. has
Arms, to make good the Trenches taught us how to biſect or divide :
againſt the Sallies of the Enemy. any given right-lined Angle into
Theſe Places of Arms are ſometimes two equal Parts, and from thence
covered by a Rideau or Riſing- it will be eaſy to divide it into 4, 8,
Ground, or elſe by a Cavin or deep 16, 32, 64, &c. equal Parts.
Valley, which ſaves the trouble of 6. But the Ancients, as we learn
fortifying them by means of Para- from Papfus, in his Mathematical
pets, Faſcines, Gabions, &c. they Collections, could not triſect or di-
are always open in the Rear, for vide an Angle given into three equal
their better Communication with the Parts, by a ſtraight Line and a Cir-
Camp. When the Trenches are cle; and when they found it could
carried on as far as to the Glacis, not be done this way, they began
chey make it very wide, that it may to conſider the Properties of other
ferve for a Place of Arms.
Curves, and found the thing could
PLACE GEOMETRIC. See Locus. be done by the Conchoid, Cifioid,
Dd 3
Or
PL A
PLA
1
1
M
or Conic Sections. But Archimedes, tial; yet by laying down Places ac-
Pappus, and Sir Iſaac Newton, ap- cordingly, and breaking a long Voy-
prove of the Conchoid for effecting age into many ſhort ones, a Voy-
this Buſineſs. And,
age may pretty well be performed
7. Sir Iſaac Newton, in Prob. 14. by it near the ſame Meridian.
Arit, univer. fhews how to divide In plain Sailing 'tis imagined, that
an Angle into any given Number by the Rhumb-Line, Meridian, and
of equal Parts, but here the follow- Parallel of Latitude, there always
ing Equations muſt be firſt ſolved. will be formed a right-angled Tri-
For if the given Angle be DAD, angle; and that fo pofited, as that
and BAD be the fought Angle the Perpendicular may repreſent
that is to be any given Part thereof, part of the Meridian or North and
South Line, containing the Diffe-
D
rence of Latitude : The Baſe of the
Triangle repreſents the Departure,
and the Hypotheneuſe the Diſtance
failed; the Angle at the top is the
B
Courſe, and the Angle at the Baſe
the Complement of the Courſe ;
D
any two of which, with the Right
A
К.
Angle being given, the Triangle
may be protracted, and the other
and the Radius AD be called r, the three parts found.
Sine DM of the given Angle ? PLAIN SCALE, is a thin Ruler,
and the Sine Complement AK of either of Wood or Braſs, whereon
the ſought Angle x: Then the Bi- are graduated the Lines of Chords,
ſection of the given Angle will be Sines, Tangents, Secants, Leagues,
had by the Reſolution of this Equa- Rhumbs, &c. and.is of ready Uſe
tion, xx—2rr=9r; the Trifection in moſt parts of the Mathematics,
by the Reſolution of this xxx chiefly in Navigation.
3rrx=9r? ; the Quadriſection, by PLAIN TABLE, is an Inſtrument
the Reſolution of this x4marrxxt uſed in ſurveying of Land.
2r4=gr3; the Quinquiſection, by 1. The Table itſelf is a Paralle-
the Reſolution of this xs — 572x3 logram of Wood, 14 Inches and a
+5r4x=qr45, &C.
half long, and 11 Inches broad, or
PLAIN CHART, is the Plot or thereabouts.
Chart, that Seamen fail by, whoſe 2. A Frame of Wood fixed to it,
Degrees of Longitude and Latitude ſo as a Sheet of Paper being laid on
are made of the ſame Length. the Table, and the Frame being
PLAIN SAILING, is the Art of forced down upon it, ſqueezeth in
finding all the Varieties of the Ship’s all the Edges, and makes it lie firm
Motion on a Plane, where all the and even, fo as a Plot may be con-
Meridians are made parallel, and veniently drawn upon it
. Upon
the Parallels at Right Angles with one ſide of this Frame ſhould be
the Meridians, and the Degrees of equal.Diviſions for drawing parallel
each Parallel equal to thoſe of the Lines both long ways and croſs-
Equinoctial ; which tho' notoriouſly ways (as occaſion may require)
falſe in itſelf, ſuppoſing the Earth over your Paper ; and on the o-
and Sea to be a plane Flatneſs, and ther ſide the 360 Degrees of a Cir-
each Parallel equal to the Equinoc- cle, projected from a Braſs Cen-
tre
:
PLA
PLA
tre conveniently placed in the PLANE of Gravitation, or Gra-
Table.
vity in any heavy Body, is a Plane
3 A Box with a Needle and ſuppoſed to paſs thro' the Centre of
Card, to be fixed with two Screws Gravity of it.
to the Table; very uſeful for plac PLANE, in Fortification, is the
ing the Inſtrument in the ſame po- Repreſentation of a Work in its
ſition upon every Remove.
Height and Breadth.
4. A three-legged Staff to fup PLANE of the Horofter, in Optics,
port, it, the Head being made ſo as is that which paſſeth thro’ the Ho-
to fill the Socket of the Table, yet ropter, and is perpendicular to the
ſo as the Table may be eaſily turn'd Plane of the two optical Axes.
round upon it, when 'tis fixed by the PLANE NUMBER, is that which
Screw,
may be produced by the Multipli-
5. An Index, which is a large cation of two Numbers one by an-
Ruler of Wood, (or Braſs) .at the other ; thus 6 is a plane Number,
leaſt 16 Inches long, and 2 Inches becauſe it may be produc'd by the
broad, and ſo thick as to make it Multiplication of 3 by 2; for twice
ſtrong and firm; having a floped 3. makes 6. So alſo 15 is a plane
Edge, call’d the Fiducial Edge, and Number, ariſing from s being mul-
two Sights of one Height, (whereof tiply'd by 3: And 9 is a plane
the one hath a Slit above, and a Number, produc'd by the Multi-
Thread below, and the other a Slit plication of 3 by 3.
below and a Thread above) ſo ſet PLANE PROBLEM, in Mathema-
in the Ruler, as to be perfectly of tics, is ſuch an one as can be folved
the ſame Diſtance from the Fiducial geometrically by the Interſection
Edge. Upon this Index 'tis uſual either of a Right Line and a Circle,
to have many Scales of equal Parts, or of the Circumferences of two
as alſo Diagonals, and Lines of Circles : As having the greater Side
Cords.
given, and the Sum of the other
PLANCERE, in Architecture, is two, of a right-angled Triangle ;
the under part of the Roof of a Co- to find the Triangle : To die
rona; which is the ſuperior part of ſcribe a Trapezium. that ſhall
the Cornice, between two Cima- make a given Area of four given
fiuins. See thoſe Words.
Lines.
Plane of a Dial, is the Surface PLANE of Refle Etion, in Catoptrics,
on which any Dial is ſuppoſed to is that which paſſes thro' the Point
be deſcribed.
of Reflection, and is always perpen-
PLANEGEOMETRICAL, in Pere dicular to the Plane of the Glaſs,
spective, is a plane Surface, parallel or reflecting body.
to the Horizon, placed lower than PLANE of Refraction, is a Sur-
the Eye; wherein the viſible Ob- face drawn thro' che incident and re-
jects are imagined without any Al- fracted Ray.
teration, except that they are ſome PLANE SURFACE, is that which
times reduced from a greater to a lies evenly between its bounding
leffer ſize.
Lines; and as a Right Line is the
PLANE HORIZONTAL, in per- Torteſt Extenſion from one Point
ſpective is a Plane which is paral- to another, ſo a plain Surface is the
lel to the Horizon, and which paffes ſhorteſt Extenſion from one Line to
thro' the Eye, or hath the Eye ſup- another.
poſed to be placed in it.
PLANE VERTICAL, in Optics
and
D A
PL A
PO I
and Perſpective, is a plane Surface PLATFORM, in Architecture, is
which paffeth along the principal a Row of Beams that ſupport the
Ray, and conſequently thro' the Timber-Work of a Roof, and lie
Eye, and is perpendicular to the on the top of the Wall, where the
geometrical Plane.
Entablature ought to be raiſed.
PLANETS, are the erratic, or Alſo a kind of Terraſs-Walk, or
wandering Stars, and which are not even Floor on the top of a Build-
like the fixed ones always in the ing; from whence we may take a
ſame poſition to one another. We
We fair proſpect of the adjacent Gar-
now number the Earth among the dens or Fields: So an Edifice is
primary Planets, becauſe we know ſaid to be covered with a Platform,
it moves round the Sun, as Saturn, when it hath no arched Roof.
Jupiter, Mars, Venus, and Mercury PLATONIC BODIBs. See Regu-
do; and that in a Path or Circle lar Bodies.
between Mars and Venus. And the PLEIADES, the fame with thoſe
Moon is accounted among the ſecon- ſeven-Stars in the Neck of the Bull,
dary Planets, or Satellites of the which are uſually thus called.
primary, fince ſhe moves round the PLINTH, in Architecture, is a
Earth, as Yupiter's four Moons or ſquare Piece, or Table, under the
Satellites do round him, and sa- Mouldings of the Baſes of Columns
turn's five round him; if Caſſini's and Pedeſtals.
Eyes may be credited. But I could Plow, is an Inſtrument made of
never ſee my ſelf, or meet with any Pear-tree, uſed by Seamen to take
body elſe, who ever did fee any but the Height of the Sun or Stars, in
the Huygenian Satellites.
order to find the Latitude: It ad-
PLANIMETRY, the ſame with mits of the Degrees to be very large,
Planometria. Which fee.
and is much etteem'd by many Ar-
PLANISPHERE, ſignifies the Cir- tiſts.
cles of the Sphere deſcrib'd in plano, PLUMB-LINE, the ſame with
or on a Plane; or it is a plane or Perpendicular.
flat Projection of the Sphere. And PNEUMATICs, is the Doctrine of
thus the Maps either of Heaven or the Gravitation and Prellure of ela-
Earth are called Planiſpheres ; as ftic or compreſſible Fluids.
alſo other aſtrolabical Inſtruments. PNEUMATIC ENGINE, the ſame
And all Charts or Maps for the Uſe with the Air-Pump.
of Mariners, are call'd the Nautical POETICAL, Riſing and Setting
Planiſpheres. See Nautical Plani- of the Stars : This is peculiar to
sphere.
the ancient poetical Writers ; for
PLAT-BASTION. See Baſtion. they refer the Riſing and Setting of
PLAT-BAND, in Architecture, is the Stars, always to that of the Sun;
a ſquare Moulding, having leſs Pro- and accordingly make three ſorts
jecture than Height : Such are the of poetical Rilings and Settings ;
Faces of an Architrave, and the Coſmical, Acronical, (or as ſome
Plat-Band of the Modillions of a write it, AcronyEtal,) and Heliacal.
Cornice,
See thoſe Words.
PLATFORM, in Fortification, is POINT, in Geometry, is that
a Place prepared on the Ramparts which is ſuppoſed to have neither
for the raiſing of a Battery of Can- Breadth, Length, or Thickneſs, but
non ; or it is the whole Piece of is indiviſible,
Fortification raiſed in a re-entring 1. The Ends or Extremities of
Angle. See Battery.
Lines are Points,
1
2.
1
Ρ ΟΙ
POL
2. If a Point be ſuppoſed to be Glaſs, which a Ray parts from,
moved any way, it will by its Mo- after its Refraction, and when 'tis
tion deſcribe a Line.
returning into the Rare Medium a-
POINT-BLANK, a Term in Gun- gain,
nery, ſignifying that a Shot or Point of Inflexion of a Curve,
Bullet goes directly forward to the See Inflexion.
Mark, and doth not move in a POLAR DIALS, are thoſe whoſe
Curve as Bombs and highly elevated Planes are parallel to ſome great
Random-Shots do.
Circle that paſſes thro' the Poles,
Point of the Compafs, in Navi or parallel to ſome one of the Hour-
gation, ſignifies 11 Degrees and 15 Circles ; ſo that the Pole is neither
Minutes, or one 320 Part of the elevated above, nor depreſſed below
Compaſs: The half of which is 5 the Plane: Therefore the Dial can
Degrees and 38 Minutes, which have no Centre, and conſequently
they call a Half-Point; and the its Stile, Subſtile, and Hour-Lines,
half of this, which is 2 Degrees are parallel. This therefore will
and 49 Minutes, they call a Quarter- be an Horizontal Dial to thoſe that
Point.
live under the Equator pr Line.
The Seamen alſo call the Extre 1. In a direct polar. Dial, the
mity of any Promontory, (which is Hour-Lines muſt be drawn all pa-
a Piece of Land running out into rallel to the Hour-Line of Twelve.
the Sea) a Point ; which is of much 2. The Style may be either a
the ſame ſenſe with them as the ſtraight Pin ſet upright, or a Wire
word Cape.
made to lie parallel to the Plane;
They ſay two Points of Land are and muſt ſtand over the Hour-Line
one in another, when the innermoſt of Twelve.
is hinder’d from being ſeen by the 3. The Length of the Plane may
outermoft.
be taken in any Inches, or Parts of
Point of Concourſe in Optics, is Inches, reckoning the Inch to be
that Point where the viſual Rays, divided into io, or 100 equal Parts
being reciprocally inclined, and fuf- of the Style.
ficiently prolonged, meet together, 4. Then for the Height.
are united in the middle, and croſs As the Tangent of the Hour-Line
the Axis. This point is moſt uſually 4 or 5, turned into Degrees, is to
called the Focus; and ſometimes the the Logarithm of their Diſtance
Point of Convergence.
from the Meridian in Inches, and
Point of Concurrence, a Term Parts :
Parts : So is the Radius to the
in Perſpective. See Principal Point. Height of the Stile in Inches and
Point of Divergence. See Vir- Parts.
tual Focus.
5. For the Hour-Lines.
Point of Diſtance, is a Point, in As the Radius is to the Logarithm
the Horizontal Line, ſo far diſtant of the Stile's Height, in Parts of
from the principal Point as the Eye Inches : So is the Tangent of any
is remote from the ſame.
Hour-Line, to the Logarithm of
Point of Sight. See Principal the Diſtance thereof from the Meri.
Point.
dian-Line.
Point of Incidence, in Optics, is POLAR PROJECTION, is a Re-
that Point on the Surface of a Glaſs, preſentation of the Earth, or of the
or other Body, on which any Ray Heavens projected on the Plane of
of Light falls: And as ſome expreſs one of the Polar Circles.
themſelves, it is that Point of the POLARITY, is the Property of
the
POL
POL
the Magnet, or of a piece of ob- Hours, he is above the Horizon
Jong Iron touched by a Magnet, to there; but is not ſo much elevated
point towards the Poles of the as under the Pole.
World.
Poles of the Ecliptic, are Points
Pole, in Meaſuring, is the ſame in the folftitial Colure 25 Degrees
with Perch or. Rod.
and 30 Minutes diſtant from the
POLE, in Mathematics, is a Point Poles of the World; and thro'
go Degrees diſtant from the Plane theſe all Circles of Longitude in
of any Circle, and in a Line per- the Heavens do paſs, as the Hour-
pendicularly erected in its Centre ; Circles do thro' the Poles of the Æ-
which Line is called the Axis. And quator.
from this polar Point may Circles POLLUX, a fixed Star in the
be deſcribed on the Globe or Sphere, Twins, of the ſecond Magnitude,
as they are on a Plane from their whoſe Longitude is 108 Degrees
Centre.
and 47 Minutes, Latitude 6 Degrees
POLE-STAR, is a Star in the and 38 Minutes.
Tail of the little Bear, (a Conſtel POLYACOUSTics, are Inſtru-
lation of ſeven Stars, which is cal- ments contrived to multiply Sounds,
led Cynoſura,) and is very near the as Multiplying-Glaffes or Polyſcopes
exact North Pole of the World. do Images of Objects.
POLE of a Glaſs, in Optics, is the POLYEDRON, the fame with Po-
thickeſt Part of a Convex, but the lyhedron,
thinneſt of a Concave Glaſs; and if POLYGON, a Term in Geometry,
the Glaſs be truly ground, will be fignifying in the general any Figure
exactly in the middle of its Surface of many Sides and Angles, tho' no
This is ſometimes called the Vertex Figure is called by that Name, un-
of the Glaſs.
leſs it have more than four or five
POLES of the World, are two Sides.
Points in the Axis of the Æquator, 1. Every Polygon may be divided
each 90 Degrees diſtant from its into as many Triangles as it hath
Plane; one pointing North, which Sides
therefore is called ahe North or Arc 2. The Angles of any Polygon
tic Pole ; the other Southward, which taken together, will make twice as
therefore is called the South, or An- many right ones, except four, as
tarctic Pole.
the Figure hath Sides.
Whether any people live directly 3. Every Polygon circumſcribed
under the Pole, or not, is a Que- about a Circle, is equal to a rect-
ftion; but Dr. Halley hath proved, angled Triangle, one of whoſe Legs
that the ſolſtitial Day under the hall be the Radius of the Circle,
Pole, is as hot as under the Equi- and the other the Perimeter (or Sum
noctial, when the Sun is vertical to of all the Sides). of the Polygon.
them, or in their Zenith becauie If you make a Table, wherein
for all the 24 Hours of that Day the firſt horizontal Row being 1,
under the Pole, the Sun's Beams are and the ſecond zml; let the third
inclin'd to the Horizon with an An zz-Z-I be equal to the Pro-
gle of 23 Degrees : Whereas un duct of the ſecond by z leſs the firſt
der the Equinoctial, tho' he be- the fourth 23 - 2z+1, equal
comes vertical, yet he ſhines no to the Product of the third by z,
more than 12 Hours, and is abſent leſs the ſecond, and ſo on : And
12 Hours. And beſides, for three then form an Equation, one ſide of
Hours eight Minutes of that, 12 which being nothing, let the other
be
;
;
1
I
POL
.POL
be that horizontal Row of Quanti- nate an Arc, whoſe Chord ſhall be
ties in the Table, whoſe Exponent the side of a Polygon, whoſe Num-
is half the Number of Sides of a ber of Sides are expreſied by the
Polygon plus 1 : I ſay, the greateſt firſt upright Row of Numbers.
Root z of this Equation ſhall termi-
A Table for the Inſcription of
regular Polygons in a Circle.
3
4
z3__ZZ-2% +1
-3zz+2z ti
24----473 +3zz+32 - 1
7 z5_524+423 +6zz 32
8
bzs +524+10z3-6zz~4% ti
91z 7–720 +ózat-1524-1023-10zzt-4% to
2
2
- I
23
5 24
5
-I
7
20
8
27
m3, and
For Example, if it be required to of which the fame Number of Terms
inſcribe an Heptagon in a Circle : muſt be taken, as there are Units
Take the 4th horizontal Row of in m+1; what follows being equal
Quantities in the Table, becauſe too: For Example, let 7 be the
four is greater than half feyen by Number of Sides of the Polygon to
plus 1, and making it equal to no- be inſcribed ; then will
thing, we have 23 2z+1=0, ſo z3~— Zz - 22tio; and the
and the greateſt Root z of this Equa- greateſt Root % of this Equation
tion ſhall expreſs the Value of the will be the Length of the Side of
Chord terminating an Arch, being the Heptagon.
the ſeventh Part of the whole Cir There are ſeveral other curious
cumference.
Theorems relating to the Chords
If the Radius of a Circle be =1, and Polygons in Circles to be found
and z be the Length of the side of at the End of the roth Book of the
a regular Polygon inſcribed in that Marquis de l'Hoſpital's Analytic
Circle, and in general mt be equal Treatiſe of Conic Sections.
to half the Number of Sides of the POLYGON EXTERIOR, in Forti-
Polygon, which is ſuppoſed to be 'fication, is the Diſtance of one Point
odd; then will 0%-2
of a Baſtion from the Point of ano-
ther, reckon'd all round the Work.
+ m
3
+ POLYGON INTERIOR, is the
4
Diſtance between the Centres of any
two Baſtions, reckoned all round as
before.
POLYGONAL NUMBERS, are
ſuch as are the Sums or Aggregates
M-3
of Series of Numbers in Arithmeti.
cal Progreſſion, beginning with U-
3
nity; and ſo placed that they re-
M-4
+
preſent the Form of a Polygon.
3
Thus,
m-4
E c. be
3
4
a general Equation for finding the
Side of a regular Polygon in a Circle;
3 6
zawy
m
M2
I 2
2.zm.
912—3
m 2
m
X
zm-4
I
2
1
mcm 3
M-5
M-4
X
z
5
1
X
2
1
"2
m6t
m-6
m5
Х
2
mm-7
M-6
x
zmeny
X
1
2
M-5
X
Х
zm8
i
10
are
1
POR
PON
I
2
Of Hexagonal, 403 +372-1.
PORES,
are triangular Numbers, becauſe ternal Space betwixt them. They
they are the Aggregates of a certain have Props and Rails on each ſide';
Number of Points plac'd in the and the whole Structure ought to be
Form of Triangles, &c.
folid, as to be able to tranſport the
Horſe, together with Cannon and
Baggage, as well as the Infantry.
PONT-VOLANT, or the Flying
Bridge uſed in Sieges, is made of
4 9
16 two ſmall Bridges laid one over
are Quadrangular Numbers, &c. another, and ſo contrived by the
If the side of a Polygonal Num- means of Cords and Pulleys placed
ber be =n, and the Number of along the Sides of the Under Bridge,
Angles be =a, and the firſt Term that the Upper can be puſh'd for-
=1; then the sum of a Series of wards till it joins the place where
it is to be fix'd ; but however the
Triangular Numbers will be,
n3 40392 +2n.
whole Length of both theſe Bridges
Triangular
muſt not be above four or five Fa-
6
thom long, left they ſhould break
13+n?.
Of Pentagonal,
with the Weight of the Men, Theſe
are chiefly uſed to ſurprize Out-
473 -12 works or Poſts that have but narrow
6
Moats.
are ſmall Interſtices,
Of Septagonal, 593 +372—21.
Spaces or Vacuities between the Par-
6
ticles of Matter that conſtitute every
2013 ton?
Of Octogonal,
Body, or between certain Aggre-
gates or Combinations of them.
POLYGRAM, is a Geometrical Fi-
Mr. Boyle has written a particular
gure confifting of many Lines.
Effay on the Poroſity of Bodies, in
POLYHEDROUS FIGURE, in which he proves, that the moſt ro-
Geometry, is a Solid contained un lid Bodies that are, have ſome kind
der or conſiſting of many Sides ; of Pores : And indeed, if they had
which if they are regular Polygons, not, all Bodies would be alike ſpeci-
all ſimilar and equal, and the Body fically weighty.
be inſcribable within the Surface of Porime, (Gr. mógua) in Geo-
a Sphere, 'tis then call'da Regular metry, is a Theorem, or Propofi-
Body. See that Word.
tion ſo eaſy to be demonſtrated, that
POLYNOMIAL,
or Multinomial 'tis almoſt felf-evident; as, that a
Roots, in Mathematicks, are ſuch as Chord is all of it within the Circle.
are compoſed of many Names, Parts, And on the contrary, they call that
or Members; as atbfd-tae. an Aporime, which is ſo difficult as
POLYSCOPES, or Multiplying Glaf- to be almoſt impoſſible to be demon-
fes, are ſuch as repreſent to the Eye ſtrated ; as the ſquaring of any al.,
one Object as many.
fign'd Portion of Hippocrates's Lunes
POLYSPASTIUM, a Term in Mie was, till a little while ago.
chanicks, the ſame with the Troch-
PORISME.
Proclus and Pappus
Tea or Pulley.
define this Geometrical Term to ſig-
PONTON, in Fortification, is a nify a kind of Theorem, in the
Bridge made of Two Boats, at ſome Form of a Corollary, which is de-
diſtance one from another, both co- pendant upon, or deduced from
ver'd with Planks ; as alſo the in- fone other Theorem already demon-
ſtrated,
2
1
1
Herse,
POR
POS
ſtrated. And 'tis commonly uſed to pey at Rome, and that of St. Peter's
ſignify fome general Theorem, Palace in the Vatican.
which is diſcover'd from finding out POSITION, or Site, is an Af-
fome Geometrical Place, or Locus: fection of Place, and expreſſes the
As, for inſtance : If a Man hath Manner of any Body's being in a
found out by Algebra, or any other Place: This therefore is not Place,
Method, how to conſtruct a Local
nor indeed hath it any Quantity; as
Problem; and from that place fo Sir Iſaac Newton well obſerves in
conſtructed and demonſtrated, hath Princip. Mathem. P. r.
deduc'd ſome general Theorem, that Position, or the Rule of Poſition,
Theorem is by the Geometrick otherwiſe called the Rule of Fallhood,
Writers call'd a Poriſme.
is a Rule in Arithmetick, wherein
Porístick METHOD, in Ma- any Number is taken to work the
thematicks, is that which deter- Queſtion by, inſtead of the Number
mines when, by what Way, and fought ; and ſo by the Error or Ér.
how many different Ways a Problem rors found, we find the Number re-
may be reſolved.
· quired.
PORTCULLICE, Herſe, or Sara This Rule of falſe Poſition is of
zine, in Fortification, ſignifies feve- two kinds, viz. Single and Double.
ral great Pieces of Wood laid or POSITION SINGLE, is when
join'd acroſs one another like an there happens in the Propofition fome
Harſon, and at the Bottom it is Partition of Numbers into Parts pro-
pointed at the End of each Bar with portional, and then at one Opera-
Iron ; theſe formerly uſed to hang tion the Queſtion may be reſolved
over the Gate-ways of fortify'd by this Rule:
Places, to be ready to let down in Imagine a Number at pleaſure,
caſe of a Surprize, when the Enemy and work therewith according to
ſhould come ſo ſoon, as that there the Tenor of the Queſtion; as if it
is no Time to ſhut up the Gates : were the true Number; and what
But now a days the Orgues are more Proportion there is between the falſe
generally uſed, as being found to be Concluſion, and the falſe Poſition ;
much better. See Orgues.
ſuch Proportion hath the given
PORTICO, in Architecture, is a Number to the Number fought:
kind of Gallery raiſed upon Arches, Therefore the Number found by Ar-
where people walk under Shelter. gumentation ſhall be the firſt Term
It has ſometimes a Soffit or Ceiling, of the Rule of Three, and the Num-
but is more commonly vaulted. ber ſuppoſed ſhall be the ſecond
Though the word Portico be de- Term, and the given Number ſhall
riv'd from Port or Gate, yet do we be the third Term.
call the whole Diſpofition of the Co POSITION DOUBLE, is when
lumns in the Gallery by this Name. there can be no Partition in the Num-
The moſt celebrated Portico's of An- bers to make a Proportion : There-
tiquity were thoſe of the Temple of fore, you muſt make a Suppofition
Solomon, that of Athens built for the twice, proceeding therein according
People to divert themſelves in, and to the Tenor of the Queſtion; and if
where the Philoſophers held their either of the ſuppoſed Numbers hap-
Converſation, that which occaſion'd pens to ſolve the Propofition, the
the Diſciples of Zeno to be callid Work is done ; but if not, obſerve
Stoicks from the Greek Stoa, a Porti- the Errors, and whether they be
co : That magnificent one of Pom- greater or leſſer than the Reſolution
requireth;
A
A
are
POW
PRE
requireth ; and mark the Errors ac. ber again ; and this third Product
cordingly, with the Signs tor by the Root again ; and ſo on ad
Then multiply contrariwiſe the infinitum; as 2, 4, 8, 16, 32, 64,
one Poſition by the other Error; 128, 256, &c. Where 2 is called
and if the Errors be both too great, the Root or firit Power, 4 is its
or both too little, ſubſtract the one Square or ſecond Power, 8 is its
Product from the other, and the one Cube or third Power, 16 its Biqua-
Error from the other, and divide drate or fourth Power, &c. Ånd
the Difference of the Products by theſe Powers in Letters or Species,
the Difference of the Errors. are expreſſed by repeating the Root
But, if the Errors be unlike, as as often as the Index of the Power
the one t, and the other add expreſſes ; thus, a is the Root or
the Products, and divide the Sum firſt Power, aa the Square or ſe-
thereof by the Sum of the Errors cond Power, aa a the Cube, a a a a
added together: For the Proportion the Biquadrate or fourth Power.
of the Errors, is the ſame with the And to avoid the tediouſneſs of re-
Proportion of the Exceſſes or De- peating the Root ſo often when the
fects of the Numbers ſuppoſed, to Powers are high, we only put down
the Numbers fought.
the Root with the Index of the
Positive QUANTITIES, in Al- Power over it, thus ; a", that is the
gebra, are ſuch as are of a real and ninth Power of a; 610, 694,
affirmative Nature, and either have, the ſixteenth and the ninety fourth
or are ſuppoſed to have the affir- Powers of b.
mative or poſitive Sign + before Power of an HYPER-BOLA, is
them, and 'tis always uſed in oppo- the 16th Part of the Square of the
fition to the negative Quantities, conjugate Axis, or the I Part of
which are defective, and have this the Square of the ſemi-conjugate
Sign - before them.
Axis ; or it is equal to a Rectangle
POSTERN, in Fortification, is a under the of the tranſverſe Axis,
Falſe-Door uſually made in the An- and . Part of the Sum of the tranſ-
gle of the Flank, and of the Curtain, verſe Axis, and Parameter.
or near the Orillon, for private Sallies. Powers of LINES, or Quantities,
Posticum, is the Poſtern-Gate, are their Squares, Cubes, &c. or o
or Back-Door of any Fabric. ther Multiplications of the Parts
POSTULATES, or DeMANDS, in into the whole, or of one Part into
Mathematics, &c. are fuch eaſy another,
and ſelf-evident Suppoſitions, as need PRACTICE, in Arithmetic, is a
no Explication or Illuſtration to Rule which expeditiouſly and com-
render them intelligible. As, modiouſly anſwers Queſtions in the
That'a Right Line may be drawn Rule of Three, when the firſt Term
from one point to another. That is 1, or Unity ; and 'tis fo called
a Circle may be deſcribed on any from its Readineſs in the Practice
Centre given, of any Magnitude, of Trade and Merchandize.
&c.
PRECESSION of the Equinox. Be-
POTANS, or Potence, a part of caule in reality the Axis of the
a Watch ; fee under Ballance. Earth doth a little vary from ſuch
Powers, in Algebra, are Num an exact Paralleliſin, and doth not
bers ariſing from the Squaring or point always preciſely to the ſame
Multiplication of any Number or Star, when it is in the ſame place;
Quantity by it felf, and then that hence it happens that the Equinoc-
Product by the Root or first Num- tial Points, or the common Inter-
3
ſection
PRI
PRI
ſection of the Equator and Ecliptic, Powder or Touch-Powder to fire off
do retrocede or move backward the Piece.
from Eaſt to Weft, about 50 Seconds PRIMUM MOBILE, in the Ptole-
each Year; and this Motion back- maic Aftronomy, is ſuppoſed to be a
wards is by ſome called the Recef- vaſt Sphere, whoſe Centre is that of
foon of the Equinox, by others the the World, and in compariſon of
Retroceffion; and the advancing of which the Earth is but a Point :
the Equinoxes forward by this This they will have to contain all
means is called the Preceſſion of other Spheres within it, and to give
them.
motion to them, turning itſelf and
Prelude, in Muſic, ſignifies any all of them quite round in twenty-
Flouriſh that is introductory to Mu. four Hours.
fic, which is to follow after.
PRINCIPAL RAY, in Perſpec-
Priest's CAP, a Term in For- tive, is the perpendicular one which
tification. See Bonnet a Pretre. goes from the Spectator's Eye to the
Prick. To prick the Chart or vertical Plane, or the Picture. And
Plot at Sea, ſignifies to make a Point the Point where this Ray falls on
in their Chart whereabout the Ship the Picture, is called from hence,
is now, or is to be at ſuch a time, the
in order to find the Courſe they are PRINCIPAL Point, and is that
to ſteer, &c.
Point of the Picture wherein a Ray
PRIMARY PLANETS, are thoſe drawn perpendicular to it, cuts it.
fix that revolve about the Sun, viz. PRISM, is a ſolid Figure, con-
Mercury, Venus, the Earth, Mars, tained under ſeveral Planes, whoſe
Jupiter, and Saturn.
Baſes are Polygons, equal, parallel,
Prime FIGURE, is that which and alike ſituated.
cannot be divided into any other 1. Priſm in Optics, is a Glaſs
Figures more ſimple than itſelf; as bounded with two equal and paral-
a Triangle in Planes, the Pyramid lel triangular Ends, and three plane
in Solids : For all Planes are made and well poliſhed Sides, which meet
of the firſt, and all Bodies or Solids in three parallel Lines, running
compounded of the ſecond. from the three Angles of one End,
PRIME NUMBERS, in Arithme- to thoſe of the other, and is uſed
tic, are thoſe made only by Addi- in Optics to make many noble and
tion, or the Collection of Units, curious Experiments about Light
and not by Multiplication : So an and Colours : For the Rays of the
Unit only can meaſure them; as Sun falling upon it at a certain An-
2, 3, 4, 5, &c. and is by ſome cal- gle, do tranſmit thro' it a Spectrum
led a ſimple, and by others an un or Appearance, coloured like the
compound Number.
Iris or Rainbow in the Heavens.
PRIME VERTICALS, or Direct, 2. The Surface of a right Priſm,
Erect, North, or South Dials, are is equal to a Parallelogram of the
thoſe whoſe Planes lie parallel to the fame Height, having for its Baſe a
prime vertical Circle, which is that right Line equal to the Periphery of
Circle perpendicular to the Horizon, the Priſm.
and paſſing thro' the Eaſt and Weſt 3. All Priſms are to one another
Points of it.
in a Ratio compounded of their
PRIMING-Iron, is a ſmall ſharp Baſes and Heights.
Iron which is thruſt into the Touch 4. All like Priſms are to one an-
hole of a great Gun, and pierces other in the triplicate Ratio of
into the Cartridge that holds the their anſwè able Sides.
5.
PRO
PRO
5. A Priſm is the triple of a Py- or middle Term, if the Number of
ramid of the fame Baſe and Height the Terms, be odd.
PRISMOID, is a ſolid Figure, 3. If the firſt and laſt Terms, and
contained under ſeveral Planes whoſe the Ratio in any Geometrical Pro-
Baſes are rectangular Parallelograms greſſion be given, and the Sum of
parallel and alike muute.
all the Terms be required, multiply
PROBLEM, is a Propoſition which the ſecond and laſt Terms together,
relates to Practice; or which pro- and from the Product ſubftract the
poſes ſomething to be done: As to Square of the firſt Term ; and then
make a Circle paſs through three divide the Remainder by the Diffe-
given Points not lying in a right rence between the firſt and ſecond
Line, &c.
Term, and the Quotient will be the
PRODUCE, a Term in Geometry, Sum of all the Terms.
fignifying to continue a right Line, 4. Any infinite Series of Fractions
or draw it out farther, till it has 'decreaſing according to the Propor-
any affigned Length.
tion of the Denominator of the laſt
PRODUCT, is the Quantity ariſ- Term, and having a common Nu-
ing from, or produced by the Mul- merator lefs by an Unit than the
tiplication of two or more Numbers, Denominator of the laſt Term, is
Lines, &c. into one another; thus, equal to Unity.
if 6 be multiplied by 8, the Product PROJECTILES, are ſuch Bodies
is 48. In Lines, 'tis always, (and as being put into a violent Motion
ſometimes in Numbers,) called the by any great force, are then caſt
ReEtangle between the two Lines off or let go from the place where
that are multiplied one by another. they received their Quantity of Mo-
See ReEtangle.
tion, and do afterwards move at a
PROFILE, in Architecture, is the diſtance from it; as a Stone thrown
Contour or Out-line of any Member, out of one's Hand by a Sling, an
as that of the Baſe, Cornice, or the Arrow from a Bow, a Bullet from
like. Or it is more properly a Pro- a Gun, &c.
ſpect of any Place, City, or Piece of 1. The Line of Motion which a
Architecture, viewed fide-ways, and Body projected deſcribes, abſtracting
expreſſed according to the Rules of from the Reſiſtance of the Medium,
Perſpective.
is, as hath been proved by Gallileus,
PROGRESSION ARITHMETI and many others, and particularly
CAL. See Arithmetical Progreffion. by Sir Iſaac Newton, Prop. 4. Cor.i.
Progression GeomeTRICAL, of his Second Book, the Curve of a
or Geometrical Proportion continued, Parabola, which Line is alſo deſcribe
is when Numbers, or other Quan- ed by every deſcending Body. He
tities, proceed by equal Proportion ſhews alſo, that if the Line of Di-
or Ratio's, (properly called,) that rection of the projectile Motion of
is, according to one common Ra- any Body, the Degree of its Velo-
tio whether increaſing or decreaſing city, and at the Beginning, the Re-
As,
fiſtance of the Medium being given,
1, 2, 4, 8, 16, 32, 64, &C.. the Curve which it will deſcribe may
2 If there are never ſo many be diſcovered, and vice verſa. He
continual Proportionals, the Product faith alſo in Schol. Prop. X. Lib. 2.
of any two Extremes is equal to the that the Line which a Projectile de-
Product of any two Means that are ſcribes in a Medium uniformly re-
equally diſtant from the Extremes, fiſting the Motion, rather approaches
as alſo to the Square of the Mean, to an Hyperbola than a Parabola.
2. The
ones.
VAH?
X AI.
PRO
PRO
2. The horizontal Diſtances of approach nearer to the Aſymptotes
Projections made with the ſame Ve- than theſe Hyperbola's; but in Prac-
locity at ſeveral Elevations of the tice theſe Hyperbola's may be uſed
Line of Direction, are as the Sines inſtead of thoſe more compounded
of the double Angles of Elevation.
And if a Body be projected
3. The Velocities of Projectiles, from the Place A, according to the
in the ſeveral Points of a Curve, are right Line AH, and AI be drawn
as the Lengths of the Tangents to parallel to the Afymptote NX, and
the Parabola in thoſe Points, inter- GT is a Tangent to the Curve, (in
cepted between any two Diameters: the Vertex :). Then the Denſity of
And theſe again are as the Secants the Medium in A will be recipro-
of the Angles, which thoſe Tangents cally as the Tangent AH, which if
continued make with the horizontal it had been a ſtanding Quantity,
Line.
the Medium would have had a gi-
4. If AGK be a Curve of the ven Denſity as our Air may be ſaid
hyperbolic kind, one of whoſe A to have, ſo far as Projectiles can
ſymptotes is NX, perpendicular to
move in it, and the Body's Velocity
the Horizon AK, and the other
IX inclin'd to the fame, where VG will be as
and the Reſi.
AL
is reciprocally as DN" whoſe Index
is n: This Curve will nearer repre-
ſtance thence to Gravity, as AH to
ſent the Path of a Projectile thrown 2nn + 2n
in the Direction AH in our Air,
2 ton
X The Doctrine of Projectiles, when
Parabolas, is very briefly and ele-
V
gantly handled by Mr. Cotes, at the
End of his Works. So it is alſo in
the French Memoirs of the Royal
Academy at Paris, for the Year
1731, or thereabouts.
I
G G
PROJECTION of the Sphere in
plano, is a true geometrical Delinea-
T
tion of the Circles of the Sphere, or
any aſligned Parts of them, upon
А
DKN the Plane of ſome one Circle ; as
on the Horizon, Meridian, Equator,
(which may be taken as a uniform Tropic, &c.
Medium, reſiſting Bodies as the The Projection of the Sphere is
Squares of their Velocities,) than a handled by Clavius, in his Treatiſe
Parabola which is only deſcrib'd by of the Afirolabe. Likewiſe very e-
a Projectile, where there is no Mé legantly by Agulonius, in his Optics.
dium refifting its Motion, Sir Ijaac See alſo Taquet, in his Optics, Witty,
Newton, in the ſecond Book of his Haines's Trigonometry, Harris's Tri-
Principia, ſays indeed, That theſe gonometry, &c.
Hyperbola's are not accurately the PROJECTION (MONSTROUS) of
Curves that a Projectile makes in an Image, in perſpective, is the De-
the Air; for the true ones are Curves formation of an Image on a Plane,
which about the Vertex are more or the Superficies of ſome Body,
diſtant
from the Afymptotes, and in which ſeen at a certain Diſtance
thoſe Parts remote from the Axis will appear formous.
H
Еe
IR
PRO
PRO
SI
1
If it be required to delineate a vided into a Number of Areola's, or
monſtrous Projection on a Plane, leſſer Squares.
proceed thus :
2. In this Square let the Image,
1. Make a Square ABCD (called to be repreſented deformed, be
the Craticular Prototype) of a Big- drawn.
D
C
3. Draw the Line ab= AB, and
divide it into the fame Number of
equal Parts, as the Side of the Pro-
totype A B is divided into.
4. In E, the middle thereof, erect
the Perpendicular EV, ſo much the
longer, as the Deformation of the
Image is to be greater.
5 Draw VS perpendicular to EV,
fo much the leſs in Length, as you
would have the Image appear more
А
B.
deformed.
neſs at pleaſure, and divide the Side 6. From each Point of Diviſion
A B into a Number of equal Parts, draw ſtraight Lines to V, and join
that fo the ſaid Square may be di- the Points a and S, as alſo the Right
Line as.
7
E
7. Thro' the Points d, e, f, g,
a
draw Right Lines parallel to a b.
Then will abcd, be the Space that
the monſtrous Projection is to be deli-
neated in, called the Craticular Eco
type.
8. In every Areola, or ſmall Tra-
09.12
pezium of this Space abcd ; let
there be drawn what appear deli-
neated in the correſpondent Areola
of the Square ABCD, and by this
means you will obtain a deformą
Image, which will appear formous
to an Eye diſtant from it by the
Length FV, and raiſed above it
the Height VS.
9. It will be very diverting to
manage it ſo, that the deformed I-
mage does not repreſent a mere
Chaos ; but ſome other Image dif-
ferent from it, which by this con-
trivance ſhall be deformed. As I
have feen a River with Soldiers,
Waggons, &c. marching along the
ſide of it, ſo drawn, that when it is
looked at by an Eye in the Point S,
appears to be the fatyrical Face of
a Man.
10. An Image may be deformed
S
mechanically, if you place theImage,
V
having
MA
3
necores...........
perronn Sur«HI HIKYUTIUNIIOITRICIA.
a Cone.
, PRO
PRO
having little Holes here and therefolded rightly up, will form the Sul-
made in it with a Needle or Pin, perficies of a Cone, whoſe Baſe is
againſt a Candle or Lamp, and ob- the Circle ABCD.
ſerve where the Rays going thro’ 4. Divide the Arch EG into the
theſe little Holes fall on a Plane, or fame Number of equal Parts, as the
Curve-Superficies ; for they will Craticular Prototype is divided into,
give the correſpondent Points of the and draw Radii from each of the
Image deformed, by which means Points of Diviſion.
the Deformation may be com 5. Produce GF to I, ſo that FI
pleated.
EFG, and from the Centre 1,
with the Radius. IF, draw the Qua-
To draw thel Deformation of an I. drant F KH, and from I to E draw
mage upon the Convex-Surface of the Right Line I E
6. Divide the Arch KF into the
From the laſt Problem it is ma- fane Number of equal Parts, as the
nifeſt enough, that all that is to be Radius of the Craticular Prototype
done here, is to make the Craticular is divided into, and draw Radii thro'
E Etype in the Superficies of the Cone, each of the Points of Diviſion from
which ſhall appear to an Eye duly the Centre I, meeting EF in 1, 2,
placed over the Vertex of it, equal to 3, &c.
the Craticular Prototype. Therefore, 7. Finally, from the Centre F
1. Let the Baſe ABCD of the with the Radii, F1, F2, F3, &c.
Cone, (Fig. 1.) be divided by Dia- deſcribe concentric Arches. Thus
meters into any Number of equal will you have the Craticular Ec-
Parts ; that is, let the Periphery be ope, whereof each Areola will ap-
thus divided.
pear equal to one another.
8. Theretur , if what is delineata
ed in every Areola of the Craticular
Prctotype be transferred into the A-
A
B
reola's of the Craticular EEype, the
Image will be deformed; but the
Eye being duly raiſed over the Ver-
2. Likewiſe let ſome one Radius
tex of the Cone, will perceive it
formous.
be divided into equal Parts, and
9. If the Chords of the Quadrants
draw
concentric Circles. And thus Thall be drawn into the Craticular Proto.
the Craticular Prototype be made.
type, and Chords of their fourth
Part in the Craticular Eclype, all
3. With the double of the Dia-
ameter A B, as a Radius, deſcribe things elſe remaining the fame; you
the Quadrant EFG, (Fig. 2.) fo that will have the Craticular Eclype in a
the Arch E G may be equal to the quadrangular Pyramid. And from
whole Periphery; then this Quadrant hence you may learn how to deform
an Image in any other Pyramid,
whoſe Baſe is any regular Polygon.
10. Becauſe the Eye will be inore
K.
deceived, if from contiguous Objects
it cannot judge of the Diſtance of
the Parts of, che deformed Image:
Therefore, theſe kind of deformed
Images muſt be looked at chro' a
{mall Hole.
F
G
Ee 2
TO
D
PRO
PRO
7
Fig2
S
T
To delineate a Figure in an horizontal
Plane, which ſhall appear by Re-
flexion on a Cylindrical Speculum
ſtanding on that Plane,“ like a
Square divided into many little
Square Areola's.
1. About EB (Fig. 2.) the Dia-
meter of the Cylindrical Speculum,
deſcribe a Circle equal to the Baſe
of the Cylinder.
2. Take the Point O under the
Eye, and draw the Tangents O E,
OB; becauſe no Ray reflected from
the Speculum beyond them, will fall
upon the Eye. Likewiſe the Right
Lines O B, O E, may be ſo drawn,
Fig. 1.
+
H
E E
B
NE34
2244
G
to the side of the Square appearing
in the Speculum, and divide the
Р
as to cut the Circle ; fince what are
Fig. 3.
perceived by the Tangents, will not
be diftinct enough.
R
3. Join the Points of Contact, or
Interſection E,B, by a ſtraight Line
EB, which muſt be taken for the
Side of the Square appearing in the
TIL II. TIR
M
Speculum: Becauſe the Image ap-
pears in a Cylindrical Speculum be- fame into the ſame Number of equal
tween che Centre and the Super- Parts, as that Side is divided into.
ficies.
8. Thro' every Point of Diviſion
4. Divide E B into any Number 1, 2, 3, &c. draw the Right Lines
of equal Parts; and from every of P.I, P.II, P.III,&C.
the Points of Diviſion, 1, 2, 3, &c. 9. From L to 1, 2, 3, &C.
draw Right Lines O1, O2, O3, c. transfer the Right Lines Li, L2,
to the Point O under the Eye. L3, &c. equal to QI, QII,
5. Let the Radii OH, OI, be QIII, &c.
reflected to the Points F,G, &C. 10. After the ſame manner, let the
that is, let HF, IG, be the Re- Lines. HF, IG, &c. be divided ;
flexions of O1, O2, &c.
and thro' the Points of Diviſion of
6. Upon the indefinite Right Line the ſame Order draw Curves : Or,
MQ.. (Fig. 3.) raiſe the Perpendi- fince there is no need of very great
cular MP, equal in Length to the Accuracy in theſe caſes, draw cir-
Height of the Eye.
cular Arches thro' three Points, as
7. From M to transfer the is done in the Figure.
Line OH, and at Q raiſe the Per I ſay the Figure STFGA, be-
pendicular QR, which let be equal ing erected upon the Circle ACDB,
will
/
1
(
с C
PRO
PRO
will appear in the Cylindrical Spe- its Axis. And in AO produced
culum, as a Square divided into ſe. take A B equal to the Height of the
veral equalſquare Areola's. Whence, Eye.
if a Square be made, whoſe Side is 3. To every of the Points 1, 2, 3,
equal to QR, and the fame be di- &c. of Diviſion from the Point B,
vided into equal Areola's, and in wherein the Eye is ſuppoſed, draw
the ſame be painted any Image, and the Right Lines, Bi, B2, B3, &c.
then what is in every Areola of it
be transferred in the correſpondent
Fig. 2.
Areola's of the deformed Square,
B
that deformed Image will by Re-
flexion appear formous in the Cy-
lindrical Speculum.
To delineate a deformed Figure upon
an horizontal Plane, that fall
0723 IIIII
appear formous by the Reflexion of
a Conical Speculum to an Eye over
4. Becauſe theſe are the reflected
the Vertex.
Rays by which the Points 1, 2, 3,
&c. are ſeen, and AE is the Inter
1. The Image to be deformed
muſt be delineated in a Circle, equal ſection of the Plane of Reflexion and
to the Baſe of the Conical Specu- IDEII, CE, equal to the Angles
the Speculum, make the Angles
jum, and the Periphery muſt be BDA, BCA, &c. then ſhall D.I,
4 b
C. II, &c. be the Rays of Incidence :
Conſequently I. II, & c. the radiat-
9
ing Points which are ſeen by Refle-
Id
xion, in 1, 2, 3, &c.
5. Therefore produce the Radii
Oa, Ob, Oc, &c. in the Craticular
divided into equal Parts by the Dia- Prototype, and transfer in them the
meters, ad, be, sf, &c. and the Diviſions 0.1, 0.II, O.III, EC.
Radii Ob, Oc, od, &c. (Fig. 1.) into And laſtly draw Concentric Circles
equal Parts Oa, 1.2, 2.3, &c. by
Concentric Circles.
you have the Craticular Ectype.
6. Therefore if in every of its
2. To get the Points I, II, III,
&c. in the Plane that the Cone's in the correſpondent Areola’s of the
Areola's you depict what you find
Baſe ſtands upon, which are ſeen by Craticular Prototype, you will have
reflected Rays within the Speculum
at the Points, 1, 2, 3, &c. make a deformed Figure, which will ap-
(Fig.2.) a right-angled Triangle, duly placed over the Vertex of the
an
O E is
Cone.
the Radius of the Speculum, and
Altitude AO equal to the Height To delineate a deformed Image upon a
of the Speculum, that is equal to Plane, that ſhall appear formous
by Reflexion to an Eye, placed over
the Vertex of a Pyramidal Speculan.
For Example. Let it be required
to delineate a deformed Image, which
will appear formous by the Reflexi-
on of a quadrangular Pyramid.
ho Y III I I
e
T
I
c
Еe 3
1. In
PRO
PRO
1
1. In this caſe, the Image to be Note, Deformed Images, that are
deformed, is to be dețineated into made by means of pyramidal Spe-
the Square ABCD, equal to the culums, are more diverting than
Baſe of the Speculum, whoſe Peri. thoſe made by others. Becauſe the
meter muſt be divided into equal Parts of the deformed Image being
Parts by Diagonals, from the Cen- disjoined, any others may be painted
tre E; and allo by Right Lines, bif- between them, forming one and the
ſame continuous thing with them
A
B
without the Speculum, which in the
Speculum will not be ſeen.
L
PROJECTURE, in Architecture,
fignifies the Prominency or Emboff-
sc
D
ment, which the Mouldings, and o-
ther Members have, beyond the
ſecting the Sides AB, BC, &c. naked Wall; and is always in pro-
Moreover, the Lines E L, E B, muft portion to its Height. The word
be divided into any Number of e. is alſo applied to Galleries, Balco-
*qual Parts; ſo that Lines drawn nies, &c. which jet beyond the
thró' the Points of Diviſion, which face of the Wall.
are parallel to the Sides of the Baſe, PROLATĘ SPHEROID, is a Solid
may include the Craticular Proto produced by the Revolution of a Se-
type.
mi-Ellipſis about its longer Diamc-
2. Now, ſince the Section of the ter or Axis ; but if a Solid be form-
Speculum thro' the Axis, and the ed by the Revolution of a Semi-El-
Right Line E L drawn in the Baſe, lipſis about its ſhorter Diameter, it
is a right-angled Triangle; and e- is then called an Oblate Spheroid :
yery Point of Diviſion of the Crati. And of this Figure is the Earth we
cular Prototype, is in the reflexed inhabit, and perhaps all the Planets
Ray, after the very fame manner are fo too, having their Equatorial
as in the laſt Problem are found Diameters longer than their Polar.
the Points I, II, III, &c. of the
PROMONTORY,
is an Hill ar
Axis LE, of the Triangle BEC, high Land running out into the Sea,
the Extremity of which towards the
Sea, is uſually called a Cape, or
Headland.
PROPORTION, is an Equality of
Ratio's.
B
1. Magnitudes are ſaid to have
G
E
Proportion to each other, which
being multiplied can exceed one an-
D
other.
2. Magnitudes are ſaid to be in
the fame Ratio, the firſt to the ſe-
cond, and the third to the fourth,
F
when the Equimultiples of the firſt
and third, compared with the Equi-
to be reflected : Which being given, multiples of the ſecond and fourth,
the Triangle itſelf may be made. according to any Multiplication
3. Laply, What elſe is to be done, whatſoever, are either both toge-
muſt be proceeded with, as in the ther greater, equal, or leſs, than
lalt Problem.
the Equimultiples of the ſecond and
fourth,
A
MU
'L
aea :
1
og ::
then oa
PRO
PRO
fourth, if thoſe be taken that an 2. Alternately, a :b :: eg : eb.
ſwer each other.
3. Compoundedly,
That is, if there be four Magni actuea : ea :: b4eb : eb.
tudes, and you take any Equimul 4. Converſby, a trea:a::bteb:b
tiples of the firſt and third, and alſo 5. Dividedly,
any Equimultiples of the ſecond and
:eb.
beb
fourth: And if the Multiple of the
firf be greater than the Multiple of
6. By a Syllepſis,
the ſecond, and alſo the Multiple of a : ea :: a+b : ea teb.
the third greater than the Multiple of 7. By a Dialepſas,
the fourch: Or, if the Multiple of the
a: Pa ::ab: eamweb.
firſt be equal to the Multiple of the 2. If in two Rows of Proportio-
ſecond ; and alſo the Multiple of the nals a : ea :: b:eb, and ea :
third equal to the Multiple of the eb: ob; then by ordinate Proportion
fourth : Or, laſtly, if the Multiple of of Equality, a : oa :: b: ob. But if
the firſt be leſs than the Multiple of they are diſorderly placed, viz. oa :
the ſecond ; and alſo that of the ea :: ob : eb; and ea: a :: eob : ob :
third leſs than that of the fourth ;
:a :: eob : eb. If there
and theſe things happen according are two Rows of Proportionals a:
to every Multiplication whatſoever; ea :: b:eb::6:00 :: d: od : then
then the four Magnitudes are in the ſhall a xc : ea xoc :: bxd: eb xod.
ſame Ratio, the firſt to the ſecond, All theſe are manifeſt by comparing
as the third to the fourth.
the Rectangles of the Means and
Expounders uſually lay down Extremes, or by dividing the Con-
here that Definition which Euclid fequents by their Antecedents.
has given for Numbers only, in his PROPORTIONALSCALES, ſome-
Yevench Book, viz. That
times alſo called Logarithmetical, are
· Magnitudes are ſaid to be Pro- only the artificial Numbers or Lo-
portionals, when the firſt is the ſame garithms placed on Lines, for the
Equimultiple of the ſecond, as the Eafe and Advantage of multiplying,
third is of the fourth, or the ſame dividing, extracting Roots, &c. by
Part or Parts.
the means of Compaſſes, or by
But this Definition appertains on- Numbers, as they are called. by Mr.
ly to Numbers and commenſurable Gunter ; but made fingle, double,
Quantities ; and ſo fince it is not u- triple, or quadruple ; beyond which
niverſal, Euclid did well to reject it they feldom go.
in his 5th Element, which treats of PROPORTIONAL Spiral Lines.
'the Properties of all Proportionals; See Spiral Lines.
and to ſubſtitute another general PROSTAPHERESIS, in Aſtrono-
one, agreeing to all kinds of Magni- my, is the ſame with the Equation
tudes. In the mean time, Expoun- of the Orbit, or ſimply the Equa-
ders very much endeavour to de- tion ; and is the Difference between
monſtrate the Definition here laid the true and mean Motion of a Pla-
down by Euclid, by the uſual re The Angle alſo made by the
ceived Definition of Proportional Lines of the Planets mean and
Numbers ; but this much eaſier Motion, is called the Profaphereſis.
flows from that, than that from this. PROTRACTING-PIN, is a fine
1. If there are four Quantities Needle faftned in a Piece of Wood,
proportional, as 'a, ea, b, eb, then Ivory, &c. uſed to prick off any
they will be alſo proporcional. Degrees and Minutes from the Pro-
1. Inverly, eaia :: eb: b. tractor.
Pron
net.
true
Ee 4
P TO
PUL
PROTRACTOR, is an Inſtrument degrees it came to be quite dif-
uſed in Surveying: It is commonly uſed.
made of a well-polished thin Piece PULLEY, is a little Wheel move-
of Braſs, and confifteth of a Semi- able about its Axis, over which,
Circle divided into Degrees, and a goes a Drawing-Rope.
Parallelogram with Scales upon it, I. In ſeveral caſes where the Axis
and may be of any bigneſs de- in Peritrochio cannot conveniently
fired.
be applied, Pulleys muſt be made
Its Uſe is chiefly to lay down an uſe of to raiſe Weights: A Machine
Angle of any aſſigned Quantity of made by combining ſeveral of them,
Degrees : Or, an Angle being pro- lies in a little compaſs, and is eafi-
tračted, to find the Quantity of De- ly carried about, if the Weight be
grees it contains readily; which is fixed to the Pulley, ſo that it may
of great uſe in plotting, and mak- be drawn up along with it : Each
ing of Draughts, @OC.
End of the Drawing or Running-
PSEUDOSTELLA, in Aftronomy, Rope ſuſtains half the Weight;
ſignifies any kind of Comet or Phæ- therefore when one End is fixed, ei-
nomenon newly appearing in the ther to a Hook, or any other way,
Heavens like a Star.
the moving Force or Power applied
PTOLEMAIC System of the Hea to the other End, if it be equal to
vens, was that invented by Ptolemy; half the Weight, will keep the
in which he ſuppoſes the Earth im- Weight in Æquilibrio.
moveable any way in the Centre of 2. Several Sheaves may be joined
the Univerſe, round about which in any manner, and the Weight be
the Moon firſt moves in a Circle ; fixed to them, then if one Ènd of
next her Mercury', then Venus : A- the Rope be fixed, and the Rope
bove which moves the Sun, then goes round all thoſe Sheaves, and
Mars; above him Jupiter, and laſt as many other fixed ones, as is ne-
of all Saturn, all in the Zodiac ceffary, a great Weight may be
from Weſt to Eaſt. Above Saturn raiſed by a ſmall Power: In that
he places the Sphere of the fixed caſe, the greater the Number of
Stars, which he ſuppoſes to move Sheaves fixed in a moveable Pulley,
ſlowly alſo, from Eaſt to Weſt, on or of moveable Wheels are (for the
the Poles of the Ecliptic. While fixed ones do not change the Action
the fixed Stars themſelves, and all of the Power,) ſo much may the
the Planets, move from Eaſt to Weſt Power be leſs, which ſuſtains the
on the Poles of the Equator, in the Weight; and a Power which is to
Space of a natural Day or twenty- the Weight, as the Number one to
four Hours. This vulgar Syſtem of twice the Number of the Sheaves,
Aftronomy, (in which I omit to will ſuſtain the Weight.
mention the Epicycles and Defe Pulse, by the Mathematical Na-
rents, &c. with which they endea- turalifts, is the Term uſed for that
voured to ſolve the Phænomena Stroke with which any Medium is
which did almoſt all of them con- affected by the Motion of Light,
tradict this Scheme) was plainly o. Sound, &c.
verturned and refuted as ſoon as e And Sir Iſaac Newton demon-
ver the Uſe of the Teleſcope ac- ftrates, Lib. 2. Prop. 48. Princip.
quainted us with the Phaſes of Venus that the Velocities of the Pulſes, in
and Mercury; for from thence it any elaſtic fluid Medium, (whoſe
was apparent, that their Orbits in- Elaſticity is proportionable to its
cluded the Sun, and therefore by Denſity,) are in a Ratio, compound-
ed
1
PU R
PYR
ed of the fubduplicate Ratio of the a-croſs the Rafters on the Inſide, to
Elaſtic Force directly, and the ſub- keep them from finking in the mid-
duplicate Ratio of the Denſity in- dle of their Length.
verſly. So that in a Medium, whoſe PYRAMID, in Geometry, is a
Elaſticity is equal to its Denſity, all folid Figure, whoſe Baſe is a Poly-
Pulſes will be equally ſwift. gon, and whoſe Sides are plain Tri-
Pulsion, is the driving or im- angles, their feveral tops meeting
pelling of any thing forward.
together in one Point.
PUNCHINS, in Architecture, are 1. The Solidity of a Pyramid is
fhort Pieces of Timber placed to of the perpendicular Altitude mul.
ſupport fome conſiderable Weight: tiplied by the Baſe.
They commonly ſtand upright be 2. The fuperficial Area of a Py-
tween the Poſts, and are ſhorter and ramid is found by adding the Area
Nighter than either the principal of all the Triangles, whereof the
Poſts or Prick-polts. Thoſe that Sides of the Pyramid confift, into
ſtand on each side of a Door are
one Sum.
called Door-Punchins.
3. The external Surface of a
PUNCTATED HYPERBOLA, is right Pyramid, that ſtands on a re-
any Hyperbola whoſe Conjugate gular Polygon-Baſe, is equal to the
Oval is infinitely ſmall, that is, a Altitude of one of the Triangles
Point.
which compoſe it, multiplied by the
PUNCTUM FORMATUM feu Ge- whole Circumference of the Baſe
NERATUM, in Conics, is a Point of the Pyramid.
determined by the Interſection of a
The Demonftrations of the three
Right Line drawn thro' the Vertex
of a Cone to a Point in the Plane following Problems being ſhort and
of the Baſe, with the Plane that eaſy, and not every where to be
conftitutes the Conic Section. See found ; I therefore thought it might
not be amiſs to inſert them here.
De la Hire's Latin Conics, p. 15,
16.
1. To find the Solidity of the Frufium
PUNCTUM EX COMPARATIONE, of a Square Pyramid.
is either Focus, in an Ellipſis and
Let A D be one of the Sides of
; ſo
Apollonius, becauſe the Rectangles the greater Baſe, which let us call
under the Segment of the Tranft b, and BC the side of the leffer
verſe Diameter in the Ellipſis, and Baſe, which call a ; and let EF be
under that and the Diſtance between the Height of the Fruftum, which
the Vertex and Focus in the Hy- let be h.
perbola, are equal to Part of what mid ASD, and draw the Line GG,
Now, compleat the whole Pyra-
he calls the Figure.
PUNCTUM LINEANS, is that
Point of the generating Circle,
S
which in the Formation of either
fimple Cycloids or Epicycloids, pro-
duces any part of a Cycloidal Line.
PURE HYPERBOLA, is one,
which, by the Impoffibility of its
B
Roots, is without any Oval, Node,
Spike, or Conjugate Point.
PURLINES, in Architecture, are
D
G
thoſe Pieces of Timber, which lie
parallel
A
F
PY R
P Y R
saa
+
saa
r
a
1
;
parallel to EF. Now, becauſe the
b
sbb
Triangles ADS, BCS, are fimi- ſquare Pyramid, will be
*+
3
lar, it will be as b-a: (2GD)
sba
bb
Whence it is manifeft,
h :: (EF=GC) b : (AD) :
b-a sbb
(ES). And in like manner, as b-a
that and is the Sum of the
: (2GD) :: (EF=GC) a : (BC) :
sba
ah
circular Baſes, and
mean
ba (FS). Therefore the Solidity
of the Pyramid ASD, will be Bafes. Therefore the Corollary is
Proportional between the circular
h 63
manifeft.
36-3a
And the Solidity of the
ha²
2. To find the Curve-Superficies of a
Pyramid BSC, will be
right Cone.
3 baza
and conſequently the Solidity of the
If a right Cone A B D lies, upon
Fruſium ABCD of the Pyramid, the Plane AC, or touches it in the
Right Line AB; and if the ſaid
563ba3
will be
and by dividing
Cone revolves upon the ſaid Plane
353a
about the Point A, until the Point
bbb haa
35-32) hb-ha3 ( +
3
3
hab
This laſt Expreſſion will
с
3
A
B
be the Solidity of the Fruſtum ,;
therefore, if the Sum of the Baſes, B, in the Periphery of the 'Baſe,
and the Rectangle under the Sides comes to touch the Plane again.
A D and BC, are added together, Then, I ſay, that the whole Super-
and multiplied by 1 of the Height ficies of the Cone will have touched
EF, the Product will be the Solidity the Plane in every Part; and con-
of the Fruftum.
ſequently, if the Lines AB, AG,
be equal to A B, the ſlant Height
COROLLA RY.
of the Cone, and about the Centre
A, be deſcribed an Arch of a Cir-
Hence the Solidity of the Fruftum cle, whoſe Length BG is equal to
of a Cone, or any other kind of the Periphery of the circular Baſe
Pyramid, may be alſo found. For
it is but adding the two circular
G
Baſes together, and to that Sum a
mean Proportional between the ſaid
circular Baſes, and then multiplying
As
the whole Sum by of the Height,
and that will be the Solidity of the
B
Fruftum of a Cone. For ļet the
Ratio of the Square of the Diameter of the Cone, that the Area of the
of a Circle to the Area thereof be circular Sector ABG will be equal
as r to s: Then the Solidity of the to the Curve Superficies of the Cone.
Pruftum of a Cone circumſcribing Therefore, if half the Periphery of
the Fruftum of the before-mentioned the Baſe of any Right Cone be mul-
tiplied
+
1
That is,
1
PY R
P Y R
tiplied by the ſlant Height, you will
pacqab
have the Area of the Curve-Surface Fruſtum will be
26-26
thereof.
pc- qb
3. To find the Area of the Curve-Sur-
Now let us ſuppoſe
cab
face of the Fruſtum of 4 Right
Cone.
n to be ſuch a Quantity, that if the
Let us call the lant Height AB, it, the Quotient will be the Dia-
Periphery of a Circle be divided by
a; the Diameter AD of one of
the Baſesc, and the Diameter of meter. This being ſuppoſed,
X
2
22
72
с
>
1
D (, and L=b. Then our laſt Theo-
P-99
S
rem will be thus,
multi-
B А
p-q
the other Baſe BC, b. Alſo let plied by will be the Area of the
us call the Periphery of the greater
Baſe p, and of the leffer g. Now, Superficies of the Fruftum ; but
if ŞB, S A, are equal to‘SB, SA, pp-99 divided by A---9, will be pta.
and about the Point S be deſcribed Therefore, to find the Curve. Super-
ficies of the Fruſtum of any Cone,
you muſt add the Circumferences of
D
the two Baſes together, and chat
Sum multiplied by į of the flant
st
Height, will be the Area of the
Curve-Superficies fought.
Having happened upon a very
B
A
eaſy way of ſquaring the Parabola,
by the Method of Indiviſibles, I
two Concentric Portions of Circles, thought it would not be amiſs to
the greater of which is equal to the inſert it here. But firſt the follow-
Periphery of the greater Circular ing Lemma muſt be demonſtrated.
Baſe D A, then the Area CDAB
will be equal to the Area of the The Sum of all the Rectangles (infi-
Curve Superficies of the Fruftum nite in Number) that can be made
ſought. Now to get this Area we by cutting the given Line AB into
muſt find SA, and SB, the former two Segments, as ACXCB, is
equal to 5 of the Cube of the ſaid
will be
and the other
ca
Line.
ab
whence the Area of the Sec-
А с
B
6
pac
tor SAD, will be
and the
DEMONSTRATION.
Area of the other lefſer Sector SCB,
Let us call the whole Line
will be
And therefore Segment AC, *, then a-* * x will
the Area of the Figure CDAB, that be the firſt Rectangle, amžx * 2 *
is the Area of the Superficies of the the ſecond, --3* X 3* the third,
and
20-26)
a, the
gab
20-26
PYR
PY T
a X I.XIXX
1
and 4* * 4*, the fourth, and equal to DC multiplied into ſome
so on. That is, the Sum of the ſtanding Quantity, as m, which is
Rectangles will ſtand thus,
C
a x ax-4xx
a x 34-9**
a x 44-16xx, &c.
From whence you may ſee that
the Sum of all the firſt Terms will
be equal to the Solidity of a trian-
gular Priſm whoſe Height is
А D
B
the Baſe the Right-angled Ifoſceles
Triangle CAB, each of whoſe equal the Latus Rectum: Therefore of
, ,
will
B
be the Area of the Parabola. Now
a,
and
аа
m
divided by GF, which
4
40
fuppoſe b, that is, 76=m. There-
1
fore the Area of the Parabola will be
enitesi
a3
aa
divided by 46
Q3
23
ő
which will be
ba, or of the circumſcribing Pa-
Sides is = 2. Therefore their Sum
rallelogram AHIB.
is
H
G
I
D
ſecond Terms, (becauſe the Co-effi-
cients are the Squares of Numbers
in Arithmetical Progreſſion,) will be
Fig za
equal to a ſquare Pyramid, having
its Baſe doubled to BAC, and the
fame Altitude a, whence their Sum A
B
a3
will be
And taking from
3
3
PYRAMIDOID, is what is fome-
times called a Parabolic Spindle ;
, you will have ada for the and is a ſolid Figure formed by the
Sum of all the Rectangles. Q.E.D. Baſe or greateſt Ordinate.
Revolution of a Parabola round its
COROLLARY I.
PYTHAGOREAN THEOREM, is
the 47th Prop. of the firſt Book of
Hence the Sum of the Squares of Euclid.
all the Sines CD drawn in a Cir PYTHAGOR EAN SYSTEM, is the
cle, is equal to of the Cube of the ſame with the Copernican, but is ſo
Diameter. And ſo the Solid called called, as being maintained by Pyo
the Hoof or Ungpla, may be ſquared. thagoras, and his Followers, and
Likewiſe from hence we may have is the moſt ancient of any. In
the Quadrature of the Apollonian Pa- this the Sun is ſuppoſed at Reſt
rabola (See Fig. 2.) For becauſe the in the Centre of our Syſtem of
Rectangle under AC X CB, is Planets, and the Earth to be car-
ried
2
1
}
O
+
QUA
QUA
ried round him annually in a Track String :) So is the Diſtance between
or Path between Venus and Mars. the Station and Foot of the Object
to its Height above the Eye.
QUADRATIC Equation, is one,
when made as ſimple as poffible,
Q:
that conſiſts of not
more than
three Terms; the third of which
UADRANGLE, or Quadran- is a known Number or Quan-
gular Figure in Geometry, is tity, and the Dimenſion or Power of
that which hath four Angles. the unknown Quantity making the
QUADRANT, is an Arch which firſt Term is the double of the un-
is the fourth Parth of a Circle, con- known Quantity (or its Power) con-
taining, 90 Degrees. And often- ftituting the ſecond Term.
times the Space contained between a All Quadratic Equations conſiſt of
quadrantal" Areh, and two Radii one of the following Forms:
perpendicular one to another in the
1. *x + a
Centre of a Circle, is called a
2. xx + ax=).
Quadrant.
3. xx + ax+b=0.
QUADRANT of Altitude, is a Part Or generally 4. *2m+_axm+b=o.
of the Furniture of an artificial
1. In the firſt Form it will be x
Globe, being a thin Braſs-Plate di
vided into 90 Degrees, and marked Fa, or x= va; that is,
upwards with 10, 20, 30, &c. being *Xa=0 has two equal real Roots,
rivetted to a Braſs Nut which is fit- and xxtam, has two equal imaa
ted to the Meridian, and bath a ginary Roots. 2. In the ſecond
Screw in it, to ſcrew upon any De. Form it will be x=Ia, and x=0.
gree of the Meridian : 'When it is 3. In the third Form it will be x
uſed, 'tis moſt commonly ſcrewed
1/2 aab
to the Zenith. Its Uſe is for mea-
that is when it
ſuring of Altitudes, to find Ampli-
-V aa-b
tudes and Azimuths, and deſcribing is axtax+b=1, the two Roots or
Almicanters.
Values of x will be negative. But
QUADRANT ASTRONOMICAL. when it is xxt-ax-b=, the two
See Atronomical Quadrant.
Roots or Values of x' will be the
QUADRANT TRIANGLB. See one affirmative, and the other ne-
Triangular Quadrant.
gative ; that is, it will be * *
QUADRANTAL TRIANGLE, is
a Spheric Triangle, one of whoſe
Sides, (at leaſt,) is a Quadrant, and
a-Vaat
one Angle Right.
a x t bo, then will *
QUADRAT, or Line of SHA-
that
dows on a Quadrant, are only a
Line of natural Tangents to the
La-Vaat6
Arches of the Limb, and are placed of x being both affirmative. When
there in order to meaſure Altitudes it is xx-ax-b=0, then will x-
readily; for it will always be, As
at Taath
Radius to the Tangent of the Angle
one Root or
{a- Vaat6
of Altitude at the Place of Obſer-
vation; (that is, to the Parts of the Value of x being the one affirma,
Quadrats or Shadows cut by the tive, and the other negative. 4.
Laſtly,
>
2
fat vaatb. When it is xx-
on
QU A
QUA
བ་པ་རྩོམ་པ་
T
U..
[L
Laſtly, in the fourth Form it will be Hb and He, it will be, As the whole
x=V+ya+ Vaa£6.
Quadrantal Arch D B is to the Part
IB: So will the whole Right Line
If the laſt Term of a Quadratic D A be to the Part of it cut off
Equation be negative, its two Roots bA, or its Equal He.
will be real; and when the laſt 2. Wherefore any Arch of the
Term is affirmative, and the Quadrant as I B, or any Angle as
Square of the Co-efficient of the IAB, may by this Quadratrix
ſecond Term be leſs than the third be eaſily divided into three e-
Term, the two Roots will be ima- qual Parts, or any other Number
ginary.
at pleaſure, or according to any gi-
QUADRATRIX, (in Geometry,) ven Ratio, by only drawing the
is a Curve-Line thus generated. Let Radius Al, and then from the
there be a Radius of a Circle, as Point of the Quadratrix H, letting
AD, which imagine to move on fall the perpendicular He.
the Centre A down the Circumfe 3. The Baſe of the Quadratrix
AE, is a third Proportional to the
D
o Radius AD, and the Quadrant
BD.
4. If on the Baſe of the Quadra-
trix A E, a Quadrantal Arch be de-
ſcribed, it will be equal in Length
h
to DA, the side of the Square :
And conſequently the Semi-Circle
F
G
will be double ; and the Periphery
А*
Quadruple of DA.
e bb B
5. If AV the baſe of a Circle
inſcribed in the Quadratrix, GV
rence of the Quadrant DB, and at
the fame time let the side of the
Square CD move equally down-
G
wards, ſo that the Radius AD, and
the side of the Square CD may
come to the Line AB together. Or
let the Right Line DA, and the
Quadrantal Arch D B, be both di-
K
vided into a like Number of equal
Parts, as in this caſe, they are cach
into 8, and to the Diviſions of the
A
Quadrant, let as many Radii be
drawn from the Centre A, and thro' be 1, and the Arch of the Circle
the Diviſions in AD as many Pa- VK be called x, then will the Area
rallels to CD; for then if a Curve. BDVA = x,
X.- X3
Line be drawn neatly connecting 7613 *-, &c.
21; -
the Points of Interſection of theſe QUADRATURE Of any Figure
Radii and Parallels, it will be that in the Mathematics, is the find-
Line which is called the Quadratrix, ing a Square equal co the Area of
(as DE.)
it.
1. If through any Point, as H in This Doctrine is as far advanced
this Quadrartix, you draw a Radius by Sir Iſaac Newton in his Quadra-
AHI, and the two Perpendiculars ture of Curves, publifhed by Mr.
Jones,
to
V
A
QUA
QUI
Fones, as the Nature of the thing Quantity of Matter in, and the Ve-
will admit with any Elegance and locity of the Motion in that Body.
Perfpecuity ; nor is every Man ca QUARTERS, in a Clock, or Move-
pable, altho' perhaps a tolerablement, are little Bells which ſound
Mathematician, of perceiving the the Quarters, or other parts of an
Progreſs this great Man has made Hour.
in this difficult part of the Science. QUARTILE, is an Aſpect of the
QUADRATUREs of the Moon, are Planets, when they are 3. Sines or
the middle Parts of her Orbit, be- 90 Degrees diſtant from each other,
tween the Points of Conjunction and is marked thus .
and Oppoſition: And they are ſo QUAVER, is a Note in Muſic fo
called, becauſe a Line drawn from called. See the Words Notes and
the Earth to the Moon, is then at Time,
Right Angles, with one drawn from Queue D'ARONDE, a Term in
the Earth to the Sun.
Fortification, being what we call
QUADRILATERAL FIGURES, Swallow's Tail ; and fignifies a De-
are thoſe whoſe Sides are four Right tached or Out-work, whoſe Sides
Lines, and thoſe making four An- open towards the Head or Campaign,
gles; and they are either a Paral- or draw narrower or cloſer towards
lelogram, Trapezium, Rectangle, the Gorge. Of this kind are either
Square, Rhomboides, or Rhumbus. fingle or double Tenailles, and ſome
QUADRIPARTITION, is to di- Horn-Works, whoſe Sides are not
vide by four, or to take the fourth parallel, but are narrow at the
Part of any Number or Quantity. Gorge, and open at the Head, like
QUALITY, ſignifies in the gene- the Figure of a Swallow's Tail.
ral the Properties or Affections of When theſe Works are caſt up be-
any Being, whereby it affects our fore the Front of a Place, they are
Senſes fo and ſo, and acquires ſuch defective in this point, that they do
and ſuch a Demonſtration.
not fufficiently cover the Flanks of
1. Senſible Qualities, are ſuch as the oppoſite Baſtions ; but then they
are the more immediate Objects are very well flanked by the Place,
of our Senſes.
which covers all the Length of
3. Occult Qualities, were by the their Sides the better.
Ancients named ſuch, of which no QUINCUNX, is that Poſition, or
rational Solution in their way, or Aſpect, that the Planets are ſaid to
according to their Principles, could be in, when they are diſtant from
be given.
each other 150 Degrees, or 3 Sines,
QUANTITY, fignifies whatſo- and is marked thus, Vc, or 0.
ever is capable of any ſort of Etti QUINDECAGON, is a plain Fi-
mation or Menſuration, and which gure of 15 Sides and Angles, which
being compared with another thing if they are all equal to one another,
of the ſame nature, may be ſaid to is called a Regular Quindecagon.
be greater or leſs, equal or unequal The Side of a Regular Quindeca-
gon, ſo deſcribed, is equal in Power
1. The Quantity of Matter in to the half Difference between the
any Body, is its Meaſure ariſing Side of the Equilateral Triangle,
from the joint Conſideration of its and the Side of the Pentagon; and
Magnitude and Denſity.
alſo to the Difference of the Perpen-
2. The Quantity of Motion in diculars let fall on both sides, taken
any Body, is its Meaſure ariſing together.
from the joint Confideration of the QUINQUE ANGLED, in Geome.
try,
to it.
1
R A D
RAI
Z ZZ
R.
try, is a Figure conſiſting of five Fluxions, and you put i for x, and
Angles.
z for j in that Equation ; and if
QUINTILE, an Aſpect of the from the Equation that ariſes, you
Planets when they are 72 Degrees find the Relation between x, j, and
diſtant from one another, and is ; and at the ſame time put i for
noted thus, C. or O.
s, and z for y, as before." By the
QUINTUPLE, five-fold or five former Operation you will obtain
times as much as another thing. the Value of z; and if A be the
QUOTIENT, is that Number in Length of a Perpendicular to the
Divition, which ariſes by dividing Point of a Curve terminating at the
the Dividend by the Diviſor : And Extremity of y, and interſecting the
is called the Quotient, becauſe it an-
A3
ſwers to the Queſtion, how often Abſciſs; then will be the
one Number is contained in ano-
2 x 93
ther.
Length of the Radius of the Curva-
Quoin, the Workman's Term ture at the Extremity of y.
For
for an Angle or Corner:
Example, let the Parabolic Equa-
tion be propoſed, then will aš to
2b Å x - 2 jy=0; and putting i
for s, and z for y, it will be zb-
23 y =o. And again,
writing i and z for å and j, by the
a+2bx
ABANET. See Rabine.
firſt we ſhall have za
RABINET, a Sort of Ord.
pance, whoſe Diameter at the Bore and i-
So that the Ra.
is 1 Inches, Weight 300 Pounds,
Length 5 Foot, Load of a Pound, dius of the Curvature of this Curve
Shot ſomething more than an Inch at the Extremity of y will be
and a Quarter Diameter, and į a
Pound Weight.
and generally the Ra-
RADIANT POINT, is the Point 92 x 6
from which the Divergent Rays dius of the Curvature of Conic Sec-
proceed.
tions is as A3. See this Subject
RADIATION, ſignifies the caſting well handled in Sir Iſaac Newton's
forth of Beams, or Rays of Light; Fluxions, and the Marquis de l'Hof
and in Optics, it is conſidered as pital's Infinimens Petits.
threefold, viz. Direet, Reflested, and RAINBOW, or Iris. The Pria
RefraEted. See Ray.
mary Iris is only the Sun's Image,
Radius, in Geometry, is the reflected from the Concave Surfaces
Semi-Diameter, or half the Diame- of an innumerable Quantity of ſmall
ter of a Circle.
ſpherical Drops of falling Rain,
RADIUS of the Curvature of a with this neceffary Circumſtance;
Curve, is the Radius of a Circle that thoſe Rays, which fall on the
that has the ſame Curvature in a Drops, parallel to each other, ſhould
given Point of the Curve, that the not after one Reflection, and two
Curve has in that Point.
Refractions, vix, at going into the
If any Equation is propoſed ex- Drop, and coming out again, be dif-
preſſing the Relation of the Abſciſs perſed, or made to diverge, but
à, and correſpondent Ordinate y, come back again, to the Eye, paral-
and the Equation be thrown into lel to each other.
Some
RRABINET,
2 Y
6--22
у
A3
ZZ
20.
Ř A Í
RAM
Some of the Ancients, as we find the Refractions, as Deſcartes did.
in Ariſtotle's Meteors, knew, that The Doctor ſhews the way; as alſo
the Rainbow was cauſed by the Re- ſome other things (in n. 14. 15, 16 y
fraction of the Sun's Light in Drops regarding the Rainbow, worth while
of falling Rain. But it was more to be peruſed, and agreeable to the
fully diſcovered and explained by elegant Genius's of thoſe two great
Antonius de Dominis, in his Book de Men.
Radiis Viſits & Lucis, publiſhed at Concerning the Rainbow, fee A-
Venice by his friend Bartolus, Anno riſtotle's Meteors, lib. 3. cap. 4, 5, 6,
1611, and written above 20 Years 7.-- Dr. Halley's Diſcourſe in the
before; wherein he ſhews how the Philoſophical Tranſactions, n. 267. -
interior Bow is made in round Drops Mr. s'Graveſande's Inſtitutions of the
of Rain by two Refractions of the Newtonian Philoſophy, lib. 3. cap.
Sun's Light, and one Reflection be-
tween them; and the exterior, by RAKED TABLE, a Term in Ar-
two Refractions and two Sorts of chitecture.
chitecture. See Table.
Reflections between them in each RAMMER, is a Staff with a round
Drop of Water, and proves his Ex- Piece of Wood at one end, in order
plications by Experiments made with to drive home the Powder to the
à Phial full of Water, and with Breech of the great Gun; as alſo
Globes of Glaſs filled with Water, the Shot and the Wadding, which
and placed in the Sun to make the keeps the Shot from rolling out.
Colours of the two Bows appear in At the other end of theſe Rammers '
them. The ſame Explication has are uſually rolled in a certain Piece
been purſued by Deſcartes, in his of Sheep's Skin fitted to the Bore .
Meteors, who mended that of the of the Piece, in order to clear her:
exterior Bow; and he indeed was after ſhe has been diſcharged; and
the firſt, that by applying Mathe- this is called Spunging the Piece.
matics towards the Inveſtigation of RAMPART, in Fortification, is
this ſurprizing Appearance, ever the Maſs of Earth, which is raiſed
gave a tolerable Theory of the Rain- about the Body of any Place, to co-
bow. But as they did not under ver it from great Shot, and confifts
ſtand the true Origin of Colours, of ſeveral Ballions and Curtains ;
Sir Iſaac Newton's Explication in his having its Parapet, Platform, inte-
Optics at Prop. 9. is the beſt by rior and exterior Talus and Berme ;,
much, where he makes the Breadth as alſo ſometimes a Stone-Wall, and
of the interior Iris to be nearly then they ſay it is lined. The Sol-
2º. 15', that of the exterior 30.40', diers continually keep Guard here,
their Diſtance 89.25', the greatelt and Pieces of Artillery are planted
Semi-diameter of the interior Iris for the Defence of the place.
42° 17', and the leaſt of the exte The Height of the Ramparts
rior 50°. 42', when their Colours muſt exceed three Fathom, as being
appear ſtrong and perfect.
fufficient to cover the Houſes from
Dr. Barrow, in his Lectiones Op- the Batteries of the Cannon : Nei .
ticæ, at Let. 12. n. 14. tells us, that ther ought its Thickneſs to be a-
a Friend of his (by whom we are to bove ten or twelve, unleſs more
underſtand Sir Iſaac Newton) com- Earth be taken out of the Ditch,
municated to him a way of deter-- than can be otherwiſ beſtowed.
mining the Angle of the Rainbow The Ramparts of Half-Moons are
(which was hinted to Newton by the better for being low, that the
Slufius) without making a Table of ſmall Fire of the Defendants may
Ff
the
R AR
R A T
the better reach the bottom of the RASANT Line of Defence. See
Ditch ; but yet it muſt be ſo high, Line of Defence Razant.
as to be commanded by the Cover'd. RASH. See Ratch.
way.
Ratch, is a fort of a Wheel of
RANDOM-SHOT, is a Shot made twelve large Fangs that runs concen-
when the Muzzel of a Gun is raiſed trical to the Dial-Wheel, and ſerves
above the horizontal Line, and is to lift up the Dentes every Hour,
not deſigned to ſhoot directly or and make the Clock ſtrike ; and
point-blank. The utmoſt Random are by ſome called Raſ.
of any Piece, is about ten times as RATCHET, in a Watch, are the
far as the Bullet will go point- ſmall Teeth at the bottom of the
blank.
Fuſee or Barrel, that ſtop it in wind-
The Diſtance of the Random is ing up.
reckoned from the Platform to the RATIO. When two Quantities
Place where the Ball firſt grazes, are compared one with another, in
RANGE, a Term in Gunnery, ſig- reſpect of their Greatneſs or Small-
nifying the Line a Shot goes in from neſs, that Compariſon is called Ratio.
the Mouth of the Piece: If the Bul Euclid, in his fifth Element, ſays,
let
goes in a Line parallel to the that Ratio is the mutual Habitude of
Horizon, that is called the Right or two Magnitudes of the ſame kind
Level Range ; if the Gun be mount- each to the other, according to
ed to 45 Degrees, then will the Ball Quantity. But I muſt confeſs this
have the higheſt or utinoſt Range ; is ſomewhat obſcure ; for the word
and ſo proportionably all others be. Habitude does not in my opinion)
tween 60 Degrees and 45, are cal- readily convey an Idea of Ratio ;
led the Intermediate Ranges. the Meaning of this Word being al-
1. If two Elevations are taken at moſt as obſcure as the thing defin'd
equal Diſtances from 45 Degrees, one by help of it. However, ſee Euclid
above, and the other below it, the defended concerning this Matter
Ranges ſhall be equal.
by Dr. Barrow, in his Mathemati.
2. The greateſt Altitude of a per- cal Leclures, wherein he explains
pendicular Projection, is equal to and clears it up, with a wonderful
half the greateit Range.
deal of Skill and profound Learn-
3. When Projectiles are thrown ing.
into the Air, the greater Range is The greater A of two unequal
at the Elevation of 44 Degrees and Magnitudes A and B, has a greater
a half; the lower Ranges go far- Ratio to the ſame third Magnitude
ther than the upper correſpondent C; and the ſame third Magnitude
Ranges, and the greateſt Height of C has a greater Ratio to the leffer
the perpendicular Projection is more B than to the greater A : for the
than half the greateſt Range. All Ratio of A to C being always ex-
theſe Irregularities are occaſioned by
A
the Reſiſtance of the Medium. prefled thus
Ć,
and of B to C thus
RARE BODIES, are ſuch as have
more Space, or take up more room
R
А
B
in proportion to their Matter, than T; it will be į greater than ē.
other Bodies do.
And the Ratio of C to A being
RAREFACTION of any Natural
с
Body, is when it takes up more Di But that if C to B equal to
menſion, or a larger Space than it
A
had before.
C
B
5
.
1
1
1
1
I
mi +7&c.
to ,
Ř AT
RAT
с
C
B ; it will be à leſs than
B
at
Both theſe follow from the Nature
of Fractions.
et
Of Magnitudes having Ratio to
the fame Magnitude, that which has
stai
the greater Ratio is the greater
Magnitude, and that Magnitude to
kt
which the fame Magnitude bears
a greater Ratio, is the leſſer Mag-
nitude.
The firſt and moſt remote Ratio
It may not be amiſs to ſet down
here the following uſeful Problem, from the given one, being the
viz.
ſecond nearer approaching is c+
To find a Series of Numerical Ra-
tio's expreſſed in leſer Numbers, con the third ſtill nearer is of
ſtantly approaching to a given Nume-
rical Ratio expreſed in greater Numa
the fourth nearer yet, is
bers, whoſe Terms are prime to each
+1,
other.
g
Let a be the lefſer Term, and bot-
the greater of the given Ratio ;
et
now proceed with theſe two Num-
and the fifth fit
bers in the ſame manner as when
sti
you want to find their greateſt
common Meaſure, by conſtantly di- nearet, is
viding the greater Term by the
leſs, and the Diviſor by the Remain- ct
der, thus
et-
I
e
I
i
st
i
Åt-
&c.
7
and ſo on.
When theſe Fractions
are each reduced on more fimple
mt,&c.
b.
f
ones.
a) b(c+
d
dý a letá
50(8+
b) flet
+ +
Forct (=) is=cts
1) m,&.
å (=e+ ) =
But (=8+)
î(=i+1)
a
where C, e, g, k, m, &c. are the But
whole Numbers ariſing from the ſe-
; , ,
Eg €the ſeveral Remainders
3
then
6
be exactly =
will
a
-=
is = 8+
But To
Ff 2
is
RAT
RA T
1
A
F
= +
isittBut +(--+)
=m1+
is = m+ 1, c. therefore
*&is
eti
1
6
I
iet
1
22
+, &c.
11
1
1
kt
*
I
An Example in Numbers. Let it to 314159, being one of the ap-
be required to find a Series of Ra proximating Ratio's of the Dia-
tio's in leſſer Numbers conſtantly meter of a Circle to the Circum-
approaching to the Ratio of 100000 ference.
100000) 314159 (3=0
d=14159) 100000 (7=
f=887) 14159 (1558
h=854) 887 (1=k
b33) 854 (15=m
29) 33 (imp
14) 29 (739
1) 4 (4=
that is 100% will be
, or that of 113 to 355 nearer
ſtill; the fifth will be
3t-
3t
7-
at
15+
16
1+
{, or that of 1702 to 5347,
nearer yet; the ſixth will be
15t
34-
it
7-
So that the firſt and moſt remote
16+
Rasio from the Truth will be , or
=fit, or that of 1815 to 5702,
that of 1 to 3; the ſecond will be ſtill nearer ;. and ſo on till you get
3+1=2, or that of 7 to 22 be- ** the given Ratio.
ing nearer; the third will be 34
Lelt this way of adding up the
333
ſeveral Diagonal Fractions may ap-
106'
or that of 106 to pear difficult to ſome, they may
333 still nearer; the fourth will bess uſe the following Rule. After you
have proceded with the given Ra-
3to
tio expreſied Fraction-wile, as if in
queſt of the greatest common Mea-
fure of the Fraction, the firit Quo-
tient divided by Unity will be the
157-
fractional
1
1
I
IS
1
1
I
I
1
1
1
1
I
it
1
1
ift Quot.
t
4th.
41 (64) 1
7th.
R A T
R A T
fractional Expreſſion of the firſt and you a third fractional Expreſſion of
moſt remote (from the Truth) ap- the Ratio. So alſo having the few
proximating Ratio, whoſe Terms cond and third fractional Expreſs
being each multiplied by the ſecond fions ; to find a fourth, multiply
Quotient, and Unity being added to the Terms of the third Fraction by
the Numerator, gives the ſecond the fourth Quotient, adding in the
fractional Expreſſion of the Ratio, Terms of the ſecond Fraction, and
approaching to the given one ; ever this gives a fractional Expreſſion for
after having two fractional Expreſ- a fourth Ratio ; and thus you may
fions found, as ſuppoſe the firſt and proceed till you have got the laſt
ſecond to find the third; multiply fractional Expreſlion of the given
the Terms of the ſecond Fraction by Ratio.
the third Quotient, and adding to For Example, let the Ratio givep
the Terms of the fame, the firſt frac- be as 5978 to 97435, or 837
tional Expreſſion, and this will give
5978) 97435 (16
1787 (5978) 3
2d.
617) 1787 (2
3d.
153 (617)
64) 553 (8
sth.
6th.
23) 41 (1
18) 23 (1 8th.
5) 18 (3 gth.
3) 5 (1. Toth.
2) 3 (1 : uth.
1) 2 (t. 12th,
on the firſt Fraction.
Dr. Wallis in a little Piece at the
End of Horrox's Aftronomy, treats
16* 3+1= 4, the ſecond.
of the Nature and Solution of this
Problem, with a great deal of tedi.
49 x 2 +16
ous Preparation and unneceſſary Cir.
the third.. cumlocution, almoſt enough to di
3 x 2 +I
courage a Perſon from attempting to
114 X 1 +49
apprehend what he has a mind to
1163 the fourth, be at. The great Mr. Huygens too
7*1 + 3
has given a Solution, and the Reaſon
163 x 8 + 114
thereof; but after a much morter and
=.
1118 the fifth,
more natural way. So alſo has Mr,
10 ~ 8 +7
Cotes at the Beginning of his Har-
mon. Menſur. But from more ina
tricate and myſtical Principles, uſing
87 *+10
therein unintelligible and unnej ary
ways of Expreſion; ſuch, as the Re.
1581 x 1 + 1418
20 the fe- tio of 1 to o; and of 2 to o The
97 X1 + 87
Problem is of much uſe, in expreſing a
yenth; and ſo on.
Ratio in ſmall Numbers, that mali
be near enough in practice, to any gi-
I * 3
II4
7
1418 x1 + 163 = "$$* the fixth.
Ff 3
R A Y
REC
A
open Ratio in great Numbers ; fuch Corpuſcles of Matter, which con-
as that of the Diameter of a Circle tinually iſſuing out of the Sun, do
to the Circumference ; of the Square thruſt on one another all round in
of the Diameter to the Area ; of the phyſical ſhort Lines ; and that this
Cube of the Diameter of a Sphere to is the right Opinion, many Expe-
the Solidity, and many other uſeful riments do evince, particularly Sir
Ratios, too many to mention here, or Ifaac Newton's about Light and Co-
even for me to think of.
lours; or elſe, as the Carteſians aſ-
RATIONAL HORIZON. See Ho- ſert, they are made by the Action
rizon.
of the Luminary on the contiguous
RATIONAL QUANTITies. Any Æther and Air, and ſo are propa-
Quantity being propoſed, for gated every way in ſtraight Lines,
which we may always put i, and through the Pores of the Medium.
which Euclid ( Book X.) calls Ra Rays CONVERGENT. See Con-
tional, there may be infinite others, verging Rays.
which are commenſurable, or in RAYS DIVERGENT, See Di-
commenſurable to it; and that verging Rays.
either ſimple, or in Power. Now, REACH, is the Diſtance between
all ſuch as are commenſurable any any two Points of Land, that lie in
how to the given Quantity, he calls a Right Line one from another.
Rational Quantities, and all the o RECESSION of the Equinoxes, is
thers Irrational.
the going back of the Equinoctial
RAVELIN, in Fortification, is a Points every Year about fifty Se-
Small Triangular Work compofed conds.
only of two Faces, which make a RECIPROCAL FIGURES, in
ſaliant Angle, without any Flanks. Geometry, are ſuch as have the An-
It is generally raiſed before the tecedents and Conſequents of the
Curtains or Counterſcarp, and com- Ratio in both Figures.
monly called a Half-Moon by the RECIPROCAL PROPORTION, is
Şoldiers.
when, in four Numbers, the fourth
A Ravelin is like the Point of a is leſſer than the ſecond, by ſo much
Baſtion with the Flanks cut off. The as the third is greater than the firſt,
Reaſon of its being placed before a and vice verſa.
Curtain, is to cover the oppoſite RECLINATION of a Plane, is the
Flanks of the two next Baſtions. Quantity of Degrees which any
?Tis uſed alſo to cover a Bridge, or Plane, on which a Dial is ſuppoſed
a Gate; and 'tis always placed to be drawn, lies or falls backwards
without the Moat.
from the truly upright or vertical
What the Engineers call a Ra. Plane.
velin, the Soldiers generally call a RECLINING, in Dialling. The
Half Moon ; which ſee,
Plane that leans from you when you
RAY of Refra£tion, or Broken Ray, ſtand before it, is ſaid to be a Re-
is a Right Line, whereby the Ray clining Plane.
of Incidence changeth its Rectitude, RECLINING DECLINING DIALS.
ar is broken in traverfing the ſe- See Declining Reclining Dials.
cond Medium, whether it be thicker RECTANGLE, in Arithmetic, is
the ſame with Product; which ſee.
RAYS, or BEAMS af the Sun, or RECTANGLES, in Geometry, are
Rays of Light, are either accord. Parallelograms, whoſe Sides are
ing to the Atomical Hypotheſis, unequal; but Angles right Their
thoſe yery minute Particles, or Area is found by multiplying the
two
3
or thinner.
REC
RED
two unequal Sides one into another, faſten'd together in their Centres,
for then the Product is the ſuperfi- that they repreſent two Compaſſes,
cial Content or Area,
one fixed, the other moveable ; each
RECTANGLED TRIANGLE ; the of them divided into the 32 Points
ſame with Right-angled Triangle. of the Compaſs, and 360 Degrees,
RECTANGULAR, or RIGHT- and number'd both ways, both from
ANGLED, is ſpoken of a plain Fin the North and the South, ending
gure in Geometry, when one or more at the Eaſt and Weſt, in 90 De-
of its Angles are right: Of Solids, grees.
'tis ſpoken in reſpect of their Situa The fixed Compaſs repreſents the
tion ; for, if their Axis be perpen- Horizon, in which the North, and,
dicular to the Plane of the Horizon, all the other Points of the Compaſs
they are therefore rectangular, as are fixed and immoveable.
right Cones, Cylinder, & c.
The Moveable Compaſs repre-
RecTANGULAR SECTION of a ſents the Mariner's Compaſs, in
CONE; by this the ancient Geome- which the North, and all the other
ters always meant a Parabola, which Points are liable to Variation.
Conic Section, before, Apollonius, was In the Centre of the moveable
only conſidered in a Cone, whoſe Compaſs is fäſten'd a Silk Thread,
Section by the Axis would be a Tri- long enough to reach the outſide of
angle, right-angled at the Vertex. the fixed Compaſs; but, if the In-
And hence it was, that Archimedes ſtrument be made of Wood, there
entitled his Book of the Quadra- is an Index inſtead of the Thread.
ture of the Parabola, (as 'tis now cal. Its Uſe is to find the Variation of
led) by the Name of Rectanguli Coni the Compaſs, to rectify the Courſe
Sectio.
at Sea, having the Amplitude or
Recripy, is a Word uſed in the Azimuth given.
· Deſcription and Uſe of the Globe, RECTIFYING of Curves, in Ma-
or Sphere. For the firſt thing to thematics, is to find a ſtraight Line,
be done before any Problems can equal to a curved one.
be wrought on the Globe, is to rec The firſt who gave the Rectifica-
tify it ; that is, to bring the Sun's tion of any Curve was Mr. Neal, a
Place in the Ecliptic on the Globe, Son of Sir Paul Neal, as we find at
to the graduated Side of the Braſs the End of Dr. Wallis's Treatiſe of
Meridian, to elevate the Pole above the Cifoid; wherein the Doctor
the Horizon, as much as is the La- fays, that Mr. Neal's Rectification
titude of the Place, and to fit the of the Curve of the ſemi-cubical Pa-
Hour-Index exactly to Twelve at rabola, was publiſhed in July or Au-
Noon, ſcrewing alſo the Quadrant guſt, 1657. Two Years after, viz.
of Altitude, (if there be occafion) to Anno 1659, Van Haureat, in Hol-
the Zenith.
land, gave the Rectification of this
All this is comprehended under Curve; as may be ſeen in Schouten's
the Word Rectify the Globe: And Commentary upon Deſcartes's Geox
when this is done, the Celeſtial metry,
Globe repreſents the true Poiture of RECTILINE AL, or RIGHT,
the Heavens, for the Noon of that LINED, in Geometry, is ſpoken of
Day it is rectified to.
ſuch Figures as have their Extremi-
RECTIFIER, in Navigation, is ties all Right-Lines.
an Inſtrument conſiſting of two Parts, REDENT, in Fortification, is a
which are two Circles, either laid Work made in form of the Teeth,
upon, or let into the other, and ſo of a Star, with ſaliant and re-entring
Angles,
Ff4
REF
R E F
1
Angles, to the end that one Part meeting of another Body, which it
may defend another. Theſe ſort of cannot penetrate. Thus the mate-
Works are uſually erected on that rial Rays of Light are reflected va-
fide of a place which looks towards riouſly from ſuch Bodies as they can-
a Marſh, or River.
not paſs through.
REDOUBT, in Fortification, is a REFLECTION of the Rays of
ſmall Fort of a ſquare Figure, hav- Light .Sir Iſaac Newton finding, by
ing no Defence but in the Front ; Experiment, that Light was an he-
its Uſe being to maintain the Lines terogeneous Body, confiſting of a
of Circumvallation, Contravallation, Mixture of differently refrangible
and Approach.
Rays; and conſequently concluding
In marſhy Grounds, theſe Re- that no further Improvement could
doubts are often made of Maſon's well be made in optical Inſtruments
Work, for the security of the Neigh- in the dioptric way, he took Re-
bourhood. Their Face conſists of flections into conſideration, and
from ten to fifteen Fathom, the tells us, that by their Help, Optic
Ditch round about being from eight Inftruments might be brought to
to nine Foot broad and deep, and any Degree of Perfection ; if we
their Parapets having the ſame could but find a reflecting Subſtance,
Thickneſs.
which would poliſh as finely as
REDUCTION, in Aſtronomy, is Glaſs, reflect as much Light as Glaſs
the Difference between the Argu- tranſmits, and be formed into a pa-
ment of Inclination, and the eccen- rabolical Figure.
trical Longitude ; that is to ſay, the An Experiment of which, he
Difference of the two Arches of made in the kind of a catoptric Te.
the Orbit, and the Ecliptic, inter- leſcope, and by which, tho' not above
cepted between the Node and the two Foot long, he could (he faith)
Circle of Inclination.
diſcern the Jovial Satellites, and
REDUCTION, in Arithmetic, is the Phaſes of Venus. Philos. Tranſ. .
the manner of converting or bring. N° 18.
ing one Species of Money,Weight, REFLECTED RAY, or Ray of Re-
or Meaſure, into another; that from flection, is that whereby the Reflec-
a greater to a leſs, being performd tion is made upon the Surface of a
by Multiplication; but from a leſs reflecting Body.
to a greater, by Diviſion.
REFLECTING, or Reflexive Dials,
REDUCTION of Equations, in Al- are made by a litýle Piece of Look-
gebra, is the clearing of them from ing-glaſs-Plate, duly placed, which
all fuperfluous Quantities, and the reflects the Sun's Rays to the top of
ſeparating of the known Quantities a Ceiling, &c. where the Dial is
from the unknown, to the end that drawn. This Glaſs ſhould be as
at length every reſpective Equation thin as can well be ground.
may remain in the feweſt and fim REFLECTING TELESCOPES.
pleit Terms; and ſo diſpoſed, that See Teleſcopes.
the known Quantities may poſſeſs REFLECTION of the Moon, is (ac-
one Part thereof, and the unknown cording to Bullialdus) her third In-
the other.
equality of Motion: This Tyche
Re-ENTRING ANGLE, a Term calls by the Name of her Variation.
in Fortification. See Angle.
Which ſee,
REFLECTION, in general, is the Reflux of the Sea, is the Ebbing
Regreſs or Return that happens to of the Water off from the Shore;
* moving Body, becauſe of the as its coming on upon it, or lide
of
R E G.
REG
of Flood, is called the Flux of the REGION ÆTHERBAL, in Cof-
Sea. See Tide.
mography, is the vaſt Extent of the
REFRACTED ANGLE, in Optics, Univerſe; wherein are comprized
is the Angle contained between the all the Heavens and Cæleftial Bo-
refracted Ray and the Perpendi- dies.
cular.
REGIONELEMENTARY, accord-
REFRACTION in general, is the ing to the Ariſtotelians, is a Sphere
Incurvation or Change of Determi- terminated by the Concavity of the
nation in the Body moved, which Moon's Orb, comprehending the
happens to it whilft it enters or pe- Earth's Atmoſphere.
netrates any Medium.
REGULAR BODY, is a Solid,
In Dioptrics, it is the Variation of whoſe Surface is compoſed of regu-
a Ray of Light, from that Right lar and equal Figures ; whoſe tolid
Line which it would have paſſed on Angles are all equal. Such as the
in, had not the Denſity of the Tetrahedron, Hexahedron, Oeto-
Medium turned it aſide.
hedron, Dodechahedron, and Icoſa-
REFRACTIONASTRONOMICAL, hedron. There can be no more re-
is that which the Atmoſphere pro- gular Bodies beſides theſe.
duceth, whereby a Star appears more REGULAR FIGUREs, in Geome-
elevated above the Horizon, than try, are ſuch whoſe Sides, and
really it is.
conſequently their Angles, are all
REFRACTION HORIZONTAL, equal to one another. Whence all
is that which cauſeth the Sun or regular multilateral Planes are cal-
Moon to appear on the Edge of the led Regular Polygons.
Horizon, when they are as yet
The Area of ſuch a Figure is
ſomewhat below it.
ſpeedily found by multiplying a
REFRACTION from the Perpen- Perpendicular let fall from the Cen-
dicular, is when a Ray falling in- tre of the inſcribed Circle to any
clined from a thicker Medium intoa Side by half that Side ; and then
thinner, in bieaking departs further that Product by the Number of
from that Perpendicular. And the sides of the Polygon.
REFRACTION to the Perpendicu REGULAR FORTIFICATION.
lar, is when it falls from a thinner See Fortification.
into a thicker, and ſo comes nearer REGULAR POLYGON. The
the Perpendicular.
Truth of the general Method of
REFRANGIBLE, is whatever is Sturmius and Renaldinus for in-
capable of being refracted. ſcribing any regular Polygon in a
Regel, or RIGEL, a fixed Star Circle may be trigonometrically exa-
of the firſt Magnitude in Orion's mined thus : Suppoſe ACGà Cir-
Left-foot, its Longitude is 72 De- cle, D the Centre, AC the Dia-
grees, 19 Minutes, Latitude 30º. meter, ABC an equilateral Tri-
jo',
angle deſcribed upon the Diameter,
REGION, is taken for our Hemi- E the ſecond Point of Diviſion of
ſphere, or the Space within the four the Diameter divided into any Num-
Cardinal Points of the Heavens, or ber of equal Parts, DB perpendicu-
of the Air, &c.
lar to AC, and the Points D, F,
In Geography, it ſignifies a large joined.
Extent of Land inhabited by many 1. Now, becauſe the Semi-
People of the fame Nation, and in- Diameter DC, and the whole Dia
closed within certain Limits or aneter B C are given ; B D may
Bounds.
be had, which is equal to the ſquare
Root
1
REG
R E G
Root of the Diameter AC (=BC,) D'É, Log. 2.000000 : So is the Ra-
fay, As DB, Log. 2.937532 is to
DCą.
2. Again, becauſe the Number of dius 10.000000 to the Logar. Tan-
equal Parts, the Diameter is divided gent of the Angle DBE, viz. 69.
into, for any given Polygon is alſo 35'.
given ; the Line CE, which is e-
3. Again, As DF, Log.2.699404,
qual to two of thoſe Parts will be is to DB, Log. 2.237532 : So is the
, , to
B.
Log. Sine of the Angle DFE,
which will be 11°. 26'. Now, the
Sum of theſe Angles taken from
GO
180°, will be the Angle FDB
161° 59'. From which, if 90° be
taken, there remains 71° 59', for
the Quantity of the Angle FDC
at the Centre ; but this wants i' on-
D
А
E
ly of being 72°, the true Angle :
and the greater the Number of Sides
is, the greater will the Error be
fo that if the Number be 20, the
Error will be half a Degree and
TOEICULAQUI:
;
more.
F
REGULATOR, is a ſmall Spring
belonging to the Balance in the
given, and conſequently the Part Pocket-Watches.
be
RELAIS, a French Term in For-
right-angled Triangle BDE, there tification ; the ſame with Berme.
are given the sides B D, and DE, RELATION INHARMONICAL, a
to find the Angle DBE, in fay- Term in Muſical Compoſition, figo
ing, As DB is to DE: So is the nifying a harſh Reflection of fat
Radius to the Tangent of DBE. againſt ſharp, in a croſs Form, viz.
Moreover, in the Triangle DBE, when ſome harſh and diſpleaſing
becauſe the sides BD, and DF, and Diſcord is produced, in comparing
the Angle DBE, being now found, the preſent Note of another Part.
are given, the Angle BFD may be
RELATIVE GRAVITY,
the
found : in faying, As DF is to DB: fame with Specific. Which fee. ,
So is the Sine of the Angle DBE RESIDUAL FIGURE, in Geome-
to the Sine of the Angle BFD, try, fignifies the remaining Figure
which being found, add it to the after the Subtraction of a leſler from
Angle D BF, and ſubtract the ſame
a greater.
from 180 Degrees; the Remainder RESIDUAL Root, in Mathema-
is the Angle BDF: then ſubtract- tics, is one compoſed of two Parts
ing the Angle BDE, which is a or Members, only connected toge-
right Angle, from the Angle BDF, ther with the Sign-: Thus amb,
and there remains the Angle at the or 5-3, is a Reſidual Root; and
Centrę CDF. For Example, to is ſo called, becauſe its true Value
examine this for a Pentagon, let us is no more than its Reſidue or Difa
fuppoſe the Diameter AC, or C B, ference between the Parts a and
to be 1000, then the Log. BD b.
2.9375 32. Again, CE = 400, RESISTANCE of a Medium, is
and conſequently DE=100. Now the Oppoſition againſt, or Hindrance
of
1.
RES
RES
of the Motion of any Body moving the tranſverſe Axis, and Latus
in a Fluid, as in the Air, the Water, Re&tum, is to the tranſverſe Axis ::
the Æther, &c. and this, together So is the Square of the Latus Rec-
with the Gravity of Bodies, is the tum to the Square of the Diameter
Cauſe of the Ceſſation of the Mo- of a certain Circle, in which Circle
tion of Projectiles, &c. This Re- apply a Tangent 'equal to half the
ſiſtance in Mediums, which are very the Baſe of the Hyperbola or El-
denſe and rigorous, ſo that Bodies lipſis.
can there move but yery ſlowly, is 7. Then ſay again, As the Sum
nearly as the Velocity of the moving and Difference of the Axis, and a-
Body : But in a Medium free from gain, as the Sum (or Difference) of
all ſuch Rigour, as the Squares of the Axis and Parameter, is to the
the Velocities. Neut. Princip. p. Axis :: So is the circular Arch cor-
245
reſponding to the aforeſaid Tangent,
If an Iſoſceles Triangle be to another Arch. This done, the
moved in a Fluid according to the Reſiſtances will be as the Tangent
Direction of a Line which is normal to the Sum (or Difference) of the
to its Baſe ; firſt with the Vertex Right Line thus found, in that Arch
foremoſt, and then with its Baſe, the laſt mentioned.
Refittances will be as the Sides.
8. In general, the Reſiſtances of
2. The Reſiſtance of a Square any Figure whatſoever, going now
moved according to the Direction with its Bafe foremoſt, and then
of its fide, and of its Diagonal, is with its Vertex, are as the Figures
as the Diagonal to the Side.
of the Baſe is to the Sum of all the
3. The Reſiſtance of a Circular Cubes of the Elementa of the Baſe
Segment, (leſs than a Semi-circle,) divided by the Squares of the Ele-
carried in a Direction perpendicular menta of the Curve-Line.
to its Baſe, when it goes with the Baſe RESOLVEND, a Term in the Ex-
foremoſt, and when with its Vertex traction of the Square and Cube-
foremoſt, (the ſame Direction and Roots, &c. fignifying that Number
Celerity continuing, which is all which ariſes from augmenting the
along ſuppoſed,) is as the Square of Remainder after Subtraction, by
the Diameter to the ſame, leſs ; drawing down the next Square,
of the Square of the Baſe of the Cube, &c. and writing it after the
Segment.
ſaid Remainder.
4. Hence the Reſiſtance of a Se ResOLUTION, in Mathematics,
mi-Circle, when its Baſe, and when is a Method of Invention, whereby
its Vertex goes foremoſt, are to one the Truth or Falſhood of a Propo-
another in a ſeſquialteral Ratio. ficion, or its Poflibility or Impoſſi-
5. A Parabola moving in the Di- bility is diſcovered, in an Order con-
rection of its Axis, with its Bafe, trary to that of Syntheſis or Com-
and then its Vertex foremoſt, hath poſition : For in this Analytical
its Reſiſtance as the Tangent to an Method, the Propoſition is propoſed
Arch of a Circle, whoſe Diameter is as already known, granted, or done;
equal to half the Baſe of the Para- and then the Conſequences thence
bola.
deducible are examived, till at laſt
6, The Reſiſtance of an Hyper- you come to ſome known Truth or
bola, or Semi-Ellipfis, when the Fallhood, or Impoſibility, whereaf
Baſe and when the Vertex goes fore- that which was propoſed is a necef-
moſt, may be thus computed : Let fary Conſequence, and from thence
it be, as the Sum (Difference) of juftly conclude the Truth or Impof-
fibility
ay
R E V
RHO
a
fibility of the Propofition; which if rea given to be cut off from any
true, may then be demonſtrated in Point in the Axis, &c.
a ſynthetical Method. This Me-
REVOLUTION : In Geometry,
thod of Reſolution confifts more in the Motion of any Figure round a
the Judgment, Penetration, and Rea. fixed Line, (which is called there-
dineſs of the Enquirer or Artiſt, fore its Axis,) is called the Revolu-
than in any particular Rules ; tho' tion of that Figure; and the Figure
thoſe of Algebra are of neceſſary fo moving is ſaid to revolve. Thus
ufe, and a good Treaſure of Geo- a right-angled Triangle revolving
metry in his Head will be of great round one of its Legs, as an Axis,
advantage to him in all manner of generates by that Revolution
Inveſtigations.
Cone. And to inſtance in a caſe
Resi, (in Muſic ) See Pauſe. very wonderful; the Body called by
RESTITUTION; the returning of TORRICELLIUS Hyperbolicum Á.
elaſtic Bodies forcibly bent to their cutum, tho' itſelf, (as he demon-
natural State, is called the Motion of ftrates,) be finite, is yet formed by
Reftitution.
the Revolution of an infinite Area.
RETIRED FLANK. See Flank.
RHOMB SOLID, is two equal
RETRENCHMENT, in Fortifica- and right Cones joined together at
tion, is a Ditch bordered with its their Baſes.
Parapet, and ſecured with Gabions RHOMBOIDES, a Figure in Geo-
or Bavins laden with Earth. It is metry. See Quadrilateral Figures.
ſometimes taken for a fimple Re RHOMBUS. See Quadrilateral
tirade in part of the Rampart, Figures.
when the Enemy is ſo far advanced, RHOMBS. See Rhumbs.
that he is no longer to be reſifted,
or beaten from his Poft.
The following Propoſitions being
RETROCESSION, of the Equinoxes, of great uſe in the Theory of Na-
is the annual going backward of the vigation, and not to be found every
Equinoctial Points about 50 Seconds. where, I thought it would not be
See Equinoxes.
amiſs to inſert them with their De-
RETROGRADE in Aftronomy, is monſtrations here.
uſually appropriated to the Planets,
PROP. I.
when by their proper Motion in the
Zodiac, they move backward or PD, &c. be at a ſmall diſtance
If the Meridians PA, PB, PC,
contrary to the Succeſſion of the
Signs : As from the ſecond Degree
P
of Aries to the firſt, c.
But this Retrogradation is only
apparent, and occafioned by the
Obſerver's Eye being placed on the
Earth: For to an Eye at the Sun,
the Planet will appear always di-
rect, and never either ſtationary or
N
G
retrograde.
F
Reversed TALON. See Talon.
ReveRSION of Series, in Alge-
E
bra, is a Method to find a Number
from its Logarithm, being given ;
or the Sine from its Arch: The
B
c
Ordinate of an Ellipſis, from an A.
3
from
M
L
H H
1
A
K
I
1
2
2
RHO
RID
from each other, then the Rhumb- So is AI+IH+GH, that is, AG,
Line AIHG is divided into equal to AB+IK+HF.
Parts AI, IH, GH, by Parallels,
PRO P. IV.
LE, MF, NG, &c. at the equal
Diſtances, BI, HK, GF from each
The Difference of Latitude DG is
to the Sum of AB+IK +HF, &c.
other.
This is plain, becauſe the Angles the Courſe PAG, or AIB.
as the Radius is to the Tangent of
B, H, F, being right ones, and PAG
=,
From the Demonſtration of the
BC,CD, being very ſmall, the Trian- ſecond Theorem, it is manifeft, that
gles A ÍB, IHK, HGF, may be the Radius is to the Tangent of the
taken for right-lined ones.
Courſe AIB, as I B to AB, HK
to KI, GF to FH. Therefore,
PRO P. II.
alſo, as the Radius is to the Tan-
The Length of the Rhumb-Line AG, gent of the Courſe, fo is IB+HK+
is to the Difference of Latitude GD, GF, that is, the Difference of La-
in the ſame Meaſure, as the Radius titude GD to AB+IK+HF.
is to the Cofine of the Courſe or An-
PROP. V.
gle PAG.
The Sum of ABRIK+HF is a
For in the Triangles AIB, IHK, mean Proportional between the Aggre-
and GHF, as the Radius is to the gate of the Diſtance AG and the
Sine of the Angles BAI, KIH, Difference of Latitude GD, and their
FGH, that is, to the Coſine of Süm.
the Courſe PAG, or PIG, or PLG,
PHG, so are the parts of the For AI - IB AB, and ſo
Rhumb-Line AI, IH, GH, to the AI+IB : AB :: AB: AI-IB.
Parts IB, KH, GF, of the Diffe- Wherefore fince after the ſame man-
rence of Latitude. Therefore Al ner it is proved that IH+HK: IK ::
+IH+GH, that is the Rhumb- IK: IH -- HK, and GH +GF:
Line AG, is to IB+KH+GF. HF :: HF : GH-GF; therefore
That is the Difference of Latitude ſhall AI+H+HG +1B+HK+
DG, as the Radius is to the Co-GF be to ABHIK+Hr, as AB+
fine of the Courſe.
IK+HF to Al+H+HG-B-
HK-GF ; that is, AG +GD:
PROP. IIT.
ABTIK+HF :: AB+IK +HF:
The Length of the Rhumb-Line AG AG-GD.
is to the sum of the Baſes of the ſmall From hence it follows, even in
Right-lined Triangles, viz. to AB+ plain Sailing, that of theſe three
IK+HF as the Radius to the Sine of things, viz. the Difference of Lati-
the Angle GAP, or Courſe.
tude, Courſe, and Distance, any two
From the Demonſtration of the being given, the other will be had
laſt Theorem, it is manifeft, that by one Operation of the Golden
the Radius is to the Sine of the Rule, to a Geomctrical Exactneſs.
Courſe, as AI to AB, IH to IK, But the Departure which is reprc-
or GH O HF: (l'hat is, fince ſented by the Line AD, will not be
IAB is the Complement of the found by the common Canon in
Courſe GAP to a right Angle PAD, plain or Mercator's Sailing.
and becauſe B is a right Angle, and RIDEAU, in Fortification, is a
alſo AIB the Complement of BAI Ditch, the Earth whcreof is raiſed
to a right Angle," and therefore on its fide, or it is a ſmall Eleva-
AIB is equal to the Courſe PAG.) tion of Earth, extending itself in
)
RID
'R IG
;
Length on a Plain, which ſerves to by 4, if odd; and if the Product be
cover a Poft ; being alſo very con- multiplied by the greater Terms
venient for thoſe that would beſiege this laſt Product added to the Nu-
a Place at a near diſtance; and to merator firſt found, and you will
ſecure the Workmen in their Ap- have a ſecond Numerator.
proaches to the Fort of a Fortreſs. 4. Laſtly, To have a ſecond Deno-
RIGHT-ANGLED, a Figure is minatot, add the Square of the
faid to be right-angled, when its Difference of the Terms; if it be
Sides are at right Angles, or ſtand even, or the double of it, if odd,
perpendicularly one upon another : to the Denominator firſt found, and
And this is ſometimes in all Angles that will be a ſecond Denominator.
of the Figures, as in Squares and 5. For Example, if the Terms of
Rectanglus ; fometimes only in the Ratio be 1 and 2, theſe multi-
part, as in right-angled Triangles. plied, make 2, and ſo 4 ſhall be
RIGHT - ANGLED TRIANGLE. the firſt Numerator: Again, ſince i
See Triangle.
and 2 added is 3, an odd Number
1. In the following two Progref- therefore, 3 multiplied by 1, the
fions, viz.
Difference of the Terms is 3, the
Whence the firſt
13. 25. 3. 4. 515. 6,, &c. Denominator.
Term of the Series will be ori
13. 21 3iš. 418. 534.63}. &c. Again, becauſe i the Difference of
If the Denominator of the Fraction the Terms is odd ; if it be multi-
be taken for the Baſe, and the Inte- plied by 4, and this Product 4 by 2,
ger multiply'd by the Denominator the greater Term; 12 the Sum of
Plus the Numerator for the Perpen- this Product, and the firſt Numera-
dicular of any right-angled Trian- tor, ſhall be the ſecond Nume-
gle, the Hypotheneule will be a
rational Number.
6. Laſtly, Becauſe 1, the Square
2. And after the following man of the Difference of the Terms, is
ner may an infinite Number of ſuch odd; therefore, if the double of it
Series of mixed Numbers, or im 2, be added to the Denominator 3
proper Fractions be found, viz. hav- before found, the Sum 5 Iliall be
ing taken two Terms of any Ratio, the ſecond Denominator, where
in order to find the Numerator, and, each of them, expreſs two
multiply one of the Terms by the Sides of a right-angled Triangle,
other, and obſerve whether the whoſe Hypotheneuſe is rational ;
Product be even or odd ; if it be and if the Terms of the Ratio, viz.
odd, it will be the Numerator it 2 to 3, 3 to 4, 4 to 5, &c. be uſed ;
ſelf; but if it be even, it will be
will be after this way you will get the
the double of the Product : But to Terms of the firſt Series above.
get the Denominator, add the ſaid Right Angle. See Angles.
Terms of the Ratio together, and RIGH'r AscENSION of the Sun,
multiply the Sum, if it be odd by or Star, is that Degree of the Equi-
the Difference of the Terms, and noctial, accounted from the Begin-
that Product will be the Denomina- ning of Aries, which riſeth with it
tor; but if that Sun be even, half in a right Sphere.
of the Sum will be the Denomi Or, 'tis that Degree and Minute
of the Equinoctial (counted as be-
3. Now, to obtain a ſecond Nu- fore) which cometh to the Méri-
merator, multiply the Difference of dian, with the Sun or Stars, or with
the Terms by 2 į if it be even, or any Point of the Heavens. The
Reaſon:
rator.
nator.
RIN
ROO
Reaſon of which referring it to the the figure of Saturn not to be
Meridian, is becauſe that is always round; but, that the Inequality was
at right Angles to the Equinoctial ; thus in the Form of a Ring, Mr.
when the Horizon only is in a right Huygens firſt found out, and publiſh-
or direct Sphere.
ed in his Syſtema Saturniana, 1659.
Right CIRCLE, in the ſtereo- 'Tis this Ring, and its various pa
graphical Projection of the Sphere, fitions in reſpect of the Sun, (whoſe
is a Circle that is at Right Angles, Light it reflects like the Body of
to the Plane of Projection, or that Saturn itſelf) and of the Eye of the
which paſſes thro' the Eye. Spectator, which occafions all the
RIGHT LINE, is the neareſt Di- various Appearances of Saturn with
ſtance between any two Points. See his Anfæ, (as they call them) or
Line.
with none; with broad or narrow
RIGHT SAILING, is when a ones, &c.
Voyage is performed on ſome one RISING of the Sun or Star, is
of the four Cardinal Points. their appearing above the Ho-
RIGHT SINE, the ſame with rizon.
Sine; which ſee.
Rod, a Meaſure of Length con-
Right, or Direct SPHERE, taining by Statute juſt fixteen Feet
is that which has the Poles of the and a half Engliſh: See Pole. This
World in its Horizon, and the Equa- muſt carefully be diſtinguiſhed from
tor in the Zenith: The Conſequence Rood, which is a ſquare Meaſure,
of living under ſuch a Poſition, (as containing the fourth Part of an
thoſe who live directly under the Line Acre.
are in,) is that they have no Latitude, ROMAN ORDER, in Architec-
nor Elevation of the Pole. They can ture, is the ſame with the Compoſite.
ſee nearly both Poles of the World ; 'Twas invented by the Romans, in the
all the Stars do riſe, culminate, and time of Auguſtus, and ſet above all
ſet with them; and the Sun always the others, to fhew that the Ro-
riſes and deſcends at Right-Angles to mans were Lords over other Na-
their Horizon, and makes their tions: 'Tis made up of the Tonic
Days and Nights equal; becauſe and Corinthian Orders, and is more
the Horizon biſfects the Circle of ornamental than either.
this Diurnal Revolution.
RONDEL, in Fortification, is a
Rim, in a Watch or Clock, is round Tower, ſometimes erected at
the Circular Part of the Balance the foot of the Baſtions.
thereof.
Rood, a ſquare Meaſure, con-
RING-DIAL. See Univerſal E- taining juſt a quarter of an Acre of
quinoctial Dial.
Land: Some confound this Mea-
RING of Saturn, is an opacous, ſure with a Rod, which is the Length
folid, circular Arch or Plane, like of fixteen Foot and a half; and o-
the Horizon of a Globe of Matter, thers with a Yard Land, or the Quar.
entirely encompaſſing round the tona Terræ, but both very erro-
Planet, and no where touching it ; neouſly.
its Plane is at this time nearly pa Root. Whatever Quantity be-
rallel to the Plane of our Earth's E- ing multiplied into itſelf
produces a
quator ; the Diameter of this Ring Square, and that Square again be-
is 2 of Saturn's Diameters, and ing multiplied by that firit Quan-
the Diſtance of the Ring from the tity produces a Cube, &c. is called
Planet, is about the Breadth of the a Root, and is either the Square,
Ring itſelf. Galilæus firſt diſcovered Cube, or Biquadrate Root, &c. ac-
cording
RUL
RUM
cording to the Multiplication. See in 6 Days eat 10 Buſhels of Oats,
Squart, Cube, &c.
eight Horſes will eat 10 Buſhels in
Root of an Equation. See E- a leffer Number of Days,, viz. 3.
guation.
g. The Double Rule of Three, both
Rota ARISTOTELICA, is the Direct and Indire&, may be com-
Confideration of a Wheel moving priſed in one Rule, with two Ope-
along a Plane, till it hath made rations, only obſerving, That the
one entire Revolucion : For then given Terms are always five, whereof
will its Centre have deſcribed a three are Conditional and Antece-
Line equal to that of the Circum- dent, or Suppofitions; the other two
ference of the Wheel, and ſo will demand the Queſtion, and are Con-
all leſſer Concentrical Circles. fequents anſwering ſome of the for-
ROYAL FORT. See Fort. mer Antecedents ; infomúch, that
ROYAL PARA PET, or PARAPÉT with the Anſwer there will be as"
of the Rampart, in Fortification, is many Conſequents as Antecedents,
a Bank about three Fathoms broad, which muſt match one another in
and fix Foot high, placed upon the the ſame Denomination exactly.
Brink of the Rampart, towards the If the Power of any Agent be gi-
Country, to cover thoſe who defend ven, and it be required to find how
th: Rampart.
many ſuch Agents can produce a
RULE of Three, or the Rule of given Effect à in a given time;
Proportion, or, as it is called from let the Power of the Agent be fuch,
its excellent Uſe, the Golden Rule, is that the Effect c may be produced
that which teaches to find a fourth thereby in the time b. Then will
Number, which ſhall have the ſame
ad
Proportion to one of the three the Number of Agents be
bc
Numbers given, as the others have
to one another. And this is RUMB, or COURSE of a Ship, is
performed by multiplying the the Angle which ſhe makes in her
ſecond Number by the third, Sailing with the Meridian of the
and dividing the Product by the Place, where ſhe is.
firft.
Complement of the Rumb, is the
This Rule of Three is, 1. Direct. Angle made with any Parallel to the
2. Indirect. 3. Double Rule Di- Equator by the Line of the Ship's
rect. 4. Double Rule Indirect.
Run.
1. Rule of Three Direct finds a RUMB, in Navigation, is
fourth Number in ſuch Proportion Point of the Compaſs, or in De-
to the third, as the ſecond is to grees and a quarter, viz. the 32d
the firſt, or as the firſt is to the fe- Part of the Circumference of the
cond, ſo is the third to the fourth. Horizon, or Compaſs Card, which
2. Rule of Three Indirect, or Back. is the Repreſentative of the Ho-
tward Rule, is known by being con-
rizon.
trary to the Direct ; for whereas the RUMB-Line, is a Line deſcribed
former required, that more ſhall by the Ship's Motion on the Surface
have more, and leſs leſs; as if of the Sea, fleered by the Compaſs,
4
Yards coft 2 s. 8 Yards will coit making the ſame, or equal Angles
more than 2 ; becauſe it is double with every Meridian.
Theſe Rumbs are Helifpherical
Yards; and fo muſt the Anſwer
4
be double to 2 s. that is, 4s. or Spiral Lines, proceeding from
But in this Rule more will require the point where we ſtand, winding
leſs, and leſs more ; as if four Horſes about the Globe' of the Earth, till
the
one
to
SA
S A K
SA T
they come to the Pole, where at four Inches Diameter at the Bore,
laſt they loſe themſelves.
1800 Pounds weight, 10 Foot long,
But in the Plane, and Mercator's its Load five Pounds, Shot three In-
Charts, they are repreſented by ches and a quarter Diameter, and
ſtraight Lines. Their Uſe is to ſomething more than ſeven Pounds
ſhew the Bearing of any Places one and a quarter weight; its Level-
from another; that is, upon what Range is 163 Paces.
Point of the Compaſs any Shore or SAKER Of the LEAST SIZE, is
Land lies from another.
three Inches and three quarters Bore,
nine Foot long, 1500 Pounds weight,
its Load near thrée Pounds and a half,
$.
Shot four Pounds and three quarters
weight, and three Inches one quar-
CCER. See Saker. .
ter Diameter
SACKS of Earth, uſed in Fer-
SAPPE, in Fortification, formerly
tification, are made of coarſe Cloth, fignifyed the Undermining, or
the largeſt of them being about a deep Digging with Pick-ax and Sho-
Cubick-Foot wide, and the leſſer vel at the foot of a Work to over-
ſomewhat more than half a Foot, throw it without Gunpowder : Now,
They are ſerviceable upon ſeveral it is uſed to ſignify a deep Trench
Occaſions, more eſpecially for mak- . carried far into the Ground, and
ing Retrenchments in haſte, to place deſcending by Steps from Top to
on Parapets, or the Head of the Bottom ; ſo that it covers the Men
Breaches, &c. or to repair them fideways; and to ſave them from
when beaten down. They are of Danger on the Top,' they uſe to lay
good uſe alſo when the Ground is a-croſs it Madriers, that is, thick
rocky, and affords not Earth to Planks, or Clugs, which are Bran-
carry on Approaches, becauſe they ches of Trees cloſe bound together,
can be eaſily brought on, and car- and then they throw Earth over alí
ried off: The ſame Bags, on occa to ſecure them from Fire.
fon, are uſed to carry Powder in; When a Cover'd-Way is well de-
of which they hold about fifty fended by Muſqueteers, the Beſiegers
Pounds a-piece
muft make their way down into it
SACE Ř. See Saker.
by Sapping
SAGITTA, a Conſtellation in
SARRASIN, in Fortification, is a
the Northern Hemiſphere, conſiſting kind of Portcullice, otherwiſe called
of eight Stars.
a Herſe, which is hung with a Cord
SAGITTA, in Mathematicks, is over the Gate of a Town or Fortreſs,
the ſame as the Verſed Sine of any and let fall in caſe of a Surprize.
Arch, and is ſo called by ſome Wri SATELLITES, by Aſtronomers,
ters, becauſe 'cis like a Dart or Ar- are taken for thoſe Planets which
row ftanding on the Cord of the are continually, as it were, waiting
Arch. See Verfed Sine.
upon, or revolving about other Pla-
SAGGITTARIUS, is the Ninth, nets; as the Moon may be called
in the Order of the twelve Signs of the Satellite of the Earth; and the
the Zodiack.
reſt of the Planets, Satellites of
SAILING. See Plain, and Mer- the Sun : but the Word is chiefly
cator's Sailing
uſed for the new diſcovered ſmall
SAKER, a ſort of Cannon, and is Planets, which make their Revolu.
either extraordinary, or leaſt Size. tion about Saturn and Jupiter.
SAKER EXTRAORDINARY, is SATELLITES of Jupiter, are
four
G&
SAT
SA T
four ſmall Moons or Planets moving They were firſt diſcovered by Gali-
round about the Body of Jupiter, as læus, by the help of the Tele-
the Moon doth round our Earth : ſcope.
1
Z
1
The Diſtances of theſe Satellites, from the Body of Jupiter, are as follows;
from the Obſervations of
3 4
Mr. Callini
8. 13. 23.
Semi.
Mr. Borellus
5. 8.3 14:
Dia-
Mr. Townley by the Micromet. 15. 51 18. 78 13. 47 24. 72
meter
Mr. Flamſtead by the Microm. 5. 31 8. 85 13. 98 24. 23
of yu-
Mr. Flamſtead by the Eclip. of Sat. 5. 578/8. 876|14. 159/24. 903
piter.
From the Periodical Times 15. 57818. 87614. 168/24. 968
24.
The Periodical Times are : Of the
Days.
Hours, Min.
18
28
3
13
Vid. Newton's Princip.
7
3
59
pag. 403.
16
IS
5
Firſt
Second
Third
Fourth
1
17
+
Mr. Flamfiead, in Philos. Tranſ. Ring, above of the apparent
Nº 154. fays, that when I upiter is Length of the fame Ring; and it
in a Quartile of the Sun, the Di. was found to make one Revolution
ſtance of the firſt Satellite from his about Saturn, in one Day, 21 Hours,
next Limb, when it falls into his and 19 Minutes ; making two Con-
Shadow, and is eclipſed, is one Se- junctions with Saturn, in leſs than
mi-Diameter of Jupiter; of the fe two Days ; one in the upper part of
cond, two, or a whole Diameter his Orb, and the other in the lower
nearly; of the third, three; of the Part. It is diſtant from the Center
fourth, five of his Semi-Diameters, of Saturn 4 of Saturn's Semi-Dia-
or ſomething better, when the Parmeter.
rallax of the Orb is greateſt : But 2. The ſecond Satellite of Saturn
theſe Quantities diminiſh gradually was obſerved to be of the Length
as he approaches the Conjunction or of his Ring diſtant therefrom, mak-
Oppoſition of the Sun ſomewhat ing his Revolution about him in two
nearly; but not exactly in the Pro- Days, 17 Hours, and 43 Minutes.
portion of Sines.
This is diftant from the Center of
SATELLITES of Saturn. Anno Saturn 5 Semi-Diameters of that
1684, in the Month of March, Planet.
Mr. Caſini, by the Help of ex 3. From a great Number of choice
cellent Object. Glaſſes of 70, 90, Obſervations he concluded, that the
100, 136, 155, and of 220 Foot, Proportion of the Digreſſion of the
diſcovered the two innermoſt ; (that ſecond to that of the firſt, counting
is, the firſt and ſecond) Satellites of both from the Centre of Saturn, is
Saturn.
as 22 to 17.
1. The firſt Satellite he obſerved 4. And the Time wherein the
to be never diſtant from Saturn's firit makes its Revolution, is to the
Time
SAT
SA T
Time wherein the firſt makes its, as may have other Satellites moving
24 to 17.
round him.
5. The third is diſtant from Sa 8. Mr. Halley, in Philos. Tranſ.
turn, eight of his Semi-Diameters, Nº 145. gives a Correction of the
and revolves round him in almoſt Theory of the Motion of the Huyge-
4 Days.
nian, or fourth Satellite of Saturn,
6. The fourth, or Huygenian Sa- and makes the true Time of its De-
tellite, as 'tis called, becauſe diſco- riod to be 15 Days, 22 Hours, 41
vered firſt by Mr. Huygens, revolves, Minutes, fix Seconds ; its Diurnal
round Saturn, in about 16 Days, Motion to be 22 Degrees, 34 Mi-
and is diſtant from his Centre about nutes, 38 Seconds, 18 Thirds, and
18 Semi-Diameters of Saturn. the Diſtance of this Satellite from
7. The fifth Satellite of Saturn is the Centre of Saturn, to be about
diftant from its Centre 54 Semi-Dia- four Diameters of the Ring, or nine
meters of Saturn; and revolves of the Globle : and the place where
round him in 79 Days. The greateſt it moves, to differ little or nothing
Diſtance between this Satellite, and from that of the Ring ; that is to ſay,
the preceding, made Mr. Huygens interſecting the Orb of Saturn with
fufpect there may be a fixth between an Angle 23 Degrees and a half; fo
theſe two; or elſe, that this fifth as to be nearly parallel to the Earth's
Equator.
1
The Periodical Times of the Satellites of Saturn, according to
Mr. Caſſini are, of the
Days. Hours. Min.
Firſt
21
19
Second
2 2
17
43
Third
4
27
Fourth
23
15
Fifth
79
I
I 2
!
I
15
22
O
:
to 1.
SATURN, is the higheſt of the 5. Mr. Huygens found the Incli-
Planets.
nation of the Ring of Saturn to the
1. The Ratio of the Body of Sa- Ecliptick, to be an Angle of 30
turn to our Earth, is about as 30 Degrees.
6. M. Azout afferts, that the re-
2. T'he Perodical Time of Saturn mote Diſtance of Saturn from the
about the Sun is in the Space of 30 Sun doth not hinder but that there
Years, or 10950 Days.
is Light enough to ſee clear there,
3. The Semi-Diameter of Saturn's and more than in our Earth in cloudy
Orbit is almoſt ten Times as big as Weather.
that of the Magnus Orbis, and there-
an Obſervation, which
fore is more than 946969690 Eng- Mr. Calini made June 19, 1692. of
liſh Miles.
a preciſe Conjunction between a fixed
4. According to Mr. Cafini, Sa- Star, and one of Saturn's Satellites,
turn's greateſt Diſtance from the he faith, that with his 39 Foota
Earth is 244330, his mean Diſtance Glaſs he could plainly ſee the Sha-
210000, and
his leaſt Diſtance dow of Saturn's Globe to be in part
175670 Semi-Diameters of the oval upon the hinder part of his
Earth.
Ring. The Diameter of Saturn.ad
Gg2
the
7. In
S A T
SCA
the Time of this Obſervation, ap 12. And the Interftice between
peared to be 45 Seconds.
the Planet and the Ring, is the
8. Che Diameter of Saturn to that Breadth of the Ring.
of the Ring, is as 4 to 9.
13. How the Ring of Saturn will
9 And the Diameter of the Ring appear in all Parts of the Orbit of
ſeen from the Sun, would be but the Planet, to an Eye placed at the
50", and therefore, the Diameter Sun, or at the Earth, the ſame
of Saturn ſeen from thence would learned Aſtronomer ſhews in his
be but 11". As Mr. Flamſtead found Aſtro. Phy. & Geometr. Lib. IV.
by meaſuring it But Sir Iſaac New Prop. 69, 70.
ton thinks it ought rather to be ac SCALE, in Mathematicks, figni-
counted but as 10". or 9". becauſe he fies any Meaſures or Numbers which
ſuppoſes the Globe of Saturn to be a are commonly uſed; or, the Degrees
little dilated by the unequal Re- of any Arch of a Circle, or of ſuch
frangibility of Light.
Right Lines as are divided from
10. The Diſtance of Saturn from thence; ſuch as Sines, Tangents,
the Sun is about ten Times as great Chords, Secants,&c. drawn or plot-
as that of our Earth from him ; and ted down upon a Ruler, for ready
therefore that Planet will not have Uſe and Practice in Geometrical, or
above the rooth Part of the Influ- other Mathematical Operations.
ence of the Sun which we have ; The Plain Scale (for Sea-Uſe) has
and conſequently cannot be habitable alſo ſet thereon the Scale of Chords,
by ſuch Creatures as live on our natural Sines, Tangents, Semi-Tan-
Globe, unleſs there be some un gents, Secants, Rhumbs, Hours,
known Way of communicating Heat Leagues, and Longitudes; with the
to him.
Diagonal Scale on the Back-Side,
11. Dr. Gregory, in his Aftronomy, and ſome others, according as there
makes the Semi-Diameter of the is Room.
Ring of Saturn to that of the Planet,
as 2 to 1.
0
1
A
1
--
SCA L E
2
1
70
+
Plain Scale
Secants
Tangents
1..
60
60
50
40
30-
20
10
:40
C20
--Yo
30
70
00
1,30
vo
Longitudes
120:
Sernitangentó
40 о
Chorito
gro
20
10
Equial para cr League
20 40 50
10 20 30 40 50 6079
Sinew'.
1
2
Hourý
Ruibu
X
4
1
SCE
SCO
SCALE of the Gamut, or Muſical Scenography is the Manner of de-
Scale, is a kind of Diagram, con- lineating the ſeveral Parts of a Build.
fiſting of certain Lines and Spaces ing or Fortreſs, as they are repre-
drawn to ſhew the ſeveral Degrees, ſented in Perſpective.
whereby a natural or artificial Voice Scheme, is the Repreſentation
or Sound may either aſcend or de- of any Geometrical or Aſtronomical
ſcend. The Name thereof is taken Figure or Problem, by Lines ſenſibly
from the Greek Letter Gamma, which to the Eye, and theſe are otherwiſe
Guido Aretinus, who reduced the called Diagrams.
Greek Scale into this Form, placed at SCHOLIUM, is a Diſcourſe ei-
the Bottom, to ſignify from whence ther declaring what Things are ob-
it was derived; ſo that ever ſince, ſcure in Definitions or Propoſitions,
this Scale or Gamut hath been taken and their Corollaries; or elſe clear-
for the Ground-Work, or firſt ing up of Doubts that may ariſe ;
Foundation of all Muſick, both Vo or fhewing the uſe of the Doctrine
cal and Inſtrumental.
in hand, or laſtly deſcribing the Hi-
But there were three different ſtory or Origin of an Invention.
Scales in uſe among the Ancients, SCIOGRAPHY, is the Art of
which had their Denoininations from Shadows, or Dialling: Alſo in Ar-
the three ſeveral ſorts of Muſick, chitecture, this Word is ſometimes
viz. the Diatonical, Chromatical, taken for the Draught of a Building
and Inharmonical. Which fee. cut in its Length or Breadth, to fhew
SCALENOUS Cones, are ſuch the Inſide of it, as alſo the Thick-
whoſe Axes are not at Right Angles neſs of the Walls, Vaults, Floors,
to their Baſe.
Timber Works, &c.
SCALENOUS TRIANGLES. See SCIOPTRICKS. See Obſcura Can
Triangles.
SCARP, in Fortification, is the SCIOTHER ICUMTEL È SCOPIUM,
Foot of the Rampart-Wall, or the is an Horizontal Dial, with a Tele:
Sloping of the Wall from the Bot- ſcope adapted for obſerving the true
tom of the Work, to the Cordon on Time both by Day and Night, to
the Side of the Moat.
regulate and adjuft Pendulum Clocks,
SCENOGRAPHY, in Perſpective, Watches, and. other. Time-Keepers ;
the Scenographick' Appearance of invented by the ingenious Mr. Moly
any Figure, Body, or Building, is that neux, who has publiſhed a Book
Side that declines from, or makes with this Title, which contains an
Angles with that ſtraight Line ina- accurate Deſcription of this Inftru-
gined to paſs through the two out- ment, and all its Uſes and Application.
ward Convex Points of the Eyes, ge SCON Es, are ſmall Forts built
nerally called by Workmen, the for Defence of ſome Paſs, River, or
Return of a Fore-right Side; and Place. Sometimes they are made
differs from the Orthographick Ap. regular of four, five, or ſix Baſtions ;
pearance in this, that the latter re others of ſmaller Dimenſions fit for
preſents the side of a Body or Build- Paſſes, or Rivers, and likewiſe for
ing as it is ſeen, when the Plane of the Field; which are,
the Glaſs ſtands parallel to that Side: 1. Triangles with Half-Baſtions,
But Scenography repreſents it, as it which may be all of equal Sides, or
ſeems, through a Glaſs not parallel they may be a little unequal. How-
10 that Side.
ever it be, divide the Sides of the
In Architecture and Fortification, Triangle into two equal Parts, one
of
3
mera.
SCO
SE C
of theſe three Parts will ſet off the Ditch of fifty or fixty Foot wide,
Capitals, and the Gorges, and the and are thus made to ſet upon Paſſes
Flanks being at Right Angles with or Rivers to endure.
the Sides, make half of the Gorge. Score, in Muſick, is the origi-
2. Squares with half Baſtions, nal Daught of the whole Compofi-
whoſe Sides may be betwixt 100 and tion, wherein the ſeveral Parts, viz.
200 Foot ; and let one Third part Treble, Second Treble, Baſe, &c.
of the Side fet off the Capital and are diftinctly ſcored or marked.
the Gorges ; but the Flank (which SCORPIO, is the Eighth Sign of
riſes at Right Angles to the Side) the Zodiack, being uſually marked
muſt be but one half of the Gorge thus (M.).
or Capital, that is, the fixth Part SCOTIA, in Architecture, is a
of the side of the Square.
certain Member hollowed in form
3. A Square with Half-Baftions of a Demi-Channel, which is placed
and Tong.
between the Torus, and the Aſtra-
4. Long Squares.
gal in the Baſes of Pillars; as alſo
5. Star Redoubt, of four Points. ſometimes under the Larmier or
6. Star Redoubt, of five or fix Drip, in the Cornice of the Dorick
Points.
Order.
7. Plain Redoubts, which are ei Screw ? is one of the mechani-
ther ſmall or great: The ſmall are SCRUE, Scal Powers, conſiſting
fit for court Guards in the Trenches, of a Cylinder fulcated or hollowed
and may be a Square of twenty foot in a Spiral Manner, and moving or
to thirty. The middle forts of Re- turning in a Box or Nut, cut ſo as
doubts may have their Sides from to anſwer to it exactly.
thirty to fifty Feet: The great ones In the Screw, the Power is to the
from fixty to eighty Feet ſquare
Reſiſtance, as
the ſaid Diſtance
The Profile (that is, the Thick between two Threads to the Peri-
nefs and Height of the Breaſt-Works) phery of a Circle, run through by
to be ſet on theſe ſeveral Works, that Point of the Handle to which
and the Ditches are alterable and un the Power is applied.
certain ; for ſometimes they are uſed
SCROWLES,
or VOLUTES,
in Approaches, and then the Wide- Term in Architecture. See Volutes.
neſs of the Breaft-Work at the Bots SEA-QUADRANT. See Back-
tom may be ſeven or eight Foot, in- Staff:
ward Height fix, and outward five
SECANT
is the Line drawn
Foot. The Ditch may be eight or from the Centre of a Circle, cut-
ten Foot, and ſometimes twelve: ting it, and meeting with a Tan-
And for the Slopes to be wrought gent without.
according to the Nature of the SECOND, is the fixtieth Part of
Earth; ſometimes they may be a Minute,
made fourteen or twenty Foot wide SECONDARY CIRCLES, in re-
at the Bottom, and the Height of ference to the Ecliptick, or Circles
ſeven, eight or nine Foot, and to of Longitude of the Stars, are ſuch
have two or three Aſcents to riſe to
as pafling through the Poles of the
the Parapet: The Ditch inay be Ecliptick, are at Right Angles to the
ſixteen or twenty-four Foot wide, Ecliptick, (as the Meridian and
and five or fix deep ; and fome- Hour-Circles are to the Equinolia?)
times they may come near the ſmall- "By the help of theſe infinitely ina-
eſt fort of Ramparts, and have a ny Circles) all Points in the Heavens
Breaft-Work Cannon-proof, with a are referred to the Ecliptick: Thic
is,
a
Gg4
SE C
SEG
are
is, any Star or Phänomenon. And turn out to make a true Square, with
if two Stars, &c. are thus referred Lines of Sines, Tangents, Secants,
to the fame Point of the Ecliptick, equal Parts, Rhumbs, Polygons,
they are ſaid to be in Conjunction; Hours, Latitudes, Metals, Solids,
ifin oppoſite Points, they are ſaid to &c. and is generally uſeful in all
be in Oppoſition: If they are re the practical Parts of the Mathema;
ferred to two Points at a Quadrant's ticks, and particularly contrived
Diſtance, they are ſaid to be in a for Navigation, Surveying, Aftro-
Quartile Aſpect; if the Points differ nomy, Dialling, Projection of the
a fixth Part of the Ecliptick, the Sphere, &c. by Gunter, Foſter, Col-
Stars are faid to be in a Sextile Af- lins, and others. There are like
pect, c.
wiſe Sectors for Fortification and
And, in general, all Circles which Gunnery, by Sir Jonas Moor.
interſect one of the fıx greater Circles The great Advantage of the Sec-
of the Sphere at Right-Angles, may tor above any Rule or Scale is, that
be called Secondary Circles ; as the all its Lines can
all its Lines can be accommodated
Azimuths or Vertical Circles in re to any Radius; which is done by
ſpect of the Horizon, c.
taking off all Diviſions parallelwiſe
SeCONDARY PLANETS, and not lengthwiſe. The Ground
ſuch as move round others, which of which Practice is this, that Pa-
they reſpect as the Centre of their rallels to the Baſe of any Plain Tri-
Motion, though they move alſo a- angle, bear the ſame Proportion to
long with the Primary Planets in it:: as the Parts of the Legs above
the annual Orbit round the Sun; the Parallel do to the whole Legs.
and theſe are otherwiſe called the Sector of a Circle, is a mixt
Satellites, ſuch as the Moon to the Triangle comprehended between
Earth: And Jupiter hath four mov two Radius's and an Arch of the
ing round him; as Saturii, accord- Circle.
ing to Caſini, hath five. Mars, Ve-
SECUNDANS, in Mathematicks,
nus and Mercury, have no Secondary is an infinite Series of Numbers,
Planets moving round them, that beginning from nothing, proceed-
have been yet diſcovered.
ing as the Squares of Nuinbers in
SECTION CONICK. See Conick Arithmetical Progreſſion. As for
Section,
Inſtance,
Sectiox, in Mathematicks, fig-
0, 1, 4, 9, 16, 25, 36, 49, 64, &C. .
nifies the cutting of one Plane by SEGMENT of a Circle, is a Fi.
another, or a Solid by a Plane.
gure contained between a Chord and
The common Section of two an Arch of the fame Circle.
Planes is always a Right-Line, be If the Altitude AB of the Segment
ing the Line ſuppoſed to be drawn DAC of a Circle be biffected in E,
on one Plane by the Section of the and the Right-Line DE be drawn
other, or by its Entrance into it.
SECTION of a Building, in Ar-
A.
chitecture, is underſtood of the Pro-
file and Delineation of its Heights
and Depths raiſed on a Plane ; as if
E
the Fabrick were cut aſunder to dif-
cover the Infide.
D
B
C
Sector, is an Inſtrument made
of Wood, Ivory, Braſs, &c. with as alſo the Chord AD; then the
a Point, and ſometimes a Piece to Area of the Segment DAC will be
nearly
X
SEG
SE M
nearly equal to 4 BD ADx AB. this Product again by the conſtant ,
.8 DE +2AD Decimal . 5236, the Sum will be
Or, nearly equal to
nearly equal to the Solidity of that
15
Segment.
2 AB. Or, if you take BE to AB The Surface of any Segment of a
as 10 to 5, twice the Rectangle Sphere generated by the Rotation of
ABXED will be to the Area of the the Semi-Segment A BE of a Circle,
Segment DAC, as 3 to 2 nearly. is equal to a Circle, the Radius of
See Newton's Fluxions at the End. whoſe Baſe is the Chord A B drawn
SEGMENT of a Sphere, is a Part from the Vertex A to the Extre-
of it cut off by a Plane; and there- mity B of the Radiąs of the Baſe of
fore the Baſe of ſuch a Segment the Segment.
muſt always be a Circle, and its Su SE M I-BR EVE, a Term in Mus
perficies a Part of the Surface of the ſick. See Notes and Time.
Sphere.
Sem 1-CIRCLE, is the figure
Its Solid Content is found by mul-
contained between the Diameter of
tiplying the Surface of the whole a Circle, and half the Circumfe-
Sphere, by the Altitude of the Seg- rence.
ment, and then dividing the Pro-
Alſo an Inftrument for ſurveying,
duct by the Diameter of the Sphere, made of Braſs, and divided into
and to the Quotient adding the Area 180 Degrees, being half the Theo-
of the Baſe of the Segment.
dolite, is ſo called.
If ACD be a Quadrant of a
SEMI-CUBICAL Parabola, is a
Circle, AFCD a Square, and EBCD
Curve as A Mm of the ſecond Or-
be the Complement of a Segment der, or one of Sir Iſaac Newton's
five diverging Parabola's, wherein
F
G
C
the Cubes of the Ordinates PM are
as the Squares of the Abſciſſes, that
B
is, ſuppoſing a an invariable Quan-
tity of a proper Magnitude, it will
be a x AP' =PM, or Pm'.
The Solid generated by the Ro-
tation of the Space APM about the
Axis AP, will be of a Cylinder
E D
circumfcribing it, and a Circle equal
of a Circle to a Semi-circle, then
to the Surface of that Solid may be
the-Segment of a Sphere generated found from the Quadrature of an
by the Rotation of the Semi-Seg Hyperbolick Space.
ment EBC D about AED, toge-
The Length of any Arch AM of
ther with the Cone generated by the
Right-angled Iſoſceles
Ifoſceles Triangle
M
E HD, are equal to a Cylinder ge-
nerated by the Rotation of the Ob.
PI
long ECD, about ED.
A
If to 3 times the Square of the
Semi-diameter of the Bare of the
Segment of a Sphere be added
m
the Square of the Segment's Alti-
tude, and the Sum be multiplied by this Curve, may be eaſily obtained
the Altitude of the Segment; and from the Quadrature of a Space
contained
H
S E M
SER
are
contained under part of the Curve SEMI-SEXTILE,
an Aſpect of
of the common Parabola, two Semi- the Planets when diſtant from one
ordinates to the Axis, and the Part of another 30 Degrees, or one Sign,
the Axis contained between them. and is noted thus, SS.
And the Curve, may be deſcribed by SEMI-TONE, a Term in Mufick,
a. continued Motion, viz. by faſten- of which there are two forts, viz.
ing the Angle of a Square, in the a greater and leffer; the Inharmo-
Vertex of a common Parabola ; nical Deiſis being the Difference be-
and then carrying the Interſection tween them.
of one ſide of this Square and a long SENSIBLE HORIZON. See Ho-
Ruler (which Ruler always moves rizon.
perpendicular to the Axis of the SeNsIBLE POINT. See Point
Parabola) along the Curve of that Senſible.
Parabola. For the. Interſection of ŠERPENTARIUS,
a Conſtella.
that Ruler, and the other ſide of the tion in the Northern Hemiſphere,
Square will deſcribe a Semicubical conſiſting of thirty Stars.
Parabola. Mr. Mac-Laurin in his SEPTENTRION AL SIGNS,
Geometr. Organica does this without the firſt fix Signs of the Zodiack, ſo
a common Parabola.
called, becauſe they decline towards
Semi-DIAMETER, or Radius, is the North from the Equinoctial,
that Line that is drawn from the and are the ſame with Boreal
Centre to the Circumference of a Signs.
Circle.
Series, properly ſpeaking, is
SEMI-DIAMETER, in Fortifica- an orderly Proceſs or Continuation
tion, is two-fold, viz. the greater of things one from another. 'Tis
and leffer : The former being a Line commonly in Algebra connected
compoſed of the Capital, and the with the word Infinite, and there by
ſmall Semi-Diameter of the Poly- Infinite Series is meant certain Pro-
gon; and the other, a Line drawn greſſions, or Ranks of Quantities, or-
to the Circumference from the derly proceeding, which make con-
Centre through the Gorges. tinual Approaches to, and if infinite-
SEMI-DIAPASON, a Term in ly continued, would become equal
Mufick, fignifying a defective or to what is inquired after.
imperfect Octave.
This Method took its Riſe from
SEMI-DIAPENTE, in Muſick, the learned Dr. Wallis's Arithme-
fignifies an imperfect Fifth.
tick of Infinites, and has been of
Semi-Ditone, in Muſick, is late ſo purſued by ſeveral worthy
the lefſer Third, having its Terms Perſons of our Nation, eſpecially
as ſix to five.
the incomparable Sir Iſaac Newton,
SEMI-QUADRATE,
the fame that it is now one of the greateſt Im-
with Quartile.
provements of Algebra.
SEMI-QUARTILE, an Aſpect of Every infinite Series may be
the Planets when diſtant from each ſummed up, if the Terms of it are
other 45 Degrees, or one Sign and expreſſed by a fraction, the Factors
of the Denominator of which are
SEMI-QUAVER, a Term in taken from any Arithmetical Pro-
Mufick. See Noies and Time.
greſſion, and the Numerator be a
SEMI-QUINTILE, an Aſpect of Multinomial, whoſe Dimenſions at
the Planets, when at the Diſtance of leait are leſs by two than thoſe of
36 Degrees from one another. the Denominator,
The
an half
1
SER
SER
The following Account of the being new, eaſy and plain, it will
Method of Increments of Mr. Cunn's not be amiſs to inſert here.
ET 2 be fome Integral Quantity, equally increaſing by the conſtant
Increment q, which let bear a finite Relation to Q.
Then if Q be the preſent Value of the Integral,
2+ , will be the Firſt fucceeding Value,
2+ 29 the Second,
2+ 39 the Third,
And
2+ n the 7tb.
L
A
And let thefe ſeveral fucceffive Values,- for Eafe and Convenience, be
denoted by the ſame Q accented underneath :
Then Q will be denoted by 2
2+
by 2
by 2
2+28
2+ 39
by 2
II
Q+*
by 2
And in like manner whilſt 2 is the preſent Value of the Integral, the
Value of , immediately preceding the preſent, which, if you pleaſe to
call the Firſt Preceding, will be
29
9
The Second 2
The Third
2-39
The nth.
2ng
1
And if you pleaſe to let theſe be denoted by the fame Q accented
above;
Then Q-9 will be denoted by
2-29
Gy V VAGY
2-39
2-19
f
N. B. Dr. B. Taylor, and others, choſe to denote the Increments by
the fame Letters with the Integrals ; only for Diſtinction ſake they point
them beneath :
So
SER
SER
* +
11
II
So
* 2
+ 21
W
4?
29 = ?
nt 3n
22
aq = 2
* + 40 mm
And then the Integral Quantities may be denoted by the ſmall Let-
ters; and the preceding and ſucceeding Values of Integrals by the Let-
ters repreſenting the preſent Values accented above and below : And to
diſtinguiſh theſe Values of the Integral from like Values of Fluents, the
Grave Accent is uſed inſtead of the Acute.
Cor. From the very Notation itſelf, it follows, that if the Value of
any given Integral be fubtracted from its next ſucceeding Value, the Re-
mainder is the Increment of that given Integral : And therefore to find the
Increment of any given Integral, this is
The Rule.
Every where in the given Expreſſion, inſtead of the Integrals, write
their next ſucceeding Values, and from the Reſult take the given Expreſ-
fion, and the Remainder is the Increment ſought.
Example I. So che Increment of nºis n-ton. And,
1
Example II. The Increment of nn is n n -
nnnnn
III
n+ 2n-1 X 12 = 2nn.
/
Example III The Increment of nnn
The Increment of nnn is nnnnnn
a III Iro
=
n+ 50-on- 2n xnn
n. 2n xnn 3 n n n.
n . X
n xnn
0 III IV
III IV
• III I
1 X 2
1,
Example IV. In like manner,
In like manner, the Increment of nn, C. till
o ati
n is nn,
E c. till 12 nn, &c. till n = N -
& C.
ß ats atz
Bti
aati
B BTI at ati atz
till n = n + B to I x n
x n , &c.
n, &c. till n
B
: atı atz
B
B = a + 1 x + x n, &c. till n.
• ati atz .
B
Example
a 2
>
--
SER
SER:
.
i
(
WWW
Example v. So the Increment of n n n is n n n - n n =
nn X n n = n nmnt 41 xn n m 3 n n no
a I
Ba-Ida 2
Example VI. And the Increment of 1 n, &c. till n is n
71,
BI a I
B B-
I 02
&c. till
n, &c. till n = net nem *n
B
2
B
B
nta 12 x
X 2 n, 80. till í
N
1
n, &c. 'till
12
I 7
aI a2
B
- B + in M, &C. till n.
22
Example VII. The Increment of n n n n n (or of its Equal in
the negative Notation n n n n n n) is -, * * nn 1972 n =
'Ill 111 lap
H/
1
N
22 X N
n n n n = n + 4 n
n + 2 + x
X N N N N N =
>
6 n n n n n n.
Cor. From the Fourth, Fifth, and Seventh Examples it follows, that
the Increment of any Expreſſion involving the ſucceffive Values of any one
variable Quantity, is had thus :
Multiply the Expreſſion by the Number of the Factors into the con-
ftant Increment of the variable Quantity ; then divide by the firſt Value,
and you have the Increment of the Expreſfion.
Cor. Hence, alſo, it naturally follows, that the Integral Correſpondent
to any Increment, will be had thus
Multiply by that Value of the variable Quantity which immediately
precedes the firſt Value given'; then divide by the new Number of factors
into the conſtant Increment, and you will have the Integral.
One Uſe of the preceding Principles, is to raiſe the Binomial Theorem.
✓ The Form may be eaſily had by Induction: The Co-efficients re-
main to be inveſtigated; which ſuppoſe to be thus :
atxilm = am tramaxx tsames 72 +tam-3x3 + vam-4 x4, &c.
Now,
SER
SER
find the following State m, that is, when m becomes m tol; or which
Now, if we make m to increaſe uniformly by the Increment 1, and
amounts to the fame Thing, multiply the preceding Suppofition by a t*,
we ſhall have the Product.
amtitrax sam-1*2 + tàm2 73.+vam 3 z4,. &c.
+ iam x + ram-1 *+-sam-2 x3 tam-3 x4, &c.
In which
ar
whence mr
I
mm
Alfo
*
2
ܪ
mmm
And
22
3.2
1
nimm
mmm
Likewiſe
mmmm
mm
4. 3. 2.
3. 2
Praesent
mmm
21-3
mm
Whende at = "tina
am names x +
am-2 x² +
43, &c.
2
2. 3
$
There are other Theorems for this Purpoſe, eafily deduced from this:
Such is this following ; where A is the firſt Term, B the ſecond, the
third, &c.
m I
m Ax
B x
C%EC,
+
+ 3* =a + +
atx
m -+2
-3
at
+ x
at x
1
Of all the Varieties for this purpoſe, every one hath ſome peculiar
Property which the reſt have not.
2. To raiſe the Infinitonomial to any Power indetermined m.
Suppoſe it to be A + By to Cyz + Dy3 + Ey4, &c.
D
A
Which call Axit by + cya + dyz to eye fys, &c.
Then for the Form of the Power, you may obſerve (by Induction) that
the firſt Term will be always r; the ſecond, where the Index of y will be
an Unit, cannot be formed by any of theſe Terms but the 1ſt and 2d; and
that the third Term, where the Index of y is 2, can be produced only by
either
Axi+ 9 + +230, &c.
SER
SER
either ſquaring the zd, or multiplying the 3d by the A Term. And the
fourth, either by multiplying the if by the 4th, the 2d by the 3d, or the
Cube of the ad; and ſo proceeding, taking all the poſſible Ways. And
then you will have, if the required Power's
Index be denoted by m,
A* *1 +gby + bcy2 + Idy3 + qey
+ kb2 gy2 + nbcy3 + rbdy
+ pb3 33 + fccut
+ tbbcy4 + vb+ j4
1
Let m increaſe to m + 1, and the Power of the Infinitonomial will
be,
it gby + hey?
ge
主​主
​+ k6²,² nbcy3 rbd
+ by
+ pbi y3
foc 74
to tbbc
bcb
064
kbs
73
gbc
d
+ nb²c
+ p64
bc² rt
k6² c
+ gbd
I 86²
+ 2db
十​十​十​十一​十​十​十​十​十​++
Here it appears, by comparing like Terms, that by is the Increment
of gby, that is,
g
m, therefore
8
And
hI Im
b m
mn
k = g = m
11
2
?
1=I=m
1=m
mm
2 min
n = hot g= m + m = 2m :: n =
mm
77 178
2
mm
mmm
mm
mmm
=
11
܀
Þ
2
2. 3.
l
---
S'ER
SÉR
1
I m
qm
t
w
=i+8+
11
2 2
I
mm
Sb=m
=
y
>
2
?
mm
mmm
mmm
mmm
>
views 年表​:
7
mim +
4
2
3
2.3
1
$
mmm.
ü
་་་་
mnmm
Ž. 3:
U=P
4. 3. 2,
Now reſtore the Values of bync, de, &c. and multiply by Am, and
reſtore the Values of gi hd, &c. and it will be
)
+ An- By
+ A*-2 Baina
m
X
2
A
$
m
+
AM-I Cyz.
1
.
:
m-2
X
AM-3 B3 y 3
+
2
3
+ 2 /
ml
X
1
Am-2 BCy?
+ Am-? Dys.
I
A
I
m 2
+
+
m 3 A-4 B4 yt
X
X
X
2
3
4
şim
1
2
th
X
X
AM-3 B2 G4
1
;
2
3
OM
1
+ ***
AM- 2 CP 94
2
1
M
m
+
X
Am-2 BD v4
I
I
m2
+
AM-1 Byt
I
&c.
3. If
S E R
SER
9
3. If the Terms of any Series be a, a-tx, a +24, &c. till a + 9%
and you require the Sum of all:
Let the Term next following the laſt (viz. a+1xx) be called m;
then will m be the Increment of the Sum : And ſo
+ A, the Inte-
2 m
gral of m, will be the Sum it felf. But when the Term next following
the laſt is a, then the Series is nothing. Therefore when m = a; then
mm
+ A, i, e. +A=0
a
2
Whence
A=-àa
dr-x x
x
2a
2
Therefore the Sum fought is m m
à
2 m
z a
Which, if c be the laſt Term, will be
'asca
ca
à a
='
contacx=aa taso
cca
1
20
2 a
2 CO
24
From the former Series a, a to *, &c. let there be found this
a xatx, a tex x a +- 2*, at 2* * a + 3 %, a + 3 * *
a to 4*, &c. till a fa 8* *aton +1%: to find the Sum of all.
Then let the Term immediately following the laſt be mm, which is the
Increment of the Sum
.
抗 ​in mi
Therefore the Sum is
fe A.
377
When that which immediately follows the laſt is a & the Sum is
nothing
Therefore, writing a a for mi, the Sum
1
à aa
and
1+A=3
ܕ܀
anum
mi
Hh
Cona
3.
34
SER
SER
5a,
M. in m
ace is the Sum fought
.
à a a
LI
3 a
Conſequently
is the Sum fought :
3 m 3 a
Where if be the laſt Term of the Arithmetick Series, it will be
cos
(3 +*
20** -a3
30
3.
Alſo from the fame Series, viz. a *, &c. let this be form'd:
a x a t * * a + 2 x x a + 3x, a + x x a + 2 x xa * 3 *
xat 4*, &c. till a t n * * a + n + 1. * * at n + 2 x
Xa+n+ 3 *.
Let the Term immediately following the laſt be m m m m, which is the
Increment of the Sum ;
in m m mm
+ A is the Sum.
And the Integral
5 m
But when the Term next -following the laft is a a a a, the Series
is 0.
аааа
à ad
a q a a a
WW
Therefore
1
+AO
54
5 a
a
ทา 1
à a
a aa a a
22 11 M M-AM
1
Conſequently
is the Sum ſought.
5 m
CCCCC
а а а а а
Or,
if c be the laſt Term in the Arithmetick
Series.
5!
5
Other Examples of putting variable Quantities into Increments.
12
7
--
n 12
1
I
Example 8. The Increment of
I
n
72
N A
nn
72
11
1
11 22
1
Example
SER
SER
1
1
Example 9. The Increment of
is
22 2 2
N N N 大
​2 2 2
1 2 2
N 22 22
2 X N 1
nn
t_n
31
nn n
2
N n² n2n
N N N N
N N N 2
3
2 N N N
Example 10. The Increment of
I
1
I
1
11
1
is
% z c. till %
% g c. till % % z z c. til
a afi
B atratz BBHI
aati
B
% % &c. till z
% &c. till &
% &c. X % &C.
aati
B ati atz BBHI
a Bti ati B
% %2, &c. till z?
% 22 &c. till 22 %
aat
BB+I
ααβι
B Bti
zt az
B + 1
O BHI
z c. till
%, &c. till z
a ati
BBHI
0
BHI
B+10xx
Z, &c. till Z
BI
man
11
11
.
Therefore to put any ſucceſſive Values, dividing Unity of a flowing
Quantity into Increments, multiply the Expreſſion by the Number of the
Factors into the conſtant Increment of the variable Quantity : Then di-
vide by that Value of the variable Quantity which next ſucceeds the laſt
which' is given, and change the Sign: So you will have the Increment.
Cor. Hence alſo it naturally follows, that the Integral of any ſuch
Increment will be had thus :
Change the Sign, multiply by the laſt given Value of the variable
Quantity, and divide by the new Number of Factors into the conſtant
Increment, and you will have the Integral,
Example 11. The Increment of any Power nn is nnnn 2 N
tonn
n n.
nnn = 3n21
Example 12. And the Increment of n nn is nnn
+ 300 + na
Hh 2
Example
Ś ER
SER
Example 13. In like manner the Increment of
(1 2 + (2x +(3%
2**+2, &c. is
(*+ (23*** + (33***, &c. – 13" (22 (32"-?, &c.
m-1
mo
mmm
m-3
x
23, &c
M m
+ (2x
mam 2 zt
mt3
z?, &c.
2
9-3 2°,&c
+ (3-*
*c.
+&c.
&c.
- From theſe Examples it is evident, that always the Index.of the higheft
Power of the variable Quantity in the Increment is leſs by an Unit than
the Index of the higheſt Power of the Integral; and that the other ſuc-
ceeding Terms deſcend as the Binomial Theorem for raiſing integral
Powers; and conſequently, the Form of the Integral of any Power is
known.
Wherefore, if it be required to find the Integral of
m-3
(1% + (2x + (3% + (4%
we may with Safety put it
n.1
m.2
,&c.
m
w
azt sz" +22" + dz &c.
where m's Increment is an Unit. And then to determine the Coefficients,
de B, Yo &c. we have
fi %
+ (2 %
+
(3 2
+ (4 272
&c.
m m m
m m m m
min
-% 23 +
mm
-% %4,&C.
z zt
= ax m % 2 +
1
2. 3
.
2. 3. 4
2
mm
m mm
m'n
+BX
m 2 %
+
+
z %3,&C.
2
2 3
m m
1
tox
22 2
+
2 z?,&C.
2
+^x
% , &c.
There
SER
SE M
Therefore
(2
9
X
B -
xxº
m
2
(3
ß m
M m
Х
gs
- ا
m
2
12
2. 3
(4
(1
mmm
X
B mm
2 m
X
2. 3
2
m
2. 3. 4
M M M M
mma
(5 (
B m m m
y mm
on m
It
x 23
2. 3. 4. 5
2. 3. 4
2. 3.
2
&c.
But if a, b, 2, &gc. denote the Coefficients, excluſive of the Powers
of %,
Men
m
(2
(1
B =
X
2
(3
(1
>
X
to BX
3
2
ma
(4
om
(1
뷰
​X
X
ху
Х
2
4
3
加
​m
ma
(5
(1
M
+ 6 x
4
х ух
tx
%
-5
3.
m
&C.
Hh3
There
SER
SER
Therefore if n denotes the Order of the Terms, the firſt will be
MI
'% and either of the following Terms will be
+
(1
mt
21
(+8xtg *
int2f 2
&c. X2
%
+8x
in orci
2
12
n
In which Theorem we go on till a Value of n = 2, and the Number of
the Terms will be m + 1.
Hence, if there be a Series of Cubes whoſe Roots are in an Arithmeti-
cal Progreſſion, and z3 be put for the Term which immediately follows
the laſt; to ſum up ſuch a Series, we muſt find the Integral of z3. In
which Cafe m= 3,1 1, and all the following Values of C, are no-
things. Therefore
The firft Term
I
(1 209
=12%,
The ſecond Term
(n
m+2 -1
The third Term
x
mf2f
z*--*=o={***
XZ
m
(
(
{n
m + 2 –
FBxnxx+2******2=-*
1
n
"
22
1 * * x=+%? %
The fourth Term
(*
m+2-1
+ Bxm +,*+ * x 2m +2-*
1.-2
2
.
Tx} + x x%7? = 0.
Therefore the Integral is 1 &4x-'-1z3+1%. % + A.
But when a, the firſt Term in the Arithmetical Progreſſion, is that
which immediately follows the laſt, the Sum is nothing.
There"
S E X
SHO
3
Therefore a**-********+A=0.
Whence A=-*a* " +
a mae
I
1
2
Therefore the Sum fought is,
I
I
1
I
3
4
Z
z²+
% %
at
*+
Q3
a
11
1
1 +
4
N
.4
2
4
2
2
2 X Z
-axa-z
7
I
xxx--|-
2
4
a²zxxa
*arpa
43
or
SerENTINE LINE, the fame they are ſtill retained in many Caſes,
with Spiral; which ſee.
though Decimal Arithmetick begins
SesQUIALTER, in Muſic. See to grow in uſe now in Aftronomical
Time.
Calculations.
SESQUIALTERALPROPORTION, SEXANGLE, in Geometry, is a
is when any Number or Quantity Figure conſiſting of fix Angles.
contains another once and an half; SexTANS, is the fixth part of
and the Number ſo contained in the any Thing : Thus, there is an
greater is ſaid to be to it in ſubfer- Aſtronomical Inſtrument called a
quialteral Proportion.
Sextant, as being the 6th Part of a
SESQUIQUADRATE, an Aſpect Circle. This hath a graduated Limb,
Poſition of the Planets, when they and is uſed like a Quadrant.
are at the Diſtance of four Signs and Sextile, the Polition or Aſpect
an half, or 135 Degrees from each of the Planets, when at 60 Degrees
other,
diſtance, or at the Diſtance of two
SES QUI QUINTILE, an Aſpect of Signs from one another; and is
the Planets, when 102 Degrees di- marked thus
Itant from each other.
SHOULDERING, in Fortification,
SESQUITERTIONAL PROPOR. is a Retrenchment oppoſed to the
TION, is when any Number or Enemies, or a Work caft up for
Quantity contains another once and Defence on one ſide, whether it be
one Third.
made of Heaps of Earth caft up, or
SexAGENARY TABLES, were of Gabions and Faſcines. A Shoul.
Tables contrived (formerly) of Parts dering alſo is a ſquare Orillon fome-
Proportional ; where, by Inſpection, times made in the Baſtions on the
you may find the Product of two Flank near the Shoulder, to cover
-Sexagenaries to be multiplied, or the che Cannon of a Caſemate. Again,
Quotient of two that are to be di- it is taken for a Demi-Baſtion, or
vided by one another, &c.
Work conſiſting of one Face, and
SEXAGESIMAL FRACTIONS, or one Flank, which ends in a Point
Sexagenaries, are ſuch as have al at the Head of a Horn-work or
ways 60 for their Denominators: Crown-work: Neither is it to be
There were antiently no others uſed underſtood only of a ſmall Flank ad-
in Alronomical Operations; and ded to the ſides of the Hornwork, to
Ih
defend
SIM
SI M
defend them when they are too long, ing inſcribed within one of them,
but alſo to the Redoubts which are we can inſcribe always a ſimilar
raiſed on a ſtrait Line.
Right-lined Figure in the other.
SIDEREALYEAR. See Solar Year. SIMILAR CONIC SECTIONS:
SILLON, in Fortification, is an Two Conic Sections are ſaid to be
Elevation of Earth, made in the fimilar, when any Segment being
Middle of a Moat, to fortify it when taken in the one, we can aſſign al-
too broad : It is otherwiſe called ways a ſimilar Segment in the other.
Envelope, which is the more com SIMILAR DIAMETERS of two
mon Name.
Conic Sektions. The Diameters in
SIMILAR, in Geometry, is the two Conic Sections are ſaid to be
ſame as like.
ſimilar, when they make the ſame
SIMILAR ARCHES of a Circle, Angles with their Ordinates.
are ſuch as are like Parts of their SIMILAR SOLIDS, are ſuch that
whole Circumferences.
are contained under equal Numbers
SIMILAR BODIES, in natural of ſimilar Planes, alike ſituated.
Philoſophy, are called ſuch as have SIMILAR TRIANGLES, are ſuch
their Particles of the ſame Kind and as have all their three Angles re-
Nature one with another.
ſpectively equal to one another.
SIMILAR Plane Numbers, are 1. All ſimilar Triangles have the
thoſe Numbers which may be ranged Sides about their equal Angles pro-
into the form of Similar Rectan. portional.
gles : That is, into Rectangles 2. All ſimilar Triangles are to one
whoſe Sides are proportional, ſuch another, as the Squares of their ho-
are 12 and 48 ; for the Sides of 12 mologous Sides.
are 6 and 2. and the sides of 48 are SIMPLE FLANK. See Flank.
32 and 4. But 6. 2 :: 12. 4. and SIMPLE PROBLEM, in Mathe-
therefore thoſe Numbers are Simi- macicks. See Linear one.
lar.
SIMPLE QUANTITIES, in Al-
SIMILAR POLYGONS, are ſuch gebra, are ſuch as have but one Sign,
as have their Angles ſeverally equal, whether poſitive or negative : Thus,
and the Sides about thoſe Angles 2a, and 3b, are ſimple Quantities.
proportional.
But at5, and +-th are com-
SIMILAR RECTANGLES, are pound ones.
thoſe which have their Sides about SIMPLETENAILLE. See Tenaille.
the equal Angles proportional. SINE, or Right Sine, is a Right
1. All Squares are Similar Rect- Line drawn from one End of an
angles.
Arch, perpendicularly upon the
2. All Similar Rectangles are to Diameter drawn from the other
each other as the Squares of their End of that Arch; or it is half the
homologous Sides.
Chord of twice the Arch.
SIMILAR Right-lind Figures, If the Radius be 1, then the
fuch as have equal Angles, and the Length of the Arch of a Quadrant
Sides about thoſe equal Angles pro- will be 1.57070, &c. and the Square
portional.
of it is 2.4694, 8c. Now iſ this
SIMILAR SEGMENTs of a Circle, Square be divided by the Square of
are ſuch as contain equal Angles. a Number expreſſing the Ratio of
SIMILAR Curves. Two S-g- 90 Degrees to any given Angle, as
ments of two Curves are called fimi- 1, and the Quotient be called z,
Jar, if any Right-lined Figure, bc- three or four Terms of this Series
1-
1
22
23
20
A.
0
SOL
SOL
The Sidereal Year, is the Space
1-5+24
+ 4
Eg c. wherein the Sun comes back to any
40320
will give the Cofine of the Angle 366 Days, eight Hours, and nine
particular fixed Star, which is about
Minutes.
Sine COMPLEMENT. See Com-
SOLID ANGLE, is an Angle
Alement.
SINGLE, or Simple Eccentricity. more Planes, and thoſe joining in a
made by the meeting of three or
See Eccentricity.
Point, like the Point of a Diamond
SINICAL QUADRANT is made of
well cur.
Braſs or Wood, with Lines drawn
SOLID BASTion. See Baſtion.
from each ſide interſecting one ano-
ther with an Index, divided by Sines, Species of Magnitude, having three
Solid, in Geometry, is the third
alſo with ninety Degrees the Dimenfions, Length, Breadth, and
Limb, and two Sights to the Edge, Thickneſs, and is frequently uſed in
to take the Altitude of the Sun.
Sometimes inſtead of Sines, 'tis di- be conceived to be formed by the
the fame Senſe with Body; it may
vided all into equal Parts : and is direct Motion, or the Revolution of
uſed by Seamen, to ſolve by Inſpec- any Superficies, of what Nature or
tion any Problem of Plain-failing.
crooked Pipe, Tube, or Cane. See Iſaac Newton, in his Principia ,
SIPHON, a Glafs or Metalline Figure ſoever.
SOLID of leaſt Reſiſtance. Sir
Syphon.
ſhews, that if there be a Curve-
SIRIUS, the Dog-Star, a bright
Star of the firſt Magnitude in the Figure, as DNFB, of ſuch a Na-
Conſtellation Canis Major. Its Lon- ture, as that from any Point, as
N, taken in its Circumference, a
gitude is 99 Degrees, 47 Minutes, Perpendicular NM be let fall to the
Latitude 39 Degrees 32 Minutes.
Axis AB: And if, from a given
SLIDING Rules, or Scales, are
Inſtruments to be uſed without Com- be drawn parallel to a Tangent to
Point, as G, the Right Line GR,
paſſes, in Gauging, Meaſuring&c.
having their Lines fitted ſo, as to
D N
anſwer Proportions by Inſpection ;
they are very ingeniouſly contrived
G
and applied by Gunter, Partridge,
A C с
Copfhall, Everard, Hunt, and others,
M
who have written particular Trea-
F
tiſes about their Uſe and Applica-
tion.
SOLAR Comer. See Diſcus.
SOLAR CYCLE. See Cycle of the
Sun.
the Curve in that Point N: And
SOLAR Spots. See Spots of the alſo, if the Axis being produced,
Sun.
till GR cut it, it will then be as
SOLAR YEAR, is either Tropical
or Sidereal.
MN: GR :: GR3 : 4BG x GR.
Tropical Year, is that Space of
Time, wherein the Sun returns a Then the Solid, which may be ge-
gain to the fame Equinoctial or Sol- nerated by the Revolution of this
iticial Point, which is always equal Curve round its Axis HB, wnen
to 365 Days, tive liours, and about moved moſt ſwiftly in a rare and
55 Minutes.
elaſtick Medium, lall meet with
del's
B
R
H
I
SQL
SOU
leſs Refiftance from the Medium, Mr. Halley ſhews in Philos. Tranfast.
than any Ciréular Solid whatſoever, Nº 188.
deſcribed after the fame Manner, SOLIDITY, (ſee Firmnes) is a
and whoſe Length and Breadth are Quality of a Natural Body contrary
the fame às that.
to Fluidity, and appears to conſiſt
As Sir Iſaac Newton did not give in the Parts of the Body's being in-
a Demonſtration of this famous The- terwoven and intangled one with an-
ořem, ſeveral have done it for him; other, ſo that they cannot diffuſe
amongſt which, Mr. Facio's is a very themſelves ſeveral ways, as Fluid
uncommon one, altho' ingenious e-
Bodies can.
nough. Mr. Bernoulli alſo has done SOLSTICE, is the time when the
it in the Aeta Eruditornm, A. 1699. Sun, entering the Tropical Points, is
p. 514. And ſo has the Marquis got furtheſt from the Equator, and
de l'Hoſpital in the French Memoirs before he returns back towards it,
of the Royal Academy of Paris. in the fame Parallel, and ſcarce mak-
See my Tranſlation of this Author's ing any other Lines than perfect
Infiniment Petit.
Circles, fo ſmall is its Progreſs.
SOLID NUMBERS, are thoſe Theſe Solſtices are two :
which ariſe from the Multiplication 1. Æſtival, or Summer Solſtice,
of a plain Number, by any other when the Sun enters Cancer, June
whatſoever ; as 18 is a Solid Num. Il, making the longeſt Day, and
ber made of 6, (which is Plane) mul- the ſhorteſt Night.
tiplied by 3 ; or of g multiplied by 2. 2. And the Hyemal, or Winter
SOLID PLACE. See Solid Locus. Solſtice, December 11, when he en-
SOLID PROBLEM, in Mathema- ters Capricorn, the Nights being
ticks, is one which cannot be geome. then at the longeſt
, and the Days at
trically ſolved, unleſs by the Interſec- the ſhorteſt, that is, in Northern Re-
tion of a Circle,and a Conick-Section; gions ; for under the Equator there
or by the Interſection of two other is no Variation, but a continual
Coniclt-Sections beſides the Circle. Equinox; and in the Southern Parts,
1. As to deſcribe an Iofceles Tri- the Sun's Entrance into Capricorn
angle on a given Right Line, whoſe makes the longeſt Day, and into
Angle at the Baſe ſhall be triple to Cancer, the longest Night.
that at the Vertex.
SOLUTION, in Mathematicks, is
2. This will help to inſcribe a the Anſwering of any Queſtion, or
Regular Heptagon, in a given Circle; the Reſolution of any Problem.
and may be reſolved by the Inter SOUND, ſeems to be produced by
ſection of a Parabola and a Circle. the ſubtiler and more ætherial Parts
3. The following Problem alſo of the Air, being formed and modi-
helps to infcribe a Nonagon in a fied into a great many ſmall Maſſes
Circle ; and may be ſolved by the In or Contextures, exactly ſimilar in
terſection of a Parabola, and an Hy- Figure; which Contextures are made
perbola between its Afymptotes, viz. by the Colliſion and peculiar Mo-
4. To deſcribe an Iſoſceles Trian- tion of the ſonorous Body, and lly-
gle, whoſe Angle at the Baſe ſhall ing off from it, are diffuſed all
be quadruple of that at the Vertex. round in the Medium, and there do
5. And ſuch a Problem as this affect the Organ of our Ear in one
hath four Solutions, and no more ; and the fame Manner.
becauſe two Conick Sections can cut Sound alſo appears not to be pro-
one another but in four Points. How duced in the Air lo much by the
all ſuch Problems are conſtructed, Swiftneſs, as by the very frequer
Reper-
SOU
SOU
Repercuſſions, and reciprocal Shak- been obſerved of Sound ; in many
ings of the ſonorous Body.
of which there is a near Relation be-
Sir Iſaac Newton demonſtrates, tween it and Light : For,
(-Prop. 43. Lib. 2. of his Principles,) 1. As Light acquaints the Eye.
that Sounds, becauſe they ariſe from with the different Qualities, Mag-
the tremulous Motion of Bodies, are nitudes, and Figures of Bodies, ſo
nothing elſe but the Propagation of Sound, in like manner, informs the
the Pulſe of the Air: And this, he Ear of many of the ſame Things in
faith, is confirmed by thoſe great the ſonorous Body.
Tremors that ſtrong and grave 2: As Light preſently vaniſhes
Sounds excite in Bodies round about, on the Removal, or total Eclipſe of
as the Ringing of Bells, Noiſe of the Radiating Body, ſo a Sound pe-
Cannon, &c.
riſhes as ſoon as the Undulation of
And in other places he concludes, the Air ceaſes, which Motion both
that sounds do not conſiſt in the produces and preſerveth all Sounds.
Motion of any Æther, or finer Air, 3. The Diffuſion of Sound from the
but in the Agitation of the whole fonorous Body is ſpherical, like the
common Air; becauſe he found by Radiation of Light from its Centre.
Experiments, that the Motion of 4. A great Sound drowns a leſs,
Sound depended on the Denſity of as a greater Light eclipſes a lefs.
the whole Air.
5. Too great, loud, or thrill a
He found by good Experiments, Sound is offenfive and injurious to
that a Sound moves 968 Foot, the Ear, as too great and bright a
Engliſh, in a Second of Time, ſup. Light is to the Eye.
poſing the Air by the Pulſe which 6. Sound alſo (like Light) moves
caufes Sound, to be in a Motion, fenfibly from Place to Place, though
like that of Water, when its Waves nothing near fo ſwift as Light: It
roll: He calculates the Breadth of is reflected like Light from all hard
the Pulſe, or the Diſtance between Bodies; it is hindered and refracted
Wave and Wave, to be in the by paſſing through a denſer Medi-
Sounds of all open Pipes double the um. But it differs from Light in
Length of thoſe Pipes; which he this, That whereas Light is always
grounds on an Experiment of Father propagated in Right-Lines, the
Merſennus, in his Harmonics, that Motion of Sound is almoſt always
an extended String made 104 Vi- curvilineal.
brations in a Second, when it was 7. Sound alſo differs much from
an uniſone with the C faut Pipe of Light in this, That it is very much
an Organ, whoſe Length was four weakened by Winds, and ſuch-like
Foot open, and two Foot ſtopped. Motions of the Air, which yet have
Why the Sound ceaſes always no Effect on Light: For Nerſennus
with the Motion of the ſonorous computes, that the Diameter of the
Body, and why they reach the Ear Sphere of a Sound heard againit che
equally ſoon, when far off or near, Wind is near a third Part leſs, than
he ſhews in Prop. 48. Cor. Where when coming with the Wind.
he proves, chat the Number of the 8. A very ſmall Quantity of Body
Pulles propagated, is always the very ferves to reflect the Rays of Light;
fame with the Number of the Vi- as we perceive manifeitly in ſmall
brations of the tremulous Body, and Pieces of Looking-Glaffes, &c. But
that they are not by any means mul- here appears to be neceflary a Body
tiplied as they go from it.
of much larger Dimenſions to re-
The following Properties have turn a Sound, or make an Echo.
Der
9.
SOU
SOU
9. As to the Reflections of the Surface of Water, which is cal-
Sounds, 'cis obſerved, that if one. led a Wave of Air.
stands near the refting Body, and 12. And the Motion of theſe
the Sound be not very far off, though Waves is the Motion of a Sphere ex-
an Echo be produced, yet it can- panding itſelf in the fame Manner
not be heard; becauſe the Direct as the Waves move circularly upon
and Reflex Sound enter the Ear al- the Surface of the Water.
moſt at the ſame time : But then 13. While a Wave moves in the
the Sound appears to be ſtronger Air, wherever it paſſes, the Parti-
than ordinary, and laſts longer, eſpe- cles are removed from their Place,
cially when the Reflection is made and return to it, running through
from divers Bodics at once; as from
a very hort Space in going and
Arches and vaulted Rooms, from coming.
whence the confuſed Sound of ſuch 14. Wherever the neighbouring
like Places ariics.
Particles are not equally diſtant, the
And from hence probably may Motion ariſing from Elaſticity cauſes
be deduced the Reaſon, why Con- the leſs diſtant Particles to move to-
cave Bodics are (cæteris paribus) wards thoſe that are moſt diftant.
fitteſt to produce great and clear 15. Therefore, the Motion of the
Sounds, ſuch as Rells, &c. For in tremoulous Body, by which the Air
fuch Bodies the found is very ſwift- is agitated, ceaſing, there are new
ly and very often reflected from fide Waves generated,
to ſide, and from one part of the 16. Waves, whether the Air be
Cavity to another, and the Bell more or leſs agitated, are equally
hanging at liberty, this produces ſwift,
great Tremblings and shakings of 17. Waves, whether equal or any
the whole Concave Body, which oc- Way unequal, move with the ſame
cafions the Sound to continue till
Velocity
they ceaſe and are quiet.
18. In Waves, the Squares of
io. There is one Phænomenon, their Celerities are inverſly as the
viz. that Sounds great or ſmall, Denſities.
with the Wind, or againſt it, from 19. When the Denſity remains the
the fame Diſtance, come to the ſame, but the Elaſticity is changed,
Ear at the ſame time.
the Squares of the Celerities of the
Dr. Holder, in his Book of the Waves are as the Degrees of the E-
Natural Grounds and Principles of laſticity.
Harmony, ſays, That if the tremu 20. If the Elaſticity and the Den-
lous Motion which cauſeth Sound fity differ, the Squares of the Velo-
be uniform, then it produces a mu. cities of the Waves will be in a Ratio
fical Note or Sound : But if it be compounded of the direct Ratio of
difform, then it produces a Noiſe. the Elaſticity, and the inverſe Ratio
The Florentine Academicks found of the Denſity.
a Sound to move one of their Miles
21. If the Denſity and the Elafli,
(viz. 3000 Braccia, or 5925 Foot) city incrcaſe or decreaſe in the ſame
in five Seconds of Time: There- Ratio, the Celerity of the Waves
fore, according to them, it moves will not be changed
1185 Foot in one Second.
22. Therefore, from the changed
But Sir Iſaac Newton found it to Height of the Pillar of Mercury,
move bue 968 Foot in one Second. which is ſuſtained in a Tube void of
11. If the Air be agitated in any Air hy the Preſſure of the Atinot
Manner, there ariſes a Motion anạ- ſphere, we muſt not judge the Ce-
logous to che Motion of a Wave on lerity of the Waves to be changed.
22,
Summer, Bodies do more eaſily tranſ- JIS
SPACE, if conſidered barely in
SOU
SOU
23. For the Waves are moved compreſſing the Air; and lafily, of
with the fame Celerity in the Top the inverſe Ratio of the Square Root
of a Mountain, as in a Valley. of the Elaſticity.
24. The Waves move faſter in 37. And the Degrees of the Sharp-
Summer than in Winter.
neſs of different Sounds are to one
25. By determining the Height another, as the Number of the
of the Atmoſphere, ſuppoſing it e. Waves which are produced in the
very where equally denſe with the Air at the ſame time.
Air near the Earth, the Velocity of 38. A Tone does not depend upon
the Waves will be the ſame as a Bo- the Intenſity of the Sound, and an
dy could acquire in falling from half agitated Cord gives the fame Sound,
that Height.
whether it vibrates through a greater
26. The Motion of Waves in the or a leſs Space.
Air produces Sound.
39. Concords ariſe from the A-
27. A Body that is ſtruck, con- greement between the different Mo-
tinues to give a Sound ſome time tions in the Air, which affect the au-
after the Blow,
ditory Nerves at the ſame time.
28. The Celerity of the Sound is 40. Cæteris paribus, if the Lengths
the ſame as the Celerity of the Waves, of two Cords are as the Number of
which Itrike the Ear.
Returns in a Conſonance, you will
29. The Celerity of Sound is e. have the Conſonance between the
quable; yet in going through a Sounds which the Strings produce.
great Space, it is ſometimes acce 41. And generally ſuppofing any
lerated or retarded.
Cords of the ſame kind, if the Ra-
30. The Celerity of Sound does tio be compounded of the direct Ra-
not much differ, whether it goes tio of the Lengths and of the Dia-
with the Wind, or againſt it. meters, and the inverſe Ratio of the
31. Therefore, Sound may be Square-Roots of the Tenfiong, (be
heard at a greater or ſmaller diſtance, the Ratio between the Numbers of
according to the Direction of the the Vibrations performed in the
Wind.
ſame time in any Conſonance what-
32. Cæteris paribus, the Intenſity ever,) you will have that Conſo-
of Sound is as the Space run through nance by the Agitation of thoſe
by the Particles in their going and Cords.
coming
42. An agitated String will com-
33. Therefore, cæteris paribus, the municate Motion to another, which
Intenſity of Sound is as the Weight performs two or three Vibrations,
by which the Air is compreſſed. whilſt the firit performs but one.
34. If all things remain as before, SOUND, in Geography, is any
and the Elaſticity be increaſed, the great Indraught of the Sea, between
Intenſity of Sound is directly as the two Headlands, where there is no
Square Root of the Elaſticity, and Paſſage through.
inverſely as the Elaſticity itſelf. SOUTH DIRECT DIALS. See
35. The Intenſity of Sound is leſs Prime Verticals.
in Summer than in Winter ; yet in SOUTHERN SIGNs. See Auftral
mit Sound.
36. The Intenſity of Sound, con- Length between any two Beings, is
ſidered in general, is in a compound the fame Idea that we have of Di-
Ratio of the Space run thro gh by flance; but if it be conſidered in
the Particles, in their going back. Length, Breadth and Thickneſs, it
ward and forward, of the Weight is properly called Capacity ; and
when
SPE
SPH
when conſidered between the Ex- Name of Species to the Letters of
tremities of Matter, which fills the the Alphabet ſubſervient to Alge-
Capacity of Space with ſomething bra, and why he calls it Arithmen
folid, tangible and moveable, or tica Speciofa, ſeems to have been in
with Body, it is then called Exten- imitation of the Civilians, who call
fion ; ſo that Extenſion is an Idea Caſes in Law, but abſtractedly, be-
belonging to Body only. But Space, tween John a Nokes and Tom a Stiles,
in a general Signification, is the between A and C; ſuppoſing thoſe
fame thing with Diſtance, conſider. Letters to ſtand for any Perſons in-
ed every way, whether there be any definitely ; ſuch Caſes, I ſay they
folid Matter in it or not.
call Species : Wherefore ſince the
Space, therefore, is either Abſo- Letters of the Alphabet will alſo
lute or Relative.
as well repreſent Quantities, as Per-
ABSOLUTE SPACE, conſidered ſons, and that too indefinitely one
in its own Nature, and without re-Quantity as well as another, they
gard to any thing external, always may properly enough be called Spe-
remains the ſame, and is immove. cies ; that is Symbols, Marks, or
able; but Relative Space is that Characters. From whence the li-
moveable Dimenſion or Meaſure of teral Algebra is frequently now-a-
the former, which our Senſes de- days called Specious Arithmetic, or
fine by its Poſitions to Bodies with- Algebra in Species.
in it: And this the Vulgar uſe for SPECIFIC, is in general what-
immoveable Space.
ever is peculiar to any diſtinct Spe-
RELATIVE SPACE, in Magni- cies of Things, and which diſtin-
tude and Figure, is always the ſame guiſhes them from all others of dif-
with Abſolute, but 'tis not neceſſary ferent Species; therefore the Logi-
it ſhould be ſo numerically. Thus, cians fay, that in every good Defi-
if you ſuppoſe a Ship to be indeed nition of any thing, the ſpecific Dif-
in abſolute Reſt, then the Places of ference ought always to be in-
all things within her will be the ferted.
fame abſolutely and relatively, and SPECIFIC GRAVITY, is the
nothing will change its Place. But appropriate and peculiar Gravity
then ſuppoſe a Ship under Sail, or or Weight which any Species of na-
in Motion, and ſhe will continually tural Bodies have, and by which
paſs through new Parts of abſolute they are plainly distinguiſhable from
Space; but all things on board con- all other Bodies of different kinds.
fidered relatively, in reſpect to the By fome 'tis not improperly called
Ship, may be notwithitanding in the Relative Gravity, to diſtinguiſh it
fame Places, or have the fame Situ. from Abſolute Gravity, which in-
ation and Poſition, in regard to one creaſes in proportion to the Bigneſs
another.
of the Body weighed.
SPECIES, in Algebra, are thoſe SPHERE, is a ſolid Body made
Letters, Notes, Marks, or Symbols, by the entire Rotation of a Semi-
which repreſent the Quantities in Circle about its Diameter.
any Equation or Demonſtration. 1. All Spheres are to one another,
This ſhort and advantageous way of as the Cubes of their Diameters.
Notation was introduced by Vieta, 2. The Solidity of a Sphere is
about the Year 1590, and by it equal to the Surface multiplied into
made many Diſcoveries in the Pro one third of the Radius.
ceſs of Algebra, not before taken no 3. The Surface of the Sphere is
tice of.
equal to four times the Area of a
The Reaſon why Victa gave this great Circle of it.
4.
1
درو
30
کو
56c5 yi
576c7 - &c.
533
&c.
20 24
x5
3x5
40624
5*?
336 caº
5625
5029
576986
SPH
SPH
4. As 2904 to 49, fo is the Cube and if it be cut by four Planes, AB
of the Circumference of a Sphere to pafling through the Axis ; DG pa-
its folid Content.
rallel to AB,C DE, perpendicular-
5. As 22 is to 7, fo is the Square ly biſfecting the Axis; and FG pa-
of the Circumference of the greateft rallel to CĚ; and if the Right Line
Circle of a Sphere to the ſuperficial CB=a, CE, CF=x, and FG=y:
Area of the Sphere.
Then the Segment CDGF of the
6. As 21 is to the Sine, fo is it Spheroid comprehended under the
times the Square of that Sine added ſaid Planes will be = 20 x
y
to 33 times the Square of half the
Chord of any Segment of a Sphere
to the folid Content of that Seg-
2003
? -
ment.
5*
២
7. As 14 is to 44 times the Dia-
meter of any Sphere, ſo is the сх3 23 *3
Length of the Sine of any Segment
3aa
18caa 40c3 aa
4003 aa 3365 aa
of it, to the Convex Superficies of
the faid Segment.
cxs
8: An entire Glaſs Sphere will
160 c3 at
&c.
unite the parallel Rays of an Object
at the Diſtance of near its Semi-Dia-
сх7
&c.
meter behind it.
SPHERE of Aktivity of any Body,
is that determinate Space or Extent
&c.
all round about it, to which, and no
farcher, the Elluviuns continually
2. Where the numeral Co-Ef-
emitted from that Body do reach, ficients of the Terms (2,
and where they operate according 0,&c.) are produced by myl-
to their Nature.
tiplying the firſt Co-Efficient 2 by
SPHERICAL NUMBERS. See the Terms of this Progreſſion
Circular Numbers.
1X2 283 385 5x7 789
SPHERIC GEOMETRY, or Pro- 2x39 4x576x7) 849) 10X11)
& C.
is the Art of defcribing and the numeral Co-Efficients
on a Plane the Circles of the Sphere, in each Column of the deſcend-
or any Parts of them in their juft ing Terms are produced, by mul-
Poſition and Proportion, and of tiplying continually the Co-EF-
meaſuring their Arches and Angies ficients of the upper Term in the
when projected.
SPHEROID, is a folid Figure fion; but in the ſecond by the.
firſt Column, by the fame Progreſ-
made by the entire Rotation of a Terms of this,
Semi-Ellipſis about its Axis.
1. If AEB be a Spheroid gene- XI 3X35X5 787
rated by the Revolution of the El. 2x3 4x576x7) 8x9?
lipſis AEB about the Axis AB, In the third, by the Terms of this,
9x73x1 543 745 9x7
E
&c.
8x9) 2x32 4x596x72.89)
In the fourth, by the Terms of
D G
581 73 985
this
c.
2x3) 4x59 6x72
A
C F B
3
3.
EC.
S PH
SPH
3. A Spheroid generated by an will be equal to the Sum of all the
Ellipfis revolving upon the Diame- Annuli, that is, the Exceſs by which
ter thereof, is of its circumſcribing the Cylinder exceeds the Spheroid.
Cylinder. Suppoſe ADLB be a Therefore, the Propoſition is mani-
Quadrant of an Ellipſis, then if the feft, that a Spheroid, generated by
whole Figure (AL) is conceived to an Ellipſis, revolving upon any Dia-
revolve upon the Semi-Diameter BL, meter thereof, is two thirds of its
the Semi-Ellipfis ALB will deſcribe circumſcribing Cylinder. Q. E. D.
a Semi-Spheroid, and the Parallelo The great Geometrician, Mr.
gram AMLB a Cylinder ; and Huygens, in his Horolog. Ofcill. gives
laſtly, the Triangle MBL a Cone, the following two moſt elegant Con-
all having the fame Bafe and Alti- ftructions for deſcribing a Circle
equal to the Superficies of an
oblong and prolate Spheroid,
M
I which, he ſays, he found out to-
wards the latter End of the Year
1657
Let an oblong Spheroid be gene-
E
DF
G
rated by the Rotation of an Ellipfis
ADBE, (Fig. 1.) about its trans-
verſe Axis AB, and let DE be its
A
B
Conjugate ; make D F equal to C.B,
or let F be one of the Foci, and
draw BG parallel to FD, and a-
tude. Now, draw any Line EG
parallel to the Baſe, then by the
Fig. I.
Nature of the Ellipſis BL : AB:
A
BL- BG: GD. But from the
fimilar Triangles BML, BFG, we
have BL : AB =ML
AB=MÃ= EG")
:: GB": GF. And (alternando )
D C
E
H
BL”: GB” :: AB" : GF". And
(dividendo) BL : AB :: BL
GB': AB’ --GF. Whence, fince
before it was B L: AB :: BL
B
BG”: GD;
: GD; therefore
therefore (11. 5.
Eucl.) AB*_ GĚ, that is EG
GF= DG”; and fo EG* =
A
DG + FGʻ. Whence ,
c
D
E
4. The Circle made by the Re-
volution of (FG) will be equal to
the Annulus deſcribed by (ED) and
the Sum of all the Circles (FG)
B
that is, the Solidity of the Cone
bout
2
2
2
G
2
2
bl
z
z
Fig. 2.
F
SUIKER
S PI
SPI
bout the Point G with the Radius the Curve-Line B, 1, 2, 3, 4, 5, &*c.
BG deſcribe an Arch BHA of a which is called an Helix, or Spirál
Circle; then between the Semi-Con- Line ; and the plain Space contained
jugate CD, and a Right Line equal between the Spiral-Line and the
to Det the Arch AHB, find a Right-Line B.A, is called the Spiral
mean Proportional, and that will be Space.
the Radius of a Circle equal to the 2. If alſo you conceive the Point
Superficies of the oblong Spheroid. B to move twice as flow as the Line
Let a prolate Spheroid be gene- AB, ſo as that it ſhall get but half-
rated by the Rotation of the Ellip- way along BA, when that Line ſhall
fis ADBE (Fig. 2.) about its con have formed the Circle, and if then
jugáte Axis A B. Let F be one you imagine a new Revolution to be
of the Foci, and biſfect CF in G, made of the Line carrying the
and let AGB be the Curve of the Point, ſo that they ſhall end their
common Parabola whoſe Baſe is the Motion at laſt together; there will
conjugate Diameter AB, and Axis be formed a double Spiral Line, and
CĠ. Then if between the tranf two Spiral Spaces, as you ſee in the
verſe Axis DE, and a Right Line Figure.
equal to the Curve AGB of the 3. The Lines B 12, Bil, B 10,
Parabola, a mean Proportional be &c. making equal Angles with the
taken, the fame will be the Radius firſt and ſecond Spiral, fas alſo B
of a Circle equal to the Surface of 12, B10, B8, &c.) are in Arith-
that prolate Spheroid.
metical Proportion.
SPIRAL Line, in Geometry, is 4. The Lines B7, B10, &c.
according to Archimedes thus ge- drawn any low to the firſt Spiral,
nerated.
are to one another as the Arches of
1. If a Right Line, as AB, hav- the Circle intercepted betwixt BA,
ing one end fixed at B, bc equally and thoſe Lines.
moved round, ſo as with the other 5. Any Lines drawn from B to
end A, to deſcribe the Periphery of the ſecond Spiral, as Bi8, B22,
a Circle; and at the ſame time a @ c. are to each ocher, as the afore-
Point be conceived to move forward ſaid Arches, together with the
equally from B towards A in the whole Periphery. added on both
Right Line BA, ſo as that the point fides.
deſcribes that Line, while the Line 6. The firſt fpiral Space, is to
generates the Circle : then will the the firſt Circle, as I to 3.
Point, with its two Mocions, deſcribe 7. The firſt Spiral Line is equal
to half the Periphery of the firſt
Circle ; for the Radii of the Sectors,
and conſequently the Arches, are in
a fimple Arithmetic Progreſſion,
while the Periphery of the Circle
contains as many Arches equal to
the greateſt; wherefore the Periphe-
D
12 24
18
A ry to all thoſe Arches is to the Spi-
ral Line, as 2 to 1.
SPIRALS (PROPORTIONAL,) are
14
ſuch Spiral Lincs as the Rhumb
15
Lines on the Terreſtrial Globe.
SPRING-ARBOR, in a Watch, is
that part in the middle of the Spring-
li
Box,
21:
22
20:
10:
23
4. MAALANIvo
13
1
16
STA
STA
STARS.
Box, which the Spring is wound or poſed of from five to eight Points,
turn'd about, and to which it is with ſaliant and re-entring Angles
hooked at one end,
flanking one another, every one of
SPRING-Box, is that Cylindrical its Sides containing from 12 to 25
Caſe or Frame that contains within Fathoms.
it the Spring of a Watch, or other STAR-FORT. See Fort.
Movement.
See Fixed Stars.
SPRING-TIDE, is the increaſing STATICAL BAROSCOPE. See
higher of a Tide after a dead Neipe : Baroſcope.
This is about three Days before the STATICAL HYGROSCOPB. See
Full or Change of the Moon; but Hygroſcope.
the top, or higheſt of the Spring STATICS, is a Science purely
Tide is three Days after the Full ſpeculative, being a Species of Me-
or Change; then the Water runs chanics converſant about Weights,
higheſt with the Flood, and loweſt and ſhewing the Properties of the
with the Ebb, and the Tides run Heavineſs and Lightneſs, or Æqui-
more ſtrong and ſwift than in the libria of Bodies: When it is re-
Neipes,
ftrained to the ſpecific Weights and
SPRINGY ; the ſame as Elaſtic. Æquilibria of Liquors, it is called
Which ſee.
Hydroſtatics. Which ſee.
SPUNGING of a great Gæn, is STATION, in Aftronomy, figni-
clearing of her Inſide, after ſhe hath fies certain Places of the Zodiac,
been diſcharged, with a Wad of where a Planet being arrived, ſeems
Sheep-Skin. or the like, rolled a to ſtand ſtill for ſome time in the
bout one end of the Rammer : Its fame Degree, either in aſcending
Deſign is to prevent any Parts of to its Apogee, or deſcending to its
Fire from remaining in her ; which Perigee.
would endanger the Life of him who Apollonius Pergæus has ſhewn how
fhould load, or charge her again. to find the Stationary Point of a
SQUARE, is an Inſtrument of Planet, according to the Old The-
Braſs or Wood, having one fide per- ory of the Planets, which ſuppoſes
pendicular, or at Right Angles to them to move in Epicycles; which
the other ; ſometimes made with a was followed by Ptolemy in his
Joint to fold for the Pocket, and Almag. lib. 12. cap. I. and others
ſometimes has a Back to uſe on a till the time of Copernicus.
See
Drawing-Board, to guide the concerning this, Regiomontanus in E-
Square.
pitome Almageſti, lib. 12. prop. I.
Square FICURE, in Geometry, Copernicus's Revolutiones Cæleft.
is one whoſe Right-lined Sides are lib. 5. cap. 35, 36.- Kepler in Ta-
all equal, and its Angles all right. bulis Rudolphinis, cap. 24. - Har .
See Quadrilateral Figure. For its Ārea, man in Miſcellan. Berolinenſ. p. 197.
ſee Area.
- Ricciolus's Almageſt. lib. 7. Sect.5.
SQUARING. By the word Sqrar- cap. 2. Dr. Halley, Mr. Facio,
ing, Mathematicians underftand the Mr. De Moivre, and Dr. Keil, have
making of a Square equal to a Cire treated of this Subject.
cle. Thus the Quadrature or Squar STATION, is a place where a
ing of the Circle, is the finding a Man fixes himſelf and his Inftru-
Square equal to the Area of a Čir- ment, to take (as in Surveying) any
cle.
Angles or Diſtances.
STAR in Fortification, is a Work STATION-LINE. See Line of
with ſeveral Faces generally com- Station.
STA-
STE
STÈ
STATION-STAFF, is an Inſtru STEREOGRAPHICK Projection of
ment conſiſting of two Rulers that the Sphere, is the Projection of the
flide to ten Foot, divided into Feet Circles of the Sphere upon the Plane
and Inches, with a moving Vane or of ſome one great Circle, the Eye
Sight, two of which are uſed with a being in the Pole of that Circle.
Level; and on the Edges, there are In this Projection, a Right Circle
the Links of Gunter's Chain divided. is projected into a Line of Half
It is uſed in Surveying, for the more Tangents.
eaſy taking Off-ſets.
The Repreſentation of a Right
ŚTATIONARY: A Planet is ſaid Circle perpendicularly oppoſed to
to be Stationary, when, to any Éye the Eye, will be a Circle in the Plane
placed on Earth, it appears for ſome of the Projection.
time to ſtand ſtill, and to have no The Repreſentation of a Circle
progreſſive Motion forward in its placed oblique to the Eye, will be
Orbit round the Sun.
a Circle in the Plane of the Pro-
STENTOREOPHONICK TUBE,or jection.
Inſtrument, is the Speaking Trum If a great Circle be to be pro-
pet, invented by Sir Samuel More- jected upon the Plane of another
land.
great Circle, its Centre fhall lie in
Mr. Durham, in his Phyfico-Theo- the Line of Meaſures, diſtant from
logy, Lib. 4. Chap. 3. fays, that the Centre of the Primitive by the
Kircher found out this Inſtrument 20 Tangent of its Elevation above the
Years before Sir Samuel Moreland, Plane of the Primitive.
and publiſhed it in his Mufurgia; If a leſler Circle, whoſe Poles lie
and Caſper Schottus is ſaid to have in the Plane of the Projection, were
feen onė at the Jeſuits College at to be projected; the Centre of its
Rome.-One Conyers, in the Philos. Repreſentation ſhall be in the Line
Tranfa&t. N° 141. gives a Deſcrip- of Meaſures, diſtant from the Centre
tion of an Inſtrument of this kind of the Primitive, by the Secant of
different from thoſe commonly made; that leffer Circle's Diſtance from its
and Mr. s'Graveſande, in his Philo- Pole, and its Semidiameter or Radius
Sophy, finds fault with the Figures of ſhall be equal to the Tangent of that
theſe Inftruments as generally made, Diſtance.
where he would have them to be If a leffer Circle were to be pro-
parabolick Conoids, with the Focus jected, whoſe Poles' lie not in the
of one of its parabolickSections, to fit Plane of the Projection, its Diameter
the Mouth. --See concerning this In- in the Projection, if it falls on each
ftrument too in Sturmy's Collegium fide of the Pole of the Primitive,
Curiofum, Part 2. Tentam. 8. will be equal to the Sum of the Half
STEREOBATA, in Architecture, Tangents of its greateſt and neareſt
is the Greek Word for the firſt Begin: Diſtance from the Pole of the Pri-
ning of the Wall of any Building, mitive, ſet each Way from the Cen-
and immediately ſtanding on the tre of the Primitive in the Line of
Foundation. This is wrongly con-· Meaſures.
founded with Stylobata, which is the If a leſſer Circle, to be projected,
Beginning of a Column, or its Pede- falls entirely on one ſide of the Pole
Ital.
of the Projection, and does not en-
STEREOGRAPHY, is the Art of compaſs it, then will itsDiameter
drawing the Forms of Solids' upon a' be equal to the Difference of the
Plane,
Half-Tangents of its greateſt and
neareſt
1
I'i 2
STY
SUB
ch
nearest Diſtance from the Pole of SUBCONTRARY Position, in
the Primitive, ſet off from the Cen- Geometry, is when two ſimilar Tri-
tre of the Primitive one and the angles are ſo placed as to have one
fame Way in the Line of Meaſures. common Angle V at the Vertex, and
In the Stereographick Projection, yet their Baſes are not parallel.
the Angles made by the Circles on And therefore if the Scalenous
the Surface of the Sphere, are equal Cone B V D be ſo cut by the Plane
to the Angles made by their Repre- CA, as that the Angle C=D, the
ſentatives in the Plane of their Pro-
jection.
V
STILE. See Style.
Strait, or Streight, in Hydro-
graphy, is a narrow Sea ſhut up be-
tween Lands on either ſide, afford-
ing a Paſſage from one great Sea
B
into another, as the Strait of Ma-
A
gellan, the Strait of Gibraltar, &c.
STRIKING-WHEEL, in a Clock,
D
is that which by ſome is called the
Pin-Wheel ; becauſe of the Pins
which are placed upon the Round Cone is then ſaid to be cut ſubcon-
or Rim, (which in Number are the trarily to its Baſe BD; and the
Quotient of the Pinion, divided by Section CA of a Cone thus cut is a
the Pinion of the Detent-Wheel.) In Circle.
16 Days Clocks, the firſt or great SUBDUCTION, the ſame with Sub-
Wheel is uſually the Pin-Wheel ; ftraétion; which ſee.
but in Pieces that go eight Days, SUBDUPLE RAT10, is when any
the ſecond Wheel is the Pin-Wheel, Number or Quantity is contained in
or ſtriking Wheel.
another twice: Thus 3 is ſaid to be
STYLE, in Dialling, is that Line Subduple of 6, as 6 is double of 3.
whoſe Shadow on the Plane of the SUBDUPLICATe Ratio of any
Dial ſhews the true Hour-Line. This two Quantities, is the Ratio of their
is always ſuppoſed to be a part of ſquare Roots.
the Axis of the Earth, and there-
SUB LUNARY, are all Things that
fore muſt always be ſo placed, as are in the Earth, or in the Atmo-
that with its two extreme Points it ſphere thereof, below the Moon.
ſhall reſpect the two Poles of the SUBMULTIPLE NUMBER,
World, and with its upper End, the Quantity, is that which is contained
elevated Pole This Line is the up- in another Number, a certain Num-
per Edge of the Cock, Gnomon, or ber of Times exactly : Thus, 3 is
Index.
Submultiple of 21, as being con-
STYLOBATA, in Architecture, tained in it 7 Times exactly.
is the Pedeſtal of a Column or Pil SUBMULTIPLE PROPORTION,
lar.
the Reverſe of Multiple. Which
STYLOBATON, or Stylobata, in fee.
Architecture, is the ſame with the SUBNORMAL, is a Line, as PC,
Pedeſtal of a Column: This is ſome- determining in any Curve the Inter-
times taken for the Trunk of the ſection of the Perpendicular to the
Pedeſtal, between the Cornice and Tangent in the Point of Contact,
the Baſe'; and then called Truncus, with the Axis. And this Subnor-
as it is alſo by the Name of Abacus. mal in the common or Apollonian
Paraboal,
or
0
1
O
2
2
2
1
manner.
SU B
SUB
Parabola, is a determinate invariable that is, the Nature of the Curve, b:
Quantity; for 'tis always equal to expreſſed by this Equation, 33
half the Parameter of the Axis. 2xxy+bxx-bbx+-byy—3=0, then
this will be the Rule of drawing a
M
Tangent to it: Multiply the Terms
of the Equation by any Arithmetical
Progreſſion ; according to the Di-
menſions of y, ſuppoſe
43--2xxy+bxx-bbx +-by-g3 ; as
3
т у Р
у в с D
alſo according to the Dimenſions
of x, as,
SUBSTITUTION, in Algebra, or *3-2xxy-+-6xx--bbxt-by--3;
Fluxions, is the putting in the room
3
of any Quantity in an Equation fome
the former Product ſhall be the Nu-
other Quantity which is really equal merator, and the latter, divided by xa
to it, but expreſſed after another
the Denominator of a Fraction ex-
preſſing the Length of the Subtan-
SUBTANGENT, in a Curve, is a
Line, as TP, which determines the
gent BD, which in this Caſe will be
Interſection of the Tangent in the -2xx3+2byy--3y3
Axis or a Diameter; and in any 3xx-4xy +2bx-bb.
Equation, if the Value of the Sub-
tangent comes out poſitive, 'tis a fign is that Line drawn on the Plane of
SUBSTYLAR LINE, in Dialling,
that the point of Interſection of the
Tangent and Axis falls on that Side the Dial, over which the Style ſtands
of the Ordinare, where the Vertex at Right-Angles with the Plane.
of the Curve lies, as in the Para-
This is always the Repreſentation of
bola and Paraboloids : But if it comes
the Meridian of that Place, where
out negative, the Point of Interſection
the Plane of the Dial is Horizontal.
will fall on the contrary Side of the
The Angle between this Line and the
Ordinate, in reſpect of the Vertex true Meridian, is the Plane's Diffe-
or Beginning of the Abfcila ; as in rence of Longitude, and is meaſured
the Hyperbola and Hyperboliform
on the Equinoctial.
Figures. And univerſally in all Pa-
SUBSUPER-PARTICULAR PRO-
raboliform and Hyperboliform Fi-
PORTION, is contrary to Super-Par-
gures, the Subtangent is equal to the ticular Proportion, which ſee.
Exponent of the Power of the Ordi SUBTENSE, or Chord of an Arch,
nate multiplied into the Abſciſſa.
is a Right Line extended from one
If CB be an Ordinate to AB in
End of that Arch to the other End
thereof.
any given Angle terminating in any
SUBSTRACTION, in general, is
taking a leſſer Quantity from a
C
greater, to find the Difference be-
tween them, which is commonly
called the Remainder, as the leſſer
Quantity to be ſubftracted is called
the Subitrahend.
D
SUBSTRACTION of whole Num-
bers is performed by placing the
Curve AC, and A B=X, BC=y, leſſer under the greater, as in Ad-
and the Relation between x and y, dition, and then beginning at the
Right
A B
Ii 3
SUB
SUN
From 945
Right Hand, taking each Figure be- to two others 74 and 1} equal
low from that above, and ſetting to them, and then their Difference
down the ſeveral Remainders, or will be 7$* Algebraick Fractions
Differences underneath, and the are ſubftracted much after the fame
Number ſubſcribed will be the Dif- way, and Algebraick Subftraction
ference, or Remainder, of the two in general is performed by connect-
Numbers. But when any one of the ing the given Quantities, as in Ad-
Figures of the under Number is dition, and changing every Sign of
greater than that of the upper, from the Quantity to be ſubftracted into
which it is to be taken, you muſt its contrary, and this Connection
add 10 (in your Mind) to that upper thus altered will be the Difference,
Figure; and having taken the an or Remainder ſought.
ſwerable under one from this Sum, The general Sign or Mark of Şub-
ſet the Difference underneath, and ſtraction is
add an Unit to the Figure next to
SUBTRIPLE Ratio, is when
be ſubtracted. Example 1. From any one Number or Quantity is con-
9758 let us ſubtract 3514. Place tained in another three times. Thus
them thus, 9758
2 is ſaid to be ſubtriple of 6, as 6 is
3514
the Triple of two.
6244 the Difference, or
SUBTRIPLICATE RATIO, is the
Ratio of the Cube-Roots.
Remainder.
Example 2. From ſubtract 608.
SUCCESSION of Signs, is that Or-
der in which they are uſually rec-
945
608
kon'd: As, firſt, Aries, next, Taurus,
then Gemini, &c. This is otherwiſe
737 the Remainder.
called Conſequence.
SUBSTRACTION in Decimal
Fractions is the fame as in whole in Mechanicks for a Bare Axis or
SUCULA, or Succula, is a Term
Numbers, always obſerving to put Cylinder, with Staves in it to move
every Figure of the ſame Place under it round, but without any Tympa-
the like Place above, and imagining num or Peritrochium,
all void Places to be ſupplied with SUN. Our excellent Sir Iſaac
Cyphers. Examples.
Newton faith, in his Principia, that
From
352.09.576 79. the Denfity of the Sun's Heat (which
Take 63.74 .0829 .2987 is proportional to his Light) is ſeven
Remains 288 35 4931 78.7013 times as great at Mercury as with us;
SUBSTRACTION of Vulgar Frac- and therefore our Water there would
tions is performed by taking the be all carried off, and boil away:
Numerator of the leſſer Fraction from For he found by Experiments of the
that of the greater, and ſetting down Thermometer, that an Heat but fe-
the Difference for the Numerator of ven times as great as that of the Sun-
the Fraction wanted, its Denomina- Beams in Summer, will ſerve to make
for being the ſame as either of the Water boil.
Denominators of the givenFractions ; 1. He proves alſo, that the Mat-
which Denominators muſt either be ter of the Sun to that of Jupiter is
equal at firſt, or elſe made ſo by re- nearly as 1100 to 1; and that the
ducing the Fractions to a common Diſtance of that Planet from the Sun,
Denominator. As if from you are is in the ſame Ratio as the Sun's
to take , then will the Remainder Semidiameter.
be. And if from ri you take 14, 2. That the Matter of the Sun to
you must firſt reduce theſe Fractions that of Saturn, is as 2360 to I; and
the
1
S UN
SUP
the Diſtance of Saturn from the Sun Equinox to the Autumnal, than from
is in a Ratio but little leſs than that the Autumnal to the Vernal.
of the Sun's Semidiameter : And 6. The Sun's Diameter is equal to
confequently, that the common Cen- an hundred Diameters of the Earth;
tre of Gravity of the Sun and Jupi- and therefore the Bo ly of the Sun
ter is nearly in the Saperficies of the muſt be 1000000 times greater than
Sun; of Saturn and the Sun, a little that of the Earth.
within it.
Mr. Azout aſſures us, that he ob-
3. And by the ſame manner of ſerved by a very exact Method the
Calculation it will be found, that Sun's Diameter to be no leſs than 21
the common Centre of Gravity of Minutes 45 Seconds in his Apogee,
all the Planets, cannot be more than and not greater than 32 Minutes 45
the Length of the Solar Diameter Seconds in his Perigee.
diſtant from the Centre of the Sun: 7. The mean apparent Diameter
This common Centre of Gravity he of the Sun, according to Sir Iſaac
proves to be at reſt, and therefore Newton, is 32 Minutes 12 Seconds,
tho' the Sun, by reaſon of the yari. in his Theory of the Moon.
ous Poſition of the Planets, may be
8. If you divide 360 Degrees
moved every way, yet it cannot re- i.e. the whole Ecliptick) by the
cede far from the common Centre of Quantity of the Solar Year, it will
Gravity, and this, he thinks, ought quote 59 Minutes 8 Seconds, &c.
to be accounted the Centre of our which therefore is the Quantity of
World. Book 3. Prop. 12.
the Sun's Diurnal Motion ; and if
4. By means of the Solar Spots it this 59 Minutes 8 Seconds be divided
hath been diſcovered, that the Sun by 24, you have the Sun's Horary
revolves round its own Axis, with. Motion, which is 2 Minutes 28 Se-
out moving (conſiderably) out of his conds ; and if you will divide this
Place, in about twenty five Days, laſt by 60, you will have his Mo-
and that the Axis of this Motion is tion in a Minute, C. And this
inclined to the Ecliptick in an An- Way are the Tables of the Sun's
gle of 87 Degrees 30 Minutes nearly. mean Motion, which you have in
The Sun's apparent Diameter being the Books of Aſtronomical Calcula-
ſenſibly ſhorter in December than in tion, conſtructed.
June, as is plain, and agreed from 9.
The Sun's Horizontal Parallax,
Obſervation, the Sun muſt be pro- Dr. Gregory and Sir Iſaac Newton
portionably nearer to the Earth in make but io Seconds.
Winter than in Summer ; in the for SUNDAY LETTER, the fame
mer of which Seaſons will be the with Dominical Letter.
Perihelion, in the latter the Aphe-
SUPERFICIAL NUMBERS, the
lion: And this is alſo confirmed by fame with Plain Numbers.
the Earth's moving ſwifter in De SUPERFICIES, the ſame with
cember, than it doth in June ; as it Surface, (which ſee,) is Length and
doth about is.
Breadth only, without Thickneſs.
5. For fince, as Sir Iſaac Newton The Notion of a Line's being
hath demonitrated, by a Line drawn made up of an infinite Number of
to the Sun, the Earth always de equidiſtant Points; of a Superficies,
ſcribes equal Areas in equal l'imes, of an infinite Number of cquidiſtant
whenever it moves ſwifter, it muſt Lines; and of a Solid's, of an in-
needs be nearer to the Sun : And for finite Number of equidiſtant Sur-
this Reaſon there are about eight faces or Superficies, is falſe, and will
Days more from the Sun's Vernal lead a Perſon into a llultitude of
lit
Ab-
SUP
I
ŞUR
3
Abſurdities in the Inveſtigation of ļ. When any Ņumber or Quag-
Proportions of the Surfaces of Bo- tity hath its Root propoſed to be
dies, &c. For if a Pyramid or Cone extracted, and yet is not a true figu-
be conceived, the one as made up of rate Number of that kind, that is,
an infinite Number of equidiſtant if its Square Root being demanded,
Squares, and the other as made up it is not a true Square, &c. then 'tiş
of an infinite Number of equidiſtanţ impoſſible to affign, either in whole
Circles parallel to their reſpective Numbers or Fractions, any exact
Baſes, continually increaſing as the Root of ſuch a Number propoſed ;
Squares of the Natural Numbers, it and whenever this happens, 'tis uſu,
will from thence follow, that the al in Mathematicks, to mark the
Surfaces of any two Pyramids, or required Root of ſuch Ņumbers or
Cones, of the fameBaſe and Altitude, Quantities, by prefixing before it the
will be equal, which every one proper Marks of Radicality, V.
knows is falſe: And the Reaſon why Thus, V ž ſignifies the Square
from this Notion a true Concluſion
is ſometimes drawn, when the Pro- Root of 2. and V 16. or v (3) 16.
portions of Plain Surfaces, or of So. fignifies the Cubical Root of 16.
lids, contain'd between the fame Which Roots, becauſe they are im-
Parallels, is fought, is becauſe the poſſible to be expreſſed in Numbers
infinite Number of Parallelograms, exactly, (for no Number, either In-
of which a Plain Figure may con- teger or Fraction, multiplied into
fift, and the infinitely ſmall Paral. itſelf, can ever produce 2, or being
lelepipedons, of which a Solid does, niultiplied Cubically, can ever pro-
when their Proportions are fought, duce 16,) are very properly called
are all of the fame infinitely ſmall Surd Roots.
Height, and ſo they are to each other 2. There is alſo another Way of
as their Baſes ; whence theſe Baſes, Notation, now much in uſe, whereby
in this Cafe, may be taken for the Roots are expreſſed, withoạt the
Correſpondent Parallelograms or Radical Sign, by their, Indexes ;
Parallelepipedons, and ſo no Error Thus, as *? *3! 45. c. fignify
will ariſe.
the Square, Cube, and fifth Power
SUPER-PARTICULAR PROPOR- of x ; fo xł. x3. x3. &c. fignify the
TiON, is when one Number or Square Root, Cube Root, Gc. of x.
Quantity contains another once, and The Reaſon of which is plain e-
one ſuch Part whoſe Numerator is
1 ; then the Number fo contained in nough; for ſince vx is a Geometri-
the greater, is ſaid to be to it in
calmean Proportional between 1 and
fuper-particular Proportion.
X, fo is an Arithmetical mean
SUPER-PARTIENT PROPORTI therefore as 2 is the Index of the
Number between 0 and I; and
On, is when one Number or Quan-
I proper
tity contains another once, and some Square of x, 1 will be the
Number of aliquot Parts remaining;
Index of its ſquare Root, &c.
3. Obſerve alſo, that for Conve-
as, 1}, I, I, C.
nience or Brevity fake, Quantities
SUPPLEMENT of an Arch, in
Geometry, or Trigonometry, is the or Numbers, which are not surds,
are often expreſſed in the Form
Number of Degrees that it wants
of Surd Roots : Thus V 49 V 2,
of being an entire Semi-Circle; as
the Complement ſignifies' what an
V 27, &c. fignify, 2, 1, 3, &c.
Arch wants of being a Quadrant. Surds are either ſimple, which
SURD Roots, or Numbers.
are expreſied by one ſingle T'es m, or
elle
3
1
3
7
k.
E
g
SUR
SOU
elſe compound, which are formed let us call the Side BC (6), tb, An-
by the Addition or Subſtraction of gle BAC (a), and the Angle CAD
fimple Surds : As, ✓ 5 + 5 (d), alſo BDA (8), and the Angle
✓ 2. or ✓ 1 + V 2. Which
B
f
с
laſt is called an Univerſal Root;
and ſignifies the Cubick Root of that
m т.
Number, which is the Reſult of ad-
ding 7 to the Square Root of z.
SURFACE, (the ſame with Su-
perficies) is the bare Outſide of any
Body; and conſidered by it ſelf, is
Quantity extended in Length and d
Breadth only, without Thickneſs.
SURSOLID Lọcus.
See Locus
А.
X
Şurſolid.
ŠURSOLID PROBLEM, in Ma. BDC (!), ard the Angle AED,
thematicks, is that which cannot (which is alſo given) (ki
, and the
be reſolved "but by Curves of a Angles B; C, (m and »), and laſtly
higher Nature than a Conick Sec- the Side AD, (x); then it will be as
the Sine of the Angle (k) is to (F) ::
tion, v.gr. in order to deſcribe a
Regular Endecagon, or Figure of Sine (g):
g*
A E. And as the Sine
eleven Sides in a Circle, 'tis re-
quired to deſcribe an Iſoſceles Tri- of the Angle (k) is to (x), ſo is the Sine
angle on a Right Line given, whoſe
xd
Angles at the Baſe ſhall be quintuple
k
to that at the Vertex; which may
eaſily be done by the Interſection of the Sine (m) : (A E) :: fo is
a Quadratrix, or any other Curve of
the ſecond Gender.
xon
the Sine of (a): to
=BE, And
mk
SURVEYING of Land, or Pla-
nometria, is the Art of meaſuring
xd
as the Sine of (n):
all manner of Plain Figures, in
(ED) ::
k
k
order to know their ſuperficial
xdh
Cotent;
which how to do, I fo is the Sine of (3) to = (CE)
nk
have ſhewn all along, under the Now as B E + EC: EC-BE::
particular Name of each Plane fo is the Tangent of half the Sun of
Figure : But how to bring this to the Angles B CE and CBE (which
Practice, ſo as to meaſure the A-
are given) to the Tangent of half
reas of Real Lands, Fields, Grounds, their Difference: Therefore
&c. by the Help of proper In-
ftruments, is what we uſually call **nag + md
(BE + EC):
The following uſeful Problem
nag
being uncommon, and the Solution ** m d h
(EC-BE).
eaſily following from the Inveſtiga-
tion, I thought it might not be a- ſo is the Tangent of half the Sum
miſs to inſert it.
of the Angles BCE, CBF, to the
The Side BC given, together Tangent of half their Difference.
with the Angles BAC, CAD,
But becauſe is in both Terms
ADB, BDC, to find the Side AD,
mnk
of
of (d) to že
E D.
Alſo as
1
as
m nk
m nk
S Y N
SYP
to.
of the Ratio, it will be as the Sines of Part ends and breaks off upon the
nag + m dh: m dh nag::fo Middle of a Note of another Part.
is the Tangent of half the Sum of the SYNCOPE, in Mufick, is the
Angles, to the Tangent of half their Driving Note, when ſome ſhorter
Difference : But becauſe the Sines Note perfixed at the Beginning of
of nag and mdg are all known, the Meaſure, or Half-Meaſure, is
therefore may the Angles BCE followed by two, three, or more
and CBE be found, and conſequent- Notes of a greater Quantity, before
ly the Sides CE and BE, as alſo you meet with another ſhort Note
Á E and ED, and thence the Side equivalent to that which began the
A D fought may be alſo found. l. Driving, to make the Number even ;
E. P.
as when an odd Crotchet comes be-
SUPERFICIAL FOURNEAU, a fore two, three, or more Minims,
Term in Fortification, the fame or an odd Quaver before two, three
with Caiſſon, which is a wooden
or more Crotchets,
Cheſt, or Box, with three, four, SYNODICAL MONTH, is the
five, or fix Bombs in it; and ſome- Space of Time (viz. 29 Days, 12
times 'tis filled only with Powder, Hours, 45 Minutes) contained be-
and is uſed in a cloſe Siege, by be tween the Moon's parting from the
ing buried under Ground with a Sun at a Conjunction, and returning
Train to it, to blow up any Lodg- to him again'; during which Time
ment that the Enemy ſhall approach the puts on all her Phaſes. And her
SYNODICAL REVOLUTION, is
SURVEYING SCALE, the fame that Motion whereby her whole Sy-
with Reducing Scale.
ftem is carried along with the Earth
SWALLOWS-I'AIL, in Fortifica- round the Sun.
tion, is a ſingle Tenaille, that is SYNTHETICAL METHOD of En-
narrower towards the Place than to- quiry, or Demonſtration, in Mathe-
wards the Country:
maticks, is when we purſue the
ŞWING-WHEEL, in a Royal Pen- Truth, chiefly by Reaſons drawn
dulum, is that Wheel which drives from Principles before eſtabliſhed,
the Pendulum. This Wheel in a and Propoſitions formerly proved,
Watch is called the Crown Wheel, and proceed by a long regular Chain,
as alſo in a Balance Clock.
till we come to the Concluſion;
SYDEREAL YEAR. See Year, as is done in the Elements of Euclid,
SYMMETRY, in Architecture, and in almoſt all the Demonſtrations
comes from the Greek Symmetria, of the Ancients. This is called
with Meaſure, and ſignifies the Re- Compoſition, and is oppoſed to the
lation of Parity, both as to Height, Analytical Method, which is called
Depth, and Breadth, which the Reſolution. Which ſee.
Parts have, in order to form a beau-
SYPHON, is a Tube or Pipe of
tiful Whole. In Architecture we Glaſs or Metal, which is uſually
have both uniform Symmetry, and bent to an Acute Angle, and hav-
reſpective Symmetry: In the for- ing one Leg ſhorter than the other,
mer, the Ordonance is purſued in They are frequently uſed to draw
the ſame manner throughout the off Liquors out of one Barrel or
whole Extent; whereas in the latter, Veſſel into another, without raiſing
only the oppoſite Sides correſpond the Lees, or Dregs, and are called
to each other.
Cranes. Soinetimes Glaſs Tubes or
SYNCOPATION, a Term in Mu. Pipes, tho' [.rait, are called Syphons.
fic':, which is when a Note of one
SYSTEM
j
TA I
T E L
SYSTEM, in Muſick, is the Ex- fquare Fillet, and a ſtreight Cyma.
tent of a certain Number of Chords, tium, and is only two Portions of a
having its Bounds toward the Grave Circle.
and Acute, which hath been diffe TALUS, in Architecture, is the
rently determined by the different fame with Aftragalus; which fee :
Progreſs made in Muſick, and ac-
But in Fortification it fignifies any
cording to the different Diviſions of Thing that goes ſloping ; or it is the
the Monochord.
French Word for a Slope.
The Syſtem of the Ancients was TANGENT of a Curve is a Right
compoſed of four Tętrachords, and Line, which fo meets a part of a
one fupernumerary Chord, the whole Curve, as not to cut that part,
making fifteen Chords,
TAPER-BORED, a Term in Gun-
SYSTEM, properly is a regular nery. A Piece of Ordnance is ſaid
orderly Collection, or orderly Dif- to be Taper-Bored when it is wider
poſition of all thoſe Planets, which at the Mouth than towards the
move round the Sun as their Centre, Breech.
in determined Orbits, and never de TELESCOPE, is a Dioptrick In-
viate farther from him than their ſtrument, compoſed of Lens's, by
proper and uſual Bounds. And a
means of which remote Objects ap-
System of Philoſophy, is a regu- pear as if they were near.
lar Collection of the Principles and It is certain that Johannes Bap-
Parts of that Science into one Body, tiſta Porta, a Neapolitan, was the
and a treating of them dogmatical- firſt that made a Teleſcope, abo ut
ly, or in a ſcholaftical Method; the Year 1594: For he ſays, in
which is called the Syſtematical Way, Magiſ. Natur. lib. 17. c. 10. Si
in contradiſtinction of the Way of utrumque (that is, a 'Concave and
Eſſay, wherein the Writer delivers Convex Glaſs) rectè conjungere no-
himſelf more looſely, eaſily and veris, & longinqua & proxima ma-
modeſtly.
jora, & clara videbis, non parum
SYSTILE, in Architecture, is multis amicis auxilii preflitimus, qui
that Manner of placing Columns & longinqua obſoleta, proxima turbi-
where the Space between the two da conſpiciebant, ut omnia perfectif-
Fuſts conſiſts of two Diameters, or me contuerentur. But Porta did not
four Modules.
well underſtand his own Invention,
SYZYGY, in Aſtronomy, is the which he had found out by Chance,
fame with the Conjunction of any and ſo had not effected it with any
two Planets or Stars, or when they great Induſtry, or applied the ſame
are both referred to the ſame point to Celeſtial Obſervations. Not long
in the Heavens ; or to the ſame De- after hiin, there were ſeveral others
gree of the Ecliptick, by a Circle that made hort Teleſcopes ; but
of Longitude paffing through them they were of ſmall Uſe, till Gali-
both.
leo applied himſelf to the making
of one, who was the firſt that made
it tolerably good.
A Teleſcope, made by a convex
and concave Lens, repreſents vaftly
T.
diſtant Objects, diftinct and erect;
and magnifies them according to the
AILIER: See Abacus. Proportion of the Focal Diſtance of
TALON, a little Member the Convex Lens, to the Focal Dif-
in Architecture, conſiſting of a tance of the Concave Len .
А
T
T E L
TEL
Van
AP
A Teleſcope, made of two Con- ture appear Curve: Therefore we
vex Lens's, repreſents vaftly diftant muſt make our Teleſcope of four
Objects, diſtinct but inverted; and Lens's, which is done after the fol-
magnifies them according to the Pro- lowing Manner:
portion of the Focal Diſtance of the The Exterior, or Object Lens,
Exterior or Object Lens, to the Fo. is A, whoſe Focal Diſtance is AB,
cal Dittance of the Interior or Ocu- and in the ſame Axis are placed
lır Léns.
three Ocular Lens's C, D, and E,
Here follows the Explanation of all equal to one another, the inmoſt
the Conſiruction of a Teleſcope com of which is placed beyond the Fo-
pounded of four Convexes, by means cus B, by its Focal Diſtance BC;
of which Objects are ſeen erect, and
very ample.
TUIN
Teleſcopes, made of two Con-
yexes, becauſe of their Inverting
G
the Poſition of the Object, are ſel-
A
dom uſed, except in obſerving the
Stars, the Poſition of which is not
regarded. The Proportion in which
this Sort magnifies the Object, has
already been ſhewn; but if we
would have theſe Images again
made erect, and at the ſame time
B
a great Share of them be repreſent-
ed to the Eye, at one View, very
ample, we muſt uſe three, four, five,
B
or more Lens's; which, however,
are not to be multiplied without
K
Ц
Cauſe, becauſe the Matter of each
I
of them, and the Reflexion of their
H
S
ſeveral Surfaces, divert Part of the
Rays : But we cannot obtain the
7
R
deſired Effect perfectly, with fewer
than four Lens's. For although, in
E
E
the ſame Length of the Teleſcope,
M N
both anerect Situation, and the ſame
Degree of magnifying, and an equal
Share of the Object, may be had as
well with three as four Lens's, yet
the Compoſition of three Lens's is and the next D, is placed beyond
much more inconvenient than that C, by twice that Diſtance B C, and
of four ; becauſe in that, the two on the laſt as far from D as that was
cular Lens's, or, at leaſt, that which from C; and laſtly, the Eye muſt
is next the Eye, muſt be made of be placed beyond this laſt by the
larger Segments of a Sphere, with Diſtance BC,
reſpect to its Diameter, or to the There is here again Occaſion for
Focal Diſtance, if the ſame Mag- two Figures ; in the firſt of which
nitude of the Viſual Angle be re are repreſented Rays proceeding from
quired: And hence the Objects a ſingle Point of the vaftly diftant
come to be Coloured; and Right Object; which, 'tis plain to any
Lines, at the Margins of the Aper- who underſtand what has gone be-
fore,
C
T EL
Τ Ε L.
fore, Firſt, fall, as it were, paral. being equal, as the Diſtance AG,
lel upon the Lens A, and are by it to the Diſtance EF; that is, as
collected at its Focus B; and thence AB, the Focal Diſtance of one of
diverging, fall upon the Lens C, the ocular Lens's. Q.E.D.
which makes thein again parallel, It appears, moreover, that the
and throws them upon the Lens D, viſual Angle MFN comprehends
which collects them at its Focus H, the fame Latitude of the Object,
the middle Point of the Diſtance with a Teleſcope made of two
DE ; from whence proceeding on Lens's, only A and C: for that
to the Lens E, they are by it made Share of the Object which is com-
a third time parallel ; and being prehended in the Angle TGV,
received ſo by the Eye F, they make would be ſeen through that Tele-
diſtinct Viſion by being collected at ſcope in the Angles KSL, equal to
its Focus which is in the bottom of the Angles MFN.
the Eye.
This incomparable Compoſition
The other Figure comiders the of Lens's was found out by I
Proportion of magnifying, which is know not whom at Rome; and may
that which AB, the Focal Diſtance be much improved by placing an
of the Object Lens, bears to BC, Annulus.or Ring either at H, the
the Focal Diſtance of one of the common Focus of the Lens's D and
Ocular Lens's, and demonſtrates E, or at B, the common Focus of
likewiſe the Amplitude of the vi- the Lens's A and C; which is e-
ſual Angle. For the Apertures of ſpecially of very great uſe in mea-
the three Ocular Lens's, being ſuring the Diameters of Planets :
ſuppoſed equal, which muſt not ex For this Annulus does therefore
ceed the Apertures of the Object exactly circumſcribe the Circle of
Lens A, draw Mn, NR, parallel the apparent Images, becauſe it
to the common Axis ; and compre cuts off thoſe irregular Rays which
hending the Diameters of the A are not collected near enough to B
pertures of the Lens's E and D, or H, and conſequently are not, by
and alſo KO, LP, parallel to the means of the ſucceeding Lens's,
fame Axis, and comprehending fent parallel to the Eye, which di-
KL the Aperture of the Lens C; ftinct Vifon requires ; and the Coe
and taking AG, equal to AB, draw lours likewiſe near the Margins are
the Lines OGV,PGT, interſecting by this contrivance taken away,
one another in G. Now, it is evi which without it are not well to be
dent, the Latitude of the Object avoided. The Proportions between
which is ſeen by the naked Eye the Focal Diſtance of the Object
from the Point G, and conſequently Lens, (which is likewiſe the Length
from F alſo, the Diſtance of the of the Teleſcope,) the Aperture of
Object being as it were infinite, the ſame Object Lens, the Focal
would appear comprehended in the Diftance of the Ocular Lens, and
Angle MFN; and conſeqnently the apparent magnified Diameter of
the Proportion of the apparent the Object ; for Teleſcopes, from
Magnitude to the true, is as the the Length of one Rhinland Foot
Angle MFN to the Angle TGV to a hundred, are expreſſed in the
or PGO; that is, PO and MN Table following.
"
4
TEL
TÉL
A Table for TÉLESCOPES.
tion of mag-
The Focal Dif|The Diameter of thelThe Focal Diſtance of The Propor-
tance of the Ob- Aperture of the ob- the Ocular Lens.
ject Lens, or the jeet Lens.
nifying con-
Length of thel
ſidered as to
Teleſcope.
the Diame-
ter.
i intilo nos do
20.
28,
34.
40.
44.
49.
53.
56.
1,80.
60.
62.
Rhinland Feet. Inches and Decimals, Inches and Decimals.
1.
0,55
0,61.
2.
0,75
0,85.
3
0,95
1,05.
4
1,09
1,20.
5
1,23
1,35.
6.
1,34
1,47.
7
1,45
1,60.
8.
1,55
1,71.
9.
1,64.
.
1,73:
1,90.
13
1,97
2,17
15
2,12.
2,33.
20.
2,45.
2,70.
25.
2,74
3,01.
30.
3,00.
3,30.
35.
3,24
45
40.
0,04:
50.
55
4,47
6o.
4,24.
65.
4:42.
70.
4,58.
5,04
75.
4,74.
5,21.
80.
4,90
5,39.
85
5,05.
90.
5,20.
5,72.
95.
5:34.
5.87.
100.
6,03.
3,56.
3,81.
72.
77.
89.
100.
109.
118.
1 26.
133
'141.
148.
154.
161.
3,46.
3,67.
3,87.
4,06.
4,26.
4,66,
4,86.
166.
172.
178.
5,56.
183.
189.
194.
199.
5,48.
Şir Iſaac Newton, in his Optics, for the Air through which we look
ſays, if the Theory of making Tes upon the Stars, is in perpetual T're-
leſcopes could, at length, be fully mor, as may be ſeen by the tremu-
brought into Practice, yet there lous Motion of Shadows caft from
would be certain Bounds beyond high Towers, and by the twinkling
which Teleſcopes could not perform of the fixed Stars. But theſe Stars
do
are
T EL
T E R
do not twinkle when viewed numb. 81. and in numb. 376. Mr.
through Teleſcopes, which have Hadley has given us a Deſcription
large Apertures ; for the Rays of of an Inſtrument of this kind of
Light, which paſs through divers five Feet one fourth in Length ;
Parts of the Aperture, tremble each which, uſed as a Night-Teleſcope,
of them apart; and, by means of will magnify about two hundred
their various, and ſometimes con- and twenty times, and, as a Day.
trary Tremors, fall at one and the one, about one hundred twenty-
fame time upon different Points in five times; and is in ſeveral reſpects
the bottom of the Eye, and their ſuperior, and in none inferior to
trembling Motions are too quick Mr. Huygens's Dioptric Telefcope
and confuſed to be perceived ſe- of one hundred and twenty-fix Feet
verally : And all theſe illuminated in Length.
Points conſtitute one broad lucid Mr. Jackſon, an ingenious Mathe-
Point, compoſed of thoſe many matical Inſtrument-Maker, has late-
trembling Points, confuſedly and in- ly made one of thoſe reflecting Te-
ſenſibly mixed with one another by leſcopes, the largeſt that I ever ſaw,
very ſort and ſwift Tremors, and being fix Feet long and ſeven Inches
thereby cauſe the Star to appear in Diameter, and magnifying the
broader than it is, and without any Objects zoo times.
trembling of the whole. Long Te TeleSCOPICAL STARS,
leſcopes may cauſe Objects to ap- thoſe that are not viſible to the
pear brighter and larger than fhort naked Eye, but diſcoverable only
ones can do ; but they cannot be ſo by the help of a Teleſcope.
formed as to take away that Con TEMPERATE ZONE. See Zone.
fuſion of the Rays which ariſes TEMPORARY FORTIFICATI-
from the Tremors of the Atmo- on. See Fortification.
ſphere. The only Remedy is a TENAILLE, in Fortification, is
moſt ſerene and quiet Air, ſuch as a kind of Out-Work reſembling a
may perhaps be found on the tops Horn-Work, but generally fome-
of the higheſt Mountains above the what different; in regard that in-
the groſſer Clouds.
ſtead of two Demi-Baſtions, it bears'
TELESCOPE (AERIAL) is one of only in Front a re-entring Angle
Mr. Huygens's, deſcribed in the Philo- between the ſame Wings without
ſophical Tranſaktions, pag. 161. made Flanks, and the Sides are parallel :
for uſing only in the Night ; and ſo But when there is more Breadth at
having no cloſe Tube, ſince there is the Head than at the Gorge, theſe
no need of one in the Night. Tenailles are called Queuë d'hironde.
TELESCOPE (REFLECTING,) All Tenailles are defective in this
conſiſts of a large Tube, open at reſpect, that they are not Aanked,
one end, being that next to the Ob- or defended towards their inward
ject, and having the other end cloſe, or dead Angle, becauſe the Height
where a Concave Metalline Specu- of the Parapet hinders ſeeing down
lum is placed ; and having near the before the Angle; ſo that the Enea
open End a flat Oval Speculum in- my can lodge himſelf there under
clined towards the upper part of the, Covert : Wherefore. Tenailles are
Tube, where is a little hole furniſh- never made, but when they wano
ed with a ſmall plane Convex Eye- time to make Horn-Works.
Glaſs. There is a full account of; TENOR, is the Name of the firſt
this Inſtrument by Sir Iluar Now. Mean or middle Part in Muſick.
ton, in the Philoſophical Tranfa&ioni, TERM, in Geometry is taken
3
fos
TER
THE
ch
15
ze S
for the Bounds and Limits of añya by the inner Talus on the other tö-
thing."
ward the Body of the Place,
TERMS of an Equation, in Alge TERRELLA : When a Loadſtone ,
bra, are the ſeveral Names or Mem- is made ſpherical, and is placed to
bers of which, it is compoſed, and that its Poles and Equator, &c. do
fuch as have the ſame unknown exactly correſpond to the Poles and
Letter, but in different Powers or Equator of the World, it is called
Degrees : For if the ſame unknown by Gilbert a Tereļla, or little Earth;
Letter be found in ſeveral Meme being in ſome meaſure a Reprefen-
bers in the fame Degree ori Power, tation of our great Globé of Earth;
they ſhall paſs but for one Term. TERRESTRIAL GLOBE. See
As, in this Equation, *x + ax Globe.
bb; the three Terms are **,
TERRESTRIAL LINE, See Liñe
ax, and bb. Moreover, in this, Terreſtrial.
ab
TETRACHORD, in Mufic, is a
Concord or
x
or
Interval of threex
cd
Tones.
the Terms are 34, 43, 4, The Tetrachord of the Ancients,
ab fP
ab was a Rank of four Strings, ac-
+ xx, and yy. Where X, counting the Tetrachord for one
Tone, as it is often taken in Muſic.
f
TETRA DIAPASON. A Qua
and x, are the ſame Terms;
druple Diapaſon is a muſical Chord;
otherwiſe called
and the firft Term in any Equation otherwiſe a quadruple,
must be that where the unknown eighth, or nine and twentieth.
Root hath the higheſt Dimenſions ;
TBTRACONIAS, a.Comet, whoſe
and that Term which hath the Root. Head is of a quadrangular Figures..
in it, of one Dimenſion of Power and its Tail or Train long, thicki
and uniform; and does not differ
lower, is called the ſecond Term,
much from the Meteor called
and ſo on.
Trabs.
Terms of Proportion, in Mathe-
maticks, are ſuch Numbers, Letters,
TETRAHEDRON, is one of the
or Quantities
, as are compared one regular Bodies, confifting of four.
with another.
equal equilateral Triangles; or it is
a triangular Pyramid of four equal
2. 4 :: 6. d.
then a, b, Faces.
a. b :: 8. 16,
TetraSTYLE, in Architecture,
c, d, or 2, 4, 8, 16, are called the is a Building which hath four Cow-
Terms; a being the firſt Term, b lumns in the Faces before and be-
the ſecond Term, &C.
hind.
TERRAQUEOUS, in Geography,
TEXTURE. The Texture of
fignifies the Globe of Earth and Wa- any natural Body, is that particu-
ter, as they both together conftitute lar Difpofition of its conſtituent
one ſpherical Body.
Particles, as makes it have ſuch
TÉRRE (PLAIN) in Fortification, Form, or be of ſuch a Nature, or
is the Platform or horizontal Sur- be endow'd with ſuch Qualities.
face of the Rampart lying level, THEODOLITE, is an Inſtrument
only with a little ſlope on the out- uſed in Surveying, and taking of
fide for the Recoil of the Cannon. Heights and Diſtances ; and con-
It is terminated by the Parapet fifteth of ſeveral Parts, as a Circle
on that fide toward the Field, and of Braſs, about one Foot Diameter,
divided
Thus if
ged Staff.
TH R
T ID
divided into four Quadrants, fome- fhut all together, and to take off in
times with a Teleſcope at the bot- the middle, for the better Carriage;
tom of it.
and on its top is uſually a Ball and
Each of the Quadrants is divided Socket to ſupport and adjuſt In-
into ninety Degrees, and ſubdivided ſtruments for Aſtronomy, Survey-
as the Largeneſs of the Inſtrument ing, &c.
will permit.
Tide. Tide fignifies as well the
A Box and Needle contrived to Ebbing as the Flowing of the Sea;
ſtand upon the Centre of the Circle, the former of which the Seamen call
upon which Centre, the Inſtrument, Tide of Ebb; the latter, Tide of
the Index, with its Sights, and Flood.
ſometimes a Teleſcope, is made to In a Lunar Day, that is, the
turn about; and yet, both the In- Time ſpent between the Moon's
goo
ftrument, and the Box and Needle, ing from the Meridian, and coming
remain firm. At the bottom of the to it again, the Sea is twice elevated,
Box, there is a Card, or Mariner's and twice depreſſed, in any aſſigned
Compaſs fix'd.
Place.
A Socket on the Backſide, to be In any Place the Water is moſt
put upon the Head of a three-leg- elevated, two or three Hours after
the Moon has paſs'd the Meridian
A Staff to ſet the Inſtrument up- of the Place, or the oppoſite Meri-
on ; the Neck, at the Head where- dian.
of, muſt be made to go into the The Elevation towards the Moon
Socket on the Backſide of the In. a little exceeds the oppofite one.
ſtrument.
The Aſcent of the Water is dimi.
N.B. I muſt do Mr. Thomas niſhed as you go towards the Poles, ,
Heath (Mathematical Inſtrument- becauſe there is no Agitation of the
Maker, near the Fountain-Tavern Water there.
in the Strand,) the Juſtice to ſay, From the Action of the Sun, every
that I have ſeen excellent Theo- natural Day the Sea is twice ele-
dolites made by him, as well vated, and twice depreſſed. This
as all other Mathematical Inſtru- Agitation is much leſs, on account
ments.
of the immenſe Diſtance of the Sun,
THBorem, is a ſpeculative Pro- than that which depends upon the
poſition, demonſtrating the Proper. Moon; yet it is ſubject to the ſame
ties of any Subject.
Laws.
THERMOSCOPE is an Inftru The Motions which depend upon
ment ſhewing the Increaſe and De- the Action of the Moon and Sun, are
creaſe of Heat and Cold in the Air : not diſtinguiſhed but confounded ;
But the
and from the Action of the Sun, the
THERMOMETER is an Inſtru- Lunar Tide is only changed; which
ment by which we can meaſure the Change varies every Day, by rea-
Heat and Cold of the Air.
ſon of the Inequality between the
It is uſually made of a Tube of Natural and Lunar Day.
Glaſs of about four Foot long, fil. In the Syzygies the Elevations
led with tinged Spirit of Wine, or from the Action of both Luminaries
ſome other proper Liquor, having a concur, and the Sea is more ele.
Bill at the bottom of it.
vated; the Sea aſcends lefs in the
THREE-LEGGBD-STAFF, is an Quadratures ; for where the Water
Inſtrument conſiſting of three is elevated by the Action of the
wooden Legs, made with Joints to Moon, it is depreſſed by the Action
Kk
of
TID
1
T I M
of the San, and ſo on the contrary come to the greateſt Height; and
Therefore, whilft the Moon paſſes every Day the greateſt Elevation of
from the Syzygy to the Quadrature, the Sea will be after the Moon has
the daily Elevations are continually paſſed thro' the oppoſite Meridian.
diminished : On the contrary, they All Things which have been hi-
are increaſed when the Moon moves therto explained would exactly ob-
from the Quadrature to the Syzygy. tain, if the whole Surface of the
At a new Moon alſo, Cæteris pari- Earth was covered with Sea ; but
bus, the Elevations are greater, and ſince the Sea is not every where,
thoſe that follow one another the ſome Changes ariſe from thence
e ;
ſame Day, are more different than not indeed in the open Sea, becauſe
at Full Moon.
the Ocean is extended enough to be
The greateſt and leaſt Elevations ſubject to the Motions we have
are not obſerv'd, till the ſecond or ſpoken of. But the Situation of the
third Day after the New or Full Shores, the Streights, and many
Moon. Íf we conſider the Lumina- other Things depending upon the
ries receding from the Plane of the particular Situation of the Places,
Equator, we fhall perceive that the diſturb theſe general Rules : Yet it
Agitation is diminiſhed, and becomes is plain from the moſt general Ob-
leſs, according as the Declination of ſervations, that the Tide follows the
the Luminaries becomes greater.
Laws which we have laid down.
In the Syzygies, near the Æqui-, The mean Force of the Sun to
noxes, the Tides are obſerved to be move the Sea, is to the mean Force
the greateſt, both Luminaries being of the Moon to move the fame, as
in or near the Equator.
I to 4.4815
The Actions of the Moon and Sun The 'Action of the Sun changes.
are greater, the leſs thoſe Bodies are the Height of the Sea two Feet;
diftant from the Earth; but when and that the Action of the Moon
the Diſtance of the Sun is lefs, and changes it 8,95: And that, from the
it is in the South Signs, often both joined Action of both, the mean
the greateſt Aquinoctial Tides are Agitation is of about eleven Feet,
obſerved in that Situation of the which agrees pretty well with Ob-
Sun, that is, before the Vernal, and fervations. For, in the open Ocean,
after the Autumnal Æquinox; which as the Sea is more or leſs open, the
yet does not happen every Year, ben Water is raiſed to the Height of fix,
cauſe ſome Variation may ariſe from nine, twelve, or fifteen Feet; in
the Situation of the Moon's Orbit, which Elevations, alſo there is a
and the Diſtance of the Syzygy from Difference arifmg from the Depth of
the Aquinox. In Places diſtant the Waters; but thoſe Elevations,
from the Equator, the Elevations which far exceed theſe, happen
that happen the fame Day are un where the Sea violently enters into
equal.
the Streights or Gulphs, where the
As long as the Moon is on the Force is not broken till the Water
fame Side of the Equator in any ariſes higher.
Place, the Elevation of the Water is Time, in Mufic, is that Quan-
obſerved to be the greateſt every tity or Length whereby is aſſigned
Day, after the Moon has paſſed the to every particular Note iis due
Meridian of the Place.
Meaſure, without making it either
But if the Equator ſeparates, or longer.or ſhorter than it ought to be;
is between the Moon and the Place and it is twofold, viz. Duple or
of which we speak, the Water will Common, and Triple.
TIME
.
TRA.
TRA
Time (DUPLE,) or Semi-breve, TRANSIT, in Aſtronomy, figni-
generally called Comman, becauſe fies the paſſing of any Planet juſt by
moſt uſed, is when all the Noteş are or under any fix'd Star; or of the
increaſed by two.
Moon, in particular, covering or
Time (Triple,) is that where- moving cloſe by any other Planet.
in the Meaſure is counted by TRANSITION, in . Muſic, is
Threes.
when a greater Note is broken into
TIMB, is a Sụcceflion of Phæno. a leſſer, to make ſmooth or ſweeten
mena, and the idea that we have the Roughneſs of a Leap by a gra-
thei eof, confifts in the Order of dual Tranſition, or pafling to the
fucceflive Perceptions : It is divided Note next following; whence it is
into Abſolute and Relative. commonly called the Breaking of a
Time (Agronomical, Mathemati. Note, being ſometimes very neceſſary
cale or Abſolute) Hows equably in in Muſical Compoſition.
it felf, without relation to any Thing TRANSMUTĀTron, in Geome-
external; and, by another Word, is try, is to reduce or change one Fi-
called Duration. But,
gure or Body into another of the
TIME ( Relative, Apparent, or fame Area or Solidity, but of a dif-
Vulgar,) is the ſenſible and external ferent Figure; as a Triangle into a
Meaſure of any Duration eſtimated Square, a Pyramid into a Parallelo-
by Motion; and this the Vulgar piped, &c.
uſe inſtead of true Timę.
TRANSPARENT, or Diaphanous
TONDINO, a Term in Archi- Bodies, are ſuch as may be ſeen
tecture. See Affragal.
through.
TONE, a Term in Muſic, figni TRANSPOSITION, in Algebra, is
fying a certain Degree of Elevation to bring any. Term of an Equation
or Depreſtion of the Voice, or fome over to the other Şide, as if at bar;
other Sound.
and you make arch, then is 6
TOPOGRAPHY, is a particular tranſpoſed,
Deſcription of ſome ſmall Quantity TRANSVERSE Axis, or Diame-
of Land, ſuch as that of a Manor, ter of an Ellipfis, is the longer Axis.
or particular Eſtate, &c.
TRAPEZIUM, in Geometry, is
TORRID ZONE. See Zone. a Plane Figure contained under four
TORUS, in Architecture, is a unequal Right Lines.
large round Moulding in the Baſes 1. Any `three Sides of a Trape.
of the Columns
zium taken together, are greater
TRABEATION, the ſame with than the third.
Entablement.
2. The twa Diagonals of any Tra-
TRAJECTORY of a Comet, is the pezium do divide it into four pro-
Line which, by its Motion, it de- portional Triangles.
fcribes.
3. If two Sides of a Trape-
TRANSCENDENTAL Curves, zium be parallel, the Rectangle un-
are ſuch as when their Nature order the Aggregate of the parallel
Property is expreſs'd by an Equa- Sides, and one half their Diſtance, is
tion, one of the variable Quantities equal to that Trapezium.
therein denotes a Curve Line; and 4. If a Parallelogram circum-
when ſuch Curve Line is a Geome- ſcribes a Trapezium, ſo that one of
trick one, or one of the firſt Degree, the sides of the Parallelogram be
or Kind, then the Tranſcendental parallel to a Diagonal of the Tra-
Curve is ſaid to be of the ſecond pezium, that Parallelogram will be
Degree or Kind, &c.
the double of the Trapezium.
5
Kk 2
5.
1
I
T I-D
TRA
5. If any Trapezium has two of from F, K, let fall the perpendi-
its
oppoſite Anglèseach a right culars FG, FH to AK, DK, and
Angle, and a Diagonal be drawn KI, KL to FC, FB then will
joining theſe”, Angles; and if from GK be the infinitely ſmall Decre-
the other two Angleš be drawń two 'ment of A Ki KH that of DK;
Perpendiculars to that Diagonal, the If the infinitely ſmall Increment
Diſtances from the Feet of theſe Per- of KC, and LF that of B K. But
pendiculars to thoſe right Angles, fince the Angle FBH is infinitely
reſpectively taken, will be equal.
ſmall, KH will differ from LF
6. If the sides of a Trapezium be only by an infinitely fmall Quantity,
cach biſſected, and the Points of which may therefore be rejected;
Biffection he joined by four right ſo that KHELF. In like, manner
Lines; theſe right Lines will form GK=FI. Therefore G K+KH
a Parallelogram, which will be one +FI+LF = 0. But it is well
half of the Trapezium.
known, that when the Aggregate of
7. If the Diagonals of a Trape- any Number of variable Lines KB,
zium be each biflected, and a right KC, KA, KD, becomes a Minimum,
Line joins thoſe Points ; the Aggre- or Maximum, the Aggregate of
gate of the Squares of the Sides is their Increments and Decrements
equal to the Aggregate of the Squares will be equal to nothing. Where,
of the Diagonals, together with four fore it is evident that the Diagonals
times the Square of the Line joining AC, BD, are either leſs or greater
the Point of Biſſection.
than thoſe of any four right Lines
8. In any Trapezium ABCD, drawn from any Point, except -K,
the Aggregate of the Diagonals within the Figure to the four An-
AC, BD, is leſs than the Aggre. gles. But they cannot be greater,
gate of four right Lines drawn from confequently, they muſt be leſs. 2,
any Point (except the Interſection of
The Truth of this Propoſition ap
the Diagonals) within the Figure. pears almoſt evident by Infpeétion:
Let K be the Interſection of the for ſuppoſe the Point P to be at any
Diagonals, and conceive the Point finite Diſtance from K, the Inter-
F to be infinitely near to K, from ſection of the Diagonals AC, BD,
which draw four right Lines' AF, and draw, as before, the Lines AF,
FB, FC, FD, to the Angles, and BF, CF, DF; then the Side-AĆ
of the Triangle ACF, is fhorter
than the two Sides AF+FC; and
C
the Side B D of the Triangle FBD
ſhorter than the sides BF+FD.
Therefore AC +BD is leſs than
B
AF+FC+BF+FD.
TRAPEZOID, is a ſolid irregular
Figure, having four Sides not paral-
K
lel to one another.
TRAVERSE, a Term in Gunnery,
I
fignifying to turn a Piece of Ord-
G
nance which way one pleaſes upon
her Platform.
Alſo the laying and removing a
Piece of Ordnance, or a great Gun,
in order to bring it to bear or lie
level with the Mark, is called Tr4-
verſing the Piece.
А
H
D
TRA-
L
TRI
TRI
TRAVERSE, in Navigation, is the Of Triangles there are ſeveral
Variation or Alteration of the Ship's forts,, as,
Courſe upon the ſhifting of Winds, 1. A Right-angled Triangle, is that
Gc.
which hath one Right Angle.
TRAVERSE, in Fortification, is 2. An Obtuſe-angled Triangle, is
a little Trench, bordered with two ſuch as hath 'one Obtuſe Angle.
Parapets, viz. one on the right Side, 3. An Acute-angled Triangle, is
and the other on the left, which the that which hath all its Angles acute.
Beſiegers make quite thwart the 4. Any Triangle that is not Right-
Moat of the Place, to paſs ſecure angled, is called Oblique-angled, or
from Flank-Shot, and to bring the Amblygonial.
Miners to the Baſtions,
5. An Equilateral Triangle, is that
TREBLE, is the laſt or higheft of which hath all its Sides equal to one
the four Parts in Muſical Proportion. another.
Trenches, in Fortification, are 6. An Iffceles, or an Equilegged
certain Moats or Ditches, which Triangle, is that which hath only
the Befiegers cut to approach more two Sides equal.
ſecurely to the Place attacked, and 7. A Scalenous Triangle, is that
are of ſeveral forts, according to the which has no two Sides equal.
different Nature of the Soil; for if In every Triangle, the Sum of all
the adjacent Territory be rocky, the the three Angles is equal to two
Trench is only an Elevation of Ba. Right ones; and the External Angle
vine, Gabions, .Wool - Packs, or made by any Side produced, is equal
Shouldrings of Earth, caft up round to the Sum of the Internal and its
about the Place: But where the Oppofite one.
Ground may be eaſily opened, the
In every Triangle, as well Plane
Trench is dug therein, and bordered as Spherical, the Sines of the Sidės
with a Parapet on the side of the are proporcional to the Şines of the
Beſieged. The Breadth of it ought opposite Angles.
to be from 8 to 10 Foot, and the If a Perpendicular bé let fall
Depth from 6 to 7.-
upon the Baſe of an Oblique-angled
Theſe Trenches are to be carried Triangle, the Difference of the
on with winding Lines, in fome Squares of the Sides is equal to the
manner parallel to the Works of Double Rectangle under the Baſe,
the Fortreſs, ſo as not to be in View, and the Diſtance of the Perpendi-
of the Enemy, nor to expoſe its cular from the Middle of the Baſe.
Length to their Shot, which they The Side of an Equilateral Tri-
call Enfilading; for then it will be angle, inſcrib'd in a Circle, is in
in danger of being enfiladed or Power triple of the Radius.
ſcoured by the Enemies Cannon : The sides of a Triangle are cut
and this carrying of the Trenches proportionably, by a Line drawn pa-
obliquely, they call
, carrying the rallel to the Baſe.
Trenches by Coudees or Traverſes. A whole Triangle, is to a Tri-
TRIANGLE, in Geometry, is a angle cut off by a Right Line, as the
Figure of three sides and three An- Rectangle under the cut Sides, is to
gles; and is either a Plane Triangle, the Rectangle of the two other Sides.
or a Spherical one.
In a Right angled Triangle, a
A Plane Triangle, is contained Line drawn from the Right Angle
under three Right Lines.
at the Top, perpendicular to the
A Spherical Triangle, is contain'd Hypothenule, divides the Triangle
under three Arches of a great Circle into two other Right-angled Tri-
of the Sphere.
angles, the which are ſimilar to the
firit
{
Kk 3
TRI'
TR'I
firſt Triangle, and to one another. out the Line BE till it cuts the
In
every Right-angled Triangle, Circle in D, and drawn the Line
the Square of the Hypothenuſe is DC, the Triangles ABE and
equal to the Sum of the Squares of BCD will be ſimilar, which may
the other two Sides.
be thus proved. The Angle A BÉ
If any Angle of a Triangle be = E BC by Conſtruction; and be
biffected, the biffecting Line will di- becauſe the Angles B AC and
vide the oppoſite Side in the fame BDC ſtand upon the ſame Arch-
Proportion as the Legs of the Angle BC, they will likewiſe be equal ;
are to one another.
and conſequently the Angles A E B,
Triangles on the ſame Baſe, and BCD, muſt be equal. Therefore,
having the ſame Height, that is as A B :BE::DB:B C, whence
between the fame Parallel Lines, AB x B C = BE x D B. But
are equal.
fince A E x EC B E x E D
Every Triangle is one half of a from the Nature of the Circle. And
Parallelogram of the fame Baſe and becauſe B Eq = DB X BE (=
Height.
A B x B-C) - ED BF, from the
The Area of any Triangle may Third of the Second of Euclid ;
be had by adding all the three Sides therefore A B x B C – A EXEC
together, and taking half the Sum ; =BEq. Q. E. D.
and, from that half Sum, ſubtracting In any Triangle any one Side is
each Side feverally, and multiply- greater than the Difference between
ing that half Sum and the Remain- the other two Sides, and two Sides
der continually one into another, is' greater than the Third.
and extracting the Square Root of In any Triangle the Difference
the Product.
of the Squares of the Sides is equal
The following uſeful Propofition, to the Difference of the Squares of
being one of thoſe mentioned by Sir the Segment of the Baſe, made by
Iſaac Newton in his Algebra, which letting fall a Perpendicular from the
is neceſſary to be known by all thoſe vertical Angle upon the Baſe; and
who intend to apply Algebra to Geo- the Square of one side, together
metry; but he neither demonftrat- with the Square of the alternate Seg-
ing it, nor directing where it is de- ment, is equal to the Square of the
monſtrated, therefore I have given other Side, together with the Square
a Demonſtration thereof.
of the other Segment.
If there be any Right Line B E In any Triangle, if the Baſe be
which biſects the Angle ABC of biflected, and a Right Line be drawn
А.
B
from the Angle oppoſite to the
Baſe to the point of Biſfection, the
Squares of the two Sides together,
are equal to twice the Square of the
E
Biffecting Line together, with twice
C the Square of -half the Baſe.
In every Triangle the Rectangle
under the Sides is equal to the Rect-
angle under the Perpendicular drawn
D
from the vertical Angle to the Baſe,
the Triangle ABC, I fry the Square and the Diameter of the circum-
of the ſaid Line BE ABXBC ſcribing Circle.
A E ⓇEC.
In every Triangle the Angle con-
Having deſcribed a Circle about tained under the Perpendicular to
the ſaid Triangle, and continued the Baſe, and the Right Line drawn
from
TRI
TŘI
from that Angle to the Middle of of, whoſe two other Sides AG, GC,
the Baſe, is 'equal to half the Diffe- make an Angle equal to one third Part
rence of the Angles at the Baſe. of four right Angles (or 120 Deg.) and
In every Triangle the Rectangle the Right Line EG be drawn, and
under thé Aggregate and Difference if any Point F be taken in the Line
of the Sides is equal to the Rectangle E G, and the Right Lines AF, FC,
under the Aggregate and
Difference be drawn : 1 ſay the Aggregate of the
of the Segment of the Baſe, made Lines AF, FC, will be greater than
by letting fall a Perpendicular from the Line EF.
the Vertical Angle to the Baſe. Conceive a Circle to be deſcribed
If the Point D be taken within a about the Triangle' A CE, which
Triangle A B C ſuch, that the An- will paſs through
G, and from the
gles A DB, B DC, ADC, formed
Fig. 2.
at the ſame by three Right Lines
Fig. I.
B
H.
)
A А
A А
411 AS
с
E
Fig. 39
역
​UITOICT«««...YIM<volvervui**dente
2
G
A
(
AD, BD, DC, drawn from the An-
gles A, B, C, of the Friangle ABC,
be equal to each other. I ſay the Ag-
gregate of thoſe three Right Lines AD,
BD, CD, will be leſs than the Ag-
gregate of three other Right Lines
E
drawn from the Angles of the Triangle Point f let fall the perpendiculars
A B C, to any 'Point beſides D, taken FH, FI, to AG, and CG. then
within tihat Triangle.
will the Angle AGE be = EGC,
Before this can be demonſtrated, we
becauſe the Arches A E, EC, are
mult lay down the following Lemma. equal. So that each of theſe will
teral Triangle, and a Triangle AGC be of one Right Angle, or 60 De-
be conftructed upon one side À C there grees. Since the whole Angle AGC
K
4
is
TRI
TRI
is of two Right Angles, or 1żo Therefore A +FC - FG.is
Degrees, and (in Fig. 3) the Angles greater than E G, and adding: F G
FGI, FGH will be each Parts to both, we hall have A F+FC
of one Right Angle, or 60 Degrees, greater than FG + EG; that is,
becauſe theſe are each Vertical to than E F.
EGA, EG Cé. Conſequently the This being granted, upon the Side
Angles H FG, IF G of the right- AC of the Triangle ABC make
angled Triangles. FHG, TH G the equilateral Triangle A E C (Fig.
will be part of a Right Angle or 1.) and conceive a Circle to be de-
30 Degrees. So that HG=GI. ſcribed about the ſame; then from
and GF will be the double of HG. what has been already faid, the
Şince, if F G be ſuppoſed the Radius, Point D will fall in the Circumfe-
FI and FH will each be the Sipe rence of that Circle, and if the Points
of 60 Degrees, and HG, or GI, E-and B be joined by a Right-Line ;
the Sine of 30 Degrees; and it is well this Line will paſs through the Point
known that the Sine of 30 Degrees D, and therefore, ſince E G = A
is equal to the Radius. Therefore · DEDC; the Line EB will be
FGHG+ GI, alſo (by n. 22. un- equal to the three Lines A D, DB,
der the word Circle) AG+GC= D C together. Now take any other
EG. Now becauſe (Fig. 2.) AH+CI Point G within the Triangle, and to
+ FG (HG+GI)= ĘG, and A the ſame draw the three Right Lines
F+F C is greater than AH+CI; AG, GB, GC, as alſo the Line
for AF, F C are the Hypothenuſes EG, then I ſay, ,AG +GB+
of the right angled Triangles A FH, G C will be greater than AD+
CFI. Therefore AF + FC + DB + DC, that is, than EB. Now
FG is greater than EG, and tak. it has been already ſhewn, that A G
ing away F G from both; it will be +GC is greater than EG, and
AF+F C greater than EG adding G B to both, it will be AG
FG, that is, than EF. After the +GC-EGB greater than E G+
like manner in Fig. 3.) becauſe Ġ B. But ſince the two Sides E Ġ
AH +CI-EG = EG. But f G B of any Triangle E G B are
AF+FC greater than AH+CI. greater than the third Side E B;
B
7
D/
H
EVG
IT
A
с
-
there-
TRI
TRI
therefore AG +GC+GB will. D, in a tight Line RG given in
be much greater than
E-B, that is, Pofitions will be a Minimum, when
than A DDB EDC.
the Angles ABCBG are equal.
Note, When the Point G is taken For if D'be taken in G'inkitely
in the Periphery of the Circle, the near to B, and AD, DC be drawn,
Demonſtration is very ſimple, there and AD Cuts BC in E) then
being then no Occaſion for the Lém- ' fince the Angles A BFZ FDP mày
ma aforegoing. Wherein
be taken for Equalss they differing
Otherwiſe : Let F be any Point from one anotketo only by can'inti-
infinitely near to D, from which nitely ſmall Anglez the Angle CBG
draw FÉ,FG perpendicular to AD, will be =AB i panduro the sides
DC, and join the Lines AF, pc, BE, ED of the imali Triangle
FB and FD, and let fall DH perpen- BED, are equal
. *** Bút thêfe Sides
dicular to BF, then will ED be the are the Fluxions of the Lines A'B,
Fluxion of AD; DG that of DC; CB; fo that ED EEBZ0, or
HF that of BD: ſo that if FD bé ED : EB; and therefore''AB
fuppoſed the Radius, ED will be the + BC is a: Minimum.
Sine of the Angle DFE, DG the Sine
Note, It is very eaſy to demon-
of the Angle DFG, and FH the Sine ſtrate this by commonGeometry
of theAngle FDH. But the Angle too : by letting fall from A a Per-
EFG is 60 Deg. Since E and Ġ, pendicular to FG; and continuing
and EDG being 180° +120°, and it down below FG till it meets
that the Complement of theſe three CB continued alfo below 'FG-for
toi360°; alſo the Angle FDH then if any other Point D be taken
EFDG +GDH, that is, FDG + in FG, and AD be drawn, as al-
30%) = 60° +EFD. But it is ſo CB, which being continued be-
demonſtrated in Trigonometrical low B, will fall in the ſame Point
Writings, (See Sherwin's Tables) with the Continuation of A Fit
That the Sum of the Sines of any will follow that AB + BC is 'leſs
two Angles making 60°, is equal than AD+DC Since two sides
to the Sine 'of an Angle as much of any Triangle are greater than
above 60°, as is the Quantity of the third.
one of thoſe Angles, that is, ED+ Equal Triangles, that have one
DG=-FH;and fo ED+DG + Angle of the one equal to one An-
FH=o. Conſequently AD+BD+ gle of the other, have the sides
DC is a Minimum ;' for when the including the equal Angles recipro-
Sum of any Number of Quantities cally proportional, and contrari.
is a Minimum, the Sum of their wiſe.
Fluxions will be equal to 0.
In any Triangle, if the three
After this
Sides be biflected, and right Lines
by Inſpection, that the Sum of two be drawn from the Angles to the
right Lines AB, BC, drawn from Points of Biffection, they will cur
two given Points A, C, to a Point one another in the ſame Point with
in the Triangle, which Point will
triſleet them; and the Squares of
A
theſe three Lines taken together,
are to the Squares of the three Sides
of the Triangle, as three to four.
E
If any Point be taken within a
TE
G ,
Triangle, and through the ſame be
BD
drawn three right Lines from the
Angles
rrrr
way
it almoſt appears
C
is
TRI
TRI
Angles of the Triangle cutting the TRIANGULAR COMPASSE S, are
Sides into fix Parts, the Parallele- ſuch as have three Legs or Feet to
pipedon contain'd under the firſt, take off any Triangle at once.
third, and fifth of theſe parts or TRIANGULAR QUARDRANT,
derly taken, is equal to the Paral. is a Sector with a looſe Piece to make
lelepipedon under the ſecond, it an Equilateral Triangle; the Ca-
fourth and fixth.
lendar is graduated on it, with the
If any three ſimilar Figures be de- Sun's Place, Declination, and many
ſcribedupon the three ſides of a right- other uſeful Lines; and by the Help
angled Triangle, the Figure upon of a String and a Plummet, and the
the Hypothenuſe will be equal to Diviſions graduated on the looſe
the Figures, taken together, upon Piece, it may be made to ſerve for
the other two fides.
a Quadrant.
If one Angle of any Triangle be TRIANGULUS SEPTENTRIONA-
two third Paris of two right Angles Lis, or Deltoton. The Triangle, a
or its Meaſure 120°, the Squares de- Northern Conſtellation conſiſting of
ſcribed upon the Sides containing fix Stars.
that Angle, and the Rectangle un TRIGLIPH, in Architecture, is
der thoſe Sides, all three taken to a Member of the Doric Freeze,
gether, are equal to the Square de- placed directly over each Column,
Scribed upon the Baſe. And if the and at equal Diſtances in the In-
Sides containing that Angle be equal, tercolumnation, having two entire
the Square of the Baſe will be three Glyphes or Channels engraven in it,
times the Square of either of the meeting in an Angle, and ſeparated
equal Sides. And the Square of the by three Legs from the two Demi-
Perpendicular let fall from the ver- Channels of the Sides.
tical Angle upon the Baſe, is one TRIGON, fignifies a Figure with
twelfth Part of the Square of the three Angles : And, in Dialling, is
Baſe.
an Inſtrument of a Triangular
In any Triangle, whoſe vertical Form.
Angle is two third Parts of two TRIGONOMETRY, is either Plain
right Angles, the Difference of the or Spherical.
Cubes of the Sides containing that TRIGONOMETRY (PLAIN,) is
Angle, is equal to a Parallelepipe- the Art of finding, from three given
don, whoſe Baſe is the Square of Parts of a Right-lin'd Triangle, the
the Baſe of the Triangle, and Altitude reft. And
the Difference of the Sides. And TRIGONOMETRY (SPHERIC
the Sum of the Cubes of the Sides, CAL,) is the Art of finding, from
is equal to a Parallelepipedon whoſe three given Parts of a Spherical Tri-
Baſe is the difference of the Squares angle, the reſt ; as from two Sides
of the Baſe, and twice the Rectangle and one Angle, the two other An-
under the Sides, and Altitude equal gles and the third Side.
to the Aggregate of the sides of the 1. In all Right-angled Plane Tri-
Triangle.
angles, if one of its Sides, be made
There are many other Properties the Radius, the other two will be
of Triangles to be found among the the Sines, Tangents, or Secants, of
Geometrical Writers; as Gregory St. the Acute Angles : And whatever
Vincent has wrote a thin Folio Proportion the Side made has to
Book upon Triangles. You have alſo the Radius, the ſame has the other
ſeveral in his Quadrature of the Sides to the vines, Tangents, or Se-
Circle.
canis, repieſented by this.
AS
TRI'
TRI'
Ć
C
A
B
If the Sides A C, BC, AB, of a
A
B
Triangle A B C, be given, and if
A B be biffected in I, and you take
As if AC be the Radius, then
C
S. BAC:BCU:: Radius : Hy-
S. ACB:ABS pothen. A C.
If the Leg AB be the Radius,
then
F A HIDEB G
Radius : ABUS:: Sec. BAC: upon it (when both Ways produced
T. BAC:BC Hypothen. A C.
the Lines AF, A E, equal to AC,
and B G, B H, equal to BC, and
If the Leg B C be made the Ra- join CE, CF, and from C let fall
dius, then
a Perpendicular CD, to AB, fap-
Radius : BC US:: Sec. ACB : poſed the Baſe, then will the Area
T. ACB: ABS Hypothen. AC.
VFGXFHX HEX E G.
And for determining the Angle A,
2. In any Right-lined Triangle there come out ſeveral Theorems :,
the Sides are proportional to the
Sines of the oppoſite Angles.
3. As 2 ABXAC:FHEG
(:: AC: DE):: Radius : verſed
Whence in the Triangle A C B. Sine of the Angle A.
B
4. 2. AB AC:VFGxFH
(:: AC:DF) :: Radius : verſed
Cofine of A.
5. 2 A BⓇAC: VFGxFHⓇ
HEXEG (::AC:CD):: Radius:
А
Sine of A.
6. VFGxFH:N HEXEG
As,
SA :S.C::
SBCT
(::CF:CE):: Radius : Tangert
S. BS
: AB.
of A.
S. C
: S. B::
SAB
7. V HEXEG:VFGXFH
S. A
AC.
BCS
(::CE:FC):: Radius : Cotangent
S. B
SACZ
of A.
S.C
ABS
8. 2VBXAC:VH EXEG
In every Right-lined Triangle, as (::FE:EC) :: Radius : Sine of
ABC, as the Sum of the Sides A B,
AC, about a given Angle A is to 9.2V ABXAC: V FGxFH
their Difference, fo is the Tangent (FE:FC):: Radius : Cofine of
of half the Sum of the remaining A.
Angles B, C, to the Tangent of half 1. In every Splerical Triangle,
their Difference.
each Side is leſs than a Se:111-Cirile.
AC
:S. A::
:C.
I A.
2.
TRI
T RI
4
2. In every Spherical Triangle, is, be both greater, or both leſs
any two Sides together are greater than a Quadrant) the Hypothe-
than the third.
neufe is leſs than a Quadrant; or if of
3. The Sum of the sides of a different Affections, then greater,
Spherical Triangle is leſs than two and the contrary.
Semi-Circles
12. In a Right-angled Spherical
4, 'If two sides of a Spherical Triangle, the Sum of the Oblique
Triangle be equal to a Semi-Circle, Angles are leſs than three Right
the two-'Angles at the Baſe ſhall be Angles.
equal to two Right Angles ; if they 14. In any Spherical Triangle
be leſs than a Semi-Circle, the two whoſe Angles are all Acute, each
Angles ſhall be leſs; but if greater Side is leſs than a Quadrant.
than a Szmi-Circle, the two Angles 15. In Spherical Triangles, there
thall be greater than two Right are twenty-eight Caſes, fixteen in
Angles.
Rectangular, and twelve in Oblique
5. The Sum of the three Angles Angular. The fixteen Caſes of Rec-
of a Spherical Triangle, are greater tangular are reſolv'd by the two firſt
than two Right Angles, and leſs than of the following Theorems.
Ix
36. Two Angles of any Spheri-
THE O. I.
cal Triangle are greater than the
Difference between the third Angle In all spherical Rectangular Tri-
and a Semi-Circle. Therefore, angles, having the fame Acute Angle
Any Side being continued, the at the Baſe, the Sines of the Hypothe-
Exterior Angle is leſs than the two neuſes are proportional to the Sines
, of
Interior oppoſite ones.
their Perpendiculars.
**.8: 'In any Spherical Triangle the
Difference of the Sam" of two An-
THE O. `II.
gles and a whole Circle, is greater
than the Difference of the third An In- all Spherical Reftangular Tri-
głe anđa Semi-Circle,
angles, having the fame Acute ingles
9. In any Spherical Triangle, at the Baſe, ihe Sines of the Bajes,
one Side being, produced, if the and the Tangents of the Perpendiculars,
other two Sides' be equal to a Semi-. are proportional.
Circle, the outward' Angle ſhall be
equal to the inward oppoſite Angle.
That all the caſes of a Right-
upon the Side produced: If they be angled Spherical Triangle may be
leſs than a Semi-Circle, the outward reſolved by theſe two Theorems.
Angle ſhall be greater than the in-, The ſeveral Parts of the Sphe-
ward oppoſite Angle; if they be rical Triangle propoſed, muft ſome-
greater than a Semi-Circle, the out- times be continued to Quadrants,
ward Angle ſhall be leſs than the that ſo the Angles may be turn'd
inward oppoſite Angle.
into Sides, -the-Hypotheneuſes into
*10. The Legs of a Right-angled Baſes and perpendiculars, and the
Spherical Triangle are of the ſame contrary. By which Means the
Affection with their oppoſite Angles. Proportions, as to the Parts of the
**11: In a Right-angled Spherical Triangle given, inſtead of Sines,
Triangle, if either Leg be a Qua- da ſometimes fall in Co-fines, and
drant, the Hypotheneule ſhall be al- ſometimes in Co-tangents, inſtead
fo a Quadrant; but if both the of Tangents. Such Parts as do
Legs be of the fame Affection that change their Proportion, are noted
3
with
T RI
TRI
with their Complements
, viz. the guiſhed into five Circular Parter
Hypotheneuſe, and both the Obli- for the more Eafe in reſolving all
que Angles; but the Sides containing Spherical Triangles, the Lord Napier
the Right Angle do not change.
invented this Catholick and Univer:
Theſe are called the Five Circu- fal Proportion, viz.
lar Parts of a Triangle, amongſt
The Sine of the Middle Part and
which the Right Angle is not rec. Radius is reciprocally proportional
koned ; and therefore the two Sides to the Tangents of the Extreams
which do contain it, are ſuppoſed to conjunct, and the Co-fines of the Ex,
be joined together.
treams disjunct.
į
Each of theſe Circular Parts, That is, as the Radius to, the
may, by Suppofition, be made thé Tangent of one of the Extreams
Middle Part; and then the two conjunct, fo is the Tangent of the
Circular Parts, which are next to other Extream conjunct to the Sine
that Middle Part, are the Extreams of the Middle Part.
conjunct; the other remote from And alſo, as the Radius, to the
the Part affumed, are the Extreams Co-line of one of the Extreams dis-
disjunct.
junct, ſo is the Co-fine of the other
As in the Triangle A BC, (ſup. Extream disjunct, to the Sine of the
poſe a Triangle to be drawn, if Middle Part.
Comp. A C be made the Middle Part, Therefore if the Middle Part be
Comp. Ą and Comp. C. are the Ex- fought, the Radius muſt be in the
treams conjunét; and the Side AB firſt place; if either of the Extreams,
and BÇare the Extreams disjunct i the other Extream muſt be in the
and ſo of the reſt, as in the Table firſt Place.
following
Only note, that if the Middle
Part, or either of the Extreams con
Mid.Part. Extr. conj. Extr. disj.
junct, be noted with its Comple,
ment in the Circular Parts of the
Triangle, inſtead of the Sine or Tan
Leg. AB Camp. A Comp. AC
gent, you muft uſe the Co-fine or
Leg: BC Comp. C
If either of the Extreams disjunct
Comp. A. Comp. AC Comp. C be noted by its Complement in
Leg. A B Leg. BC the Circular Parts of the Triangle,
inftead of the Co-fine you muſt uſe
Comp. AC Comp. A Leg. AB
the Sine of ſuch Extream disjunct.
Comp. C. Leg. BC
That the Directions may be bet-
ter underſtood, there is in the Table
following, the Circular Parts of a
Comp. Comp. AC Comp. A
Leg. BC Leg. AB
Triangle under their reſpective Ti.
cles, whether they be taken for the
Middle Part, or for the Extreams;
Leg. BC Comp. C Comp. A
whether conjunct or disjunct; and
Leg. A B Comp. AC
unto thofe Parts there is prefixed the
Sine and Co fine, the Tangent or Co-
The Parts of a Right-angled Sphe tangent, as it ought to be by the Ca-
rical Triangle, being chus diftin- tholick Proportion.
T
Co-tangent.
$
Mid.
TRI
TRI
7
rence;
THE O. V.
Mid.Part. Extr.conj. Extr. dist.
In all Oblique - angled Spherical
Sine. AB Go-tan. A Sine AC Triangles, in which two Angles are
Tan. BC Sine C leſs than two Right Angles :
As the Sine of half the Sum of two
Angles,
Co-line A Co-tan AC Sine C
To the Sine of half their Diffe-
Tang. A B Co-fine DC
Ca-fane AC Co-tan. A Co-fine AB jacent Side,
So is the Tangent of half the inter-
Ca-tan. C. Ca-6ne BC
To the Tangent of half the Diffe-
rence of the Oppoſite Sides.
Co-fine C Co-tan AC Sine A And, as the Co-fine of half the Sum
Tan. BC Co-fine AB of the Angles,
To the Co-fine of half their Diffe-
Sine BC Co-tan. C Sine A
cs
rence,
Tan. AB Sine AC
So is the Tangent of half the inter-
jacent Side,
To the Tangent of half the Sum of
THE Q. III.
the Oppoſite Sides.
THE O. VI.
In all Spherical Triangles, the Sinęs
of the sides are in direct Proportion
to the Sines of their Oppoſite Angles, containing Sides,
As the Rectangle of the Sines of the
and the contrary.
To the Square of the Radius ;
Sa is the Rectangle of the Sines of
T H E 0. IV.
half the Sum of the three Sides, and
of the Difference of the Oppoſite Side
In all Oblique - angled Spherical therefrom,
Triangles, in which two Sides are To the Square of the Co-fine of half
leſs than a Semi-Circle :
an Angle ſought.
As the Sine of half the Sum of the
wo Sides
Ptolemy is the firſt Trigonometri-
To the Sine of half their Diffea cal Writer. Hipparchus alſo wrote
12 Books concerning Triangles, but
So is the Co-tangent of half the con- they are loft. Amongſt the more
tained Angle,
modern, there are Regio Montanus's
To the Tangent of half the Diffe- Libri Quinque de Triangulis
, wrote
rence of the Oppoſite Angles.
Anno 1464, and publiſhed by Scho-
And, as the Co-fine of half the Sum ner, Anno 1533. You have Pitiſcus,
of the Sides,
Snell, Napier, Clavius, Urfinus, Gelle-
To the Co-fine of half their Diffe- brand, John Newton, Seth Ward,
Oughtred, Gooden the Jeſuit, Nor-
So is the Co-tangent of half the con- wood, the Wilfons (two different
tained Angle,
Perſons, one in England, and another
To the Tangent of half the Sum of in Holland,) Ozanam, Dechales,
the Oppoſite Angles
Wolfius, Harris, Hains, Dr. Keil,
&c.
!
Tence ;
rence ;
TRO
TWI
&c. who have all wrote of Trigono- pick of Capricorn, becauſe they lie
metry.
under theſe Signs.
TRILATERAL, in Geometry, is TRUCKS of the Carriage of a
the ſame with Three-fided.
Piece of Ordnance, are the Wheels
TRINE, is an Aſpect of the Pla- which are on the Axle-tree to move
nets, when at the Diſtance of 120 the Piece.
Degrees, or four Signs, from each TRUE CONJUNCTION. See Con-
other, and is noted thus A. junction True.
TRIANGLB, in Architecture, is a True Place of a Planet or Star,
little Member fixed exactly upon is a Point of the Heavens ſhewn by.
every Triglyph under the Plat-Band a Right Line drawn from the Centre
of the Architrave, from whence hang of the Earth, through the Centre of
down the Guttæ, or pendant Drops, the Planet or Star.
in the Dorick Order.
TRUNCATED PYRAMID, or Cone,
TRINOMIAL Root, in Mathe- is one whoſe Top is cut off by a
maticks, is a Root conſiſting of three Plane parallel to its Baſe.
Parts connected together by the Sign A Truncated Cone, or the Fruf-
+; as x+y+z.
tum of that Body, is called fome-
TRIPARTITION, is the Divi- times a Curtie Cone.
fion by 3, or taking the third Part TRUNNIONS of a Piece of Ord-
of any Number or Quantity. nance, are thoſe Nobs or Branches
TRIPLICATE RATIO, is the of the Gun's Metal which bear her
Ratho of the Cubes.
up upon the Cheeks of the Carri-
TRIS - DIAPASON, or Triple ages.
Diapaſon Chord, in Muſic, is what TURN, a Term belonging to the
is otherwiſe called a Triple Eighth, Movement of a Watch, and figni-
or Fifteenth.
fies the entire Revolution af any
TRITONE, a Term in Muſic, Wheel or Pinion.
which fignifies a great Fourth. TUSCAN ORDER, in Architec-
TROCHILE, in Architecture, is ture, is the firlt, the moſt fimple,
that Hollow Ring, or Cavity, which and the ſtrongeft: Its Column has
runs round a Column next to the ſeven Diameters in Height ; and its
Tore.
Capital, Baſe, and Entablement, have
TROCHLEA, is one of the Me- no Ornaments, and but few Mould-
chanic Powers, and is what we ings.
uſually call the Pulley.
TWILIGHT, is that faint Light
TROCHOID, the ſame with Cy- which we perceive before the Sun-
cloid. Which fee,
Riſing, and after Sun-Setting. 'Tis
TROPICAL YEAR. See Year. occafioned by the Earth's Atmo-
Trophy, in Architecture, is an ſphere refracting the Rays of the
Ornament which repreſents the Sun, and reflecting them from the
Trunk of a Tree charged or encom- Particles thereof.
paſſed all round about with Arms 1 he Sun's Depreſſion below the
or Military Weapons, both offenſive Horizon, at the Beginning and End
and defenſive.
of the Morning and Evening Tw!-
TROPICKS, are Circles ſuppoſed light, was obſerved by Alhazın 19º.
to be drawn parallel to the Equi- Tycho 17º. Rothmau 24°. Stevini s
noctial at 230 30' Diſtance from it ; 18º. Caffini 15º. Ricciolus, at the
one towards the North, is called the Time of the Equinox in the Morn-
Tropick of Cancer; and the other to. ing 16°. in the Evening 20°. 30' In
wards the South, is called the Tro- the Summer Solſtice in the Morning
2007
V A R
VER
1
1
!
V
2rº.25'. and in the Winter 170, 15'. monly called by the Seamen the
Whence it appears that the Cauſe of North-Eaſting, or North-Weſting of
the Twilight is inconftant: But a- the Needle.
bout 18 Degrees of the Sun's De VECTIS, or the Lever, is the
preſion will, in our Latitude, be Firſt of the Mechanic Powers, as
the Beginning and End of the Twi- they are uſually called.
light.
VENT, in Gunnery, fignifies the
TYMPAN, in Architecturë, is that Diſtance between the Diameter of a
Part of the Bottom of the Frontons, Bullet, and the Diameter of the Bore
which is encloſed between the core of the Piece, and muſt be one twen-.
nices, and anſwers the. Naked of tieth Part of the Diameter of the
the Freze.
Bore.
TYMPAN of an Arch, is a Tri-, VELOCITY. See Celerity.
angular Table placed in its Corners. VENUS, the Name of one of the
Planets, being the ſecond from the
Sun.
The Diſtance of Venus from the
.;
V.
Sun is 723, of ſuch Parts, of which
the Earth's are 1000, its Excentricity
ACUUM, his by Philoſo. 5, the Inclination of its Orbit 3 De-
phers ſuppoſed to be a Space grees, and 23 Minutes : It performs
devoid of all Body ; and this they its Periodical Motion in 224 Days,
điftinguiſh into a Vacuum Diffemina- 17 Hours ; and its Motion round its
tum, or Interſperſum, i. e. ſmall void Axis is performed in 23 Hours.
Spaces interſperſed about between The Diameter of it is almoſt equal
the Particles of Bodies ; or, a Vacu- to the Earth's Diameter.
sm Coacervatum, which is a larger In the Years 1672 and 1686, Mr.
void Space made by the meeting Caflini, with a Teleſcope of 34 Foot
together of the ſeveral interſperſed long, believes he ſaw a Satellite
or diſſeminate Vacuities before men- moving round this Planet, and di-
tioned.
ftant from it about y of Venus's Di-
VANE. Thoſe Sights which are ameter. It had the fame, Phafis,
made to move and ſide upon Croſs- with Venus, but was without any
Staves, Fore-Staves, Davis's Qua well defined Form, and its Dia-
drants, &c.
meter ſcarce exceeded of that of.
VAPOURS, are Watry Exhala- Venus.
tions raiſed up either by the Heat Versen SINE of an Arch, is a
of the Sun, the fubterraneal, or any Segment of the Diameter of a Cir-
other accidental Heat, Fire, &C. cle, lying between the foot of the
VARIATION, is, according to Right Sine and the Lower Extre-
Tycho, the third Inequality in the mity of the Arch.
Motion of the Moon ; and ariſes As AC is the Verſed Sine of the
from her Apogæum being changed
as her Syſtem is carried round the
D
Sun by the Earth.
VARIATION of the Needle, or
Compaſs, is the Deviation or Turn-
ing of the Magnetical Needle in the
Mariner's Compaſs, from the true
North Point, which happens more
A
C B
or leſs in moſt Places; and is com-
Arch
1
)
1
(
V "R
VIT
Arch AD, and C B the Verſed Sine Divergence in a Concave Glaſs is the
of the Arch BD.
Point E in the following Figure.
Vertex, is that Point of the Let the Concavity of the Glafu
Heaven juſt over our Heads, and be A BC, and its Axi& DE: Let
the ſame with the Zenith; which
ſee.
The Point of any Angle is called
K
А
alſo its Vertex, and that Point of HS
the Curve' of a Conick Section,
where the Axis cuts it, is called
В.
alſo the Vertex of that Section.
D c
VERTEX, of a Cone, or Pyra-
mid, &c. is the Point of the upper FG be a Ray of Light falling on
Extremity of the Axis, or the Top the Glaſs, parallel to the Axis DE:
of the Figure: So the Vertex of an and let D be the Centre of the Arch
Angle, is the Angular Point. ABC: This Ray FG, after it hath
ERTEX of a Glafs, in Opticks,, paſſed the Glaſs, at its Emerſion at
is the ſame with its Poles; which" G will not proceed directly to H,
fee.
but be refracted from the Perpen-
VERTICAL CIRCLES: See Axia . dicular DG, and will become the
muths.
Ray GK; draw directly GK, fo
VERTICAL LINB: See Line Vor- as that it may croſs the Axis in E,
sical.
ſo found. Mr. Molyneux calls it the
VERTICAL OPPOSITEANGLES: Virtual Focus, or Point Of Diver-
See Angles.
gence.
VERTICAL PLANE, in Perſpec VISIBLE HORIZON: See Hori-
tive: See Plane.
VERTICAL POINT, the fame VISIBLE PLACE of a Star : Sec
with · Vertex : So that in Atro- Apparent Place.
nomy, a Star is ſaid to be Vertical, Vision, is a Senſation in the
when it happens to be in that Point Brain, proceeding from a due and
which is juſt over any Place. various Motion of the OptickNerves,
VERTICITY, the Property of produced in the Bottom of the Eye,
the Loadſtone, or a touched Needle, by the Rays of Light coming from
to point North and South, or to any Object : by which means the.
wards the Poles of the World: See Soul perceives theilluminated Thing,
Magnet and Magnetiſm.
together with its Quantity, Quality,
VIA LACTEA: See Milky Way. and Modification.
VIBRATION, is the Swing or VISUAL POINT, in Perſpective,
Motion of a Perpendicular, or of a is a Point in the Horizontal Line,
Weight hung by a String on a Pin. wherein all the Ocular Rays unite,
VINDIMATRIX, a fixed Star of VISUAL RAYS: See Rays.
the third Magnitude, in the Con VITREOUS HUMOUR, or Glaſy
kellation Virgo, whoſe Longitude is Humour of the Eye, is the third
185 Degrees and 23 Minutes, Lati- Humour of the Eye, ſo called from
tude 10 Degrees and 15 Minutes. its Reſemblance of a melted Glaſs :
VIRGO, one of the twelve Signs It is chicker than the Aqueous, bus
of the Zodiac, being the Sixth ac not fo folid as the Cryſtalline : It is
cording to Order.
round or conyex, behind, and ſome-
VIRTUAL Focus, or Point of what plain before, only hollowed a
LI
litcls
zon.
U NI
U NI
little in the Middle, where it re- Sections of the Solid, made by Planes
ceives the Cryftalline. It.exceeds perpendicular to the Horizon, and
both the other Hamours in Qgan- parallel to one of its Sides, be Semi-
tity:
Elliptick Spaces of the fame Mag-
UMBILICUS, the fame as Focus. nitude, whoſe tranſverſe Axes are
DUMBILICK POINTS, the ſame as the Lengths of the Solid, and Semi-
Foci.
Conjugates the Height in the Mid-
: UNCIE, in Algebra, are thoſe dle! i'his Solid will have the ſame
Numbers which are prefixed before Diſpoſition to break in all its Parts ;
the Letters of the Members of any and ſo Joifts, &c. cut after this Fi-
Power produced from 'a Binomial, gure, will be as ſtrong as when they.
Reſidual, or Multinomial Root. are of the ſame Height all the way
Thus in the fourth Power of as this Solid has in the Middle; and
atb, that is, aaaa + 4 aaab + 6 conſequently the Timber ſaved by
aabb + 4 abbb + bóbb, the Unciæ cutting a joilt in Figure of this So-
are, 4, 6, 4.
lid,, will be about three Parts out of
UNIFORM MATTER, is that foarteen.
which is all of the fame Kind and If a folid Parallelepipedon of
Texture.
Uniform Matter be ſupported Hori-
If there be a Right-angled Paral- zontally, as a Prominent Beam in
Jelepipedón of Uniform Matter, fup- the side of a Wall, the Difpofition
ported horizontally by two Fulcrums to break of that Part coming out of
at its Ends, its Diſpoſition to break the Wall in any Place by the
Weight
in any Part (or Point) of it by its of the whole Prominent Párt, will
own Gravity, will be as the Rectan- always be as the Diftance of that
gle under the Diſtanee of that Part Place from the End of the Promi-
(or Point) from each Fulcrum ; and nent Part; and ſo its Diſpoñion to
to its Difpofition to break in the break at the Wall will be greateſt,
Middle, will be greateſt, ſince the And if the Upper" Surface of the
Rectangle there becomes a' Maxi- Prominent Part be changed into a
mum.
Curve Surface, ſuch that all Sections
This is true of Cylinders and of it, by Planes parallel to the up-
right Faces of the Solid, and pera
The fame Thing being ſuppoſed pendicular to the Horizon, are equal
when the Length and Breadth, and Semi-Parabolas, having their Axes
the Parallelepipedon remain the in the under Surface, and Vertexes
ſame, its Diſpoſition to break in the in the lower Side of the End-Fáce
Middle (or at any other Point at the of the Solid, which is parallel to
fame Diſtance from the Fulcrums) the Wall, then this Prominent So-
will always be as the Square of the lid will have the ſame Diſpoſition
Height; and ſo the Strength of a to break in any Part of it, that is,
Parallelepipedon, laid edge-ways it will as ſoon break in one Part as
upon the Fulcrums to its Strength the other; and ſo there may be
when laid flat-ways, will be as the Part of the Matter ſaved by cutting
Height in one Caſe, is to the Height it into this Solid ; and yet it will
in the other.
as ſtrong as a Parallelepipedon, of
From what has been faid, if the the ſame Length, Breadth, and
Upper Face of the Parallelepipedon, Height (that it has at the Wall)
lying horizontally upon the two Ful- with itſelf, provided it be of the
cums, be changed into a Curve (ame Uniform Matter.
Surface, being ſuch that all the
3
UNI-
Priſms likewife.
*
one, or 1.
1
UNI
v Ú
UNIFORM Motions, are the parallel to the Plane of the ſaid.
fame with equal, or rather equable Equinoctial Circe: Bui this is but
ones; which fee.
about one 'Hour every Day, and
UNISON, is one and the ſame four Days in the Year,
Sound.
UNIVERSAL PROBLEM, the
UNIT, or Unity, is the ſame as fame as Indeterminate Problem.
VOLUTE, in Architecture, is one
UNIVERSAL EQUINOCTIAL of the principal Ornaments of the
Dial, is one confiſting of two Ionic and Compoſite Capitals, repre-
Rings of Braſs, or Silver, that open ſenting a kind of Bark wreathed, or
and fold together, with a Bridge or twiſted into a Spiral Scroll. There
Axis, and a Slider, and a little Ring are eight Angular Volutes in the
to hang or hold it up by: It is divided Corinthian Capital, and theſe are
on one side of the great Ring into accompanied with eight other little
90 Degrees, and ſometimes on the ones, called Helices.
other into two Quadrants, or 180 VORTEX, according to the Carte-
Degrees, but one is enough. The fian Philoſophy, is a Syſtem of Par-
innermoſt Ring is divided into 24 ticles of Matter moving round like
Hours, ſubdivided on the Face, and a Whirl-Pool, and having no void
on the Outſide of the Ring, into Interſtices, or Vacuities between the
every 5 Minutes. The Axis has the Particles.
Sun's Declination on one side, and One would have thought that this
the Day of the Month, and the fooliſh Fable of Vortexes had been
Sun's Place on the other.
ſufficiently exploded many Years
To uſe it for the Hour, the Per- ago, by the great Newton, Cotes,
pendicular Line, or Stroke, which and other ingenious Perſons, (lee
is.on the Slider, wlich moves on the Mr. Cotes's Preface to the ad Edition
outer Ring, muſt be ſet to the La- of Newton's Principia, as alſo the
titude of the Place, and the Hole in last Section of the Second Book of
the Slider, or the Bridge, either to the ſaid Principia;) but füch is the
the Sun's Place in the Ecliptick, the Obſtinacy, Partiality, and perhaps
Day of the Month, or bis Declina- Folly of fome of the French, that
tion; and then the Rings being within this few Years they have
open'd, and ſet fquare to one ano again renewed the Controverſy, and
ther, move the Dial about, to and endeavoureď to account for the Cele
fro, till the Sun ſhines through the țial Phänomena from this vain and
Hole, and on the inner Edge of the imaginary. Hypotheſis. See the
innermoſt Ring it will shew
the true Tranſactions of ibe Royal Academy of
Hour.
Sciences at Paris, Anna 1715,1728,
The Hour of 12 cannot be ſhewn 1729...
by this Dial, becauſe the outermoſt UPRIGHT SOUTH DIALS. See
Circle, or Ring, being then in the Prime Verticals.
Plane of the Meridian, it hinders URSA MAJOR, a Northern Con-
the Rays of the Sun from falling ſtellation, contifting of 27 Stars, and
upon the innermoſt or Equinoctial is otherwiſe called Charles's Wain,
Circle. And when the Sun is in the and the Great Bear,
Equinoctial, you cannot tell the VULGAR FRACTIONS, See
Hour of the Day by this Inſtrument, Fractions.
becauſe at thac Time his Rays fall
*
; 1ܐ
W A T
W A T
WA
G
the Height to which the Water
: W.
aroſe at that time: He then hung it
up, and 80 Years after it was found
ADHOOK, among the in Kircher's Study, juſt as full as it
Gunners, is a Rod or great was at firſt. So that from hence it
Wire of Iron, turned in a Serpen- would ſeem, that Wates will not
tine Manner; and its end is put paſs thro' Glaſs.
upon a Handle or Staff, to draw out
The Motion of Water running
Wads, or Okum, that the Piece out of a Hole in a Veſſel may be
may be unloaded.
thus defined.
WAGONER. See Charles's Wain.
Let SAS be an infinite Superfi-
WARNING-WHEEL, in a Clock, cies of Water, CC a Circular Hole
is the third or fourth Wheel, ac made in the Bottom of a Veſtel,
cording to its Diſtance from the firſt AB a ftrait Line drawn perpendi-
Wheel.
cular through the Hole. SGCC
WATER, is a very fluid volatile a Column or Cataract of Water run-
and taſteleſs Subſtance, very proba- ning out through the Hole CC,
bly conſiſting of Hard, Smooth, SGC a Curve, by the Rotation of
Ponderous, Spherical Particles, of e- which, about the Axis AB, the
qual Diameters, and of equal Speci- Solid or Cataract SGCCS is gene-
fical Gravities.
The Poroſity of Water is ſo very S A
S
great, that there is at leaſt forty
Times as much Space as Matter in
it; for Water is nineteen Times
G
fpecifically lighter than Gold, and
ſo rarer in the ſame Proportion ;
E
but Water can be preſſed through
the Pores of Gold, and therefore
CBC E
may be ſuppoſed, at leaſt, to have
more Pores than Solid Matter. raied; for ſince the.Water deſcends
Mr. Boerhaave, in his Chemiſtry, freely, and with an accelerate Mo-
defines Water to be a Liquor very tion, it must of neceſſity be con-
fluid, inodorous, inſipid, pellucid, tracted into a leſs Breadth, accord-
and colourleſs, which in a certain ing as in falling it requires a greater
Degree of Cold freezes into a brittle, Velocity, and will run out through
hard, glaſſy Ice.
the llole C C with thie fame Velo-
Well Water, which is eſteemed city that it would have in falling
the moſt pure of all, is in Weight, the Height AB.
to pure Gold, as I 10 19 $. An Now the Velocity that a heavy
Engliſh Cubick Inch, Mr. Boyle fays, Body acquires by falling is in the
weighed 252, 256, 260 Grains. fubduplicate Ratio of the Heigho
Heat eaſily makes Waters lighter, from whence it falls : Wherefore if
and the heavier they are, the more any Ordinate D E be drawn to the
they are to be ſuſpected of having Curve SGC, and D E be called
Jy
hetereogenous Matter within them. and AD, x; then the Velocity of
Berhaave ſays, Clavius che Ma- the Water, in the Section ED, will
thematician, poured ſome Water into be expreíted by X, and the Pro
a Bolt-head," and then ſealed the duct of that Velocity drawn into the
Mouths of its long Neck herme- ſaid Section by ✓ * *y2.
Bilally, and marked with a Diamond
Whic!
i
W A T
WE DO
Which Product is as the Quan- Quantity of Water which runs out
tity of Water paſſing through that in a given Time, with ſuch a Velo-
Section in a given Space i of Time, city as is acquired in that ſame gi-
and -becauſe the fame Quantity of ven Time to move through a Space
Water paſſes through each Se&tion equal to the Height AB; and if
of the Cataract in a given Time, the Veſſel and the Hole be of any
has Product will be always equal other Figure, the Motion of the
to it felf; andfo . x X:32 1, Cataract of Water will be the famė,
and xx 24 = I which is an Equa- uſing a Proportion of Water' of the
tion of the Curve SG C, being an Height A B for a Cylinder.
Hyperbola of the 5th Order, one of .: War of the Rounds, in Fortifica-
the Affymptotes being the Right tion is a Space left for the Paffage
Line AS parallel to the Horizon, of the Rounds between the Ram-
and the other the Line AB.perpen. part and the Wall' of a fortified
dicular to it;
Town: But it is not fo much in
The Power of it is the Quadrata- uſe, becauſe not having a Parapet
Cube of the, Ordinate ED drawn above a Foot thick, it may be ſoon
torthe Point G, where the Right overthrown by the Enemies Cannon.
Line AG biſfecting the Angle, form WEDGB, is a Priſm of a ſmall
ed by the Affymptotes, meets the Height, whoſe Baſes are Æquicrural
Curve,
Triangles, as A.
If Water runs out through a Cir The Height of the Triangle is
cular Hole in the Bottom of a Veſ- the Height of the Wedge, asdb.
fel of an Infinite Breadth, the Mo The Baſe of the Triangle is called
tion of the whole Cataract of Water the baſe of the Wedge, as cei
towards the Horizon, is equal to the
Motion of a Cylinder of Water under
the Hole itſelf, and Height of the
Water; the Velocity of which will
el
be equal to the Water running out
through the Hole, or equal to the
Motion of a Quantity of Water,
which runs out in a given time, the
Velocity of which will be equal to
that which is acquired in that ſame
given Time by the Motion, through
f
a Space, equal to the Height of the
Water.
16
IF BA:BD::D G4:D G4..
B C4. and Water runs out through The Edge of the Wedge is a Riga
CC, a Circular Hole made in the Line, which joins the Vertices of
Middle of the Bottom of a Cylin- the Triangles, as bf.
drical Veſſel G GEE conſtantly full The Edge of the Wedge is appli-
of Water, 'the Motion of a Cataract ed for cleaving of Wood, and the
of Water towards the Horizon, fhall Power is the Blow of a Hammer,
be equal the Motion of a Cylinder or Mallet, which drives the Wedge
of Water under the Hole, and the into the Wood.
Height AB, whoſe Velocity ſhall The Power is to the Reſiſtance of
be equal to the Velocity of the the Wood, when its Action is equal
Water running out through the to it, as the Half-Baſe of the Wedge
Hole; or it thall be equal to the is to its Height
A
..:
1
}
Ll 3
WIX : 2
Y E A
C
ZOD
1
>
ing for the feveral phenomena ZENITH. If we conceive a
I
WIND, is any ſenſible Agitation and contains 365 Days, 5 Hours,
of. Dhe Air, andris caufed by the cand 12 Minutės...
Days,
Action of the Sun's Beams upon the The Sydereal Year is that Time
Air and Water, as he paffes every in which the Sun, departing from
day-over the Ocean, conſidered to any fixed Star, comes to it again;
gether with the Nature of the Soil and this is in. 365 Days; 6 Hours,
and Situation of the adjoining Con- and almoſt 10 Minutes ! But accord-
tinents.
ing to Sir Iſaac Newton's Theory of
Itis found by Experience that the the Moon, the Sydereal Year is 365
Velocity of the Wind in a great Days, 6 Hours, 9 Minutes, 14
Storm is not more thango or 60 Miles Seconds; and the Tropical, 365
id' án Haur, and that a common Days, 5 Hours, 48 Minutes, 57
brik Wind moves about. 15 Miles Seconds.
an Hour :. And the Courſe of
many
Winds is fo Now, as to be leſs than
one Mile in an Hour.
Z
Concerning the Cauſe of the
Winds, and the manner, of account-
Line drawn through the Ob-
thereof, in the different Parts of the ſerver and the Center of the Earth,
World, fee Dr. Halley's Diſcourſe which muſt neceſſarily be perpen-
upon this Subjeet, Philof. Tranſ. dicular to the Horizon, it will reach
N° 183. as alfo Varenius's General to a Point among the fixed Stars,
Geography (Part abfol.) Sect. 6. which is called the Zenith.
Cap. 20. The Lord Bacon too has ZENITH DISTANCE, is the
a little Treatiſe upon the Complement of the Sun, or Star's
Winds.
Meridian Altitude, or what the
WINGS, in Fortification, are the Meridian Altitude wants of go! De-
large Sides of Horn-Works, Crown-
grees.
Works, Tenails, and the like Out Zetetick METHOD, in Ma-
Works ; that is to ſay, the Ram- thematicks, is the Analytick or
parts and Parapets, with which Algebraick Way, whereby the Na-
they are bounded on the Right and ture and Reaſon of the Thing is
Left, from their Gorge to their primarily inveſtigated and diſco-
Front. Theſe Wings or Sides are vered.
capable of being flanked either with
Zocco.' See Plinthus.
the Body of the Place, if they ftand ZOCLB, in Architecture, is a
not too far diſtant, or with certain ſquare Body, leſs in Height than
Redoubts, or with a Traverſe, made Breadth, and placed under the Baſes
in their Ditch.
of the Pedeſtals of Statues, Vafes,
WINTER SOLSTICE,
See Sola GC.
fice.
ZODIACK, is a Zone or Belt
which is imagined in the Heavens,
which the Ecliptick Line divides
y
into two equal Parts ; and which,
on either șide, is terminated by a
TEAR, is the Time the Sun Circle parallel to the Ecliptick
takes to go through the twelve Line, and eight Degrees diftant
Signs of the Zodiack. This is pro- from it, on account of the ſmall
perly the Natural or Tropical Year, Inclinations of the Orbits of the
Planets,
wrote
1
Y
mo
Z O D
ZOD
Planets, to the Plane of the Eclip- Day the Sun riſes and ſets ; for the
tick: No Bodies of the Planetary Dittance of the Sun from the Pole
Syſtem appear without the Zodiack. always exceeds the Height of the
ZODIACK of the Comets, is a cer Pole ; yet every where, but under
tain Tract in the Heavens, within the Equator, the Artificial Days are
whoſe Boundś moft Comets are ob- unequal to one another, which in-
ſerved to keep their Courfe. equality is ſo much the greater, the
ZONE, in Geography, is a Space leſs the place is diftant from a Fria
contained between two Parallels.
gid Zone.''
The Whole Surface of the Earth But in the Polar Circles, juſt
is divided into five Zones. The where the Temperate Zones are ſe-
Firſt is contain'd between the two parated from the Frigid ones, the
Tropicks, and is called the Torrid Height of the Pole is equal to the
Zane. There are two Temperate Diſtance of the Sun from the Pole,
Zones, and two Frigid Zones. The when it is in the Neighbouring
Northern Temperate Zone is ter- Tropick ; and therefore, in that
minated by the Tropick of Cancer; Caſe, once a year, the Sun in its
and the Arctick Põlar Circle, the Diurnal Motion performs one entire
Southern Temperate Zone is con- Revolution, without going down
tained between the Tropick of Ca- under the Horizon.
pricorn, and the Polar Circle: The But every where in a Frozen
Frigid Zones are circumſcribed by Zone, the Height of the Pole is
the Polar Circles; and the Poles are greater than the leaſt Diſtance of
in the Centres of them.
the Sun from the Pole; therefore,
In the Torrid Zone, twice a Year during ſome Revolutions of the
the Sun goes through the Zenith at Earth, the Sun is at a Diſtance from
Noon; for the Elevation of the Pole the Pole, which is leſs than the
is leſs than 23 Degrees, 30 Minutes; Pole's Height; and, during all that
and the Diſtance of the Sun from the Time, it does not fet, nor ſo much
Equator towards the Pole, which is as touch the Horizon ; but where
above the Horizon, is twice in a the Diſtance from the Pole, as the
-Year equal to the Height of the Sun recedes from it, does exceed the
Pole ;' for which Reaſon alſo, in the Height of the Pole or Latitude of
Limits of that Zone, namely under the Place, the Sun riſes or ſets every
the Tropicks, the Sun comes to the Natural Day. Then in its Motion
Zenith only once in a whole Year. towards the oppoſite Pole, it stays
In the Temperate and Frigid in the Same Manner below the Ho-
Zones, the leaſt Height of the Pole rizon, as was ſaid of the Motion
exceeds the greateſt Diſtance of the above the Horizon.
Sun from the Equator, and there Theſe Times in which the Sun
fore, to their Inhabitants, the Sun makes entire Revolutions above th
never goes through the Zenith; yet Horizon, and below it in its Diurnal
if on the ſame Day the Sun riſes at Motion, are ſo much the greater,
the fame Time to a greater Height, that is, the longeſt Day and Night
the leſs the Height of the Pole is, laſt the longeſt, the leſs the Place in
becauſe thereby the Inclination of the Frigid Zone is diftant from the
the Circles of the Diurnal Revolu- Pole, till, at laſt, at the Pole itíelf,
tion with the Horizon is leſs. they take up the Time of the whole
In the Torrid Zone, and in the Year.
Temperate Zones, every Natural
FINI S.
E QU
E QU
To be added to the Head of RootS OF EQUATIONS.
T.
HERE have been many En- reaſon of their unelegance and
deavours to find the Roots length; for over encreaſing with
of Equations in finite Terms by Au. the Number of Dimenſions of the
thors. 'Sir Iſaac Newton himſelf has Equation, whoſe Roots are to be
given a tentative Method of finding fought. I ſhall only mention a way
whether an Equation of four, fix, of Sir Iſaac Newton's of finding the
or more Dimenſions may be divided Roots of Numerical Equations by
fo into two equal Parts, as that the means of Gunter's Lines ſliding by
Root of each may be extracted. But one another.
his Rule for ſo doing it, is very long Take as many Gunter's Lines, (up-
and troubleſome, ſerving more, as on narrow Rules) all of the ſame
he himſelf owns, to fhew the pos- Length, ſliding in Dove-tail Cavi-
sibility of doing the thing than forties, made in a broad oblong Piece
any real uſe. See his Algebra to. of Wood, or Metal, as the Equa-
wards the End. You have alſo in tion whoſe Roots you want the Di-
the Aita Eruditorum, an. 1683. p. menfions of, having a Slider carrying
204. an univerſal Way of Mr. Tſchirn- a Thread or Hair backward or for-
haufen's, of finding the Roots of E- wards at right Angles over all theſe
quations, by throwing out all the Lines, and let theſe Gunter's con
intermediate Terms. But this is fiſt of two ſingle ones, and a dou-
both tedious and falſe, when the ble, triple, quadruple, &c.
Equation has more than three Di. fitted to them; that is, let there be
menſions. The Ingenious Mr. De a fixed ſingle one a top, and the
Moivre, likewiſe in the Philofophi- firſt ſliding one next that, let be a
cal Tranfa&tions, Nº. 309, has ſingle one, equal to it, each Num-
a given way of reſolving in finite ber from 1 to 10. Let the ſecond
Terms, particular Equations of ſliding one be a double Line of
the odd Dimenſions ad infinitum Numbers, number'd 1, 2, 3, 4, 5,
contained in this general Equá 6, 7, 8, 9, to 10, in the Middle,
tion, when ne is an odd Number, and from í in the Middle, to 1, 2,
3, 8c. to 10, at the End. Let the
third ſliding one be a triple Line
not of
3 x3+
of Numbers, numbered 1, 2, 3, 4,
2 X 3
5, 6, 7, 8, 9, 1, and again 2,
-9
nn -9 3, 4,
3, 4, c. to 10, and again 2, 3,
nast
4, &c. to 100 at the End. The
4Xs
4 * 5 4* 5
Diſtance from i to i, i to jo, and
25
10 to 100, being the ſame ; lct
12 *? &C. a. Being
a. Being the fourth ſliding one, be numbered
1, 2, 3, 4, 5, 6, 7, 8, 9, 1; and
a Series for dividing an Arch of a again 2, 3, 4, &c. to 10; and a-
Circle into any odd Number of equal gain 2, 3, 4, &c. 100; and again
Parts. Buc to ſpeak Truth, what 2, 3, 4, &c. 1000. The Diſtance
ever Arithmetical Rules have hither from i to i, i to 10, 10 to 100,
to, or ever will be given for fir.d. and 100 to 1000, being the ſame,
ing the Roots of Equation of more and ſo on.
than three Dimenſions in finite This being done, take the co-
Terids, muft from the Naiure of the efficient prefixed to the ſingle Value
'Thing be not worth the Pains and of the unknown Quantity upon the
Perplexity in computing them by fixed ſingle Line of Nurnbers; the
CO-
nn I
7 1 -
X
2 X 3
nnl
X
6x7
N
E QU
E QU
Co-efficient of the Square of the un. Dimenſions leſs, and by a Repeti
known Quantity, upon the double tion of the Operation you will get
Line of Numbers ; the Co-efficient a third Root, and ſo a fourth, fifth,
of the Cube of the unknown Quanti. &c. if the given Equation has ſo
ty, upon the tripleLine of Numbers ; many, and if any of the interme-
the Co-efficient of the Biquadrate of diate Terms are wanting, the Gun.
the unknown Quantity, upon the ter's expreſsd by the Dimenſions of
Quadruple Line of Numbers; and thoſe Terms, muſt be omitted.
so on. And the Co-efficient of the But this Method only gives the
firſt or higheſt Term (being always Roots of Equations the signs of all
Unity) take upon that Line of Num- the Terms whereof, except the
bers expreſſed by its Dimenſion, that known one, are Affirmative; that
is, if a Square, upou the double Line; is, of ſuch that have all Negative
a Cube, upon the triple Line, &c. Roots, but one, which laſt, the faid
I ſay, when this is done, Nide all Method finds. Therefore when an
thele Lines of Numbers for that Equation is given, to find its Roots
theſe Co-efficients be all in a right after this manner, whoſe Signs have
Line directly over one another, and other Diſpoſitions, it muſt be firſt
keeping the Rulers in this Situa. changed into another Equation,
tion, nide the Thread or Hair in whoſe Signs are all Affirmative; but
ſuch manner, that the Sum of all the that of the known Term, which may
Numbers upon the fixed fingle Line, be done by putting ſome unknown
the double Line, the triple Line, Quantity y Plus or Minus, fome gi-
&c. which the. Thread or Hair ven Number or Fraction, for the Va-
cuts, be equal to the known Term · lue of the unknown Quantity x in
of the Equation, which may be the propoſed Equation.
readily enough done with a little Note, Inſtead of freight Parallel
practice; and then the Number un. Sliding-Rules, you may have fo
der the Thread upon that Line of many Gunter's Lines graduated upon
Numbers of the ſame name with the Concentrick Circles, each moving
higheſt Power of the unknown Quan-, under one another, by which Con-
tity of the Equation, will be the trivance, you will have as large Di.
pure Power of the unknown Quan- viſions for your Logarithm within the
tity, whoſe Root may be had by Compaſs of one Foot, as you have
bringing Unity on the ſingle Slid. upon a ſtreight Ruler of more than
ing-Line directly over Unity upon three Feet in Length. Although per-
this Line. After this, if you divide haps by thefe Sliding-Rules, you
the Equation by this Root, you will cannot get all the Signs of the Roots
have another, one Dimenſion leſs; exactly, for want of ſufficient Sub-
and thus you may proceed to find diviſions of the Gunter's Lines, yet
a Root of this laſt'Equasion which if we can get two or three of the
done, if it be divided by this last firſt Figures, it will be of good 'ule to
Root, you will get an Equation two find the Roots by Approximation :
1
1
E RR AT A.
A
r. to.
.
I.
F.
Bacus, in the Figure for e, read c. Acceſlible Altitude, in the Fi-
gure for 663, read 66.3. Acceſſible Depth, for 16 Feet, r. 16 Feet
and 1 Inch. Addition of Decimal Fractions, for the Sum of 36.24,
&c. 1. 907.023. Algebra Specious, line 2, for formed r. perform-
ed. Alternation, line 10, for Alteration r. Alternation. Altitude Inaccel-
fible, I. 28. for BAСr. BCA. Line 103, for take it from, r. take
from it. Line 115, for HI, ř. HC. Altitude of a Figure, 1. 3. for of
Altitude Meridian, l. 25, for 20 r. 200. Angles Curved Lin'd,
1. 30, for DCÉ, 8. DCA. Angles Equal, 1. 10, for DF r. DE. An-
gle of Emergence, the Letter E wanted in the Figure where the Line AB
interſects the Parabola. Ibidem, 1. 15, for CD s. CB. Angle Right-
line, r. Angle Right-lin'd. Angle in a Segment, for ACB r. ADC.
ibid. Fig. 2. for Dr. B. Angle Spherical, 1. 5. dele the Angle re-entring,
Line 5. for thoſe r. whole. Anomaly Coequate, or True, 1.50. for Keil,
r. Keil's. Apparent Magnitude, n. 1. 1.9. add afterwards. Ibid. n. 2.
1. 1. for CH, r. CD. And l. 3. r. CAD.
And l. 3. r. CAD. N.9. 1. 8. after AC add
and AB-BC. lbid. n. 13. for MH, r. QH. Antiparallels, 1. 6. add
that cut them the ſame way. Antipodes, 1. 26. 1. St. Auguftine in. Apo-
tome, the Letter G is miſplaced. Aftronomy, 1. 20. for forty-ſeven, r.
four hundred and ſeventy. *Axis Conjugate, for EF, r. FF. Baſe the
leaſt Sort of Ordnance, for 1 , r. iš Binomial Root 1. 21. for next, r.
bbx bxx
teft. Ibidem, 1. 24. 1. Dimenſion. Ibid. 1. 32. for
Ibid.
Axta c4xxs
N
1. 50. for
Ibid. 1. 57. for
-65
a²,
N
and for Nx33 say! 1. Nxy3 mo azyl Ibid.
Vytay
1.6z. for B.C, r. B. lbid. 1.65. for Point, r. Perver. Ibid. 1.71. for
fourth, r. fifth. Ibid. l. 121. for ventured, r. entered. Under this Word
in Sir Iſaac Newton's Example of extracting the Root, for 1-X, s.
Biquadratic Equation, n. 2. 1. 55. for sxx, r. xx.
under the word Conſtruction, for C, 1. 6. Ibid. n. 7. 1. 49. for Af=
-- AG“ ,. AF* -- AG". Biquadratic Parabola, 1. 15. dele or AC.
Ibid. 1. 147. for zihere, r. when. Biffextile, 1. 9. for 24th, r. 28th.
Bombe, Paragr. 9. at the End, add according to the parabolic Hypotheſis.
Ibid. 1. 30. for Impreſjes, r. Impulfes. N. 4. 1. 16. dele plane. Calculus
Differentialis, parag: 5. 1.43, for | ABC, r. axx. 'Centre Common
of Gravity, in the parag. next n. 13. 1. 30. for Ž AB, 1. A B. Ibid.
1. 34. for the Rudius, r. the Radius. lbid. in the Fig. n. 6. inſtead of
the Letter P at the End of the Line EP, ſhould be F. Circle, n. 19. E is
the Centre. Ibid. n. 25. 1. 3. for Theorems, r. Propoſitions. Ibid. n. 30.
1. 30, after 14. put a Commia. Ibid. Prob. s. for D at the bottom of
Fig. 1. read G. The Letter M near the bottom of Fig. 2. is wanted.
And for F near the bottom of Fig. 3. r. P. Alſo l. 21. for ME, r. MP.
Conchoid, the Letter A is wanted in Fig. 4. Cone, n. 4. 1. 16. for
Plane Curve Superficies, r. Plape Superficies terminated by a Curve. Croſs-
Multiplication,
: در
3.
it
XX.
Ibid. n 2.1.7.
2
E RR A T A.
Multiplication, 1. 13. for Inches, r. Feet. Curves in Geometry, parag. 4.
1. 3. for Diameters, r. Dimenſions. Cycloid, parag. 7. 1. 43. for Farnat,
r. Fermat. Decimal Fractions, 1. 10. for r. 146. Fluent, pa
ragr. 4. I. 20. for fzn. r. fzn. Ibid. l. 22. for Terms, f. Forms. Ibid.
parag: 7. 1.7. for axxm. r. a ***. Fluxions ſecond, parag. 2. 1.4, 5
for Quantities *** r. Quantity *** Heptagon, 1. 2. for fe-
veral, r. ſeven. Hyperbola, n. 2. parag. 2. 1.6. for D, r. G. lbid.
1. 12. for Points, r. Pins. Ibid. n. 5. 1. 8. for ROI. AR. Ibid. n. 6.
the Letter T is wanted in the Figure. Hyperbolic Cylindroid, 1. 4. for
there, r. they. Hyperbolic Space, the Letter F miſplaced, it fhould be
where the Line drawn from L cuts the Curve. Ibid. 1.5. for CF, r. LF.
Ibid. 1. 6. for a”, r. ab. Imaginary Root, parag, 5. 1. 13. for Square,
1. Squares. Ibid. parag. 6. 1. 8. dele an Ordinate. Impervious, l. §. for
21 x 17 x
do, r. 10. Indetermined Problem, n. 4. parag. 5. 1.31. for
4
21 x 17 +11
= 92. Ibid. 1. 36. for 94, s. 92. Index, 1.
4
94. r.
1
penult. for
1.
3
Intereſt, l. 2. for Lot, 1. Loan. Ma-
*s ง
♡ X5
thematics, parag: 9. 1. 20. for aſaults, r. afaults with. Mercator's Chart,
n.4. 1. 10. for AB, r. Ab. Oſcillation, n. 4: for three Feet, 3.125. r.
3.125 Feet. Parabola, n. 4. for the Right Line FM, r. the Square of the
Right Line FM. Ibid. 1.4. for the Abfifa, r. the Squares of the Abfcifs.
Parabola Carteſian, l. 8. for DN, r. BN. Perfect Number, 1. 23. for y, x.
r. yx. Quadratic Equation, n. 3. for-Jaamb.r. Jamvaab.
Ibid. n. 11. for vaatb, r. Jaam. Ratio, n. 7. 1: 38. for on, r. to,
.
BOOKS printed for W. Innys and T. LONGMAN.
A
;
N Analytic Treatiſe of Conic Sections, and their Uſe for. re-
ſolving of Equations in determinate and Indeterminate Pro-
blems. Being the Pofthumous Work of the Marquis De
l'Hoſpital, Honorary Fellow of the Academy of Sciences. Made Engliſh
by E. Stone.
2. An Introduction to Natural Philoſophy: or Philoſophical Lectures
read in the Univerſity of Oxford, Anno Dom. 1700. To which are
added, the Demonſtrations of Monſieur Huygens's Theorems, concerning
the Centrifugal Force and Circular Motion. By John Keil, M. D. Sa-
vilian Profeffor of Aſtronomy, F.R.S. Tranſlated from the laſt Edition
of the Latin. The Third Edition.
3. The Philoſophical Works of the Honourable Robert Boyle Efq;
Abridged, Methodized, and Diſpoſed under the general Heads of Phy-
fics, Statics, Pneumatics, Natural Hiftory, Chymiſtry, and Medicine :
The whole illuſtrated with Notes, containing the Improvements made
in the ſeveral Parts of Natural and Experimental Knowledge ſince his
Time. In Three Volumes. By Peter Shaw, M. D. The Second Edi.
rion, corrected.
4. A Treatiſe of the Five Orders in Architecture. To which is an-
nexed, a Diſcourſe concerning Pilálters, and of ſeveral Abuſes introduced
into Architecture. Written in French by Claude Perrault, of the Royal
Academy of Paris, and made Engliſh by Mr. John James of Green-
wich, The Second Edition.
5. A Courſe of Experimental Philoſophy, by 7. T. Defaguliers, LL.D.
F. R.S. Vol. I. with 32 Copper Plates in Quarto.
N. B. The Second Volume is preparing for the Preſs; and will ſpeedi-
ly be publiſhed.
6. Mathematical Elements of Natural Philoſophy, confirmed by Ex-
periments : or an Introduction to Sir Iſaac Newton's Philoſophy. In two
Volumes. Written in Latin by William James 'sGraveſände, LL. D.
Tranſlated into Engliſh by Dr. Defaguliers. The fifth Edition.
7. An Eſſay on Perfection. Written in French by William James
Graveſande, Doctor of Laws and Philoſophy, Profeſſor of Mathema-
tics and Altronomy at Leyden, and Fellow of the Royal Society at Lon-
don ; and now tranſlated into Engliſh.
8. Rules and Examples of Perſpective, proper for Painters and Archi-
tects, C. In Engliſ and Latin : Containing a moſt eaſy and expedi-
tious Method to delineate in Perſpective all Deſigns relating to Archi-
tecture, after a new manner, wholly free from the Confuſion of Occult
Lines : By that great Mafter thercof, Andrea Pozzo, Soc. Jes. Engraven
in 105 ample Folio Plates, and adorned with 200 Initial Letters to the
Explanatory Diſcourſes. Printed from Copper-Plates on the beſt Paper;
by John Sturl. Done into Engliſh from the Original printed at Rome
in 1093, in Latin and Italian. By Mr. John James of Greenruich.
2/4 جدول
- الم
UNIVERSITY OF MICHIGAN
3 9015 05979 2195
NON
CIRCULATING
2
A 3 9015 00387 146 7
University of Michigan
BUHR
9
܀