A 547273
DUPL
T
1
ARTES
LIBRARY
1837
SCIENTIA
VERITAS
OF THE
UNIVERSITY OF MICHIGAN
ZIMURIOUS UMUM
TUEBOR
SI-QUÆRIS-PENINSULAM-AMŒNAME
CIRCUMSPICE
:
THE
Young Gentleman's
TRIGONOMETRY,
MECHANICKS,
AND
Un
OPTICK S.
Containing fuch
ELEMENTS of the faid Arts
or Sciences, as are moſt Uſeful
and Eaſy to be known.
By EDWARD WELLS, D.D.
Rector of Cotesbach in Leicester-
ſhire.
LONDON,
Printed for James Knapton, at the Crown
in St. Paul's Church-Yard. 1714.
THE
Young Gentleman's
TRIGONOMETRY
CONTAINING
Such ELEMENTS of
Trigonometry, as are moft feful
and Eaſy to be known,
By EDWARD WELLS, D. D.
Rector of Cotesbach in Leicester-
Shire.
LONDO N
Printed for James Knapton, at the Crown
in St. Paul's Church-Yard. 1714.
THE
PREFACE
B
EING once prevail'd upon to
draw up the Young Gentle-
man's Geometry, I judg'd it
requiſite alſo to draw up this Trea-
tife of Trigonometry; forafmuch as
Trigonometry is really no other than
one Branch of Geometry, and that too
Branch, which is of principal Vſe
in the common Concerns of Life; at
leaſt that Part of it, which is call'd
Plain Trigonometry. As for Spheri-
cal Trigonometry, it carrying in it
much more of Difficulty, and at the
fame time being of much less Ufe in
the common Affairs of Life, (as re-
lating chiefly to Aftronomical Calcula-
tions or the like,) I have omitted it,
A 2
as
The Preface.
as not agreeabls to the Defign of this
Treatife.
On the like Account, tho' I judg'd
it not proper wholly to omit the Fun-
damental Theorems of plain Trigo-
nometry; yet I have plac'd them ſo in
Chap. II. that the young Student
may pass them over, without
any A-
bruptness, if the Demonftration of the
two latter prove too difficult, or at
leaft if he cares not to take the Pains
to make himſelf Mafter of the faid
Demonftrations.
?
THE
THE
CONTENTS.
CH
HAP. I. Containing the Explicati-
on of Trigonometrical Terms Pag. I
Chap. II. Of the Solution of plain Triangles
by Calculation
II
29
Chap. III. Of the Solution of plain Trian-
gles by Protraction
Chap. IV. Of Taking the Altitude or
Height of an Object, viz. Tower, Tree,
&c.
32
Chap. V. Of Taking the Distance between
two Object's
47
Chap. VI. Of Surveying or Meaſuring of
Land
Chap. VII. Of Fortification
58
7.2
Chap. VIII. Of the Tables of Logarithms
and Artificial Sines and Tangents, ad-
join'd to the End of this Treatife
82
THE
"
{
The CONTENTS
Of the Young Gentleman's Mechanicks.
CH
HA P. I. Wherein are explain'd fuch
Terms as relate to Mechanicks in ge-
neral
Chap. II. Of the Leaver
Chap. III. Of the Balance
Pag. 1
12
22
Chap. IV. Of the Axle in the Wheel 27
Chap. VI. Of the Screw
Chap. V. Of the Pulley
Chap. VII. Of the Wedge
Chap. VIII. Of Staticks
Chap. IX. Of Hydroſtaticks
31
35
38
41
54
The CONTENTS
Of the Young Gentleman's Opticks.
CHAP.
A P. I. Of Opticks in general
Pag. 83
Chap. II. Of Opticks more properly fo
call'd, or Direct Vifion
90
Chap. III. Of Captoptricks, or Reflex Vi-
fion
99
Chap. IV. Of Dioptricks, or Refracted
Vision
Chap. V. Of Perſpective
123
157
THE
(1)
!
THE
Young Gentleman's
TRIGONOMETRY.
+
CHAP. I.
Containing the Explication of Trigo-
nometrical Terms.
F
I.
Trigono
metry,
VERY Triangle confifts in all
of ſeven Parts, viz. three Sides,
three Angles, and the Area or what.
Space comprehended by the Sides.
Now, although the Word Trigonometry
do's in the Greek Language literally fig-
nity the Meaſuring of a Triangle; yet, in
its proper Senfe, or as it is diftinguiſh'd
from Geometry, it (do's not denote the
Meaſuring of the Area of a Triangle, this
being deduced from (*) other than what
are
(*) The Meaſuring of the Area of a plain right-lin'd
Triangle, is deduced from the Principles of Geometry,
(as
B
2
The Young Gentleman's
:
2.
The Solu-
tion of a
are properly call'd Trigonometrical Prin-
ciples, but it) denotes only the Meafu-
ring, or rather the Art of finding the
Meaſure of, the unknown Sides and An-
gles of a Triangle, by the Help of thoſe
that are given or known.
And the actual finding of the un-
known Sides and Angles of a Triangle,
Triangle, is call'd, in one Word, the Solution or Re-
folving of a Triangle.
what.
3.
The Solu-
tion of
on what
founded.
The Solution of Triangles is founded
on that mutual Proportion, which is be-
Triangles tween the Sides and Angles of any Tri-
angle. And this Proportion is known
by finding the Proportion, which the
Ray of a Circle has to certain other
right Lines apply'd to the fame Circle.
And theſe right Lines are diftinguifh'd in
general by the Names of Chords, Sines,
Tangents, and Secants.
4.
A Chord,
what.
Fig. 1.
A (†) Chord is a right Line drawn
from one Extremity of any Arch to the
other. Thus, Fig. 1, CH is the Chord,
(as diftinguifh'd from Trigonometry,) viz. from Theorem 7,
Chap. 1, of the Young Gentleman's Geometry. And the
Meaſuring of the Area of a fpherical Triangle, is dedu-
ced from the Meafuring of a fpherical Surface.
(†) As an Arch is fo call'd, becauſe it reſembles a bent
Bow, fo a Chord is fo call'd, becauſe it reſembles the
String of a bent Bow. It is otherwiſe call'd a Subtenje,
becauſe it fubtends, (i. e. is extended under) the Arch
from one End of it to the other.
both
A
3
Trigonometry.
both of the leffer Arch CTH, and alſo of
the greater Arch CDH. And it is to be
noted, that a Chord of 60 Degrees, is
always equal to the Ray of the fame
Circle.
5.
A Sine, or
right Sine,
A (1) Sine is either a Right or Verſed
Sine. A right Sine is one Half CA of a
Chord CH, bifected by the Diameter what.
DT; or, it is a Perpendicular CA, let
fall from Cone End of an Arch CT, upon
a Diameter DT, drawn from T the other
End of the Arch. A right Sine is fre-
quently ftil'd fimply a Sine; and it is to
be obferv'd, that by the Word Sine, put
fimply or by it. felf, is always to be un-
derſtood a right Sine.
6.
whole
Sine,what.
It is alſo to be obferv'd, that, accor-
ding to the fore-going Definition of a 4 total or
Sine, the Sine BK of an Arch KT of 90
Degrees, is always equal to the Ray of
the fame Circle; and fo the greateſt
Sine that can be. Whence it is call'd,
the total or whole Sine.
7.
A (*) vers'd Sine is the Segment of
the Ray intercepted between the right 4 vers'd
Sine,what.
(1) The Ufe of Sines is faid to be introduced by the
Arabs; and for what Reafon a Sine is fo call'd, I know no
good Account.
(*) A vers❜d Sine is otherwife call'd, at least by fome
ancient Writers, the Sagitta or Arrow, becauſe it ſtands
to its Arch and Chord, as an Arrow do's to its Bow
(when bent) and String.
B 2
Sine
།
The Young Gentleman's
4
1
8.
gent, and
Secant,
what.
Sine and the Arch. Thus, AT is the
vers'd Sine of the Arch CT, and AD the
vers'd Sine of the Arch CD.
A (†) Tangent and a Secant mutually
A Tan- explain one the other. For, if a right
Line BN of an indeterminate Length be
drawn from the Center B, through C
one End of an Arch CT, and from T
the End of the Diameter DT, there be
erected a Perpendicular TN, extended
till it meets with BN; the faid Perpen-
dicular TN will be the Tangent of the
Arch CT, and the Line BN will be the
Secant of the fame Arch. And it is to
be noted, that the Tangent of 45 De-
grees, is always equal to the Ray of the
'fame Circle; as may be ſeen, Fig. 2.
9.
4 Cofine,
Coran-
gent, Cofe-
As when the Arch CT is less than 90
Degrees, the Arch CK (which, being ad-
ded to the Arch CT, do's compleat or
Cant, and make it up an Arch of 90 Degrees,
Covers'd, thence) is call'd the (1) Complement of the
Arch CT; fo (Fig. 1,) the Sine CQ, the
Tangent KO, the Secant BO, and the
vers'd Sine KQ of the Arch CK, are re-
what.
{
(†) A Tangent is fo nam d, becauſe it touches (not cuts)
one End T of its Arch CT; and a Secant is fo nam'd,
becauſe it cuts or croffes the other End C of its Arch
CT.
(1) It is not to be omitted, that the Excefs CK of the
Arch CD above 90 Degrees, is by fome Writers call'd,
the Complement of the faid Arch CD.
fpectively
Trigonometry.
in
ſpectively call'd the Sine, the Tangent,
the Secant, and the vers'd Sine of the Com-
plement of the Arch CT; or in fhort, the
(*) Cofine, the Cotangent, the Cofecant,
and the Covers'd of the faid Arch CT.
10.
The Sup-
plement or
what.
The Arch CD, (which, being added
to the Arch CT, makes it up a Semicir-
cle,) is peculiarly (†) ftil'd the Supple- Conjunct
ment or Conjunct of the Arch Cr. And of an Arcb,
it is to be noted, that both the Arch CT,
and its Conjun& CD, have the "fame
Sine CA, the fame Tangent TN, and the
fame Secant BN.
11,
Tables of
Sines and
Tables of (||) Sines and Tangents are fo
call'd, becauſe in them is fet down the
Ratio or Proportion, which Sines and Tangents,
Tangents have to their Ray.
B 3
what.
And
(*) It is of Uſe to obferve, that the Cofine CQ is al-
ways equal to BA, the Segment of the Ray between the
Center B, and the Sine CA. Wherefore, if the Arch CD
be greater than 90 Degrees, then the Sum of the Ray and
Cofine is equal to the vers'd Sine; viz. DB+BA DA.
But, if the Arch CT be less than 90 Degrees, then the
Difference of the Ray and Cofine is equal to the vers’d
Sine, viz. BT-BAZAT,
(†) Namely, to avoid Confufion by giving the fame
Name of Complement, both to CK and to CD, as is done by
fome.
(It appears from §. 5, of this Chapter, that every
Sine, (i. e. right Sine) is the Half of a Chord, and that
the Arch of the Sine, is the Half of the Arch of the ſaid
Chord.. Wherefore, becauſe there is the fame Propor-
tion between two Halves, as there is between their
Wholes, and the faid Proportion between the Halves
(as
The Young Gentleman's
Y
12.
Natural
and artifi-
cial Sines
and Tan
gents,
what.
13.
Chords,
And it is to be noted, that, if the faid
Proportion be exprefs'd in the Tables by
natural or common Numbers, then the
Table is call'd, a Table of natural Sines
and Tangents. But, if the faid Propor-
tion be express'd by artificial Numbers,
i. e. Logarithms, then the Table is call'd,
a Table of (*) artificial Sines and Tan-
gents.
If the faid Proportion be not exprefs'd
A Scale or by Numbers, but the feveral Sines and
Line of Tangents, as alfo (†) Chords and Se-
Sines, Tan- cants, be themfelves actually transferr'd
gents, and into, (i. e. their feveral Lengths fet off
&c. what. upon) feveral diftinct Scales or Lines, (as
Fig. 2. Fig. 2); then the faid Scales or Lines are
reſpectively
Secants,
t
(as being expreffible in lefs Numbers) is much easier to
be calculated, than the Proportion between their Wholes;
hence, Sines are used rather than Chords in Trigonome
trical Calculations, and the Proportion of Sines (not of
Chords) to the Ray is fet forth in Trigonometrical Ta-
bles.
(*) It has been afore obferv'd in my Arithmetick,
that the Multiplication and Divifion of great Numbers
is much easier to be perform'd by Logarithms than na-
tural Numbers. Hence it is that Tables of artificial
Sines and Tangents, are more uſed than thofe of natural
Sines and Tangents, and confequently adjoin'd to moſt
Trigonometrical Treatifes.
(†) The Reafon, why there are no Tables of Chords,
has been obferv'd already in the Note on §. II. There
are however Scales of Chords, thefe being as easily made
as Scales of Sines, and as eafy or eafter for Uſe in fome
Refpects. And the like is to be understood of Secants,
!
viz:
Trigonometry.
7
reſpectively ftil'd, one a Scale or Line of
Chords, the other of Sines, the other of
Tangents, and the other of Secants. And
the like is to be underſtood of the Scales
of any other Sort of Trigonometrical
Lines, as of vers'd Sines, Cofines, &c.
14.
A Scale of
Beſides the fore-mention'd Scales, there
is also another Sort of Scale, call'd a
Scale of equal Parts ; which, although it Parts,
equal
what.
Fig. 3, &
4.
viz. as to the Reafon, why there are few or no Tables of
them, but there are ufually, if not always, Scales of
them adjoin'd to the other Scales. If it be ask'd, why
the Scale of Chords is never continued beyond 90 De-
grees, when the greatest Chord is the Diameter or
Chord of 180 Degrees? The Reaſon is, becauſe the
Chords above 90 Degrees, may be very eafily fupply'd by
a Scale of Chords to 90 Degrees, viz. Chord of 105 De-
grees Chord 90 Degrees + Chord 15 Degrees. And fo
of any other to 180 Degrees. The Reafon, why Tables
and Scales of Sines go no further than 90 Degrees, is
obvious, viz. becauſe a Sine of 90 Degrees, is the grea-
teſt Sine that can be, as appears from §. 5, and 6, of
this Chapter. In like manner, the greateſt Tangent and
Secant is of 89 Degrees, 59 Minutes. For, at 90 De-
grees, Lines drawn in like manner as a Tangent and
Secant, become parallel one to the other, and fo never
meet to conſtitute a Tangent and Secant, at leaſt pro-
perly fo call'd. Hence Tables of Tangents proceed on-
ly to 89 Degrees, 59 Minutes. And as for Scales of
Tangents, and alfo Secants, they are uſually continued
no farther than to about 70 or 75 Degrees; becauſe,
after that they extend to a vaft Length, if the Scales
be of any confiderable Size, fo as to be fit for Uſe.
Which may be perceiv'd by Fig. 2, where the Scales are
of a very ſmall Size, as being defign'd not fo much for
practical. Ufe, as to fhew only the Manner of Making fuch
Scales. Such Scales, as are fit for Practice, are to be bought
ready made of fuch as fell Mathematical Inftruments.
B 4
be
8
The Young Gentleman's
Scale of
equal
Parts,
what.
be not deduced from Principles of Tri-
gonometry, yet is of fo great Ufe there-
in, (as well as in other Parts of the
Mathematicks,) that it will be requifite
to deſcribe it here, eſpecially, fince it
has not been afore deſcrib'd. It is then
call'd, a Scale of equal Parts, becauſe it
fhews the Meaſure of any Line or Ex-
tent, apply'd thereto, or taken there-
from, in fome Number of equal Parts,
denoting either Inches, or Feet, or Yards,
&c.
IS.
It is ufually diſtinguiſh'd into the fim-
Afimple ple or plain Scale, and the diagonal Scale.
or plain
The fimple or plain Scale, is no other
than a right Line divided into any Num-
ber of equal Segments or greater Parts:
One of which greater Parts, (viz. at ei-
ther End of the faid Line,) is fub-divi-
ded into Ten other leffer and equal Parts,
as Fig. 3; So, that this Scale repreſents
any two Degrees of decuple Proportion,
viz. if the Sub-divifions of the Segment
BA be taken to denote Units, then the
other equal Segments CB, DC, &c. will
denote Tens. If the aforefaid Sub-divi-
Gions denote Tens, then the Segments
will denote Hundreds, &c. Hence, the
Extent from C to 4 will denote either 14
or 140 (c.) Inches, or Feet, or Yards,
&c.
The
1
Trigonometry.
9
16.
Scale
what.
The diagonal Scale will be better ap-
prehended by 'Fig. 4, than by Words. A diagonal
It thence appears to differ from the fim-
ple Scale principally in this, viz. that
one of its extreme Segments (being like
the fimple Scale) fubdivided into ten e-
qual Parts, each of the faid ten equal
Parts, are again ſubdivided into ten o-
ther ſtill lefs and proportional Particles,
by ten right Lines obliquely croffing the
Scale, which are the Diagonals of fo ma-
ny (1) Parallelograms; whence this Scale
takes its Name. Hence, by the Divifions
and Sub-divifions of this Scale, are re-
preſented three Degrees of decuple Pro-
(1) This is illuftrated, Fig. 5, which is divided after
the fame manner, as is the Segment KL of the diagonal
Scale, Fig. 4; and has withal the Sides of each Paralle-
logram defcrib'd by prickt Lines, fo that the faid Figure
contains five Parallelograms, viz. ABCD, and CDEF, and
EFGH, and GHIK, and IKLM; which are each croſs'd
by Diagonals, viz. the firft by the Diagonal CB, the ſe-
cond by the Diagonal ED, the third by the Diagonal
GF, the fourth by the Diagonal IH, and the fifth by the
Diagonal LK. And thefe Diagonals anfwer the right
Lines obliquely croffing the Segment KL of the diagonal
Scale, Fig. 4, and which divide each Part of the faid
Segment into lefs and proportional Particles, as is illu
ftrated by Fig. 6. For the Lines bk, and cl, and dm,
&c. being all parallel one to the other, hence (by Theo-
rem 9, Chap. 1, of the Young Gentleman's Geometry) ab:
bk : : ac : el :: ad : dm :: ae : en: : af: fo, &c. But ab—
to ax; ac ax, adax, &c. therefore bk === ux,
cl_² ux, dm2ux, &c. And confequently, the Line
ax is divided by the Diagonal au, into ten leffer and
proportional Parts, viz. bk, cl, Sic.
HX
portion
10
The Young Gentleman's
17.
Trigonome-
portion. For Inftance: If you would
denote the Extent of 245 equal Parts,
(viz. Inches or Feet, &c.) one Foot of
your Compaffes being placed in the mid-
dle or prickt Line over-againft 200, open
the other Foot to that Point of the Sub-
divided Segment KL, wherein meet the
two Lines mark'd, one with 40 on the
Side of the Scale, the other with 5 on
the Top of the Scale. That Extent of
your Compaffes will denote 245 equal
Parts, (viz. Inches or Feet, &c.) accor-
ding to the faid Scale.
Further, It is to be here obferv'd, that
Plain and Trigonometry is two-fold, Plain and Sphe-
ſpherical rical. Plain Trigonometry is fo call'd, be-
try, what. caufe it teaches how to refolve any (*)
plain (i. e. right-lin'd) Triangle: Sphe-
rical Trigonometry is fo call'd, becauſe it
teaches how to refolve any Spherical Tri-
angle, i. e. any Triangle whoſe Sides
are made by the Concurrence of Lines on
a fpherical Surface, and are no other
than the Arches of Circles. It will be
fufficient to the (†) Deſign of this Trea-
tife, to ſpeak only of plain Trigono-
metry.
(*) Hence, by a plain Triangle is to be understood
throughout this Treatife a right-lin'd Triangle.
(†) See the Preface.
Lastly,
1
Trigonometry.
II
18.
on and
Lastly, The Solution of plain (as alſo
of fpherical) Triangles is two-fold, ei- Calculati,
ther by Calculation, i. e. Working by Protracti
Numbers, to which are fubfervient the on, what.
Tables of Sines and Tangents; or by
Protraction, i. e. drawing Lines and An-
gles, to which are fubfervient the Scales
of Chords, Sines, &c. of each in its
Order; and firſt of the Solution of plain
Triangles by Calculation.
CHA P. II.
Of the Solution of plain Triangles by
Calculation,
I.
reſolve a
always gi-
F the fix Trigonometrical Parts of
a plain Triangle, viz. its three In order to
Sides, and three Angles, any three being plain Tri-
given, excepting the three Angles alone, angle,there
the reſt may be found either by Calcula- must be
tion or Protraction. From the three An- ven one
gles given by themſelves, can be inferr'd Side, and
or found out, only the Proportion of the Fig. 7, &
Sides, not the Sides themselves, or their 8.
respective determinate Lengths. For In-
ſtance: Suppofing the three Angles of
the Triangle BCD, Fig. 7, to be given,
viz. B 27 Degrees, D 38
Degrees, and
confe-
why.
1
.
་
1
1
12
The Young Gentleman's
confequently, C 114 Degrees; from
hence it may be known, that the Side
CD=1BD, and the Side CB BD. But
from hence can never be diſcover'd the
determinate Length of BD, (or any other
Side,) and confequently not of the Reft
CB and CD. Namely, becauſe ÷=÷
흥
​÷÷÷÷÷÷÷, &c. hence the Proportion of
CD to BD will be the fame, whether BD
be 2 Inches (or Feet, &c.) and CD one;
or BD 4 Inches, and CD 2; or BD 6
Inches, and CD 3; or BD 20 or 200 In-
ches, and CD 10 or 100, &c. That is
in ſhort, it can never be difcover'd by
the three Angles alone, whether the Tri-
angle fought be BCD, Fig. 7, or BCD,
Fig. 8, (or any other equiangular Trian-
gle,) forafmuch as both theſe Triangles
have the fame Angles, and confequently
their Sides alike proportional one to the
other; though the Sides of the one are
much longer than the Sides of the other.
Wherefore, to determine the Length of
the Sides fought, there must always be gi-
ven one Side, or (which is the fame) its
determinate Length; and hereby will
be determin'd the Length of each Side
fought. Thus, the Side BD, (Fig. 7,)
'being given, whofe Length according to
the diagonal Scale, (Fig. 4.) is 324 (e-
qual Parts, fuppofe) Inches; hence, it
being found, that CD=BD, and CB=
'BD,
Trigonometry.
13
E
BD, thereby it is known, that CD-
162 Inches, and CB=216 Inches. But,
if the Side BD (Fig. 8,) be given,
whoſe Length according to the fame
Scale is 135 (equal Parts, fuppoſe) In-
ches; then in the fame, Fig. 8, CD=
67 Inches, and CB=45 Inches.
2
2.
The Solu
tion of
plain Tri-
angles re-
It evidently appearing from what has
been faid, that, in order to refolve a
plain Triangle, of the three Parts given,
one muſt be always a Side; it hence fol-
lows, that all the Cafes, that can be pro- three gene
pos'd, are reducible in general to theſe ral Cafes.
Three, viz.
1. One Side and two Angles being gi-
ven, to find the other two Sides, and the
(*) third Angle : Or,
2. Two Sides and one Angle being gi-
ven, to find the third Side, and the two
other Angles: Or,
3. and lastly, All the three Sides be-
ing given, to find the three Angles.
ducible to
Theſe three general Cafes are diſtin- 3.
guifh'd into a great Variety of particular All the
Cafes, (according to the particular Sides Cafes are
particular
(*) Two Angles being given, the third may be known
(without Trigonometry) by Corol: 1, Theorem 4, of the Young
Gentleman's Geometry. And by the fame Corollary, one
Angle being given in Caſe 2, and one of the fought be-
ing found, the other fought is known. And fo in Cafe
3, two of the Angles ſought being found, the third foughc
is known.
and
contain'd
in the twe
following
Tables.
14
•The Young Gentleman's
angle
10.
4.
and Angles given or fought,) which are
all contain'd in the two following Ta-
bles. Whereof, the former contains all
the particular Cafes, that relate to the
Solution of plain rectangular Triangles;
and the latter contains all the particular
Cafes, that belong to the Solution of
plain obliquangular Triangles.
In Reference to which Tables it is to
The literal be obferv'd, that for Brevity's fake it is
Symbols ufual to denote the feveral Parts of a
of the Se-
veral plain Triangle by Letters, instead of
Parts of a Words. In order whereto a rectangular
plain Tri-
Triangle is wont to be denoted by A, B,
Fig. 9, & and C, fo placed, as (Fig. 9, viz.) that
A always denotes the right Angle, BA
the Bafis, CA the Cathetus or Perpendi-
cular, which falling upon the Bafis makes
the right Angle; BC the Hypotenufe; B
the Angle between the Bafis and Hypo-
thenufe, call'd in ſhort, the Angle at the
Bafis; C the Angle between the Cathe-
tus and Hypotenufe, call'd in ſhort, the
Angle at the Cathetus. And an obliquan-
gular Triangle is wont to be denoted by
B, C, and D, fo placed, as (Fig. 10,
viz.) that by a Cathetus CA, let fall
from C, it may be divided into two Tri-
angles, the Cathetus CA being a common
Side to both Triangles. And then, as
BD is eſteem'd the Baſis of the whole
Triangle BCD, fo BA and DA are the
particular
Trigonometry.
15
particular Baſes, and BC and DC the
particular Hypotenufes of the two Tri-
angles BCA and DCA. Hence, B. and
D denote the two Angles at the Bafis BD
of the whole Triangle BCD; whereof B
is always the Angle oppofite to the Side
CD, and D. the Angle oppofite to the,
Side CB, as C is always the Angle oppo-
fite to the Bafis BD.
5.
Other
It is likewife to be obferv'd, that in
the following Tables, the Letters de-
Symbols
notes a Sinę, ta Tangent, and the Greek used in the
a Cofine. Thus, Tab. I, Cafe I, B, following
t tB. Tables.
denotes the Tangent of the Angle B.
And Cafe 3, sC denotes the Sine of the
Angle C. And Tab. II, Cafe 16, ºB and
D denote the Cofines of the Angles B
and D. Alfo note, that Z do's (in this
Treatife, as in Algebra) denote the Sum
of two Things; X their Difference. Laft-
ly, This Character denotes an Angle,
24 two Angles; la Leg or Side, ll two
Legs or Sides. And confequently, Zll
denotes the Sum of two Legs of a Trian-
gle; Z44 the Sum of two Angles, &c.
Obferva-
tions in
It is alfo to be obferv'd, that a right 6.
Angle being always 90 Degrees, hence it Some other
is put down, neither among the Parts
given, nor the Parts fought, in the feveral Reference
Cafes of Tab. I. As for the other two to the twe
following
oblique Angles of a plain rectangular Tables.
Triangle, they are both put down among
the
條
​16
The Young Gentleman's
the Parts given in each of the first fix
Cafes in Tab. I, (not (†) becauſe it is
neceffary both the faid Angles ſhould be
actually given, but) becauſe one of them
being given, the other is in effect given
alfo, its Quantity being known (by Corol.
1, Theorem 4, of the Young Gentleman's
Geometry,) to be the Complement of the
given oblique Angle. On this and the
like Accounts, fome Tables are drawn up
in a much more contracted or ſhorter
Form, than the two which follow. But
I judg'd it more agreeably to my Defign,
to have all the particular Cafes diftinctly
fet down: So, that the young Student
may, both more readily find the Cafe
propos'd to him; and alſo, having found
it, may more readily perform the Calcu-
lation, having Nothing elſe to do, but
to Work according to the Proportion,
which anſwers to the Cafe propos'd.
(†) Hence it is faid in the firft fix Cafes of Table I,
thus, viz. B or C, denotifig, that if either be given, it is
fufficient.
F
TAB.
Trigonometry,
#7
TABLE I.
Shewing the Solution of plain rectangular
Triangles.
*
Cafe. Parts given. Bought.
I B or C. BA CA
Proportions.
R: tB: BA : CA.
:
2 B or C. CA
BA
_3
B or C. BA
BC
4 B or C. CA
BC
5 B or C. BC
BA
:
6 B or C. BC
CA
CA C
9 BC
•
CA B
10 BC
BA C
7 BA CA B
8 BA
R: tC CA: BA.
SC: R: BA : BÇ.
B: R:: CA : BC.
R: SC BC: BA.
R: SB BC: CA.
BA:CA:: R: tB.
CA:BA:: R : tC.
I BA
12 BC
13 BC
CA BC
BA CA
CA' BA
BC: R:: CA: sB.
BC; R :: BA : sC.
CA:R: BA: TC; & then sC: BẠ :: R: BC.
BC R::BA: SC; & then CC: BA:: R: CA
BC: K:: CA: SB; & then tB : CA ::R: BA.
C
TAB.
18
The Young Gentleman's
II.
TABLE
Shewing the Solution of plain obliquangn-
lar Triangles.
Cafe Given Sought
Proportions.
1 BC. BD. D.
C BC: BD:: SD: (||) C•
sD
2 BC.BD.C.
D
BD: BC:: SC:
SD
3
BC.CD.D.
B
BC: CD::sD:
sB.
4
BC. CD. B.
D
CD: BC :: SB:
SD.
5 BD. CD. C.
B
BD: CD :: sC :~
a sB.
6 BD. CD.B
C
CD: BD:: sB :
.
sC.
L
7 B.C.D. BC
BD
SD: BC :: SC:
BD
8 B.C.D. BC
CD
SD: BC:: SB:
sB
CD
9 B.C.D. BD
BC
SC: BD:: SD:
BC
10 B.C.D. BD
CD
SC: BD :: SB:
CD.
--II B.C.D, CD
BC
·sB : CD :: sD :
BC.
12 B.C.D.CD
BD
sB : CD :: sC :
BD
(1) It is to be noted, that in this Cafe it may be doubtful, whe-
The Reafon of which
ther the Angle fought C be Acute or Obtufe.
Doubtfulneſs arifes from what is faid, §. 10, Chap. 1. This happens,
when two Sides being given, whereof one is the greateft Side, toge-
ther with the Angle oppofite to the leffer of the given Sides, the An-
gle oppofite to the greater of the given Sides is demanded. And,
in this Cafe, the Doubtfulneſs can't be better taken away, than by Pro-
trafting the Triangle; as is fhewn, Chap. 3.
Cafe.
Trigonometry.
19
#
Cate Given
Sought
C
·(t).
of
13 EC. BD.B
Proporcions.
BC+BD BC BD D+C
2:
DDCCDC, and
2
2
(*) ==
C-D
Then
2
D+C C-D
2
BC+CD BC-CD B+D
2
2
2
B-D.
::t
: C
Then
2
2
B+D+ B-DB, and
+
B.
(+)
or
14 BC. CD.C
D
2
B
(+)
2
or*
15 BD. CD.D
C+B
C
3
4
2
BD+CD BD-CD
2.
B+D ___ B-D—B.
2
2
2
C-B. Then
C+B
:: [
: t
2.
C+B_C-B
B.
2
2
C-BC, and
2
E or DB: DG:: DF: DE. Then DB-DE—2BA.
And DB-BA-DA. Then
D
16 BC. CD. BD
or
C.
CB: R :: BÃ: ☛B; and CD: R:: DA : σD.
And Complement of B+D to 180=C.
1
}
(*) It is to bé obferv'd, that, in this and the like Cafes, the Sum
of the two Angles fought, C and D, is found by fubftracting the Quan-
tity of B, the Angle given, out of 180, according to Theorem 4, of the
Young Gentleman's Geometry.
(tt) It is alſo to be obferv'd in Reference to Cafe 13th, 14th,
and 15th, that having found the three Angles B, C, and D, the third
Side not given in either Cafe may be found by fome of the twelve
fore-going Cafes.
C A
The
20
The Toung Gentleman's
7.
The Ufe
The Ufe of the fore-going Tables fhall
be illuftrated by one or two Examples.
of the fore-
going T4- And firft, fuppofing of the rectangular
bles illu Triangle ABC, (Fig. 9.) I have the Side
ftrated, as
to artifi-" BA given, viz. 8 (I) Foot long, and the
cial Sines, Angle B, viz. 36 Degrees, 52 Minutes;
&c. and it is requir'd to find the Side CA. I
confult the fore-going Tables, and find
that this is Cafe 1, of Tab. I, and conſe-
quently, that R:tB :: BA: CA, the
Side requir'd. Hereupon I turn to my
Table of artificial Sines, &c. whofe Ray
I find to be 10.000000, and the Tan-
gent of 36 Degrees, 52 Minutes, to be
9.875010. Then in my Table of Laga-
rithms, I find the Logarithm of 8 to be
0.903090. Having thus found the Num-
bers, that are to conftitute the three first
Terms of the Golden Rule in my
Calcu-
lation, I then (according to the Method
of Working the Golden Rule by Loga-
rithms) add
the 2d and 3d 36 Degr. 52 Min.
Terms, and BA 8 f. long.
Tangent of B } 9.875010
0.903090
out of the
Sum fubftra&t
Sum
10.778100
the ift Term,
The Ray
10.000000
and the Refi-
Refidue
0.778100
due is the 4th
(1) This Extent of BA in Fig. 9, anfwers to the dia
gonal Scale, Fig. 4, and fo of BC.
Term
1
Trigonometry.
21
}
Term of my Proportion, and fo the
Number fought, viz. 0.778100. I look
for this Number in my Table of Loga-
rithms, and (though I find not exacly
the fame Number, yet) I find one very
near it, viz. 0.778151, the Logarithm of
the Number 6. Wherefore, the Side re-
quir'd CA is 6 Foot long very nearly,
viz. wanting only of a Foot.
5 I
8.
to natural
The Reaſon, why the fore-going Ex-
ample is wrought by artificial rather than The Ufe
of the Ta-
natural Numbers, is, becauſe (as has been bles illu-
(*) afore obferv'd) Tables of artificial frated, as
Sines, &c. are more eaſy to work by, Sines, &c.
and fo are generally made Ufe of, and
therefore ufually adjoin'd to Books of
Trigonometry. Whereas few Books have
Tables of natural Sines, c. Howfoever
I fhall here ſhew (for the Satisfaction of
the more Curious) how natural Sines,
c. may be found by artificial Sines,
and alfo, how the fore-going Example
is to be wrought by natural Sines or
Tangents, &c. when they are found.
If then you would have the natural
Sine or Tangent of any Angle, and want
a Table of them, but have a Table of
artificial Ones; then look out the artifi-
cial Sine or Tangent in the Table, and
(*) See the fecond Note on §. 12, Chap. I.
C 3
not
22
The Young Gentleman's
not much regarding the Characteriſtick
thereof, fee what Number anfwers to the
other Part thereof in the Table of Loga-
rithms, and that will fhew you the natu-
ral Sine or Tangent defir'd. For In-
ſtance: Not regarding the Characteriſtick
of the artificial Tangent (above-found)
of B, viz. 9.875010, I look in the Table
of Logarithms for the other Part, viz.
875010; and the nearest to it, there to
be found,is the Logarithm 875061, which
anfwers to the Number 750, which,
therefore may be taken as the natural
Tangent fought of the Angle E, the Dif
ference being only ...
S I
Having thus found the natural Tan-
gent of B, if thereby you would find the
Side CA requir'd; then you muſt place
(as afore) the Terms according to the
Direction anſwering to Cafe 1, Tab. I,
and must work the Golden Rule, (as in
natural Numbers, i. e.) by multiplying
the 2d and 3d Terms together, and di
viding the Product by the ft Term.
The Quotient will be the Number Sought
or CA, viz.
Ray. (†) Nat. Tang. BA.
1000: 750 : : 8
1
CA.
: 6.
(+) The Ray always confifts of an Unit, and fo many
Cyphers, as there are Figures in the (natural or artificial)
Sine or Tangent.
;
For
Trigonometry.
23
For 750×8=6000, and 1000) 6000 (6.
And thus the Length of the Side CA re-
quir'd, hath been found to be 6 Foot by
both Sort of Operations.
Again, of the fore-mention'd Trian-
gle ABC, fuppofe CA and B to be given,
and BC to be requir'd. I find this to be
Caſe 4, of Tak. I, and confequently, the
Rule for finding BC to be this, viz. SB:
R: CA: BC. Wherefore,
As artifi. Sine of B 36 Deg. 52 Min. 9.778119
To the Ray
So CA 6 Foot long
To BC the Side requir'd (10 f.)
10.000000
0.778151
1.000032.
For 10.000000+0.778151=10.778151;
and 10.778151-9.778119-1.000032,
which is (but, i, e. is nothing
different from 1000000) the Logarithm
of the Number 10, which, therefore is
the Meaſure of the Side requir'd BC. And
thus having given the Side BA (=8 f.)
of the Triangle ABC, the Side CA has
been found equal to 6 Feet, and BC to
ten Feet; agreeable to what is obferv'd
in the Ufe of Theorem 12, of the Young
Gentleman's Geometry; whence may be
obferv'd the Truth of trigonometrical,
as well as geometrical Principles,
CA
I pro-
9.
Another
Example.
24
The Young Gentleman's
10.
Theo-
rems, on
I proceed now to ſpeak of the funda
Of the four mental Theorems of Trigonometry, fo call'd,
becauſe on them is founded the Solution
which plain of plain Triangles, in all the Variety of
Trigono- Cafes contain'd in the two fore-going
founded. Tables. And theſe Theorems are four う
​metry is
Fig. 11.
on the first of which is founded, Tab. I;
or the Solution of plain rectangular Tri-
angles; and on the other three is foun-
ded, Tab. II, or the Solution of plain
obliquangular Triangles.
THEOREM I.
Whereon is founded the Solution of
plain rectangular Triangles.
In every plain rectangular Triangle
ABC, if BC be the Ray, then BA is the
Sine of the Angle C, and CA the Sine of
the Angle B. If BA be the Ray, then
BC is the Secant, and CA the Tangent of
the Angle B. But, if CA be the Ray,
then BC is the Secant, and BA the Tan-
gent of the Angle C; as is illuftrated,
Fig. 11.
Whence it follows,
1. R:sB :: BC: CA.
2. RSC :: BC: BA.
3. RtB:: BA: CA.
4. R tC :: CA: BA.
R:
And
1
25
Trigonometry.
And theſe four Proportions are the
fame with thofe appertaining to Cafe 6,
5, 1, and 2, of Tab. I, and from thefe
are deduced all the other Proportions,
belonging to the feveral other Cafes of
Tab. I, as may be feen by comparing
them together. And therefore it evident-
ly appears, how on this fundamental
Theorem I, is founded Tab. I. Proceed
we to Theorem 2.
THEOREM II.
Whereon is founded the Solution of
plain obliquangular Triangles,
when, there being given either
all the Angles with one Side, or
two Angles with the Side oppo-
fite to one of the faid Angles,
the Reft are fought.
$
In every plain Triangle, the Sides are
proportional to the Sines of the oppoſite Fig. 12.
Angles; that is, sB: DC:: SC: BD::
sD: BC. For
BC BD
2
2
2
:
2
BC CD
::sD: SB, and
CD BD
:: SD: SC, and
2
SB: SC; as is illuftrated, Fig. 12.
sC
2
THE-
:
26
The Young Gentleman's
THEOREM III.
Whereon is founded the Solution
of plain obliquangular Trian-
gles, when, there being given
two Sides with the Angle con-
tain'd between them, the Reft
are fought.
ZII XII
2
2
::t, ZL Lop
ZLLop: XLLOP; that is
Lop
2
2
in Words, as Half the Sum of the (Legs
or) Sides given, is to Half the Difference
of the Sides given; fo is the Tangent of
Half the Sum of the oppofite Angles
fought, to the Tangent of Half the Dif
ference of the oppofite Angles fought:
Fig. 13. That is, in Fig. 13, GE: BE:: GH:
FH.
Demonftration. In the Triangle BCD,
Fig. 13, let the Legs or Sides given be
BD and BC. Draw BG BC; and fo
XII
ZIL
there will be GE:
and BE
2
2
and the (1) extern Angle GBC=BCD+,
BDC, the two intern and oppofite An-
Ey Theorem 4, Chap. 1, of the Young Gentleman's
Geometry.
gles
}
Trigonometry.
27
!
gles of the Triangle BCD. Then draw
CG, and after that draw EH, and alfo
BF parallel to CD; whence will arife
the Angle (*) GBF=BDC, and (†) FBC
BCD. Then bife& CG by BH, and
there will arife the Angle (|) GBH:
HBC. But GBH+HBC=GBC=GBF+
FBC=BDC+BCD, i. e. ZLLopp. And
therefore, (becaufe GBH HBC, and
GBH+HBC=GBC,) GBH=
ZL Lopp ;
2
and ſo GH the Tangent of ZLLopp. Then
2
make the Angle HBK FBH. Now, be-
caufe (as afore has been fhewn) the An-
gle GBH HBC, therefore GBH-FBH
HBC—HBK, i. e. GBF=KBC. Where-
fore FBC (BCD) -KBC (GBF or BDC)
=FBK or XL Lopp. Wherefore FBH=
XLLOPP, and fo FH is the Tangent of
Lopp ;
2
XL Lopp. But now GE: BE:: GH: IH,
BE::GH:
2
that is,
ZII XII
:: t,ZLLop: + XL Lop.
t₂XL
2
2
2
(*) By Theorem 3, Chup. I, of the Young Gentleman's
Geometry.
(†) By Corol. 2, Theorem 3, Chap. 1, of the Young Gen-
tleman's Geometry.
(1) By Theorem 5, Chap. 1. of the Young Gentleman's
Geometry.
THE
28
The Young Gentleman's
Fig. 14.
THEOREM IV.
1
On which is founded the Solution
of plain obliquangular Trian-
gles, when, the three Sides being
given, the Angles are fought.
DB: DG :: DF: DE, that is in Words
The greateſt Side DB of any plain Tri-
angle, is to the Sum DG. of the other two
Sides, as is the Difference DF of thoſe
Sides, to that Segment DE of the greateſt
Side, which being taken away, the Ca-
thetus CA will bifect the remaining Seg-
ment EB of the greateſt Side.
Demonftration. By Prop. 36, 3 Euclid,
DG×DF=DTq=DB×DE.
Wherefore,
by Prop. 15, 6 Euclid, DB: DG :: DF:
DE.
From the three laſt Theorems is dedu-
ced, Tab. II, or the Solution of plain
obliquangular Triangles, in all the Varie-
ty of Caſes that can be propos'd. Nay,
the faid Theorems may ferve for the Re-
folving of all plain Triangles whatever,
Rectangular as well as Obliquangular.
But forafmuch as the first Theorem do's
fuffice to fhew the Solution of plain
rectangular Triangles, and that after a
more cafy Manner in fome Cafes, hence
the
Trigonometry.
29
A
the three laft Theorems are wont to be
reſtrain'd to the Solution only of plain
obliquangular Triangles. And confe-
quently, Tab. II, is accommodated on-
ly to obliquangular Triangles. Of which
Table the twelve firft Cafes are founded
on the 2d Theorem, the three next
Cafes, (viz. 13th, 14th, and 15th,) on
the 3d Theorem; and Cafe the 16th or
laft on the 4th and laſt Theorem. And
thus much for the Solution of plain Tri-
angles by Calculation.
CHA P. III.
Of the Solution of plain Triangles
by Protraction.
HER
+
I.
to be here
taught.
ERE are two Things to be fhewn; Two
ift, How to protract or deſcribe Things
any plain Triangle requir'd. 2dly, How
to refolve the faid Triangle, when it is
protracted i. e. how to find the Mea-
fure of the unknown Angles or Sides in
the faid Triangle, when protracted.
;
2.
to protract
Protraction is either of fome Side, or
of fome Angle. A given Side is
A given Side is protra- First, how
cted by drawing a right Line BC, Fig. any given
15, taken off from the (fimple or diago. Side.
nal) Fig. 15.
30
The Young Gentleman's
3.
how to
Angle.
nal) Scale of equal Parts, at fuch a
Length, as to contain the fame Number
of equal Parts according to the Scale, as
is the Number of Inches or Feet, &c. in
the Side given, viz. 254.
A given Angle B=48 Degrees, is pro-
Secondly, tracted, by (taking a Chord of 60 De-
protract grees for your Ray, i. e. by) opening
any given your Compaffes to the Extent of 60 De-
grees on your Scale of Chords, and then,
one Foot being faſtened on one End B of
the given Side BC, drawing with the o-
ther Foot the Arch CA; and on the faid
Arch fetting off from A to E, the Extent
of 48 Degrees taken from your Scale of
Chords; and after that drawing the right
Line BD.
4.
A plain
how to be
when the
After this manner is any given Side and
Angle to be protracted, and fo any plain
Triangle Triangle to be defcrib'd, when either one
defcrib'd, Side and two Angles, or two Sides and
one Angle are given. But, if the three
are given, Parts given be the three Sides; then the
and no An- Triangle is to be defcrib'd, as is fhewn,
Chap. 2, Problem 2, Cafe 1ft, or 2d, or
3d, of the Young Gentleman's Geometry.
three Sides
gle.
5.
How to
The Triangle BCD, (Fig. 15,) being
protracted, the Meaſure of any Side not
find the given, is found, by applying the Extent
4 Side not thereof with your Compafles to your
(fimple or diagonal) Scale of equal Parts,
and fo finding how many fuch equal
Meafure of
a
given.
·
he
Parts
Trigonometry.
3.
Parrs are contain'd in the faid Side.
Which equal Parts are to be eſteem'd In-
ches, or Feet, or Yards, accordingly as
the Meaſure of the given Side is denoted
by Inches, or Feet, or Yards, &c. Thus,
fuppofing BC to be 254 Feet long, CD
will be 198 Feet, and BD 204 Feet.
6.
find the
not given
In like Manner, the faid Triangle be-
ing protracted, the Meaſure of any An- How to
gle C or D not given, is found, by open- Measure of
ing your Compafs to the Extent of 60 an Angle
Degrees on your Scale of Chords, and
then fixing one Foot on the Point of the
Angle fought, and making an Arch from
one Leg to the other of the faid Angle.
The Extent of the Arch apply'd to your
Scale of Chords, will fhew the Degrees
of the reſpective Angle. Thus, Fig. 15,
the Extent of the Arch FG, fhews C to
be an Angle of 52 Degrees; and the
Extent of the Arch EG, fhews D to be
an Angle of 80 Degrees.
7.
tion of
plain Tri-
Protrachi-
It remains to be obſerv'd, that the So-
lution of plain Triangles by Protraction, The Solu-
is founded on Theorem 9, Chap. I, of the
Young Gentleman's Geometry. Namely, angles by
becauſe the protracted Triangle is equi-
angular to the real Triangle, (i. e. the what
triangular Ground, or the like,) there- founded.
fore, the Sides of that have the fame
Proportion one to the other, as have the
Sides
on, on
32
The Young Gentleman's
8.
plain Tri-
Sides of this; albeit thoſe are in a man-
ner infinitely leſs than theſe:
And thus having fhewn, how to re-
The Use of folve any plain Triangle, either by Cal:
culation or Protraction; I proceed now
to illuſtrate the great Uſefulneſs of plain
Trigonometry, in taking Heights and Di-
gonome-
try ap-
pears in
taking
Heights Stances, as alfo in Surveying and Fortifi-
and Di-
ftances,
cation.
dc.
I.
The Height
CHA P. IV.
Of Taking the Altitude or Height
of an Object, viz. Tower, Tree,
IT
&c.
that
T is to be remember'd, that (according
to Definition 42, Chap. 1, of the Young
of an Ob- Gentleman's Geometry) the Altitude or
ject, what. Height of an Object, viz. Tower, Tree,
and the (*) like, is meafur'd by a Per-
pendicular from its Top to its Bottom.
Thus, the Height of the Tower, Fig.
16, and of the Tree, Fig. 17, is denot-
(*) For as to the Height of the Sun, or any other
Celestial Light, that is computed by its Diſtance above
the Horizon on a vertical Circle: Of which in Aftro-
nomy.
ed
Trigonometry.
33
ed by the Perpendicular CA; and fo
in all other fuch Cafes through this Trea-
tife.
3.
bols made
Uſe
For the Line CA of Altitude, and the
Line BA of Diſtance, (viz. between the The Sym
Obferver B, and the Foot of the Object Use of in.
A,) and the Line BC of Viſion do always taking
conſtitute the three Sides of a rectangular Heights,
Triangle ABC. Wherein the Height of
the Object is the Cathetus or Perpendi-
cular, and therefore, (according to the
receiv'd Method of denoting the feveral
Parts of a rectangular Triangle in Trigo-
nometry,) is aptly denoted by CA. In
like manner, the Diſtance of the Ob-
ferver from the Object, making always
the Baſis, is aptly denoted by BA, and
the Line of Vifion, making always the
Hypotenuse, is aptly denoted by BC.
And fo as to the feveral Angles, A de-
notes the right Angle between the Line
of Altitude and the Bafis, B the Angle
between the Bafis and Hypotenufe, which
may be call'd, the Angle of Altitude, be-
cauſe the Altitude of the Object is al-
ways either (†) its Sine or Tangent; and
fo C will denote the remaining Angle
(†) Namely, if BA be taken for the Ray, then CA is
the Tangent of B; but if BC be taken for the Ray,
then CA is the Sine of B.
D
between
34
The Young Gentleman's
3.
How to
take the
Angle of
Altitude.
4.
An acceffi-
between the Lines of Viſion and Alti-
tude.
The beſt Inftrument for taking the
Angle of Altitude is the Quadrant, thence
commonly diſtinguiſh'd by the Name of
the Quadrant of Altitude; by which it is
done thus. Hold your Quadrant up-
right, fo as the String may have Liberty
to flip to either Edge, and move the
Quadrant fo held, 'till through the Sights
you fee the Top of the Object, whofe
Height you would take. Then the De-
grees, intercepted between the Beginning
of the Quadrant and the String, are the
Degrees of the Angle of Altitude; name-
ly, of the Altitude of the Object above
your Eye. Wherefore, if you ftand up-
right, when you take the Angle of Al-
titude, to the Altitude found by the faid
Angle of Altitude, there muſt be always
added the Height of your Eye above the
Ground, to know the Altitude requir'd
of the Object.
If one may come to meaſure from the
Place B of Obfervation, to the Foot A
ble and in- of the Object, (i. e. to the very Bottom
acceffible of the Perpendicular CA,) then it is
call'd an acceffible Height. But, if any
Thing hinders from coming to A the
Foot of the Object, then it is call'd an
inacceffible Height.
Height,
what.
PRO
Trigonometry.
35
PROBLEM I
To find an acceffible Height.
5.
Calculati
Meaſure your Line BA of Diſtance,
and find the Meaſure of the Angle B First, by
by your Quadrant. Theſe two being
known, CA is found by Tab. 1, Cafe 1, Fig. 16.
viz. RtB:: BA: CA. For Inftance:
R:tB::BA:CA.
Suppofe BA 347 Feet, and B-35 De-
grees. Wherefore,
As the Ray
To the Logarithm of BA 347 Feet
So the artificial Tangent of B 35 Degr.
To the Logarithm
of the Side fought
CA 243 Feet very
10,000000
2.540329
9.845227 S
}Add
Add
12.385556
10.000000 Subſt.
on.
near.
2.385556 Refidue.
by Pro-
If you would find the Height fought 6.
CA by Protraction, work thus. Draw Secondly,
(according to Chap. 3, §. 2,) the Line traction.
of Diſtance BA=347 (equal Parts or) Fig. 16.
Feet. Then, (by Chap. 3. §. 3.) make
the Angle B 350, and the Angle A=
90 Degrees, or a right Angle. Then
draw the Sides BC and CA, 'till they
meet in C. Apply CA to your diagonal
Scale, and it will be equal to 243 (equal
Parts thereof, or) Feet.
D
To
36
The Young Gentleman's
7
Some O-
ceffible
Heights.
To the Method of taking Heights by
Protraction, may be referr'd the Method
ther Me-
thods of
of taking Heights by a Staff, or a Bowl
taking ac- of Water, or the like. Forafmuch as by
Help of the Staff or Bowl of Water,
(c.) there is as it were protracted a
Triangle, equiangular to the Triangle
made by the three Lines of Viſion, of
Diſtance, and of the Altitude fought.
Namely, in Fig. 17, or 18, the Tri-
angle bac, is equiangular to the Triangle
BAC, (as fhall be fhewn by and by.) and
therefore, the Meaſure of the Sides ba
and ca being known, as alfo of BA,
thereby is found (according to the Rule
of Proportion) CA the Altitude fought.
For, (according to Theorem 9, Chap. 1,
of the Young Gentleman's Geometry,) ba:
ca:: BA: CA.
8.
Now the Way to find the Altitude of
Acceffible an Object by a Staff, is two-fold, either
Heights
may be ta- by the Staff and its Shade, when the
ben by a Sun fhines; or elfe by the Staff alone,
at any Time, or whether the Sun fhines
Staff two
Ways.
Height by
or Not.
9. To find the Altitude of an Object by
To take an a Staff and its Shade, work thus. At B,
acceffible the End of the Shadow caft by the Top
* Staff and of the Object, fet up your Staff ca per-
pendicularly, (as Fig. 17,) and mark
the End b of the Shadow made by your
Staff. For, as baca :: BA: CA, i. 1.
its Sha-
dow.
Fig. 17.
As
Trigonometry.
37
As the Length of the Staff's Shadow, is
to the Height of the Staff; fo is the
Length of the Object's Shadow, to the
Height of the Object. For Inftance:
Suppoſe your Staff ca to be three Foot
long, or juſt a Yard, and its Shadow ba
to be nine Foot long, and the Shadow
BA of the Object to be 60 Foot long;
then the Height of the Object will be
twenty Foot. For 9: 3 :: 60: 20. This
Method depends (as is afore faid) on
Theorem 9, Chap. 1, of my Geometry, or)
on the Triangles bac, and BAC being
mutually equiangular. And that they
are fo, is thus fhewn, viz. the Staff ca
being erected perpendicularly, therefore,
the Angle a is a right Angle, and fo e-
qual to A. Alfo, becauſe the (D) Ray
of the Sun, (or, which comes to the
fame, the Extremity of the fhaded Air)
bc, is always parallel to the other Ray
BC, therefore, (by Theorem 3, Chap. 1,
of my Geometry,) the Angle b-B, and
confequently, the Angle C; and fo
(Although the Rays of the Sun, as proceeding from
the Sun as their Center, are in Strictnefs, not parallel, yet
by Reaſon of their immenfe Length or Distance from
the Sun, wherein they concur, their Obliquity is not dif
cernible in fuch Cafes, but they may be efteem'd as par-
allel, efpecially as to fuch fmall Segments or Portions of
them as are herein concern'd.
D 3
the
$8
The Young Gentleman's
a
10.
lone.
Fig. 18.
•
the Triangle abc, equiangular to the Tri-
angle ABC.
If the Sun do's not fhine, then you
To take an may nevertheless find the Height of the
acceffible
Height by Object by your Staff alone, thus. Ha-
Staff a ving fet up (as afore) your Staff ca per-
pendicularly, go backward 'till you can
fee the top C of your Object, in a Line
with the top c of your Staff, as is repre-
fented, Fig. 18, by the Line bC or BC.
Then (*) baca :: BA (or bA): CA,
that is, as the Distance ba of the Place b
where you faw the Top C of the Object,
and the Top c of your Staff in one Line
be (or Be) from the Place a of your Staff,
is to the Height ca of the Staff; fo is the
Diſtance of the former Place (b or) B
from the Foot A of the Object, to the
Height CA of the Object.
II.
Height by
a Dith or
If you would for Curiofity Sake know,
To take an how to find the Height of an Object by
acceffible
a Bowl or Difh of Water, it is done thus.
Go backward from the Bowl, 'till ftand-
ing upright, you can fee the Top Cof
the Object in the Water. Then ba: ca ::
BA: CA, i. e. as is the Diſtance ba of
Bowl of
Water.
Fig. 19.
(*) For here alfo bca and BCA are equiangular Tri-
angles. For CA and ca being both Perpendiculars, are
therefore Parallel. And confequently, (by Theor. 4, Chap.
I, of my Geometry, the Angle C, as well as aA;
and the Angle Borbis common to both Triangles.
your
Trigonometry.
39
your Station a from (†) the Bowl b, to
your own Height ca as far as to your
Eyes; fo is the Diſtance BA of the Foot
A of the Object, to the Height CA of
the Object. For, according to that prin-
cipal Theorem of Catoptricks, the Angle
of Reflexion Pbc, is equal to the Angle
of Incidence PBC. And therefore, the
Angle abc (=1 Right-Pbc=1 Right—
PBC) ABC. And the Angle Aa,
becauſe both Right; and therefore, the
remaining Angles C and c are equal;
and fo the Triangles ABC and abc equi-
angular.
20.
only.
Fig. 29.
Laftly, There is one Way more to take
an acceffible Height, which I ſhall here To take an
mention by Reafon of its Eafinefs and Height by
acceffible
Expediteneſs, when it may be us'd. It Quadranc
is thus. Go towards or from
your ob.
ject CA, if you have Room, 'till the
String of your Quadrant falls juſt on the
45th Degree. Then the Height CA of
your Object, will be equal to BA the
Diſtance between your Place B, and the
Foot A of the Object. For the Angles
B and C will be equal, (viz. each 45
Degrees,) and therefore, the Triangle
A
(t) If you would be exact, your Distance, and the
Diſtance of the Object must be both taken from the
very Point of the Water, whereon you fee the Top of
the Object.
D 4.
ABC
40
The Young Gentleman's
1
13.
lation.
Fig. 21.
ABC will be an Ifofceles, i. e. BA=CA,
by Theorem 5, Corol. 2, of the Young Gen-
tleman's Geometry.
PROBLEM II.
To find an inacceffible Height.
Take two Stations B and b lying in a
By Calcu- direct Line with the Foot A of the Ob-
ject. The Diſtance Bb between thefe
two Stations, with the two Lines of Vi-
fion BC and bC, make an obliquangular
Triangle BCb. All whofe Angles are
known, by taking the two Angles of Al-
titude CBA and CbA, in their respective
Triangles BCA and CA. For 180-An-
gle CBA CBb; and 180-CBb+b=
BCb. The three Angles of the Triangle
BCB being thus known, and the Diſtance
Bb between the two Stations being mea-
fur'd, the Side BC is found by Cafe 11,
Tab. II, (viz. SB: CD:: SD: BC. For,
that is in Reſpect of the Notation of the
feveral Sides of the Triangle BCb, Fig.
21,) sBCb: Bb: sCbB BC. But now
BC is alfo the Hypotenufe of the rectan-
gular Triangle ABC; and the Meaſure
of BC being known, as alſo of the
Angle CBA, (viz. CBA=180-CBb,)
hence may be found CA the Height of
1
کیا ہم کچھ
the
Trigonometry.
41
the Object, by Cafe 6, Tab. I, viz. R:
SB:: BC: CA.
For Inftance: I would know the
Height of the Church-Steeple, Fig. 21,
between A the Bottom of which, and b
the Place where I am, runs a River, fo
that it is an inacceffible Height. Where-
fore, by my Quadrant, I take the Angle
GbA of Altitude at my Station b, which
I find to be 27 Degrees. Then meafu-
ring in a ſtraight Line from b to B my
fecond Station, (which I find to be 92.7
Foot,) I take the Angle CBA of Altitude
again at my fecond Station B, and find
it there to be 52 Degrees. Hence it
follows, that the Angle CBb=127 De-
grees, (for CBb-180-CBA ;) and al-
fo, that the Angle BCb=25 Degrees,
(for BCb-180-: CBb+CbB:=180
-155) Having thus in the Triangle
BCb the three Angles, and one Side Bb
given, thereby is found the Side BC, by
Cafe 11, Tab. II, Viz.
1
As
2.
41
The Young Gentleman's
As the Sine of BCb 25 D. 30 M.
To the Log. of Bb 92.7 Foot
9.633984
1.967080
So the Sine of CbB 27 D. 30 M.
·9.664406
211.631486
14.
To the Log. of BC fomewhat above 9.633984
99.4 f. viz. 100 f. very nearly
1.997502
3
Having thus found the Meaſure of BC,
and having afore found the Angle CBA,
thereby is found CA the Height fought
by Cafe 6. Tab. I. viz.
As the Ray
To the Log. of BC 100
So the Sine of CBA 52 D. 30 M.
To the Log.of CA=793 and fome-
what more, viz. 80.3 very nearly
›
*
10.000000
2.000000
9.899467
11.899467
10.000000
1.899467
If you would work by Protraction, you
To take an muft proceed thus. Having (as afore)
inacce fible
Height by at the two Stations B and b, found the
Protracti- Angles of Altitude, viz. CbB and CBA,
on. and meafur'd the Length of Bb, then
Fig. 21.
draw a right Line Bb of as many Parts on
the Scale, as the Meaſure of Bb is found
to be of Feet (or Yards, c.) At the
two Points B and b make two An-
gles CBA and CbA equal to the Angles
of
'
Trigonometry.
43
of Altitude taken by you at B and b.
From C the Point, where CB and Cb in-
terfect or croſs one the other being
drawn, let fall the Perpendicular CA to
the Line of Diſtance Bb extended, if need
be, 'till it meets with CA. The Perpen-
dicular CA apply'd to the Scale of equal
Parts will ſhew the Number of Parts, i. e.
of Feet or Yards, &c.
Height by
dow.
15.
An inacceffible Height may alſo be To take an
found by the Help of a Staff, and its inacceffible
Shade, thus. Set up the Staff ca perpen- a Staff and
dicularly, and mark the End b of its Sha- its Sha-
dow Fig. 23; and at the very fame time Fig. 22,
mark the End B of the Shade of the inac- & 23.
ceffible Object CA Fig. 22. Then three
or four Hours after mark again the End d
of the Staff's Shadow, and alſo the End D
of the Shadow of the inacceffible Object.
Then meaſure the Diſtance between the
two mark'd Ends b and d of the Shade of
the Staff, and alfo between the two
mark'd Ends B and D of the Shade of the
inacceffible Object. As bd to ca, fo BD
to CA, i. e. As the Diſtance between the
two different Ends of the Staff's Shadow,
to the Height of the Staff; fo will the
Diſtance between the two different Ends
of the Object's Shadow, be to the Height
of the Object. For (by Theor. 9. Chap. 1.
of my Geom.) da: ca :: DA: CA. Where-
fore
ļ
44
The Young Gentleman's
fore da― (†).ba: ca :: DA—(†) BA:
CA. i, e. bdca :: BD: CA.
16. If it should happen, that you have not
To take an Room to take two Stations, as Fig. 24.
nacceffable then an inacceffible Height may be found
when there thus. First take the Arch CBA of Alti-
Height,
is not
Room to
take two
Stations
on the
Ground.
tude on the Ground B; and then go up
to the Top of the Houfe (or Tree) where
you are, and there take the Angle CDB.
Draw an indeterminate Line BA, and
Fig. 24. thereon at Berect the Perpendicular BD,
of an equal Length with the Height of
the Houfe DB, (which is fuppos'd to be
known,) and thereon at D make an Angle
CDB equal to the Angle taken on the Top
of the Houfe, and at B make an Angle
CBA equal to the Angle of Altitude ta-
ken on the Ground. Then extend BC
and DC 'till they meet in C. A Perpen-
dicular CA, let fall from C upon the in-
determinate Line BA, being apply'd to
the Scale of equal Parts, will ſhew the
Height of the Object CA.
(tt) For as the Lines da and DA are proportional by
the fore-mention'd Theorem, fo will their Parts ba and
BA; becauſe, the Shade of the Staff will be proportion-
ably lengthened or fhortened, as is the Shade of the
Object.
PRO.
T
لا
Trigonometry.
45
PROBLEM III.
3
A
To meaſure the Height of an Ob-
ject, which ftands on the Top of
another Object.
I would know the Height of the Spire Fig. 25,
Cd above the Tower Aa. Find the whole & 26.
Height CA (by Prob. 1. if A be acceffible,
as Fig. 24; if inacceffible, as Fig.25, then
by Probl. 2.) Then find the Height Ad
(or ca) of the Tower alone (as afore by
Probl. 1. or 2.) Then CA-Ad (or ca)
Cd the Height of the Spire.
PROBLEM IV.
To find the Height of an Hill or
Mountain.
This may be done in fome Manner by Fig. 27.
Probl. 2.
But becauſe the Tops of Hills
or Mountains, if of any confiderable
Height, do appear higher than really
they are, by reafon of the Refraction of
the Ray of Vifion; therefore there is an-
other more exact Method to find the
Height of Hills and Mountains, by the
help of an Inftrument called the Brazen T,
from
1
!
F
{
46
1
The Toung Gentleman's
from its reſembling the Letter of that
Name, as may be ſeen Fig. 27. For this
Inftrument being plac'd between two
Staves (or Spikes, divided into a certain
Number of Feet, and thoſe ſubdivided
into Inches) thereby may be obferv'd
(at least by the Help of flipping Marks,
to be mov'd upwards or downwards)
what Number of Feet and Inches the
Hill rifes or the Aſcent increaſes at all the
feveral Poſitions of the ſaid Staves, from
the Bottom to the Top of the Hill or
Mountain. And confequently, by ad-
ding together the feveral Feet and Inches,
which at the feveral Pofitions of the
two Staves, have been obferv'd to be
contain'd, between the horizontal or le-
vel Lines, thence will ariſe the Sum of
all the Feet and Inches from the Bottom
to the Top of the Hill, and fo the Height
fought. Thus the Height CA of the
Hill or Mountain Fig. 27. is equal to
the Sum of ab+cd+ef+gh. And this
Method is called Cultellation.
CHAP.
Trigonometry.
47
CHAP. V.
Of Taking the Distance between twe
Objects.
IN tak
F
I.
rent gene-
taking Diſtances (as well as Heights)
there are in general two different Ca- Two diffe-
fes. For either we may come to (*) One, ral Cafes
or Neither of the Two Objects, whofe in taking
Distances.
Diſtance one from the other we would
take.
The Angle which is oppofite to the 2.
Line of Diſtance (between the two Ob- The Angle
jects) may be call'd, the Angle of Di- of Di-
ftance,
Stance, and is here denoted by the Letter what.
D: Forafmuch as the Triangles arifing in
taking Diſtances are generally obliquan-
gular Triangles.
The most commodious Inſtrument for 3.
taking Angles of Diſtance is the Theodo- How to
lite or Circumferentor, i. e. a Braſs Circle take the
Angle of
divided into 360 Degrees, and fitted with Diftance.
(*) As for the third Cafe, viz. wherein both the Ob.
jets, whofe Diſtance is requir'd, may be come at, that
de's not ftand in need of Trigonometry to find the Di-
ance, becauſe it may be meafur'd (without any more
do) by a Chain or other long Meature.
four
J
48
The Young Gentleman's
1
40, of
Geometry.
four Sights. For the Sights being fo
mov'd, as that you can through two of
See Fig. them (lying in a ſtrait Line AB) ſee one,
A, of the Objects, and through the other
two (on the Line DE) fee the other, E, of
the Objects, the Angle ACE will be the
Angle of Diſtance; whofe Meaſure will
be the Degrees intercepted between CA
and CE ; and are always equal to the
Degrees between CD and CB, becauſe
the Angles ACE and DCB are verti
cal Angles, and confequently equal.
Whence BCD as well as ACE may be e-
ſteem'd the Angle of Diſtance.
lation.
Fig. 28.
PROBLEM I.
To take the Distance between two
Objects, when One of the Ob-
jects is acceffible.
I would know the Diſtance of the Ob-
By Calcu- ject B (Fig. 28.) from the Object C 49
one, B, of which Objects is acceffible, but
the other, C, is not. On one Side of B, I
chooſe a Station D, at a (†) confiderable
Diſtance from B. The Line BC of Di-
(†) Note, That the greater the Distance BD is be-
tween the acceffible Object B, and the Station D, fo
much more exact will the Operation be.
ftance
Trigonometry.
49.
ftance between the two Objects B and C,
and the Lines BD and CD of Distance be-
tween each Object and my Station, make
an obliquangular Triangle, BCD. Now the
Angles CBD and BDC being found by the
Theodolite, and confequently the Angle
DCB being known, (viz. DCB=180—
CBD+BDC) and the Side BD being
meafur'd, hereby may be found BC the
Diſtance requir'd between the two Ob-
jects B and C, by Tab. II, Cafe 9, viz.
SC: BD:: SD: BC.
ple.
5.
For Inftance: Suppofe BD, the Diſtance
of my Station D from the acceffible Ob- An Exam-
ject B, to be one hundred and four Yards;
and the Angle BDC=86 Degr. and the
Angle DBC=63 Degr. and confequently
the Angle BCD=31 Degr. It follows,
according to the foremention'd Caſe 9,
Tab. II, that
As the Sine of C 31 Deg.
9.711839
To the Logarithm of BD 104 Yards 2017033
1
So the Sine of D 86 Deg.
9 999404
12.016437
To the Log.of BC 202 very nearly
9.711839
2.304598
6.
if you would take the Distance by Pro-
traction, you must work thus. Draw By Pro-
BD containing 104 equal Parts upon your Fig. 28.
E
dia-
traction.
50
The Young Gentleman's
7.
or Pole
only.
diagonal Scale, equal to the Number of
Yards found upon meaſuring BD. And
then at one End B defcribe the Angle B
equal to its Meaſure of 63 Degrees found
by the Theodolite; and likewife at the
other End D defcribe the Angle 'D=86
Deg. as found by the Theodolite. Then
draw the Lines BC and DC, 'till they meet
in C, the Line BC apply'd to your di-
agonal Scale, will fhew the Diſtance of
the two Objects B and C, viz. about 202
Yards.
If you have no Theodolite, or fuch
By a Chain like Inftrument by you, the Diſtance BC
may be taken only by a Chain or Pole
Fig. 29. (or the like) thus: Meaſure with the
Chain the Diſtance BD between the ac-
ceffible Object B and your Station D;
which your Station chooſe ſo in this Cafe,
as that it may be (as near as you can
guefs) at an equal Diftance with B from
the inacceffible Object C. Then in the
Lines of Viſion DC and BC mark out any
two Points d and b, and meaſure with the
Chain the feveral Lines DB, Eb, bd, đ□,
and dB. Then protract a Trapezium
BDdb of the fame Sides ; the two Sides
Bb and Dd of which being produc'd, will
meet in C, which will denote the Place
of your inacceffible Object. And confe-
quently the Line BC, being apply'd to
the Scale of equal Parts, will thereon fhew
the
Trigonometry.
5x
the Diſtance fought. For by meaſuring
and protracting the Sides of the Trapezi-
um BDdb, you do in effect find and pro-
tract the Angles B and D of the obliquan-
gular Triangle BCD, as well as the Side
BD of the faid Triangle. And therefore
the protracted Triangle BCD is equian-
gular to the Triangle made by the Lines
of Viſion at B and D, and the Line BD
;
and confequently the Sides of the protra-
&ted Triangle are proportional to the Sides
of the real Triangle, reprefented by the
protracted one.
By this laſt Method may the Breadth
of a River be taken. For if you fuppofe
BD to be one Bank of the River, and C
to denote a Point of the Bank on the o-
ther Side of the River; then the Diſtance
of the Banks or Breadth of the River will
be equal to BC. Where it is to be noted,
that in this Method the Trapezium BDbd
may be taken or meafur'd out, on either
Side of BD, viz. either towards the inac-
ceffible Object C, or on the other Side of
BD, as you have Room or moſt Conve-
niency.
8.
To take the
Breadth of
a River.
Fig. 29.
9.
like Di-
The fame may be performed alſo after To take the
this Manner. Set up a Staff perpendicu- a River,
larly on the Bank where you are, and or any fuch
thereon place a fquare Rule with its An-
gle downwards on the Head of the Staff.
Move the fquare Rule, 'till the Line BA
E 2
a
tance, by
Staff and
a Square-
rule.
of Fig. 30.
52
The Young Gentleman's
IO.
the Di
as of a
of Viſion from B a Point in the oppofite
Bank of the River, falls in directly with
the upper Edge of one Side of the fquare
Rule; and then from the Point C (where
the upper Edges of the two Sides of the
fquare Rule meet) draw, along the upper
Edge of the other Side, a String, or Rope,
or Chain, 'till it comes in a ſtraight Line
to the Ground at b. As (†) bA : CA ::
CA: BA, i. e. As the Diſtance bA be-
tween the Point b, where the String came
to the Ground, and the Foot A of your
Staff, is to CA the Height of your Staff;
fo is the Height CA of your Staff to the
Diſtance BA of A (the Foot of your Staff,
or) the Bank where your Staff is, from B
the other Bank. And after the fame Man-
ner may any other Diſtance be taken,
where only one Object is inacceffible.
If the inacceffible Object be an Obje&
To take of Height, viz. a Tower, or Tree, or the
ftance of like, then having found (by the Problem
an Object for taking an inacceffible Height) the
of Height,
Height CA of the faid Object, and its
Tower, &c. Angle B of Altitude, thereby may be
Fig. 21, found the Diſtance fought BA by Tab. I,
or 22, & Cafe 2; viz. R : tC :: CA: BA. For the
Angle B being known, thereby is the An-
gle C known; viz. C=90-B.
23.
(†) Namely, by Corol. I, Theorem 11, of the Young Gen-
tleman's Geometry.
་
PRO
Trigonometry.
53
PROBLEM II.
To take the Distance between two
Objects, when both are inac-
ceffible.
11.
Take your firſt Station D as you pleaſe;
and on the Side of the former chooſe a By Calcu-
fecond Station d, fo as Dd may be paral- lation.
lel to BC as nearly as you can. Then with
Fig. 31.
your Theodolite at D one of your Stati-
ons, take the Angles BDC and CDd,
which together make the Angle BDd.
Then at your other Station d take the
Angles BdC and CdF, (the Mark F being
ſet up in a right Line with your two Sta-
tions D and d) which together make the
Angle BdF. Now in the Triangle BDd
the Angles BDd+DBd=BdF. Where-
fore the Meaſure of the Angle BɗF being
found by the Theodolite, DBd=BdF
-BDd. And therefore the ſtationary Di-
ftance Dd being meaſur'd, sDBd: Ďd ::
sBDd: Bd by Cafe 12, Tab. II; viz. sB ;
CD::sC: BD. In like manner in the
Triangle CdD, the Angles CDd+DCd
CdF, Wherefore the Meaſure of CdF
being found by the Theodolite, the An-
gle DCd CdF-CDd. And therefore
by Cafe 12, Tab. II; sDCd: Dd: sCDd;
E 3
محمد
Cd,
54
The Young Gentleman's
12.
By Pro-
traction.
Fig. 31.
*
Fig. 32,
& 33.
Cd. Wherefore the Angle CdB of the
Triangle BCd being found afore, and the
Angle CBd CDd found alfo afore, it
follows by Cafe 12, Tab. II, that sCBd:
Cd: sCdB: BC the Diſtance fought.
To take the like Diſtance by Protracti-
on, you muſt work thus. Take (as afore)
at your two Stations, D and d, the feve-
ral Angles BDC and Cdd, and CdB and
BdD. Then having meafur'd your ſtati-
onary Distance Dd, and protracted it, on
one End D thereof protract its reſpective
Angles BDC and CDd; and on the other
Ended its reſpective Angles CdB and
BdD. Draw out the feveral Sides of the
Angles meet refpectively, two BD and
Bd in B, the other two, Cd and CD, in
B. The Diſtance BC apply'd to your
Scale of equal Parts, will give you the
Diſtance fought.
PROBLEM III.
To take the Draught of a City, or
Port, or the like.
This is done after the fame Manner as
is taken the Diſtance between two Ob-
jects, when both are inacceffible; it be-
ing no other than a Repetition of the faid
Operation, according to the Variety of
the
Trigonometry.
35
•
the remarkable Objects which you would
delineate or defcribe in the Draught. For
Inſtance: Let A, B, C, D, E and F de-
note ſo many remarkable Buildings (viz.
Churches, or Pillars, or other publick E-
difices) in a City; or elfe let them de-
note fo many Ships lying in a Port or
Harbour. I take two Stations M and N
at ſo confiderable a Diſtance, as that the
Line MN of the faid Diſtance may be
long enough to ſerve conveniently for the
Baſis of all the Triangles to be taken.
Then at M placing the fixt Index of my
Theodolite in a right Line with MN, I
turn the moveable Index, as Occafion re-
quires ; and ſo take in order the feveral
Angles MNA, MNB, MNC, &c. and fo
to the laſt MNF. Then removing to my
other Station N, and placing (as afore)
the fixt Index of my Inftrument in a Line
with MN, I take the feveral Angles
NMA, NMB, NMC, &c.
Having done this, I (†) protract the
Line MN of Distance between my two
Stations M and N, taking from my Scale
of equal Parts the Line mn anfwerable to
(†) Having the Bafis MN, and the two Angles at the
Bafis in each Triangle, the other Sides in each Triangle
may be found by Calculation; but this Method being
tedious, therefore Protraction is generally made Uſe of
in this Cafe.
E 4
MN
56
The Young Gentleman's
}
Fig. 34.
mn,
MN. And then at the End m of the Line
I protract the Angles mna=MNA,
nb MNB, mnc-MNC, &c. Like-
wife at the other End n of the Line mn,
I protract the Angles nma=NMA, nmb
NMB, nmc-NMC, &c. The Legs
of the feveral Angles being produced, will
fhew in the feveral Points of Interfecti-
on a, b, c, &c. the refpective Places of
the ſeveral Objects A, B, C (c); and
confequently their refpective Distances,
both one from the other, and alfo from
each of my Stations M and N ; viz. the
Extents ab, or ac, &c, or bc, or bd,
&c. being apply'd to my Scale of equal
Parts, give the Meaſure of the ſaid Di-
ſtances.
PROBLEM IV.
To find the Horizontal Line or
Length of an Hill or Moun-
tain.
On the Top of the Hill, at C, fet up
a Staff or Mark of an equal Height with
your Eye (above the Ground) or with
the Staff to which the Quadrant is faſt-
ned at B the Bottom of the Hill. Then
move the Quadrant 'till you fee the Top
of the (Staff or) Mark at C, whereby
you'll
Trigonometry.
57
}
You'll take the Angle CBA. Then mea-
fure up the Side BC of the Hill.
4
Being
come up to C, meaſure the Level Cc on
the Top of the Hill. Then meaſure the
other Side be of the Hill, and take the
Angle cba, as you did afore the Angle
CBA. Having done this, you may find
the Length of the Hill, either by Calcu-
lation or Protraction.
If you work by Calculation, then in
the Triangle ABC, having the Angle
(CBA or) B, and the Hypotenuſe BC,
the Bafis BA is found by Cafe 5, Tab. İ,
viz. R: SC: BC: CA. In like manner,
in the Triangle abc having the Angle b,
and the Hypotenufe be, the Bafis ba is
found by Cafe 5, Tab. II.
If you work by Protraction, then ha-
ving protracted the Hypotenufe BC, and
on one End B thereof protracted alſo the
Angle (CBA or) B; from the End C of the
Hypotenufe let fall a Perpendicular CA
upon the other Leg of the Angle B. The
Extent BA will be the Bafis of the Trian-
gle ABC. In like manner will be found
ba, the Bafis of the other Triangle abc.
Having thus found (either by Calcula-
tion or Protraction) the Meaſure of the
two Baſes BA and ba, theſe two added
together, and alfo to the Length of the
Level-Line Ce on the Top of the Hill,
will give the total Meaſure of the hori-
zontal
58
The Young Gentleman's
zontal Line or Length of the whole Hill.
Namely, Suppofing BA=160 Yards or
Chains, &c. and ba=182, and the Le-
vel-Line Co=122, the total Meaſure of
the Length of the Hill will be 464. And
thus much for taking of Diftances. Pro-
ceed we next to Surveying.
I.
Surveying,
why e-
CHAP. VI.
Of Surveying or Meaſuring of Land.
ΤΗ
HE Surveying or Meaſuring of Land
is eſteem'd as belonging to Tri-
gonometry, and fo ufually taught in
Steem'd as Treatifes of Trigonometry, becauſe the
to Trigo- Way formerly us'd to ſurvey, was by ta-
nometry. king the Angles and Sides of the feveral
2.
Triangles, into which any Spot or Tract
of Ground may be divided; and then
protracting the faid Angles and Sides on
Paper; and after that finding the Area
or Content of the feveral Triangles ſo pro-
tracted.
This Method is illuftrated Fig. 35.
An Exam- wherein the feveral Lines AB, BC, CD,
ple of the c. reprefent the feveral Sides or Hed
ges of a Ground
thod of
and the ſmall Cir-
Surveying. cle T reprefents the Place of the Theodo-
old Me-
3
lite.
Trigonometry.
59
•
lite. Which being there plac'd, by turn-
ing the moveable Index thereof as Occafi-
on requires, the feveral Angles ATB,
BTC, CTD, &c. are taken. And then
by meaſuring with a Chain (or the like)
from T respectively to A, B, C (&c.)
thereby is found the Length of the feve-
ral Sides TA, TB, TC, TD, &c. of the
feveral Triangles ATB, BTC, CTD, &c.
And fo by the Help of the Angles and
Sides thus found, the other unknown An-
gles and Sides of the feveral Triangles may
be found, (either by Calculation or Protra-
&tion ;) and fo the feveral Triangles may
be protracted, and confequently the whole
Ground, made up of the faid Triangles.
The Ground being thus protracted, its A-
rea or Content is found (as has been ob-
ferv'd in the Ufe of Theor. 7, Chap. 1, of
the Young Gentleman's Geometry) by let-
ting fall a Perpendicular from any Angle
in each Triangle to its oppofite Side, and
multiplying the faid oppofite Side in-
to half the Perpendicular. Thus the
Content of the Triangle ATB may be
found, by letting fall a Perpendicular
from the Angle ABT to the Side AT,
and multiplying the Side AT into
half the faid Perpendicular, (or, which
comes to the fame, by multiplying half
the Side AT into the whole Perpendicu-
Jar.) And in like manner may be fonnd
the
6.0
The Young Gentleman's
3.
the Contents of the other feveral Trians
gles BTC, GTD (c.) and confequent-
ly of the whole Ground ABCDEF.
But now forafmuch as the Finding of
A better the Angles and Sides in each Triangle,
Method of
Surveying.
into which any Ground may be divided,
is only in order to protract the Triangles,
and fo to find the Contents of the ſaid
Triangles when protracted; hence there
is found out another and much more ex-
pedite and exact Way for finding the faid
Contents, (without being at the Trouble,
firft to find the Angles and Sides of each
Triangle, and then to protract them
when found, viz.) by finding the Mea-
fure of the Perpendicular and its Bafis in
the Field or Ground it felf. I fhall there-
fore fay no more of the old Method of.
Surveying, (which properly belongs to
Trigonometry) but. fhall forthwith pro-
ceed to ſhew the other Method, as being
much to be preferr'd before the old Me-
thod, for the Reaſons already mention'd.
And this Method I fhall illuftrate, by
fhewing how to furvey thereby the fame
Ground ABCDEF.
ple.
4.
An Exam
Fig. 36.
And this Method is illuftrated Fig. 36.
For being come into the faid Ground,
after viewing it, I perceive it may be con-
veniently diſtinguiſh'd into two Trapezi-
um's ABCD and ADEF. Wherefore ha-
ving
Trigonometry.
61
ving fet up (*) Marks at the feveral An-
gles or Corners of the Ground ABCDEF,
the ſtrait Line AD (fuppos'd to be drawn
between the Angles A and D) will divide
the Ground into the two Trapezium's a-
foremention'd. Then I (†) meaſure with
my Chain from A to C for the Diagonal
AC, whereby the Trapezium ABCD is di-
ftinguiſh'd into two Triangles ABC and
ADC. And (either () as I go along mea-
furing the Diagonal AC, or elſe after I
have meaſur'd it) I find the Perpendicu-
lars PB and PD by the Inftrument for that
Purpoſe, which from its Uſe and Shape
(*) Theſe need be no other than Sticks, with fome
white Thing (as linnen Rags or Paper) faſtened on theis
Tops.
(t) In order to meafure aright the Diagonal AC, (or
any other ftraight Line or Side of your Ground,) with
the Chain, you must take great Care, that you keep in
a ftraight Line (as you go meaſuring) from A to C. In
order whereto, he that carries the End of the Chain,
must take Care that he that carries the Beginning of it, is
always (at leaſt that Hand of his which carries the Be-
ginning of the Chain) directly between him and the
mark C they are meaſuring to, every Time the Chain
is laid down, And in order to prevent any Miftake as
to the Number of Chains, it is advifable for him that
goes firft at the Head of every Chain's Length to thruſt
or faften a Stick into the Ground, to be taken up by
him that carries the End of the Chain.
And fo ac
laft the Number of Sticks will fhew the Number of
whole Chains.
(1) It ſeems most adviſable to measure the whole Dia-
gonal firit, becaufe, hereby there is the lefs Danger of
miſtaking the Length of the Diagonal,
may
62
The Young Gentleman's
may be call'd the (*) Perpendicular-Top.
In like manner I meaſure from A to E for
the Diagonal AE, which diſtinguiſhes the
Trapezium ADEF into two Triangles
ADE and AEF. And then by my Per-
pendicular-Top, I find (one after the o-
ther) the two Perpendiculars PD and PF,
and meaſure them with my Chain. And
thus, the outer Sides or Hedges AB, BC,
CD, DE, EF, FA, being all of them
ſtraight or right Lines, the Work is done
in the Field; all that remains to be done,
being only to multiply half the Perpendi-
culars into their reſpective Diagonals, in
order to know the Area or Content of
each Triangle or Trapezium; and ſo of
the whole Ground: Which may be done
at Home, the Numbers of the Meaſure
(*) For this Inftrument is beft made in the Shape of
a common Top, which Boys ufe to whip. It is beft
made of a dried Piece of Box or Pear-tree, that will
bear three Inches, or three Inches and an half Diameter.
This being turn'd flat on the upper Round, (with a
Neck to fit to the Head of a Staff,) find the Center of
the faid upper Round or circular Surface, and thereon
draw two or three concentrical Circles, (as you fee, Fig.
38,) divide the faid Circles exactly into four equal
Parts, (as in Fig. 38,) and then, with a Whipfaw very
thin, faw by the Marks a, b, c, d, the two Lines ab
and cd at right Angles pretty Deep. Thus, your Inftru-
ment is made; only in order to uſe it, it muſt be ſet
on the Head of a Staff, as has been above intimated. To
which End it is to be made with an Hole at the Bottom
to flip on upon the Head of a Staff.
of
Trigonometry.
63
of the feveral Diagonals and Perpendicu-
Jars being fet down in Writing, as foon
as taken.
5.
Chain,
and the
by Num-
Content of
a Ground
by the faid
Chain.
And here it muſt be fhewn, how the
Numbers, denoting the Number of whole of Gunter's
Chains or of fingle Links, are to be mul-
tiplied in order to find the Content. The Way of
Chain then here fuppos'd to be made ufe Working
of by the Surveyor is that, which from bers, to
its Contriver goes by the Name of Gun- find the
ter's Chain, and which is beft contrived
for Surveying. It contains four Statute-
Poles or Perches, each Perch containing
fifteen Feet and an half, or five Yards and
an half: fo that the whole Chain is fixty-
fix Feet or twenty-two Yards long. The
whole Chain is divided into an hundred
equal Links, fo that twenty-five Links
are juſt a Pole or Perch. And for the
more ready counting the Links, it is uſu-
al to diſtinguiſh every Ten with a Braſs
Plate, (or the like) fo contriv'd as to
fhew, fome way or other, whether it be
the first, or fecond, or third, &c. Length
(i. e. the 10th, or 2cth, or 30th, &c.)
Link from either End; the 50th or Mid-
dle Link having a more remarkable Plate
than the rest.
In meaſuring any Line or Extent by
this Chain, regard need be had only to
whole Chains or whole Links; which are
to be writ down with a feparating Note,
(whe-
64
The Young Gentleman's
:
(whether Full-point or Comma, or any o
ther Mark) between the Chains and
Links, (as is ufual to do between Integers
and Decimals,) viz. 6.35 denotes 6
Chains and 35 Links. If the Links be
under 10, a Cypher must be prefix'd, viz.
7.09 denotes 7 Chains and 9 Links.
;
The Numbers denoting the Chains and
Links being fet down, as has been direct-
ed; to find the Content (of an exact
Square or other rectangular Parallelogram,
you multiply (*) the Length by the
Breadth but it feldom hap'ning that
Grounds are of fuch a Shape, hence it is
generally requifite to diftinguish your
Ground into Trapeziums, as has been a-
fore fhewn, and then to find the Con-
tent) you multiply the Diagonal by (†)
half the Perpendicular, (or (1) the like)
and from the Product cut off five Figures
(accounting Cyphers for fuch,) reckon'd
from the right Hand, with a Daſh of
(*) Agreeably to what has been taught in Defin. 41,
Chap. 1, of the Young Gentleman's Geometry.
(†) Agreeably to what has been taught in the Ufe of
Theorem 7, Chap. I, of the Young Gentleman's Geometry.
() Namely, the Product will be the fame, whether
you multiply the whole Diagonal (or Bafis) by Half the
Perpendicular, or the whole Perpendicular by Half the
Diagonal, or the whole Diagonal by the whole Perpen-
dicular; and take Half the faid Product for the Product
fought.
your
Trigonometry.
65
your Pen: So that the Figures to the left
Hand denote the Acres.
!
If the five Figures cut off to the right
Hand were not all Cyphers, multiply
them by 4; and from the Product cut
off again five Figures towards the right
Hand, and the reft will denote Roods or
Quarters of an Acre.
Laftly, if among the five Figures of the
laft Product, cut off towards the right
Hand, there be any Figures befides Cy-
phers, multiply all the five Figures by
4c. From the Product cut off again five
Figures to the right Hand; and thofe of
the left Hand will denote fquare Perches
or Poles. I ſhall now illuftrate the fore-
going Rules by fome Examples.
Suppoſe then in Fig. 36, the Diagonal
AE to be juſt 7 Chains long, and the Per-
pendicular PF to be 2 Chains and 88
Links, and the other fhort Perpendicular
pD to be 60 Links. I (†) add the two
Perpendiculars together, and the Sum of
them makes 3 Chains and 48 Links.
(†) For this is the most expedite Way, hereby ha
ving fav'd the Trouble of two diftinct Multiplications,
firft of PF by 70%, and then of
pD.
F
1
Then
66
The Young Gentleman's
Then the Half of the faid Sum
of my Perpendiculars,viz. 174,
I multiply by the whole of my
Diagonal (700.
I cut off
the five Figures to the right
Hand, and the remaining 1 de-
notes one Acre.
174
700
121800
Then I multiply the five Figures cut off
by 4, and the Product, (viz. 21800×4=)
87200, not amounting to above five Fi-
gures, I perceive the Trapezium ADEF
do's not contain a full Rood (or Quarter
of an Acre) above the whole Acre found
afore.
87200
40
Therefore I multiply again the fecond
Product 87200 by 40; and
cutting off again the five Fi-
gures to the right Hand in this
third Product, the two remain-
ing Figures 34 denote fo ma-
ny fquare Perches or Poles.
Which wants almoſt fix fquare Perches of
a Rood, or Quarter of an Acre, this con-
taining 40 fquare Perches. For a Sta-
34/88000
tute-Acre contains 160 fquare Statute-
Perches.
(1) I chooſe my Diagonal for the Multiplier, becauſe
it being 700, I need multiply only by the Figure 7,
and to the Product thereof add the two Cyphers.
Upon
Trigonometry.
67
}
Upon the Whole therefore I find, that
the Trapezium ABEF contains one Acre,
and a Quarter of an Acre over, wanting
fix fquare Perches. I proceed then to find
the Content of the other Trapezium
ABCD; whofe Diameter AC fuppoſe
to be 5 Chains and 15 Links; the Perpen-
dicular PB to be 1 Chain and 61 Links ;
and the other Perpendicular PD to be 1
Chain and 89 Links. Where-
}
fore 1 multiply the Diagonal
5.15 by Half the Sum of the
two Perpendiculars, viz. 175;
and the Product not arifing to
above five Figures, I perceive
that the faid Trapezium does
not contain a Whole Acre.
515
175
2575
3605
515
90125
90125
Then I multiply the Product found by
4; and from the ſecond Pro-
duct cutting off five Figures to
the right Hand, there remains
the Figure 3, which denotes the
4
Contents of the Trapezium to 360500
be 3 Roods or three Quarters of
an Acre, and ſomewhat more.
60500
I proceed therefore to find what more
it is, which I do by multiplying the 5 Fi-
gures cut off by 40. From
this third Product I cut off a-
gain the five Figures to the
right Hand, and the remain-
ing 24 denote fo many fquare
F 2
40
24/20000
Perches;
68
The Young Gentleman's
Perches which is a little above Half a
;
Quarter of an Acre.
So that upon the
Whole I find the Trapezium ABCD to
contain 3 Roods and a little (viz. 4 ſquare
Perches) above half a Rood.
Acres Roods Sq. Per.
I . 00. 34
0. 03. 24
In the last Place then, by adding the
Contents of the two Trapeziums ABCD
and ADEF toge-
gether, I find the
whole Ground
ABCDEF to con-
tain one Acre,
three Roods, and
fifty-eight ſquare
Perches: Which
ſquare Perches a-
1.
3. 58, that is,
2. 0.18
mounting to a Rood more, and eighteen
fquare Perches over, therefore the Con-
tent of the faid Ground is in all, (when
exprefs'd after the more proper Manner)
two Acres, and eighteen odd fquare Per-
ches.
After the fame Manner may the Con-
tent of the Ground ABCDEFGHIKL, re-
prefented Fig. 37, be found. Name-
ly, firſt, ſet out in it as large a Trape-
zium ABGL as you can; and find the
Content of the ſaid Trapezium, by mul
tiplying Half the Diagonal, AG into the
Whole Sum of the two Perpendiculars PB
and PL. Then find the Content of the
Triangle BFG by multiplying its Bafis
BG
Trigonometry.
69
BG into Half the Perpendicular PF. Then
find the Content of the Nook CDE by
diſtinguiſhing it into as many Trian-
gles (*) as you fee requifite, viz. CDc,
Ded, and dcE; and finding the Content
of each Triangle, as you did of the Tri-
angle BFG.
And in like manner is the
Content of the other two Nooks GHI and
IKL to be found. And then at laſt, the
feveral Contents of the feveral Parts of
the faid Ground being added all together,
will give the Content of the Whole
Ground. And after this Method may a-
ny Piece of Land be ſurvey'd, the feveral
Cafes that can happen in meaſuring the
Sides or Hedges of any Ground being re-
preſented in Fig. 37.
6.
or Setting
Hitherto it has been fhewn, how to
furvey or meaſure a Piece of Land already of Laying
Set or laid out within its reſpective out a Piece
Bounds. It may be of uſe to obſerve of Land
here next, how to fet or lay out a Piece requir'd.
of Land or Ground of fuch a Meaſure,
as fhall be requir'd, or containing ſuch or
fuch a Number of Acres, &c.
(*) Namely, by this Method circular Nooks may be
meaſur'd as exactly as by any other; at least fo exactly, as
that what is wanting of Exactnefs, fhall be fo little as
to be altogether inconfiderable. But it must be noted,
the more Triangles fuch circular Nooks or Windings are
divided into, the nearer you'll come to the exact Meaſure
of it.
F 3
In
70
The Young Gentleman's
7.
Acre,
what.
In order hereto, it is to be known in
A Statute the first Place, that an Acre according
to Statute-Law does contain one hundred
and fixty fquare Statute-Poles or Perches ;
each fuch Pole or Perch containing (as
has been above obſerv'd) fifteen Feet and
an Half, or five Yards and an Half.
Such being the Content of an Acre, it
follows, that all rectangular Pieces of
Ground,any of whoſe contiguous Sides are
of any of the
two Lengths fet
oppofite here-
to, will contain
juſt an Acre.
Whence it is
obvious, how a
Ground of a fin-
Perches
80 *
40 ×
32 *
2
4/ Square Perches
5=160=1 Acre:
8
20 x
16 × 10
&c.
gle Acre may be laid out, if there be
Room to make its contiguous Sides per-
pendicular one to the other, viz. by ma-.
king the two contiguous Sides of fome of
the two Lengths ſet together above,
Hence alſo it is obvious, that Grounds
which have very different Sides, may yet
have the very fame Content.
If there be not Conveniency to lay out
the Acre of Ground in a rectangular
Shape, then it may be laid out it in any
other Shape or Figure, by confidering
what Proportions are to be taken, to
make the Content of the faid Figure juft
equal
Trigonometry.
7 1
equal to 160 equal Perches. For In-
ftance; if I can lay out the Acre in
a Triangular Bafis Perpend.
Shape; then
the Baſe and
Perpendicular
of the faid
Triangle muſt
have ſome of
oppofite
the
160 & 2
160 × I
80 × 2
80 & 4
64 &
5 for
64 ×21=160
40 &
8
40 × 4
32 & 10
32 × 5
&c.
(&c.)
Perpend. Bafis
(or fuch like) Proportions one to the o-
ther; forafmuch as the Bafis multiplied
into Half the Perpendicular (or vice ver
fa) gives the Content of a Triangle.
The Way, how to lay out an Acre of
Ground in a triangular Shape, being
known, thereby may be known the Way
to lay out an Acre of Ground in any múl-
tangular Figure; forafmuch as all Poly-
gons may be conftituted by adjoining to-
gether feveral Triangles. And it being
thus known how to lay out a fingle Acre
of Ground in any Shape or Figure;
thence may be known how to lay out
Grounds of more Acres than one in any
Shape, by proportionably encreafing the
Lengths of the reſpective Parts. And thus
I have taken notice of ſo much of Sur-
veying, as feems requifite to the Defign
of this Treatife.
F A
CHAP.
The Young Gentleman's
I.
A Fort,
what.
2.
B
CHA P. VII.
Of Fortification.
Ya Fort or Fortress is fignify'd a
Place fortify'd, i. e. made ftrong by
the Art of Military Architecture, or For-
tification.
The principal Part of a Fort, is that
A Ram- which is called the Rampart or Rampire,
part,what. being a Wall of a good Height and
Breadth, compos'd chiefly of Earth, and
encompaffing the Place fortify'd. It is re-
prefented Fig. 39, by DEWXYZ.
3.
The Bulwark or Bastion of a Rampart
Bulwark is that Part of it, which runs farther out
Baſtion, than the reft, ending in an Angle. It is
reprefented Fig. 39. by IFEGH, &c.
bat.
4.
·oulder,
on, what.
The feveral Parts of a Baftion are its
be Face, Face or Front FE; its Gorge (or Neck)
:orge, IH; its Shoulder F; and its Flank (or
d Flank Wing) FI; which, according to the new
oj a Baſti. Method of the French, is not built in a
ftrait Line, but rounding, as is repreſent-
ed in fome of the other Baftions, by the
Prick'd-circular Lines about the Flank.
The Part of the Rampart between two
A Curtain, Baſtions is called the Curtain KI, which
xbat.
joyns together the two Baftions IFEGH
5.
and KLDMN.
+
1
More-
{
Trigonometry.
73
Moreover, forafmuch as (according
6.
outer and
what:
to the Rules of Fortification) every Ram- The Side
part is either a (*) Polygon, or at leaſt of the
a Tetragon; hence the Line DE is cal- inner Po-
led the Side of the outer Polygon; BC the lygon, &c.
Side of the inner Polygon; AC the Semi-
diameter of the inner Polygon; AE the
Semidiameter of the outer Polygon; pA
the Cathetus of the inner Polygon; PA
the Cathetus of the outer Polygon; Pp the
Distance of the Polygons.
Befides which there are wont to be 7.
peculiarly diſtinguiſh'd alſo theſe feveral TheGorge-
Lines, viz. the Gorge-line IC; the Head- Line,
Line CE; the (†) Longest Line of De- Line, &c.
fence KE, the Shortest Line of Defence what.
kE.
Head-
8.
Angles in
By the foremention'd Lines and Parts
of the Rampire are constituted feveral of the
Angles, whereof the Chief are theſe ; viz. principal
BAC the Angle of the Center; BCA or Fortifica
CBA the Angle of the Polygon; FEG the tion.
Angle of the Baftion, otherwife called the
Diamond Point, and the Flanked Angle;
DOE the outer Flanking Angle; Fkl the
(*) The Reaſon of this Rule of Fortification depends
upon the due Proportion there ought to be between the
feveral Parts of the Rampart. And hence, according to
the faid Rules, a regular Polygon is to be preierr'd before
an Irregular. The Rampart, Fig. 39, is a regular Hexa-
gon.
(†) See Set. 30, 31, 32, of this Chapter.
1
inner
74
The Young Gentleman's
9.
inner Flanking Angle; FCE the Front-
angle, or Angle forming the Face; FCI
the Flank-angle, or Angle forming the
Flank; CEF the Head-line Angle, &c.
It is alfo obvious, that by the foremen-
How For- tion'd Lines are made feveral plain Trian-
belongs to gles, viz. ABC, ADE, APD, APE, ApB,
Trigono. ApC, pFl, kFI, FIC, FCE, &c. Hence
tification
metry.
10.
any three Parts of each of the faid Trian-
gles (except the (1) three Angles) being
known, the other three unknown may be
found by Trigonometry, as has been a-
fore fhewn. Such as would have parti-
cular Illuſtrations by Examples, may con-
fult fuch Treatifes as are profeffedly writ-
ten of Fortification. Thofe young Gen-
tlemen, for whoſe Uſe this Treatife is e-
fpecially defign'd, being fuch as are not
likely to concern themſelves in the practi-
cal Part of Fortification, it feems abun-
dantly fufficient to my Defign, to have
thus fhewn in general how Fortification
depends upon Trigonometry.
However being thus led, by (the Sub-
The Expli- je&t of this Treatife) Trigonometry, to
cation of take notice of the foremention'd Parts
Terms of and Terms of Fortification, becauſe it
Fortifica- may be of good Ufe, in order to make
other
tion.
(1) The Reafon of this Exception is largely given, Chap.
2. §. 1, of this Trigonometrical Treatife.
young
Trigonometry.
75
young Gentlemen underſtand, in a com-
petent meaſure, Relations given of Sieges,
I fhall therefore here add the Explication
of fuch other Parts and Terms of Fortifi-
cation, as are of principal Note and Ufe,
and ſo moſt frequently occur in the Ac-
counts of Sieges.
II.
A Parapet then is (properly) a Work
rais'd on the Rampart, behind which the A Parapet,
Soldiers ftand, and where the Cannon is what.
plac'd for the Defence of the Fort. As in
Fig. 51 and 52, ab denotes the Breadth
of the Rampart at its Bafis or Bottom, fo
de denotes the Breadth of the Parapet at
its Bottom, or where it is rais'd on the
Rampart.
I 2.
A Banquet is no other than a Foot-
pace at the Bottom of a Parapet, upon A Ban-
which the Soldiers ftand up to fire into quet,what.
the Ditch, or upon the Covert-way be-
yond the Ditch. Some Parapets have on-
ly one Foot-pace, as is reprefented by
de, Fig. 51. But it is the more commo-
dious to have two, one for the taller Sol-
diers, and another (fomewhat above the
former) for the fhorter, as is reprefented
between d and e, Fig. 52.
13.
According to the old Way of Fortifica-
tion, there is erected between the Ram- 4 Fauffe-
parts and the Ditch, a Work called the bray,
Fauffe-braye, or Falf-brage, being a fmal-
ler Sort of a Rampart, having a Parapet,
and
whats
76
The Young Gentleman's
14.
The Co-
vert-way,
what.
15.
The Gla-
and running round the whole main Ram-
part of the Fort. It is repreſented Fig.
51, by xyz.
But the French omit this
Work now-a-days in their Fortification,
leaving between the Rampart and the
Ditch only a Space about three or four
Foot wide, to receive the Earth which
falls from the Parapet. This Space
they call the Berme.
See Fig. 52.
Wherein, as alfo Fig. 51, klmn denote
the Ditch.
On the outer-fide of the Ditch is the
Covert-way, being a Way or Walk (round
the out-fide of the Ditch) cover'd or de-
fended by the Glacis. It is reprefented
Fig. 51 and 52, by np.
The Glacis ferves as a Parapet to the
cis, what. Covert-way, and lofes it felf infenfibly
in the Field. Hence the Largeſt are e-
ſteem'd the Beft, becauſe they lofe them-
felves more infenfibly than fhorter ones.
It is denoted Fig. 51 and 52, by prs.
16.
The Counterfcarp is properly the Slope
The Coun- nm of the Ditch on its Out fide, as being
terfcarp, oppofite to the Scarp or Slope kl of the
what.
17.
Ditch on its Infide. But by this Term
now-a days is generally understood the
Covert-way with its Parapet.
Befides the Works hitherto ſpoken of,
of the ad- and which are requifite to all Fortificati-
Works of ons, there are alfo other additional Works
Fortificati- contrived by Maſters of Military Architec-
ditional
n.
ture.
Trigonometry.
77
ture,for to further fortify the feveral Parts
of Fortification already defcrib'd. And
fuch are Half-Moons, Horn-works, Crown-
works, Star-works, Ravelins, Tenailles, &c.
Of which the four former take their
Names from their Shape. They will all
be better apprehended by their refpective
Draughts, than by verbal Deſcriptions.
18.
what.
I begin with the Tenaille; of which
there are three Sorts, reprefented Fig. 40, ATenaille,
41, and 42. They are placed in the
Ditch, and before the Curtains of the
Rampart. They are faid to be contriv'd
by the French inſtead of the Fauffe-braye.
The Tenaille, Fig. 41, is diftinguifh'd
by the Name of the fingle Tenaille; and
that Fig. 42, by the Name of the double
Tenaille.
19.
what.
A Ravelin is reprefented Fig. 43; be-
ing a ſmall triangular Work. It is gene- A Ravelin,
rally rais'd alſo before fome Curtain, ei-
ther of the main Rampart, as Fig. 43,
and 46, or of fome Out-work, as the
Ravelin, Fig. 44. before the Curtain of
the Horn-work, Fig. 47, or elſe before
ſome other Part analagous to a Curtain,
as the Ravelin, Fig. 45, before the fingle
Tenaille, Fig. 41.
20.
Moon,
An Half-Moon (Fig. 53.) is generally
confounded with a Ravelin, its Diffe- An Half-
rence being principally (if not only) what'
this, that its Gorge df is an Arch in Shape
of
78
The Young Gentleman's
21.
work,
what.
of the Concave Part of an Half-Moon,
and it is placed overagainst Z, the Point
of a Baſtion to cover it; whereas a Rave-
lin confiſts of two ſtraight Demigorges,
as de ef, Fig. 46, and is placed overagainſt
fome Curtain, as is afore obferv'd.
An Horn-work is reprefented Fig. 47.
An Horn. The Fore-part of it confifts of a Curtain
between two Half Baftions, which run out
like Horns, whence it takes its Denomi-
nation. It is placed either before a Cur-
tain or a Baſtion.
22.
work,
what.
A Crown-work is reprefented Fig. 48. It
A Crown is fo called, becauſe the Fore-part of it
reſembles a Crown. It is ufually made
before an Horn-work, to keep the Ene-
my at a Diſtance. It is alfo placed imme-
diately before a Curtain or Baſtion of the
Rampart.
23.
A Star-
work,
what.
24.
A Re-
doubt,
what.
Out-
25.
works,
what.
A Star is a Work with feveral Fa-
ces, which flank one the other, as Fig.
49.
A Redoubt is a ſmall fquare Fort, or a
Guard-houſe, having no Defence but in
Front. It is ufually defigned for main-
taining the Lines of Circumvallation, or
Countervallation. In watery Places they
are often made of Mafon's Work, It is
repreſented Fig. 50.
All the foremention'd Works, which
lie without the Rampart of the Forts, are
in general call'd the Out-works,
A
Trigonometry.
79
26.
A Battery is a Place rais'd whereon to
plant the great Cannon, in order to bat- A Battery,
ter or play upon the Place befieg'd.
what.
what.
Cavaliers are Works rais'd on the Para- 27.
pet of a Rampart, whereon to plant Can- Cavalliers,
non in order to fcour the Field, and to
oppoſe any Work rais'd by the Befiegers,
which commands the Place befieg'd.
Fafcins are a Sort of Faggots, made to 28.
throw into the Ditch, where there is Fafcins,
much Water, in order to make the Paf- what.
fage over to the Wall or Rampart of the
Place befieg'd, more eafie.
29.
Gabions are Baskets filled with Earth,
and uſually placed upon Batteries, and Gabions,
Parapets, that have fuffer'd very much, what.
and alſo before other Places, to ſecure
them from the Enemies Shot. Leffer Ga-
bions are call'd Corbeilles.
30.
A Gallery is a made Way or Walk, co-
ver'd with Earth or Turf, and upheld A Galery,
with Planks. They are made in the Ditch what.
already fill'd with Fafcins, to the end
the Miners may approach fafe to the Ba-
ftion.
The Line of Defence is that, which is
31.
reprefented by the Diſcharge of the fmall Line of
Shot towards the Face of a Baftion. It Defence,
is diftinguifh'd into the Longest and Short-
est, otherwife called Fichant and Ra-
Sant.
what.
The
80
The Toung Gentleman's
32.
Line of
Defence
Fichant,
what.
33.
Defence
Razant,
what.
The Line of Defence Fichant (or Long-
est Line of Defence) is the Line KE, fo
called, becauſe the Bullets or Shot, which
paſs along this Line, may fix on the Face
of the oppofite Baſtion.
The Line of Defence Rafant (or the
Line of Shortest Line of Defence) is the Line kE,
drawn from the Face of the Baſtion, to
that Point k, of the adjoining Cur-
tain, where the Face of the faid Baſtion
begins to be diſcovered. It is fo call'd,
becauſe the Shot, that paffes along the
faid Line, only rafes the Face of the ſaid
Baſtion.
34.
A Lodg-
ment,
what.
35.
A Lodgment is a Work caft up in a
dangerous Poft won by Attack, to ſecure
the Befiegers againſt the Enemies Fire. It
is made of any Materials, that are capable
of making Reſiſtance.
To nail up a Cannon, is to drive a
To Nail up Nail (or the like) into its Touch-
a Cannon, hole.
what.
36.
To Sap,
what.
To Sap, is to dig deep into the Earth of
the Covert-way and Glacis in the Form
of a Trench. The Earth dug out ſerves
for a Security on the Right and Left, and
covers above, the Soldiers (or others) in
the Trench againſt the Enemies Fire-
works, by the help of Hurdles laden with
the faid Earth.
Military
Trigonometry.
8:
what.
Military Trenches are of the fame Kind 37.
with common Trenches, only larger, be- Trenches,
ing uſually from fix to feven Foot in
Depth, and from eight to ten Foot in
Breadth. They are made by the Befieg-
ers, to approach more fecurely the Place
befieged whence they are otherwife cal-
;
led the Lines of Approaches, or fometimes
the Approaches. And they are therefore
fo carried on, as not to be in View of
the Enemy, or preſently and eafily difco-
verable by the Enemy or Befieg'd.
Circum-
vallation,
Lines of Circumvallation are Trenches 38.
made by the Befiegers about their Camp: Lines of
as well to hinder the Relief of the Be-
fieged, as to ftop Deferters from their what.
own Army. They are guarded with a Pa-
rapet.
Lines of Countervallation are Trenches
39.
which the Befiegers make to fecure then- Lines of
felves from the Sallies of the Befieg'd. Counter-
They are alfo guarded with a Para- what.
pet.
vallation,
Lines of Communication are Trenches 40.
made from one Work to another by the Lines of
Befiegers, for the better holding Commu- Commu-
nication one with the other, and helping what.
one the other, in cafe of an Attack by
the Enemy, in any of their Works.
nication,
41.
A Retrenchment is a Work made by the
Befieg'd to fecure themſelves, and fo to 4 Re-
enable them to hold out fome time lon- trench-
C
menc,
ger what.
$ 2
The Young Gentleman's
ger againſt the Enemy, who is fo far ad-
vanc'd as to be lodg'd on the Rampart.
It has either a good Parapet, or inſtead
thereof Gabions. And thus I have ex-
plain'd the more ufual Terms of Fortifi-
cation, and fuch as more frequently oc-
cur in Relations of Sieges.
I.
The Me-
adjoin'd to
tife.
CHA P. VIII.
Of the Tables of Logarithms, and
Artificial Sines and Tangents.
T
HE Tables of Artificial Sines and
Tangents belonging properly to
i hod of the Trigonometry, they are therefore here
artificial adjoin'd to the End of this Treatiſe.
Sines and They are ſo contriv'd, by placing the
Tangents, Sines and Cofines by the Side one of the
this Trea- other, and likewife the Tangents and Co-
tangents, that each Page fhews the Sine
and Cofine of two Angles, one whereof
is fet down on the Top of the Pages,
the other (being the Complement of the
former to 99) at the Bottom of the
Page. Thus the Page (or Pages,) that
hath Degree 2 at the
gree 87 at the Bottom.
and Cofines, Tangents
Top, hath De-
And as the Sines
and Cortangents
belonging
Trigonometry.
83
belonging to the Degrees fpecified at
the Top, are reſpectively fhewn by the
refpective Words fet at the Top of the
feveral Columns in each Page; fo the
Sines and Colines, &c. belonging to the
Degrees ſpecified at the Bottom, are re-
ſpectively fhewn by the refpective Words
fet at the Bottom of the feveral Columns
in each Page.
2.
The Minutes between each two De-
grees are fet down on the Sides of the
Pages, thoſe appertaining to the Degrees
at the Top increafing downwards ; the
other appertaining to the Degrees at the
Bottom increaſing upwards. And on the
Side of each Minute ſtands its reſpective
Sine and Cofine, Tangent or Cotangent.
And, although Logarithms are of Ufe,
in other Parts of Mathematicks as well The Table
as in Trigonometry, yet becauſe they rithms,
of Loga-
muſt always be uſed with Artificial Sines why ad-
and Tangents; I have therefore judg'd join alf
this the most proper Place alfo for the Treatiſe.
Table of Logarithms belonging to my
Mathematical Treatifes: of which I need
fay no more here, the Ufe of the ſaid Ta-
ble of Logarithms having been already
explain'd, in the Appendix to the Young
Gentleman's Arithmetick.
to this
G 2
A TA-
A
TABLE
OF
Logarithm's,
From One to Ten Thoufand
་
N Log. N Log. |N| Log.
110.000000 13411.531479 671.826075
20.301030 351.544068 681.832509
30.477121 361.556303 691.838849
40.602060 371.568202 70,1.845098
50.69897381.579783 711.851258
60.778151 391.591064 1721.857332
70.84509814c1.602060 73 1.863323
80.903090||41|1.612784 74|1.869232.
90.954242 42 1.623249 751.875061
101 000000431.633468 761.880813
111.041393 44 1.643452 77 1.886491
121.079181|45|1.6532-1-2) 1781.892094
131.113943 161.662758 179 1.897627
141.146128 471.672098801.903090
151.176091 48 1.681 241 811.908485
161.204120 19 1.690196 821.913814
171.230449 50 1.698970 831.919078
181.255272511.707579 841.924279
191.278753 521.71600385|1.929419.
201.301030 531.724276 86 1.934498
211.322219 541.732394 87 1.939519
221.342422551.740362 881.944482
231.361728 561.748188 89|1.949390
241.380211 571.7558751 1901.954242
251.397940 1582.763428) 1911.9590417
261.414973 591.770852 92 1.963788
271.431364 601.778151931.968483
281.447158611.785330 941.973128
291.462398621.792391 951.977723
301.477121 6311.799240 1961.982271
311.491361| |64.1.806180| 1971.986772
321.505150 651.812913 981.991226
331.518514 661.819544] 1991.995635
!
The Table of Logarithm's.
N|| 。 |
이 ​1 2 | 3
34
100/1000000000434000868001301001734
101004321004751005181005609006038
102 008600009026009451009876010299
103 012837013259013679014100014521
10401703301745017868 018284018700
105|021189 021603022016022428022841
106 025306025715026125026533026942
107029384029789030195030599031004
108 033424033826034227034628035029
1090374261037825038223038620039017
110041393041787042182042576042969
111045323045714046105546495046885
112049218049606049993050379050766
113053078053463053846054229054613]
114056905051286057666058046058426
115||060698061075061452061829062206
116064458064832065 206065579065953
117068186058557068928069298069668
118071882072249072617072985073352
1190755470759120762760766401077004
1200791811079543079904080266080526
121 082785083144083503083861084219
122086359086716087071087426087781)
123089905090258090610090963091315
12409342209377209412209447109482c
125||0969100972570976040979511098298
126 100371 100715 101059101403101747
127 103804104146 104487104828 105169
128 107209107549107888 108227 108565
129 110589110926 111263111599111934
The Table of Logarithmes.,
1
5 | 6 | 7 | 8 | 9 |D
002166002598003029003461003891 | 432
006466006894007321,0077480c8174
010724011147011570011993012415 424
014940015359015779016197016516419
428
01911601953201 99470203610207751416
0232520236640240750244861024896||412
027349227757028164028571028978408
031408031812232216232619033021||404
035429035829036229036629037028400
03941403981104020406020409981496
04336210437550441480445390449321393
047275047664048053048442048830389
05115305153805192405230905 2694 386
054996055378055760056142056524 382
058805059185059563059942c60320379
06258206295806 3 3 3 30 637091064083|376
066326 066699 067071067443067815372
070039070407070776 071145071514 369
366
073718074085074451074816075182
077368 077731078094|078457078819363
108 0987 081347 | 281707|082067082426||360
1080987
084576084934285291085647086004||357
688136c88490088845 089198 009552 355
091667092018 092369092721 092071 251
095169095518095866 096215096562349
098644098989099335099681|100026||346
102091,102434 102777103119103462 342)
105510 105851106191 106531106871 340
108903 109241 109579 109916110253 338
112269112605112939 113275113609335
H
The Table of Logarit hmes.
N|| 0 | 1 | 2 | 3 | 4
130|113943114277114611|114944|115278
131 17271117603117934118265 118595
132120574120903121231121559121888
123 1238521241781 24504124830125156
134 127105127429127753128076128399
1 35||
130334|130655|130977|131298|131619
136 133539133858134177134496 134814
137 136721137037137354137671 137987
138 1398791401941 40508 140822141136
139143015143327143139143951144263
1
140||146128 146438|146748|147058147367
141 149219 149527 149835150142150449
142 152288152594 152899153205153509
143155336 155639155943 156246156549
144158362158664158965159266 159567
145||161368161667|161967|162266|162564
146 164353164650164947165244165541
147 167317167613167908 168202168497
48 170262170555170848 171141171434
149173186173478173769174059174351
150||175091|176381|176669|176959177248
151|17897717926479552179829|180126
1
152181844182129 82415182699182985
153 184691 18497518525985542185825
154||187521|187802|188084|188316|18864.
17
55||190332,190612|190892|191171191451
156||193125|193403|193681|193959194237
157 195899 196176196453967291 97005.
158 198657198932199206 99481199755
158 201297 201670201942 202216202488
The Table of Logarithmes.
|
5 | |
6 | 7 | 8 | 9 || D
1156 11 || 1 5943|116276|116608116939 333
118926 119256 119586119915120245 330
122216122544122871123198 123525 328
325
12548125806 12613112645612678
128722/129045 12926812968930012 323
1319391322591132579132899133219||321
135133135451135769 1360861 36403318
138303 138618138934139249139564 315
141449141763142076 142389142702314
144574144885|145196|145507145818|3 I I
147676 14798511482941148603148911309
150756157154151369151676151982 307
153815154119154423 154728155032 305
156852157154 157457 157759158061 303
159868 16016160469160769161068||301
162862163161|163459|163758164055 259
65838166134 166430 166726167022 297
168792169086|169380169674169968 295
171726 172019 172311 172603 172895 293
174641 174932175222 175512175802 291
177536 177825 178113,178401 178689||289
180413180699180986 181272181558 287
183269183555183839 184122 184407 285
186108186674 186674186956187239 283
188928189209189490 189771190051281
191739|192009|192289|192567|192846|| 279
194514 194792195069195346195623278
197281 197556197832198107198382 276
200029 200303 200577 200850201124 274
202761 203033 203303120357712038481272
H 2
The Table of Logarithmes.
|N|| 0 | | 2 | 3 | 4
t
1601204119/204391 204663 204924 205 204
161 206826207096 207365 207364207904
162 209515209782210051 210319210586
163 212187212454 212720212986 213252
164 214844215109 215373215638/215902
16521748421774721801c21827 2 | 218536
165 22010822036922063 220892 221153
167222716222976223236223496 223755
16225209 225568 225826226084 226342
16522788722814222840c 228657 | 2 2 8 9 1 3
170 | | 2 3 0 4 4 9 1 2 3 0704 230959 231 21 5 2 3 1 469
171 232996233250 233504233757234011
172 235528 235781 236033236285 239537
173 238046238297238548238799239049
174240549240799241048241297 241546
175|| 243038/243286 243534 243782244027
176 245513245759 246C06246252246499
177 247973 248219248464248709 248954
178 250420 250664250908251151 251395
179125285325309512533342535S253822
180/2552731255514|255755 255996 256237
181257679'257918258158 258398258637
182 260071 260309 260548 260787261025
182 262451 262688 262925 263162263399
184264818 265054265289 265525265761
1851267172 267406267641 267875/268109
186 269513269746269979270213270446
187 271842272074 272206272538 272769
188 27415274389 274619274850275081
184 27462276692 276921277151277372
The Table of Logarithmes.
5 | 6 | 7 | 8 | 9 ||D
2054751205746 206016 206286 206556||27|
208173 208441 208710,208978 209247 269
210853 211121 211388 211654211921 267
213518 213783 214049 214314214579 266
216166 216429 216694 216957217221| |264
218798 212060 219323 219585 219846 267
221414 221675 221936 222196 222456 [261
224015224274 224533|224791 225051259
226599 226858 227115 227372 227629 258
229169 229426 229682 229939230193 256
231724 231979 232234232483 232746 254
234264 2345 17|234770|235023 235276||253
236789 237041237202 237544 237795252
239299239549239799 240049 240299 250
241795|242044|242293|242441 242789249
244277 244525 24+772 245019245266248
246745 246991247237 247482,247728 246
248198 249443 249687 249932250176245
351638 251881 2521 25/252268|252610243
454064/254306 25+548/254789 255031| |242
256477 2356718 256958|257198 257438 241
258877 259116 259355 25959+,259833, 239
261 263 261501 261739 261976 262214
263636 263873|264109|264:46'264582||237
255996 256232|266467|2667c2|266937||235
238
268341268578268812 269046 269279 234
|270679 270912271144 271377 271609 233
273001273233 273464 273696 273927 232
275311275542 275772 276002 276232 230
277609277838|278067|278296278525||229|
The Table of Logarithmes.
1
I
N|| 0 | 1 | 2 | 3
| 4
190278754278982 2792111279439279667
191 281033 281 261 281488 281714 281942
192 283301 283527 283753 283976284205
193 285557 285782286007286232 286456
194287802288026288249 288 4 7 3 2 8 8 6 9 6
195||290035|290257|290479|290702|290925
196 292256 292478 292695 292920|293141
197394466294687 294907 295127295347
198 296665 296884 297104 297323 2975421
199 298853299071 2.99289|2995071299725
2001 3010293012473014641301581301898
201 303196303412303628|303844304059)
202 305351305566|30578 305996 306211
203307496 307709307924308137 308351
204|1309630309843|310056|3 10268 310421
205|||3117541311956;3 1 2177|31 238931260c
206 313867314078314289314499314709
207 3 159703161803 163893.16 599 316809
208|| 318063318270318481|318689318898|
209 3201463 203543 2056 23 20769320977
وم
2 1 0 || 3 2 2 2 1 9 3 2 2 4 26,32 26 3 3 1 3 2 2 835 | 323046
211 3242823244883246943 24899|325105
212 326336326541 3267453 26945 327155
213 328379328583 328787328991329194
213330414330617330819331022331225
21133243833264033284213330441333246
21
216 334454334655 334856335057 335257
336459336659336856337059337259
218 338456338656338856339054 339253
219340444 340642 340841341039 341237
The Table of Logarithmes.
-
r
5 | 6 | 7 | 8 | 9 | D
1 1
279895280123|÷80351| 28057828089 ‹ || 228
282169282396282622 282849 233075 227
284431 284656284882285107 285332||226
286681 286905287125287354 28757225
288919/289143 289366| 289585|289812|| 223
2911471291369|291591/291813|292034||222
293362 293584293804 294025 294246221
295567 295787|296007|296226 296446|| 220
297761 297979298198298416298635||219
299943 3001613303780000595/300813218
3021141302331302547302764302979 217
304275304491304706304921305136 216
306425 306639306854307068307282 215
308564308778308991 309204309417313
310693310906311118311329311542212
312812313023313234313445313656|| 211
3149103151303 15349 315551315760 21C
3170183172273 17436317646 317854 209
3191063193143195223197203 1993 208
321184321391321598321805 322012||207
32325232345 3 23665 3 2 38711324077206
325310325516325721325926326131205
327359 327563327767527972328176204
329398 329601329805330008330211 203
331427 331629|331832/33 2034332256202
333447 333649333850 334051334253||202
335458335658335859336059336259 201
3374593376593785833805338257 200
339451 3396503984934004740246 199
341425341631|341830342028|342225||198
1
The Table of Logarithmes.
N|| 0 | 1
| I
1
| 2
| 2 | 3 | 4
220134242234262C
4281713430143212)
221
223
344392344589|344785344981345178
222 346353346549346744346939 347135
34830524849534869434888249082
224350258350442350636350829351023)
2 2 5 | | 3 5 2 1 8 3 3 5 2 3 7 5 1 3 5 2 5 6 8 1 3 5 2761352954
226354108 354301 354493354685354876
227 35602635621735640835659935679c
228 357935 358125|358316358506 358696
229||359835360025360215360404360593
239136 17 28 3619173621051362294362482
231363612 363799363988364176 364361
232 36548 365645 365862 366049366236
233 367356 3675423677293679153 68
234|| 369216369401|369587|369772369958
ICI
235371068371253, 271437 371622371806
236 372912373094373279373464373647
237 376577374932375115375298 375481
238 376577 3767593769423771 24377336
39|378358 378579378761378942379124
240 13802113803921380573 380754380934
241 282017 382197382377382557 382737
242 383815383995 28417438435 3 3 8 4 5 3 3
243 385606385785 385964 386142386321
2443873891387568287746|38792338810
245 38916613893431389529138 9 6 9 8 3 89 875
246 39093539111239128839146439164
247 392697392873393048393224 393399
248 394452394627394802394977395152
249 396199'396374 396548396722 196896
'
!
}
The Table of Logarithmes.
'
5 | 6 | 7 | 8 | 9 || Dj
||DI
343409343606343802343999344196||197
345373 345569345766 345962 346157 196
347330 347525 347720 347915 348110|| 195
349278349472349659 349860 350054 194
351216 351409|351603351796 351989 193
353147 353339 353532353724353916193
355068355239355452355642355834 192
356981 357172357363357554357744 191
358886359076359266359456 35964 190
360783 360972361161 361350361539|189
362671 362859363048363236|363424||188
364551 364739 364926 365113365301 188
366423 366609366796366982 367169 187
368287358473368659368845 369030 186
370143 3703281370513370698370882||185
371991 372175372359372544372728||184
373831374015374198 37438: 374565 184
375664375846 376029 376212376394 183
377488 377670377852 378034378216 182
379306379487379668379849 380030181
3 8 11 15 3 8 12 9 6381476|381656 381837|| 181
382917383097383277383456 3 8 36 3
384712 384891385c69385249385428 179
3 86 199 386677386856387034387212 178
388279388456388634388811388989 178
180
390051390228390405 390582390759177
391817391993392169392345 392521 179
393575393751393926394101 394177 17t
395326395501395676 395850 96025 175
1297071297245397419397592397766174
(
The Table of Logarithmes.
Ꮓ
| | 0
N|| 0 | I
250
| 2 | 3 | 4
397940398114[398287 398461398634
251 399674399847 400019400192400365
252 401401401573 401745 401917402089
253 403121403292403464403635403807
254 404834405005 405176405346405517
25511406540406710406881407051407221
256 408239408409 408579408749408918
257 409933410102 410271410439410609
258 +11619411788 411956412124412293
259413299413467413635413803413969
26C1414973/4151404153071415474415641
261416641416807416973 417139417306
262 418301 418467418933418798418964
253 419956420121 420286 420451420616
264 421604421768421933 422097422251
26511423246,423 409142357414237371423901
266 424882 425045425208425371 425534
267426411426674426836426999 427161
268428135428297428499428621 428783
265429752429914430075 430236430398
270 | | 431364431525|431685,431846432007
271 432969433129433289433449433609
2721434569434729434888 435048435207
273 436163436322 436481 436639 436799
274437751437909438067438226438384
275 439333 4394911396484398061439964
276 440909441066 441224 441381441538
277 442479442637 442793442949 443106
278 444045 444201 444357 444513444669
279 1445604/445759445915446071446226
The Table of Logarithmes.
1
5 | 6 | 7 | 8 | 9 || D
39880398981399154399328399501 || 173
400538 40071 1400883401056401228 173
402261 402433402605402777402949 172
403978 10414940432040449 404662 171
405688 405 858 406029|406199405869|| 171
107391 407561407731407901408079|| 170
409087409257 409426 409595409764169
410777 410946 411114411283411451
169
412461412629412796412964413132 168
414137414405414472414639414806||167
415808415974416141416308416474 167
417472417638417804417969418135||166
419129419295419460 419625419791 165
420781 420945421110421275 421439 165
422426422589422754422918423082164
4240651424228424392 424555|424718||164
+25697425860426023126186426349 163
162
427324427486427548 +27811427973
428944 429106429265429429 129591 162
4305594307191430881431042431203161
+32167432328 432488;432649432809 || 161
+33769 +33929434089434249434409160
+35366 +35526 +35685435844 +36004159
+36957 +371 16 37275437433437592 159
+38542 +38701438859439017439175|148|
440122
440279440437440594440752 | | 158
441695441852442009 442166442323157
443263 443419443576 443732 443889157
444825 444981445137445293445449156
| 4 4 6 38 2 4 4 6 5 37 416692 416848447003||155
I 2
The Table of Logarithmes.
No 1 | 2 | 3 | 4
1
2801 +47158447313447468|4476231447778
281 448705448851449015449169449324
282 450249450403450557450711450865
283451786 451939 +52093452247452399
284 453318453471453624453777454929
285454845 45+997455149455302455454
286 +56366 4 5 6 5 1 8 4 5 6 6 6 9 4568 1456973
287457889458033 453184458336458487
288459392459543459694459845459995
2891460898461048,461196461348 461499
2901462398462548462697462847462997
291 +63893464042 +64191464340 464489
292 +65383465532 +6568046829465977
293 +66868 46701646716446731246746ċ
| 294|1458347 1468495+68643468790 468938
295469822469969470116 +70263 +70410
296471292471438471585 +71732471878
297 472756472903473049 +73195 473341
298 +74216474362474508 +74653 +74799
299 +75671475816475962 +76107 +76252
300 +77121+77266477411477555 477699
201 +78566 +78711478855478999479123
302 +80007180151480294480438480582
202481443481586481729 481872482016
304482874 +83016483159483302483445
305||484299484442,484585|484727/484869
306485721485853486005485147486289
307 48713848727948742 487564487704
208488551488592488832 +88974 +89114
209 489958490099'490229 490279 1905 20
›
The Table of Logarithmes.
5 | 6 | 7 | 8 | 9 || D
447932448088/448242,44839714485521155
44947449633 449787 449941450095154
451018451173451326451479451633154
452553 452705152859453012453165153
4550 2454235/454387454539454692|| 463
455606455758
455910456062456214 152
45712545727045742845757945773152
4 5 8 638 45 87 8 9 +58939459091459242 15 1
460146460296460447 460597460748|151
461649461799461948 4620984622481|150
463146463296463445 463594 463744|150
464639454788464936465085 465234 149
466126 466274466423 466571166719 149
467608 467756467904 468052 +68199 148
459085 469233469380 469527469675447
|
146
4705571+70704470851 +70998471145147
472025472171 472318172464 47261C 140
47 3487 ± 7 3 6 3 3 473775 +7392597407:
474944 +7508175235 +75381 475528 146|
476367476542476687 +76833476976 145
145
477844477989 +78133478278 | 478422
479287479431479575 479719 179863 144
480725 480865481012481156 481299
144
4821594823048244548258848273143
4835874837 29 483872 184015484157143
4850 185153 185295/485437 +85575 | | 142
4864356572486714 186855486997142
487845.87986488127 +88269 +88409141
48925 +8939648953 189677 489818141
490661190801490941 +91081491122||14C
The Table of Logarithmes.
N|| 0 | 1 | 2 | 3 | 4
I
3101 491362 491502491642191782191922
311 492760492900 493039 193 179 +933 19
312 494155|494294494433|49457249471
313 495544495683495822 195960496099
314 +96929497068497206+9734497483)
3151498311/4984481198586498724/198862
31649968749.9824 499962500099500236
317 501059401196501333501470501607
818502427502564502700502837502973
31915037911503927150406315041991504335.
3201505149505286505421 | 5055 5 7 1 5 0 5 6 9 3
321 506505506640506776506911507046
322 50785650799.508126508260508395
323 509203509337 509471509606|50974C
3241 1510545 5106795 10813510947511081
3255118835120175121515122841512418
326 513218513351513484513617513750
327 514584514681514813514946515075
328 515874516006516139516271516403
329 5171955 17 328517459|517592517724
3 39 15 1 8 5 1 41 5 18646518777518909|519040
33151982851995952009 520221 520353
1 3 2 || 5 2 1 1 3 8 5 2 1269 521399 52153952166
333522444522575 522705522835522966
334 523746523876523876524136524266
3351 52504515 25 17 4 5 25 304|5254341525563
336 526339526469526598 526727526856
337 527629 527759527888528016528145
338 528916529045 529174529302529430
33953019953032815304565305841530711
I
The Table of Logarithmes.
1
5 | 6 | 7 | 8 | 9 ||D
4920624922011492341492481 492621||140
49345 493597493737 493876 494015 139
494850 +94989495128 +95267 495406 139
49623 496376496515 +96653 196791 139
497621 497759497897 498035 498173 138
498999 49913749927514994121499549; 1138
50037 5005 10500648500785500922137
501744 50188c502017 502154502291 137
503109 503246503382503518503655 136
504471504607504743504878 505014136
505828505964|506099|5062341506369
507181 5073 16 507451 507586 507721 135
508529508664508799508934509068 135
509874 510009510143 510277510411 134
511215511349511482511616511749 134
36
512551512684512818512951 | 513084 || 133
5138835140165 14149 514282514415 133
515211 515344515475 515609 515741 133
5165355166685167995 16931517064 132
5178555 17987518119518251518382||132
51917 1519303519434 519566,51969||131
520484 520615520745520876521007 131
521792 521922 522053 52218352231413
523096 523226 523356523586523616 13C
523096 523226 5233565
52429615245261524656'424785 524915130
}
5 25 692 | 525 822 | 52595 1|526081|5 2621c 129
526985527114527243 527372527501129
528274528402 528531528659528788 129
529559 529687529815529943530072 128
53283「7356s8[5ལྟïo96l;༢222「5313;1[ 125
The Table of Logarithmes.
N|| 0 1 | 2 | 3 | 4
340 531479531607|531734|53 1862 | 5 3 1989
341 532754532882533009533136533264
342 534026534153534280534407534534
343 535294535+21 535547535674535800
344 536558536685536811536937537063
345537819537945538071538197538322
346 539076539202539327539452539578
347 540329540455540579540705540829
348 541579541704 541829541953542078
349542825542929543072543199543323
350544008544192|544316|544440 544562
351 545307545431545555545678 545802
352 546543546666546789546913547036
353 547775 547898 548021 548144548267
354549003549126549229549371549492
3 5 5 5 5 0 2 285 503555047355059555071-7
356 551449551572551694 551816551938
357 552668552789552911553033553155
358 553883 554004554126554247554368
359155509455521515553361555457555578
360 | | 556303 | 556423155654 4 5 5 6 6 6 4 5 5 6 7 8 5
361 557507557627557748557868557988
362558709558829558948559068559188
363 559907560026560146 560265560385
364561101561221 561339561459561578
365562293562412562531 562649 | 562769
366 563481563599563718 563837 563955
367 564666564784564903 565921 565139
368 565848 565966566084 566202566319
369567026567144 567262 567379567497
The Table of Logarithmes.
5 | 6 | 7 | 8 | 9 || D|
522117532245532372 532499532617) j128
533391533518533645533772523099 127
53 4661534787534914535041 53 5167|| 127
535927536053536179536304536432||120|
5371895373155374415375671537693, 12.
538448538574538699538825538951126
5-970353982953995454 07954-204125
5409,5541079541205541329541454125
542203542327542452 542576542701125
543447543571543696543819543944|12.
544688544812544934 545059 545183 || 1
545925546049546172546296,546419124
54715954728254740547529547652 123
548389548512548635548758 548881||123
4.
549616549739 549861549984550106122
550839.550962551084|551206|551328| 122
552059,552181 552303552425552547 122
553276 55338655351955364055375212
55448955461c554731554852554973121
5556991555819555940556061556182121
20
556905 557026|5571465572671:573871
558108558228568349558469558589|| 120
$59308559428559548559667559787||120
560504 560624560743 560863 560982 115
561698 561817561936 | 562055 56 2174|
562887 563006563125563244563362119
564074564192564311564429564548 119
565257565376 565494565612565729118
566437566555 566673566791566909||118
567614567732 567849567967:68084118
K
The Table of Logarithmes.
X
}
No
| 0 | 0
| 1 | 2 | 3 | 4
370|| 56820256831956843656 8 5 5 4 1 5 6 8 6 7 1
371569374569491 569608569725 569842
37257054357065957077657089: 571009
373 571709 571825 571942 572058 72174
374|572872 572988573104573219573336
375574041 574147 574263574379574494
376 575188 575303575419575534575649
377 576341 576457 5765725766875,6802
378 577492577607577722577836 577951
379 5786395787545788681578983579097
3801579784579898 580012580126580241
381580925581039'581153581267581381
382 582063582177582291582404582518
383 583199583312583426583539:58365
384||584331584444584587|584670584783
385585461585574 585686585799585912
386585587586699586812586925587037
387 587711 587823587935 588047588 59
288 588832588944589056 589167589279
3891589949590061590173590284590396
390 | | 59 10 65 59117659128 1591399 591509
391592177592268592399592509592621
392 593286593397593508593618593729
393594393 94503594614594724594834
394595496595606595717595827595937
395 596 597 5967 07596817|596927|597037
396 597695 597805 597914598024598134
397 59879 598899599009599119599228
399599883599992600101600210600319
399|60097360108260119160129901408
The Table of Logarithmes.
5 | 6 | 7 | 8 | 9 || D
ליי
568788 | 5689051569023| 569139 | 5 6 9 2 57
569959570076570193 570309570426 117
57 126 571243571359571476571592117
57229 572407572523572639572755 115
57 245 2573568573684573799573915||116
57460947+726574841 574957|575072||116|
575765 575880 575996576111576226 115
576917577032 5771475772625773 75
578066 578181578295.578409578525115
579212 579326 579441 5795551579669114
580355 5804 691580583 580697580811114
581495581608581722 581836 581.949 114
582631 582745582858582972 583085 114
583765583879583992 584105 584218 113
848965850095851221585235585348113
こ
​5860245861375862495863621586475
587149587262587374587486587599||1 12
588272588384588496588608588719 112
589391589503589615589726589838||112
590507590619590750590842590953||112
591621 59173259184359195559206.
592732592843592954593064593175
1
593839593950 594061594171594282 11
594945 595055 59516559527559538611C
596047 596167596157596377.596487||118
597146 597256 5973665 7476 597586||110
598243 598353598462 598572 598681 110
599337599446599556599665599774|109
6004286 0537600646,600755600864 109
601517601625601734601843 601951109
K 2
2:
The Table of Logarithmes.
N| O 1 | 2 | 3 | 4
405020596021696022771602386,602494
40 503144603253603361603409503577
1403
40
60422660433460 +442604550604658
505 205, 60541 265521 605628605736
40606381606489606596606704606811
405 607455607562507669607777607884
406 6085261608633608739 608847608954
407 609594609701 609808,609914610021
408 610660510767610873610979611086
409 61172361 829611936612042612148
41061278461 2889612996|613102|613207
41 16 2842613947614053 614159614264
412614897615003615108615213615319
41 31 615950616055616160616265616370
414617000617105 617210617315 617419
4.15 618048618153518257 618362618466
4166190935191986 9302619406619511
417620136620240620344620448620552
418621176621280621384621448 621592
419622214622318622421 622525622628
+20 | | 62 324 96 23 3 5 31623456 623559623663
421624282624385624488 624591624695
+22625312625415 625518625621625724
4236263406264436 26546 625 6 48 626751
424627 36 6 6 27468527571 527673 627775
+25628389628491162859362869568797
426294096 951262961362971629817
427630428630529630631630733630835
428 63144463 164763 1647,631748631849
42916324576325591632559163276632862
The Table of Logarithmes.
718
110
5 | 6 | 7 | 8 | 9 || D
6026036021176028191602928603036|108
603686,603794|603902|604009|604118||108
604766604874604982 50508005197 08
60584460595 506059606166 606274 108
606919607026607133607241607348107
607991608098608205601312608419 107
609061 6091676092746~9381609488 107
610128610234610341610447610554
107
61119261129861 140561151161 1617 106
612254612359612466612572|612678||106|
613313613414613525|613630613736||106
614369614475514581614686 614792 06
615424 615529 615634615739615845||105
C5
616476616581616686616790616895
617525 617625|517734|6178396 17943 || 105
618571;618676618780|61 834618989|| 105
619615 619719619824619928620022104
620656620760620864620968621072||| 104
521695621799 52190262200 221 C 04
62273262283562293962204262314304
6 2 3 7 6 6 6 2 3 8 6 9 | 6 2 3973|624076|6 ≥4179||103
624798624901625004625
625827625929526032626135526237
107625209 10
103
626853 62695652705862716162726103
627878627979628082628185628287 102
628899629002 629104 | 6292066293081103
629919 630021530123530224620326 102;
630936631038631139531241631342|| (02:
531951 632052632153632255632356 101
6320636230641633165633266623267101
The Table of Logarithmes.
}
!
3 |
N|| 0 | 1 1 2 | 3 | 4
IN
4
43063346 16 33 569 6336706337711633872
431634477534578634679634779634880
432635484 635584 635685635785635806
433636488536588636688636789636889
434637489537589637689637789637889
4356384896 3 8 5 8 3 | 538 68 91 63 87 8 9 6 3 5 888
436 6394866395803968653978; 639085
437 640481640581540680640779640879
438 641475641573641672641771 641871
43964216542563 64266254276164 2860
44 || 5 4 3 4 5 3 64 355 15436505437 49 1 0 4 3 8 47
441544439644537044636644734544832
444045422645521645619 5457. 545815
44 54540464650264659964669 646796
4441547383 647481647579 647670547774
44554836064845864855564865 2 1 5 4 8 7 5 0
446549335649432549529649627549724
447550308650405550502,650599550696
551278651375147265156651666
44
4 4 9 11 5 5 2 2 4 6 6 5 2 3 4 652439652536652633
1;
8.
4 5 0 1 1 5 5 3 2 1 5 5 3 3 0 9 6 5 3 4051553502 55359
45 6541775427365436955446555450
45 55512 555235 55331555427555523
456560956194 55628 656386556482
45455705657152 657247557343 557438
455 | | 558011558107658202|65 8 2 9 8 1 658 39:
456 55896565906065915565925065934
45 559916560011660106 56020166029
458 660865660960361055 661149561245
459661813661907 66200266209666219!
The Table of Logarithmes.
5 | 6 | 7
1617
| 8 | 9 || D
633973634075634015342766343761|100
6351835283635383 100
634981635081
6 359 8 6 6 3 6 0 8 763618-535288636388 10
636989637089637189 637289637389|| 100|
637989638089628 893828963838
婆
​6 3 8 9 8 8 6 3 9 0 8 0 | 639188639287|639387
639984640084640183 64028364038
640978641077641177641276641375
641969642069642168642267642360
99
99
99
99
99
642959643058 6431566432556433541 99
643946164404416441431644242 544340 ၄င်
644931645029645127645226645324
98
93
645913646014610964620546306
64689464699254708964718 517285 98
647872647969648067548165548262
6488 486 48945|549043649140549237
6498216499195500 6650113550210
98
97
97
97
65079365089065098765108455118:
65176265185965195665205 552145 97
652729652826155292365301; 553116
653695 65379165388853984554080
97
96
96
6546585547546548555 9+6555042
55619655715655819 559-6556002 96
656 577 556672556769568 6 4 5 5 6 9 6 0
657534657629557725657820557916
96
96
95
6584 8 7 1558584155867955877-1558869195
659441 559536659630659726559821
660391 66048666058166067 560771
661335561434661529661623 561718 9
662286 562380662475662569 66266 95
95
95
The Table of Logarithmes.
N| O| I 1 | 2 | 3 | 4
+6 01166275 8 6 6 2852662947663041663135
+61 663701 66379 663889663983664078
462 664642664736664829664924665018
463 665581565675665769665862665956
464 666518666611666705 666799 | 666892
4656674536675461667639667733667826
568386668479668572668665668759
669317669409669503669596 669 689
468 670246670339670431670524670617
469671173671265671358671451671543
466
467
4701 6720986721906722836723751672467
471 673021673113673205673297673389
472673942674034074126674218674309
473 674861674953675045675137675228
474 675778675869675962676053676145
475 | | 6766941 6767851676876676968 | 677059
476 677607677698677789677881 677972
477 67851867860y678700678791678882
478 579428679519679609679700679791
479 580336680426680517680607680698
48
4801681241681332|681422681513581603
481 682145682235682326682416682506
482 583047683137683227683317683407
483 583947684037 84127684217584307
484684845 684935685025585114/685204
485 16857425858315859211586010686099
486 586636 58672668685685904 686994
487 687529 587618687707687790 687885
488 68841958850968859858687688776
489|1689309589398689486689575689664
The Table of Logarithmes.
1
5 1 6 1 7 | 8 | 9 ||D
6632296 63324663418663512,663607|| 94
664172664266664359664454664548||
665112665206665299665393665487|| 94
6660496661436662376063315664.4
666986667079667173667266 6673 9 94
94
94
6679196680 13668106668199668293 | 93
668852668945669038669131669224
669782 669875669967670060670153 93
670709670802670895670988671080
671636671728671821671913672005 93
93
93
672559672652672744672836672929|| 92
673482673574673666673758673849 92
674402674494674586674677674769 92
675319675412675503675595075687 92
676236676328676419676511676602 92
6771516772426773336774246775161 91
678063 678154678245678335678427 91
678973679064679155679246679337 91
679882579972680063680154680245 91
680789680879680969681060681151 91
681693681784681874581964682055 || 90
682596682686682777682867682957 90
683497683587683677683767683857 90
684396684486684576684666684756 90
685294685383685473685563685652 90
1686189686279/686368,586458686547 89
387083 687172687261 687351687439 30
587975688064688153 688242688331
688865688953689042 689131689220 89
689753689841689930690019 6901071 891
L
89
The Table of Logarithmes.
N|| 0 | 1 | 2 | 3 | 4
}
490||690196690285690373690462690550
491691081691169691258691347691435
192691965 692053692142692229692318
493 692847692935693023693111693199
49469372769381.5693903|6939911694078
495/169460516946931694781694868694956
496 1695482695569695657695744695832
497 696356696444696531696618696706
498 97229697317697404697491697578
499 698101698188698275698362698449
700184
50011698970,6990576991446992311699317
301 699838 699924 700011700098
502 700704700790700877700963701049
503 701568701654701741701827 701913
504702430702517702603702689702775
5051703291 703377|703463703549703635
506 704151704236 704322 704408 704494
507 705008705094705179705265 705350
508 705863 705949706035 706120706206
509 706718706803706888 706974707059
510707570707655707740707826 707911
511708421 708506 708591 708676 708761
512 709269709355709439 709524 709509
513 710117710202 710287710371 710456
514710963711048711132711217711301
515||711807711892|711976 7120601712144
616712649 712734712818712902712986
51773491 713575715659713742713826
518 714329714414714497 714581714665
1519715167715251715335 715418715501
f
The Table of Logarithmes..
5 | 6 | 7 | 8 | 9 || D
1
690639|690728|690816'690905|690993!| 89
6915246916126917 0691789691877
692496692494692583 692671692759
88
88
692287693375,693463693551693639 88
694166 694254694342694429|694517|| 88
695044 695131 695219|695307,695394|| 88
695919696007645094696182696269 87
696793696880696068697055 697142|| 87
697665697752 697829697926 698014 87
698535698622698709698706698883
87
87
699404699491699578,699664 699751
70027170035870044470053700617|| 87
701136701222701309701395701482|| 86
701999 702086 702172702258702344
702861702947 703033703119 703205|
86
86
86
703721|703807|703893703979704065|| 86
704579 704665704751704837704922
705436 705522705607|705693|705778
706291706376 706462706547706632 85
7071441707219707315707399707485
86
85
85
85
707996 708081 708166708251708336|| 85
708846 708931709015709100709185
709694 709779 709863709948710033
710540710625710709 710794710879 85
711385711469|7115547116391711723|| 84
712229|712313712397|712481|712566|| 84
7130707131547132387 3223713407
7139107139947140787 4162714246
714749714833714916714999715084
84
84
84
715586 715669715753 715836|715919|| 84
L 2
The Table of Logarithmes.
N!! 0 | | 2 | 3 | 4
52
716003 716087716170716254716337
521 716838716921717004717088 717171
522 717671717754717837717920718003
52371850278585718668718751718834
524 719331|719414719497719579719663
525720159720242720325720407720490
526720986721068
721151721233721316
527721811721893 721975722058722140
28722634 722716722798722881722963
529 17 2 3 4 5 6 7 23538|723619|723702723784
5301724276,7243581724439|724522724604
531725095725176725258725339725422
532 725912725993726075726156726238
333 726727726809726890726972727053
$34 727541 727623727704727785727866
5 3 5 || 728 3 5 4 7 28435728516728597728678
536729165729246729327729408729489
537 729974730055 730136730217730298
538 730782730863730944731824731105
539 731589 731669731749731832731911
540732394732474732555 7 3 2 6 3 5 | 73 27 15
541 733197733278733358733438733518
542 733999734079734159734239734319
543 734799734879734959 735039735119
5+4 7355997356797357591735838735918
$45 736397736476|736556|736635736715
546 737192737272737352737431737511
547 737987738067738145738225 738305
548 738781 738859738939739018 739097
549739572739651|739731|739809639886|
The Table of Logarithmes.
5 | 6 | 7 | 8 | 9 ||D
7164217165041716588|716671|716754|| 83
717254 717338 717421717504717587 83
718086 718169718253718336718419 83
718917718999719083719165719248|| 83
719745 719828719911719994720077 83
720573720655720738720821720903|| 83
721398 721481 721563 721646721728| 82
722222722305722387722469722552|| 82
723045723127723209723291723374 82
723866723948724029|724112|724194|| 82
82
724685724767|724849|724931|725013||
725503725585725667725748725829|| 82
726319726401 726483 726564726646|| 82
727134727216727297 727379727459 81)
727948728029728110728191728273 8:
728759728841 728922 729003729084|| 8:
729569729651729732 729813729893
81
730378730459730540730621730702|| 81
731186 731 266731347731428731508 8
731991 732072/732152732233732313
381
732796 7328767329:6733637 733117 80
733598 733679733759733839733919
734399734479 734559734639 7347!!
735199 735279735359735439735516
735998736078736157736237 / 7 3 6 3 17
80
80
80
80
80
736795 73687417369547370341737113
737590737669737749737829737908 19
738384 738463 738543738622738701 79
739177739256 739335 739414739493 79
739968740047|740126|740205740284|79|
The Table of Logarithmes.
N|| 0 | 1 | 2 | 3 | 4
|
559 7403631740442740521|740599 740678
551741152741230741309741388 741467
552 741939742018742096 742175742254
553 742725 742802742882742961743039
554743509743588743667|743745743823
555 744293744371|744449|74452 × 1744606
556 745075745153745231 745203745387
557 745855745933745011746089 746167
558 74663474671274678974686 746945
8
559 747412 747489747567|747645 747722
560748188748266 7483431748421748498
561748963 749040 749118749195749272
562 7497361498 14 749891749968750045
563750508750586750663750739750817
564751279751356751433751510751587
565||7520481752125752202752279|752356
566 752816752893752969753047753123
567 753583753659753736753813753889
568 754348754425754501754578754654
569 755112755189755265 755341755417
5701755875 7559517560277561031756179
571 756636756712756788756864756940
572757396757472757548757627 757699
573758155758230758306758382758458
574758912758988 759063759139759214
5751759668 759743759819759894759969
576 760422760498760573 760649 760723
577 761176 761251761326761402761477
578 761928 762003 762078762153 76222
579 752675 762754762829762904762978!
28
The Table of Logarithmes.
5|6| 7 1 8 1
| 8 | 9 || D
79
79
74075774083674091574099474107:
741546741624 741703 74178274186c
742332742411742485742568742647 79
743118743196 743275 743353743431 78
743902743979744058744136744215 78
74468474 47621744840744919|744997|| 78
745465 745543 745621 745699 745777 78
746245 746323746401 746479746556 78
747023 747 101 747179 747256747334
747800 747878747955748033|748110
74811c
78
78
74857674865317487317488087488851 77
749349 749427 7495047495821749659 77
750123 750199 750277 75035475043 77
750894 75097751048 751125751202 77
751664 751751751818751895 751972 77
752433752509752586|752663752735|| 77
77
77
753199 753277 753353753429753506
753966754042 754119754195754272
754730754807 754883754959755036 76
7554947555691755646 755722755799|| 76
7 5 6 2 5 6 1 7 5 6 3 3 217564281756484175656|| 76
1757016757092 757168 757244 757320 76
757775 757851757927 758003758079 76
75 85 33 758609758685 758761 758836
76
759290759366759441759517759592|| 76
760045760121 760196|760272|760347|| 75
760799 760875 760949 761025 761101 75
761552761627761702761778 761853 75
762303 762378 762453762529762604||||75|
763053 763128763203|763279763356
The Table of Logarithmes.
|D|||
I | 2 | 3 | 4 |
5801763 42 8176350317635787636531763727:
581 764176764251764326764400764475
582764923764998765072765147 765221
583 765669 765743765818765892765966
584 | | 766413766487766562766636
766710
58 57 6 7 1 5 6 76 7230767304767379|767453
586 767898767972 768046768119768194
587 76863876871276878676886068934
588 769377769451769525 76959 769673
589770115770189770263 770336770410
5901 77085217709261770999|771073|771146
591 771587 771661771734771808771881
592 772322772395 772468772542772615
593 773055773128 773201 773274773348
594 773786 773859 773933774006774079
5957745171774585 7746637747367748
596 775246775319775392775465775538
597775974776047776119776193776265
809
598 776701776774 776846776919776992
599||777427777499777572777644777717
60011778151778224177829617783681778441
601778874778947779019779091779163
602779596779669779741779813779885
603||780317780389780461|780533780605
6041781037781109781181781253781324
605|17817551781827781899781971782042
606782473782544782616 782688 782759
607 783189 783260 783332 783403 783475
608 7839047839757840467841 18784189
609|784617784689784759784831784902
The Table of Logarithmes.
T71
5 | 6 | 7 | 8 | 9 ||D
75
763802763877763952764027|764101
754549764624764699764774764848 75
765296765370765445765519765594
766041 766115766189766264766338
766785766859766933767007767682 74
767527767601 767675767749767823||
74
74
74
74
768268768342768416768490768564
769008769082769156769229769302
769746 769820769894769968770042
770484 770557770631770705770778|| 74
74
74
73
771219771293771367771440971514||||
771955772028772102772175772248
772688772762772835772908772981
773421 773494773567773640773713
774152774225774298774371|774444 | 73
73
73
73
774882 774955775028 775100775173
775610 775683775756775829775902 73
77633877641177648377655776629
73
777064777137777209 777282777354 73
777789777862777934778006778079 72
72
72
72
778513778585778658778729,778802
77923677930877938077945277-524
779957780029 700101 780173780245
780677780749780821780893 780965 72
781396781468781539781612781684 72
N N
72
7 72
71
782114782186782258782329780401
782831782902782974783046|783
783546783618783689783761 783832
784261784332784403784475 74546 71
78 497 478 50 45 785116785187785259|| 78
M
kring.
The Table of Logarithmes.
| 1 | 0 ||N
2 | 3 | 4
610|| 7853291785401|785472|785543|7856×5
611786041 786112786183785254 786325
612 786751786822786893|786964787035
613 787460 707531787602787673787744
614708164788239788309|788381788451
615|| 7888751788946|789016|789087789157
616 789581 789651789722789792,789863
617 790285790356790426790496790567
6.879098879105979112979119979:269
619791691791761791831791901|791971
620||792392792462792532792602792672
621 793092793162793231 793301793371
622 793791793860 79393 793999794069
623794488794558794627794697794767
624|| 7951851795254795324795393795463
625795880795949 796019|796088196158
626 796574796644796713 796782796852
627 797268797337797406797475 797545
728 797959798029798098798167798236
629 798651798719798789798858798927
630||799341 799409799478799547799616
931800029 800098800167800236 800305
532800717800786800854800922 800992
533801404 801472801541601609 801678
634802089802158802226802295|802363
635802774802842802910802979803047
536 803457003525 803594803662803730
637 804139804208804276804344804412
638 804821804889 804957805025805093
3918055011805569805637805705|805773)
The Table of Logarithmes.
7 | 8 | 9 || D
||D
5 | 6 |
16171
7856861785757|7858281785899785979|| 71
786396786467786538786609786680|| 71
787106787177787248787319787389||
71
787815787885787956788027788098|| 71
788522788593788663788734788804 71
789228789299789369,789439,789510 71
789933790004 790074790144790215
70
790637790707790778790848790918|| 70
791339 791409 791480 79155c791620 70
792041792111|792181|7922527923221! 70)
792742.7928121792882 792952793022|| 70
793441 793511793581,793651793721
794139 794209794279794349794418
794836794900794976 795045795115
70
70
70
795532 7956027956727957417958r0|| 70
69
796227796297796366|796436|796505: 59
796921796990 797059797129 797198
797614 7976837977;2797821797890 69
798305798374798443798513798582|| 69
7989961799065799134799203799272|| 69
799685799754799823|799892,799961|| 69
800373800442800511800579800648 69.
801061801129801 19880126680 335 69
801747 801815801884801952802021|| 69
802432802500802568802637802705 68
803116803184 803252 803321803389|| 68
803798803867 803935 804003 804071 68
804480804548804616 804685 S04752 68
805161805229805297 805365805433 68
Bo5841 8o5908805976 8o6o44 8o61126
M 2
The Table of Logarithmes.
N|| 0 | 1 | 2 | 3 | 4
540061798062481806316806384806451
641 806858806926 806994807061807129
642 807535807603 807670807738807806
643 808211808279808346808414808481
644808886808953809021809088809156
6451809559 809627809694 809762809829
646 810233810299810367810434810501
647810904810971811039811106 811173
648 8115758.16428 1709811776811843
64981224512312812379812445812512
650812913812980,813047813114813181
551 813581813648 813714813781813848
652814248814314814381 814447814514
653814913814979815046815113815129
654 815578815644815711 815777815843
6558162418163088163748164401816506
656 816904816910817036817102817169
657 817565817631817698817764817829
658318226818292 818358818424818489
659318885818951819017819083|819149
66c|| 819543|819609819676|8197411819807
661 820201 820267820333820399820464
662820258820924820989821055821120
663 821514821509821645821709821775
664822168822233822299822364822429
1660
6 6 5 1 3 2 282 2822887822952823018823083
666 823474823539823605823669823735
567824126 824191824256824321824386
668824776824841 824906824971825036
669325426825491825556825621825686
1
The Table of Logarithmes.
5 1 6 | 7 | 8 | 9 D
1
8065198065878066558067238067901 68
807197807264807332807399807467
68
68
807873807941808008 808076808143
808549 808616808684 208751808818 67
809223809290809358 809425809492|| 67
8098961809964810031810098810165
810569810636810703810770810837
67
811239 811307811374811441811508 6
811909811977S12044 812111812178
812579812646812713812779812847 67
8132478133148133811813448813514|| 67
81391481398181404881411481418 87
814581814647814714814780814847 67
815246815312815378815445815911
66
815909 815976 816042816109816175
66
816573816639816705 816771816838|| 66
817235817301817367817433817499
66
81789681796281828818094 818159
818556818622818688 818754 818819
66
66
819215819281819346819412819478 66
819873819939|820004820074|820136|| 66
820529 820595 820661820727820792 66
821186821251821317821382821448 66
82 841821906 821972822037822103 65
822495822560 822626|822691 822756|| 65
65
823148823213823279 8233441823409|| 65
823800823865823930823996824061 65
824451824516824581824646824711
825101825166825231825296|825361|| 65
825751825815825880825945826009|| 65
ず
​The Table of Logarithmes.
1
靠
​N|| o | I 1 2 1 3 1 4
67082607518261398262041826269826334
671 826723826787 826852826917826981
672827369827434 827499 827563827628
673 828015 828079828144 828209828273
674||
|1828659|828724828789 828853828918
6.751182930418293681829432829497 829561
676 829947830011830075830139830204
677 830589830653 830717830781830845
678 831229831294831358831422831486
679 || 831869 831934831998 832062832126
6891832509832573822637832700832764
681833147833211 833275 833338833402
682 83378483384883391 2833975834039
683 834421834484834548 834611834675
684835056835119.835 183835247 835310
6851 83 5691835754183581718358811835944
68 836324836387836451836514836577
687 836957 837019837083 837146837209
688 83758883765283771583777783784
689 83821983828282834583840883847
169011838849 | 83891218389751839038 | 839101
691 839478839541|839604839667839729
692840106840169840232840394840357
693 840733 840796 840859840921840984
594 841359 841423841485 041547841509
69518419558420471842109|842172 | 342235
596 842609,842672 842734842796842859
697 843233843295 843357843415343482
698 843855843918843979 844042844104)
6991844477844539844601844664844726
The Table of Logarithmes.
Y
5 | 6 | 7 | 8 | 9 || D
826399826464|82652882659382665865
827046827111827175 827239827305
65
827692827757827822827886827951
65
828338828402828467 828531828595 64
828982829046829111829175829239
64
82962582968982975482981882988211 64
830268830332830396 830460830525 64
830909830973 831037831102831166 64
821614831678831742831806
832185832253832317832381832445
831549
8 3 2 8 2 8 1 8 3 2 8 9 2 | 0 3 29 5 6 | 8 3 3 0 1 9833083
833466 833525 833593833657 8337 2
834103834166 834229834294834357
34739834802834866 834929834993
835373835437835500335564835627
836007 33607|836134|8 3619 7 | 8 36 26 1
83664183670483676783683c | 836894
837273837336837399837463837525
837904837967838030838093838156||
64
64
64
64
64
6.4
62
63
63
63
63
838534183859783866083 872 318 38786|| 63
8391648392278392898393528394151 63
839792 839855 839918839981840043 63
840419840482840545840608840671
841049 8411c9|841172841224841297
841672841736841797 841859 841922
63
63
63.
842-97842359 8424321842484842547|| 62
842921842983843046 843108843170 62
62
843544843606843669843731843893
844166844229844291844353844415 62
1844788844849844912844974845026
624
The Table of Logarithmes.
N|| 0 | 1
이
​I 1 2 3 4
700 845098845160845222845284845346
701 845718 845779 345842845904845966
702 846337846397846461 846523846585
703846955847017847079847141 847202
704847573847634847696847758847819
7251 8481898482518483 12/848374848435
06 848805 848866848928848989849051
707 849419 849481 849542 849604 849665
708 8500338500958501568502 17 850279
709 850646 8507078507698508291850891
710185125885 1 3 191851 38 1185 14421851503
711 851869851931851992 852053 852114
712852479852541 852602852663 852724
713853089 853150853211853272853333
714853698853759853819853881'853941
7 +5 1854 30 6 1 8 5 4 3 67|85 44 281854488;854549
716 854913854974855034855095 855156
717 855519855579855640855701855761
718 856124856185856245856306 856366
719 | 856729856789|856849|856910 856970
72018573321357393857453857513857574
721 857935357995 858058858116858176
722858537358597 858657858718858778
723 859138859198859258859318859379
724859739859799 859859859918859978
72586033886039818604581860518860578
726 860937 860996 861056851116861176
727861534 851594861654861714861773
728 86213186219186225 18623 10 862369
729862728862787852817862906 862966
The Table of Logarithmes:
5 | 6 | 7 | 8 |
1 61
9 || 1
9 D
845408845470845532845594845655| 62
8460288460898461518462138.6275 62
846646846708846769846832846894
62
847254847326847388 847449 84751 62
84788118479431848004848067848128
62
84849-18485598486 20 848682848743, 62
849112 849174849235 849297849358|| 61
849726849788849849849911849972 61
850339850401850462850524850585 61
85095:1851014851075851136851197
61
85156485162518516868517471851809 61
852175 852236852297852358852419 61
852785 852846 852907852968853029 61
853394853455853516853577853637 61
854002854063 854124854185854245|| 61
8546098546708547318547921854852 | 61
855216855277855337855398855459|| 61
8558228558828559438560038,6064 61
856427 856487856548856608856668 60
857031857091857152857212857272 60
85763485769418577551857815857875 | 60
88236858297858357858417858477 60
858838858898858958859018859078
6c
859439859499859559859619859679 60
8600388600981860158860218860278
60
860637|860697|86c757|8608171860877
861236861295861355861415 861475
60
861833 861893861952862012862072|| 60
862429 86.489862545862608862668 60
863025 863085 | 863144863204863263|| 601
N
The Table of Logarithmes.
|N|| 0 |
1
| 2 | 3 | 4
730 | | 85 3 3 2 3 | 8 6 3 3 8 286 3442|863 5 0 1 1 863561
863917863977864036864096864155
864629864689864748
731
733
86510486516386522:865282865341
732864511864570
7341865696,865755 1865814865874865933
735 || 86628 7 | 866346,8664051866465866524
736 866878866937 866996 857055867114
737 367467 867526 867585867644867703
738 368056868115868174868233868292
739 3656438687031868762868821868879
740 86923286929|869349|869408|869466|
741 869818869877869935 869994870073
742 870404870462870521870579870638!
743870989871047271106871164871223
744|871573871631871689871748871806
745872156872215872273|8723311872389
746 872739872797872855 872912872972
747873321873379873437873495873553
748 873902 873959874018874076874134
749874482874539874598 874656874714
750|1875061|875119875177875235|875293
751 875639875698875756875813875871
752 876218876276876333876391876449
753 876795876853876910876968877026
754877371877429877487877544877602
5518779478780041878062|878119878177
756878522878579878637878694878752
757 879096879152879211879268879325
758 879669879726879784879841 879898
$7598802428802998803568804131880471
The Table of Logarithmes.
51617
।
8 | 9 ||D
1 63620 | 863 6 79 | 8637391363799|863858|| 59
364214864274864333 86439286445 59
164808864857864925864985865045 59
365400865459865519 865578865637
59
365992|866051866119866169|866228 59
366583 1866642|866701|866759866819|| 59
367173867232867291867349867409
59
867762867821867879867939 86799
868350868409868468868527368586
868938868997869056869114 869173
59
59
59
869525 18595841869642|869701|369759 | 59
870111870169870228870287870345 59
870695870755 870813870872870930 58
871281871339371398871456871515 58
871865871923 8719811871039|872098|| 58
872448137250687256418726221872681|| 58
87302987308887314687220+873264
58
873611873669873727873785873844 58
874192874249874308874366874424|| 58
8747728748298748888749458750021
58
187 5 3 5 11 8 7 5 4 99,875466875524875582|| 58
875929875987876045876102876160
876507876564876622876679876737
58
58
877083 877141 877199877256877314 58
877659877717|877774877832877889|| 58
8782341878292 87834918784071878464|| 57
678808878866878924878981879039 57
879382879459879497-79555879612 57
379955880013880070 8801 27880185 57
1880527380585880642880699880756||
N 2
57
The Table of Logarithmes.
*
N|| 0 | 1 | 2 | 3 | 4
760 880814880877,88092818809851881042
761 081385 881442881499881556381613
762 881955 882012882 69882126 38218
762 882525 882581882638882695 882752
764 883093 883050883207883264383321
765 | | 883661 | 883718883775883832883888
766 884229884285 884342884399884455
767 884795884852884909884965 885022
768 885361885418885474885531 885587
769188592688598388603918860961886152
7701886491886547886604886659 886716
771 887054887111 887167 387223.887279
772 88761788767488772 887786 887842
773 888179888236888 92888348 888404
774888741 888797 888853388909.888965
775889302889358889414889459889526
776 889862 889918889974890029890086
777 890421890477890533890589890645
778 890979891035891091891147891203
779 891537891592891649891705891760
78018920958921508922061892262892317
781 892651892707892762892818 892873
782893207893262893318893373893429
783 893762,893817893873893928893984
7841894316894371894427894482894538
7851894869894925894980895036|395091
786 8754238954788955339558 895644
787 895975896029 896085896140 96195
788 896526896581896635396692896747
1789 897077897132897187397242897297
The Table of Logarithmes:
5 | 6 | 7 | 8 9 || D
081099881156881212881271 881328 57
881669581727881784881841 881898
882239882297882354882411882468 57
57
882809882866882923882979883037
883377883434883491883549 883605
57
57
8839458840021884059|884115|884172 57
884512884569884625884582084739 57
885078 885135885192885248885305 57
885644885700885757 885812885869
57
8862c88626588622188637858643456
8 8 6 773 8 8 6 829 8868≥ 5886941 886981 56
887336887392887449587505 88,561 56
887898887955888011388067868123 5 5.0
88846088851688857368862888685 56
889021 88907788913488918889246 56
889 5 8 2 8 8 9 6 34 | 389694889745089806 ||56|
890141890197 390253890309 390365 56
890700 890756 390812890868890924 56
891259 89131439137089142691482 56
89181 8918739192889198:1892039|| 56
89 2 373 39 2 4 2 5 189248418925391 8 9 2 5 9 5
1892925 392985893040 893096 893151
89348489353993595893651893706
894039594094894149 894205894261
894595 59464889470-189475989481
56
56
56
55
551
55
551
8951461895201 1895257895312 9536, 55,
895699095754375809.895864895919
896251 896300596361896416896471
896802896857896912896967897022|| 55
897352897407897462897517897572 55
The Table of Logarithmes.
N || 0 | 1 | 2 | 3 | 4
1 1
7900976278976828977-7897792897847
79 598176898221598286 898341898396
792
598725898780898835898889898944
79399273899328899383 899437899492
7948998218998751899929|899985500039
795 900367900422190047619005 311900586
796 900913 900958 9010 22 901077 901131
797,901458901513901567901622 901676
798 902003 902057902112902166 902221
799 902547 7026019026551902709902764
800 19030890314490319919032531903307
801 903632903687|903741|903795903849
802 904174904229904283904337904391
805 904716904769904824704878 904932
80411905256'9053107053641905418905472
805119057961905849 305904905958906012
806 906335906389906443 906497 90655.1
807 906874906927 907981 907035907089
808 907411907465 90751907573 907626
80919079491908002 908057708109908163
1810 908485908 5 3 919085 9219086461908699
811909021909074909128909181909235
812 909556909609909663 909716 909769
31391COS1910144910197910251910304
31491062491067910731910784910838
315191115819112119 112639 1 1 3 17|9 11 371
316 911690911743911797 911849911903
317912222912275 912323912381912435
318 912753912806912859912913912966
31 0191 32841 913 3 3 7|913389912443912496
The Table of Logarithmes.
1
5 | 6 | 7 | 8 | 9 | D
397902897957 898012898067898122
898451898506898561898615898670
898999899054899109399164899218
899547 899602 899656899711 899766
900094/900149900 203 9002581900312
55
551
55
55
55
900640900695 900749900804900859 55
901186 901240 901295 901349 901404 55
90173190178590183901894 901948
902275902329 90238490243902492
54
54
9028189028731902927|902981|903036 54
9033611903416903469/90352490357811 54
903904903956904012904066 904120
904445 904499904553 904607904661
904986 905039905094705148905202
9055259055809056349056881905742 54
541
54
54
54
906066|906119906172|906227|906281||54|
906604 906658906712906766906819
907143 907196907250907304907358
907680 907734907787 907841 907895
90821719082709083249083781908421
54
54
54
90875319088071908860 1909 190890711 54
909289 909342 909396109449909503|| 54
909823909877909930|909984910037| 53
910358910411910464910518910571|| 53
910891910944|7109989110511911104|| 53
911424911477911539911584|911637|| 53
119569120099 1 206 39 12 116912169 53
912488912541912594912647912700
913019913072913125913178913231
53
53
913549/9136029136551912708912761|| <?
The Table of Logarithmes:
N| O|I| 2 | 3 | 4
T
820913
820 9138141913867913919913973 914026
821 914343 914396 914449914502 914555
822 914872914925 914977915030915083
823 91539991545391505915558915611
82419159279159791916033|916035916138
825916454916507 916559916612916664
826 916980917033 917085 917138917190
827 917506917558917611 917663917716
828918030918083918135 918188918240
829918555918607918659918712918764
830919078919130919183 919235|919287
831 919601 919653,919706 919758|919810
832 920123920176 920228 920279 920332
833920645'920697920749′920801 920653
834||921166921218 921270921322 921374
∞
921686,921738 921790921842921894
836 922206 922258922310922362922414
837 922725 922777922529 9228: 1922933
838 92324492329692334: 923399923451
839||923762|9238149238651922917|923969
840924279924331|924383924434924486
841 924796924848 924899924951925003
842 925312925364925415 925467 925518
843 925828925879925931925982'926034
8449263429263949264451926497926548
8451926857926908/926959927011927062
846 927370927422927473 927524927570
847 927883927935 927986 928037928088
848 928396 928447928498 928549 928601
8491928908928959929009929061929112
The Table of Logarithmes.
5 | 6 | 7 | 8 | 9 ||D
1 6171
11
9140799141329141841914237914290|| 53
9145089146609147139 4766914819
91513691518995241915294915347
915669157:6915769915822 915 875
9161913162439 6296916349|916401
53
mm mm m
LA LA
53
53
53
916717916769916822916875916927|| 53
917243917295917348917400917453 53
917768917820917873917925917978||
918293918345918397918449918502
52
52
9188169188699189219·8973 919025|| 52
919339919392919444|919496|919549 52
919862919914919967920019920071 52
920384920436920489920541920593|| 52
920906920958921009'92 1062 921114 52
921426921478921530921582 921634 53
92194619219981922050922102922154
|
52
52
922466 922518922569922622922674 52
922985 923037923089 923140923192
923503 923555923607923658923710 52
924021924072914124924176924228
9245381924589 924641924693924744
52
2 2
52
5 21
925054925106925157925209925251
925569925621 925673 92.725 925776 52
926085926137926188926239926291
926599926651925702926754926805 51
9271149271659272161927:68 | 927319| 51
92762792767892 729927781 92783 SI
9281399281919282+2928293 928345
928652928703928754928805 928857
929163929215929266 929317929368
51
51
51
The Table of Logarithmes.
O 1 | 2 | 3 | 4
N|| 0 |
850929419929470 929521|929572929623
851 929929 929981|930032930283,930134
852 930439:930491930542930592930643
853 930949 93099993105193110293153
85493145093150993155993161c 931661
8 5 5 9 3 1 9 6 6 1 9 3 2017|932068
856932474932524932575
9321181932169
932626932677
857 93291 933031933052933133932103
858 933487933538933589933639,933659
8599339939340441934094 9341451934195.
860193449819345491934599 9346492 4700
861935003 935056935104935154935205
8629355079355589356c8935658935709
863||936011|936061936111,936162936212
864 936514936564936614936665 936715
865|1937016937066|937117|937167937217
866937510937568 937618 93766893771
8671938019938069938119'938169938219
868 93851993856993 619938669938719
8691939019939069 939119 939169939219
870 || 939514 939569939619939669939719
87194001 940068940118940168940218
872940516940566 940616940666940716
873 941014941054 941114941163941213
874 941511941561941611941660941710
8751942008 942058942107942157942207
8769+2504942554942603 942653 942702
377 942999943049 943099943148 943198
878 943495943544.943594943643943692
187919439899440359440
The Table of Logarithmes.
5 16 1
1 6 | 7 | 8 | 9 || D
|
||D
929674929725929776|929827|929879
930185930236930287930338930389
930694930745930796930847930898
51
921204931254931305931356931407
931712921762931814931865 | 9 319 15
51
51
932220922271932322932372932423,
93272793277893282993287993293
933234 9:3285933335933386933437
933740933791933841933892 933943
|9342469342961934347|934397934448|
51
51
51
SI
51
93475119348019348529349021934953)
50
935255935306935356935406935457 So
93575993580993585993591093596c 50
|936262|936313|936363|936413936462 50
936765936815|936865936916936966|| 50
937267,9373171937367937418937468| 50
937769937819937869937919 937969 SC
9382699383199393699284199384-9
938769938819938869938919 938969 50
939269939319939369939419939469|| 50
Sc
50
50
939769939819|929869|9399181939968,
940267940317940367940417940467
940765940815940865940915 940964 50
941263941313 94136294141 2941462
9417599418099+1856941909 941958 50
SC
942256 9423061942355|942405|9424551, 50,
942752 9428019428519+2901 942950
943247943297 943346943396743445
50
49
94374294379194384194389 943939 49
944325944283|9443351944384744433
O 2
46
The Table of Logarithmes.
15
No
2 | 3 | 4
1
8801194448319445321944581|944621944680
88944976 945025 94507494512494517.3
882 945468945518945567 945616 945665
883945961 946009946059946108946157
88 419 46552 946501194655 1946599946649
8859469439469529470419470901947139
886947+34 947483 947532947581 947629
889479249479 3948022 48070948119
888 948413948462 94851 948559948609
883 1948902948951|948999949048949067
89c||949398'9494399494889495361949585
891 949878949926949975 950024×50073
892 950365950414950462950511 950559
893950851950900950949950997951046
8941951338951386951435 951483951532
8951 951823951872951920951969 952017
896||952308,95235695 405952453952502
897 952 92952841952889952938952986
898 953276953325 953373953421 953469
8999537599538089538561953905|953953
9001954243954291|954339'954387954435
901 954725954773 954821 95469954918
902 955207955255955303.95 351955399
903 955688955736955784 955832955880
904 1956168956216956265 956313956361
|905||95664995669-956745956793956840
906 957128957176957224 957272957319
907 957607957655957703957751957799
908 958086958134958181958229958277
1909 195856 495861295865919587071958755
The Table of Logarithmes.
5|6།
| 7 | 8 | 9 ||
D
94 47291944779944828,944877|944927|| 49
945222 945272945321945370945419 49.
945715 945764 94581394586245912 49
946207 946256946305 946354946403 49
946698 9467471946796:946845946894|| 49
94718919472894728794 732694735|| 49
947679 947728947777 9478269-7875 49
94 8168 948217 94826694831;948364|| 49
948657 94870694875 5948804 948853 49
194914619491959492449492921949:41
49
44
49
9496331949683|949731194978c194982 49
650121950170950219950267950216
950608950657950706950754950803|| 49
951095 951143951192 951240 951289 49
951580951629951677|951729|951775)|
49
9 5 2000 | 95 2 1 1 49 52163952211|952259|| 4
952550952599952647952696952744
953034953083953131953179953228
48
48
95351 953566953615953663 953711 48
954001540499540991954146954194
954484954534954589954628954677
48
48
48
954966955014955062955110955158
955447955595 955543 955592955639 48
955928955976 956024956072956120
95640295645719565051956553956601
9 5 6 8 8 8 9 5 6 9 3 619569841957032957080
957368957416957464957512957559
957847957894957942957990958038
958325958373958421 958468958516
Co Co 00
48
48
48
48
481
48
958803958850/9588981958946958994|| 48
The Table of Logarithmes.
N|| 0 | I
910
༢།
2 3 4
95904199089|959137|959185 | 959232
911 959518959566959614959661 959709
912 959995960042 960090960138960185
913 960471 960518960566 960512960661
914 960946 9609919610419610894961136
915961421961469 9615169615 6 3 | 96 16 1
9161961895 961943 961690962038962085
917 9623699624179624649625 11 962559
918 962842962889'962937962985 963032
91996 3315963 363963410 96345, 1963504
920 9637889638351963882953929 963977
921 964259964307964354964401 964448
922 964731964778964825964872964919
923 965202 965249965296965343 965389
924965672965719965766965813965859
925||96614296618919662391966283,966329
926 966611966658|966705966752966799
927 967079 967127 967173 967220967267
928 967548967595 967642967688 967735
929 968016|968062|968109|968156|968202
9301968483968529968576968623 | 9 68 669
931||968949968996 969043 969089969136
932 969416969463 969509969556969602
933 969882969928 969975 970021 970068
934|| 9703479703939704391970486970533
935 97081297085819709041970951 970997
936 971276971322971369971415971461
937 971739 971786 971832971879971925
938 972203 972249 972295 972342972388
939||97266697 2 7 1 29 7 27 5 8197280497 285 1
The Table of Logarithmes.
5 | 6 | 7 | 8 | 9 || D
959279 959328 959375 95942395947
95975 959804959852959899959947
960233 960260,960328 960376960423
960709960786960804960851 960899
96118496123961279 961326961374
48
48
48
48
47
47
47
961658 961706|961753|961801196184
962132962179962227962275962322
962606 962653 962701 962748962795 47
963079 963126 963174 963221963268|| 47
9 63552963599963646363693963741|| 47
9640241954071964118|964165964212
47
964495964542964585 964637 964684|||| 47
964966 965013965061 965 108,965155 47
965437965484965531 965578965624 47
965906 965954 966001 966048 966095 47
47
47
9663761966423|966470|966517 | 96 65 64
966845 965892 966939966986967033
967314967361 967408 967454967501
967782 967829 967875967922967969
968249 968286968343′968389′998436|| 47
47
47
968719|968763|96880919688561968902|| 47
969183969229969276969323969369
47
47
969649969695969741 969789969835
970114970161 970207970254970300 47.
970579970626970672/970719/970765 46
971044 9710919711371971 1831971229|| 46|
971508971554971601971647971693 49
971971972018 972064 972110972157 46
972434972481 972527 972573972619|| 46
972 8 67 972 9 4 3'972989973039973082|| 46|
The Table of Logarithmes.
No.
0.1
2 1 31 4
9409731289731749732201973 266 973313
941 973589973636973682973728973774
942974050974097974143974189974235
94 974512 974558974604974649974695
944974972 975018975064975109975156
945 975432975478975524975569975616
946 975891975937975983976029 976075
947 976349976396976442976488976533
948 976808976854976899976946976992
949 9772669773129773589774031977449
95019777249777691977815977861977906
951 978181978226 978272978317978363
952 978637978683978728978774978819
953 979093979138979184979229979275
9541979548979594979639979685979730
955 980003980049 9800941980139980185
956 980458980503980549 980594980639
957980912980957 981003981048981093
958 981366981411 981456981501 981547
959 981819981864981909981954981999
950||9822719823161982362|982407982452
961 98272398276998214982859982904|
962 983175983220933265 983310983356
963 98362698367198371698376298,807
964984077984122984167 984212984257
965984527984572984617984662984707
966 9849779850229 5067985112985157
967 985426985471985516985561985606
968 985875985920985965 986009986055
9691986324986369986429864589865041
The Table of Logarithmes.
1
5 | 6 | 7 | 8 | 9 ||D
973359973405973451|973497973543
46
46
45
46
973820973866 973913973959974005
974281 974327974374974419974466|| 46
974742974788 974834974819974926
46
975202975248 975294975339975386|| 46
975662975707975753975799975845|| 46
976121976167976212976258976304
97657 976625 976671 976717976763
977037977082 977129977175 977220 461
977495 97754977586977632 977678|| 46
977952 977998978042 978089 978135 46
978409 978454978500 978546 978591 46
978865 978911 978956979002979047 46
979321 97936679412979457979503 45
979776 979821 979867979912 79958 46
98023119802761980322980367980412 45
1980685 980730980776980821980867
981139 981184 981229981275981320 45
981592981637981683 981728981773 45
9820451982090982135982181 98222645
1982497,982543,982588982633982678|| 45
982949982994 983039983085983129 45
983401 983446983490983536983581 45
45
45
983352983897983942 983987984032
984202984347|984392|984+27984 482 45
984752984797,984842984887984932 451
985202 985247 985292985337985382 45
if 5
985651|9856969857419857879858301
986099986144986189986234986279 45
9 8 6 5 4 8 9 8 6 5 9 219866279866829867:7 49
{
The Table of Logarithmes.
N|| 0 | 1
0 | 1 | 2 | 3 1.4
979||966772986817|986861 986906|986951
971 987219987264987309 987353987398
972 967666987711987756 987800987845
988113988157988202 98824798829
27419885599886041958748 98869388737
973
975 9890051989049 989094989138989183
976 989449 989494 989539989584989628
977 989895 9899399899839900-8990072
97 990339990383 990 28990472990516
979 199078399082799087119 90916, 90960
8c||991226|991270991315|991359991403
81991669991713991758 991802 991846
282992111992156992199 992244992288
283992554992598992642992686|992730
084992995993039993083993127 993172
9851993436993480993524 993568993613
986 993877993921 993965,994009994053
87 994117994361994405 994449994493
988 994356994801 994845,994 89994933
289995196995240995 284 995328995372
999 1995 635995679995723995764995811
991 996074-96117 996161996205 996249
992 996512996555 996599996643 996687
993 996949996993 997037997080 997124
994 1997 389997430997474997517 997561
995||997823997867|997910997954 997998
|996 998259998303998347998390 998434
|997||998695998739998783998826998869
998 999133999174999218999261 999305
1999||999565|999609999652999696999739
The Table of Logarithmes.
5 16 17 1
| 6 | 7 | 8 | 9
8 | 9 ||D
906996987040987085|987129987175|| 4
987443987488987532937577987622
45
4
45
987889987934 98797998802988068
988336 988381 988425988469 9 8 8 5 1 4 45
988782988826988871988916988960|
989227989272|989316|989361|989405|
45
45
989672989717989761 9898c6 989850 44
44
44
199011799016199020699025c990294
1990561990605 990649 990694990738
991004991049991093 9911379911821 44
991448991492991536|991580|991625|| 44
99189991935991979992022992067
992333992377992421 992465992509
44
44
44
992774992819992863992907992951
99321699325919933041993348 993392 44
993657 993701 1993745993789993833 | 44
994097 994141994185 994229994273 44
994537 994581994625 994669994713 44
994977 995021 995065|995108|995152 44
9954169954599955049955479-5591
9958541995898|9959421995986|996029|
996293996337996380996424996468
44
4+
44
996731 996774 995818996862996906| 44
997168997212997255997299997343
44
997605997648 997692997736|997779|| 44
998041 9980851998129|998172|998216
998477,998521998564998608998652
99891299895999899999043999087
999348999392999435999479999522
44
44
44
44
999783929826999869 999913999957 43
P 2
A
TABLE
OF
Artificial Sines
AND
TANGENTS
To Every
DEGREE and MINUTE
OF THE
QUADRANT.
The Common Radius being 10,000,000.
LONDON,
Printed in the Year MDCCXIV.
Degree o.
M Sine | Co-Jine
F
00.000000 10.0000col
16.463726 9.999999
26.764756 9 999999
36.9408+7 9.999999
47.065786 9.999999
57.162696 9.999999
617.241879 9•999999|
77.308824 9.999999,
87.3668169.999999
97.417968 9.999999
107.463726 9.9999981
7.505118 9.999998
127.442906 9,699997
137.5776€8| 9.99999*
147.609853| 9.99999
Tangent, Co-tang.\
0.000000 Infinita. 160
[5.46 372611 3.5 36274159
.764756 13.23524458
•940847 13.059153 57
7.065786 12.93421456
17.16269612-83730455
7.241878112-75812254
7.30882512.691175 53
7.366817 12.63318352
7.417970 12.58203051
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428.086965 9.999968
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8.619313 11.38068737
8.622343 11.37765736
8 62535211.374648 35
18.628340 11.37166034
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8.554352.11 345648125
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28 8.9789419.99802c
29 8.980259 9.998008
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8.966019 11.033981 +3
8.967 394 11.03260642
8.968766 11.0312344
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18.971495 11.02850539
8.972855 11.02714538
8.974209 11.02579137
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18.978248 11.021752 34
8.979586 11.020414 33
8.98092 11.019079 32
3.932251|11.017749,31
18.9835711.01642330
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Degree 84.
Degree 5.
MI Sine Co Sine I Tangent Co-tang. |
3018.9815739-997996|
18.983577 11.01642330
368.9893749 9979221
378.9906609-997910
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18.997908 11.00209219
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8.999188 11.000812 18
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19.00300710.996993 15
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9.013031 10.986969
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9.014268 10.985732
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9.015502 10.984408
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19.016732|10.983268 4
579.0156139.997654
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599.0180319.997628
9.020403 10.979597 1
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9.021620 10.97838c| c
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| Co-tang. | Tangent. M
Degree 84.
R
Degree 6.
M
Sine | Co-fine 1
Tangeut Co-tang. I
1
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19.0204359.997601
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9.024044 10.97595658
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19.02885210.97114854
9.030046 10.96995453
9.03123710.96876352
9.03242510.96757551
19.033609/10.966391 50
139.0345829.997439
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189.040342 9.997369
9.042973 10.95702742
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19.045284 10.954716 40
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29 9.0527499.992714
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1
9.047582 10.952418 38
9.04872710.95127337
9.049869 10.950131 36
19.051008 10.94899235
19.052144|10.947856134
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9.054408 10.94559232
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Co-tang. Tangent M
Degree 83.
*
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|
M Sine Co-fine |
3019.0538599.997199|
3119.0549669.997185|
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339.0571729.997156
349.0582719.997141
359.05936719.997127
36 9.0604609 997112
379.0615519.997098
38 9.0626389.997083
39 9.06372319 997068
40 9.0648069.997053
Tangent Co-tang. I
19.05664010.943340130|
19.057781 10.942219;29
9.058900 10.94110028
9.06001610.93998427
9.061130'10.93887026
19.062240 10.937760|25|
19 063348110 936652/24
9.064453 10.93554723
9.065556 10.93444522
19.066655 10 933345 21
19.067752|10.932248 20
4119.0658859.997039|
429.0669629 997024
9.068847 10 93115319
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9.071027 10.928973 17
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9.07211310.92788716
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549.079676 9.996843
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9.082833 10.917167 6
9.08389110.916109
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9.085999 10.914000| 3
9.087050 10.912950 2
9.088098 10.911902
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Co-fine Siae
Sine | Co-tang. | Tangent. M
Degree 83.
R 2
Degree 7.
Sine Co-fine |
09.0858949.9967511
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9.0879.479.996720
39.088970 9.996704
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109 096062 9.995594
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189.104025 9.996465
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2119 10697319.99641
229 1079519.996400
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249.109901 9.996358
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2619.1118429.9963351
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1
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9.091228 10 908772 58
9.092266 10.907734 57
9.093302 10.90669856
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9.096395 10.90360453
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9.101504 10.898496 48
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9.103532 (0.896468 46
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9.107559 10.892441 42
9.108560 10 8914401
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9.110556 10.889544139
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9.112542 10.887457 37
9.113533 10.88646-36
9.114521 10.885478 35
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Degree 82.
M
Degree 7.
M Sine Co fine. I
309.1156989.996269|
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349,1195 99,996 202
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362,1214179,996168,
372,1223629,996151
Tangent Co-tang. I
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9,12040410,87959629
9,121377 10,87862328
9,12234810,877652.27
9,12331710,87668326
19,12428410,87571625
9,12524810,874751|24
9,12621110,87378923
389,1233069,996134 9,12717210,872828 22
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9,130994 10,8690c618
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9,131944 10,86805617
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9,132893 10,86710716
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479,1317069,995,80
489,132630 9,995963
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9,135726 10.86427413
9,136666|10,863334
9,137605 10,86239511
19,13854210,86145810
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5119,1353879,995912|
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529,1363039,995894
9,14040910,859591 8
539,1372169,995876
9,141340 10,858660 7
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9,14246910,85773 6
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579,140850 9,995806
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529.1426559,995770
609,1425559,995753
9,14688510,853115 1
19,14780310,852197 0
| Co-fine | Sine | Co-tang. | Tangent |
Degree 82.
9,14412110,85587 4
9,145044 10,854956 3
9,145965 10,854035 2
}
F
Degree 8.
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MI Sine Co-fine |
09.1435559.995753
19.14445319-995735)
Tangent | Co tang. I
19.14780310-852197|60|
19• 148718 | 10.851282/59
9.4963210.85036858
9.159544 10.84945057
29.145349|9·995717|
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49.1471369.995681
9.151454 10.84854656
59.14802619.9956641
9.1523631c.84763755
69.1489159.995646|
9.153269 10.846731|54|
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89.150686'9.995610
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119.1533309.995555 |
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9.160457 10.839543 46
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219.162025 9.995372
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9.167532 10.832468|38|
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Co-fine | Sine | Co-tang. Tangent M
Degree 81.
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I
My Sine | Co-fine |
3919.16970 219.995 2051
3119.1705469.995 184
329.1713899.995 165
Tangent | Co-tang. I
9.174495 10.825501130
[9.17536210.824638 29
9.176224 10.823776 28
9.177084 10.82291627
339.1722309.995146
349.1730709.995127
9.177942 10.82205826
359.173908 9.995108
19.1787991c.821202/25
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9.180508 10.81949223
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409.178072 9.99501 2|
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19.183907 10.81609319
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19.190629 10.80937111
9.19146210.808538 10
519.1870929•994798
9.192294 10 2077061 9
529.1879039.994779
9.193124 10.806876 8
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9.193953 10.806047 7
549.1895199.994739
5519.1903259.994719|
2.194780 10.805220 6
7.195606|10.804394| 5
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579.1919339.99468c
589.1927349.994660
599.1935349.994640
609.1943329.994620
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9.197253 10.802747 3
9.198674 10.801926 2
9.198894 10.801106 1
19.199712 10.800287
| Co-fine Sine I | Co-tang. | Tangent |
Degree 81.
Degree 9.
M Sine Co-fine | Tangent Co-tang. I
09.19433219.994620|
19.199712 10.800287160
19.1951299.994500
9.200529 10.79947959
29.1959259.994580
9.201345 10.79865558
39.1967189.99+56
9.202159 10.79784157
49.1975119•994540
9.20297110.79702956
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19.20378210.79621855
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89.200660 3.994459
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9.205400 10.794600 5 3
9.206207 :0.793793 52
9.207013 10.792987|51
19.20781710.792183|50
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129.2037979-994377
139.2045779.994357
149.2053549.994336
159'2051319.994316|
169.2069069.994295
179.1076799.99427+
189.2084529.994254
199.2092229.994233
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119.21076019.994191
229.2115269.994171
9.2122919.994150
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259.2138189.994108
269.214579,9.9940871
27 9.2153389.994066
289.2160979 994044
29 9.2168549.994024
3019.2176099.994003|
Co-fine Sine |
19.20861910.79138149
9.20942010.79058048
9 210220 10.78978047
9.211018 10.78898-46
9.211815 10.78818545
9.21261110.78738944
9.213405 10.78659543
9.214198 10.78580242
9.214989 0.78501141
9.215780 10.784220 +0
1
19.216568 10.78 3 4 3 2 1 39
9.217356 10.78264438
9.218142 19.781758 37
9.218929 10.78107436
9.219710 10.78029035
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9.22127210.77872833
9.22205210.777948 32
9.2228310 77717031
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|
Co-tang. Tangent (M)
}
Degree 80.
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Degree 9.
+
M Sine | Co-fine |
3019.2176099.994003,
Tangent Co-tang. |
19.223607|10.776393130
31|9.21836319.993982|
329.2191169.993960
19.224382'10.775618129
9.225156 10.774844 28
339.2198689.993939
9.225929 10.774071 27
349.2206189.993918
9.226704 10.773300 26
359.2213679993897
19.227471 10.77252925
369.2221159.993875
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379.222861 9.993854
9.229007 10.770993 23
289.223606 9.993832
9.229774 10.770226 22
399.2243499.993811
499.2250929.993789
9.230539 10.76946121
19.231302 10.768698 20
4119.2258339.993768|
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429.2265739.993746
9.232826 10176717418
439.2273119.993725
449.2280489-993703
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9.233586 10.76641417
9.234345 10.76565516.
9.235103 10.76489715
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529.2338999.993528
53 9.2346259.993506
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559.2360739.993462|
19.235859|10.764141|14|
9.236614 10.76338613
9.237368 10.76263212
9.238120 10.761880'11
9.23887210.761128 10
19.239622 10.7603781 9
9.240371 10.79629 8
9.241118 10.758882 7
9.241865 10.758135 6
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56,9.23579519.993440
57 9.2375159.993418
589.2388359.993396
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19.24335410.756646 4
9.244097 10.755903 3
9.244839 10.7551612
9.245579 10 754421| I
9.24631910.753681 0
Co-fine | Sine | | Co-tang. Tangent. [M
Degree 80.
S
Degree 10.
M: Sine Ce-fine 1
Tangent Co-tang. I
19.2396709 99335 |
19.246319|10.75368 160
119.240386|9.993329|
19.247057110.752943159
29.2411019.993307
39.2418149.993284
9.247794 10 75220658
9.248530 10.75147057
49.2425269.993262
9.249264|10.750736 36
5'9.24323719.993240
9.249998/10.75000255
69.2439479.993117|
79.2446569.993195
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9.251461 10.74853953
9.252191 10.74780952
9.252920 10.747680 51
19.253648|10.74635250
1119.24747819.993104
9.254374110.745626149
129.2481819.993011
9.255200 10.744900|48|
13.9.2488839.993059
9.255824 10.74417647
149.2495839.993036
9.256547 10.74345346
159.2502829.993013|
19.257269|10.7427345
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19.257990110.742010144
179.2516779.992967
9.258710 10.741290 43
189.2523739.992944
9.25942910.74057142
199.2530679.992921
9.260146 10.739854|41|
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19.260863 10.739137,40
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19.261578 10.738422139
249.255144 9.99282
239.2558349-992829
249.2565239.992806
259.2572119.992783|
9.262292 10.737708 38
9.263005 10.73699537
9.26371710.73628 36
9.26442810.735572 35
2619.257897.9.992759
279.2585839.992736
28'9.2592689 992713
29 9.2599519.992690
309.26063319.992666|
|
| Co-fine Sine |
(9.265138/10.734865|34
9.265847 10.73415333)
9.266555 10.73344532
9.26726110.7327393
9.267967 10.73203330
Co-tang. Tangent M
Degree 79.
Degree 10.
MJ Sine Co-fine | 1 Tangent Co-tang. I
3019.2606359.992666|
3119.2613149.992643
19.267967|10.732033 30
19.26867110.731329129
329 2619949 992616
9.26937510.73062528
339.2626739.992596
9.270778 10.72992327
349.2633919.992572
19.27147910.729221 26
35.9.2640279.992549
9.271479 10.72852125
9.272178 10.727822/24
409.267395'9.992430
4119.2680659.992406
429.2687349.992382
36,9.264703,9.992525
37.9.265378 9.992501
38,9.266051 9.992478|
399.266723.9.992454
9.272876 10.727124 23
9.273573 10.72642722
9 274269 10.72573121
19.274964 10 725036|2€
19.275658/10.724342115
9.276351|10.723649|1
439:2694029.952362
9.277043 10.72295711
449.2700699.902335
459.2707359.992311|
9.277734 10.72226716
19,27842410.72157615
+49.27140019.992287
47 9.2720639.992263
489.2727269.992239|
49 9.273388 9.992214
5019.2740499.992190|
19.279113[10.72088714
9.279801 10.720199|13|
9.280488/10.71951212
9.281174 10.7188-26 11
9.281858 10 718142 10
$9.274708,9.992166|
529.2753679.992142
539.2760259.992118
549.276681 9.992093
559.2773379-992069|
69:27799119.9920451
379.2786859.99202
$58 912792979.991996)
599-2799489.991971
609.2805999.991947
|9.282542|10.717458, 9
9.283225 10.716775 8
9.28390710.716093
9284588 10.715412 6
9.285268 10.714732 5
19.285946110.714053
9.286624 10.713376 3
9.28730110.712699
9.287977 10.712023
9.288652 10.711348 c
| |
Co-fine Sine I Co-tang | Tangent ↑
Degree 79.
S 2
Degree 11.
?
MI Sine Co fine |
09.28059919.991947
119.2812299.991922
29.281897 9.991897
9.2825449.991873
49.283190|9.991848
59.28383619.991823
69.284480|9.991799
79.2851249.991774
Tangent Co-tang. I
19.288652110.711348160
19.28932610,710674159
9289999 10.710001 58
9.290671 10.709329 57
9.291342 10.70865856
19.292013110.707987155
1
89.2857669.991749
99.2864089 991724
109.287048 9.991699
11 9.28768819.991674
12 9.2883269.991649
13 9.2889649.991624
14 9.289500'9.991592
15 9.290236 9.991574
19.292682,10.70731854
9:293350 10.706650 53
9:29401710.70598352
9:294684 10.705316|51
.295349/10.704651150
19.296013|10.703987149
9 29:67710.70332348
9.29733910.702661|47|
9.298001 10.701999 46
9.298662 10.70133845
9 29932210.70067844
9.299980 10.70002043
9.300638 10.69936242
9.301295 10 69870541
1619.2908709.991 549)
179 2915049.991524
189,2921379.991498
199.292768 9.991473
2019.29339919.991448]
9.301951 10.69804940
2119.29402919.991422|
229.2946589.991397
239.2952869.991373
249.2959139.991346
259.2965399.991321
19.302607|10.697393 39
9.303261 10.696739 38
9.303914 10.696086 37
9.304567 10.69543336
19.30521810.69478235
269.29716419.991295
279.2977889.991270
289.2984129.991244
29 9.299034′9.991218
30 9.2996559.9911931
1
| fine Sine
19.305867 10.694131|3
9 306519 10.69348133
9.307168 10.69283232
9.307816 10.69218431
9.308463110.69153730
Co-tang. Tangent. M
Degree 78.
={4» c%eu{gy&%.Y==v、t、༨༧་
Degree II.
MI Sine | Co-fine. | Tangent Co-tang. I
30 9.2996559.991193|
19.308463 10.691537130
319,2002769,991167
9,309109 10,690891129
329,3008959,991141
9,30975410,690246 28
339,3015149,991115
9,31039910,689601 27
349,3021329,991090
359,3027499,991064
9,311042 10,688958 26,
9,311685 10,68831525
369,3033649,991038|
19,312327|10,687673124
379,3039799,991012
9,312968 10,687032 23
389,3045939,990986 9,313608 10,686392 22
399,3052079,990960
9,314247 10,685753|21
409,3058199,990934!
19,31488510,685115 20
4119,3064309,990908
19,315523|10,68447719
429,3070419,990882
9,316159 10,68384118
439,3076509,990855
2,316795 10,68320517
449,3082599,9908 29
2,317430 10,68257016
45.9,3088679,9908031
31806410,681936|15
469,3094749,990777
479,3100809990750
48 9,3106859,990724
499,312899,990697
|50|9,3118999,990671|
51|9,312495|9,990645
529,3130979,990618
53 2,3136989,990591
549,3142979.990565
5519,3148979,990538|
5619,31549517,990512
579,3160929,990485
589,3166899,990458
599,317284'9,990431
609,317879,9,990404
|9,318697|10,68130314
9,319330 10,68067013
9,319961 10,68003; 12
9,320592 10,6794081 1
19,32122210,678778|10|
9,321851 10,678149 9
9,322479 10,677521 8
9,323106 10,676894 7
9,323733 10,676267 6
19,324358 10,675642
|9,324983|10,675017 4
9,32560710,674393
9,32623110,672769
9,32685310,673147
19,327+7510,672525
| Co-fine | Sine || Co-tang. | Tangent |
Degree 78.
Degree 12.
ቲ
i
My Sine Co-fine |
|
Tangent Co-tang. I
0.9.31787919-99049419.3274750.672525160)
19.328095110.671905159
19.3184739.999377
29.3190669.990351
39.3196589.990374
9.328715 10.67:28558
|9 • 329 33 4 1 0.670656 57
49.3202509.990397
59.3208409.990370
|9.329953| 10.670047 56
19.320570|10.669430'5 5
69.3214309,990247|
79.3220199.998215
19.331187 10.66882354
9.331803 to.66819753
89.3226079.990188
9.33241810.66758252
99.323194 9.990$61
9.333033 10.6669675 1
1019.32378c19.9901341
19.333646 10.66635450
119.3243669-990197
19.33425910.665741149
$29.3249509·990079] 9.334871 10.66512948
139-3255349.990052]
19.335482 10.66451847
149.3261179.990025
9.336093 10.663907 46
15,9.326699'9.989997
19.336702 10.663298 49
169.3272819-9899701
19.337311[10.66268944
179-327862 9.989943
9.33791910.66208143
189.3284419-9899) 5
9.33852710.66147342
199.3290209.98988;
9.339133 10.56086741
2019.32959919.989860
19.339739/10.66626140
219-33017619.989832
19.34034410.65965639
229-3307539.989804
9.34094810.65905238
23 9.33 13289,989777
249•331903 9.989749
2519.3324789•989721}
2619-3330519-989693
279•3336249•989 565
28
9.33419 9.989637
299.334766 9.989609
399-3353379.989581]
9.34155210.55844837
9.34215 10.657845 36
(9.342757|10.65724335
9.34335810.656642|34|
9.3+395810-65604233
9.344554 10.65544232
9.345157 10.65484331
9.34575510.654245/30
Co-fine Sine I
| Co-tang. I Tangent IM
Degree 77.
Degree 12.
M Sine | Co-fine I
Tangent Co tang. I
19.345755|10.65424×130
309.3353379•989581]
319-3359069-9895531
329-3364759-989525
339.3370439.989597
|9.346353|10.653647|29|
|9.346949 10.65305128
9.347545 10.652455 27
349.3376 109.989469
9.34814110.651859 26
359-3381769.989441|
19.348735 10.651265|25
369.3387429.989413
19.349329 10.650671124
[379.3393069.989384
9.349922 10.65007823
389.3398709.989356
9.350514 10.649486 22
399.3404349 989328
9.351106 10.648894 21
4019.34099619.9892991
19.351697 10.648303 20
4119.34155819-9892711
42 9.342119 9•989243|
439.3426799.989214
+49.3432399.98918.
(+59•3437979 989157|
19.352287 10.647713|19
9-352876 10.64712418
9.353465 10.64653517
|9•354053|10.645947|16
9.354640 10.64536015
46,9•3443559.989128)
19.35522; 10.544773|14
+79.3449129.989100
9.35581210.64418713
489•3454699.989071
9.356398 10.64360212
499.3460249.989042
9.356982 10.643018'11
509.3465799.989014
9.357566 10.642434 10
519.3471349.988985)
19.358149 10.641851 9
529.3476879.988956
539.3482409.988927
549.3487929.988898
559-3493439 988869
9.35873110.541269 8
9.35931310.640587 7
9.35989; 10.64 107 6
9.560474 10.639526 5
5619.34989319.988840
579-3504439.9888 1
589.3509929.988782
9.361053[10.635947| 4
9.361632 10.638369 3
9.362210 10.637790 2
599.351540 9.988754
609.3520889.988724
9.362787 10.637213 1
9.363364 10.636636 0
1 Co-fine Sine I | Co-tang. Tangent M
Degree 77.
1
Degree 13.
09.35208819-9887241
19.35263519-988695
29.35318.9.988666
33.3537269.988636)
M Sine Co-fine
1
Tangent Co-tang. I
19.363364|10.63663660
19.363940 10.636060159
9.364515 10.635485 58
9.365090 10.63491957
49.354271'9.988607
9.36566410.63433656
5/9.354185 9.988578
19.36623715.63376355
69.355358 9.988548|
9.36681c[10.633190154
79-3559019-9885 19
9.367382 10.63261853
89.3564439.988489
9.367953 10.632047 52
99.3569849.988460
1019.3575249.98843°
9.368524 10.63147651
19.569094 10.630906150
119.35806419.988401
129.358603 9.988371
139.359141 9.988341
149.3596799.988312
[159.3502159.988282
19.369663|10.630337|49|
9.370232 10 629768 48
9.370799 10.629201 47
9.371367 10.62863346
9.37193310.62806745
16.9.3607529.988252
[9.372499¸10.627501|44|
179.3612879.988223
9.37306410-62693043
189.3618229.988193
9.373629|10-62637142
199.3623569.988163
20 9.362889'9.988133
9.374193 10.62580741
9.374756 10.62524440
|21|9.36342219.988103
229.36395.988573
19.375319|10.524681|39
9.37588110.62411638
239.3644859.988043
9.376442 10.62358 37
249.36501.988013
9.377003 c.62299; 36
259.3655469.987983
9.377563|10.622437:5
|26|9.36607519.987953| 19.378 12 2 1 1 0,621878 34
279.3666049.987922
9.37868110,62131933
289.3671329.987892
9.379239 10,620761 32
29.3676599.987862
9.379797 10,62020331
309.3681859.987832
9.380354 10,619646 30
| Co-fine Sine | Co-tang. | Tangent |
Degree 76.
Degree 13.
M Sine | Co-fine |
3019.368185 9.98 78321
31/9.36871119.987801|
329.3692369.987771
Tangent Co-tang. I
19.380354|10.619646130
19.380910 10.619090|29|
9.381466 10.618534 28
339.369761 9.987740
349.3702859.987710
359.370808 9.987679
9.382021 10.617980 27
9.382575 10.617425 26
19.383129 10.61687125
369-371330 9.987649
379.3718529.987618
389.3723739.987588
399.3728949.987557
19.383682 10.61631824
9.384234 10.6157662
9.384786 10.615214|22|
9.385337 10.61466321
>
409.373414'9.987526
9.385888 10.61411220
41|9.3739339.987496|
19.386438 10.613562|19
429.3744529.987465
9.386987,10.61301318
439.374970 9.987434
9.387536 10.61246417
449.3754879 987403
9.388084 10.61191616
459.3760039.987372
9.388631 10.61136915
4619.37651919.987341|
47 9.3770359.987310
48 9.3775499.987279
499.3780639.987248
509.3785779.987217
5119.379089,9.987186|
529.379601 9.987155
19.389178 10.610822,14
9.389724 10.61027613
9.390270 10.60973012
9.390815 10 60918511
19.391360 10.608640 10
19.391907|10.608097 9
9.39246710.607553 8
539.3801139.987124
9.39298910.637011 7
549.3806249.987092
9.39353110.606469 6
559.3811349.9870611
9 394074 10.605927 5
5619.3816439.987030|
19.394614 10.6053861 4
579.3821529.986998
9.395154 10.604846 3
589.3826619.986967 9.395694 10.604306
9.396233 10.603767 I
599.3831689.986936
609.3836759.9869041 396771 10.603229 0
| Co-fine | Sine || Co-tang. | Tangent. [M
Degree 76.
T
Degree 14.
M Sine Co-fine |
Tangent i Colang. I
19:39677110603229|60
109.38367519.9869041
19.38418119.986873)
29.3846879.986841
|9.397309|10.6q2694159|
9.397846 10.6021 54 58
39.3851929.98680g
9'398383 10.60161757
49.3856979.986778
9.39891910.601081 56
59.3862019.986746|
[9.39945510.600549155
(59.3867049.986714
9.399990|10.60001054
1
7.9.3872079.986682
18 9.3877099.986651
99.3882109.986619]
9.400524 10.59947653
10 9.3887119.9865871
9.401058 10.59894452
9.401591 10.59840951
19.402124 10.597876 50
119.38921149.9865551
129.3897119.986523
139.3902 109.986491
149.390708 9.986459
1519.391206 9.986427
(9.402656|10.597344149
9.408187 10.59681348
9.403718 10.59628247
9.404249 10.59575146
9.40477810.595222|4
169.39170319.9863951
179-3921999.986363
[9.405306|10.594692144
9.405836 10.59416443
189.3926959.986331
9.406364 10.59363642
199.3931909.986299
9.406892|10.59360841
2019.3936859.986266
19.40741910.59258140
2119.39417919.9862341
229.3946739.986201
239.395166 9.986169
19.40794510.592055139
9.408471 10.591529 38
9.408996 10.59100137
249.395654 9.986137
9.409521 10.59047936
259.3961509.986104
9.410045 10.59995435
2619.39564119.986072|
19.410569|10.58.9431134
279.3971319.986039
9.411092 10.58890833
289.3976219.086007
9.411615 10.588385 32
29 9.3981119.985974
9.41213710.587863 31
30 9.3986009.9859421
| Co-fine
9.412658 10.58734230
Sine
| Co-tang. | Tangent. M
Degree 75.
1
Degree 14.
Mp Sine Co-fine |
3019.39860019.985942
3119.3990879.9859091
329:3995759.985876
339.4000629.985843
349:4005499.985811
35.9.40 0359.985778
36.9.401520|9.98′5745
719.402005 9.985712
389.4024899.985679
399.402972 9.985646
4019.4034559.985613|
Tangent Co-tang. [
19.412658|10.5873423c
19.413179 10.586821129
9.413699 10.586301 28
9.414219 10.58578127
9414738 10.58526225
9.415257.10.584742 25
A
415775|10.58422 5/2 4
9.416293 10.58370722
9.416810 10.58319022
9.417326 10.58267421
19.41784210.58215720
4119:40393819198558.
429:4044209.985547
439.404901 9.985513
449.405382 9.985480
459:4058629.985447
9.418357|10.58164219
9.418873 10.58112718
9.419387 10.58061317
9:419901 10.58009916
9 420415'10.579585|1'5
4619.406341 9.985414
479.4068209.985380
489.4072999.985347
49 9.407776 9.985314
5019.4082549.9852801
$119.40873119.985247
$29.4092079.985213
539.4096829.985180
549.4101579.985146
559.4106329:985112
5619.411106 9.985079
579.4115799.985045
589.4120529.985011
599.4125249.984977
609.41299619.984943
| Co-fine \ Sine 1
19.420927110.579072 14
9.4214400.578560 1
9.42195110.578048 12
9.422463 10.5775371
9.422973 10.5770261
19:423484110.575516 9
9.423993 10.576007 8
9.424503 10.5754977
9.425011 10.574989€
9.425518 10.574480
19.426027|10.573973.4
9.426534 10.573466 3
9.42704110.572959 2
9.427547 10.572453 1
9.428052 10.571947 c
Co-tang Tangent | M
Degree 75.
T 2
Degree 15.
M Sine Co-fine |
09.4129969.984944
Tangent Co-tang. |
19.42805210.571947160
9.4134679.98491c
19.428557110.571442159
29.413938 9.984876
9.429062 10.570938 58
39.4144089.984842
9.429566 10.570434 57
49.4148789.984808
9.420070 10,579939 56
59.4153479.984774
9.42057310.57942755
69.4158159.98474
9.431075|10.568925|54|
79.4162839.984706
9.431577 10.56842353
89.4168509.98467:
9.432079 10.567921 52
99.4172179.984637
9.432580 10.567420|51|
1919.457684'9.984603|
9.43308010.566920 50
11194181499.984569
19.433580/10.566419149
129.4186159.984535
139.4190799.984500
149.4195449.984466
159.4200079.984431
9.43408c 10.56592048
9.434579 10.56542147
9.435078 10.56492246
9 435576 10.56442445
1619.420470,9.984397)
179.420933 9.984363
189.4213959.984328
199.421856'9.984293
19.436073|10.56392744
9.436570 10.563430 43
9.437067 10.562933 42
9.437563 10.562437 41
2019.422317,9.984259.
19.438059 10.561941 40
2119.42277819.984224|
19.439554|10.56144639
229.4232389.984189
239.423697 9.984155
249.424156 9.984120
2519 424615|9 984085|
9.439548 10.56095238
9.439543 10.56045737)
9.440036 10.559964 36
19.440529|10.55947135
2619.42507219.9840501
279.4255309.984015
28 9.4259879.983980
299.4264439.983945
39.426899 9.983910)
Co-fine Sine |
19.441022110.558978 34
9.441514 10.55848633
9.441006 10.55799432
9.442497 10.55750331
9.442988 10.55701130
Co-tang. Tangent/M
Degree 74.
Degree 15.
MI Sine 1 Co-fine. |
Tangent Co-tang. I
3019.42689919.983910
19.442988 10.557011/30
3119.4273549.983875
9.443479110.55652129
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9.454148 10.545852 7
9.454625 10.545372 6
19.45510710.5448931 5
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9.45654210.543458 2
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Degree 74.
Degree 16.
19.4407789.982805
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Degree 73.
Degree 16.
319-4537689.981699|
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58 9.465108 9.980672
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9.482167 10.517833
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59.491922 9.978000
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9.51263510.48736558
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9.513493 10.48650756
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9.519882 10.480118 41
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9 525778 10.47422227
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9.5758110.424189 22
6.57619310.42380/21
4019.5476899.971112)
19.576576) 10.423424/20
4119.5480249.971065)
429.5483589.971018
439486939·9:0970
|9.576958 10.423041 | 19
3.577341 10.42265918
9.577723 10.42227717
9.578104 10.42189616
9.578486|10.421514/15
44 9.5490269-970922
459.5493609.970874
46 9.549693 9.970826
147:9.5500269.970779
489.5503599.970731
499.5506929.970683
5 19.5 0249.970634
15-19-551355-970586
529.551687970538
539.5520189 970490
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[56]y•$$3©£Q9•970345]
5755334970247
589.55367-970249
599.55+000).970200
6019.554329.970 152
Co-fine Sine
14
19.578867|10.42
19.5 78867 | 10.421133|1
9.579248 10.42075213
9.579628 10.42037112
|9.58000910.419991|11
2.580389|10.41961 1 1 0
{
9.580769/10.419231 9
9.58149 10.4188518
9.581528 10-4:84727
9.581907 10.418092 6
9.582286 10.417713
9.582665110.417335 4
9.5.8304310.416956 3
9.583422 10.416578 2
9.583300 10.410200]
9.584177 10.415723) 0
Co-tang. Tangent M
Degree 69.
ཏྭཱ
Degree 21.
MJ Sine Co-fine I
$9,5543299701528
19,5546589,970103]
29,5549879,97005
39,5553159,97000(
49,555643 9,969957
59,5559719,9699-91
69,5562999,969860 |
79,5566269,969611
89,556953 9,969762
99,5572799,969713
109,5576069,969665)
Tangent Co.tang. |
19,584177|10,415822160
|9, 5 8 4 5 5 5 || 1 0,4 1544559
9,5849210,41 5068 58
9,585308 10,414691|57|
9,58568610,414314 16
19,58606210,41393855
19,58643910,41356154
9,586815 10,41318553
9,587190 10,412800/52
9,587566 10,41243451
19,587941 10,41205950
|1 1|9,5579329,969616|
$29,5582589,969567
19,588316/10,411684149)
9,58869110,41130948
139,5585839,969518
149,5589099,969469|
1519,559 2349,969419
169,5595589,969370|
179,5 59883 9,96932 1
9,58906610,41093447
9,589440 10,410560 46
19,58981410,410285(45)
9,590188110,409812/44
189,56c2079,969272
199,5605319,969223
2019,560855'9,969173
9,590561 10,409438 43
9,590935 10,40906542
9,591308 10,40869241
19,591681 10,40831940
219,5611789,969124
229,5615019,969075
239,5618249,969025
249,5625469,968976
259,5624689,998926
19,592054110,40794639
9,592426 10,407574 38
9,592798 10,407201 37
9,593170 10,40682936
19,593542|10,40545735
269,5627909,968877|
279,5631129.968827
|28|9,5634339,968777
299,5637549,968728|
309,5640769,958678,
| Co-fine | Sine |
59391410,40608634
9,594285 10540571533
9,594656|10,4°534432
9,595027 10,405073 31
19,595397|10,404602 30
Co-t ang. Tangent M
Degree 68.
Degree 21.
MI Sine Co-fine I
3019.56407519.968678|
319.56439619.968628|
329.5647169.968578
339.5650369.968528
349.5653569.968478
359.56567519.968428
3619.56599519.968378
379.5663149.968328
389.566632 9.968278
399.566951 9.968228
409.567269'9.968178
Tangent Co-tang. |
19.595397|10.404602130
19.59576810.40423229
9.596138 10.403862 28
9 59650810.40349227
9.596878 10.403122|26|
9.597247 10.402753125
9.597616|1.4-2384124)
9.597985 10.4020152;
9 598354 10.40164622
9.598722 10.40127721
9'599091 10.400909|20|
419.5675879.968128
429.5679049.968078
19.599459|10.400541|19||
9.599827 10.40017318
439.5682229.968027
9.500194 10.39980617
449.5685399.967977
9.50056210.399438|16||
59.5688559.967927
9.500929 10.39907115
4619.56917219.967876||
9.60129610.398704 14
479.5694889.967826
9.60166210.39833713
89.5698049.967775
9.60202910.39797112
99.5701209.967725
9.602395 10.39760511
5019.57043519.967674
9.60276110.39723510
$19.5707519.967623
9.603127,10.396873 9
$29.5710659.967573
9.603493 10.396507 8
$39.5713809.967522
$49.5716959.967471
$59.5720099.967420|
5.603858 10.396142 7
9.604223|10.395777 6
9.604588 10.395412
69.5723229.967370,
19.604953 10.395047 4
79.5726369-967319
9.605317,10.394683
$8 9.572949 9.967368
|9.605681|10.39431 2
99.5732639.967217
9.60604610.393954 I
09.5735759.967166
9.60640910.39359 0
Co-fine Sine | | Co-tang. Tangent. M
Degree 68.
#
Degree 22.
Sine | Co-fine |
Tangent Co-tang. |
09-573575′9.967165
19.606409|io 393590 60
119.5738889.967115)
9.666773110.39322715
29.5742009.967064)
9.607136.39286358
39.5745129.967012
9.607500 10.39250057
49.5748249.966961
9.60786210.392137:56
59.5751359.966910
19.60822510.391774 55
6,9.57544719.9668591
9.608588 10.391412154
7.9.5757589.966807.
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99.5763799.966705
10 9.5766899.966653
1119 576999 9.966602
129.577309:9.966550
139.577618,9.966499
149.5779279.966447
159 57823619.9663951
169.5785459.9663441
79.5788539.966292
£8 9.5791619.966240
199.579469).966188
9.608950 10.391050ls
9.609312 10.390688:52
9.609674 10.39032651
19.600036|10.399964,50
9.610397,10.389603149
9.610758 10.38924148
9.61111910.38888047
9.611480 10.38852046
9.61184-10.38815945
[9.612201|10.387799144
9.61256110.387438 43
9.612921 10.387078 42
9613281 10.38671941
2019.5797779.966136
9.61364110.386359'40
219.58008419.966084
229.5803929.966032
23'9.5806989.965980
19.614000|10.386000139
9.61435910.385641 38
9614718 10.38528237
24.9.5810059.965928
9.615077 10.384923|36|
25.9.5813119.965876
9.615435 10.38456535
2619.581618 9.965824
279.5819239.965772
289.582229 9.965720
9.615793|10.384207134
9.61615110.383+48:33
9.616509 0 38349132
299.5825349.96566;
3019.582840 9.965615
9.616867 10.3831 33 31
9.61722410.382776 30
1 Co-tine
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Sine
| Co-tang. Tangent. M
Degree 67.
Degree 2 2.
MJ Sine i Co-fine |
3019.582840 9.965615.
3119.58314419.965563!
329.5834499.965511
339 5837539.965458
| Tangent Co-tang. |
19.617224110 382776130
19.61758110.382418129
9.617938 10.382061 28
9.618295 10.381705 27
349.5840589 965406
9.618652 10.381348 20
9.619008 10.380992 25
19.619364 10.38063524
9.619720 10.38027923
9.62007610.37992422
9.62043210.379568 21
9.62078710.379213/20
19.62114210.37885819
9.621497 10.378503 18
9.621852 10.378148 17
9.622206 10.37779316
9.622561 10.37743915
359.584361 9.9-5353
36 9.58466519.965301
379.584968,9 965248
389.4852719 965195
39 9.5855749.965 43
40 9.5858779.965090
;
4119.5861799.965037
429.586481 9.964984
439.586783 9.964931
44.9.587085 9.964878
459.58738619.964825
4619.587687 9.964772)
479.587988 9.964719
489.5882899.954666
499 588519 9.964613
5019.5888909.964560
519.589190 9 964507)
529.5894899.964454
53 9.5897899.964400
549.5900889 964347
55.9.5903879.964:94
569.5906869.964240|
579.5909849.964167
589.5912829.964133
599.5915809.96408c
6019.5918789.964026
Co-fine Sine
9.62291510.377085,14
9.6232691 .37673113
9.623623 10.37637712
9.62397610.376024 11
9.62433010.375670 10
19 624683110.375317
9.625036 10.374964
9.625388 10.374612 7
9.625741 10.374259 6
|9.626093|10.373907 5
19 626445 10.373555) 4
9.626797 10 373203
9.627149 10.37-850 3
9.627501 10.372499
19.627852 10.372148
| Co-tang | Tangent | M
Degree 67.
X 2
Degree 23.
MI Sine Co-fine |
Tangent Co-tang. I
09.591878 9.964026||
'9.62785210.37214860
19.592175 9.963972|
19.628203/10.371797,59
29.5924739.963919
9.62855410.371446 58
39.5927709.963865
9.828905 10.371095 57
49.5930679.963811
9.62925510.370744 56
59.593363.9.963757
19.629606 10.370394 55
69.59365819.9637031
19.629956 10.370044154
9.5939559.963650
8.9.5942519.963596
99.5945479.963542
109.5948429.9634881
I 119.595137|9.9634331
129.5954329.963379
139.5957279.963325
149 5960219.963271
630306 10.369694 53
9.630655 10.369344 52
9.631005 10.368995|51|
9.631354'10.368645 50
15.9.596315'9.963217
[9.631704|10.368296149
9.632053 10.367947 48
9.632401 10.367598 47
9.632750 10.36725046
9.633098 10.36690145
169.59661019.963102}
179.596903 9.963108
189.5971969.963054
19.633447110.366553|44|
9.633795 10.36620543
9.634143 10.365857 42
199.597490′9.962999
9.634490 10.36551041
209.5977839.962945
9.634838 10.36516240
219.5980759.9628921
9.63518510.364815139
229.5983689.962836
9.635530 10.364468 38
239.5986609.962781
9.635879 10.364121 37
249.5989529 952726
9.636226 10.36377436
259.5992449.962672
9.636572 10.36342835
269.59953 19.962617
279.5998279.962562
289.6001189,962507
299.6004099.962453
309.600700 9.962398
19.636918|10.363081134
9.637205 10 36273533
|9.637610|10.362389|32|
9.637956 10.36204431
9.638302 10.361698 30
¡Co-fine Sine
1
1
Co-tang. Tangent M
Degree 66.
Degree 23.
M Sine Co-fine. | |Tangent Co-tang.
19.638302|10.361698|30|
309.600700.9.9623981
319.60099019.9623431
9.638647 10.361353|29|
329.6012809.962288
9.638992 10.36100728
339.6015709.962233
9.639337 10.350662 27
349.6018609 962178
9.639682 10.360318 26
359 6021499.962122)
9.640027 10.369973/25
369.6024399.962067|
|9.640371|10.359624|24|
37.9.6027289.962012
9.640716 10.359284 23
389.6030179.961957
9.641060 10.358940 22
309.6023059.961902
9.64140410.35859621
40 9.6035949.961846
9.641747 10.35825320
419.60388219.9617911
429.6041709.961735
43 6044579.961680
449 604745 9.961624
459.6050329.961569|
19.64209110.35790919
9.64243410.3575568
9.642777 10.35722217
9.643120 10.35698016
19.643463 10.356537 15
4619.6053199.961513
479.605605 9.961458
489.6058929.961402
499.6061799.961346
sol9.6064659.361290
5119.60675019.961235|
529.6070369.961179
539.6073229.961123|
549.6076079.96
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19.643804110.35519414)
9.644148 10.35585213
9.644490 10.35551012
9 644832 10.355168 11
19.64517410.354826/10
9.645516 10.354484) 9
9.645857 10.554142 8
9.64619910.353801 7
9.546540 10.353460 6
559.6078929.96 101!
19.646881|10.353119) 5
5619.6081759.9609551
9.64722210.352778 4
579.6084619.960899
9.647562 10.352438 3
589.6087459.960842
9.547903 10.352097 2
599.6090299.960785
6019.6093139.960730
9.648243 10.351757 1
19.648583 10.351417 0
| Co-fine | Sine | | Co-tang. | Tangent
Degree 66.
赛季
​"
Degree 24.
M Sine Co-fine |
ol9.60931319.960730|
Tangent Co-tang. \
(9.6 48 5 8 3 | 10. 3 5 1417 60
9.64892, 10.35107759
9.649263 10.35 737 58
9. 49602.0.35039857
9.649942 10.350058.6
9.650281|10.3+9719′5 5
1|9.609597|9.96067+
29.6098809.960617
3.60169.960561
4.6104469.960ses
59.6107299.960448)
69.6110129.960392
9.550620 10.349380 54
79.6112949.960335
89.6115769.960279
39.6118-89.960222
10.612 409.960165)
119.61242 9.960 19
125.5127 29.9 0052
139.6129839.959995
1 4'9.6 1 3 264 9·9 9938
159.6354 19.9598811
169.6138:59.959824)
179.6141059.959768
189.6143859.959710
1199.614665.91965;
|20|9.5149449.959596|
|21|9.5 15 22 3 9.95953
22.5155029.959482
239.6157819.959425
249.6160609.959;67
25.9.6163389.959310
2619.616…. . É 19.959253
279.6168949.959195
289.61172 y•959138
299•6174509•959~80
30.6177279.95902;
Co-fine Sine i
1
[9.650959 10.34904.53
19.65129710.34870352
9.65163 10.34836451
9.65 97410.348026|50|
2.552312 10.347688 49
•651650 10.34735048
).652988 10.34701247
9.653326 10.346674|46|
1.6:3563110.
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19.654000 10.34 99944)
9.65433710.34566243
9. 5407410-34532542
9.655011|10.144989′41
[9.65-348 10.344052 40
[9.655684] 1 0·3+43 1 639
9.656020 10.34398038
9.656356|10.343643|37
9.656692 10.34310836
19.657-2810.34297735
[9.6 <7,63 | 10.34263634
9.65769910.34230133
9.65303 +10.34 1966 3 2
9.558369 10•3 + * 5 3 1 3 4
9.65870434129630
Co-tang. Tangent M
Degree 65.
Degree 24.
369-6193869.958 6 7 7 1
M] Sine | Co-fine |
3 cl. 61772719-959023'
301-6177
319.6180049.958965
329.6.8281 9.9⋅ 8908
339.0185589.95 88 €
349.61883 +9.958792
|35/9.6191,0′9.9 < 873 4|
9.9586771
Tangent Co-tang.
19.658704'10-341296130
|9.659039|10.340926|29|
9.659373 10.34062; 28
9.659708 10.34029227
9.66004210.33995826
9.66037610.33962425
19.660710 10.33929024
379.61966 9.958619
9.661043 10.33895723
389.619939.958561
9.661377 1-3862322
399.6202139.958503
9.661710 10-338290:21
40.9.6204889.9-8445
19662043 10.337956 20
4119.62076319.958387
9.662375|10.337623119
429.6210389.95 8 3 29
9.662709 10.3372918
439.6213139•958271
9.662042 0.33695817
449.6215879.958212
9.6633741-33662516)
459.6218619-958154
19.66370710.33529315
46 9.622 13:19.958096|
479.6224099.958038
489.6226829.957979
499.6229569.957921
509.623229.957862
519.6235029.957804
529.6237769.957745
539.6240479.957687
549.6243199-957628
559.624-919-95-5:0
5619,6248639,95 7 5 × 1|
579,6251349,957452
589,625 +06:9,957393
599,6256779,95 7 3 3 4
609,625948 9,957276
| Co-fine
Sine
19.66403511r.33596 1/1
9.664371 10.335 129!
9.664793 0.33529712
9.6653510.334965 11
9.66536010.33453410
1966;66710-334302 9)
$9.666029|10.333971, 8
9.666;60 10.333640
기
​9.666691 10.333309 6
19.66702 Pro.13-979
31 3
| 9, 6673 5 2 1 0,1 3 26
1 132643; 4
9,66,682 10,33 23
9,668012|10,33 19
9,6683+3 0,331657
|9,663672|·0,33 132¯| 0
Co-tang. 、 Tangent !
Degree 65.
1
+ m
Degree 25.
MI Sine | Co-fine |
+
Tangent Co tang. |
|9,668672|10,331327|60|
019,62594819,957276|
19,6262199,957217
19,669002|10,330998/59
29,6264909,957158
9,669332 10,33066858
39,6267609,957099
9,669661|10,33033957
49,6270309,957040
9,669990 10,33000956
59,6273009,956981|
19,670320|10,329680|55|
69,627570 9,956922||
|9,670649|10.329351 54
79,6278409,956862
9,67097710,32902253
89,6281099,956803
9,67130610,328694|52|
99,9283789,956744
9,671634|10,328 36 551|
1019,62869:19,9566841
|9,671963|10;328037150
119,6289109,956625)
19,672291|10,32770949
129,6291849,956565|
9,672619 10,327381 48
13,6294539.956506
9,67294710;32705347
149,6297219,956449
9,573274 10,326721|46|
159,52998919,956387 | 5, 673 503 | 10,326398|45
9,673929|10,32607044
9,67425610,32574343
9,67458410,325 416 42
9,674910 10,32508941
19,67523710,32476340|
16 9,6302579,956327
176,6305249,956267
189,630792 9,956208
199,6310599,956148
209,63132619,9560881
21 2,631592 9,9560291
221,6318569,955969
269,63 29239,955739
239,5321259,95;909
249,6323929,955849
259,6326579,955789
27.9,6331899,955665
289,6334549,955609
299.6337199,955548
309,6339849,955488
| Co-fine | Sine 1.
2,67556410,324436|39|
9,675890 10,32411c|38|
7,6762 16 10,323783 37
|9,676543 10,32345736
9,676869 10,323 13 13 5
9,67719411,322805134
9,677520 (0,322480 33
9,67784510,32215432
9,67817110,321829 31
|9,678496|10,321504 30
Co-tang. Tangent (M
Degree 64.
Degree 25.
MI Sine Co-fine
309.6339849.9554881
Tangent Co-tang. I
19.678496|10.321504/30
319.6342499.9554281
329.6345149 955367
19.678821 10.3-1179129
9.679145 10 320854 28
339.6347789.955307
9.679471|10.320529|27|
349.6350429.955246
9.679795 10 32020526
359.6393069.955186
19.68012010 31988025
3619.63557019.955125|
19.680444110.31955624
379.63.58339.95 5065
9.680768|10.31923223
389.6360979.955004
9.681092 10.318908 22
399.6363609.954944
9.681416 10.31858421
409.6366239.954883
9.681740 10.31826020
4119.63688619.954823)
9.682063|10.31793719
429.637148 9.954762
9.682386 10.31761318
439.6374119.954701
9.682710 10.317290 17
449.6376739.954640
9.683033 10.31696716
459.637935′9.954579!
19.683356 10.31664415
4619.6381979-954518
19.683678|20.316321(14
479.6384589.954457
489.6387209.954396
4919.638981.9.954335
9.684001 10.31599913
9.684324 10.315676 12
9.684646 10.31535411
50 9.6392429.954274|
9.684968 10 315632!10
}
5119.639503 9.954213
9.68529c|10.314710
529.6397649.954152
9.68561216314388 8
539.6400249.954090
549.6402849.954029!
9.685934 10.314066 7
9.68625510 313745 6
559.6405449 9549681
9.686577|10.313423 5
569.6408049.953906|
9.686898 10.31310 | 4
9.687219 10.31278
579.6410649.953845
58 9.6413239.953783
599.6415839.953722
60 9.64184219.9536601
| Co-fine Sine
9.687540 10.312460
9.687861 10.312138
9 688182 10.311818
Co-tang Tangent | M
Degree 64.
Y
Degree 26.
M Sine | Co-fine
Tangent | Co-tang.
09.64184219.953660
19.688182f10.31181860
119.6421019.953598;
29.6423609.953537
39.642618 9.953475
49.6428769.953413
59.6431359.953351|
19.688502110.311498159
9.688823|10.317758
9.689143 10.31085757
9.68946310.31053756
9.689783 10.31021755
6 9.643393 9.953290|
79.6436509.953228
89.643908 9 953166
99.6441659.953104
10 9.6444239.953042|
1119.64468019 952980
129.6449369.952917
19.690103|10.30989754
9.690423 10.30957753
9.690742 10.30925852
9.691063 10.308938 51
9.691381|10.30861950
19.691700|10.30830049
9.69201910.0798148
139.6451939 952855.
9.692338 10.30766247
149.6454499.952793
9.692656 10.30734346
159.6457-69.952731
[9.692975|10.30;02545
16|9.645962|9.952668
179.6462189.952606
189.6464739.952544
199.6467299 95248 i
2019.64698419.952419
2119.64723919.952355
229.6474949.952294
239.6477499.952231
19.693293|10.306706144
9.693612 10.306388'43
9 693930 10.30607042
9.694248 10.30575241
9 694566 10.305434 40
|9.694883|10.30511739
9.69520110.304799 38
9.69551810.30448237
249.6480049.952168
9 695835 10.304164|36|
259.64825819.952105
19.69615310.30384735
26.9.64851019.9520431
279.648766 9.951980
28 9.6480209.951917
29 9.9492749.951854
309.6495279.951791
Co-fine Sine
Sine
9.697420 10.30258031
9 697738 10.302264301
Co-tang. Tangent, M
Degree 63.
19.696+7910.303539134
9.696786|10.30321333
9.697103|10.30289732
4
Degree 26.
MI Sine Co-fine | | Tangent Co-tang. |
3cl2.649527.9.951791
19.69773810.39225430
319.64978119.951728
19.69805-10.39.94729
329.500349.951665
9.6983691 30163128
339.6502879.9516021
9.698-8510.301315 27
349.6505199.951539
9.699001 10.300999 26
359.6504989.951476
19.6993 610.300684 25
369.651044,9.951412)
379.6512969.951349
389.6516489.951286
399 6518009.951222
40 9.6520529.951159
19.69963210.300358,24
9.59994710.300:52 23
9.70025310.299737 22
9.700578 10.29942221
4119.65230319.951095
429.65255 9.951032
439.6528069.950968
9.700893 10.299107|20
9.71208110.29874219
9.70152210.298477
9.701837.10.29816317
449.6530579.950905
9.7 2152 10.297848116
459.6533079.950841
19.702466.10.297534 15
4619.65355819.950777
479:6538089.950714
489.6540599.950650
499.6543099.950589!
50 9.6545589 950522
519.654808 9.950458
529.6550579.950394
$39.6553079.950330
549.6555569 95-266
559 6558059.950202
5619.6560539.950138
589.6565509.950009,
599.6567999.949945
60 9.6563479.919881|
1
| Co-fine Sine [
579.65630.9.950074)
I
9.70278210.29721614
9.703095 10.29695513
9.703499 10.290591 12
9.703722 10.2962; 711
9.704026 10.295964 10
19.704350 10.2956501 9
9.704663 10.295337 8
9.704976 10 295 23 7
9.705290 10.294710 6
19.70570310.294397 5
3
9.705915 10.2940841 4
9.706228 10.29377
9.706541 10 2934-92
9.706853.10.2931+
9.707100 10.292834
Co-tang. Tangent
Degree 63.
Y 2
1
1
Degree 27.
}
1
M
Sine | Co-fine
Tangent | Co-tang |
o'9.65704719.9408 cl 19.707166 10.292834160
IC
11{9.659763|9.949170
129.6600099.949105,
139.6602559.949040
14:9.660 009.948976
15'9.6607469-948910l
119.6572959.949816)
29.657542 9.949,52
39.6577909.949687
4.6580379.949623
519.6582849.949598
€19.65853819.949494!
79.6587779.949429
89.6590249 949364)
90 6592719.949300
c|o6595179.949225
9.707790 10.29221058
9.708102 10.291897 57
9.70841410.291 586 56
9.70872610.29127455
19 709037.10.290962154
9.709349.10.290651 53
9 709660 10.290340 52
9.709971 0.2929|51
9.710382 10.28971850
19.710593] .289407149
9 710904 10.28909648
9.711214 10.28378547
9.711525 10.28847546
9.711836 10.28816445
9.707478 10.29252359
169.66099119.9488451
179.6612369.948760
|9.712146}10.287854 44
9.712456 10.287544 43
189.661481 9.948715
9.712766 10.287234 42
199.6617269.948650
9.71307610.28692441
pol9.661970 9.948584)
9.713386 10.28661440
219.6622149.948519|
19.713695 10.28630539
229.6624599.948453
9.714005 10.28599538
239.662702 9.948388
9.714314 10.28568637
249.662947 9.948323
9.714624 10.28537636
259.66319 9.948257
19.71493310.28506735
269.6634239-948191|
9.71524210.28465834
279.6636777.948126
9.715550 10.28444933
289.6639.948060
9.715859 10.28414032
299.6 41639.947995
9.716168 10.28383231
309.66440019.947929
19.716477 10.283523/30
| Cofine Sine | Co-tang. Tangent M
Degree 62,
Degree 27.
|Tangent I, Go-tang.
19.7.1647.7110.283523130
MI Sine Co-fine. |
309.6644069.947929
3119.6646489.947863)
329.654891 9.947797
19.71678510,283215129
9.717093 10.28290728
339.6651339.947731
9.717401 10.282598 27
34 9.6653759.947665
9.717709 10.28229026
359.6656179.947599]
9.71801710.281983 25
3619.66585819.947533
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379.6661009.947467
9.718633 10.28136723
389.666341 9.947401
9.718940 10.28106022
399.666583 9.947335
9.719248 10.28075231
4019.66682419.947269
19.719555110.28044520
4119.66706519.9472031
19.71986210.280138/19
429.6673059.947136
9.720169 10.279831 18
439.6675469.947070
9.720476 10.279524 17
449.667786 9.947004
9.720783 10.279217|16|
4519.6680259.946937
19.721089|10.27891115
4619.668266,9.94687!
9.72139510.27860414
479.6685069.946804 9.721702 10.278298 13
489.6687469.946738
9.722008 10.277991 12
499.6689869.946671
9.72231510.27768511
509.6692259.946604
9.722621j10.277379 10
$119.6694649.945537
19.722927|10.2770731 9
$29.667039.946471
9.72323210.276768 8
$39.6690429.946404
9.723538 10.276462 7
549.670181 9.946337
9.72384310.275156 6
$59.6704199.946270
19.72414910.275851 | 5
569.6706579.946203
$79.6708969.946136|
$89.6711349.946069
$99.6713729.946002
6019.67160919.945935|
| Co-fine | Sine |
19.72445410.275546|
9.724759 10.275240 3
9.725035 10.274935 2
7.725369 10.274630|| 1
19.725674 10.274326 0
Co-tang. | Tangent M
Degree 62.
Degree 28.
MI Sine Co-fine I
| Co-fine Tangent | Co tang.
09,67169019,945935!
19,725674|10,274326|60|
119,671847 91945 868
29,6720849,945800
9,7259790,274021|59
9,72528410,27381658
39,6723214,94573 1
9,726581,27341257
49,5725589,945666
9,72689: 10,27310756
59,67279519,945598
9,72719710,27280355
619, 6730329,945531
79,6732689,9454
89,673509,94396
99,673741 9,945328
109,5739779,945261|
19,7275010,27249954
9,7278050,27219553
9,728109 10,271891 52
9,72841219,27158751,
|9,728716 10,271284150
119,6742139,945193}
9,729020|10,270980149
129,6744489,945125
9,72932310,270677|48|
A
139,6:46849,945058
9,729626 0,270374 47
149,6749199,944990
159,675 (5419,944922
9,7299290,270070|46|
9,730232110,26976745
169,6753899,944854|
|9,730535|10,269464|44|
179,6756239,944786
189,6758599,944718
199,6760949,944650
|20|9,676328|9,94458 2
9,730838 10,269162 43
9,73114110,26885942
9,731443 10,26855641
19,73 1746' 10,26825440
219,6765629,944514
229,676 796 9,944446|
239,6770309,944377
249,5772649,944309
2519,6774979,9442411
9,732048 10,26795239
9,73235110,267649|38|
9,732653 10,267347 37
9,732955 10,267045|36|
19,733 257|10, 2667435
269,67773119,944172
|27|9,677964|9,944104
289,6781979,94401€
19,733558/10,266441134
9,733860|10,26614033
9,73416210,26583832
92734463110,265537 31
299,678430 9,943967
3019,678663 9,943898|
| Co-fine | Sine I
19,734764 10,265236 30
Co-tang. Tangent |
Degree 61.
Degree 28.
M Sine Co-fine | Tangent Co-tang. I
309,67866319,9438981
19,734764110,265236130
|31|9,67889)943830
|9,735666|10,264934|29|
329,6791289,943761
|9,735362|10,264633 28
339,67936c9,943692
9,735668 10,26433227
349,6 9592 9,943624
9,735968 10,264031|26|
359,67982419,94355
9,736÷69|10,26373125
369,6800,69,943486]
37,6802889 4 3 4 17
389,6805199,943348
399,68075´ |9,943279
4019,68098219,9 +3 210!
4119,6812139,943141|
420,681443 9,943-72
43 9,6816749,943003
449,681904 9,742933
45}9,682135′9,942864
19,736570|10,263430124
9,736870 10,26313023
9,7371710,262829 22
9,73747262529,21
9,7377710,262229'20
19,738071 10,26192919
9,738371 10,26629 18
9,738671 10,26132917
9,738971|10,261029|16
19,73927110,16072915
4619,682365 9,942795|
9,739570 10,260430|14|
479,6825959,942725
9,739870 10,26013013
489,682825 9,942656|
499,683055 9,942587|
509.683284'9,94 25 17
9,740169 10,259821 12
|9,740468|10,259 53211
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19,740767|10,25923310
5 19,68 35 1419,942448
529,6837439:942378
539,683972 9,942308
549,684201 9,942239
5519,6844309,942169
19,741066|10,258934 9
9,741365|10,258635 8
9,74166410,258336 7
|9,741962|10.258038 6
19,7422610,257739 5
569,6846589,942099
579,6848879,942029
[9,74 2559|10, 257441 4
9,742858 10,257142 3
589,6851159,941059
9,743156 10,256844 2
599,68 53439,941889
9,74345410,256546
1
609,6855719,941819
9.74375 10,256248
Co-fine Sine | | Co-tang. | Tangent. M
Degree 61.
Degree 29.
1
M Sine Co-fine
n 9,685579,9+18191
1|9,6857999,941749|
29,6860279,941679
39,6862549,941609
49,686489,941539
59,6867cy 9,941468
69,6869369,941398
7:9,6871039,941328
89,6873899,941257
99,6876169,941187
109,6878429,941116
119,6880699,941046|
129,68 8295 9,94097
Tangent Co-tang. I
9,74 375 2|10,25 6248 16 07
9744050 10,255950
9,744348 10,255652 158
9,744645 10,255355 57
9,744943 10,255057 56
9,745240 10,254760'55
19,7 +55 38; 10,254462|54
9,745835 10,25426553
9,7461321025386852
9,746429 10,2535751
|9,746726′ 10, 25327450
9,747023|10,252977149
9,747319 10,25268048
99747616 10,25238447
9174791210,252087146
19,748209 10,251791 45
136,6885239,9409°5
49,6887479940834
159,68897219,940763|
169,6891959,940693|
9748505 10,251495 44
179,6894219,940622|
9,748801 10,251199 43
189,6896489,940551
199,6898739,940480
20 9,6900989,940409!
219,69032319,94 338|
·
9,749097 10,25090242
91749393 10,25060741
|9,749689 10,25031140
229,6905489,940267
239,6907729,940196
249,5909969,940125
259,6912209,940053}
269,691444 919 39y8 2|
279,091668|9,939911
2289,691892 ,9 ;9840
299,69211,939768
309,6923359,939697)
| |
Co-fine Sine
9.749985110,25001539)
9,75028110,249719|38
9,7505 76 10,24942437
9,750872 0,249128|36|
9,75116710,248833′35
•
19,7514620,-4853834
6,751757 10,24824333
9,75205210,24794832
9,752347 10,24765331
9,7526+210,247358130
| Co-tang. Tangent. M
Degree 60.
1
Degree 29.
MI Sine Co-fine I
Tangent Co-tang. I
3' 19.692339:9.9396971
19.752642 10.24:358130
31/9.5925629.939625)
329.6927859.939554
339.6930089.939+82
19.75293710.247063/29
9.75323110.246769 28
9.75352610.246474 27
349.693231 9.939410
359.6934539.939339|
9.753820 10.246180 26
19.754115 10.245885/25
36:9.69367619.939267|
379.693898'9 939195
389.6941209 939123
399.694342 9.939051
4019.694664 9.938980
19.754409|10.24559124
9.754703 10.24529723
9.754997 10.24500322
9.755291 10.244709|21
9.75558410.24441520
4119.694786 9.9389081
429.695007.9.938835
439.6952299.938763
44.9.695450 9.938691
459.695671 9.9385,9
4619.69589219.938547.
479.6961139.938475
48 9.6y63349.938402
1499965549938330
509. 967749.938257
519.09699519.938185|
529.69721 2.938112
539.69743593804
549.697659-937967
559.697874 9.937895
19.75587810.244122|19|
9.75617210.243828 18
9.756465 10.24353517
9.756759 10.243241|1
9.75705210.242948 15
9.75734510.2426551
9.757-48 10.242362/13
9.75793110.2420691
9 75822410.24177511
9.758517 10.241483 10
758810 10.241190 9
9.759102 10.24 898 8
9 759395110.240605 7
9.759087·10.2403136
19.7599-9 16.240021) 5
5619.69809319.9378221
19.760271110.239728 14
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9.760564 10.2394363
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9.720856|10.239144 2
599.6987519.937603
9.76114710.238852 1
609.6989709.937535
19.751439 10.238561 0
| Co-fine
Sine i Cotang. Tangent M
Degree 60.
Z
Degree 30.
C
MI Sine | Co-fine |
o/9.6989799.937531
1
Tangent Co-tang.
19.761439 10.23856160
119.699189 9.937458
29.6994079.937385
39.6996269.937312
49 6998449.937238
19.761731110.238269159
9.762023 10.237977 58
9.762314 10.237686 57
9.762606 10.237394 56
59.7000629.937165
9.762897 10.237103 55
69.70028019.937092:
79.70@498 9.937019
89.7007169.936945
99.700933.9.936872
10 9.701151 9.936799|
1119.7015679.936725
129.7015859.936652
139.7018029-936578
149.702019 9.936505
|9.763188|10.23681254
9.753479 10.23652153
9.763770 10.23623052
9.764061 10.23593951
9.764352 10.235648 50
19.764643|10.235357149
9.764933 10.23506748
9.765224 10.23477647
9.765514 10.23448646
159.7022369.936431
9.765805 10.23419545
16|9.7024529.935357)
179.7026699.935284
19.766095 10.23390544
9.766385 10.23361543
189.7038859.935210
9.766675 10.23332542
199.7031019.936136
9.766965 10.233035|41
209.7033179.936062
9.76725510.232:45/40
21|9.703533,9.9359881
229.7037489.935914
19.767545|10.23245539
9.767834 10.232166 38
23 9.7039649.935840
9.76812410.231876 37
249.7041799.935700
259.7043959.935692
9.7684 310.23158736
9.768703|10.23129735
2619.704610,9.9356181
279.7048259.935543
28'9.705040 9.935469
29 9.705254 9.935395
30 9.705469 9.935320
| |
Co-fine Sine I
9.770148|10.22985230
| Co-tang. Tangent. IM
Degree 59.
19.768992/10.231008134
9.769281|10.23071933
9.769570 10.23043032
9.76985910.23014131
}
Degree 30.
MI Sine Co-fine | Tangent Co-tang. |
3019.70546919-935320
19.77014810.229852130
3119.70568319.935246)
329.705897 9.935171
33 9.7061129.935097
349.706326 9.935022
359.70653919-934948
19.770437110.229553120
9.77072610.22927428
9.771015 10.228985|2.
9.771303 10.22869726
19.77159210.228408/2
5
3619.70675319.934873|
19.771880 10.22812024
379.7069679.934798
9.77216810.22783223
389.7071809.934723
9.77245610.22754322
399-707393 9.934649
9.77274510.22725521
409.70760619.934574
19.77303310.226967|20|
4119.707819 9.9344991
19.773321110.226679119
429.7080329.934424
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449.7084579.934274
459.7086709.934199|
9.773608 10.22639118
9.773896 10.226104 17
9.774184 10.22581616
9.774471 10.22552915
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479.7090949.934048
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19.774759 10.225241 14
9.775046 10.22495413
9.775333 10.224666 12
9.775621 10.22437911
5019.7097309.933322
9.775908 10.224092|10
5419.709949.933747
529.7101539.933671
539.7103649.933596
549.7505759.933520
559.7107869.933444
19.77619510.223805, 9
9.776482 10.223518 8
9.776768 10.223232 7
9.77705510.222945
19.77734210.222658 5
569.710997;9-933369
19.777628|10.222372
$79.7112089.933293
9.77791510.222085
589.7114189.933217
9.778201 10.221799
3
$99.7116299.933141
609.7118399.9330651
9.77848710.221513
19.77877410.221226
| Co-fine | Sine I | Co-tang. Tangent
Degree 59.
Z 2
1
x
Degree 31.
1 Sine Co fine)
Tangent Co-tang I
19.778774 10.22122660
019.7118399.93306
19.71204919.932990|
-9.7122599.932914
9.712469'9.932838
49.7126799.932761
59.7128899 932685
9.779060 10.220940159
9.779346.10.22065458
9.77963210.220368 57
9.779918 10.220082 56
19.780203 10.219796 55
19.780489 10.219511154
9 71309819.932609
9.713308 9.932533
3.7135179.932457
2.713726 9.932380
2.713935 9.9323041
9.780775 10.219225 53
9.781060 10.21894 52
9.781346 10.21865451
19.781631|10.218369|50
{119.71414419.932227
29.7143529.932151
!39.7145619.9320-4
1499.7147699.931998,
15 9.7149779.9319211
519.715185,9.931845
79.715394 9.931768
189.715601 9.931691
'99.715809 9.931614
2019.7160179.931537
19.781916 10.218084149
9.782202 10.21779948
9.782486 10.21751447
9.782771 10.21722946
|9.783056|10.216944 45
9.783341|10.21655944
19.78,626 10.21637443
19.783910 10.21609c42
9.78419510.21580541
9.84479 10.2155240
19.7162249.931460
29.7164319.931383
33.7166399.931306
9.7168459.931229
·19.7-70539·931152
9.7172599.931075
9.717456 9.930998
89.7176729.930920
299.717 699.930843
9.718 859 930766
| Co-fine
19.784764110.21523639
9.785048 10.21495438
9.78533210.2 4668 37
9 78561610.214384 36
9.785900 10.21409935
9.786184|10.21381634
9.786468 10.21353233
9.786752 10.2:3248 32
9.787036 10.21296431
19.7873 910.21268130
1
Sine |
Co-tang. Tangent j M
Degree 58.
Degree 31.
[M] Sine | Co-fine. |
309.71808 519.930766
3119.71829 9.9306881
329.7184979 930.11
339.7187039 930533
349.7189299.930456|
359.7191149.9303781
Tangent Co-tang. |
19.78731910.212681/30
9.787603|10.212397|29|
9.787886 10.21211428
9.788170 10.21183027
9.788453 10.21154726
19.78873610.211264 25
369.7193209.930300|
19.78901910.210981|24
379.7195259.930223
9:789302 10.210698 23
389.7197309.93014- 9.789585 10.21041522
399.7199359.930057
9.789868 10.210132|21|
409.72014019.929989
9.79015110.209849 20
419.7203459.929911
429.7205499.929833
439.720754 9.929755
449.720958 9.929677
45.9.7211629.929599|
469.721366 9.929521|
19.790433 10.209565|19|
9.790716 10.20928418
9.790999.10.20900117
1
9.790281 10.20871916
9.791563 10.20843615)
479.7215709.929442
489.721774.9.929364
499.721978 9.929286
509.7221819.929207
19.791846|10.208154|14
9.792128 10.20787213
9.792410 10.207590 12
9.79269210.20730811
19.792974 0.20702410
፡
5119.722385 3.929129
529.7225889.929050
539-722791|9.928972
54 9.7229949.928893
5519-7231979 928814)
$19.7234009.928736
57 9.723603 9.928657
589 7238059.928578
599.7240079.928499
609.7242109.928420
Co-fine | Sine |
9.794945 10.205054 3
9.79522710.204773 2
9.795508 10 204492 I
19.79578910.20421110
Co-tang. Tangent M
Degree 58.
19.79325610.206744| 9
9.79353810.206462| 8
9.79381910.2061807
9.794101 10.205899 6
9.79438310.205617 5
19.794664|10.205336| 4
Degree 31.
M] Sine Co-fine |
Tangent | Co-tang. |
19,795789/10, 204211†60
09,7242109,928420|
19,7244129,928341
29,7246149,928262
[9,796070|10,203930|59
9,796351|10,20364958
39,7248169,928183
9,796632 10,203368 57
49,7250179,928104
59,7252199,928025
9,79691310,203087 56
19,797194 10,2028-6′55
69,72542019,927946)
9,797474 10,202523|54|
79,725622 9,927867
9,797755 10,20224553
89,7258239,927787
9,798036 10,20196452
99,7260249,927703
9,798316 10,201684|51
109,7262259,927528
9,798596 10,201404 50
[119,726426|9,927549
19,79887710,20112349
129,7266269,927469
9.79915710,200843 48
139,7268279,927390
9,79943710,20056347
9,799717 10,20028346
9,79999710,20000345
19,800277,10,19972344
9,80055710,199443 43
149,727027927310
159,7272289,927231|
16 9,7274289,927428
179,7276289,927628|
189,7278289,926828
199,7280279,926027
209,7282279,926227|
219,728437 9,926751
229,728626 9,926671
239-728825 9,926591
24917290249,926511
2519,729 22319,926431|
269,7294229,926351 |
27'9,7296219,926270
289,7298209,926190
299,7300189,926110
309,7302169,926029
| Co-fine | Sine
9,80083610,199163 42
9,801116 10,19888441
9,801396 10,19860440
|9,801675|10,198325|39|
9,801955 10,198045 38
9,802234 10,19776637
9,802513 10,19748736
9,80279210,19720735
19,80307210,19692834
9,80 3351 10,19664933
9,803630 10,19637032
9,803908 10,19609131
|9,804187|10,195813′30
| Co-tang. | Tangent. M
Degree 57.
Degree 32.
M Sine Co-fine I
1
Tangent Co-tang. I
3019,7302169,9260291 19,804187/10,19581330
319,7304159,925949
329,730613 9,925868
339,7308119,925787
349,731009 9,925707
359,73120619,925626
9,804466 10,19553429
9,804745 10,195255 28
9,80502310,194977 27
9,805302|10,194698|26|
19,805580 10,19442025
369731404/9,9255451
379,7316019,925464|
9,805859110,194141|24
9,806137 10,193863|23
389,731799 9,925384
|39|9,73 1995 9,925303
409,73219319,925222|
4119,732390 9,925141|
42 9,7325879,925060
43 9,732784 9,924978
449,7329809,924897
459,7331779,9248:6|
9,806415 10,193585 22
9,80669310,19330921
19,806971|10,193018|20|
9,807249|10,192751|19|
9,807527 10,192433 18
9,80780510,192195|17
9,808083 10,191917 16
19,808361|10,191639|15
4619,733373,9,924735
19,808638|10,191362|14
479,7335699.924653
9,808916 10,191084 13
489,7337659,924572 9,809193 10,190807|12
499,7339619,924491
509,7341979,924409|
9,809471 10,19052911
9,809748 10,190252'10
519,7343539,924328
529,7345489,924246|
5 39,7347449,924164|
549,7349399,924083
5519,73513419,934001|
5619,73533019,923919
579,7355259,923837
589,7357199,923755
599,7359149,923673
|60|9,736109|9,92359:
| Co-fine Sine
19,810025|10,189975| 9
9,810302 10,189697 8
9,810580 10,189420 7
9,810857 10,189143 6
9,811134 10,188866 5
9,81141010,:88,891
9,811687 10,188313 3
9,811964 10,188036
9,812241 10,187759
9,81251710,187483
| Co-tang | Tangent | M
Degree 57.
1
Degree 33.
39,736692 9,923 345
49,7368869,923263
519,737-809,923180
69,7372749,923098|
M Sine Co-fine I
09,73610919,923591|
19,736309 9,923509
29,786497 9,923427
.
Tangent Co-tang. |
19,812517 | 10,18748360
[9,812794 10,18720659
9,813070 10,8693058
9,8133470,18665357
9,81362310,186 77
19,813899 10,18610: 55.
i
9,81417510,18,824
79,7374679,923016
9,814452 10,18554853
89,737661,92 2933|
9,814728 10,18527252
99,7378549,922851
9,815004 10,18499651
109,7380489,922768
9,815279 10,18472050.
119,7382419,922686)
19,81555510,18444549
129,7384349,922603
9,815831 10,184169 48
139,7386279,922520
A
149,7388209,922438
9,81610710,183893 47
9,81638:10, 18361746
1519,73901319,922 35 51
9,816658 10,18334245
16|9,739205|9,92 2 2 7 2 |
179,7393989,922189
19,816933|10,183066|44|
9,817209 10,18279143
189,739590 9,922106
9,81748410,12251642
199,7397839,922023
9,817759 10,18224041
2019,73997519,921940|
19,81803510,18196540
219,7401679,921857
229,7403599,921774
239,7405509,921691
249,7407429,921607
|2519,7409349,921524
|26|9,7411259,9214411
279,7413169,921357
289,7415079,921274
299,7416989,921198
309,74188919,921107|
| Co-fine |
Sine I
9,818310f1c,18169039
9,818585 10, 18
1415 38
9,818860 10,18114037
9,81913510,18086536
19,819410 10,18059035
9,819684|10,180315134
9,819959 10,180041 33
9,820234 10,17976632
9,82050810,179492 31
19,820783110,179217 30
| Co-tang. Tangent
Degree 56.
2
Degree 33.
M Sine Co-fine I
3919.7418899.921107|
Tangent Co-tang. I
9.820783 10.179217|30
3119.74208019.921023]
329.742271 9.920939
339.7424619.920855
349.7426529.920772
19.821057|10.178943|29|
9.821332 10.178658 28
9.821606 10.178394 27
9.821880 10.17812026
359.742842′9.920688)
19.822154 10.17784625)
3619.743032 9 920604
9.822429 10.17757124
379.743223 9.920520
9.822703 10.177297|23|
389.7434129.920436
9.822977 10.177023 22
1399.7436029.920352
9.823250 10.17673921
4019.7437929.920268
9.82352410.176476 20
4119.7439829.920184|
$429.744171 9:920099
439.744361 9.920015
449.7445509.919931
14519-74473919.919846
46|9.7449289.919762|
47 9.7451179.919677
48 9.745306 9.919593
49 9.745494‍9.919508
sol9.7456839 919424
5119.74587119.9193391
529.7460599.919254
539.746248 9.919169
19.823798,10.17620219
9.824072 10.175928 18
9.82434510.17565517
9.82461910.17538116
(9.82489210.175108 15
19.825166|10.17483414
9.825439 10.17456013
9.82571310.17428712
9.825986 10.17401411
9.82625910.17374110
19.82653210.1734681 9
9.826805 10.173195 8
9.827078 10.12922 7
549.7464369.919084
|55|9.7466249.918999|
9.827351 10.172649 6
9.827624 10.172376
569.74681119.918915) 19.827897 10.172103
$79.7469999.918830
589.7471879.918744 9.828442 10.171558
9.828170 10.171830
599.7473749.918659
9.82871510.171285
16019.74756219.918574)
9.82898710.171012
| Co-fine | Sine
| Co-tang. ↑ Tangent.
Degree 56.
A a
Degree 34.
'
MI Sine | Co-fine | | Tangent Co-tang.
ol9.7475629.918574
19.828987 10.17101260
119.7477499.918489
19.829260 10.170740159
29.7479369.918404
9.829532 10.170468 58
39.7481239.918318 9.829805 10.17019557
49.7483109.918233 9.830077 10.169923|56|
59.7484979.918147 19.830349 10.16965155
6|9.748683|9.9180621
19.830621|10.16937954
79.7488709.917976 9.830893 10.15910653
89.7499569.917891
9.831165 10.168834 52
99.7492429.917805
9.831437 10.16856351
10 9.7494299.917719
9.831709 10.168291 50
1119.74961519.917634
19.831981|10.16801949
129.7498019.917548
9.832253 10.167747 48
139.749986 9.917462
9.832525 10.16747547
149.750172 9.917376
9.832796 10.16720446
159.750358 9.917290
9.833068 10.16693245
16|9.75054319.917204|
179.7507299.917118
18 9.7509149.917032
199.7510999.916945
2019.751 284'9.916859
19.833339|10.166660144
9.833611 10.16638943
9.833882 10.16611842
9.834154'10.16584641
9.834425 10.165575 40
219.7514699.916773
|9.834696|10.165304139
229.7516549.916686
9.834967 10.165033 38
239.7518389.916600
9.835238 10.16476237
249.7520239.916514
9.835509 10.16449136
259.7522079.916427
9.835780 10.16422035
|26|9.752392|9.916340
279.7525769.916254
19.836051|10.163949134
9.836322 10.16367833
289.7527609.916167
9.836593 10.16340732
299.7529449.916080
3019.7531289.915994
9.83686410.16313631
9.83713410.102866 30
| Co-fine
Sine
| Co-tang | Tangent |M
Degree 55.
Degree 34.
MJ Sine Co-fine
309.75312819.915994
| Tangent [- Co-tang. [
9.837134 10.162866130
319.7533129.91 5907|
9.837405 10.162595129
329.7534959.915820
9.837675 10.162325/28
339.7536799.915733
9.837946 10.162054 27
349.7538629.915646
9.838216 10.161784|26|
13519.75404619.915559
19.83848710.16151325
369.754229.9.915472|
[9.83875710.161243 24
379.7544129.915385
9.839027 10.16097323
38 9.7545959.915297
9.83929710.16070222
399.7547789.91 5210
9.839568 10.160432'31
4019.7549609.915123
9.839838,10.160162 20
4119.75514319.915035
19.840108/10.159892/19
429.7553259.914948
439.755508 9.914860
9.840378 10.15962218
9.840647 10.15935217
449.755690 9.914773
9,840917 10.159083 16
4519.7558729.914685
9.841187/10.15881315
4619.7560549.914597|
841457,10.158543 14
476.7562369.914510
48 9.7564189.914422
499.7566009.914334
5019.7567819.914246
9.841726 10.15827313
9.841996 10.158004 12
9.842266 10.15773411
19.84253510157465 10
5119.7569639.914158
19.842804|10.157195, 9
529.7571449.914070
9.843074 10.156926 8
539-7573269.913982
9.843343 10.156657 7
549.7575079.913894
9.843612 10.156387 6
5519-7576889.9:3806
9.843882110.156118 5
.019.75786919.913718
19.844151|10.155849 4
$79.75804919.913630
9.844420 10.155580 3
89.7582399.913541
9.844689 10.155311 2
599.7584119.913453
9.841958 10.155042 I
6019.758591|9.913364|
9.84522710.154774 o
| Co-fine Sine
| |
1
Co-tang. Tangent M
Degree 55.
A a 2
Degree 35.
Sine Co-fine 】
Tangent | Co-tang I
919.758591|9.9133641
19.84522710.15477460
19.758772;9.913276
19.845496 10.154504159
29.7589529.913187
6.84576410.154235 58
39.7591329.913099
9.84603310.153967 57
49.7593129.913010
519.7594929.912921|
9.846302 10.15369856
9.84657010.153429|55|
619.7592729.912833)
19.846839|10.153161|54|
79.7598519.912744
89.7600319.912655
9.847107 10.15289-53
9.847376 10.152624 52
99.7602109.912566
9.847644 10.15235651
10 9.7603909.912477
9.84791310.15208750
1119.76056919.912388
19.848181110.15181949
129.7607489.912299
9.848449 10.15155148
139.7609279.912210
9.84871710.15128347
149.7611069 912121
9.84898510.15101546
159.7512859.912030
[9.849254/10 15074645
16|9.7614649.911942
19.849522 10.15047844
179.7616429.911853
9.84978910.15021443
189.7618219.911763
9.850057 10.14994342
199.7619999.911674
9.85032510.14967541
2019.7621779.911584
19.850593 10.149467140
21|9.762356|9.911495|
19.85086110.14913939
229.7625349.911405 9.851128 10.14887238
239.7627129.911315
9.851395 10.14860437
249.7628899.911226
9.851664 10.14833636
2519.763067|9.911136|
9.851931 10.14806935
269.7632459.911046|
19.85219910.14780134
279.7634229.920956
9.852466 10.14753433
289.7635999.910866
9.852731 10.14726732
299.7637779.910776
9.853001 10.14699931
3019.7639549.910686
9.853268 10.14673230
| Co-fine | Sine |
| Co-tang. Tangent. M
Degree 54.
Degree 35.
M Sine Co-fine |
Tangent Co-tang. |
309.76395419.910686)
3119.7641319.910596,
329.7643089.910506
339.764485
19.853268.10.1466732'30
19.853532/10.14646529
9.853802 10.14619828
9.910415
9.854069 10.14597027
349.764662
9.910325
9.854336 10.145664 26
359.7648389.910235
19.85460310.14539725
369.7650159.9101441
19.854870 10.14513024
379.7651919.910054
9.85513710.144863|23
389.7653679.909963
9.855404 10.144596 22
1399.7655449.909873
9.855671 10.144329|21
4019.76572019.9097821
19.855937!10.144063|20
419.7658969.909591|
9.85620410.143795|19|
429.7660719.909601
439.7662479.909510
9.856471 10.14352918
9.856737 10.143268|17
449.7664239.909419 9.857004 10.14299616
459.7665989.909328|
9.857270|10.14273015
4619.7667749.509237|
479.7669499.909146
489.7671249.909055
9.858069 10.1419311
499.7672999.908964
5019.7674749.908873
19.85753710.14246314
9.85833610.14166411
9.858602 10.14139810
9.857803 10.14219713
5119.7676499.908 7811
529.7678249.908690
53 9.767997.9.908599
19.858868 10.141132 0
9.859134 10.140866 8
9.859400 10.140600 7
549.768173 9.908 507
5519.768348 9.908416
9.859666 10.140334 6
9.85993210.140068 5
15619,768522|9.908324)
19.860198|10.139802|4)
$79.7686969.908233
9.86046410.139536 3.
$89.76887119.908141
599-7690459.908049
609.7692199.907958|
9.860730 10.139270 2
9.860995 10.139005 I
19.86126110.138739 0
| Co-fine | Sine
| Co-tang. | Tangent. |
Degree 54.
Degree 36.
MI Sine | Co-fine |
| Tangent | Co-tang. \
19,76922919,907958
19,861261|10,13873960
19,769392|9,907866
19,861527 10,138473159
29,7695669,907774
9,86179210,138208 58
39,7697409,907682
9,862058 10,137942 57
49,7699139,907590
59,77008719,907498|
9,86232310,137677 56
19,862589′10,137411'55
69,7702609,907406|
19,86285410,13714654
79,7704339,907314
9,863119 10,13688053
89,7706069,907221
9,863385 10,13661552
99,7707799,907129
9,863650 10,13635051
109,7709529,907037|
|9,863915|10,136085 50
119,7711259,906945
19,864180|10,13582049
129,7712989,906852
9,86444510,13555448
139,7714709,906760
9,864710 10,135289 47
149,771643 9,906667
9,864975 10,13502446
1519,7718159,906574
9,865240 10,134759'45
|16|9,77198719,9064821
19,865505|10,134495|44
179,7721599,906389
9,865770 10,13423043
189,7723319,906296
9,866035|10,13396542
199,7725039,906203
9,866300 10,13370041
209,7726759,906111
9,86656410,13343640
219,7728479,906018
19,866829 10,13317139
229,7730189,905925
9,86709410,132906 38
239,7731909,905832|
9,867358 10,132642|37
9,867623 10,13237736
19,86788710,13211335
249,7733619,905738
|25|9,773 5 3 39,905645|
269,7737049,9055521
279,773875 9,905459
289,7740469,905365
299,7742179,905272
309,77438819,905179
Co-fine Sine I
{
19,868152|10, 13184834
9,868416 10,13158433
9,868680 10,13132032
9,868945|10,131055 31
19,869209|10,13079130
Co-tang. Tangent M
Degree 53.
t
Degree 36.
My Sine Co-fine I Tangent Co-tang. I
3019,774388/9,905179|
19,869209/10,130791/30
319,77455819,9050851
19,86977310,13052729
329,7747299,904992
9,869337 10,130263|28|
339,7748999,904898
9,870001 10,129999|27|
349,7750709,904804
9,870265 10, 2973526
359,77524019,90471 1
9,870529|10,129471|25|
369,7754109,904617
19,870793110, 129207|24
379,7755809,904523
9,871057|10,128943|23
389,7757509,904429
9,871321 10,128679 22
399,7759209,904335
9,871585 10,128415 21
4019,7760909,904241
19,871849 0,128151|20
4119,7762599,9≈4147|
429,7764299,904053.
439,7765989,903959
9,872112110,127888119
9,872376 10,12762418
9,872640 10,127360|17|
|449,7767689,903864
9,872903|10,127097 16
459,7769379,903770
|9,873167|10, 126833|15
469,77710619,903676
19,873430 10,126570|14
47 9,7772759,903 581
489,7774449,903486
499,7776139,903392
9,873694 10,12630613
9,873957 10,126043 12
9,874220 10,12578011
509,7777819,903 298
9,87448410,12551610
519,7779509,903203|
19,
529,7781199,90 3108
,874747 | 10, 125253]
9,87501010, 124990
9
8
539,7782879,9°3013
9,87527310,124727
549,7784559,902919
$519,778623'9,902824)
5619,778792 9,902729|
579,7789609,902634
589,7791299,902539
599,7792959,902444
609,7794639,902349
Co-fine Sine
9,87553610,124464 6
9,875799 10,124201|
19,876063|10,123937| 4
9,876326 10,123674 3
9,876589 10,123411 2
9,876851|10,123149|| 1
19,87711410, 122885 0
Co-tang. Tangent M
Degree 53.
Degree 37.
MI
Sine | Co-fine. | |Tangenti Co-tang. |
99,77946319,9023491
19,877114/10,12288560
19,7796319,902253
9,87737710,122623/59
29,7797989,90 2 158
9,877640 10,12236058
39,7799659,902063|
9,877903 10,122097 57.
49,7801339,901967
9,878165 10,12183456
$19,7803009,901872)
9,878428/10, 12157255
6|9,780467|9,901776
9,87869110, 12130954
79,7806349,901681
9,878953|10,121047 53
89,7808019,901585
9,87921610,120784 52
99,7809689,901488
9,879478 10,12052051
109,7811349,9013911
19,879741105:20259150
149,781301|9,901298
19,880003|10,119997|49
$29,7814679,901202
9,880265 10,11973448
139,7816349,901106
9,880528 10,119472 47
149,781800 9,901010
9,880790|1c,r1921046
159,7819669,900914
19,88105210,11894845
169,7821329,900828|
9,88131410,118686|44
179,7822989,900722
19,8
189,7824649,900626
199,7826909,9005 29||
881576 10,11842443
9,881839 10,11816142
9,882101 10,11789941
pol9,78 279619,9004331
19,882363/10,117637140
219,78296119,900337
9,882625 10,117375 39
229,7831279,900240
9,882886 10,11711438
239,783292 9,900144
9,883148 10,11685237)
249,7834579,900047
9,88341010,11659036
259,7836239,899951:
19,88367210,11632835
2619,78378819,8998541.
(279,7839539,899757
289,784118 9,899660
299,7842829,899563
309,7844479,699467|
Co-fine Sine |
9883934|10,116066|34
9,884195 10,115805 33
9,88445710,11543 32
9,884719 10,1 152813
19,884980/10,11502030
31
Co-tang. Tangent M
Degree 52.
Degree 37.
M Sine Co-fine
3019.78444719.899267|
Tangent Co-tang. I
19.88498010.115020'30
9.885242/10.114758129
3119.78461619.89937c
329.7847769.899273
9.885503 10.114497 28
339.784941 9.899175
9.885765 10.114235 27
349.7851059.899078
9.886029 10.113974 26
3519.78526919.898981
19.886288 10.11371225
3619.7854339.898884
19.886549110.113451124
379.7855919.898784
9.886810 10.11319023
389.7857619.898689
9.887072 10.11292822
399.7859259.898592
9.887333 10.112667|21
409.7860889.898494|
9.887594 10.112406 20
4119.78625219.898397
19.88785510.112145|19|
429.7864169.898 299
9.888116 10.11188418
439.7865799.898201
449.786742 9.898104
459.7869099.8980061
9.88837710 111623 17
9.888638 10.111362 16
19.888899.10.11
4619.7870199.897908
479.7872329.897810
48 9.7873959.897112
49 9.7875579.897614
5019.7877209.897519
5119.7878839.897418
$29.7880459.897320
$39.788208 9.897222
549.788370 9.897123
559.7885329.897025
9.88916010.110840114
9.889421 10.11057913
9.889682 10.11031812
9.889943 10.11005711
9.88020410.109796 10
19.890965,10.109535 9
9890725 10.109275 8
9.890986|10.109014 7
9.891247 10.108753 6
19.891507, 10.108493 5
5619.78869419.896926]
19.891768|10.108232
579.7888569.896828
$89.7890189.896729
9.892028 10.107972 3
9.892289 10.107711
599.7891809.896631
9.892549 10.107451
609.7893429.896532
19.892810/10.107190
Co-fine Sine | | Co-tang. Tangent M
Degree 52.
Bb
1
Degree 38.
M Sine | Co-fine | Tangent Co-tang. I
09.7893429.896532
19.892810|10.10719960
119.7895049.896433
[9.893070|10.106939159
29.7896659.896335
9.893330/10.10666958
39.7898279.896236
9.893591|10.106409 $7
49.7899889.895137
9.893851 10.106149 56
59.7901499.8960381
9.894111 10.10588955
69.7903109.895939
19.894371|10.105628154
79-7904719.895840
9.894632 10.105368 53
89.79063 29.895741
9.894892 10.10510852
99.7907939.895641
9.895152 10.10484851
10 9.7909549.895542)
9.895412 10.104588 50
1119.79111519.8954431
19.895672 10.104328149
129.7912759.895343)
9.895932 10.10406848
139.7914369.895244
9.896192 10.10380847
149.7915969.895144
9.896452 10.10354846
159.7917569.8950451
9.896712 10.10328845
1619.7919179.894945
9.896971|10.103028144
79.7920779.894846
9.897231 10.10276943
189.7922379.894746
9.897491 10.102 50942
199.7923979.894646
9.897751 10.10224941
2019.79255719.894546
9.898010 10.10199040
2119.7927169.894446
19.89827910.101.739139
229.7928769.894346
9.898530 10.101470 38
239.7930359.894245
9.89878910.10121137
249.7931959.894146
9.899049 10.100951 36
259-7933549.894046
19.899308|10.10069235
2619.7935139.893946||
279.793673,9.893845
289.7938329.893745
1299-793991 9.893645
30′9.7941499.893444)
| Co-fine Sine |
19.899568 10.100432134
9.899827 10.10017333
9.900086 10.09991332
9.900346 10.09965431
19.900605|10.099395 30
Co-tang. Tangent M
Degree 51.
Degree 38.
MI Sine Co-fine | Tangent Co-tang. |
3©19.794149|9,893444)
19.90060510.099395|30
3119.79430819,8934441
19.900864|10.099135|29|
329.7944679.893343
9.901124 10.09887628
339.7946269.893243 9.901383 10.09861727
349.7947849.893142
9.901642 10.09835826
35.9.7949429.893041 19.901991 10.098099 25
3619.79510119.892940|
19.902160 10.09783-124
379.7952599.892839
9.902419 10.097580 22
389.7954179.892738 9.902678 10.097321 22
399.7955759.892637
9.902937 10.09706221
4019.795733'9.892536)
9.903196 10.096803/20
4119.79589119.8924351
429-7960499.892334
439.796206 9.892233
44 9.7963649.892132
459.7965219.892030
9.903973 10.09602717
9.904232 10.095768 16
19 904491|10.09550915
19.903455|10.096544|19
9.903714 10.09628518
4619.79667819.8919281
479.7968369.891827
19.90475010.09525014
9.90500810.094991|13
489.7969939.891726
49 9.7971509.891624
sol9.79730719.891522
9.90526710.094733|12
9.905526 10.094474|11
9.905784 10.094215|10
519-79746719.891421
529.797621 9.891319
539-7977779.891217
549.797934 9.891115
559.7980919.891013|
19.906043|10.093957, S
9.906302 10.093698 &
9.90651010.093440| 7
9.906819 10.093181 6
9 907077 10.092923
5619.79824719.8909111
$79.7984039.890809
589.7985609.890707
$99.7987169.890605
609.1988729.890503
| Co-fine | Sine |
19.907336110.0926641
9.907594 10.092406 3
9.90785210.0921472
9.908111 10.091889 1
9.908369 10.091631| (
Co-tang. Tangent: N
Degree 51.
Bb 2
Degree 39.
MI Sine | Co-fine
Tangent | Co-tang |
019.79887219.8905031 19.908369|10.091631|60|
119.79902819.8904001
19.908627/10.091373159
29.7991849.890298
9.908886 10.09111458
39.7993369.890195
9.909144 10 090856 57
49.7994959.890093
9.909402 10.090598 56
$19.7996519.889990!
19.909660 10.090840155
69.799806|9.889888
9.909918|10.090081154
7.9-799961 9.889785
9.910176 10.089823 53
8.9.8001179.889682
(9.910435 10.089565 52
99.8002729.889579
9.910693 10.08930751
10 9.8004279.889476
19.910951 10.089049 50
1119.80058219.8893741
19.911209|10.088791149
129.8007379.889271
9.911467 10.08853348
139.8008929.889167
9.911724 10.088275 47
149.8010479.889064 9.911982 10.08801746
159.801209.888961
9.912240 10.08776045
16|9.80135619.888858
19.912498'10.087502144
179.8015109.888755
9.912756 10.087244 43
189.8016659.888651
9.913014 10.086986 42
199.8018199.888548
9.913271 10.08672941
2019.801973 9.888444
9.913529 10.086471 40
219.80212719.888341
19-913787|10.08621339
229.8022829.888237
9.914044 10.085956 38
239.8024359.888133.
9.914302 10.08569837
249.8025899.888030
9.914560 10.085440 36
2519.8027439.887926
9.91481710.08518335
269.8028979.887822
19.915075 10.08492534
279.8030509.8877189.915332 10.08466833
289.803204 9.887614
9,91559010.08441032
299.8033579.887510
9.915847 10.08415331
3
019.80351019.887406
19.916104 10.083895/30
| Co-fine Sine | Co-tang | Tangent M
Degree 50.
'
Degree 39.
IM Sine Co-fine
Tangent | Co-tang. \
3019.8035109.887406)
19.916104|10.083895 30
3419.80366419.8873021
329.8038179.887198
19.916362|10.083638|29|
9.916619 10.083381 28
339.8039709.887093 9.916876 10.083123 27
349.8041239.887909
9.917134 10.082866 26
359.80427619.886884
9.917391 10.082609 25
3619.80442819.886780
|9.917648|10.082352|24|
379.8045869.886675
9.917905 10.082094 23
389.8047349.886561
9.91816210.081837 22
399.8048869.886466
9.91842010.081580 21
409.805038 9.886361
9.918677 10.081323 20
419.8051919.896257,
19.918934|10.081~66|19|
429.8053439.886152
9.919191 10.08080918
439.8054959.886047
9.919448 10.08055217
449.8056479.88 5642)
9.919705 10.08029516
459.8057999.88 5837
9.919962 10.08003815
4619.80595119.885732
479.8061039.885627
489.8062549.885521
499.806406 9.885416
09.8065579.885311
|9.920219|10.07978114
9.920476 10.07952413
9.920733 10.07926712
9.920990 10.07901011
19.921247 10.07875310
5119.8067099.8852051
9.921503|10.078496) 9
529.8068609.885100
9.921760 10.078240 8
$39.8070119.884994
9.92201710.077983 7
549.8071629.884889
9.922274 10.077726 6
5519.8073149.884783
19.92253010.077469 5
569.8074649.884677
579.8076159.884572
589.8077669.884466
599.8079179.884360
609.800679.884254
| Co-fine |
|9.922787|10.077213|
9.923044 10.076956 31
9.923300 10.076699 2
9.923557 10.076443| 1
(9.923813|10 076186 c
Sine | Co-tang. | Tangent. M
Degree 50.
Degree 40.
"
MJ Sine Co-fine
Tangent Co-tang. I
.0|9,808067|9,884254|
19,923813|10,076186|60|
119,80821819,884188)
|9,924070|10,075930159
29,8083689,884049
9,924327 10,07567358
39,8085199,883936
9,92458310,075417 57
49,8086699,883829
9,924839 10,07516056
519,80881919,883723)
19,925096|10,074904′55
69,8089699,883617
|9,925352 10,074647 54
79,8091 199,8835 10
9,925609 10,07439153
89,8092699,883404
9,925865 10,074135 52
99,8094199,883297
109,8095699,883:91
9,926121|10,073878 51
19,926378 10,07362250
119,80971819,883084|
9,92663410,07336649
129,8098689,882977
9,926890 10,073110 48
139,8100179,882871
9,927147 10,07285347
149,810166 9,882764
9,927403 10,07259740
1519,8103169,882659
9,92765910,072341′45
1619,81046519,882550
(9,927915|10,072085|44
179,8106149,88 2443
9,928171 10,07182943
189,81076; 9,882336
9,928427 10,071573 42
199,8109129,882228
9,928683 10,07131741
209,81006119,882121
(9,928940|10,07106040
219,8112109,882014)
9,92919610,070804|39|
229,8113589,881907
9,92945210,070540 38
239,8115069,881799
9,92970810,07029237
249,8116559,881692)
|9,929964|10,070036|36|
2519,8118549,881584]
19,910219 10,069781135
26,9,8119529,881477)
9,930475110,06952534
279,8121009,881369|
9,930731 10,069269'33
289,8122489,881261
9,930987 10,06901332
299,8123969,881153
9,931243 10,06875731
309,8125449,881045
9,931499 10,068501 30
| Co-fine | Sine | | Co-tang. | Tangent. IM
Degree 49.
Degree 40.
M Sine | Co-fine | Tangent Co.tang.|
3019,81254419,881045|
19,931499110,068501|30
319,81269219,880937
|9,931755|10,068245|29
329,8128409,880829
9,93201010,067989|28|
339,8129889,880721
9,932266|10,067734 27
349,8131359,88-613
9,932522 10,067478 26
359,81328319,880505
|9,932778|10,067223|25|
369,8134309,880397
[9,933033|10,066967|24|
37 9,813578 9,880289
9,933289|10,066711|23
389,8137259,880180
9,933545 10,066455 22
399,8138729,880072
4019,81401919,879963|
9,933800 10,06620021
19,93405610,065944|20|
4119,814166 9,879855
429,8143139,879746
43 9,8144609,879637
44 9,8146079,879529
14519,81475319,879420
|9,934311|10,065688|19|
9,934567 10,065433 18
9,93482210,065177 17
2935078 10,064922 16
19,93533310,064666|15
4619,814900 9,879311
19,935589 10,064411|14
479,8150469,879202
9,93584410,064156 13
489,8151939,879093
9,936100 10,063900|12
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599,816797 9,877890
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9,938397 10,061602
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ICo-tang. Tangent M
Degree 49.
Degree 41.
MI Sine 1 Co-fine. I Tangent Co-tang. I
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19,939163|10,060837160
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9,940948 10,059052 53
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9,941458 10,05854251
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19,941713 10,058287150
119,81853619,876568|
19,441968 10,058032|49|
129,8186819,876457
9,942223 10,057777|48|
139,8188259,876347
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149,818969 9,876256
9,942733 10,057267 46
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19,942488130,057012|45|
169,8192579,876014
179,8194019,875904
189,819545 9,875793
199,8196899,875682
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239,820263 9,875237
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9,945026 10,054974 37
9,945281 10,05471936
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19,945790 10,054210134
9,946045 10,053955 33
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299,821122 9,874568
309,8212649,874456|
9,946299 10,053701 32
9,94655410,053446 31
19,946808 10,053192 30
| Co-fine | Sine | Co-tang. | Tangent.
Degree 48.
Degree 41.
M Sine Co-fine |
309.8212649.874456|
Tangent Co-tang. I
19.946808|10.05319230
3119.82140719.8743441
19.947063|10.052937129
329.8215509.874232
9.947317 10.052682 28
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9.947572 10.052428 27
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9.948590 10.05141025
9.948844 10.05115622
9.949099 10.05090121
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19.949607|10.05039319
429.8229729.873110
9.949862 10.05013818
439.8231149.872998
9.95011610.04988417
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4519.8233979.872772
9.950370 10.04963016
19.950625 10.04937515
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489.8238219.872434
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9.951388 10.04861212
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589.8252309.871301
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609.825511 9.871073
| Co-fine | Sine |
19.953421110.046579 14
9:953675 10.046325 3
9.953929|10.046071
9.95418310.045817 1
|9.954437|10.045562 0
| Co-tang. | Tangent. IM
Degree 48.
Сс
1
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!
M Sine Ce-fine | Tangent Co-tang. I
0*9.8255119.871073
19.954437 10.045562160
189.825651|9.870960|
|9.954691|10.045308159
29.8257919.870846
9.954945 10.045054 58
39.8259319.870732
9.955199 10.044800|57|
49.8260719.870618
9.955453 10.044546 56
59.826211|9.8.70504
9.955707 10.04429255
69.8263519.8703901
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79.8264919.870271
9.956215 10.04378453
89.8266319.870161
9.956469 10.043531 52
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9.956723 10.04327651
Fo 9.8269109.8699331
19.956977 10.04302350
119.827049 9.869818
9.95723110.042769149
129.8271899.869704
139.827328 9.869589
149.827467 9.869474
159.827606 9.869360
9.957485 10.04251548
9.957739 10.04226147
9.957993 10.042007 46
9.958246 10.04175345
169.827745 9.869245
179.827884 9.869130
19.95850010.041 500144
9.958754 10.04124643
189.828023 9.869015
9.959008 10.04099242
199.8281629.868900
9.956262 10.04073841
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24 9.8288559.868324
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9.960023 10.03997738
9.960277 10.039723 37
9.960530 10.03946936
19.960784/10.03921635
26 9.82913119.8680931
279.8292699.867978
289.8294079.867862
299.8295459.867747
3019.82968319.867631
| |
Co-fine Sine
19.961038|10.038962134
9.961291 10.038608 33
9.961545 10.03845132
9.961799 10.03820131
9.96205210.03794730
Co-tang. Tangent. M
Degree 47.
Degree 42.
MI Sine Co-fine I
|
Tangent Co-tang. 1
3019.82968319.867631|
19.962052/10.037947|30
3119.82982119.867515|
19.962306|10.037694129
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9.96256010.03744028
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9.962813 10.037187|27
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9.963827 10.03617323
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9.964081 10.03591922
399.8309219.866586
9.964335 10.03566521
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19.964842110.035158/19
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9.965095 10.034905 18
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449.831606 9.866004
9.965602 10.034398 16
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469.83187919.865770
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589.833512 9.864363
599.83364819.864245
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19.966109 10.033891|14|
9.966362 10.033638 13
9.966616 10.033384
9.966869 10.03313111
19.96712210.032878 10
19.967376|10.0326241 9
9.967629 10.0323711 8
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9.968136 10.031864
19.968389 10.031611
19.968643 10.031357 4
9.968896 10.031104 3
9.969146 10.030851
9.969403 10.030597
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Co-tang. Tangent | M
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9.97016210.02983858
39.8341899.863774
9.970416 10.029584 57
49.8343249.863656
9.970669 10.02933156
519.83446019.863537
19.970922|10.029078 55
619.8345959.863419
9.971175 10.028827154
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9.971428 10.028571 53
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9.971682 10.028318 52
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119.8352699.862827
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149.8355729.862471
9.973201 10.02679946
159.835806 9.862353
19.973454 10.026546145
1619.83594119.862234,
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19.973707110.02629344
9.973960 10.026040 43
189.835209 9.861996
9.974213 10.025987 42
199.8363439.861877
9.974466 10.02553341
2019.8364779.861757
9.974719 10.02528040
2119.83661119.861638
19.974973110.025027/39
229.836745 9.851519
9.975229 10.024774 38
239.8368789 861399
9.975479 10.024521 37
249.8370129.861280
9.975732 10.024268 36
259.837146 9.861161
9.975985 10.02401535
269.83727919.861041,
19.976238|10.023762|34
279.8374129.860921,
9.976491 10.02350933
289.8375459.860802
9.976744 10.02325632
299.8376799.860682
9.976997 10.02300331
3019 8378129.860562
9.977250 10.022750|30|
Co-fine 1 Sine | Co-tang | Tangent [M
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3119.8379459.860442
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9.977503|10.022497 29
9.977756 10.022244 28
9.978009 10.02199127
349.8383449.86008 2
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399.8390059.859580
409.8391409.859360
9.979768 10.02023224
9.979021 10.020979 21
9.979274 10.020726 22
9 979527 10.020473 21
9.979780 10.020220 20
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429.8394849.859118
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9.980285 10.0197148
9.980538 0.019461|17|
9.980791 10.01920916
459.839800 9.858756
9.981044 10.01895615
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9.981550 10.01845013
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9.98281410.017185 8
9.98306710.016933 7
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9.984079 10.015921 3
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19.98433710.015163 0
Co-tang. Tangent M
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9,985090|10,014910159
29,8420339,856690
9,985343 10,014657 58
39,8421639,856 568
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139,8434659,855342
9,988123 10,011877 47
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249,844889 9,853986
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279,8452769,853614
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9,992672|10,007328 29
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549,846175 9,852745
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9,992925|10,007075 28
9,993178 10,006822 27
9,993430 10,006564|26
9,993683|10,006317|25|
369,84643219,842496
9,993936|10,006064|24
379,8465609,852371
9,994189|10,005811 23
389,8466889,852240
9,994441|10,005559|22|
399,8468169,852122
9,994694|10,005306 21
4019,8469449,851997
9,994947|10,005053/20
4119,8470719,851872|
9,99519910,004801|19
429,8471999,851747
9,99545210,00454818
439,8473279,851622
9,995701 10,004295 17
44 9,847454 9,851497
9,995957 10,004043|16|
4519,847582|9,851372]
9,996210|10,00379015
40 9,84770919,851246|
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489,8479649,850996
499,8480919,850870
509,8482189,8507451
9,996463 10,00353714
9,996715 10,00328513
9,996968|10,00303212
9,997220 10,002779′11
9,997473 10,00252710
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9,99772610,002274 9
529,8484729,850493
9,997979 10,002021 8
539,8485999,850367
9,998231 10,001769 7
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9,998484 10,001516 6
5519,84885219,850116
9,998737 10,001263 5
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45
THE
Young Gentleman's
MECHANICKS
CONTAINING
The more Uſeful and Eaſy
ELEMENTS of Mechanicks,
more properly fo called, of Sta-
ticks, and Hydrostaticks.
By EDWARD WELLS, D. D.
Rector of Cotesbach in Leicester-
Shire.
豬豬​粉
​** sit si
STS
* SHE SKE
LONDON,
Printed for James Knapton, at the Crown
in St. Paul's Church-Yard. 1713.
1
THE
PREFACE.
N
Othing more needs be ſaid, to
(hew the great Uſefulneſs of
the Science, denoted by the
general Name of Mechanicks, than
that it teaches, and explains in a
geometrical Manner, thoſe Principles,
whereon is founded the Power or
Force of the ſeveral Machines or En-
gins, made use of either in Civil or
Military Affairs; and alſo that it ex-
plains in like manner the various Mo-
tion of heavy Bodies, confider'd either
in the Air or in the Water. The bare
making of Machins is indeed an illi-
beral or fervile Employ, and fit only
for Perfons of the meaner Sort; whence
Such Handy craft Perfons are fome-
times called, by way of Degradation,
(A 2)
Mecha-
The Preface.
Mechanicks. But the Knowledge of
the Principles, whereon depend mecha-
nical Arts or Trade, is justly esteem'd
a Liberal Science, not below the Stu-
dy of Perfons of the highest Abilities
and Rank. Forafmuch as this Sci-
ence both gives a very confiderable In-
fight into the Works of Nature; (not
only with respect to Sublunary Bodies,
but even to the Frame, Order, and
Preſervation of the Celestial System;)
and also may be infinitely ferviceable
to the Concerns of Humane Life, by
the Invention of new Machines or In-
Struments especially if it be cultiva-
ted by Perfons, who have both Abili
ty of Parts to invent, and alfo Abi-
lity of Pocket actually to experiment
their Inventions, and by this Means
to compleat and perfect them. For it
is to be fear'd, that many an happy
Invention has unfortunately mifcarried,
becauſe the Inventer has not had a
good Purſe, as well as a good Head.
THE
سمجھے
5
THE
Young Gentleman's
MECHANICKS.
CHAP. I.
Wherein are explain'd fuch Terms as
relate to Mechanicks in general.
T
I.
what, and
HAT Art or Science, which
teaches how to move any Mecha-
given Weight, by any gi- nicks,
ven Power we call in one why so cal
Word (†) Mechanicks becauſe it is led.
chiefly ſerviceable to the contriving of
thoſe Machines or Engines, which are us'd
;
(†) The Word Mechanicks is deriv'd from the Greek
Word Maxa; from which is made likewife the corre
fpondent Latin Word Machina; and from this is form'd
our English Word Machine. From the fame original Greek
Word is deriv'd alſo the Word Mechanism, i. e. the Arti
ficial Contrivance of a Thing.
(A 3)
in
The Young Gentleman's
}
2.
and a Pow-
in the Concerns of Life, whether Civil
or Military.
By a Weight is underſtood in Mecha-
A Weight, nicks any heavy Body; by a Power (*),
whatever can move an heavy Body;
tion, what and by Motion, Local Motion or the
in Mecha- Change of Place.
er,and Mo-
nicks.
3.
of Motion,
what.
The Axis of Motion is that right Line,
The Axis round which a Body being mov'd, every
and Center Point of the Body defcribes Circles,
whoſe Centers are all in the ſaid right
Line ; and the Mid-point of the Axis is
called the Center of Motion. Thus Fig. 1
the Sphere or Bowl ADBE,being contriv'd
to move round the right Line AB, the
faid AB will be the Axis, and the Point C
the Center of its Motion.
4.
That right Line, wherein any Body or
The Direc- Power endeavours to move, is call'd the
tion of Mo-
tion, what. Line of Direction, or fimply, the Directi-
on of the Motion of the refpective Body
or Power. And if a Body moves in a
Curve Line, then the Direction of its Mo-
tion in each Point of the Curve is the
Tangent of the faid Point.
AB is the Direction of the
Thus Fig. 2,
Weight A,
and BC the Direction of the Power at C.
(*) Hence any Weight or heavy Body is or may be e-
fteem'd a Power, in reference to any other Weight or hea-
vy Body, which it do's or can move.
And
Mechanicks.
And Fig. 3, the Tangents TA, TB, TC,
are reſpectively the feveral Directions of
a Body mov'd round the Curve ABC, at
the ſeveral Points A, B, and C.
5.
a Body,
That Force, which is in Bodies mov'd,
and whereby they continually move, or The Mo-
endeavour to move, is.call'd their Mo- mentum of
mentum, and arifes from the Weight of what.
(or Quantity of Matter in) the Body
mov'd, and the Velocity or Swiftnefs
wherewith it moves.
The different Weight of Bodies arifes 6.
from the different Quantity of Matter in The Weight
the faid Bodies. Whence the Quantity proportio-
of Bodies is
of Matter in any Body may be eſtimated nal to their
by its Weight. For instance,
For instance, if a cubi- Matter.
cal Inch of Lead be fix times heavier than
a cubical Inch of Wood, it may thence
be inferr'd, that the Quantity of Matter
in ſuch a Piece of Lead is fix times as
much, as the Quantity of Matter in a like
Bulk of Wood.
7.
Gravity,
That Gravity or Weight, which arifes
from the more or lefs Compactness of Specifick
Matter naturally belonging to the ſeveral what.
Species or Sorts of Bodies, is called their
Specifick Gravity. Thus, although a
Pound of Feathers be equal in Weight
to a Pound of Lead, yet Lead is faid to
be specifically heavier than Feathers, be-
caufe there is not fo much Matter, nor
confequently fo much Weight, in a Fea-
(A 4)
ther,
.
The Young Gentleman's
8.
ther, as in a Piece of Lead of the fame
Bulk with the Feather.
Gravity or Weight being that, which
Centrum caufes in Bodies a Tendency to move (if
Gravium, not hinder'd by fome external Force)
what.
9.
of Gravity,
what.
•
downwards, or towards the Center of the
Earth; hence the Center of the Earth is
ftil'd Centrum Gravium, or the Center to
which heavy Bodies tend.
The Center of Gravity in any Body is
The Center that Point, through which a Plane paf-
fing divides the Body into two Segments
or Parts of equal Weight. It is this Cen-
ter of Gravity, which is of chief Regard
in Bodies, a Body being not properly
faid to defcend or afcend, but when
its Center of Gravity deſcends or af
cends.
10.
what.
Velocity (which is the other Ingredi-
Velocity, ent, that together with Weight makes up
the Momentum of a Body) is that Affec-
tion of Motion, which is meafur'd by
comparing together the Length and Time
of Motion. Thus Equal Velocity is that,
whereby equal Length is pafs'd over in e-
qual Time: Greater Velocity is that,
whereby either a greater Length is paſs'd
over in equal Time, or an equal Length
in lefs Time: Lefs Velocity, when con-
trariwife. Further, a Body that moves
an hundred Yards, may be faid to move
with double the Velocity of a Body, that
moves
Mechanicks.
9
moves but fifty Yards in the fame Time
and with triple or thrice the Velocity of
another Body, that moves but thirty-
three Yards and one third in the fame
Time ; and with quadruple or four-
times the Velocity of a Body that moves
but twenty-five Yards in the fame Time.
And ſo on.
II.
a Body,
The Momentum of a Body, arifing (as
has been obferv'd §. 5,) from its Weight The Mo-
and Velocity, may be known by multi- mentum of
plying one of its Ingredients into the o- how to be
ther. And confequently the different Mo- known.
mentums of different Bodies may be
known by multiplying the Weight of
each Body into its refpective Velocity.
Thus fuppofe the Weight of a Body A to
be ten Pound, and of any other Body B
to be four Pound; and the Velocity of
B to be five times as much as that of A.
The Momentum of A will be 10; viz.
10 Pound, the Weight of A, being mul-
tiplied into 1 for its Velocity, (the Velo-
city of A to B being fuppos'd as I to 5,)
the Product will be 10 for the Momen-
tum of A. And in like manner, 4 Pound,
the Weight of B, being multiply'd into 5
as the Velocity of B. (for the Velocity of
B to A is fuppos'd as 5 to 1) the Product
will be 20 for the Momentum of B.
Whence it appears that the Momentum of
I
B is
10
The Young Gentleman's
12.
The Fun-
Theorem
nicks.
B is double to that of A, being as 20 to 10.
Hence is deduced the following Theo-
rem, which may be ftiled the Fundamen-
damental tal Theorem of Mechanicks, viz. When
of Mecha Bodies have their Velocities reciprocally pro-
portional to their Weights, their Momen-
tum's will be equal. Let the Weight of
a great Body A be 500 Pound, and
the Weight of a little Body B be 50
Pound 5 and let their reſpective Veloci-
ties be reciprocally proportional, (i. e. as
the Weight of A is to that of B as 10 to 1,
fo let the Velocity of B be to that of A as
10 to 1.) It will hence follow, that the
Momentum of B will be juft equal to that
of A. Namely, 500 Weight 1 Veloci-
ty = 500 the Momentum of A; and 50
Weight * 10 Velocity 500 the Mo-
mentum of B.
13.
ral End
gin's.
X
Hence it appears, not only that any
The gene- little may have its Momentum equal to the
and Effect Momentum of any great Body which is
of Mecha- mov'd with a given Velocity; but alfo
nical En. what Method is to be taken, in order to
make it actually have fo; namely, by con-
triving that the Velocity of the Power
may be fo much greater than the Veloci-
ty of the Body to be mov'd, as the
Weight of the Body is greater than the
Weight (or fomewhat equivalent there-
to) of the Power. On which Account
it is, that the foremention'd Theorem may
be
Mechanicks:
11
!
be juſtly eſteem'd the Fundamental Theo-
rem of Mechanicks; forafmuch as there-
on is founded the principal Mechaniſm
or Contrivance of Machin's or mechanical
Engin's, made ufe of to draw or raiſe
heavy Bodies; the whole End and Ef-
fect of theſe being to give fuch a Veloci-
ty to the Power, or (which comes to
the fame) ſo to diminiſh the Velocity of
the Weight or heavy Body, as that the
Momentum of the Power may be equal,
or fomewhat exceed, the Momentum of
the Body ſo to be drawn or rais'd.
mechanical
And this is what will be illuſtrated in 14.
the fix following Chapters, in reference The fimple
to the fix fimple and therefore primary Powers, fix
Machin's, or (as they are otherwife cal- in Number.
led) Mechanical Powers; to fome one or
more of which may be reduc'd all o-
ther mechanical Engin's. And I fhall
ſpeak of them in the Order, wherein
they here follow (*) reckon'd up, viz.
I Vectis
(*) Whereas fome reckon but five fimple Machines or
mechanical Powers, comprehending the Balance under
the Leaver; and others that reckon fix timple Machines, u-
fually reckon the Balance as the firft, and the Leaver as the
fecond; I have judg'd it more proper to agree with the
latter, in reckoning fix fimple or primary Machines, and
to fall in ſo far with the former, as to reckon the Leaver
as the firſt, and the Balance as the ſecond fimple Machine,
inaſmuch as the Leaver is a more fimple Machine than the
Balance;
12
Mechanicks.
i. Vedis or the Leaver.
2. Libra or the Balance.
3. Axis in Peritrochio or the Axle in
the Wheel.
4. Trochlea or the Pulley.
5. Cochlea or the Screw.
6. Cuneus or the Wedge.
Balance; and the Balance is no other than a particular
Kind of Leaver. In like manner, whereas fome reckon
the Wedge as the fifth, and the Screw as the fixth fimple
Machine, I have rather follow'd them, who have inverted
the faid Order, forafmuch as the Screw partakes more of
the Faculty of the Leaver than the Wedge.
I.
A Leaver,
what.
CHAP.
II.
Of the Leaver.
HE Leaver is fo call'd from the
principal Uſe thereof, which is le-
vare, i. e. to lift or raiſe up heavy Bo-
dies. For the more eafy explaining the
Mechaniſm thereof, it is here conceiv'd
as a Line, which will not bend, and may
be turn'd round about an unmoveable
Point; which is therefore the Center of
its
Mechanicks.
13
its Motion, and rests upon fomewhat
thence call'd the Fulcrum, i. e. Prop of
the Leaver. Thus Fig. 4, AB is the Lea-
ver, C the Center of its Motion, and F
the Fulcrum or Prop.
may
2.
The main Doctrine of the Leaver Theorem
the first.
be comprifed under two Theorems, where-
of the firft is this: The Space (or Arch)
defcrib'd by each Point of a Leaver, and
confequently the Velocity of each Point of a
Leaver, is, as its Distance from the Ful-
crum or Prop. For inftance, the Laver
AB (Fig.4,) being mov'd from ACB to
aCb, the Point A will defcribe the Arch
Aa, and the Point B will deſcribe the
Arch Bb; and by reafon of the like
Sectors ACa and BCb, Aa is to Bb as AC
to BC, that is, the Spaces defcrib'd by
the Points A and B are as their Diſtances
from C the Center of Motion, or (which
comes to the fame) from F the Fulcrum.
Now Powers being apply'd to the Points
A and B, which draw the Arms CA and
CB of the Leaver AB perpendicularly,
the Spaces defcrib'd by them will be (not
the Arches Aa and Bb, but (the Perpen-
diculars AD and EB, theſe being the
Meaſures of the Progrefs made by the
Powers according to their reſpective
Direction. But the Triangles ACD
and BCE being equiangular, therefore
(*) AD
14
Mechanicks.
JA
(*) AD is to BE as AC to BC or 4C to
bć, that is, the Lengths gone over by
the Powers according to their respective
Directions, are as their Diſtances from
C the Center of Motion, or (which
comes to the fame) from F the Fulcrum.
Now Velocity being (according to Chap.
1. S. 10) that Affection of Motion,
which is meafur'd by comparing together
Length and Time, and confequently (one
Sort of) greater Velocity being that,
whereby a greater Length is paft over in
equal Time, as in the Cafe before us;
hence it follows, that the Velocity either
of the Point B, or of the Power apply'd
thereto, is fo much greater than the Ve-
locity of the Point A or the Power ap-
ply'd thereto; (as the Arch Bb or the
Line BE is greater than the Arch Aa or
the Line AD; and confequently fo much
greater) as BC or bC is than AC or aC
that is, the Velocity of each Point in the
Leaver, or of any Power apply'd thereto,
is as its Diſtance from (C the Center of
Motion, or from) F the Fulcrum (†).
(*) By Theorem 9, of the Young Gentleman's Geo-
metry.
(†) For the Sake of fuch as may not readily apprehend
the foregoing Geometrical Demonftration of the first Theo-
rem, the fame is ſenſibly demonſtrated in Fig. 4, by actu-
ally dividing BC, ¿C, Bb, and BE into fix equal Parts, each
of the fame Length or Compafs as AC, aC, Aa, and AD re-
Spectively.
{
The
Mechanicks.
15
The Truth of the firft Theorem being 2.
demonftrated and apprehended, it will Theorem
the second.
be eaſy to apprehend the Truth of the
fecond Theorem, which is this: In a
Leaver the Power, which is to the Weight
(or Body) to be rais'd, as the Distance of
the Weight (according to its Line of
Direction) from the Fulcrum is to the Di-
Stance of the Power (according to its Line
of Direction) from the Fulcrum, will coun-
terpoife or hold up the Weight; and there-
fore being a little increas'd, will raiſe the
Weight.
For it has been fhewn under
Theorem 1, that the Spaces defcrib'd by
(the Powers apply'd to each End of the
Leaver, i. e. by) the Power apply'd
to one End and the Weight apply'd to
the other, are proportional to their
faid Diſtances from the Fulcrum; and
confequently that their Velocities, be-
ing proportional to the ſaid Squares,
are alſo proportional to the laid Di-
ftances. Wherefore if (Fig. 5) CA the
Diſtance of the Weight from the Ful-
crum be to CB the Diſtance of the Power
from the Fulcrum, as the Power P is to
the Weight W, it will follow, that
likewife the Velocity of the Weight
(1) Thus Fig. 5, Bb the Line of Direction of the Power
is equidiftant from C to A4, the Line of the Weight's Di
rection.
will
**
16
The Toung Gentleman's
4.
Theorem
the second
illuftrated
by an Ex-
ample.
will be to the Velocity of the Power, as
the Power is to the Weight. And con-
fequently the Momentum of the Power
will (by the Fundamental Theorem Chap.
1. S. 12) be equal to the Momentum of
that Weight; and therefore the Power
will be able to fuftain the Weight; and
if it be but a little increas'd, (either by
the Addition of fome new Force, or elſe
by increasing its Diſtance from the Ful-
crum) it will be able to raiſe the
Weight.
An Example will illuftrate the Matter.
Let W the Weight of the Body to be
rais'd be 120 Pound, and the Force of
the given Power P be equivalent only to
20 Pound, and confequently the Weight
be to the Power as 6 to 1. Wherefore P
the Power being fo plac'd at B (Fig. 5)
and W the Weight at A, as that BC may
be fix times as long as AC, in fuch a Po-
fition the Power will have a Force fuffici-
ent to counterpoife, and fo hold up the
Weights. For the Velocity of P to that
of W being proportional to their Diſtan-
ces from C, will be in this Cafe as 6 to
I ; and confequently the Momentum of
P, viz. 20 Pound x 6 Velocity = 120=
120 Pound × 1 Velocity, the Momentum
of W. Wherefore the Power P (having a
little new Force added to its own, or)
being remov'd a little farther diſtant
*
from
Mechanicks.
if
from C, viz. to b, and ſo having its Ve-
locity a little more increas'd, it will raiſe
W the Weight. For fuppofing bC to be
to AC as 7 to 1, the Momentum of P will
be 20 Pound × 7 Velocity 140; which
is confiderably greater than the Momen-
tum of Wat A, this being (as afore
found) but 120.
the ad exe
Having thus fhewn the Truth of the
5.
two foregoing Theorems, and thefe be- Theorem
ing applicable to the other following me-
preſs'd in
chanical Powers, as well as to other Ca- short by.
fes in refpect of the Leaver, it may be of Letters.
Ufe to obferve here, that the Import of
the fecond Theorem here laid down will
for the future be exprefs'd in ſhort thus ;
P: W:: CW: CP, that is, As much
as the Force of the Power P is less than
the Weight W, fo much lefs muft be
CW the Diſtance of the Weight from the
Fulcrum, than CP the Diſtance of the
Power from the Fulcrum.
And as this has been fhewn to hold
good in refpect of the Leaver (as it is cal-
led by fome) of the firft Kind, that is,
when the Fulcrum is between the Power
and the Weight; fo does it hold good al-
fo in respect of the Leaver of the fecond
Kind, when the Weight is between the
Power and the Fulcrum, and alfo in the
5
Leaver of the third Kind, when the Pow-
er is between the Weight and the Ful
(B)
crum.
6.
The Kinds
of Leavers.
18
The Young Gentleman's
!
7.
Pinchers,
of Lea-
vers.
crum.
For in a Leaver of the ſecond
Kind, the Weight W at D (Fig. 6.) has
(as is obvious) just the fame Effect, as it
would have on the other Side of the Ful-
crum and at the fame Diſtance from it,
viz. at E. And in like manner the Pow-
er P at D (Fig. 7.) has juſt the ſame Ef-
fect, as it would have at E, CE being e-
qual to CD. Wherefore the Leaver of
the ſecond and third Kind being thus in
effect no other than that of the first, the
Mechaniſm of theſe depends on the fame
Theorem or Rule, as the Mechaniſm of
that, viz. P: W: CW: CP.
To the Leaver of the firſt Kind are re-
Seiffers, ferr'd Sciffers, Pinchers, Snuffers, and
c. Kinds fuch-like Inftruments, they confifting as
it were of two fuch Leavers join'd toge-
ther, but acting contrary Ways; as may
be eafily apprehended by Fig. 8 and 9.
To the Leaver of the fecond Kind are re-
ferr'd the Oars of a Boat, as alfo fuch
Cutting-Knives as are fix'd at one End.
To the Leaver of the third Kind may be
referr'd a Ladder, lifted up between the
two Ends, in order to rear it againſt at
Wall. Laftly, an Hammer drawing a
Nail is referr'd to a fourth Kind of Leaver,
call'd a Bended Leaver. For the Nail is
as the Weight at one End A, and the
Hand of him that draws the Nail is as
the Power at the other End B of the Lea-
ver ?
Mechanicks.
19-
ver ; and the Point C (Fig. 10) is as the
Fulcrum. And this fourth Kind of Lea-
ver is likewife reducible to the First, the
Weight or Nail at A having the fame Ef-
fect, as if it were at a, Fig. 10.
1
8.
proportio
As a Weight carried on a Pole or Stick
between two Perfons may alſo be referr'd How to
to a Leaver of the ſecond Kind, fo it may weight or
place a
be of good Uſe to obferve, that agreea- Body on a
bly to the foremention'd Theorem P: Stick or
the like, fo
W:: CW: CP, the Weight may be fo as to make
plac'd, as that the Burden thereof fhall the Burden
be proportion'd to the different Strengths nal to the
of the two Perfons; Namely, by taking different
Strengths
P to denote the Strongest, W the Weak- of them
eft, and C the Weight. For then the Im- that carry
port of the faid Theorem will be this: the Stick-
As much as the Strength of the Strongest
P is greater than the Strength of the
Weakeſt W, ſo much muft CW the Di-
ftance of the Weight from the Weakeft,
be greater than CP the Diſtance of the
Weight from the Strongeft. For Exam-
ple, fuppofe the Leaver or Stick PW
(Fig. 11) to be divided into 18 equal
Parts, and the Perfon at P to be twice as
ftrong as the other at W, then C the Bo-
dy to be carried muſt be plac'd at 6. For
then the Diſtance CW will contain 12 of
the equal Parts, into which the Stick is
divided, and the Distance CP will con-
tain but 6 of them 30 and fo CW 2CP,
(B 2)
and
20
The Young Gentleman's
9.
The like
explain'd
as to draw-
ing any
Weight.
and confequently the weakest Perfon at
W will fuftain but half ſo much as the
ſtrongeſt at P. In like manner, if the Bo-
dy c were placed at 3, then the Perfon at
P would fuftain 5 times fo much, as the
other at W; becauſe then the Difpro-
portion of the Diſtances would be as 15
to 3, cW containing 15 of the equal
Parts, whereof cP would contain but 3.
Laftly, if the Body c were placed at 9,
that is, in the Mid-point of the Stick or
Pole, then each Perfon P and W would
bear equal Burden, becauſe the Diſtance
9P= Diſtance 9W.
And the fame Theorem holds good in
refpect of Drawing (as well as Carrying)
any Burden. For (Fig. 12) let WC re-
prefent the Pole of any Waggon or
Coach, and AB the Crofs-bar, at whoſe
Extremities are Spring-trees DE and FG,
to which are to be faften'd the Horfes or
the like. Becaufe A and B are equidiftant
from C, therefore the Horfes at A and B
will draw equal Weight. Whereas if
one of the Spring-trees be faften'd at 1,
then the ſaid Horfe would draw a fourth
Part more of the Weight W, than the o-
ther Horſe at B; forafmuch as the Di-
ftance Ci is a fourth Part lefs than the
Distance CB. And fo if one Horfe was
faften'd to 2, he would draw twice as
other at B; becauſe Ca
much as the
is
Mechanicks.
20
is but the Half of CB. And fo
on.
10.
ticulars
folv❜d.
By the Mechaniſm of the Leaver may
we alſo account for the Reafon, why it Other Par-
is more difficult to lift up a Staff (or the
like) of any Length, at or near one of
its Ends, than nearer the Middle: Name-
ly, becauſe the Momentum of any Thing
fo fuftain'd increaſes proportionably to
its Diſtance from its Fulcrum or Prop.
And the fame is the Reaſon, why it is
more difficult to lift up a long Staff or
the like, at Arm's Length, or keeping the
whole Arm from Hand to Shoulder ex-
tended out, than when we bend the Arm
at the Elbow. Namely, becauſe in the
former Cafe the Shoulder is the Ful-
crum, or Prop, in the latter Cafe the El-
bow 30 and therefore in the former the
Diſtance of the Weight from the Prop, is
greater than in the latter. And fo much
for the Leaver.
(B 3)
CHAP.
22
The Young Gentleman's
I.
The Beam
of a Ba-·
Lance no o-
T
CHA P. III.
Of the Balance.
HE Beam AB (Fig. 13 and 14) is
the principal Part of a Balance,
and is no other than a Leaver of the firſt
ther than Kind, which (inftead of refting on a
Fulcrum at C the Center of its Mo-
tion) is held up from above by fome-
what faſtened to C its Center of Mo-
tion.
a Sort of
Leaver.
2.
The Me-
chanifm
lance ex-
plain'd in
Hence the Mechaniſm of the Balance
depends on the fame Theorem as that of
of the Ba- the Leaver, viz. P: W:: CW: CP.
Where, taking P to denote the known
general. Weight, W the unknown Weight, and
C the Center of the Balance's Motion, the
Import of the Theorem is this: As much
as the known Weight is less than the un-
known Weight, (i. e. the Weight fought of
the Body weigh'd) fo much will the Di
ftance of the unknown Weight from the Cen-
ter of Motion be less than the Distance of
the known Weight from the Center of Moti-
on, where the faid two Weights will counter-
poife one the other ; and confequently the
known Weight shew the Quantity of the un-
known Weight.
For
Mechanicks.
23
ed.
For Inftance, on the (*) Roman Ba- 3.
lance or Steel-yard, (Fig. 13) the un- The Mecha-
nism of the
known Weight or Body to be weigh'd be- Roman Ba-
ing apply'd to A, and the known Weight lances or
P being mov'd up and down, nearer and Steel-yard
particular-
farther (from the Point C) upon the Arm ly explain-
BC, and being found at the Diſtance 5
to counterpoife W the unknown Weight,
thence it follows, (viz. becauſe the Di-
ſtance CP is 5 times greater than the Di-
ftance CW, or which is the fame, CA)
that the Quantity of the unknown Weight
W is 5 times as much as the Quantity of
P the known Weight. And therefore,
fuppofing P to be one Pound Weight, W
will be five Pound Weight; or fuppofing
P to be two or three Pound Weight, W
will be reſpectively ten or fifteen Pound
Weight.
4.
But now, becauſe in a common Balance
or a (†) Pair of Scales, (Fig. 14) the The Mecha-
two Arms CA and CB of the Beam AB nim of the
are (or ought to be) always of an equal Balance or
Length; hence the Quantity of the of a Pair
common
of Scales,
explain'de
(*) This Sort of Balance is call'd the Roman Balance,from
its being much us'd at Rome. And it is commonly call'd a
Steel-yard, as being a Sort of Yard made of Steel.
(†) So the Latins call this Libra or Balance by the pe-
culiar Name of Bilanx (namely à binis Laucibus, i. e.
from the two Scales belonging thereto) whence is made
our English Word Balance.
(B 4)
known
34
The Young Gentleman's
5.
Bare
Weight is
Strictly true
Weight.
known Weight P in one Scale muft in it
felf be equal to the Quantity of the un-
known Weight W put in the other Scale.
And therefore the Quantity of the un
known Weight on this Sort of Balance is
found out, (not as on the Steel-Yard, by
moving one and the fame Weight nearer
or farther from C the Center of Motion,
but) by putting in various Weights of
known Quantity, 'till the known Weight
counterpoifes the unknown; and there-
by fhews the Quantity of the unknown
Weight.
For it is to be obferved, that the
known Weight then fhews the exactly
true Quantity of the unknown Weight,
when the one counterpoifes the other.
For the uſual Cuftom of making the
Weight of the Thing fomewhat more
than counterpoife the known Weight by
which it is weigh'd, (even fo as to bring
the Scale, wherein the Thing bought is
put, to the Ground or the like) has aroſe
from Tradesmen being willing to get Cu-
ftomers, by allowing them fomewhat more
than ftrictly true,or (as it is commonly and
(*) properly enough call'd) bare Weight.
And confequently, when we fay of a
}
(*) Becauſe hereby is denoted true Weight, confider'd
barely as to Weight, without any refpect to Cuſtom, &c.
Tradef
A
Mechanicks.
25
Tradefman, who is more free than ordi-
hary in what he allows over bare Weight,
that he makes good Weight, or better
Weight than another, that makes only
bare Weight; thereby is to be under-
ftood, not that the Weight he makes is
true, or truer than the other's, in reſpect
of its intrinfick Quantity; but that it is
good, or better than the other's, in reſpect
of the Advantage accruing to the Buyer,
by what is allow'd over bare or true
Weight.
to be in Æ.
It is alſo not to be omitted, that as one 6.
Body is faid to counterpoife (or to be of Standing
equal Weight with) another, when they Weight, and
being feverally apply'd to the two diffe- quilibrio,
rent Arms of the Balance, the Beam lies what.
(†) horizontally or equally level; fo this
is call'd in our common Speech (not on-
ly bare Weight for the Reafon afore af-
fign'd, but allo) ftanding Weight, foraf-
much as in this Cafe' the Beam ftands e-
ven, neither Arm afcending or defcend-
ing more than the other and for the
fame Reafon the Bodies in this Cafe
are faid by the Learned to be in Æquili-
brio.
;
(†) This is fhewn by the Piece of Iron CE, call'd the
Tongue, being in a Line or directly between TX and
TS.
Laftly,
26
The Young Gentleman's
7.
cover a
falfe Ba-
Lance or
Scales.
Laſtly, it is well to be obferv'd, that
How to dif- the Truth or Exactneſs of (a common
Balance or) a Pair of Scales greatly de-
pending on its two Arms being of equal
Pair of Length, as well as of equal Weight in
themſelves, hence it follows, that if one
Arm be longer than the other, it will be
a falfe Balance, or not fhew the true
Quantity of the unknown Weight, or
(which is the fame) the true Weight
of the Thing weigh'd. For inftance,
fuppofe Fig. 14, the Arm AC to be divi-
ded into ten equal Parts, and the other
Arm BC to contain but nine fuch other
Parts; and confequently AC to be longer
than BC by a tenth Part. It will follow
by the Fundamental Theorem (Chap. 1.
S. 12, or which comes to the fame, by
this Theorem, P: W:: CW: CP,) that
a known Weight of nine Pound ap-
ply'd to A, will have its Momentum e-
qual to a Body of ten Pound Weight ap-
ply'd to B; for 9 Pound x 10 Velocity
=90 Momentum 10 Pound × 9 Velo-
city,) and conſequently will counterpoiſe
one the other in fuch a falfe Balance,
though their Weights be not equal in
themſelves. The Way to diſcover ſuch
falfe Balances, is by changing the known
and unknown Weight into the contrary
Scales. For upon fuch a Change, the In-
equality of the Weights will manifeft-
X
ly
Mechanicks.
27
ly (*) appear; whereas in a true Balance
or Pair of Scales, fuch a Change of the
Scales will make no Difference, there be-
ing no Difference in the Length of the
Arms, and confequently no Difference in
the Velocity of the known or unknown
Weight, to which ever Arm they be ap-
ply'd.
(*) Namely, the Body being of 10 Pound Weight in ic
ſelf, and when apply'd to A, its Velocity being in reſpect
to the Velocity of the known Weight of 9 Pound in it felt,
as 10 to 9, the Momentum of the faid Body will be (10
Pound 10 Velocity) 100; whereas the Momentum of
×
the known Weight will be (9 Pound × 9 Velocity =) 81.
And confequently the known Weight will not counterpoife
the Body by a great deal put in the Scale of the shorter
Arm BC, though it did fo in the Scale of the longer Arm
AC.
1
CHA P. IV.
Of the Axle in the Wheel.
T
HIS
I.
fo called.
Machine is fo called, as con-
fifting principally of a Wheel BGHI, This Ma
and a Cylinder or Axle ADEF. A- chine, why
bout which Axle goes the Rope, which
draws the Weight W, when the Axle is
turn'd round by means of the Wheel; as
Fig. 15.
As
28
Mechanicks.
2.
ing perpetu-
Fulcrum.
As the Balance is no other than a Lea-
It is in ef- ver, particularly apply'd to finding out
felt no o-
ther than a
the Quantity of an unknown Weight, fo
Leaver, ca- this mechanical Power is no other than a
pable of be- Leaver, apply'd to the fame Uſe as other
ally turn'd Leavers are, (viz. to the railing up of
round its great Weights) with this particular Ad-
vantage added thereunto: Namely, where-
as in the Leaver it ſelf the Motion can be
continu'd only for fo fhort a Space Aa,
Fig. 4 and 5) as is anfwerable to the
little Distance CA between the Fulcrum
and the Weight; in this Invention that
Inconvenience is remedy'd, foraſmuch as
by continual turning round the Wheel,
and fo (together with it) the Axle, the
Weight may be rais'd to any Height, as
Occafion fhall require.
3.
niſm ex-
plain'd.
As this Machine is no other than a
Its Mecha- Leaver AB capable of being perpetually
turn'd round, fo the Fulcrum of it or
Center of its Motion is C, and the End
B is that to which the Power P is to be
apply'd, and A the other End, to which
is to be apply'd the Weight W to be rai-
fed. Hence the Mechaniſm of this En-
gine depends on the fame Theorem, as
that of the Leaver, viz. P: W:: CW:
CP; that is with particular reference to
this Machine; If as much as the Power
is lefs in it ſelf than the Weight, fo much
the Radius of the Axle be less than the
C
Radius
Mechanicks.
29
Radius of the Wheel, (and confequent-
ly fo much the Circumference of the Axle
be leſs than the Circumference of the
Wheel; and confequently fo much the
(†) Velocity of the Weight apply'd to
the Axle be less than the Velocity of the
Power apply'd to the Wheel) by the Fun-
damental Rule the Momentum of the
Power will be equal to that of the
Weight, and confequently the Power will
counterpoife and fo hold up the Weight;
and therefore the Power being a little in-
creas'd will raiſe the Weight.
>
4.
An Obfer.
The Momentum of the Power thus in-
creafing according to the Difproportionation
between the Circumference of the Wheel,
and the Circumference of the Axle T
hence it follows, that the Line of the
Power's Direction must always touch the
Circumference of the Wheel, fo as to
make a(*)right Angle with the Ray, that
fo the fame Diſtance between the Pow-
er and the Center of Motion, and confe-
quently the fame Momentum of the Pow-
er, may be every where alike preſerv❜d ;
(†) For the Weight, as is evident from the Make of this
Machine, is rais'd up no more than is anfwerable to the
Length of the Rope that goes round the Axle in one turn-
ing round of the Wheel.
(*) That is, fo as to be the Tangent of the Point of the
Circumference it touches, according to Chap. 1. §. 4.
which
30
Mechanicks.
।
which otherwife would not be fo. For
fuppofing (Fig. 16) the Power P to be
fo apply'd to any Spoke, viz. G, as that
the Line of its Direction be GH, then the
Diſtance of the Power is to be eſteem'd
IC, and fo leſs than BC, and confequent-
ly the Momentum of the Power will be
fo much leffen'd, as IC is less than BC.
Whereas if the Line of the Power's Di-
rection were
then the Momentum
of the Power at C would be the fame
as at
>
5.
To this third fimple Machine are re-
Machines ferr'd the Crane Fig. 17, and all Engines
and Inftru which confift of Wheels with Teeth, as
ments refe-
rable Fig. 18, and alſo the Machine repreſent-
thereunto. ed Fig. 19, and the feveral other Inftru-
ments repreſented Fig. 20, 21, 22, 23
24 and 25.
In each of which A repre-
fents the Place of the Weight, B the Place
of the Power, and C the Center of Moti-
on or Fulcrum.
СНАР.
Mechanicks.
31
T
CHAP. V.
Of the Pulley.
I.
what.
HE Pulley is a Machine confifting
of one or more little Wheels, a- A Pulley,
bout which a Rope being put and (*)
pull'd at one End, the faid Rope makes
the ſaid Wheels turn round upon their
proper Axles, and thereby at the fame
time moves the Weight or Body to be
drawn or rais'd, as Fig. 26, 27, 28.
2.
If the Pulley it felf be faften'd up to
fome Place or Mark H, and the Weight An upper
W be apply'd to one End of the Rope, Pulley,
and the Power P to the other, (as Fig. its Ufe ex-
what, and
26,) then it is evident, that as much as plaind.
the Weight W afcends, juſt ſo much the
Power P defcends; and confequently that
they both move with the fame Velocity.
Wherefore in this Cafe the Force of the
Power muſt in it felf be equivalent to the
Weight of the Body rais'd, in order to
(*) The Effect of this Machine being thus produc'd by
pulling the Rope, hence it takes the Name of the Pulley á-
mong us.
make
32
The Toung Gentleman's
3.
'A lower
Pulley,
1
make its Momentum equal to the Mo-
mentum of the Body. And hence it fol-
lows, that fuch a Pulley does not con-
duce to the increafing of the Momentum
of the Power, or (which comes to the
fame) to the decreafing the Momentum
of the Body or Weight to be rais'd; but
only to the keeping of the Cords from
fretting; and fo to the making them
move more eaſily than otherwife they
would. And, becauſe when there is Oc-
cafion to apply ſeveral Pulleys together,
fuch as are of this Sort, (or which the
Ropes go over,) are plac'd uppermoſt ;
hence theſe are diftinguifh'd by the Name
of Upper Pulleys.
But if (not the Pulley it felf, but) one
End of the Rope be faſten'd to an Hook
what, and H or the like, and the other End be pul--
its Ufe ex- led by the Power P, and the Weight W
plain'd. hangs on to the Pulley, (as Fig. 27.)
then it is evident, that in order to raiſe
the Weight W one Foot, each Part AH
and Bh of the Rope inuft be ſhorten'd one
Foot, (reckoning downwards from the
Hook H and its oppofite Point b) and
confequently that in the fame Time the
Power P must move two Foot. Where-
fore in this Cafe the Velocity of the Pow
er will be double to that of the Weight,
and confequently if the Power be fub-
duple to the Weight, that is, as I to 2,
its
Mechanicks.
33
Its Momentum will be equal to that of
the Weight, and fo will ſuſtain the
Weight. And therefore if the Power be
a little increaſed, it will be able to raiſe
the Weight. And the fame holds when
there are two Wheels or Pulleys, one
Upper and one Lower, as Fig. 28.
If (as Fig. 29) there be three Wheels,
and fo the Rope goes from Wheel to
Wheel three times, viz. from A to B, and
from C to D, and from E to F; and
confequently in order to raiſe the
Weight W one Foot, each of the three
Parts AB, CD, and EF of the Rope
muſt be ſhorten'd one Foot, which
can't be done without the Power P mo-
ving forward three Foot in the fame
time ;
in this Cafe the Velocity of the
Power will be triple or thrice as much,
as the Velocity of the Weight. And
therefore, though the Power be to the
Weight but as 1 to 3, yet its Momentum
will be equal to the Momentum of the
Weight, and fo will fuftain it; and if the
Power be but a little increaſed, will raiſe
the Weight.
4.
Further of
the fame.
In like mariner it is evident from Fig.
30, that if there be four Wheels, and fo The fame
the Rope goes four times frome Wheel to yer further
exempli
Wheel; then, while the Weight W is fyd.
rais'd one Foot, the Power P will move
forward four Foot; and confequently
(C)
(by
34
The Young Gentleman's
relating to
(by the fundamental Theorem) a Pow-
er P, which is to the Weight W but as
to 4, will have its Momentum equal to
the Momentum of the Weight; and
therefore being a little increas'd, will raiſe
it.
6. Hence the Mechaniſm of the Pulley de-
The general pends on this univerfal Theorem, viz.
Theorem If a Power is to the Weight, as I to the
the Pulley. Number of the Ropes (i. e. of the Parts
of the Rope, viz. AH and Bh, Fig. 27,
and 28; and AB, CD, EF, Fig. 29, and
AB, CD, EF, GH, Fig. 30,) apply'd to
the lower Pulleys, then the Power will be
able to hold up the Weight
the Weight; and there-
fore being a little increas'd, will raiſe the
Weight.
7.
For Inftance, Suppofe the Weight to
Examples. be rais'd 360 Pound, and the Power in it
felf to be equivalent only to 180 Pound,
(i. e. the Power to be to the Weight as 1
to 2.) therefore one lower Pulley (ei-
ther with, as Fig. or without an up-
per Pulley) will render the Momentum
of the Power equal to that of the
Weight, namely, by rend'ring the Velo-
city of the Power double to that of the
Weight. For 180 p. * 2 Velocity 360
Momentum
360 p. 1 Velocity. If the
Power be to the Weight as 1 to 3, or e-
quivalent only to 120 Pound, then there
muſt be three Wheels, that fo the Velo-
X
city
Mechanicks.
35
*1 *
X
city of the Power may be triple to that
of the Weight; Namely, 120 p. × 3 Ve-
360 Momentum 360 p. × 1
And fo on.
locity
Velocity.
!
CHAP. VI.
Of the Screw.
HE Screw, which makes here the
TH
1.
fifth fimple mechanical Power, is The Screw,
a Right Cylinder, having feveral Rifings what.
and Hollows alternately winding round
it in a ſpiral manner. It confifts
It confifts proper-
ly of two Parts, one call'd the outward
Screw, the other the inward; namely,
becauſe the former receives the latter in-
to it.
Whence the inward Screw is o-
therwiſe call'd the Male, and the out-
ward the Female Screw. The Male or
inward is denoted Fig. 31, by MS, the
Female or outward by TS.
2.
This Machine is chiefly us'd to fqueeze
or prefs Things, and by that Means, in Its Uses:
fome Cafes, to break Things. It is alfo
fometimes uſed to raiſe up any Weight,
as to thrust it forward.
(C 2)
For
36
The Young Gentleman's
{
3.
This in
compound
For the more eafie Ufe of this Ma-
chine, (i. e. for the more eafie moving
fome fort a the Male Screw through the Female, or
Machine. the Female along the Male) it is uſual to
apply thereto an Handle or round Stick,
AB; and fo to the Power of the Screw
to adjoin alſo the Power of the Axle with
the Wheel: The Handle or Stick AB by
its turning round, fupplying the Place of
a Wheel, and the Cylinder of the Screw
being as the Axle to the faid Wheel. So
that this Machine is in fome fort a com-
pound Machine.
4.
"
Now the Effect of this Machine de-
The funda- pends upon the fame fundamental Theo-
mental rem, as the other; which fundamental
particular- Theorem particularly apply'd to the Fa-
ly apply'd brick or Make of this Machine, ftands
Theorem
chine.
to this Ma- thus: If, as the Compass defcrib'd by the
Power in one Turn of the Screw, is to the
Interval or Distance between any two imme-
diate Spiral Windings (meafur'd according
to the Length of the Screw) fo is the Weight
or Refiftance to the Power; then the Power
and the Refiftance will be equivalent one to
the other, and confequently the Power be-
ing a little increafed will move the Refi-
fiance.
5.
The fume
further il
lu rated or
prov'd.
For it is evident from the bare Infpe-
&tion of this Machine, that in one Turn
of the Screw, the Weight P (Fig. 33) is
fo much lifted up, or the Refiſtance O
(Fig.
Mechanicks.
37
Fig. 31 and 32) is fo much remov'd, or
the Thing to be prefs'd is ſqueez'd fo
much cloſer together, as is EG the Di-
ſtance between two immediate Spirals;
and in the fame Time the Power (ap-
ply'd to A or B) to be mov'd fo much,
as is the Compaſs AB, defcrib'd by the
faid Power in one Turn of the Screw.
Wherefore the Velocity of the Weight (or
whatever anſwers thereto) will be to the
Velocity of the Power, as is the faid Di-
ftance between the Spirals to the Com-
pafs defcrib'd by the Power, in one Re-
volution or Turning round of the Screw.
And therefore by the fundamental Theo-
rem, if as the Power is to the Weight, fo
the Diſtance between the Spirals to the
Compaſs deſcrib'd by the Power, the
Power will have its Mo nent equal to
the Moment of the Weight or Re-
fistance. And therefore the tower, be-
ing a little increas'd, will overcome the
Refiftance.
(C 3)
CHAP.
2
38
The Young Gentleman's
I.
A Denniti-
on of a
Wedge,
and its fe-
veral
Parts:
2.
Its Effect
accounted
for by the
fundamen-
tal Theo-
rem.
TH
CHAP. VII.
Of the Wedge.
HE Wedge is commonly made of
Iron or fome harder Sort of Mat-
ter, in Shape of a Prifm not very high,
whofe oppofite Bafes are Ifofcelar Trian-
gles. The Height of either of theſe Tri-
angles is eſteem'd the Height of the
Wedge, and the Bafis of either Triangle
is eſteem'd the Thickness of the Wedge;
and the right Line which joins together
the Vertexes of both the Triangles is cal-
led the Edge of the Wedge; and the Pa-
rallelogram which joins together the Bafes
of the two Triangles, is call'd the Back
of the Wedge.
The Effect of this Machine depends
likewife upon the fame fundamental
Theorem as the former which being
particularly apply'd hereto will ftand
thus: If the Power directly apply'd to the
Back of the Wedge be to the Refiftance to be
overcome by the Wedge, as the Thickness of
the Wedge is to its Height; then the faid
Power will be equivalent to the Refiftance;
and therefore being increas'd will overcome it,
That
Mechanicks.
39
That Tenaciouſneſs or Firmnefs, where- 3.
by the Parts of the Wood (or the like) The fame
further il
adhere one to the other, is the Refiftance luftrated,or
to be overcome by the Wedge. Now it prov'd.
is evident, that while the Wedge is drove
into the Wood, the Way or Length
which it has gone according to its own
Propenfity, is (Fig. 34) BA; and in
like manner DC is the Way or Length
gone in the fame Time by the Impedi-
ment, i, e. the Parts C and D of the
Wood are ſo far divided aſunder. And
accordingly as the Wedge is drove down
further and further along its Height, fo
the Parts C and D of the Wood are divi-
ded more and more along the Thickneſs
of the Wedge; and in the whole Pro-
grefs proportionably, as is evident from
the Nature of a Triangle. Whence it
comes to pafs, that if as the Thicknefs
of the Wedge (i. c. the Way of the Im-
pediment, and confequently its Veloci-
ty) is to the Height of the Wedge (i. c.
the Way, and confequently the Velocity
of the Power,) fo the Power to the
Impediment or Refiftance, then the Mo-
mentum of the Power and the Impe-
diment will be equal the one to the o-
ther; and confequently the Power be-
ing increas'd will overcome the Refift-
ance,
(C 4)
To
40
The Young Gentleman's
4. To the Wedge may be referr'd all
Other In Edge-Tools, and Tools that have a
firuments
reducible to harp Point, in order to cut, cleave,
the Wedge. flit, chop, pierce, bore, or the like: As
Knives, Hatchets, Swords, Bodkins.
5.
c. And thus I have gone through the
fix fimple Machines, or primary mecha-
nical Powers, fo far forth as feems re-
quifite to the Defign of this Trea-
tiſe.
It remains only to obferve here, that
Mechanicks as the Word Mechanicks in its largeſt
what, in the Acceptation comprehends the whole Do-
more re-
ftrain'd
Sense of
the Word.
&rine or Science of Local Motion; ſo in
a reſtrain'd Senſe it is taken particularly
to denote the Doctrine of Motion, as
apply'd to Machines or mechanical En-
gines, the more fimple of which we have
here defcrib'd.
CHAP
}
Mechanicks.
41
TH
CHA P. VIII.
of Staticks.
1.
what.
HE Word Staticks denotes in the
Greek Language the Science or Do- Staticks,
&trine of Weights, and confequently in
its largest Acceptation, is of the fame Ex-
tent with Mechanicks in the largeſt Senfe
thereof, both comprehending the whole
Doctrine or Science of Motion with refe-
rence to the Weights of Bodies; and fo
being fynonymous Words, or Words of
the fame Import.
But as the Word Me-
chanicks in a reſtrain'd Senfe is taken to
denore more particularly the Doctrine of
Motion and Weights in reference to Me-
chanical Engines; fo the Word Staticks
in a reſtrain'd Senfe is taken to denote
more eſpecially that Part of the Doctrine
of Motion, which arifes from the Weight
of natural Bodies confider'd in themfelves
or compar'd together, and which confe
quently has not at leaſt ſo immediate a
reſpect to mechanical Engines. In this
Chapter therefore I fhall take notice of
fuch Particulars as relate to Staticks in its
reſtrain'd Senſe, and are agreeable to the
Defign of this Treatife.
It
42
The Young Gentleman's
2.
Bodies
It is then obfervable in the firſt Place,
The Motion that the natural Motion of heavy Bodies
of heavy (i. c. That Motion of them downwards,
which arifes from the natural Principle of
wards is their Gravity or Weight) is found by
uniformly
accelera- Experience to be a Motion uniformly acce-
ted. lerated, i. e. a Motion which acquires in
down-
༡.
equal Times equal Degrees of Velocity.
For when an heavy Body defcends, the
fame Gravity, which gave it a Motion
downwards at first, will all along conti-
nue to act alike upon it, and ſo to give
it in the ſecond Inftant of Time a Moti-
on equal to what it gave it at the firſt, and
again in the third Inſtant or Minute to give
it likewife a Motion equal to that at firſt,
So that at the third Minute the Sum of the
whole Motion will be the Triple of the
first; and fo at the fourth Minute the Sum
of the whole Motion will be Quadruple
to what it was at the first. And fo the
Degrees of Motion, or Velocity of the
Body moved, will always increaſe, as
does the Time wherein it moves.
Hence it comes to pafs, that the Spa-
The Spaces ces, through which Bodies defcend in
thro' which their Fall, are as the Squares of the Times
Bodies fall,
are as the they take up in Falling, always account-
Squares of ing from the Beginning of the Fall. For
bercin Inftance, if a Body be ten Minutes a fal-
they fall. ling, and the firft Minute it falls a Mile,
the fecond Minute it will have fall'n four
he Time
Miles,
}
Mechanicks.
43
Miles, and the third Minute nine Miles; and
fo on: According to which Proportion at
the tenth Minute, the Body will have
fall'n an hundred Miles. Whence it ap-
pears, that if the Times be taken in the
arithmetical Progreffion of natural Num-
bers, viz. 1, 2, 3, 4, 5, &c. the spa-
ces defcrib'd in thoſe Times, reckoning
from the Beginning of the Motion, will
be as 1, 4. 9. 16, 25, &c. This is wont
to be demonſtrated or at leaſt illuſtrated,
by confidering (Fig. 35.) AB to repre-
fent the first equal Part of Time, and BC
to reprefent the first Degree of Velocity
acquir'd in the first Part of Time: And
likewife BD (AB) to reprefent the fe-
cond equal Part of Time, and FE (=BC
-DF) the fecond Degree of Velocity, ac-
quir'd in the fecond Part of Time, and e-
qual in it felf to that Degree of Velocity
acquir'd in the first Part of Time: and con-
fequently AD to reprefent two equal Parts
of Time, and DE to reprefent two equal
Degrees of Velocity. Wherefore the Tri-
angle ABC will reprefent the Space gone
thro' in AB, the first equal Part of Time,
with the Velocity BC; and the Triangle
ADE will reprefent the Space gone thro'
in AD two fuch equal Parts of Time, with
the Velocity DE (the Double of BC.)
But the faid fimilar Triangles ABC and
ADE are (by Theor. 10 of Young Gen-
tleman's
44
Mechanicks.
4.
the Fall of
der'd in
Part of
Time di-
it felf.
tleman's Geometry) as the Squares of
their homologous Sides AB and AD;
that is, the Spaces gone through in the
Times AB and AD are as the Squares of
the faid Times. Thus (Fig. 35) ADE
is actually divided into four Parts, each
equal to ABC.
Hence it is obvious, that the Spaces
The Spaces defcrib'd or gone through by a Body in
deferib'd by its Fall, in each (equal Part or) Minute
bevy Bo- of Time, are according to the Series of
dies, confi- the natural Numbers, 1, 3, 5, 7, 9, &c ;
each equal thefe being the Differences of the squares
I, 4, 9, 16, &c. Thus if a Body falls
ftinctly by One Mile in the firſt Minute, and four
Miles in the ſecond Minute, it will there-
fore, during the fecond Minute, confi-
der'd diftinctly from the first, fall three
Miles. And in like manner, if a Body
falls at the End of two Minutes four
Miles, and at the End of three Minutes
nine Miles, it will, during the third Mi-
nute confider'd by it ſelf, fall five Miles,
and ſo five Times as far as it did the firft
Minute.
5.
How to
find, what
Space a
Body will
go thro' in
a given
Time.
Hence it is obvious, that the Space
gone through in a determinate Time by
any Body being known, thereby may
be found out, what Space it will go
through in a given Time う ​viz. by find-
ing a fourth Proportional. For Exam-
ple, fuppofe a Body to fall twenty-four
Foot
Mechanicks.
45
:
Foot in one Minute, and it be requir'd
to tell, How far the fame Body will fall
in the fame Medium in Three Minutes ;
the Anſwer will be, 216 Foot. For as I
to 9 (i. e. as the Square of one Minute
is to the Square of three Minutes) fo is
24 (the Space gone through in the firſt
Minute) to 216, which therefore is the
Space the Body will go through in three
Minutes.
6.
what Time
Body will
take up in
defcending
ven Spase.
thro' a gi-
In like Manner, the Time being known,
wherein an heavy Body defcends through To find,
a determinate Space, it is obvious to find, a
in how long Time it will defcend thro' an-
other given Space, viz.by finding a fourth
Proportional. For Example, fuppofe a
Body has taken up one Minute in falling
twenty-four Foot, and it be requir'd to
tell, what Time it will take up in falling
216 Foot in the fame Medium; the An-
fwer will be three Minutes.
For as 24
to 216 (ie. as the first Space given is to
the ſecond Space given) fo is the
Square of the given Time to 9 the Square
of the Time requir'd, which therefore is
three Minutes.
As a Body in falling acquires equal
Degrees of Velocity in equal Times, fo
on the contrary, in rifing it lofes equal
Degrees of Velocity in equal Times. For
the Gravity of the Body thrown up con-
tinually drawing it downwards, its Moti-
On
7.
of the Af-
cent of Bo-
dies.
46
The Young Gentleman's
on upwards will continually decreaſe, in
proportion to the Force downwards pro-
duc'd in equal Times by the Gravity of
the Body. But it has been fhewn afore
(§. 2.) that the Motion downwards is u-
niformly accelerated; and therefore the
Motion upwards must be uniformly re-
tarded. Whence it follows, that fuch a
Body goes through the fame Spaces in e-
qual Times, in rifing as it does in falling,
but in an inverted Order. Namely, if
the Body be five Seconds in rifing to the
Height of twenty-five Foot, and the Space
afcended through the first Second of
Time be nine Foot, the faid Body during
the fecond Second of Time, will afcend
through a Space of feven Foot; and in
the third Second a Space of five, and
the fourth a Space of three, and in the
fifth and laſt a Space of one Foot. Then
it will be as it were in Equilibrio, i. e.
the Impetus upwards being now juft
equal to the Impetus downwards, the
Body will neither rife nor fall for fome
Time. After which it will begin to fall
in the like inverfe Proportion viz. fo as
that the first Second it will fall a Foot,
the fecond three Foot, the third five the
fourth ſeven, and the fifth and laſt nine
Foot, thus taking up five Scones of Time
to defcend twenty five Foot, as it took up
the fame Time to afcend twenty-five Foot.
By
Mechanicks.
47
rife up a
Height,
By the fame Principles it comes to 8.
paſs, that the Force which a Body ac- Why Bodies
quires in falling, will make it rife up a- gain to the
gain to the fame Height which it fell fame
from. For Inſtance, fuppofe (Fig. 36) from which
the (†) Pendulum, i. e. the Body or they tell.
Weight fafter'd to one End A of the
String AB, whoſe other End is faſten'd
to the fix'd Point C, as the Center of its
Motion: Suppofe, I fay, the Body to
fall from D to A, it will rife again to E,
which is of an equal Height with D. For
divide DA into any Number of equal
Parts, fuppofe three, markt a, b, c, and
EF into as many markt d, e, f, E.
the foregoing Principles it is obvious,
that whatever Force or Degree of Velo-
city the Body acquires by its Gravity in
deſcending from D to a, it will acquire
twice as much in defcending from a to b
and thrice as much in defcending from b
to c; with which triple Force it will be-
gin to aſcend from c towards d.
C
But in
afcending from c to d the Body will by
its Gravity drawing downwards lofe one
of the three acquir'd Degrees of Velocity
or Force; and in afcending from d to f
it will loſe two of the faid acquir'd De-
By
(†) It is fo call'd from the Weight's hanging on the
Thread or Wire, to which it is faften'd; the Latin Word
Fendulus denoting any Thing that thus hangs.
grees
48
The Young Gentleman's
*
9.
The Refift-
ance of the
Air not con-
fider'd in
the fore,
mention'd
Cafes.
10.
grees, the Force of Gravity downwards
at f being double to its Force at d, and
fo deftroying two of the acquir'd Degrees
of Velocity or Force, wherewith the Bo-
dy at c began to afcend. And likewiſe at
E, the Force of Gravity downwards will
be triple to that at d, and ſo deſtroy all
the three acquir'd Degrees of Force,
wherewith the Body began at c to af
cend: And therefore the Body will af
cend no higher, but begin to fall down-
wards again towards e or A.
But now it is to be obferv'd, that in
this and the foregoing Cafes, the Refift-
ance of the Air, wherein the Bodies are
conceiv'd to move, is not confider'd
which is the Cauſe why the foregoing
Theorems will not exactly agree with Ex-
periments; and particularly as to the
Pendulum, which, by means of the Re-
fiftance of the Air, does not return ex-
actly to the fame Height from which they
fell, and confequently have not a perpe-
tual Vibration or Motion forwards and
backwards, but ſtand ſtill in fome Time.
Abſtracting from or fetting afide this
Pendulums Refiftance of the Air, it follows from
afcend and the foremention'd Principles, that the
defcend
thro' equal Pendulum does afcend and defcend thro'
Spaces in e- the equal Spaces DA and AE in equal
Times. To which I ſhall here add as
worth remarking, that the Longitudes
qual Times,
&c.
of
Mechanicks.
49
of Pendulums are as the Squares of the
Times, wherein. they perform their Vibra-
tions.
II.
dies de-
ter of Gra
As Gravity is that natural Principle
which caufes Bodies to defcend, fo no How the
Body will defcend, but when its Center Fall of Bo-
of Gravity can defcend. Thus the in- pends on
clin❜d Body ABDE, (Fig. 37.) which is their Cen-
fet on the Horizontal Plane HP, cannot vity.
fall toward the Part D, to which it in-
clines, becauſe by fuch a Change of Situ-
ation its Center of Gravity C would riſe;
as may be known by defcribing from A
(as the Center) the Arch CF, which is
the fame, as C the Center of Gravity
would make about the Point A, if the
Body ABDE could fall; for it is evident.
that one Part of that Arch rifes above
the Point C, and confequently the Cen-
ter of Gravity in fuch a Cafe muft for
fome Time afcend of its own Accord,
which is contrary to the Nature of heavy
Bodies.
Infrated.
But on the contrary, the inclin'd Body II.
ABDE (Fig. 38) mult of neceffity fall The fame
towards the Part D, which it inclines to, further il-
becauſe its Center of Gravity C may de-
ſcend in fuch a situation; as will appear
by drawing as afore) from A through C
the Arch CF, which is the fame that will
be defcrib'd by C the Center of Gravity
about A, when the Body ABED falls
(D)
For
:
50
The Young Gentleman's
13.
of Bodies
Standing
firmly.
14.
For all the Points of the ſaid Arch fall
below the Point C.
Hence it appears, that to have a Body
keep firm upon any Thing, that fupports
it, and is not it felf inclin'd, the right
Line CF drawn from the Center of Gra-
vity in the faid Body toward the Center
of the Earth, and wherein the ſaid Body
endeavours to defcend (i. e. in fhort the
Line of Direction) must neceffarily fall
in fome Part of the Bole or Foot AB of
the ſaid Body ; as Fig. 37. Otherwiſe,
if the Line of Direction CG falls without
the Bafis, the Body will neceffarily fall
down, as in Fig. 38.
Whence it follows, that by how much
Further of lefs the Bafis of a Body is even though
the fame.
it be not inclin'd, fo much more eaſily
will it be mov'd out of its Place; becauſe
the leffer Change of Pofition is capable to
make the Line of Direction be without its
Bafis or Foot. Which is the Reafon that
a Bowl rolls eafily upon a Plane, and
that a Needle will not ftand upon its
Point. It is obvious, that agreeably to
what has been faid, it follows alfo, that
the wider the Foot of a Body is, the
more firmly will it ftand, becauſe fo
much the greater Change is neceffary to
cauſe the Line of Direction to fall with-
out the Foot of the faid Body.
I'
Mechanicks.
51
1.5.
It appears alſo from what has been
faid, that if the Plane OP, which fuftains of the Sli-
ding and
the Body ABDE (Fig. 39) be inclin'd, Falling of
that Body will flide, when its Line of Bodies.
Direction CG falls upon any Part of its
Baſe AD And that a Body abde will fall,
when its Line of Direction
out the faid Bafe ad.
1 cg
falls with-
Whence it follows, that a Bowl fet up-
16.
on an inclining Plane, as the Roof of an Further of
Houfe, will roll off it, becauſe its Line the fame.
of Direction not being perpendicular to
the Plane, (but to the Horizon,) can-
not pass through its Foot where it touches
the Plane, it being almoſt an indivifible
Point.
17.
bends,
vie
when he ris
It is remarkable, that we naturally
obferve this Law of Staticks, we are Why a Man
here ſpeaking of, to keep our felves from
falling. Thus, when we would rife from fes from a
a Seat, we bend our Body, by thruft- Seat
ing forward our Head and Back HB,
and putting backwards our Feet and Legs
FK (Fig. 40) fo as the Line of Directi-
on LG of our Center of Gravity may paſs
through our Footing F or Bafis. And it
is obvious, that if we bend not enough,
we fall backwards or towards B; but if
we bend too much, we fall forwards or
towards F or K.
(D 2)
In
52
The Young Gentleman's
1
18.
In like Manner, when we go up Stairs,
Why a Man as we lift one Foot up, fo we bend our
ward when Body forward, that we may (both go up
he goes up eafier, and alfo) may preſerve our ſelves
leans for-
Stairs.
19.
mon Pof-
tures ac-
from falling beckwards, by having the
Line of Direction of our Center of
Gravity too far from our Footing, which
in this Cafe is chiefly the Foot lifted
up upon the Stair, when we are get ing
up.
On the fame Account it is, that when
Other com- a Man carries a Burthen on his Back,
he leans forward; but if he carries a
counted for. Burthen before him, he leans backward.
If he carries it with his right Hand, he
leans to his Left ; if with his left Hand,
he leans to his Right. Namely that
hereby he (may not only more eafily
carry the Burden, but alfo) nay pre-
ferve himſelf from falling that Way
which the Burden' he carries, pulls
him.
20.
The like
And that we learn this, not from
Reaſon, but rather from natural Inſtinct,
Rules ob..
is evident, not only from Mens doing fo
Jerv'd by
irrational without knowing the Reafon of it, but
Creatures. alfo from irrational Creatures doing the
like. For a Goofe going through a Barn
or other Door, be it never to high,
does thruft forth, and fo downwards his
Head, not becauſe he is afraid of ſtriking
it
Mechanicks.
53
*
but
it againſt what is ſo much above it, but)
in conformity to this Law of Staticks.
Namely at a Barn Door there is ufually
a Threshold or the like, to be gone o-
ver. Wherefore the Goofe having put
one Foot on the Threfhold, thrufts
forward his Head, that his Center of
Gravity may not be too far behind his
Feet, and ſo pull him backwards
that by thruſting his Head and Neck fo
far as is requifite beyond the Threſhold,
the Direction of his Center of Gravity
may be brought fo near to his Footing,
as that he may go over the Threſhold
more easily, and without falling either
backward or forward. So that it appears,
the Gooſe is not to be laugh'd at for fo
doing, but rather they, who neither
know the Reafon of his ſo doing, nor at
leaſt that he does no other than they do
themſelves in like Cafes. And this is fuf-
ficient to our Defign, to have taken no-
tice of concerning Staticks, fo call'd in a
more reſtrain'd or proper Senfe.
(D 3)
CHAP.
54
The Young Gentleman's
45
I.
Hydrofta-
T
CHA P. IX.
?
Of Hydroftaticks.
HO' the Word Staticks, both as to
its own literal Import, and alſo in
ticks,what its largest Acceptation, denotes the Sci-
ence of the Weights of Bodies in gene-
ral, whether fluid (or liquid) or not flu-
id, and whether weigh'd in the Air or
Water; yet it is ufual to reftrain Staticks
in a more appropriated Senfe to denote the
Science of the Weights only of folid (or
not fluid) Bodies, and that too when
weigh'd in the Air; and to give the pe-
culiar Name of (*) Hydrostaticks to that
Part of Staticks largely taken, which
treats of the Weight of fluid or liquid
Bodies, efpecially of Water, and of fo-
lid Bodies put into (†) liquid Bodies, e-
Specially into Water.
The
(*) It is a Greek Word, compounded of sanny. the
Science of Weights, and dwg, Water.
(t) There is this Difference between a fluid and a liquid
Body. A Fluid is a Body, whofe Parts are eafily fepára-
red, and being feparated join together immediately, as
fif
Mechanicks.
55
4
A
; nor
ry Theo-
rems or
The primary Property of heavy Flu- 2.
ids are thefe; that if an heavy Fluid The prima-
ABCD (Fig. 41) be, either not prefs'd
at all, or equally prefs'd from above; its Properties
of Fluids.
upper Surface AB will lie horizontal or
level. And if the ſaid Fluid be diſturb-
ed, it will by its own Gravity return to
the fame Level of its upper Surface. For
the lower Particles of the Fluid will not,
by Reaſon of their Gravity, afcend up-
ward of themſelves, to raiſe any Part of
the faid Surface above the other
can the upper Particles of the Fluid de-
fcend, fince there is no Room for them,
in the lower Part of the Veffel, it being
taken up by other Particles of equal
Weight. And therefore if there be no
Preffure at all from above, or an equal
Preffure, then there is nothing to diſturb
or alter the Level of the upper Surface.
But if it be diſturb'd, the fluid will re-
turn to its Level, becauſe the higher Part
HI (Fig. 41) being heavier, will partly
deprefs the Particles under it, and will
partly run down into the lower Part
Air, Flame, Water, Oil, and other Liquors. A Liquid is
a Body whofe Parts are alfo eafily feparated, and again
come together immediately, and withal will continually
flow or fpread themſelves, 'till their upper Surface becomes
level, as Water, Oil, but not Flame. Hence it appears,
that all Liquids are Fluids, but all Fluids are not Liquids.
(D 4)
and
·56
The Young Gentleman's
3.
and that fo long, as any one Part of the
Surface is higher than the others.
If the Fluid (Fig. 42) be from above
Further of unequally prefs'd, it is obvious, that the
the fame. Part thereof AE, which is moft prefs'd,
will defcend, the Particles fo prefs'd,
thrusting out of their Places the other
Particles, that are either not prefs'd at
all, or lefs prefs'd; which therefore will
afcend to HI in proportion to the De-
fcent FG of the Particles prefs'd AE,
Namely, as the Part prefs'd AE is equal
to the Part not or lefs prefs'd EB, fo
the Afcent EH or BI will be aqual to the
Defcent AF or EG.
What has been faid of the upper Sur-
face AB, holds good of any other Paral-
lel Surface within the Fluid, as EF Fig.
43. Namely, if it be equally prefs'd by
the upper Part of the Fluid, and withal
by any Thing elſe fwimming either on
the Top or within the Fluid, it will re-
tain its horizontal Situation or Level.
But if it be any where prefs'd more than
in other Parts of it, there it will fink,
the Parts of it which are lefs prefs'd af-
cending as that defcends. And from
what has been hitherto faid, are inferr'd
the following Propofitions, as fo many
Corollaries.
If
Mechanicks.
57
the first.
If an heavy Body EFHB (Fig. 44) 4.
Iving upon a Fluid AB, be of an equal Corollary
Weight with the Air that takes up an e-
qual Space AHEG, the Surface AB of the
Fluid will retain its Level, as being eve-
ry where equally prefs'd.
5.
If the Body EFIK (Fig. 45) be lighter
than the Air which takes up the like Corollary
the second.
Space, then the faid Body, and the Par-
ticles of the Fluid under it, will afcend,
'till the Aggregate (of the Body IKEF and
of the Parts EFHB of the Fluid fo af-
cending, viz.) IHBK be of equal Weight
with the Air, that takes up the like Space
GAHI. .For then the Surface AB will be
equally prefs'd every where, both in AH
and HB.
6.
the third:
If the Body EFAB, (Fig. 46,) be hea-
vier than the Air of a like Bulk, then Corollary
the Body will fwim in the Fluid, de-
fcending fo far into it, 'till the Aggre-
gate (of the Air GEIK and the Fluid
IKAH, viz.) AHEG, which take up to-
gether like Space with the faid Body, be
of equal Weight with the Body. For
the Surface AB will become equally
prefs'd, both in AH and HB.
If the Body EGDF (Fig. 47) be hea- 7.
vier (not only than the Air, but also) Corollary
the fourth
than the Water (or other Fluid) that
takes up a like Space HCGE, then the
Body will fink down quite to the Bottom
CD,
58
The Young Gentleman's
A
8.
Corollary
CD, there being nothing to counterpoiſe
or bear it up, 'till it comes to the Bot-
tom. For the Body being fuppos'd to be
put on the upper Surface AB, and to lie
upon the Part IB thereof, it being heavi
er than a like Bulk of Air, the Part of
the Fluid IBDE will be more prefs'd
thereby, than the other Part of the Flu-
id AIEC by the Air, and therefore the
Particles in IBDE will defcend. And in
like Manner when the Body is come to
GF, it being heavier alfo than a like
Bulk AHGI of Water, it will ſtill prefs
the Parts of the fluid Water GF more
than are thoſe that are under GH, and
therefore will caufe the Parts under GF
ftill to give way, 'till it comes to the Bot-
tom CD.
Upon the fame Principles, if a Body
GHIK (Fig. 48) be immerg'd into the
the fifth. Fluid, and be (of equal Weight with fo
much EFGH of the Fluid as takes up the
like Space, i. e. in ſhort, be) of the fame
fpecifick Gravity with the Fluid, then
the ſaid Body will ſtay where it is plac'd
in the Fluid, without either rifing or fink-
ing. For the Surface EI being equally
prefs'd all along, the faid Body cannot
rife. And the Surface FK being all along
equally prefs'd, neither can the faid Body
fink.
But
Mechanicks.
59
But if the faid Body be fpecifically
lighter than the Fluid and confequently
the Bulk GHIK be lighter than the Bulk
EFGH, it evidently follows, that the
Surface FK will be more prefs'd in FH
than in HK, and confequently the Body
GHIK will be thruft up and rife ftill high-
er and higher for the fame Reaſon, 'till it
comes to fuch a Situation, as that it is of
equal Gravity with fo much Air (which
is in this Cafe fuppos'd fpecifically lighter
than the Body) and fo much of the Flu-
id, as take up equal Space with it; (as
is reprefented Fig. 46) according to S.
6. But lastly, if the Bodv be of the fame
Specifick Gravity with the Air, then it
will rife juft to the very Top of the Flu-
id; as is repreſented Fig. 44. according
to S. 4.
9.
der Water.
From what has been faid, it is eaſe to
account for feveral Occurrences in Flu- Why a Buc-
ket of Wa-
ids, as particularly in Water; of which ter has lit-
the more remarkable or common fhall be tle or no
Weight une
taken Notice of. Hence then it is eaſie
to tell, why we don't feel the Weight
of the Bucket and the Water in it,
when we draw Water, 'till it begins
to rife out of the Water. Namely,
becauſe fuppofing the Bucket with
the Water in it to be reprefented,
Fig. 48, by GHIK, it being of the fame
† Gra-
60
The Young Gentleman's
IS.
Why the
Depths of
;
(†) Gravity with the like Bulk of Water
EFGH, the Surface FK will be equally
prefs'd, both in FH, and HK ; and there-
fore the Bucket of Water can't defcend,
but is held up by the Water under it
and confequently the Weight of it is no
more felt, than is a Weight which is
counterpois'd by an equal Weight upon
a Balance. And the fame holds all the
while the Bucket is drawing up to the
Surface AB. Where it, and the Water
in it, coming into the Air ſo much ſpeci-
fically lighter than the Water or Wood,
c. of the Bucket, the Air can't ſuſtain
it, and fo the Weight of it muſt be ſuſ-
tain'd then by him that draws up the
Bucket.
On the fame Principles the Reafon,
why one cannot fathom the Depth of the
Sea in fome Places, is this: Becauſe tho'
fome Seas
cannot be the Plummet or Lead made ufe of be fpe-
fathom'd. cifically heavier than Water, yet the Line
to which the Plummet is faſten'd, is (in
this Cafe) fpecifically lighter than the
(+) The Water in the Bucket is certainly of the fame
Specifick Gravity with the Reft of the Water. And the
Bucket, if all of Word, is much lighter. And if other-
wife, the fpecifick Lightness of the Wood does reduce
the fpecifick Gravity of the Iron about the Bucket, almof
to an Equilibrium with the Water.
Water,
f
Mechanicks.
61
1
Water. Wherefore when the Depth is
fo great, that fuch a Quantity of Line is
requir'd, as that the Weight of the Line
and Lead together is no greater than the
Weight of a like Bulk of Water, then the
Lead or Plummet will fink no deeper;
and confequently will not fhew the Depth
of the Sea in that Place.
II.
According to the fame Principles we
may eafily tell, why one Liquor will fwim Why fome
upon another, or remain at Top without Liquors
Swim upan
mixing, if it be pour'd on gently; as Oil others.
in reſpect of Water; and Water in re-
ſpect of Quickfilver. Namely, becauſe
Oil is fpecifically lighter than Water, and
Water than Quickfilver.
12.
Why Ships,
that fail is
the Ocean,
ter.
Hence likewife it ceafes to be a Won-
der, that a Ship, which has fail'd in the
Open Seas, has funk and been loft in the
Mouth of a River of fresh Water; name- fink in
ly, becauſe Sea-water is much heavier fresh -
than freſh Water. And therefore though
the Ship with its Burden was lighter than
a like Bulk of Salt-water, and fo was
fuftain'd by the Salt-water in the Sea, yet
it was heavier than a like Bulk of frefh
Water, and therefore funk in the River.
It is obvious, that for the fame Rea-
fon any Ship or Boat will draw more
Water (that is, though it does not fink
quite down to the Bottom, yet will fink
deeper) in freſh Water than in Salt-wa-
ter.
More-
*
61
The Young Gentleman's
13.
that will
(wim for
Some time,
finks after-
wards.
Moreover the Reafon, why a Log of
Why Wood, Wood, that has been fwimming a long
time upon Water, will fink at laft, is
this, viz. becaufe fuch a Piece of Wood
may be of the fame or greater fpecifick
Gravity than Water, fetting afides the
Pores of the Wood; but by reaſon of its
Pores, the Air which fills the faid Pores
and the Wood together, make up one
Whole lighter than an equal Bulk of Wa-
ter, and therefore the Wood will not fink
at firft. But after fome Time the Water
being got into the Pores, and fo the Air
excluded, then the Water in the Pores
and the Wood make up one Whole hea-
vier than the Water of a like Bulk, and
fo muſt fink according to the Principles
of Hydroſtaticks aforemention'd.
14.
fwim,when
From which Principles it follows alfo,
Why hollow that although a Lump of Brafs or Iron
Weffels of
Brass or I- or other Metal will fink in Water, as be-
ron will ing fpecifically heavier, yet a Kettle or
the like of Brafs or Iron (c.) will not
fink but fwim; becaufe the Kettle with
the Air contain'd in its Hollow, make up
one Whole, which is much lighter than
an equal Bulk of Water.
Lumps of
the fame
Metal will
fink.
It follows alfo, that a Body will weigh
15.
A Body will lefs in Water than in Air, by juft fo
weigh less much as is the Weight of an equal Bulk
thanin air. of Water. Whence it comes to pafs, that
in Water,
two Pieces of Metal of different ſpecifick
Gravi-
Mechanicks.
63
Gravities, fuppofe one of Gold and the
other of Silver, which fhall be of equal
Weight in the Air, will not be ſo in the
Water, that Metal whofe fpecifick Gra-
vity is greateſt, (viz. the Gold) lo-
fing less of its Weight in the Water
than the other (viz. the Silver,) be-
cauſe it takes up the Room of a lefs
Bulk of Water.
,
If (Fig. 49) a Fluid, particularly
(Viercury or) Quickfilver contain'd in
the Tube CD preffes on the Part C of the
Surface AB, more than the Air preffes on
the other Parts, C will defcend, and the
Quickfilver run down out of the Tube,
'till what is left CI in the Tube preffes C,
juſt as much as the other Parts are preſs'd
by the Air. And then no more will run
out of the Tube, becauſe of the equal
Preffure in all Parts of the Surface AB.
Where it is to be noted, that the Weight
of the Quickfilver CI is equivalent to the
Weight of a Column of Air, extended
from the Surface AB up to the Top of
the Atmoſphere, and of the fame Bafis
or Diameter of the Tube CD.
}
16.
Why a Flu-
Tube will
Stand up
higher than
Fluidwick.
id in a
the ſame
out the
Tube.
17.
id will rife
to
Why a Flu
the fame
Height in
a floping
Tube, as in
And it is obfervable (Fig. 50) that
the Quickfilver will rife up juft to the
fame Height in the oblique or floping
Tube EG, as it does in the upright Tube
DC, although the Quick-filver EF in the
Tube EG be more, and confequently an upright
heavier,
6A
The Toung Gentleman's
18.
heavier, than the Quickfilver CI in the
Tube CD. The Reafon whereof is this,
that as much as the Quickfilver EF is
more than the Quickfilver CI, fo much
greater is the Bafis E than the Bafis C,
(this being equal to the Bafis of the Cy-
linder EK, wherein the Quickfilver EH
will be found to be juſt ſo much as the
Quickfilver EF, by reaſon of the two
Tubes EK and EG being Cylinders of
Equal Height upon the fame Bafes, and
therefore Equal one to the other.) Now
it is obvious that fuppofing the Bafis E to
be Twice as big as the Bafis C, and
confequently the Quickfilver EH in the
Tube EK (or the Quicksilver EF in the
Tube EG) to be Twice as much as the
Quickfilver CI in the Tube CD, it evi-
dently follows that as the Quicksilver
CI do's equiponderate to a Column of
Air of the fame Bafis C, fo the Quick-
filver EH (or EG) which is the Dou-
ble Quantity of Ci, do's but equipon-
derate to a Double Quantity of Air,
i. e. to a Column of Air, whofe Bafis
E is the Double of the Bafis C.
Since according to the Rules of Sta-
Further of ticks, Bodies gravirate or weigh hea-
the fame. vier in a Perpendicular. than in any
Oblique or floping Situation, it may
be ask'd, why the Quickfilver EH gravi-
tating or prefling upon the Bafis E more,
than
Mechanicks.
65
than do's the Quickfilver EF, (that be-
ing perpendicular to E, this oblique.)
the Quickfilver do's not defcend lower in
the Tube EK, and afcend higher in the
Tube EG. The Reafon then is this,
that as much as EH preffes downwards
more than EF, fo much do's the Force
at E, which (arifes from the Preffure of
the Air on the other Parts of the Surface
AB, and) fuftains the Quickfilver in each
Tube, prefs upwards more in the up-
right, than in the oblique Tube; and
both for the fame Reaſon, viz. becauſe
the Preffure, whether upward or down-
ward, is in the Tube EK perpendicular,
whereas in the Tube EG it is oblique,
and therefore not fo great. Now the
Force at E, which holds up the Quick-
filver in the upright Tube; being as to
Effort fo much greater than the Force at
E which holds up the Quickfilver in the
oblique, as the Weight of the Quick-
filver in the upright Tube, is greater
than the Weight of the other; it fol-
lows, that the faid Force at E acting
upwards according to the Perpendicular
EH, is as able to fuftain the Quickfilver
in the upright Tube, as is the fame
Force acting upwards according to the
oblique Line EF to fuftain the fame
Quantity of Quickfilver in the oblique
Tube. And therefore the Quickfilver in
the
(E)
66
The Young Gentleman's
19.
of the
Fluid.
the upright Tube will not defcend low-
er by Running any of it out of the faid
Tube, nor will the Quickfilver in the
oblique Tube afcend any higher, by
any more Running into it: but the
Quickſilver in both Tubes will keep at an
equal Height.
And the fame will hold as to any o-
The Shape ther Tube (or Veffel) of any other
of the Vef-
fel makes Shape: Namely, the Height of the Quick-
no Diffe filver will be the fame in any Veffel, as
rence as to it is in a Cylinder of the fame Height
the Rifing
and Bafis with the Veffel. Thus, for
Inftance, Fig. 51, the Height of the
Quickfilver is the fame, both in the
round-headed Tube DEHF, and the
fharp-headed KLB, as it is in the Cy-
linder infcrib'd in the former, viz. CI,
or the Cylinder BLOP, circumfcrib'd a-
bout the latter. For as to the round-
headed Tube DEHF, as much of the
Quickfilver as lies without the Cylinder
CI, is fuftain'd not fo much by C, as by
the Parts MT and NS of the Tube's Head
which are under them, and no more of
the Quickfilver is properly or directly
fuftain'd by C, than what lies within the
Cylinder CI. And confequently the Effect
will be the fame, as if the Quickfilver was
put in the Cylinder CI.
As
1
Mechanicks.
64
20.
As to the tharp-headed Veffel LBK,
and the Cylinder BLOP, the Quickfilver Further of
the fames
will likewiſe rife as high in the one as
the other. For as to fo much of the
faid Veffel, as allows a free Afcent from
the Surface AB, the Quickfilver will there
rife to its juft Height CI, for the fame
Reaſon as it will rife to the fame Height
in the Cylinder BLOP. As to the other
Parts of the faid Veffel, where the
Quickfilver is hinder'd by the Sides of
the Veffel to rife up to the Height CI,
as much as is there wanting in the Weight
of the Quick-filver, which is over the
refpective fubjacent Parts of the Surface
AB, fo much is fupply'd by the Sides of
the Veffel. For although the Sides of
the Veffel do not prefs downwards as a
contrary Force, yet they keep the Quick-
filver from being thruft Higher by the
Force that preffes upwards. And confe-
quently the feveral Parts of the Sides
may be look'd on as fo many Impedi-
ments, which are equivalent to the Weight
of fo much Quickfilver, as would be
contain'd between the refpective Parts of
the Sides, and the Parallel to I. And fo
the Surface AB will be equally preſt all
along under the Veffel B.
&
(E 2)
The
68
The Young Gentleman's
21.
Why the
Fluid rifes
The Preffure of the Air being that
which makes the Quickfilver rife in the
higher in a Tube; it follows, that the greater is the
The at the Preffure of the Air, the higher will the
the Quickfilver rife; which is the Reaſon,
Top of a that the Quickfilver will rife higher in
Mountain. the fame Tube at the Bottom of a Moun-
Bottom,
them at
22.
of the
Weather-
Glaís.
tain or confiderable Hill, than at the Top.
Namely, becauſe the Column of Air pref-
fing the Surface AB will be ſo much
longer at the Bottom, than at the Top
of the Mountain, as is the Height of the
Mountain; and confequently the faid
Column containing fo much more Air,
will prefs fo much more.
So as to the (*) Barometer or Weather-
Glafs, the Quickfilver rifes and falls in
the Tube, as the Air is heavier or
lighter, and fo more or lefs preſſes the
Quickfilver that is in the Well (as it is
call'd) of the Barometer. And thus far
is agreed by all: but what it is that
cauſes the Air to be fometimes heavier,
fometimes lighter, is not fo agreed.
That the Vapours do not, (as is, or has
been, commonly fuppos'd,) increaſe the
Gravity of the Air, feems more than
(*) It is fo call'd as being a Meaſure of the Weight of the
Air, the Word Begs fignifying Weight, and uergov a
Measure, in the Greek Tongue.
probable
Mechanicks.
69
probable on this Account: Namely, that
when the Vapours are in the lower Re-
gion of the Air, as in rainy Weather,
than the Air is lighteft, as appears by
the Falling of the Mercury (or Quick-
filver) in the Barometer. And when the
Vapours are in the middle Region of the
Áir, (i. e. when the Clouds are high.)
they do not increaſe the Preffure of the
Air. For although the Air be heavier
then, it is not becauſe the Clouds are
high; but the Clouds are high, becaufe
the Air is heavy. For when the Air
near the Earth is more denfe than ufual,
then it becomes heavier than the Va-
pours, which therefore muft afcend, and
at last fettle in that Region of the Air,
which is of the fame Gravity with them-
felves. If the Vapours were the Caufe
of Increaſing the Air's Gravity, there
muſt be as many Vapours in the Air at a
Time, as are equal in Weight to three
Inches of Mercury, for fo much we find
the Mercury rifes or falls in the Weather-
Glaſs. Now Mercury is about 14 Times
heavier than Water, and confequently
there must be in the Air at once fo many
Vapours, as are equal in Weight to a
Column of Water of 42 Inches in Height,
and whofe Bafis is equal to the Surface.
of the Earth; which is much more than
falls down in Rain, during a whole Year.
(E 3)
For
79
The Young Gentleman's
23.
The prova
ble Cante
of the Ri-
fing and
For a whole Year's Rain do's not fill a
Veſſel above 14 or 15 Inches high, as is
obferv'd in the Hiftory of the Royal Socie
ty at Paris.
The Reason then, why the Air is
heavier at one Time than another, arifes
more probably from there being more
Air on that Part of the Earth's Surface,
when the Air grows heavier. And this
proceeds from Winds. For Example: If
Weather the Wind, which is nothing but a Stream
Glass. of Air, thould blow on any Place, and
Falling of
the Mercu-
ry in st
24.
Of the Si-
as'd to
the Air thus mov'd fhould be kept in
that Place by Mountains or Hills: or,
if two contrary Winds fhould blow on
the fame Place; in either Cafe the Air
will be heaped up in the Middle; and
confequently there being more Air its
Gravity will be increas'd. But if the
aforefaid Circumftances, or the like do
not happen in any particular Country,
then the Air which is over it, will grow
lefs in Quantity, and confequently ligh-
ter. Whence it is probable, that the
Winds are the Caufes of the Variation of
the Air's Gravity.
It was afore (§. 16.) intimated, that
phon, or what has been faid in that and the fol
Inftrument lowing Paragraphs concerning Mercury
draw off or Quickfilver, holds true likewife in
refpe&t of other Fluids. But the fore-
quors out of mention'd Affections have been illuftra-
Wine or c-
ther Li
Vaffelis
1
ted
Mechanicks.
71
ted, particularly as to Quickfilver, foraf-
much as it is that which is us'd in Wea-
ther-Glaffes, wherein the faid Affections
are more frequently feen by Us: It be-
ing the Defign of this Treatife by the
Laws or Principles of Mechanicks to ex-
plain chiefly fuch Things as more fre-
quently occur in common Ufe. On
which Account I fhall add here the Ex-
plication of the Effect of the Siphon, (or
Inftrument us'd by Vintners, &c. in
drawing off Wine, or the like,) accor-
ding to the forementioned Principles of
(Mechanicks, or, more particularly of)
Hydrostaticks. It is then obvious, that
(Fig. 52.) the Air being fuck'd out of
the Siphon at E, the Wine (or other Li-
quor) in the Veffel ABFG will be thruſt
up by the Preffure of the Air upon the
Surface AB, into the other End H of the
Siphon apply'd to the faid Surface. And
the Wine will be thruft upwards in the
Siphon, 'till the Weight thereof comes
to an Equilibrium with the Preſſure of
the external Air, fuppofe 'till it comes
to the Height CI. If therefore the Top
D of the Siphon be not higher than I,
the Wine will rife to the faid Top of it 33
and there finding a Vent or Paffage down
the other Leg DE of the Siphon, it will
go that way, and fo run out at E; con-
tinuing fo to do, the Caufes ftill conti-
(E.4)
nuing,
72
The Young Gentleman's
:
25.
The Me-
chanism of
a common
nuing, 'till the Veffel is emptied. But it
is to be obferv'd, that the Mouth E of
the Siphon must be lower than the
Mouth H. For elfe, E and H being e-
qually prefs'd by the Atmoſphere, if the
Leg DE be but of an equal Length with
DH, and confequently the Fluid in one
Leg of equal Gravity, with the Fluid in
the other; the Fluid would ftay in the
Siphon, not moving one Way or the
other, without fome other external Force
befide the Air. And by Parity of Rea-
fon, if DE be shorter than DH, and fo
the Fluid contain'd in DE of leſs Weight
than that in DH, the Fluid would run
(not from H to E, but) from E to H,
the Air coming up at E, and thruſting
the Fluid forward the contrary Way.
This would happen, except at leaſt the
Siphon be fo wide, as that the Air may
afcend by the Sides of the Fluid, while
the Fluid it felf defcends in DE.
Proceed we next to explain the Me-
chaniſm of a Common Pump (Fig. 53)
by the Principles of Hydroftaticks. The
Pump, ex- Handle KL being thruft down, the Sucker
plain'd.
GF is drawn upwards from E, and toge-
ther with it the Air that lies upon it.
Wherefore the Water at the Bottom, be-
ing without the Hollow of the Pump at
A and B prefs'd by the Air, but within
the Hollow of the Pump at C being not
Cat
Mechanicks.
73
(at leaſt, not ſo much) prefs'd by the
Air, will therefore afcend from C to D,
and thruſting open the Valve E, (as ha-
ving no Weight or Preffure upon it,) will
afcend ſtill higher to the Bottom of the
Sucker, as meeting with no Preffure from
above, and being continually prefs'd up-
wards from Beneath. The Water ha-
ving thus fill'd the hollow Part EF of
the Pump, and the Sucker FG (by lifting
up the Handle KL) being forced down-
wards, do's force alfo downwards the
Water under it between F and E, and
by this Means forces down and keeps
fhut down the Valve E. Wherefore the
Water between E and F being thus
prefs'd by the Sucker; and having no
Paffage downwards through E, it lifts up
the Valve G of the Sucker FG, this be-
ing the only Way it can make for it felf.
The Water thus arifing above the Sucker,
is together with it (the Valve G being
kept down by the Weight of the Water
lying upon it, fo that the Water can't
run back that way again; and the Han-
dle KL being thruft downwards again)
lifted upwards: where finding an open
Paffage, it runs out at the Mouth H of
the Spout, other Water afcending up in
the mean while through D,
when the Sucker is drawn up.
continually, as long as the
(as afore,)
And thus
Handle is
lifted
74
The Young Gentleman's
*
Wells will
Pump-
Wells.
lifted up and down, and confequently
the Sucker is mov'd up and down, the
fame Cauſes being thus continued.
26. It is obvious from the Principles of
Why fome Hydrostaticks afore laid down, that the
not be made Distance between the Water at G in the
Well, and the Valve E, (or Bottom of
the Sucker when at the Lowelt, viz. at
E,) must not be greater than the Alti-
tude, whereto the Preffure of the Air is
fufficient to make the Water rife, which
is found by Experience to be about 33
Foot. Hence fuch Wells, as have their
Water lying fo deep, as that (a Pump
can't be made, but) the Diftance will be
greater than the aforefaid Altitude, can't
be Pump-Wells, but muſt be Draw-Wells,
i. e. can't have their Water drawn up by
a Pump, but by a Bucket and Rope.
27.
I ſhall conclude this Chapter, with
The Tide fhewing how that Rifing and Sink-
by the Prin- ing of the Sea-Water, which we call
ciples of the Tide, (and which has been of
Hydrofta- old obferv'd, but is by none of the
ticks.
old Philofophers, as I know of, tole
rably accounted for,) may be very na-
furally accounted for by the Princi-
ples of Hydrostaticks, after fuch a man-
ner as is very eafy to be compre-
hended.
The
1
Mechanicks.
75
28.
Tide al-
ways comes
The Moon's Orbit being over thoſe
Parts of the terraqueous Globe, which Why the
lie between the Tropicks, and where are
the largeſt Seas, as the paffes over to Us from
theſe Seas, the preffes (thofe Particles of the South,
Matter which are next under Her, and
they the next underneath, and fo the
whole intermediate Air, and thereby at
laft) the Water of the Sea, and ſo cauſes
the faid Water to rife, on each Side of
that Part of the Sea which lies under her
Orbit, that is, toward the North and
South Parts of the World. Hence to us
who live in the northern Parts of the
World, the Tide always comes from the
South, because the Moon's Orbit is South
of us.
And on the like Account, the
Tides comes from the North, to thofe
who live in the fouthern Parts of the
World, becauſe the Moon's Orbit is
North of them.
alfo, that the Tide muſt come fo much
ſooner or later to any Place, as it lies
more or leſs diſtant from the Tropicks,
or that Track of the Sea which is under
the Moon's Orbit.
And hence it is evident
29.
Why the
The Tide being thus caus'd by the
Preffure of the Moon, it plainly follows,
that becauſe the Moon comes every Day
to the Meridian of any Place fifty Mi-
nutes later than the Day before; there- nutes La-
fore the Tide likewiſe muſt fall
every Day
Tide comes
fifty Mi-
there for than
out fifty the Day
Minutes afore.
76
The Young Gentleman's
30.
Why the
Tide is fix
Hours co-
and fix
Hours go-
ing out.
Minutes later every Day, than the Day
before, in Refpect of any particular
Place.
Alfo, becauſe the Moon is fix Hours
in coming from the Horizon of any
Place to its Meridian, therefore (it is ob-
ming in, vious, that) the Tide muſt be fix Hours
coming in. And in like manner, becauſe
the Moon is fix Hours in going from the
Meridian to the Horizon of any Place,
therefore the Tide muſt be the fame
Time in going out. Where it is obvi-
ous, that as the Preffure of the Moon
begins in respect of any particular Place,
fo foon as the comes to its Horizon; fo
her Preſſure continually increaſes 'till fhe
comes to the Meridian of that Place, and
confequently the Tide rifes ftill higher
and higher, 'till fhe comes to the faid
Meridian. Which as the Moon leaves
again, fo her Preffure decreaſes, and
confequently the Tide finks lower and
31.
Why the
Tides are
Bigger
at the
Change and
lower.
Moreover it being agreeable to the
Laws of Staticks, (as is found by Expe-
riments,) for heavy Bodies to gravitate
fo much more, as they are nearer to the
Full of the Center of the Earth; hence the Reafon,
Moon, than why the Tides are Bigger at the New
at other and Full Moons, than at the Quarters,
Parts of
the Month. is easily to be affign'd; namely, the
Moon's being at her Change and Full
C
mearer
Mechanicks.
77
nearer to the Earth, than at her Quar-
ters.
the Change
about the
Equinox's
other Parts
Again it is obferv'd, that as the Tides
32.
at the Change and Full of the Moon are Why the
bigger, than at any other Time of the Tides at
Month; fo the Tides at the Change and and Full of
Full of the Moon about the Equinox's the Moon
are bigger, than thofe at any other Time
of the Year. Which may be accounted are bigger,
for by the Moon's being then over the than at
Middle Part of the Ocean, which is un- of the Year.
der the Celeſtial Equator, and confe-
quently by its Preffure thereon making
a greater Quantity of Water recede to
each Side, viz. Northward and South-
ward, than when the preffes on one Side
of the Equator, and fo not in the Middle
of the Ocean.
be Two
It remains to be obferv'd, that it is 33.
demonftrable (†) by the Principles of Why there
Staticks, that the Moon do's cauſe (the Tides in
Sea to fwell, or in one Word) the Tide, every
twenty-five
not only in that Part of the Sea which Hours.
fhe is over, but alfo in the Part oppofite
thereto. And hence it comes to paſs,
(†) Such as would have a more accurate Account of the
Tide, let him confult Newton's Phyf. Mathem. Lib. 3.
Prof. 24 & 37, and Numb. 16. of the Philofoph. Tranfa&t.
Lond. 1666; as alfo Gregory's Aftron. Phyf. and Geom. Lib.
4. Prop. 65. I have here contented my felf with a more
general Account, as being more Easy to be under.
ftood.
that
78
The Young Gentleman's, &c.
1
1
34.
that there be two Tides every twenty
five Hours; namely, becauſe in that
Time the Moon going from any Meri-
dian, returns to it again; and confe-
quently caufes one Tide, while fhe is in
the upper Hemifphere, and another Tide
while the is in the lower Hemifphere.
And thus I have taken Notice of fo
much of Mechanicks (in general, or in-
cluding Mechanicks, Staticks, and Hy-
drostaticks properly fo called,) as is an
ſwerable to the Defign of this Trea
tife.
FINI S
i
Mechan, Plate, 1.
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*
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THE
Young Gentleman's
OPTICKS:
CONTAINING
The more Ufeful and Eaſy
ELEMENTS of Opticks, more
properly fo called, of Catoptricks,
Dioptricks, and Perfpective.
By EDWARD WELLS, D. D.
Rector of Cotesbach in Leicester-
ſhire.
LONDON,
Printed for James Knapton, at the Crowu
in St. Paul's Church-Yard. 1713.
THE
PREFACE.
A
S Vifion or Sight is the
most useful and pleasant
Senfation of the Body, fo
Opticks must confequently be
very useful and pleasant, as not
only explaining the Manner of
Viſion in a Geometrical Way,
but also by the fame Principles
teaching how to ſupply or re-
medy the feveral Defects of
the Sight, by the Help of pro-
per Inſtruments, ſuch as Spec-
(F) tacles,
The Preface.
tacles, Microſcopes, Telef
copes, &c.
Add bereto, that to this Sci-
ence is owing the Invention of
Several Inftruments, which ferve
to pleaſe and divert the Mind
by curious and furprizing Ope-
rations; fuch as are the Dark-
Chamber, the Magick-Lan-
tern, &c. All which are ac-
counted for in the following
Treatiſe.
*
THE
-
:
83
THE
Young Gentleman's
OPTICK S.
CHA P. I.
of Opticks in general.
T
what,
言​。
HE Word Opticks, as it lite-
rally fignifies in the Greek Opticks in
Tongue any Thing that relates general,
to the Sight; fo in its largeſt
Acceptation it is ufed by the Learned. to
denote the whole Compafs of Science or
Knowledge (Phyfical as well as Mathe-
matical) relating to Vifion or Sight.
2.
call'd,
It is now agreed by the Learned, that
Vifion is caus'd by the Eye's receiving in- Opticks,
to it Rays, which iffue from the Object properly fo
feen. Thefe Rays, if they pafs from the whats
Object to the Eye through a Medium all
(F 2)
along
હૈ
$4
The Young Gentleman's
!
3.
Catop-
tricks,
what.
along of the fame Nature, diffuſe them-
felves all along after the fame Manner by
right (becauſe the ſhorteft) Lines. Whence
this Sort of Viſion is call'd Direct Viſion ;
and the Word Opticks is taken in a more
reſtrain❜d Senfe, to denote that Part of
Opticks in general, which treats of direct
Vifion.
If the Rays come not directly from
the Objects to the Eye, but fall first up-
on fome impenetrable Body, and are
thence reflected or beaten back to the
Eye, then the Vifion is called Reflex Vi-
fion; and not the Object it ſelf, but its
Image is feen. A rough Surface fo breaks
and ſcatters the Rays, that they render
no Image at all, or at beft but a very im-
perfect Image, of the Object; and there-
fore thoſe Bodies, which are capable of
being moſt poliſh'd, or made moſt ſmooth,
make the beſt Reflexion. Such is Glaſs,
us'd therefore now-a-days for Looking-
glaffes; but Glafs being of it felf pellu-
cid or tranfparent, there is ufually put,on
the Back-fide of Looking-glaffes, Quick-
filver to render it impenetrable by the
Rays. Before Looking-glaffes were in-
vented, the Ancients made ufe of polifh'd
Metal; and fuch a Piece of Metal, made
uſe of to ſee one's felf in, the Greeks cal
led KάTOTTO; and from thence that Part
Κάτοπτρον
of Opticks in general, which treats of
reflex
Opticks.
85
reflex Viſion, is diftinguiſh'd by the
Name of atoptricks.
what.
If the Rays of the visible Object pafs 4.
through different Mediums, fo that they Dioptricks
are more or leſs diffus'd in the one than
in the other, then each Ray will be re-
fracted or broken in one Medium, from
that right Line wherein it diffus'd it ſelf
in the other Medium; whence this is cal-
led Refracted Viſion. Thus a Ray paffing
out of Water into the Air appears refrac-
ted, i. e. broken or bent, from that right
Line it made in the Water. And becauſe
the Knowledge of refracted Viſion is moſt
uſeful in order to the making of Telef-
copes, and fuch other Inftruments ; by
looking through which, Objects are more
clearly feen; hence that Part of Opticks
in general, which treats of refracted Vi-
fion is diſtinguiſh'd by the Name of Diop-
tricks, Aiorov being a Word capable of
denoting in the Greek Language any In-
ftrument made for to look through.
5.
To the foremention'd Parts of Opticks
in general, may be added Perspective, or Perfpec
the Art of repreſenting vifible Objects in tive, what.
a Picture, juſt as they appear to the
Eye. This Art is fo call'd, becauſe
the Picture is to be fuppos'd between the
Eye and the Object, and tranſparent
that ſo the Object may be ſeen through
its the Word perfpicere in the Latin
Tongue
(F 3)
86
The Young Gentleman's
;
Tongue fignifying to fee through a Thing
I hall ſpeak of each of the forementi-
on'd ſeveral Parts of Opticks, fo far as
is requifite to the Defign of this Trea-
tife after that I have here adjoin'd the
Definitions or Explications of fuch Terms
or Words, as are of principal Ufe in Op-
ticks; as alfo fuch Theorems, as are a-
greeable both to the Principles of Geo-
metry and Experience, and therefore are
us'd as Axioms in demonftrating the The-
orems of this Treatife relating to Catop
tricks and Dioptricks.
DEFINITIONS.
1. A Radiant is whatever fends forth
Rays from its feveral Points. Hence not.
only the Object, fuppofe a Candle, is
called the Radiant; but alſo its Image,
fuppofe the Image of a Candle feen in a
Glafs, is call'd fo likewife. As is alfo
the Image of an Image, and fo on
Where note, that for Diftin&tion's Sake,
the Object is ufually ſpecify'd by the
Name of the prime (or primary) Radiant.
And fo the immediate Image of an Ob-
ject may be diftinguifh'd by the Name of
a fecond Radiant; the Image of the faid
Image, by the Name of the third Radi-
ant, and fo on; that which is feen by
the Eye being always diftinguishable
(where
Opticks.
87
(where there are many Images) by the
Name of the last Radiant.
2. Parallel Rays are fuch, as keep all
along at an equal Diſtance one from the
other.
3. Diverging Rays are fuch, as being
produced both Ways, concur or meet in
one Point that Way, which is contrary
to their Motion. Thus Fig. 10, the Rays.
AD, AB, and AF, which are fuppos'd to
come from A, are diverging Rays.
4. Converging Rays are fuch, as being
produced both Ways, concur the fame
Way they move. Thus Fig. 11, the Rays
ED, OB, and HF, which are conceiv'd
to move toward a,, are converging Rays.
N. B. Parallelifm, Divergency, and
Convergency are to be underſtood of Rays.
of the fame Point..
*
5. The Focus is that Point, where the
Rays of the fame Point being produced
concur. Thus Fig. 10, A is the Focus of
the diverging Rays AD, AB, and AF
and a is the Focus of the converging
Rays, ED, OB. and HF. Whence it is
obvious, that the Focus of parallel Rays.
muſt be conceiv'd to be at an infinite Di-
ſtance, i. e. at ſo great a Diſtance, as that
the Rays of the fame Point, though they
are ſtrictly diverging or converging, yet
diverge or converge fo very little,
(at
leaft in respect of that Length and Porti-
(F 4)
on
{
88
The Young Gentleman's
1
>
on of them, wherein they are confider'd,)
as that they may be look'd upon as pa-
rallèl,
6. The Angle of Incidence is naturally
and ſo properly the Angle ABC (Fig. 8.)
comprehended between the incident Ray
AB and the Surface BC which it (incides
or) falls upon. But the Angle ABC be-
ing always the Complement of the An-
gle ABP to a Right; and becauſe what
is to be prov'd of the Angle ABC may
be primarily, and fo more cafily prov'd,
of the Angle ABP, hence for Conveniens
cy fake Writers of Opticks do ufually
take ABP for the Angle of Incidence,
and agreeably define it to be the Angle
comprehended under the incident Ray
AB, and a right Line PB perpendicular
to B the Point of Incidence. Agreeably
alfo hereto,
7. A Reflex Angle is that, which is
comprehended under the reflex Ray BE
and the forefaid Perpendicular PB, viz.
the Angle EBP Fig. 8: Though more
properly or naturally EBG be the Angle
of Reflexion.
8. A Refracted Angle is that, which is
comprehended under the refracted Ray
BG and the Perpendicular KF or BF,
(Fig.) viz, the Angle GBF.
9. By
Opticks.
89
9. By Inflection, as a general Word, is
denoted either Reflection or Refraction.
And confequently both the reflecting Sur-
face BC (Fig. 8) and the refracting Sur-
face BD (Fig.) are ftiled inflecting
Surfaces. And both a reflected Ray BE
(Fig. 8) and a refracted Ray BG (Fig. )
are comprehended under the Name of in-
flected Rays.
AXIOM S.
1, A Ray of Light falling perpendicu-
larly on an inflecting Surface, either goes
forward in a right Line, or is reflected in-
to its felf. Thus (Fig. ) the Ray AC
is to be conceiv'd as reflected by the re-
flecting Surface BD into it felf. And the
Ray AD (Fig. ) is to be conceiv'd to
pafs forward beyond the refringent Sur-
face BD in a right Line DE or DN.
2. The Inflection of any Ray is per-
formed in a Surface perpendicular to the
inflecting Surface. Thus (Fig. ) the Ray
AB is reflected into BE in the Surface
BCDE perpendicular to the reflecting Sur-
face BC. And likewife (Fig. ) the Ray
AB is refracted into BG in the Sur-
face BGND perpendicular to the re-
fringent Surface BD. And note, that
the faid Surface wherein any Ray is in-
flected,
90
The Young Gentleman's
Fig. 1.
flected, is thence ftil'd the Plane of In-
flection.
1
CHAP. II.
Of Opticks properly fo call'd, or
Direct Vision,
THEOREM I.
THE
fione
HYVIN A
HE Eye is always placed in the
Center of its own Sphere of Vi-
Demonftration. At what Diſtance fo-
ever; fuppofe Oa, the Eye O can fee one
Way, it is felf-evident, that the fame
Eye, if by no means hinder'd, can fee at
the fame Diſtance any other Way, viz.
towards bit can fee at the Distance Ob;
towards c at the Diſtance Oc, and fo
quite round. Wherefore the ſeveral Di-
ftances Oa, Ob, Oc, &c. being equal one
to the other, and all proceeding from the
fame Point O, it follows that they are
the Rays of a Circle, whofe Center is O,
and Circumference abcdefgh. Whence al-
fo it follows, that the faid right Lines Oa,
Ob, Oc, &c. are the Rays likewiſe of a
j
Sphere
Opticks
918
Sphere generated by the Turning of the
foremention'd Circle round about any
of its Diameters, fuppofe ae, as an Ax-
is; and confèquently that. O the Eye
will be the Center of the ſaid Sphere: Q-
E. D.
Corel, r. As it is evident from what has
been faid, that could the Eye be placed
where it might fee every Way, the Space,
which would then fall within its Sight,
would repreſent an entire Sphere; fo it
follows, that by the Interpofition of the
Earth, the Space or Compafs of actual
Vifion, even when the Eye has a full
View, is equal but to an Hemifphere.
Namely, fuppofing ae to reprefent the up-
per Surface of the Earth whereon the Eye
is placed, the Extent of actual Viſion will
be equal to the Hemifphere Qabcde, the
Line EE denoting the Horizon or the
Boundary of the Sight downwards.^.90
Corol 2. From this Theorem it follows
alſo, that whatever Portion of the Hea-
vens, (whether All abcde, or only Part:
of it, as bd, or bc, &c.) is feen by usit
appears as a Portion of the eoncave Sur-
face of a Sphere.
Corol. 3. Becauſe the Sun and Moon':
and Stars are beyond the Extent of di-
ftinct Vision, and confequently the Eye
can't difcern, how far one is more diſtant
from it than the other; hence it comes to
país,
92
The Young Gentleman's
paſs, that the Sun, Moon, and Stars,
do appear to us, as all fituated in the
Surface aceg of one and the fame Sphere
Oaceg, and fo equally diftant from us,
though they are in reality at ſeveral Di-
ftances from us.
Corol. 4. So much of the Heavens as is
feen by us, viz. abcde, appearing of a
ſpherical Figure only for the Reaſon
contain'd in this Theorem; it thence fol-
lows, that the Body of the Heavens
may be of a Quadrilateral Figure, as
ABCD, or of any befides a Spheri-
cal.
Scholium. This first Theorem with its
Corollaries is of good Uſe to prepare
young Students for Aftronomy. And I
have choſen to place this Theorem under
direct Viſion, partly becauſe it will ſo be
the more easily understood, and partly
becauſe, (though it be not one Homo-
geneous Matter contain'd in the Expanfe
or Space which we call Heaven, and con-
ſequently what we ſee therein at any
confiderable Distance, we fee not by di-
rect, but refracted Viſion, yet) it comes
to the fame in refpect as to the Cafe be-
fore us, whether Things in the Heavens.
are viſible to us by direct or refracted Vi-
fion.
THEO
Opticks.
93
THEOREM II.
The Eye can ſee at one View (i. e. at Fig. 2.
one Time and Place) no more Objects,
than whoſe Rays fall within the Com-
paſs of a Semicircle, to which the Eye,
or rather its Pupil, is the Center.
Dem. For the Eye ocaf can fee only
thoſe Objects, whofe Rays, paffing thro'
O its Pupil, reach to its Retina or Optick
Nerve placed at the Bottom of the Eye.
But now it is evident from the bare In-
ſpection of Fig. 2. that no Ray can en-
ter the Pupil O, but what falls within
the Semicircle DHE, whofe Center is Q.
For fuppofing a Ray to pafs along ED
the Diameter of the Semicircle, it will
paſs over the Pupil O, and not enter it.
And confequently all Rays falling with-
out the faid Semicircle, will much lefs
enter the Pupil. Q. E. D.
Corol. Rays of Light (as well as other
Things, whether natural or artificial) ha-
ving a greater or lefs Force, as they are
leſs or more oblique, it follows that ſuch
Rays as pafs through the very Middle of
the Semicircle DHE, are of moſt Efficacy
in caufing Viſion, as entring the Pupil of
the Eye perpendicularly, and confequent-
ly falling perpendicularly upon the Reti-
na at the Bottom of the Eye. Thus a
the
94
The Young Gentleman's
!
Fig. 2.
the Image of the Object A will be
painted moſt lively and diftinctly on
the Retina. The Image b and g of
the Objects B and G will be painted lefs
lively and diftin&t than the former Image
4, as being made by oblique Rays: but
alike lively and diſtinct in reſpect of one
another, as being made by the Rays Bb
and Gg equally oblique and more lively
and diſtinct than the Images c and fof
the Objects C and F, becauſe the Rays
Bb and Gg are leſs oblique than the Rays
Ce and Ff.
Schol. Agreeably hereto we find by
Experience, that we ſee moſt diſtinctly
fuch Objects, as (ceteris paribus) are ex-
actly before our Eyes. And upon this
Account it is, that, in all the feveral
Parts of Opticks largely fo called, prin-
cipal Regard is to be had to fuch Rays,
as enter the Eye perpendicularly.
THEOREM III
An Object appears greater or lefs, ac-
cording as it is feen under a greater or lefs
Angle.
Demonft. Let the Lines AB and CD (as
they are of equal Length, fo) repreſent
two Objects of equal Bignefs. It is evi-
dent that the Angle AOB, under which
is ſeen the Object AB, is only a Part of
the Angle COD, under which is feen the
Ob.
Opticks.
95
3
C
Object CD. It is alſo evident, that the
Rays AO and BO being extended be-
yond the Pupil of the Eye ODC
will fall respectively upon the Points
a and b of the Retina in the Bot-
tom of the Eye; and likewife the Rays
CO and DO being extended beyond O,
will fall upon the Points c and d of the
Retina. Whence it follows, that ab the
Image of the Object AB will be fo much
in proportion lefs than the Image cd of
the Object CD, that is, the Obje& AB
will appear to a Spectator fo much leſs
than the Object CD, (though they be
both of an equal Bigneſs in reality,) as the
Angle AOB is lefs than the Angle COD.
Q. E. D.:
Schol. Hence it appears, why the Dif- Fig. 4:
tance between the fame two Objects A
and B appears to be greater or leſs, ac-
cording as the Diſtance of the Spectator
from the faid Objects is greater or lefs;
namely, becauſe the Diſtance AB between
the ſaid two Objects is feen under an An-
gle proportionably greater or lefs, as the
Diſtance of the Eye from the faid Objects
is leffer or greater; that is, the Angle
AOB is proportionably greater than the
Angle AEB, as the Distance OD is lefs
than the Diſtance ED. And confequent-
ly (according to this Theorem III.) the
Diſtance AB will appear fo much in pro-
1
}
portion
96
The Young Gentleman's
Fig. 5.
portion greater to a Spectator at O, than
at E.
THEOREM IV.
The Rays of an Object being admitted
through a narrow Hole, do therein croſs
one the other, ſo as that the Image made
by the ſaid crofs Rays will be inverted in
reſpect of its Object.
Ray Bb, they muſt
being extended be-
And therefore the
3
Demonft. The Ray A proceeding from
the Point A of the Object or Arrow AB,
being (in dire& Vifion) a right Line,
and fo likewiſe the
croſs one the other,
yond the Hole O.
upper Point A of the Arrow AB will be-
come the lower Point a of the Image ab
and the lower Point B of the Arrow AB
will become the upper Point b of its Image
ab.And the like will hold as to all the inter-
mediate Points, (except the very middle
Point C, which being perpendicular or in
a right Line with O, its Image c will like-
wife be the middle Point of the Image
ab) and confequently the Image ba will
be inverted in refpect of its Obje& AB.
Q. E. D.
Schol. 1. Agreeable hereto are Expe-
ments of this Nature. Namely, a Room
being darken'd by putting together its
Window-fhuts, and a fmall Hole being
made with a common Gimblet or Nail-
paffer
Opticks
97
paffer in one of the Window-fhuts, and
the Glaſs of the Window being remo-
ved from before the Hole by open-
ing the Cafement or the like fuch
Objects as are without, and fo fituated
as that their Rays will pafs through
the Hole, will appear upon a Sheet of
white Paper, held on the Inſide overa-
gainſt the Hole, in an inverted Manner
according to this Theorem. And the
nearer the Paper is held to the Hole, fo
much leſs will the Image be; becauſe as
much leſs as the Rays Aa and Bb are ex-
tended beyond O the Hole or Point of
their Meeting, fo much lefs is the Dif
tance between them: Namely, the Dif
tance cd is less than the Diſtance ab, and
confequently the Image of the Object
(ſuppoſe the Arrow) AB will appear lefs
on the Paper held at the Diſtance cd from
the Hole, than if the Paper be held at the
Diſtance ab.
2. And the fame has been found, by
Experiments made for this Purpoſe, to
hold good in refpect of the Eye it felf,
this being as it were a ſmall dark Cham-
ber, into which are let pafs through the
Pupil the Rays of outward Objects,
which paint their Images on the Retina in
the Bottom of the Eye, agreeably to this
Theorem, that is, in an inverted Man-
ner with reſpect to its Object. As is il
(G)
luftrated
98
The Young Gentleman's
luftrated Fig. 6, which reprefents an Eye
having on its hinder Part a little Piece of
each of its Coats or Skins cut away, 'till
you come to the vitreous Humour. A
Candle being held before the Eye, and a
Piece of white Paper being apply'd to its
hinder Part prepared as aforemention'd,
the Image of the Candle has been found
to be made on the faid Paper in an in-
verted Manner; All other Rays of Light
being kept from falling on the Paper by
dark'ning the Room where the Experi-
ment was made. Concerning which more
ſhall be ſaid in Chapter IV, foraſmuch
as the Rays paffing through the Humours
of the Eye are refracted, and confequent-
ly it is requifite to underſtand the Laws
of Refraction, before we can underſtand
aright, how Viſion is caus'd in the Eye.
And for the fame Reafon I omit here
faying any thing of the Ufe of ſpherical
Glaffes in the Hole to a dark Cham-
ber, for rendring the Images more lively
and diſtinct.
3. I fhall add here (as being of great
Uſe for the clearer Underſtanding of what
has been or fhall be faid in this Treatife
of Opticks) a Defcription of the feveral
Parts of an Eye, reprefented Fig. 7,
where A denotes the Chryftalline Humour,
B the Vitreous, C the Aqueons: D the
Cornea Tunica; E the Retina
E the Retina; F the
Coat
Opticks.
99
Coat call'd Choroides; G the Coat call'd
Sclerotica, being a Part of the Dura Mater
continued about the Ball of the Eye on its
inner Part; H the Optick Nerve without
the Ball of the Eye; I the Coat or Skin
call'd the Uvea; K the Hole of the U
vea, call'd in one Word the Pupil.
It remains only to obſerve further, that
moſt, if not all the Particulars taken no-
tice of in this Chapter, belong not to di
rect Vifion alone, but are applicable alfo
to reflected and refracted Viſion ; which
we proceed to ſpeak of in their Órder.
СНАР.
III.
Of Catoptricks, or Reflex Viſion.
THEOREM I
Fa Ray AB of Light be reflected by a Fig. 81]
plain Surface GC, the Angle PBE of
Reflexion will be equal to the Angle PBA
of Incidence.
Demonft. Draw BP, and alfo AC, per-
pendicular to GC. The Medium GCde
being fuppos'd homogeneous to (i. e. of
the fame Nature with) the Medium
(G 2) EBCD,
J
The Young Gentleman's
100
EBCD, it follows that all the Rays com-
prehended between AB and AC, and
which take up or ſpread all over the right
Line BC, were they not hinder'd (by GC
the reflecting Plane) from paffing thro'
the Medium GCde, would therein diffufe
themſelves after the fame Manner, as in
the other Medium EBCD, that is, (the
right Line AB being continued to e, and
AC to d,) at the Diſtance Be they would
diffuſe or ſpread themſelves all over de.
And whereas the reflecting Plane BC does
neither increaſe nor diminiſh the ſpread-
ing of the Rays, but only turns them
back into the fame Medium they were in
afore; hence it is obvious, that after the
Reflection the Rays will fpread them-
felves in the fame Manner, asthey would
have ſpread, had they not fell upon the
Surface GC, but (that being remov❜d)
gone on in the fame homogeneous Medi-
um. But in that Cafe, at the Diſtance
Be, the Rays would (as is afore fhewn)
ſpread themſelves all over ed.
fore in the Diſtance BE Be, they will
fpread themſelves all over DE-De: and
confequently the Figure BCDE will be in
all Reſpects like and equal to the Figure
'BCde; and indeed is no other than the
Figure BCde reflected, or turn'd about (as
it were) half round upon BC. Where-
fore, if from the equal Angles EBC and
Where-
eBC
Opticks.
10.1
F
*BC there be taken away the right Angles
PBC and pBC, there will remain the An-
gles EBP and eBp equal one to the other.
But ABP eBp, therefore ABP=EBP.
Q. E. D.
Corol. 1. Hence the Angles EBG and
ABC (which, as has been Defin. 6 and 7
obferv'd, are more properly call'd the
Angles of Incidence and Reflection) are
equal one to the other. Namely, be.
cauſe PBG PBC, and PBE=PBA,
therefore (PBG-PBE=)EBG=(PBC
-PBA=)ABC.
Cor. 2. If the reflex Ray BE be taken
for the Incident, then the incident Ray
AB will be its Reflex.
Cor. 3. If the Angle PBE PBA, or
GBE CBA, then BE is the Reflex of
AB.
Schol. What has been prov'd as to a
Fig. 9.
reflecting Plane GC, the fame holds good
also in respect of any reflecting Curve Sur-
face PQ. For the Inclination or Directi-
on of the Curve, at B the Point of In-
cidence, is the fame with the Inclination
of the Plane DE that touches it there.
And therefore the Reflection following
always the Inclination, will be the fame,
whether it be fuppos'd to be made by B
as a Particle of the Curve Surface PBQ,
or of the Plain DBE. And the fame
holds good as to refracting Surfaces.
(G 3) THEO.
-
102
The Young Gentleman's
Fig, 10.
THEOREM II.
At what Diſtance CA an Object A is
placed before a reflecting Plane CF (whe-
ther of Metal or Glafs, &c.) at the
fame Diſtance Ca will a the Image of
the Object appear behind the ſaid reflec-
ting Plane, each Diſtance CA and Ca be-
ing meaſur'd upon one and the fame Per-
pendicular to the faid Plane, viz. upon
Aa.
Demonft. Let AD, AB, and AH be
Rays proceeding from the Object (or a-
ny Point of the Object) A; and let DE,
BO, and FH be their refpective reflex
Rays. It is evident, that theſe being ex-
tended beyond CF (i. e. behind the re-
flecting Plane) will all meet in a, which
therefore will be the Focus of the ſaid
Rays, and confequently the Place where
the Image of a will appear. But now the
Angle aDC EDF, and EDF ADC
by Theorem I. of this Chapter; where-
fore aDC=ADC. And becauſe in the
Triangles ACD and CD, the Angle
ADC=aDC, and alfo ACD=aCD, and
the Side CD is common to both Trian-
gles, therefore CA-Ca. And after the
very fame Manner may it be provid,
that in the other Triangles, ACB and
CB, as alfo ACF and CF, the Di-
ftance
Opticks.
103
ſtance CA of (any Point in the Object,
and (*) confequently of all) the Object
A is equal to the Diſtance Ca of the Image
a. Q. E. D.
Corol. 1. Hence it follows, that fuppo-
fing the reflex Ray OB to be that which
comes to the Eye plac'd at O, though all
the rest of the Looking-glaſs or other re-
flecting Plane, CF be cover'd (or taken
away) but the Part B, yet the Image a
will be ſeen by the Eye; whereas if the
Part B be hid, and all the reft uncove-
red, the Image can't be feen by the Eye
at O.
Cor. 2. After the fame Manner as Ob-
jects are inclin❜d towards the reflecting
Plane, will alſo their Images be inclin'd,
And confequently in fuch a Plane or
Looking-glafs plac'd horizontally, Objects
will appear inverted, or with their upper
Parts downward.
Cor. 3. Becaufe B AB, therefore
@O=AB+BO, that is, the Diſtance of
the Image from the Eye is equal to the
(*) Namely, as the vifible Object is conceiv'd to con
fift of feveral Radiant Points, fo the Image of the whole
Object is made up of all the Images of the feveral Radi-
ant Points, taken together, as is more fully illuftrated
by Fig. 11 and 12, of which fee more in the following
Scholium.
(G 4)
inci-
104 The Young Gentleman's
incident Ray and reflected Ray taken to
gether.
Cor. 4. Whatever has been faid of
the Image of any Object or prime Ra-
diant, the fame holds true alfo of the
Image of an Image, Whence arifes
the multiplying of Images by Means of
two or more reflecting Planes duly pla-
ced. Wherein this is moft remarkable,
the Diſtance of every Image is equal to
the Ray propagated from the prime Ra-
diant to the Eye, through all the inter-
mediate Reflexions.
{
Schol. It is obfervable, that by this
Theorem, the Focus A of Rays AD,
AB, and AF falling on the reflecting
Plane CF being given; thereby may be
found the Focus a after Reflection; name-
ly, by drawing AC perpendicular to CD,
and extending it to a fo, as that Ca may
be equal to CA. For then a will be the
Focus fought.
Schol. 2. For the Sake of fuch as may
not fo readily apprehend from Fig ro,'
how the Image of a whole Object is ſeen
by Reflection, I have therefore added Fig.
11 and 12. In each of which AB de-
notes a Crofs, and ab its Image ſeen by
the incident Rays reflected to the Eye at
E by the reflecting Plane CF. Namely, Fig.
fhews exprefsly, how the Rays AG
and
Opticks.
105
and BI that come from the two Extremi-
ties aB of the Object are reflected to the
Eye, and fo caufe the Image of A and B
to appear at a and b; and Fig. 12 fhews
how in like Manner the Images of the
Points D and H of the Object appear at
d and b; namely, in both Figures the I-
mages appear, where the reflex Rays EG
and EI being extended, meet with the
Perpendiculars Aa. Bb, Hb, and Dd; and
where Ca CA, Fb FB, Ch⇒CH, and
Fd=FD. And the fame may be ſhewn.
as to any other Points.
THEOREM III.
If a reflecting Glafs (or other Surface) Fig. 13.
BD be ſpherical, either concave or con-
vex, and the Object E be far distant
from the Glafs; its Image will, to the
Eye placed in the Axis AB of Radiation,
appear at C the Middle of the Semidia-
meter AB of the Glafs. But if the Ob- Fig. 14,
ject E be not far diftant, then its Image 15, 16.
will, to the Eye in the Axis, appear at
that Place C, where (BC: CA:: BE:
EA, that is, where) the Diſtance of the
Image from B the Vertex of the Glafs will
be to its Diſtance from A the Center of
the Glafs, as the Diſtance of the Object
from the faid Vertex is to its Distance
from the faid Center.
I
106
The Young Gentleman's
Fg. 13.
I have choſen to comprehend all that
relates to reflecting fpherical Glaffes, with
reſpect to the Eye placed in the Axis of
Radiation, under one Theorem, that
thereby may the more eafily be diſcern'd
the uniform Manner, whereby Viſion is
perform'd in fuch Cafes.
In reference to which it is to be ob-
fery'd, that (although in the Figures be-
longing to this Theorem, the Point D be
placed at fome confiderable Diſtance from
B, for the more diftinctly perceiving the
Demonſtrations, yet) D is to be fuppos'd
very near to B, becauſe otherwife the Re-
flex CD of the Incident Ray ED will not
affect the Eye placed in AB. For the re-
flex Rays of all fuch Rays as fall more
remote from B, will pafs by on one Side
of the Pupil of the Eye, and fo contri-
bute nothing to the Sight of the Image.
Add hereto, what has been obſerv'd in
the Corollary of Theorem 2, Chap. 2,
Hence Notice is here taken only of fuch
Rays as fall upon the Glafs very near to
its Vertex B. Theſe Particulars being
premis'd, I come now to the Demonftra-
tion of this Theorem, in refpect of its
four principal Cafes following.
Cafe 1. When the Object is far diftant
from, the Glaſs, ſo that the Rays coming
from any one Point thereof may be con-
ceiv'd parallel one to the other, as well
as
Opticks.
107
In
as to the Axis AB, as the Ray ED.
fuch a Cafe the Image will appear, to the
Eye placed fomewhere in the Axis, at C
the Middle of the Semidiameter of the
Glaſs, For bifect AB in C, and draw
AD and CD, and produce them on the
other Side of BD the Glafs. Becauſe
the Point D is look'd to be ſo very
near to B, as in a Manner to coin-
cide with it, therefore we may alſo
look upon CD=CB; and by Conftruc-
tion CA CB, therefore CA-CD, and
confequently the Angle CAD CDA. But
the Angle CAD-EDA the Angle of Inci-
dence of the Ray ED, (for AD is perpen-
dicular to the Surface of the ſpherical
Glaſs,) wherefore the Angle CDA is e-
qual to the Angle of Incidence of ED,
and therefore (by Cor. 2, Theorem I. Chap.
3.) DC is the Reflex of the Incident ED
upon the Concave Surface of the Glafs
BD. And the like may be demonſtrated
as to the Convex Surface of the Glafs
BD. For, becauſe the Angle EDA=eDO,
and CDA NDO, therefore the Line
DN will be the Reflex of the Ray eD,
and being produced backwards (or on the
Concave Side of the Glafs) will touch
AB alfo at C. And what has been de-
monftrated of any Ray ED taken at plea-
fure, the fame holds true of all Rays in
the fame Circumſtances. Wherefore Rays
parallel
J
108
The Young Gentleman's
Fig. 14.
parallel to AB, and whoſe reflex Rays
contribute to the Sight of the Image, if
they fall upon a Concave Glaſs, after
their Reflexion will concur at C; and
from thence diverging again, will render
the Image viſible at C, to the Eye placed
fomewhere in AB. And the reflex Rays
of fuch parallel Rays, as fall upon a
Convex Glaſs, do diverge from C, and
confequently render the Image vifible at
C, to the Eye plac'd in AB produc'd on
the convex Side of the Glaſs, that is, if
D be the incident parallel Ray, ND will
be the Convex, which will form the I-
mage at C. Q. E. D.
Corol. Hence, and from Corol. 2. Theo-
rem I. of this Chapter, it follows, that
Rays diverging from C and reflected by
a Concave Surface, or converging at C
and reflected by a Convex Surface, are
parallel to AB; that is, if CD be the In-
cident, DE is the Reflex; or if ND be
the Incident, De is the Reflex.
Cafe 2. If the Object E be not far dif
tant from BD the Glaſs, and fo the Rays
proceeding from any fingle Point thereof
are (not parallel, but) diverging, then
the Image will, to the Eye in the Axis,
appear at that Place C, where BC: CA ::
BE: EA. And this ſhall be demonſtra-
ted first in reſpect of Concave Glaffes,
and when the Diſtance of the Object is
greater
Opticks.
109
greater than half the semidiameter of the
Glaſs. For let the Point C be ſo taken in
AB, as is requir'd; and draw AD and
CD. Becauſe (for the Reaſon above-
mention'd) ED EB, and CD=CB,
therefore by Conſtruction CD (=CB)
: CA :: DE(=BE): EA. Wherefore al-
ternately CD: DE:: CA: EA. And fo
by (Prop. 3. Euclid. 6.) the Angle ADE
ADC. But ADE is the Angle of In-
cidence of the Ray ED therefore (by
Corol. 2. Theorem I. of this Chapter) DC
is the Reflex of the Ray ED. And ED
being taken at Pleaſure, it follows that
the Focus of all other Rays diverging
from E will, after their Reflection by a
concave Glaſs, be at C, and confequent-
ly the Image of the Object will appear
at C, to the Eye placed in the Axis
(AB produced, or) EB. Q. E. D.
;
Cor. 1. Hence C will be the Focus of
Rays eD converging to E in the like Cir-
cumſtances, and reflected by a ſpherical
convex Reflecter.
Cor. 2. Hence and from Cor. 2. Theo-
rem I. of this Chapter, if CD be the Inci-
dent, DE will be the Reflex; and if ND
be the Incident, De will be the Re-
flex.
Cafe 3. And the fame holds true, when Fig. 15.
the Diſtance of the Object is lefs than
half the Semidiameter of the Glaſs. For
"
in
110
The Young Gentleman's
Fig. 16.
S
in AB extended, let C be taken accord-
ing to the Prop. be requir'd; and let
AD and CD be drawn and produced
as alſo let ER be drawn parallel to CD.
The Arch BD being as Nothing, there-
fore ED=EB, and CD=CB. Whence
ED: EA :: CD: CA; and fo (by Rea-
fon of DC being parallel to ER) ED:
EA:: ER: EA. Wherefore ER=ED,
and the Angle ERD=EDR. But EDR
is the Angle of Incidence of ED, and
ERD=NDA its Altern, therefore (by
Cor. 2. Theorem I. of this Chapter,) DN
is the Reflex of the Incident ED. And
ED being taken at pleaſure, it is manifeſt,
that all Rays proceeding from E, and re-
flected by a concave Glafs, being extend-
ed backwards, will concur in C; and
confequently the Image of the Object will
appear at C, to the Eye placed in the Ax-
is. Q. E. D.
Corol. 1. Hence if CD be the Incident
on a ſpherical Convex Reflecter, De will
be the Reflex, and fo the Image will ap-
pear at E.
Cor. 2. Hence and from Cor. 2. Theorem
I. of this Chapter, if ND be the Incident,
DE will be the Reflex; and if eD be the
Incident, DC will be the Reflex.
Cafe 4. The like holds good in reſpect
of convex Glaſſes. For in AE let C be
taken according to the Proportion re-
quir'd
Opticks.
ΙΙΙ
quir'd in this Theorem, or (which comes
to the fame) fo as that AC: CB :: AE :
EB, and let CD be drawn and extended;
and parallel thereto let ER be drawn,
'till it meets in R with AD produced. The
Arch BD being as Nothing, therefore
CD=CB, and ED=EB. And therefore
AC: CD::AE: ED.
AE: ED. But becauſe the
Triangles ACD and AER are equiangu-
lar, therefore AC: CD :: AE: ER; and
fo AE: ED:: AE: ER; and therefore
ER ED. And confequently the Angle
(ERD, or its Altern, and fo Equal)
NDR EDR the Angle of Incidence of
ED. Wherefore DN is the Reflex of the
Incident ED. But ED being taken at
pleaſure, it is evident, that all the Rays
diverging from E, and entring the Eye
placed in the Axis, after they are reflec-
ted by the convex Surface BD, will di-
verge from the Focus C; and confe-
quently the Image of the Object E will
appear at C to the Eye fo placed. Q
E. D.
Coroll. Hence if ND be the Incident,
DE will be the Reflex; and if in a con-
cave Reflecter eD be the Incident, DC will
be the Reflex; or if CD be the Incident,
De will be the Reflex.
Schol. From this Theorem is deduced
the Way of finding the Focus of paral-
lef
112
The Young Gentleman's
•
lel or other Rays, falling upon fpheri-
cal Surfaces, after their Reflection, viz.
The Focus C of fuch parallel Rays
ED after Reflection is found by bifect-
ing the Semidiameter AB of the Sphere
in C.
And the Focus C of other Rays falling
upon a ſpherical Surface, is found after
Reflexion, by taking C fo, as that BC:
AC::BE:AE.
And thus it has been fhewn, both by
Way of Theorem and Problem, where
the Image of an Object will appear in a-
ny ſpherical Glafs, (or other like reflect-
ing Plane) to the Eye placed in the Axis
of Radiation. And this is fufficient to
our preſent Purpoſe, not only becauſe
moſt optical Inftruments are made after
this manner, but alſo becauſe the Image,
only in this Situation of the Eye, appears
fo lively and diftinct, as to deferve the
Name of an Image. Hitherto both Object
and Image have been denoted only by fin-
gle Points, as being fufficient to illuſtrate
what has been afore deliver'd. I now
proceed to ſpeak of Object and Image
sonfider'd as Lines or Surfaces.
THEO
Opticks.
113
THEOREM ÍÍ.
The Image CT of a radiant plain Sur-
face FE, made by a ſpherical Glaſs BD, Fig. 17
is alſo a plain Surface.
From A the Center of the Glafs, draw
AE perpendicular to FE, and meeting the
Glafs in its Vertex B. And (according
to Theorem III, of this Chapter) let C be
the Image of the Point E, through which
draw parallel to FE the Plane CT. In
the Plane CT will appear the Image of
the Plane FE made by the Glaſs.
Thus let AF be the Ray, drawn from
any other Point F of the radiant Plane
FE to the Center of the Glafs, and meet-
ing with the Glaſs in D, and the Plane
CT in T. And by Conſtruction let CD
be the Reflex of the Incident ED and
;
let the Reflex of the Incident FB be BH,
meeting the Ray FD in H. Now, be-
cauſe we fuppofe in the prefent Cafe the
Angle FAE to be fmall, therefore the
Arch BD will be alſo ſmall, and in effect
no other than a little or fhort right Line.
And the Circumference of the Circle de-
fcrib'd with the Diameter BF will pafs
through the Points D and E, becauſe the
Angles FEB and FDB are right. Whence
the Angles BFD and BED being in the
fame Segment, are equal. And when
the Angles at Aare Verticals or the fame,
(H)
it
& 18.
114
The Young Gentleman s
it will follow that FBA=EDA. And
(becauſe by Theorem I, of this Chapter)
ABH-FBA, and ADC=EDA, there-
fore ABH EDC. Whence the Trian-
gles ABH and ADC are equiangular,
and fo fimilar. Wherefore AB: AH::
AD: AC; and whereas ABAD, there-
fore AH-AC. But by reaſon of the
Smalneſs of the Angle CAT, we may
look upon AT-AC. and therefore we
may alſo look on AHAT. But H is the
Image of the Point F in the radiant Plane,
whence the faid Image is plac'd in the
Plane CT. And the fame may be fhewn
the fame Way as to the Image of any o-
ther Point in the Plane FE. Wherefore
the Image of the Plane FE is placed in
the Plane CT. Q. E. D.
་
If the radiant Plane be far diftant, then
the Images of feveral radiant Points
will be (by Theorem III, of this Chapter)
in the Mid-points of the ſeveral Semidi-
ameters of the Glafs tending to the faid
radiant Points, that is, the Image of a radi-
ant Plane far diftant will have the Figure
of a ſpherical Surface concentrical to the
Glafs. But becauſe the faid Radiant is (ac-
cording to our Suppoſition) feen under a
fmall Angle, that fmall Portion of a ſphe-
rical Surface, which is taken up by the faid
Image, will ſcarcely differ from a plain
Surface, to which AB is perpendicular.
Whence
Opticks.
115
Whence this Theorem holds good in all
Cafes for the Demonſtration will ferve
for any other Cafe, as well as thoſe two
reprefented in Fig. 17 and 18.
Cor. If the Angle EAF be too great,
the right Line AH will be fenfibly leſs
than AT. Whence an Obje& being ſeen
under fuch an Angle, its Image, if made
by a concave Glafs, will appear concave;
if made by a convex Glafs, will appear
convex..
THEOREM V.
The Object HF, and its Image bf, are Fig. 19
ſeen under equal Angles HBF and hBf x 29.
from the Vertex B of the reflecting Curve
BD, whether convex (as Fig. 19) or con-
cave (as Fig. 20)
Demonft. Draw AE perpendicular to
HF, (which by Theorem IV, of this Chap-
ter, will confequently be perpendicular to
hf.) and meeting the reflecting Curve BD
at B. Draw alfo BF, BH, Bƒ, Bh, and
Ff, which laft will pafs through the Cen-
ter A; forafmuch as according to our
Suppofition the Image of every Point is
placed in the Axis of Radiation. In like
Manner the Points H, A, and b will be
in the fame Axis. Now by Theorem III,
of this Chapter, BE: EA:: BC: CA.
Wherefore BE: BC:: EA: CA. But (by
BC::EA:
reafon of the equiangular Triangles AEF
and ACf) EA: "CA :: EF: Cf. Where-
(H 2)
fore
116
The Young Gentleman's
fore BE: EF :: BC: Cf
And whereas
the Angles BEF and BCƒ are right, there'
fore the Angle EBF-CBf. And in like
Manner the Angle EBH-CBb. Wherefore
HBF=bBF. Q. E D
Corol. The Obj‹ & FH (if it be a Line)
and its Image fb are one to the other, as
their Diſtances from the Vertex, BE and
BC. But if the Oje& be a Surface, it and
its Image will be one to the other, as the
Squares of their Distances aforefaid.
Whence, the Distance of the Object and
the Image from the Reflecter being given,
thereby is given alſo the Ratio or Proporti-
on of the Image to the Obje&. And in like
Manner may be found the laſt Image of
an Obje&, when it is propagated by ma-
ny Refle&ters. Laftly, it is evident,
that the fame will hold true in all re-
fpecs, if fb be the Object, and HF the
Image.
Schol. By what has been faid, it is ea-
fie to account for and illuſtrate the Parti-
culars relating to Images made by fphe-
rical Reflecters. Of which the more re-
markable are thefe that follow, viz.
1. If the Refle&er be concave, and
the Diſtance BC of the Object hf from B
the Vertex of the Reflecter, be lefs than
half the emidiameter, then Its Image HF
will appear beyond or behind the Reflec-
ter, and in a right Pofition, (as in a plain
Re-
Opticks.
117
Reflecter, namely the upper Part of the
Object will appear the upper Part of the
Image; the lower Part of that, the low-
er Part of this. c) and as the Obje&t is
placed nearer or farther from the Reflec-
ter, within the Diſtance of the Semidia-
meter, fo will the Image be nearer or far-
ther from the fame, and appear lefs or
more magnify'd. All which Particulars
will be illuftrated by comparing together
Fig. 19 and 21 and 22. Namely in Fig.
21, the Obj-& bf being placed a fourth
Part BC of the Semidiameter BA from
the reflecting Concave BD and in Fig.
19, the Object hf being placed a third
Part BC of the Semidiameter BA from the
fame concave Reflecter, it is evident that
in the former the Image HF (as well as
the Object hf) is nearer to the Reflecter,
and lefs proportionably, than in the lat-
ter. And in both Figures the Image ap-
pears in a due Pofition, viz. F is on the
fame Side of the Axis AB as f, and Has
h. And therefore if f be the upper and
h the lower Part of the Obje&, F will
be likewiſe the upper, and H the lower
Part of the Image. Laftly, it is evident,
that in both Figures bf and HF, i. e. the
Object and its Image, are one to the other,
as their Diſtances from the Vertex BC and
BE; viz. in Fig. 19, BC: BE::1:3::
bf: HF. And in Fig. 21. BC: BE
2:: bj: HF. (H 3)
::1:
3. It
312
The Young Gentleman's
2. It is evident from comparing toge-
ther Fig. 19 and 22, that if the Reflec-
ter be convex, and fo HF the Object,
and hf its Image, then likewife the Image
will appear in its due Pofition; and alfo
as the Obje& is brought nearer to the
Refle&er, fo will the Image come nearer.
But in this a convex Reflecter differs from
a concave, viz. that (whereas in a con-
cave, the nearer the Object hf is to the
Vertex B, the lefs will the Image HF ap-
pear, as is evident from comparing, as
has been faid, Fig. 19 and 21, on the
contrary) in a convex, the nearer the Ob-
ject HF is to B the Vertex, the greater
will the Image hf appear, as is evident
from comparing Fig. 19 and 22. It is al-
fo remarkable, that be the Obje& HF re-
moved never fo far diftant from the re-
flecting Convex beyond the half Semidia-
meter, yet its Image will fix it felf at the
half Semidiameter.
3
If the Refle&er be a Concave, and
the O ject HF be more diftant from it
than the length of the Semidiameter AB,
(as Fig 20 and 23) then the Image hf will
be more diftant from it than the half Semi-
diameter,and yet lefs than the whole Semi-
diameter, that is, it will appear in the Air
between the Object and the Reflecter; and
alſo it will appear in an inverted Poſition,
that is, the Point F and its Image f will ap
pear
Opticks.
119
Pear on contrary Sides of AB the Axis, as
will the Point H and its Image h; (and
fo of any other Point of the Object, and
its Image,) whence it F be the upper or
right Hand Part of the Objc&, f will be
the lower or left hand Part of the Image,
&c. Likewiſe it is evident from compa-
ring Fig. 20 and 23, that in this Cafe, as
the Obj.& HF is remov'd farther from
the Reflecter, fo its Image bf will come
nearer and appear lefs; and on the con-
trary, as the Object comes nearer, the
Image will go off farther, (and appear
bigger) 'till they meet one the other at
the Center A. In like Manner if fb be
the Obj&, then HF will be the Image;
the Affctions whereof are eafie to be ac-
counted for from what has been faid.
It is alſo obfervable that if the Object HF
be remov'd fo far from the Reflecter, as
that the Rays of it falling near the Vertex
B of the Reflecter, may be efteen,'d pa-
rallel to AB, then (according to the for-
mer Part of Theorem II, of this Chapter)
the Image bf will appear at the Mid point
of the Semidiameter of the Reflecter.
And if the Sun be the faid far diſtant
Object, then the Place of the Image (or
Mid-point of the Semidiameter) will be
the burning-point of the Reflecter, if it be
confiderably larger than the Image of the
Sun. But if a lucid Body, as a Candle,
(H 4)
be
126
The Young Gentleman's
be plac'd in the faid Mid-point of the
Semidiameter, it will caft a Light to a
great Diſtance.
Diſtance. Add as in this Cafe, no-
thing but Light will appear on the Sur-
face of the Reflecter; fo when other Ob-
jects are placed at the half Semidiameter,
nothing but their Colour will appear on
the Surface of the concave Reflecter.
4. It is obfervable, that fuppofing the
Object to ſtand ſtill, and the Eye to
change its Place, the fame Alterations will
be produc'd in refpect to the Image, as if
the Object had chang'd its Place in a cor-
reſpondent Manner.
5. It is evident from Fig. 19-23,
that a reflecting Concave magnifies, that
a reflecting Convex leffens an Object.
Namely, if in the faid Figures BDG be
confider'd as a Concave, then hf will be
the Object and HF the Image, which is
in all the faid Figures much larger than
hf the Object. But if DBG be confider'd
as a Convex, then HF will be the Ob
ject, and hf the Image, which is in all
the ſaid Figures much leſs than HF.
6 It is obfervable, that cylindrical Re-
flecters, whether concave or convex, muſt
render the Image deform'd.
For a Cy-
Jinder being generated by the Motion of
a right Line round two parallel Circles,
hence fuch a Reflecter muſt partake, part-
of the Properties of a plain Reflecter,
partly
Opticks.
partly of a ſpherical. Of that, in re-
fpect of its right Lines, in Longitude; of
this, in reſpect of its circular Lines,
Round-ways. Hence the Longitude or
Height of an Object being repreſented
justly, as by a reflecting plain Surface; but
its Latitude magnify'd or diminiſhed, ac-
cording as the Reflecter is a Concave, or
Convex; the Image appears difproporti-
onable. But on the fame Account, if a
Picture be drawn proportionably bigger
in breadth than in Length, its Image will
be reprefented by a reflecting Cylinder in
its jult Dimenſions. Which is illuſtrated
Fig 24, 25, 26. Namely, in Fig. 24, is
drawn the Picture in its juft Dimenfi-
ons, and divided into Parallelograms.
In Fig. 25 are drawn as many circular
Lines, as there are (in Fig. 24) horizon-
tal Lines parallel one to the other, viz,
7; all the faid circular Lines being con-
centrical to the circular Bafis of the Cy-
linder C; and diftinguifh'd proportiona-
bly by right Lines drawn a-crofs them in-
to as many Divifions, as the Ficture (Fig.
24) is divided into Parallelograms: So
much of the Picture, as is contain❜d in each
Parallelogram being to be drawn in each
of the foremention'd Divifions anfwering
thereunto. Which being done, and the
reflecting Cylinder being plac'd on C, the
Image of the Picture will appear in the
faid
$22
The Young Gentleman's
faid Cylinder according to its juft Dimen-
fions, but less than the Picture, if the re-
flecting Cylinder be convex ; which is ge-
nerally us'd in this Cafe, and is reprefent-
ed Fig. 26.
ter.
Laftly, It remains only to obferve, that
by the foregoing Theorems may be found
fuch a Pofition of the Radiant, as that
the faid Radiant may have a given Pro-
portion to its Image made by the Reflec
For Inftance, let BD (Fig. 27) be
the Refle&er, and the given Proportion
be that of BA to AM. If the Obje& be
a Line, bifect BM in E, and by Theorem
III, of this Chapter find the Focus Can-
fwering to the Focus E, which C will be
the Place of the Radiant. For becauſe
BC: CA :: BE: EA; therefore BC:
BC+CA :: BE: BE+EA; that is (be.
caufe BE=EM) BC: BA:: BE : AM
and therefore BC: BE:: BA: AM. But
by the Corollary of this Theorem V, BC is
to BE, as the radiant Line in C is to its
Image in E
therefore the Radiant Line
in C, is to its Image made by BD in E,
as BA to AM, i. e. in the Proportion gi-
ven. If the Radiant be a Surface, then
inſtead of AM, a right Line muſt be taken,
to which BA has a fubdulicate Ratio of
that which it has to AM. And this is
fufficient to our Defign, concerning Ca-
toptricks or reflex Vifion.
;
;
CHAP.
Opticks.
123T
>
1
CHA P. IV.
of Dioptricks, or Refracted
Viſion.
THEOREM I.
Fa Ray AB of Light be refracted by Fig. 28,
the plain Surface BD of a Medium of 29.
different Thickneſs, the right Sine of ABK
the Angle of Incidence will, in reſpect of
the fame different Mediums, always have
the fame (Ratio or) Proportion to the
right Sine of GBF the Angle of Refrac-
tion, however the Ray AB be inclin❜d to
the Surface BD.
Demonft. Let BG be the Refract of the
Ray AB, whether AB paffes out of a
thinner Medium into a thicker, (as Fig.
28) or out of a thicker into a thinner,
(as Fig. 29.) Now as the Light proceed-
ing from A, and contain'd between AB
and AD, will be diffus'd within the Me-
dium DK all over BD; fo after it has en-
ter'd the new Medium DG, it will at the
Length of BG diffuſe it ſelf all over GN:
Whereas had the fame Medium DK been
con-
72 4
The Young Gentleman's
continued on each Side of BD, the Light
at the Length of BC (=BG) would have
been diffus'd all over CE. But in both Ca-
fes (namely, whether the Medium DG be
different from or the fame with the Me-
dium DK) FN=ME, and conſequently
both the faid Lines may be look'd upon,
as not at all depending on the (*) Facili-
ties of the Medium's, and fo of no Confi-
deration in the prefent Cafe. Wherefore
GF (i. e. GN-FN) and CM (i. e. CE
-ME) are to be look'd on, as the genu-
ine Effects of the Facilities of the two Me
diums DG and (DC, or which is the
fame) DK; that is, GF is to CM, as the
Facility of the Medium DG is to the Fa-
cility of the Medium DC or DK. But
CM is the right Sine of the Angle
(CBM) ABK, which is the Angle of
Incidence of the Ray AB; and according
to the fame Semidiameter, GF is the
right Sine of GBF the Angle of Refracti-
on of the fame Ray AB. Wherefore
fince the Angle DAB (=ABK) is taken
at pleaſure, the like will hold good of a-
ny other Angle ; that is, in the Refrac-
tion of any Ray AB, let it fall how it
will on the Plain whereby it is refracted,
the Sine of the Angle of Incidence is to
(*) This Word is us'd to denote the feveral Virtues,
which different Mediums have to diffuſe the Light.
the
Opticks.
125
the fine of the Angle of Refraction, as
the Facility of the Medium DK is to
the Facility of the Medium DG, whereby
the Ray AB is refracted into BG. But in
respect of the fame Mediums. the Thick-
neffes of the Mediums, and the Facilities
thence arifing, and their Ratio will abide
the fame. Wherefore in the Refraction
of any Ray, the Sine of the Angle of In-
cidence always keeps, with refpect to the
fame Mediums, the fame Ratio or Pro-
portion to the Sine of the Angle of Re-
fraction. Q. E. D.
Corol. 1. If BG be the refracted Ray of
the Incident AB, the reft remaining the
fame, BA will be the refracted Ray of the
Incident GB.
Cor. 2. The refracted Ray BG of any
Incident AB is rightly affign'd, when the
Sine CM (of the Angle CBM) ABK the
Angle of Incidence, is to the Sine GF of
the Angle GBF contain'd under the af-
fign'd Ray BG, and the right Line BM
perpendicular to the refracting Surface
BD, as the Facilities of the given Medi-
ums are one to the other. Namely, this
Corollary is no other than the Converſe
of this first Theorem.
Schol. 1. In a Ray's paffing out of a
thicker Medium into a thinner (as Fig.
if it be fo inclin'd to the refracting Sur-
face, that GF (which is to CM, as the
Faci-
126
The Young Gentleman's
Facilities of the given Mediums) does ex
ceed the Semidiameter BC; then the in-
cident Ray AB will have no refracted
Ray, (after the Manner of an impoffible
Cafe in a geometrical Problem ;) that
which ought to be the refracted Ray not
entring the thinner Medium DG, but be-
ing reflected by its Surface, according to
Theorem I. of Catoptricks. And this Caſe
is illuftrated Fig. 30.
Schol. 2. When a Ray of Light paffes
out of the Air into Glaſs, it is obferv'd
that the Sine of the Angle of Incidence is
to the Sine of the Angle of Refraction,
as 3 to 2 in whole Numbers; and in paf-
fing from the Air into Water, I is to R,
(by which Symbols henceforward is de-
noted the Ratio of the Sine of the Angle of
Incidence to the Sine of the refracted An-
gle, which meaſures the Refraction) as 4 to
3. And on the other hand, in the paffing of
the Ray out of Glafs into Air, I will be
to R as 2 to 3 ; and in its paffing out of
Water into the Air, I will be to R as 3 to
4. Whence the Reason is manifeft, why
a Ray paffing out of Glafs, and falling on
the Surface of the Air fo, as to make the
Angle of Incidence more than about 42
Degrees, will not enter into the Air, but
is reflected by its Surface. Namely, be-
cauſe 2 : 3 :: 42: 63; that is, the Ra-
tio fefquialtera (as it is called) of 42 is
633
Opticks.
127
63; and confequently the right Line GF,
which is to denote the faid Ratio of the
Sine of 42 Degrees muft exceed the Ray,
this being the Sine only of 60 Degrees.
In which Cafe the Ray is reflected accord-
ing to Schol. I.
Schol. 3. It is demonftrable that if
Rays diverging (Fig. 30 and 31) from
E, and refracted by the Surface BD, do
diverge from C, the Mediums being tranf-
pos'd, but the fame intermediate Surface
BD remaining, the Rays converging to
E, will after Refraction converge to C.
Wherefore what fhall be demonftrated
in the following Theorems concerning di-
verging Rays, may alfo be apply'd to
converging Rays.
THEOREM II
If an Object E be feen through diffe- Fig. 30,
rent Mediums BDA and BDO, the Image & 31.
of the faid Object will appear, at that
Place C in the right Line EB (perpendi
cular to the plain Surface BD of the re-
fracting Medium, whether thicker or
thinner) where CB : EB :: I: R, that is,
where the Diſtance of the Image from the
plain Surface of the refracting Medium
is to the Diſtance of the Obje& from the
faid Surface, as the Sine of the Angle of
Inci-
128
The Young Gentleman's
M
Incidence is to the Sine of the Angle of
Refraction.
Demonft. Let ED be any Ray of the
Object E falling upon the Surface BD.
Thro' D draw AO parallel to EB. Join
C and D, and produce CD to F. Thoſe
Rays only being here (as well as in the
foregoing Chapter) confider'd, that fall
near EB the Axis of Radiation, and
wherein the Eye is fuppos'd to be plac'd,
it follows (for the Reafons mention'd in
the foregoing Chapter) that we may
look on ED=EB, and CD=CB. Where-
fore CD: ED::1: R.
I: But CD is to
ED, as the Sine of the Angle BED or its
Equal EDA, to the Sine of the Angle
BCD or its Equal ODF. Therefore the
Sine of the Angle EDA is to the Sine of
the Angle ODF, as I to R. But EDA is
Í
the Angle of Incidence of the Ray ED,
therefore ODF is the refracted Angle an-
fwering thereto; that is, DF is the re-
fracted Ray of the incident Rav ED. And
the ſame may be fhewn of any other
Ray diverging from E. Wherefore C
will be the Focus of all Rays diverging
from E and refracted by ED. that is, the
Image of the Object E will appear after
Refraction at C. Q. E. D.
Cerol.
Opticks.
129
A
S
Theo
Corol. 1. Hence and from Cor. I,
rem I, of this Chapter it follows that
(cæteris paribus) Rays in the Medium
BDO converging to C, will after Refrac-
tion' converge to E. And if the Medi-
ums be tranfpos'd, then from hence and
Schol. 3, Theorem I, of this Chapter it fol-
lows, that Rays in the Medium BDO
converging to E, will, after Refraction,
converge to C; and that parallel Rays
will, after Refraction at the plain Sur-
face of any Medium, ftill continue
rallel.
pa-
Corol. 2. If BDA be Water, and BDO
Air, then CB : EB :: 3 : 4.
Whence it
comes to pafs, that Water appears not
to be fo deep, as it really is, by a fourth
Part: But if BDA be Glafs, then CB :
EB.:: 2 : 3.
Schol. From this Theorem is deduced
the Method of finding the Focus C of
Rays refracted by a plain Surface, viz.
by drawing EB perpendicular to the re-
fracting Surface DD, and therein taking
C fo, as that CB :: EB :: I: R.
THEOREM III.
If the Surface BD of the Medium of Fig. 32,
different Thickness be fpherical, and the 33.
Object E be very far diftant, then its I-
mage will, to the Eye plac'd in AB the
(D)
Axis
130
The Young Gentleman's
$1
Fig. 34,
35, 36,
37.
Axis of Radiation, and parallel to the
incident Rays, (as ED,) appear in that
Place C of the Axis AB produced as
there is Occafion, where BC: CA::I: R.
But if the Object E be not very far di-
ftant, then its Image will, to the Eye
placed in AB produced if there be Occafi
on, appear in that Place C of AB, where
the Ratio compounded of the Ratio's of
EA to AC and of CB to BE is equal to
the Ratio of I to R. (*)
Demonft. The first Part of this Theo-
rem (as it relates to Objects at ſo great
diftance from the refracting Surface BD, as
that the Rays proceeding from each fingle
Point therein, may, confider'd by them-
felves, be eſteem'd parallel one to the o-
ther, fo it) is thus demonftrated. Let ED
denote any Ray falling on BD, and
draw AD and CD, and produce them.
The Sine of the Angle BAD or its Equal
EDO is to the Sine of the Angle CDA
(or its Complement to two Right) as CD
to AC, or (by reafon of the Smalnefs of
the Arch BD) as CB to AC, that is, by
(*) I have comprehended all here relating to refracting
fpherical Surtaces under this one Theoreni for the like
Reafon, as I comprehended all relating to reflecting fphe-
rical Surfaces under one Theorem; and alſo that the Har-
mony between reflected and refracted Viſion may the bet
ter appear.
Con-
Opticks.
131
Conftruction, as I to R. But EDO is the
Angle of Incidence of the Ray ED, there-
fore DC is its refracted Ray. In like
manner may it be fhewn, that any o-
ther of the parallel Rays proceeding
from (any Point of) the Object E after
Refraction will pass through C. There-
fore C will be the Focus of all fuch Rays,
or which is the fame, at C will appear
in this Cafe the Image of an Object very
far diftant. Q. E. D.
The latter Part of this Theorem is thus
demonſtrated. Let ED be the incident
Ray; draw AD and DC, and through
C draw CH parallel to AD, and meeting
ED in H. By reafon of the ſuppos'd
Smalness of the Arch BD, we may look
on CD CB, and ED EB. Wherefore
CD=CB, ED—EB.
the Ratio of I to R is equal to the (Ratio
of EA to AC and the Ratio of CD and
ED together; that is, by reafon of AD
being parallel to CH, is equal to the Ra-
tio of CD to ED and the Ratio of ED
to DH together, or which is the fame, to
the) Ratio of CD to DH. But CD is to
DH, as the Sine of the Angle (DHC, or
its Complement to two right,viz.ADH=)
EDO is to the Sine of the Angle (DCH
=) ADC, or in fome Cafes to the Sine
of the Complement of ADC to two Right.
Wherefore EDO being the Angle of In-
cidence of the Ray ED, it follows that I
(1 2)
is
132
The Young Gentleman's
Fig. 32.
is to R, as the Sine of the Angle of In-
cidence of ED to the Sine of the Angle
ADC. Wherefore (by Theorem I, of this
Chapter) DC is the refracted Ray of the
incident Ray ED. And ED being taken
at pleaſure, it is evident that C will be
the Focus of all Rays proceeding from E
after their aforefaid Refraction
; and
'confequently at C will appear the Image
in fuch Cafes. Q. E, D. ·
瑞
​From the first Part of this Theorem
and the Corollary of Theorem I, arife fe-
veral Corollaries; among which we
fhall here take notice of theſe following,
viz.
Corol. 1. Rays ED in the Air parallel
to AB the Axis, after their Refraction at
the ſpherical convex Surface DB of Glafs,
do converge to that Point, C, whofe Di-
ftance BC from the Vertex. B is equal to
3AB, three Semidiameters of the Sphere.
Corol. 2. Rays ED in Glafs parallel to
the Axis, after their Refraction at the
ſpherical convex Surface of the Air, do
diverge from C that Point, whofe Di-
ftance BC from the Vertex, is equal to
(3AB or) the Diameter of the Sphere: as
Fig. 33.
…
Corol. 3. Rays eD in the Air parallel to
the Axis, after Refraction at DB the
concave Surface of the Glafs, do di-
verge from C a Point three Semidia-
meters
Opticks.
133
meters 3 AB diſtant from the Vertex as
Fig. 32.
Corol. 4. Rays eD in Glafs parallel to
the Axis, after Refraction at a ſpherical
concave Surface of Air, do converge to
Ca Point diftant from the Vertex B the
Diameter of the Sphere (*): that is, BC=
2AB, as Fig. 33.
Corol. 5. In all the four Cafes, which
are the Converſe of thoſe mention'd in
theſe four Corollaries, the Rays after Re-
fraction will become parallel to (the Ax-
is, i, e. to) a right Line drawn through
the radiant Point and the Center of the
Sphere.
As to the latter Part of this Theorem,
there are eight principal Cafes relating
thereto, which may all be demonftrated
after the fame Manner, as thofe four of
them, which are here mention'd, as be-
ing fufficient to our Purpofe; and are ex-
prefs'd by Fig. 34, 35, 36, 37.
Of
which Fig. 34 and 35 relate to Rays pro-
ceeding out of a thinner into a thicker
Medium, the other two, Fig. 36 and 37,
to Rays proceeding out of a thicker Me-
dium into a thinner. And in both Cafes
the former Figure relates to concave Re-
fringents, the latter to convex.
And in
Fig. 36, is reprefented the Cafe, wherein
the incident Ray ED is not refracted,
but reflected into DC, according to what
(13)
was
134
The Young Gentleman's
I
was obferv'd above, in Schol. 1 and 2 of
Theorem I, of this Chapter,
Schol. 1. From the former Part of this
Theorem is deducible the Way, how to
find the Focus of parallel Rays falling on
ſpherical Glaffes, viz. by taking C in
AB fo,as that BC: CA::I: R. And from
the latter Part of this Theorem is dedu-
ced the Way, how to find the Focus of
diverging Rays falling upon a fpherical
Glafs, viz. by taking C in AB fo, as that
the Ratio compounded of the Ratio's of
EA to AC, and CB to BE may be equal
to the Ratio of I to R.
Schol. 2. If the Mediums propos'd be
Air and Glaſs,and the Ray falls out of the
Air upon the Glafs (as Fig.34 and 35) then
the Focus C may be very eafily found,
namely, by making EA : AB; : EC: BC.
For by tripling the Antecedents, EA;
(AB=) AD : ; 3EC: (BC=) DC. But
EA is to AD as the Sine of the Angle
EDO is to the Sine of the Angle DEA;
and 3EC is to DC, as the Triple of the
Sine of the Angle EDC or HDC is to the
Sine of the Angle DEA. Wherefore the
Sine of the Angle EDO is the Triple of
the Sine of the Angle HDC, and confe-
quently is to the Sine of the Angle ADC,
as 3 to 2. Therefore DC, or De lying in
the fame direct Pofition, is the Refract of
the Ray ED falling out of the Air upon
Glafs.
When
*
Opticks.
135
When the Ray paffes from Glaſs into
the Air (as Fig. 36 and 37.) then the
Focus may be found, according to the
fame Principles, by taking C fo, as that
EA: AB:: EC: BC.
Schol. 3. By what has been faid un- Fig. 36.
der the foregoing Theorems, there being
given one Focus of a given Lens, the
other may be found. Where it is to be
noted, that by a Lens is denoted a (dia-
phanous or) tranfparent Body (fuch as
Spectacles, Burning-glaffes, &c.) of a
different Thickneſs from the ambient
Tranſparent, terminated by two Surfaces,
which are either both ſpherical, or the
one plain and the other ſpherical. A
right Line perpendicular to both its Sur-
faces, is call'd the Axis of the Lens. The
Points, where the Axis interfects the Sur-
faces, are call'd the Vertex's; one the
Vertex of Incidence or Immerſion, the o-
ther of Emersion, accordingly as it is in
that Surface, on which the Rays first fall,
or through which they pafs out again.
The Thickness of the Lens, is the Diſtance
between its Vertex's. To which add,
that where more than one Lens is us'd,
that which is next to the Object, is thence
call'd the Object-Lens; and that which is
next to the Eye, is thence diftinguiſh'd
by the Name of the Eye-Lens.
(I 4)
Now
136
The Young Gentleman's
Fig. 38.
Now if the Focus before Incidence or
Immerfion be given, and the Focus after
Immersion be fought; let there be found
first the Focus of the Rays after their Re-
fraction at that Surface of the Lens, on
Fig. 38. which they first fall; and this is done
by Theorem II, if the Surface of the Lens
be plain; but if it be fpherical by Theo-
rem III, namely, by the former Part of
it, if the Rays are parallel; and by the
latter Part, if diverging or (*) converg-
ing. Having thus found the Focus of the
Rays after Refraction at the firft Surface
of the Lens, (that is, while they pafs
through the Lens, whence the faid Fo-
cus is call'd the Focus of Tranfition) in
like Manner may be found their Focus
after Refraction at the ſecond Surface of
the Lens, or rather at that Surface of
the ambient Medium, which is contigu-
ous to this fecond Surface; that is, the
Focus after Emerfion.
If there be more than one Lens pro-
pos'd after the like Manner is the O-
peration to be carried on in refpect to
each.
(*) Namely, according to what has been obferv'd of
Converging Rays in Schol 3. of Theorem I. of this Chap
per.
}
In
Opticks.
137
In like Manner, there being given the
Focus of one given Lens or more, there-
by may be found the Focus before Inci-
dence, that is, the optical Inftrument
being given, thereby may be determin'd
the Diſtance of the viſible Object.
THEOREM IV.
The Image CT of a radiant plain Sur- Fig. 39.
face EF, made by a refracting Surface ·
BD, whether plain or fpherical, will be
alſo a plain Surface.
Demonft. Let A be the Center of BD
the refracting Surface, and AE be perpen-
dicular to EF, meeting with BD in B.
Let C be the Focus of the Rays diverging
from E and refracted by BD. From F,
any Point taken at Pleaſure in the radi-
ant Plain draw, FA meeting with BD in
D, and the Plain CT in T. The Ray
DC will be the Refract of ED; and let
BH be the Refract of FB. Becauſe the
Angle EAF is fuppos'd to be ſmall, there-
fore the Arch BD may be look'd on
as a ſhort right Line; and a Circle drawn
with the Diameter BF will pals through
the Points D and E, becauſe of the right
Angles BEF and BDF. Whence the An-
gles EBF and EDF, (viz. the Angles of
Incidence of the Rays FB and ED) being
in the fame Segment, are equal; and con-
fequently
138
The Young Gentleman's
ſequently their refracted Angles ABH and
ADC will be equal. Therefore by rea-
fon of the equal Angles at A, the Trian-
gles BAH and DAC are equiangular, and
BA is to AH as DA to AC; and, where-
as BA DA, therefore AH-AC. But
becauſe the Angle EAF or TAC is very
fmall, therefore (in a manner) AT=AC;
and therefore AH may be look'd on as e-
qual to AC, that is, the Focus of the ra-
diant Point F, is placed as near as may
be in the Plane CT. And the Point F
being taken at pleaſure, it holds good al-
fo in all other Points of the Plane EF, that
their Focus's, are in the Plane CT; that is,
that the Image of the radiant Plane EF
will in this Cafe be alſo a plain Surface.
Q. E. D.
Corol. 1. Hence it follows, that the I-
mage of the radiant Plane EF, whereto
the Axis of the Lens is perpendicular,
will alſo be a Plane parallel to the for-
mer. For the Image CT is to be con-
ceived as the plain Surface, which fends
forth Rays upon the fecond Surface of the
Lens.
Corol. 2. If the Angle EAF be too
great, fo that AT is fo much longer than
AC, as not to be capable of being e-
fteem'd equal thereto, then the Image of
the Plane EF will be concave towards A,
if BD be the convex refracting Surface of
a thick-
Opticks.
139
}
a thicker Medium, or the concave refrac-
ting Surface of a thinner Medium; and
on the contrary.
THEOREM V.
The Angle fGb, under which the Image Fig. 38.
fCh appears from G the Vertex of Emerfi-
on, in a Lens, is equal to the Angle FBH
under which the Object appears from B
the Vertex of Incidence in a Lens.
Demonft. Let GB be the Axis of the
Lens, and perpendicular to the vifible
Plane or Object FEH at E, and confe-
quently (by Corol: 1, Theorem IV,) per-
pendicular to its Image fCh. Draw BF,
BH, Gf, and Gh. Of the innumerable
Rays flowing from the Point F, and af-
ter Refraction at the Lens again united
at f the Image of the Point F, let two
be chofen, whereof one FB meets the
Lens at Bits Vertex of Incidence, and
being there refracted tends to p the Fo-
cus of Tranſition in reſpect of F; and be-
ing again refracted at L tends towards f
The other Ray FD being first refracted at
D, tends directly to p, 'till paffing out of
the Lens at G the Vertex of Emerfion, it
is there refracted again, and thence pro-
ceeds directly to f. The Angle of Inci-
dence of the Ray DG is DGB, and its
Refract is CGf; and the Angle of Inci-
dence
149
The Young Gentleman's
dence of the Ray LB, which would (by
the Corollary of Theorem I,) be refracted
into BF, is the Angle LBG, and its Re-
fract EBF. Now becaufe the right Lines
PB and pG are to be efteem'd equal, (by
reafon that the Thickneſs of the Lens is
not to be confider'd,) therefore the Sines
of the Angles DBG and LBG being pro-
portional to the faid Lines pB and pG,
and confequently the Angles themfelves
DGB and LBG, and therefore their re-
fracted Angles CGf and EBF will be e-
qual. In like manner it may be prov'd,
that the Angles CGh and EBH are equal.
Therefore the Angles FBH and fGh are
equal. (And the like may be fhewn in any
other Cafe.) Q. E. D,
Corol. Hence it follows, that a radiant
Line is to its Image made by a Lens, as
the Diſtance of That from the Vertex of
Incidence is to the Diſtance of This from
the Vertex of Emerfion, or (the Thick-
nefs of the Lens being not regarded) as
their Diſtances from the Lens. But if the
Radiant be a Surface (the homologous
Lines will keep in the fame Ratio, but) the
faid Surface will be to its Image (in the ſaid
Ratio duplicated, or) as the Squares of
their Diſtances from the Lens. Hence
may be determin'd the Ratio, which the
laſt Image, (i. e. that which is immedi-
ately feen by the Eye) made by the Means
of
Opticks.
141
of one Lens or more, has to the viſible
Object, whoſe Image it is.
Schol. From what has been faid, may
be accounted for and illuſtrated the Affec-
tions of (or feveral Particulars relating
to) refracting Surfaces, whether plain
or fpherical. Among which Affections
the more remarkable are thefe that fol-
low, viz.
1. If the prime Radiant or Object be Fig. 40.
far diftant, and the Surface PCQ of the
Lens be plain, the Surface PDQ be of
Glaſs and convex, and the ambient Bo-
dy be the Air, then (by Corol. 4, Theorem
III) Db will be equal to the Diameter of
the Sphere PDQ; (for by Axiom 1, the
Rays parallel to BC pafs through the
plain Surface PCQ, whereon they fall
perpendicularly, without being refrac-
ted :) and confequently by Theorem IV
and V, fbe will be the Image of the far
diftant Objec, from which proceed the
Rays FBE; and therefore the ſaid Image
will appear in an inverted Situation with
refpect to its Object: Namely, the Point
F of the Object, and the correfpondent
Point f of the Image, will be on different
Sides of Bp the Axis; and fo of the o-
ther Points B and b, E and e. And
therefore, if F be the lower Part of the
Object, f will be the upper Part of the
Image; and fo as to the reft agreeably.
2. But
142
The Young Gentleman's
Fig. 41.
Fig. 42.
Fig: 43.
of Burn-
ing-glaf
1es.
2. But if the Surface PCQ be convex,
and PDQ plain, then (by Corol. 1, Theo-
rem II, and Corol. 1, Theorem III of Diop-
tricks) Cb will be equal to the Diameter
of the Sphere PCQ,`and a third Part of
the Thickness of the Lens over.
Or neg-
lecting the Thickneſs of the Lens, (as is
ufual in the Object Lens of a Telef
cope,) in this as well as the former Cafe,
and confequently in general of any plain
convex Lens, it may be faid, that the I-
mage of an Object far diftant, made by
fuch a Lens, will be equal to the Diame-
ter of the Sphere.
3. If the Lens PQ be of Glafs, and e-
qually convex on both Sides, then (by
Corol. 1. Theorem III, and Theorem IV of
Dioptricks) Db will be equal to the Se-
midiameter of either Sphere. Laftly, in
a whole Glafs Sphere, the Image of an
Object far diftant will appear at the Di-
ftance of half the Semidiameter from the
Sphere: For in this Cafe Regard muſt
be had to the Thickneſs of the Lens.
4. In all theſe or the like Cafes, if the
Sun be the far diftant Radiant or Obje&,
and the Lens be confiderably larger than
the Sun's Image, in the Place of the I-
mage will be the Burning point: And a-
ny fuch convex Lens will burn ftronger
than a concave Reflecter (cæteris pari-
bus)
1
Opticks.
143
bus) by reaſon more Rays are loft by the
latter than by the former.
with Con-
5. If in the foremention'd Place of the of Dark-
Image there be placed a lucid Body it felf; Lanterns
then the Image of the faid Body will ap- vex Glaf
pear at a great Distance ; that is, the ſe:.'
faid lucid or fhining Body will caft a
Light at a great Diſtance, fo as that
Things fo far diftant may be clearly ſeen
by the faid Light. And this is the Manner,
whereby a Candle, put into a Dark Lan-
tern with a convex Chryftal, gives a
Light at fo great a Diſtance as ufual
as
6. If the Radiant or Object be not
afore) far diftant, but yet is more diftant
from the Lens than is the Place where
would appear the Image of a far diftant
Object; befides the Affection aforemen-
tion'd, (viz. that the Image will appear
inverted) there will be in this. Cafe alſo
theſe other Affections. Namely, as the
Radiant comes to the Lens, the Image
will go from it, and contrariwiſe 'till
the Radiant being come into the Place,
where would appear the Image of a far
diſtant Radiant, its own Image becomes
far diftant. All which Affections may be
obferv'd in a Dark-Chamber, into which
Objects at feveral Diſtances fend their
Rays through a convex Lens or Glafs.
The Place of the Image of each Radiant
Object may be diftinctly known, by ob-
ferving
144
The Young Gentleman's
Of a Ma-
tern.
ferving where the Image of the faid Object
does appear moſt diftinctly on a Sheet of
White Paper or the like.
7. And on theſe fame Principles de-
gick Lan pend the Mechanifm of (what is call'd)
a Magick-Lantern; namely, whereby
the Images of little Pictures are repreſent-
eds on a white Wall or the like.
Fig. 44,
& 45.
..8. If the Radiant be nearer to a con-
vex Lens, than the Place where would
appear the Image of a far diftant Radi-
ant or Object, then its Image will appear
on the fame Side of the Lens with the
Object it felf; and its Place may be de-
termin'd from the given Place of the Ra-
diant, as afore. The Image in this Cafe
will always appear in the like Situation
with the Object, and greater than it; and
if the Object comes to the Lens, the I-
mage will alfo; if that goes from the
Lens, this will alfo; only the Image will
in both Cafes move quicker.
9. If the Lens be concave, then (by
the refpective Corollaries of Theorem III
of Dioptricks) the Image ebf made by a
plain concave Lens, PQ, will be diftant
the Diameter of the Sphere PDQ, and
will be erect or in the like Pofition with
the Object, and on the fame Side with
the Object reckoning from the Vertex D.
If the Surface PCQ be alfo concave, and
the Object not far diftant, its Image
may
Opticks.
145
may be determin'd after the like Method,
by finding (according to Theorem IV of
Dioptricks) the Image of that Point of
the Object which is fituated in the Axis
of the Lens. In refpect to which Image,
befides what has been already faid, it is
to be obſerv'd, that it will together with
the Object come to or from the Lens, but
flower.
10. Further, it is here obfervable, that of Telef
on the foremention'd Theorems depends copes, Mi-
croſcopes,
the Mechaniſm or Contrivance of the fe- and Spec-
veral optick Inftruments, made ufe of to tacles.
help the Eye, or to fupply the ſeveral
Defects of our Sight. For, as the Eye is
a natural Organ or Inftrument, defign'd
to this End, namely, that on the Bottom
thereof may be painted the diftant Ima-
ges of cutward Objects; ſo it is manifeft
that one and the fame Eye can fee diftinct-
ly only at a certain Diſtance from the vi-
fible Object. Could the Eye always place
it felf at fuch a Distance from the Object,
as is neceffary to diftinct Vilion, then
(fuppofing the other Requifites to Sight
to be given, fuch as a due Degree of Light,
&c.) the Eye might always fee diftinct-
ly. 'Tis true indeed that the Diſtance re-
quifite to diftin&t Viſion is not rigidly de-
termin'd to one fingle Point, but admits
of fome Latitude; forafinuch as the fame
Eye can fo alter its Figure according to
(K)
the
146
The Young Gentleman's
the various Diſtance given, that it may
be look'd on not as one Eye but ſeveral.,
But then, fince this Latitude of Distance
is contain'd within certain Bounds, be-
yond which the Eye can by no natural
Means fee diftinctly, hence arifes the
Need of artificial Inftruments.
And any
Lens, or ſpherical reflecting Glafs, is fuffi-
cient to remedy this; forafmuch as there-
by the Image of fuch an Object, as we
can't come to as we would, may (by
Help of the Theorems above demonftra-
ted) be brought near us, fo as that we
may have a diſtinct View thereof.
of this Kind are thoſe fingle Glaffes, us'd
by fuch as can't fee well at a common
Diſtance, and apply'd by them to one of
their Eyes, when they would difcern the
Face or Picture of a Perfon (or the like)
that is farther diftant from them than
they can diftinctly fee with the bare Eye.
As the Word Telefcope does in the Greek
Tongue denote an Inftrument which helps
us to fee far off, fo thefe Gingle Glaffes
may be call'd fingle Teleſcopes, to diftin-
guish them from fuch Teleſcopes as are
made of two or more Glaffes.
And
11. But becauſe (befide diftin& Vifi-
on) it is often requifite in humane Affairs,
that the more minute Parts of an Object
fhould be diſcern'd; and becauſe it is
found by Experience, that an Object
which
Opticks.
147
•
{
}
f
}
<
3
{
which appears under a lefs Angle than of
one Minute, appears but as a Point, or
fo as that its Parts can't be diftinguiſh'd
one from the other: Hence it often comes
to pass, that the Obje&t being brought fo
near to the Eye, as that its Particles to
be examin'd may be feen under a fenfible
Angle, thereby the Object it felf becomes
too near to the Eye, fo as not to be with-
in the Bounds requifite to diftinct Viſion.
This Inconveniency, when alone, may al-
fo be remedied by any given Lens or re-
flecting Glaſs; forafmuch as the Image
of the faid Object (*) may by the faid
Lens or Glaſs be reprefented (in any gi-
ven Meaſure, i. e.) fo as to have any gi-
ven Ratio or Proportion to its Object.
(*) How this may be done in refpect of a reflecting
Glafs, is fhewn above in Schol 7, to Theorem V, or laft of
Catoptricks. How the fame may be done in refpect of a
Lens or retracting Glafs, fhall be here fhetvn. Let a Glafs
Lens be given (Fig. 45) the Semidiameter of whofe fore- Fig. 46.
moft Surface is AB, the Semidiameter of the hindermoft
Surface CB. Draw CM as you pleafe, and let the Ratio
affign'd be fuch as ought to be between the homologous
Lines of the Radiant and its Image, and be denoted by the
Ratio of MD to DC. Draw AM, whereto through B draw
the Parallel BE, meeting CM in E; ME doubled will be the
fought Diſtance of the Radiant from the Lens. And after
the fame Manner may be found, in refpect of any given
Lens, the Pofition, or Place where any Radiant is to be
put, that (its Image made by the Lens may be equal to a
given Figure, which is fimilar or like to the Radiant, i..
in fhort, that) the Radiant may have a given Ratio (to
its Image made by the Lens.
(K 2)
And
1
148
The Young Gentleman's
Fig. 46.
And of this Kind are thofe fingle Glaffes
us'd to ſee the Shape or Parts of ſmall
Infects, and the like, and thence call'd
Microſcopes, the Word importing in the
Greek Language an Inftrument, where-
by we fee Small Things. To theſe may
be referr❜d Spectacles, of which more by
and by.
But if both the foremention'd Inconve-
niencies concur, then they are not to be
remov'd but by the joynt Help, either of
a Lens and Reflecter together, or elſe of
more than one Lens, or laftly of more
than one Reflecter. And how Inftru-
ments confifting of two or more of thefe
are made, fhall be next illuftrated.
13. Let R denote the Object given, S
the given Angle under which the Object
is to be reprefented, D the given Di-
ftance of the Sight, and L the given Di-
ftance of the Eye from the Lens. Make
the Triangle AOB, fo as that the vertical
Angle at O may be equal to the given
Angle S, and OE the Perpendicular from
the Vertex to the Bafis may be equal to
the given Line D. If you'd have the
middle Point of the Object be in the
Axis of the Lens or Reflecter, (which is
moft convenient for Practice) the Trian-
angle AOB must be made an Ifofceles.
Take OV L, and at V place a Lens or
Re-
Opticks.
149
J
5
Reflecter fo, as that its Axis may fall
in with OE.
A
Then having E thus given for one Fo-
cus of the Lens or Reflecter, by the re-
ſpective Theorem III of Catoptricks or
Dioptricks or their Scholium, find the o-
ther Focus e; namely, that fo taking e for
the Focus before Incidence, E may be the
Focus after Inflection at the Lens or Re-
flecter. Through e draw aeb parallel to
AEB, meeting with Va and Vb (drawn
through V, and parallel to VA and VB)
in a and b.
It follows from Theorem V, of Catop-
tricks or Dioptricks (refpectively, as you
calculate for a reflecting or refracting
Glaſs,) that if aeb be the Radiant, AEB
will be its Image. Wherefore if the Ob-
ject given R, (which is here fuppos'd to
be near at hand) and a fecond Lens or
Reflecter be ſo plac'd, according to Schol.
7, Theorem V of Catoptricks, or accord-
ing to the Note before added to Schol. II
of this Theorem V of Dioptricks,) that the
Image of the Object made by this fecond
Lens or Reflecter, may appear just in the
Place, Situation, and Bignefs of aeb
there will be thus made fuch a Microf-
cope as is requir'd. For aeb will be the
Image of the given Object R, the Image
of which Image, and that feen by the
Eye, will be AEB. And this will appear
(K 3)
ذ
under
150
The Young Gentleman's
under the Angle AOB equal to the given
Angle S; and in the Diſtance OE equal
to the given Distance D, and confequent-
ly will appear diftinct; and laftly, the
Diſtance of the Eye from the Lens or Re-
flecter next to it in V, will be equal to L,
the given Diſtance in this refpect. And
fo the Inftrument will be in all Refpects
fuch as is requir'd.
14. If the Inftrument be made for the
Eye of an aged Perfon, then the right
Line OE becomes infinite, and confe-
quently the Point e is to be found by
Theorem III of Catoptricks and Dioptricks.
As for the reſt of the Work, it is the ſame
as afore.
If the Object propos'd to be diſtinctly
feen be far diftant, then having all (as a-
fore) given, but the Angle S, a Telef-
cope may be made fit for the Purpoſe thus.
In a right Line extended toward the re-
mote Object propos'd, take from O for-
ward OE and OV equal to the given D
and L; and in V place a Reflecter or
Lens having its Axis in VE. The Focus
E of the Lens or Reflecter plac'd at V be-
ing thus given, find the other Focus e,
namely, that this being taken for the
Focus before Inflection, E may be the
Focus after Inflection. In the right Line
OE fo place a fecond Lens or Reflecter,
having alfo its Axis in OE, that the I-
mage
Opticks.
151
3
mage of the propos'd Object made there-
by, may appear in the right Line aeb
perpendicular to OF; and fo there will
be made fuch a Telefcope as is requir'd.
For the first Image of the far diftant Obje&
propos'd will be in aeb and the Image
of this Image will be in AEB, the Distance
of which AEB from O is by Conftructi-
on equal to the given D, and conſequent-
ly diftinct to the Eye; and. alfo the Di-
ftance of the Eye from V is equal to the
given L. And from hence may be found
the Angle, under which the laſt Image is
feen.
16. By the Help of three or more Lens's
or Reflecters, an Inftrument may be made
to repreſent even a far diftant Object un-
der a given Angle. Namely, let any one
Lens or Reflecter form the Image of the
Object propos'd; and then with the o-
ther two make (as afore) a Microſcope,
repreſenting the firft Image (as being a
given Radiant nigh at hand) according to
the Conditions propos'd; and fo there
will be made fuch a Teleſcope as is re-
quir'd. In like Manner, a Microſcope be-
ing made of three Lens's or Reflecters, by
adding a fourth it will become a Telef
cope. In all which Cafes is known the
Ratio or Proportion, which the Image
ſeen by the Eye has to the Obje&, or
(which comes to the fame) the Proporti-
(K 4)
Oil,
;
*
152
The Young Gentleman's
Fig. 48,
& 49.
on, which the Angle under which the
faid Image is feen, has to the Angle un
der which the Object is ſeen without the
Inftrument, (which, in refpect to far
diſtant Objects, is as the Diſtances of the
Lens's or Reflecters from the common Fo-
cus e.) And confequently from hence.
(according to the Corollary of Theorem
IV, Chap. III and IV,) may be computed
how much the Inftrument does help the
Sight.
17. I fhall next adjoin fome Obferva-
tions in reference to a dark Chamber, as
was abovemention'd in Schol. 2, Theorem
IV, Chap. II. Namely, it was there ob-
fervid, that Rays let into a dark Cham-
ber through a naked little Hole, will
paint the Images of their respective Ra-
diants on a Sheet of white Paper held a-
gainst the Hole. To which it is here to be
added, that the fame will be done thro'
a much larger Hole, if a Lens or ſpheri-
cal Glafs be placed therein, with this Dif-
ference, that the Images of outward Ob-
jects will be painted much more lively
and diftinctly this Way than the other
without a Lens; which is thus account-
ed for. Through the little Hole (Fig.
48) muft neceffarily pafs fewer Rays
from the outward Objects than through a
much larger Hole (Fig. 49.) And not on-
17 fo, but thofe fewer Rays, which pafs
through
Opticks.
153
through the bare Hole are difpers'd ;
whereas by a convex Lens or Glafs all the
Rays proceeding from any one Point of
the Object, are united again at a certain
Diſtance, and fo render. the Image
much more lively and diftin&t, than the
other Way, This is illuftrated Fig. 48
and 49. Whence it is alfo obvious, why
in both Cafes, the farther the Paper is
held from the Hole, the bigger the Image
will appear; as alfo why the farther the
outward Objects are from the Hole, the
lefs their Images will be; and why the
Image made by a Glafs will appear moſt
diftinctly at the Diſtance ac (Fig. 49) with
other fuch like Particulars.
18. But it is obfervable, that in both
Cafes, either with or without a Glafs,
the Image will appear inverted in respect
of the Object, as has been afore obſerv-
ed under Theorem IV, Chap. II, and is e-
vident from Fig. 48 and 49. And from
Fig. 48 it is evident, that when the Image
as is painted erect on the Bottom of the
Eye, then the immediate Radiant abc it
felf is feen inverted by us; and on the
contrary, when the Image abc is painted
inverted on the Bottom of the Eye, then
we ſee the immediate Radiant ABC e-
rect. For as in the faid Figure the fe-
cond Image aby is erect in the Eye, fo if
the Eye were put in the Place of the Pa-
per,
154
The Young Gentleman's
per, the firft Image abc would be pain-
ted therein inverted, in reſpect of the
Object ABC.
19. This being fo, the Manner how
that Senſation, which we call Vifion, is.
perform'd, feems to be the fame, as that
wherby a blind Man by the Thruſt of the
End of feveral Sticks against his Hand,
can thereby diftinguifh, which Stick
thrufts from above, which from below.
Namely, fo by the Senfation caus'd by
the Ends a and b of the Rays Aa and Bb
and Cc, touching the Retina, we are na-
turally endued with a Faculty of readily
diſtinguiſhing, from whence each Ray
proceeds, viz. that the Ray Aa, as touch-
ing the lower Part of the Retina, does
proceed from the upper Point A of the
Object; and the Ray Cc as touching a
the upper Point of the Retina, does pro-
ceed from the lower Point c of the Ob-
ject; and the Ray Bb as touching b, the
Middle of the Retina, does proceed from
B the Middle of the Object. And confe-
quently we are enabled to judge of the
Situation of Things aright, by receiving
their Rays after fuch a Manner as (it ap-
pears from the forefaid Experiments) we
do: Which is the fame as to ſay, that
we fee Things in their Erect or due Si-
tuation, by having their Images painted
on the Retina in an inverted Manner
and
Opticks.
155
and contrariwise, we fee Things invert-
ed, by having their Images painted on
the Retina in an erect Manner.
cles.
20. I ſhall now, in the last Place, ex- of Specta-
plain the Mechanism of Spectacles, or how
they come fo to help the Eye. It is then
evident from the Rules of Dioptricks,
that convex Glaffes turn the Rays in-
ward, or towards the Axis of the Glaſs
and the more convex they are, ſo much
the more inward the Ray is turn'd. It is al-
fo found by Obfervation or Experiments,
that the Chryſtalline Humour has the
fame Effect as a convex Glaſs. Where-
fore, when the Surface or Coat of the Fig. 56.
Chryſtalline Humour is of fuch a due
Convexity, as that it makes the Rays of
the feveral Points of an Object to unite
again in fo many feveral Points juſt up-
on the Retina, then Vifion is duly per-
form'd, as Fig. 50.
21. But when the Chryftalline Humour
is not convex enough, but too flat, (as it Fig. 51.
grows through Age,) then the Rays paf-
fing through it are not united foon e-
nough, i. e. are not united at the Retina,
but their Point of Union is further be-
hind the Retina, and confequently then
the Eye can't fee duly or diftinctly, but
only confuſedly, the Objects, from
whence proceed the Rays. Wherfore to
help this, convex Glaffes, (call'd Specta-
cles)
156
The Young Gentleman's
cles) are us'd, whereby the Rays are
made to unite in their proper Place, or
at the Retina. In order thereto, the more
flat the Chryftalline Humour is, that is,
the more aged the Perfon is, the more
convex must the Spectacles be.
22. On the other Hand, if the Surface
of the Chayſtalline Humour be too con-
Fig. 52. vex, (as is the Cafe of fhort-fighted Per-
fons.) then the Rays are thereby united
too ſoon, or before they come to the Re-
tina, and fo Viſion is not duly perform'd.
Hence fuch Perfons hold Things very
near to their Eyes, to fee them the better,
becauſe (according to the Rules of Diop-
tricks) the nearer the Object is to the
(Lens, or in this Cafe) the Chryſtalline
Humour, the further will be the Union
of its Rays on the other Side. Hence al-
fo concave Spectacles helps fuch Perfons
Sight, by making the Rays unite in their
proper Place, viz. at the Retina. Hence,
laftly, fuch Perſons ſee better and better,
as they grow elder; becauſe the Chryſtal-
line Humour grows more and more flat,
or less convex; fo as to come but to a due
Convexity in their old Age: Which by
the way is the Reaſon, that fuch People,
as are fhort-fighted when Young, want
not Spectacles when they are Old. And
this is fufficient to our Purpofe concerning
Dioptricks.
CHAP,
Opticks.
157
CHAP. V.
Of Perſpective.
I.
N Perſpective the chief Things (befide
the Eye and the Object) to be con- Definitions
fider'd, and therefore explain'd, are thofe of Terms
which are contain'd in the following De- Perſpective
finitions.
us'd in
2.
metrical
The (†) Geometrical Plane is a plain
Surface parallel to the Horizon, and pla- The Geo-
ced lower than the Eye, in which we plane,
imagine the viſible Objects without any what.
Change, (except it be reducing a great
Figure into a ſmall one,) and upon which
the Situation of the Object to be repre-
fented in Perſpective is defcrib'd.
3.
The Situation (call'd alfo the Ich-
nography or Plan) of an Object is its The Situa-
Orthographical Projection upon the tion of an
Object,
wh.it.
(†) It is alfo called the Bafis. This and the following
Planes, c. here denoted, are not reprefented in Figures
or Cuts, they being not cafie to be fo apprehended by
young Students; but they may be more eafily apprehended
by the bare Imagination, or elfe by actually placing fo ma-
ny Plains in ſuch a Manner.
(*) geo-
158
The Young Gentleman's
4.
The Pic
ture, what.
The
5.
(*) geometrical Plane, or, that Figure
which is defcrib'd upon the Gemetrical
Plane, when from all the Points of the
Object right Lines are drawn perpendicu-
lar to the faid Geometrical Plane. Thus
the (Plan or) Situation of a right Cylin-
der is a Circle, and the Situation of a
right Cube is a Square.
The Picture is a plain Surface, fup-
pos'd as tranfparent as Glafs, and al-
ways plac'd at fome Distance between the
Eye and the Object, to repreſent in it
the Object in Perfpective; whence the
Picture is alfo call'd the Perspective
Plane.
1
The Picture ftands (uſually (†) per-
pendicular) upon the geometrical Plane.
Ground- Whence the common Section of the Pic-
line, what.
ture and the geometrical Plané, is call'd
the Baſe of the Picture, or the Bafe-line,
and alſo the Ground-line.
1
(*) If the Orthographical Projection be (not upon the
geometrical Plane, but) upon a Plane parallel to the Pic-
ture, then it is call'd Front; and if it be upon a Plane pa-
rallel to the vertical Plane (defcrib'd §. 11) it is term'd
Profil.
(†) Sometimes the Picture is inclin'd to the geometrical
Plane or Horizon, and has alſo a curve Surface, as when
the Surface of an arch'd Roof is to be drawn. But this
being not common, the Picture is look'd upon throughout
this Chapter as a Plane perpendicular to the geometrical
Plane or Horizon.
The
Opticks.
159
6.
zontal
The Horizontal Plane is a plain Sur-
face parallel to the Horizon, and which The Hori
is conceiv'd to pass through the Eye, and Plane,
to be perpendicular to the Plane of the what.
Picture.
1
zontal
The Horizontal Line is that right Line, 7.
in which the horizontal Plane and the The Hori-
Plane of the Picture interfect, and which Line,what.
therefore is parallel to the Ground-
line.
8.
The Prin-
The Principal Ray is a right Line
drawn from the Eye, perpendicular to
cipal Ray,
the Plane of the Picture, which there- what.
fore muſt run along the horizontal Plane.
The Point of Sight, call'd alſo the
Principal Point, is the Point where the of Sight,
Picture is cut by the principal Ray, what.
which muſt neceffarily be in the horizon-
tal Line.
9.
The Point
10.
of Dif-
The Point of Distance is a Point in the
horizontal Line, whofe Distance from The Point
the Point of Sight, is equal to the Length
of the principal Ray.
The Vertical Flare is a plain Surface,
which going along the principal Ray, is
perpendicular to the Horizon, and fo to
the geometrical Plane, and alfo to the
Picture: The common Section of the Pic-
ture and vertical Plane being call'd the
Vertical Line, and being alfo the Mea-
fure of the Height of the Eye; and the
common.Section of the geometrical and
vertical
(
tance,
what.
1.1.
The Verti-
cal Plane
and Line,
whit
•
160
The Young Gentleman's
12.
vertical Planes being call'd the Line of
Station.
The Repreſentation or Appearance of a-
The Ap- ny Point of an Object is that Point, where
pearance, the Picture is cut by a right Line drawn
of a Point,
from the Eye to the Point propos'd of the
Object.
what.
13.
dental
Point,
what.
The Accidental Point of a right Line is
The Acci- the Point, where the Picture is cut by a
right Line drawn from the Eye parallel to
the Line propos'd. Whence it is eafie to
conclude, that all Lines, which are paral-
lel to the Picture, have no accidental
Point; and that all other Lines, which
are parallel one to another, have the
fame accidental Point. It is alfo obferv-
able, that all right Lines, which are per-
pendicular to the Picture, have the prin-
cipal Point for their accidental Point ;
and that all thofe, which make an half-
right Angle with the Picture, have one
of the Points of Diſtance for their acciden-
tal Point.
14.
Obferva-
bles from
what bar
been faid.
It is alſo obfervable from what has
been faid, that all the Parts of an Object,
which are lower than the Eye or horizon-
tal Plane, ought to be reprefented in the
Picture below the horizontal Line. And
on the contrary, all above the horizon-
tal Plane, or higher than the Eye, ought
to be reprefented in the Picture above the
faid horizontal Line. In like Manner, all
thofe
Opticks.
161
"S
hoſe Objects, which with refpect to the
Eye are on the right Hand of the verti-
cal Plane muſt in the Picture be repre-
fented on the right Side of the vertical
Line; and fuch as are on the Left, muſt
be reprefented on the left Side of the ſaid
vertical Line.
the Picture,
In order to perform any Operation,
15.
the Sheet of Paper, which you would How to de-
work upon, muſt be divided into two Paper the
fcribe on
Parts, by the Line AB (Fig. 53) which Plane of
will repreſent the Ground-line. The up- thei
per Part of the Paper, viz. ABDD, muft
be taken for the Plane of the Picture cal Plane.
and the lower Part ABYZ of this Paper
Fig. 53.
for the geometrical Plane.
;
and the
Geometri-
find or fet
Distance.
The next Thing is to know, (under 16.
what Angle the Picture ought to be feen, How to
that whatever is reprefented therein may off the
appear in juft Proportion, and may eafi- Points of
ly be taken in by the Eye at one View ;
that is in fhort,) how far the Points of
Diſtance ought to be from the Point of
Sight. Let there be propos'd a Square,
which takes in all that you would repre-
fent in Perſpective. And let the Ground-
line AB be of equal Length with one Side
of the Square propos'd. Wherefore, if
the Eye of the Spectator was at G, he
could not fee the two Ends A and B, be-
cauſe the Eye could fee, (f) at most, but
(†). According to Theorem II, of Chip. II.
(L)
fo
*
162
The Young Gentleman's
17.
fo much as is comprehended under the
right Angle GHI. But if the Eye were at
K, it could fee the Ends A and B, be-
cauſe the Angle AKB is leſs than a Right,
viz. 80 Degrees. And the Eye would
fee the Ends A and B better at L, and
better yet at M, where the Angle AMB
is 60 Degrees. And from fuch a Diſtance
the Objects (to be repreſented in Perfpec-
tive, and contain'd in the foremention'd
Square) may be feen in Perfection, the
viſual Angle AMB being neither too great
nor too little. The Angle ANB would
alfo be very well. Wherefore having
bifected AB in F, and from F drawn the
Line FV perpendicular to DD for the
vertical Line, and confequently V being
the viſual Point or Point of Sight, from
the faid Point V fet off the Diſtance
((*) FK_or rather FL, or) FM, or FN
on each side upon the horizontal Line
DD, viz. to D and D, which will be the
Points of Diſtance requir'd or fought.
And this may be look'd on as a gene-
A general ral Maxim, viz. that the Distance be-
cerning the twixt the Point of Sight V and the Points
Rule con-
Jame.
(*) If the Diſtance VD were lefs than FK, then the Per-
ípective Square ABCE (concerning which fee §'17) would
appear irregular, becaufe it would be fren under an ob.
tule, and fo too great an Augle.
of
Opticks.
163
of Diſtance D and D muſt be (at leaſt)
equal to FA or FB. And it will not be
amiſs to let this Diſtance be ſomewhat
greater. And, if in this Cafe the Plane
of the Picture be not wide enough to
receive two Points of Diſtance, there
mut only one be mark'd; and the prin-
cipal Point or Point of Sight V may be
plac'd on one side, as Fig. 59, &c.
18.
The Points of Distance D and D (Fig.
53) being found, draw the Lines AV The Per-
ſpective
and BV, as alfo AD and BD. And the Square,
Figure ABCE will be the Appearance (in what.
the Picture) of the Square, propos'd to
be repreſented in Perſpective; which Ap-
pearance of the faid Square is in fhort
call'd the Perspective Square.
19.
How to find
the Pic-
ture the Ap
ven in the
Proceed we now to the Reprefentation
(in the Picture) of Points, Lines, and
Planes. And first, how to find in the in
Picture the Repreſentation or Appear- pearance of
ance of a Point given in the geometrical a Point gi-
Plane. From C the given Point (Fig. geometrical
54) let fall upon the Ground-line AB the Plane.
Perpendicular CE; and from the Point Fig. 54
E draw to the principal Point V the Line
VE. Upon AB fet off EF (or EA)
CE; and then draw FD (or AD either
of) which will upon VE give at c the Ap-
pearance of the given Point C.
(L2)
C
Second-
164
The Young Gentleman's
20.
To find in
the Picture
the Appear.
Secondly, how to find in the Picture
the Appearance of a right Line given up-
on the geometrical Plane. Suppofe MN
Fig. 55 and 56) to be the right Line gi-
ven. Find (as afore) the Appearance of
geometrical the Point M, viz. at m, and then the
Plane. Appearance of the Point N of the given
Fig. 55, Line MN, viz. at n.
Then join
ance of a
right Line
given in the
56.
21.
the Picture
and
n, the Line mn will be in the Picture
the Appearance of MN the Line propo·
fed (*).
When the Line given on the geometri-
To find in cal Plane is (not a right Line, but) a
the Aptear- Curve Line, then from feveral Points of
ance of a it draw Perpendiculars to the Ground-
eurve Line line; by means of which, and what has
geometrical been faid, may be found in the Picture
given on the
Plane.
the Appearances of all thofe ſeveral
Points, which being join'd by a curve
Line, that Curve will be the Appearance
of the Line propos'd.
(*) That every Thing you defign to reprefent in Per-
fpective, may appear in juft Proportion, it is requifite
fometimes to have the Eye at a great Diſtance from the
Picture, which may hinder from marking either Point of
Distance upon the horizontal Line.
In which Cafe you
muft mark only half of the Diſtance of the Eye from the
Picture upon the horizontal Line, as (Fig. 57) from V to
Q, which will reprefent one of the Points of Diſtance.
And then you muſt only ſet off half the Perpendicular HI
upon the Ground-line AB, from I to P. Then draw QP,
and the Point b will be the Appearance of H the Point pro-
pos'd.
1
Thirdly,
Opticks.
165
22.
the Pille
gure given
Plane.
Thirdly, To find in the Picture the
Appearance of a plain Figure given on To find in
the geometrical Plane, fuppofe the Square the Appear-
CEGH Fig. 58. Find the Appearance ance of a
(as has been afore fhewn) of its four an- plain Fi-
gular Points C, E, G and H; which will in the geo-
be c, e, g and h. Join the faid (laft) metrical
Points, and the Perfpective Square cegh will Fig. 58,
be the Appearance of the given Square 59.
CEGH, when feen foreright, or when
the Line KL, of the given Square falls in
with VF the vertical Line. If the given
Square be all on one Side of the vertical
Line VF, (as Fig. 59) then the Perſpec-
tive Square cegh (in Fig. 59) will be the
Appearance of the given Square CEGH.
Where alſo it is to be noted, that the
Appearance g of the Point G may be A Remark.
found alſo after this Method.
in the Ground-line AB take
at pleaſure, and draw BV.
BM equal to the Perpendicular IG, and
draw MD, which upon the Ray VB will
give the Point N. Through which NC
being drawn parallel to AB, will upon
the Ray VI give the Point requir'd g
And this Method may be uſed, when the
Perpendicular IG is fo long, that the
Length thereof can't be fet off upon the
Ground-line AB wlthin the Picture.
23.
Namely,
Namely, Fig. 59.
the Point B
Then fet off
In the fame Manner may be found in 24.
the Picture the Appearance of any other Another
plain
Remark.
166
The Young Gentleman's
25.
Pilure the
plain Figure, viz. Triangle, Pentagon,
Heptagon, &c. Namely, by distinctly
finding the Appearance of each angular
Point of the plain Figure propos'd, and
then joining the Appearances fo found by
right Lines.
It fhall here be only obſerv'd farther
Fo find in a in reference to plain Figures, that to find
Appearance in the Picture the Appearance of a Cir-
of a Circle cle given in the geometrical Plane, there
given in the
muſt be defcrib'd about it a Square, as
geometrical
Plane. (Fig. 60,) the Square EFGH about the
Fig. 60. Circle ILKM. One Side of the Square,
viz. EF (or GH) muſt be parallel to the
Ground-line AB, and confequently the o-
ther Side EH (or FG) must be perpendi-
dicular to AB. Draw the two Diagonals
EG and FH, which interfecting at O the
Center of the Circle, will give upon IKL,
M the Circumference of the Circle, theſe
four Points, P, Q, R and N. Then find
in the Picture the Appearance of the
Square EFGH, which will be the Per-
fpective Square efgh, and alfo the Appear-
ances of the Points L, Q, K, R, M, N,
I and P, which will be in the Perfpec-
tive Square the Points l, q, k, r, m, n,
i and P. A Curve drawn through theſe
laft Points will determine the Appearance
of IKLM the Circle propos'd. Where
note, that the Appearance of the Circle
will alſo be a true Circle, if the Circle
•
pro-
1
Opticks.
167
***
Propos'd be ſeen fore-right, that is, if its
Line (*) LM, being perpendicular to
AB, make one right Line with VF the
vertical Line. But if the Circle propos'd
IKLM be fituated (as Fig. 61,) then its Ap-
pearance will be (as there) elliptical.
26.
fent aFloor
Proceed we now to fhew, how to re-
preſent in Perfpective a Floor of equal To repre-
Squares feen fore-right, without a geo- of equal
metrical Plane. Divide the Ground-line Squares
AB (Fig. 62) into as many equal Parts right.
as you pleaſe, each of which will repre- Fig. 62
(*), To have the Appearance of a Circle in the Picture,
to be likewiſe a Circle, the Diſtance between the Eye and
the Picture muſt be of a determinate Length, which may
be found thus. Having produc'd the Side of the circum-
fcrib'd Square EFGH, 'till it meets at right Angles with
the horizontal Line DD ia a Point as S, draw SL, and
cut off from it SCSV, and the remaining Part LC (of SL)
wil be the Diſtance between the Eye and the Picture, or
VD equal to the ſaid Diſtance. If VD be given, and you
would find VE (or the Distance between the horizontal
Line DD and the Ground-line AB, that is) the Height of
the Eye requifite, that the Appearance may be alfo a true
Circle; the rectangular Triangle SVL fhews, that the
Square of the Line EL (or of the Ray of the given Circle
IKLM) muſt be taken from the Square of the Line LS, and
the Side ES of the remaining Square will be the Height re-
quir'd. Note further, that the Perfpective Square efg
being found, the Mid-point X of the Line mL (or ml) will
be the Center of the Perspective Circle; upon which there-
fore Center the faid Circle may be drawn without any
more ado, or without finding the Points 1, q. k, &c. Laftly,
note, that in Fig. 60, the Perpendiculars Ef and TN are
not fet off on the Ground-line AB, partly to avoid Confuſion
by defcribing the Arch of many Circles, partly and chief-
ly, because it is evident (from the fecting off FK and QR)
that EL-El and TF TK.
fent
feen fore.
168
The Young Gentleman's
27.
ſent the Side of a Square. Draw through
each Point of Divifion to the principal
Point V, as many right Lines or Rays,
the two laſt of which will be VA and
VB. Then draw the Ray AD (or BD ;)
which will cut the foregoing Rays in
Points, through which you must draw
Parallels to the Ground-line AB. The
laſt of theſe Parallels will be EF, which
will have its Divifions equal between G
and H. And the like equal Divifions
may be continued from G to E, and
from H to F, in order to draw through
theſe new Points of Divifion, from the
principal Point V, other Rays which
will upon the Lines DA and DB meet the
foregoing Parallels reſpectively belonging
to them. Thus you'll have in Perfpec-
tive all the squares, which can be com-
prehended in the Space ABFE.
And if
you would have more, draw DE, which
will cut all the Rays that proceed from
V in Points, thro' which (as afore) you
muſt draw Parallels to AB, the laſt of
which will be KL. And fo, go on as a-
fore.
If
!
you would reprefent in Perfpective a
Toreprefent Floor feen fore right, and containing e-
a Floor of qual Squares feen Cornerwife, work thus.
The Side AB of the fquare Floor being
determin'd upon the Ground-line AB, de-
ſcribe the Appearance of the fquare Floor,
Equal
Squares
feen Cor- determin'd
ner-wife.
Fig. 63.
by
Opticks.
169
叉
​by drawing from the two Ends A and B
of the Side given, to the principal Point
V, the Rays VA and VB; and to the
two Points of Distance the Rays DA and
DE. Theſe Rays will cut the former in
the Points E and F, which being join'd,
the Line EF will be parallel to the Ground-
line; and ABEF will be the Perſpective
Square. Having divided AB (or the
Side of the Perfpective Square) into e-
qual Parts, to each Point of Diſtance D,
draw Rays through each Polnt of Divi-
fion; which Rays by their common In-
terfections, will form the Appearances of
the Squares feen Corner-wife. With
which Squares it will be eafie to fill the
Perfpe&ive Square ABFE, becauſe all
the Rays that proceed from the two
Points of Diſtance, divide equally in
the fame Points the Side EF parallel
to AB; which is alfo divided equal-
ly and in the fame Points, by the faid
Rays which go from the two Points of
Diſtance.
28.
lent a Floor
To repreſent in Perfpective a Floor
made of equal Squares feen fore-right, To repre-
and encompaſs'd with a Border or Frame, of Shares
without a geometrical Plane; work thus.
feen fore-
Divide the Ground-line AB into as many right, and
Parts, alternatively equal and unequal, fed with a
encompaj.
as you would have Squares and Borders, Border.
at the Points 1, 2, 3, 4, 5, 6, 7, 8. Fig. 54.
(M)
From
1.14
170
The Young Gentleman's
29.
To repre-
ſent a Floor
of equal
Squares
feen Cor-
From all theſe Points draw Rays to V;
the first Ray will be VA, the laſt VB.
Then draw the Ray DA, which will cut
thofe that go from V in Points, through
which you muſt draw Parallels to AB
the laſt of theſe Parallels will be EF,
which terminates the Perfpective Square
ABFE. And ſo you will have the Re-
preſentation of the Floor requir'd. And
you may continue it, if you draw ano-
ther Ray through the Point E and the
Point D; and then proceed to work as
afore.
To repreſent ſuch a Floor as is laſt
mention'd, but feen corner-wife, work
thus. Divide the Ground-line AB into
Parts as afore, viz.alternatively equal and
unequal. Then draw thro' the Points of
and with a Division, to the Points of Diſtance, Rays
or Lines; which will (as afore) termi-
nate in the reſpective Square, this allo
being to be defcrib'd as is afore taught.
ner-wife,
Border,
bc.
Fig. 65.
one or
30.
Laſtly, It is obfervable, that if you
An useful defcribe Squares given in a geometrical
Way to re- Plane, then the Ground-line AB muſt be
preſent in
Perfpective divided into Parts equal to the Sides of
the given Squares. And the littlePerfpec-
tive Squares, not only will be the Ap-
pearances of thofe in the geometrical
Plane, but alſo may be very uſeful to re-
preſent in Perspective one or more Fi-
gures made up of feveral Lines; as for
moré Fi-
gures.
Inftance,
Opticks.
171
*
Inſtance, a fortified Polygon. Namely,
the faid Polygon (or Wall of a fortified
Town) being deſcrib'd on the geometri-
cal Plane with the Squares, it will not be
difficult to defcribe the fame likewife in
Perſpective with the Squares of the Pic-
ture, drawing fuch a Part of it in each
Square of the Picture, as is deſcrib'd in
the correſpondent Square of the geome-
trical Plane. All which will be fuffici-
ently illuſtrated by a bare Sight of Fig.
66.
31.
thod.
And in like Manner, Fig. 63, if up-
on AB a Square be drawn in the geome- A like Me
trical Plane, as ABGH, whofe Reprefen-
tation is the perfpective Square ABEF;
and that Square ABGH on the geometri-
cal Plane be divided into other little
Squares by Lines parallel to the Diago-
nals AG and BH, the faid Square ABGH
may be made uſe of, to put or repreſent
in Perſpective ſeveral Things at once, the
Draught of the faid Things being de-
fcrib'd upon the ſaid Square.
33.
clufion
And thus I have taken notice of fo
much of Perſpective, as is agreeable to The Con-
the Defign of this Treatife. And there-
fore ſhall here put an End to this Trea-
tife of Opticks.
FINI S.
ERRATA to Trigonometry.
AGE 7. Line 33. For fit for Use. Read not fit for Use. p. și l. 33. f. to ax.
PAGE
r. — tux p. 19. 1. 5. f.
ΤΟ
ĘD
2
B-D
3. r. --=D. p. 30. 1. 13. f. CA. г.
2
EA. p. 35. 1. 22. f. E=350. г. B=35 degr. p. 42. l. 13. f. CA —–793. t.
CA=793. P. 44. 1. 6. t. Arch (BA. 1. Angle CBA. p. 51. 1. ult. f. Line
BA. 1. Line 85. p. 54. 1. 16 f. Angles meet. r. Angles, 'till they meet. p.
55. 1. 13. f. at M. r. at N. ibid. 1. 19. f. at N. r. at M. p. 56. l. 1. f.
at the End m. r. at the End n. ibid. 1. 4. f. as the End n. r. at the End m.
p. 65. l. 1. f. fo that. I. ſo ſhall. p. 82. 1. 22. f. to 99. г, to 89. '
PA
•
ERRATA to Mechanicks.
AGE 6. Line 14. For contriv'd. Read conceiv'd, p. 16. 1. 21. f. little may.
1. little Body mag. p. 13. 1. 12. f. Laver. T. Leaver. ibid. 1. 26. f.
bust (the. x. but) the. p. 51. 1. 21. f. faid Squares. r. faid Spaces. ibid.
penult. f. is equidiſtant from C to Aa, r. is fix times as diſtant from Cas is As.
p. 17. l. 1. f. to b. v. to B. ibid. 1. 3. f. ¿C. r. BC. ibid. I. 7. £. (this being.
r. this being p. 20. ult. t. G2 1. C2. p, 3º. 1. 10. f. were. r. were FD.
ibid. 1. 11, f, at C r. ar G, ibid. 1. 12. f. at r. at B. p. 31. 1. 12. f. or
Mark. I. or Hok. p. 34. 1. 23. f. as Fig. 1. as Fig. 28, 29, 30. ibid. 1, 24-
£. Pulley) r. Pullen as Fig. 27.) p 35. 1. 19. f. by TS. 1. by FS. p. 47. 1.15.
f. d, e, f, E. By. r. d, c, f. By. p. 5o. 1. 6 f CE. t. CG. p. 56. 1.
14. F
aqua!. t. equal. p. 57. 1. 18, £. EFAB. r. EFHB. p. 65. 1. 17. £. How the
г. Now the.
1.
ERRATA to Optické.
PAGE 87. Line 14. For Fig. 11. Read Fig. 10
p. 88. ult. £. (Eig. ) r.
(Fig. 28.) p. 89.1. 5, 7.17, 25. f. (Fig. ) r. (Fig. 28.) ibid. 1. 14. 27.
f. (Fig. ) r. (Fig. 8) ibid. 1. 16. f. Surface BD. r. BC. p. 102. 1. 11. £.
AH. 1, AF. p. 107. 1. 8. 1. ss look'd to be, x. is look'd on to be. p. 108. 1. 14. f, the
Conver. 1. the Reflex, p. 110. 1. 2. E to the Prop. be requir'd. I. to the Proportion
here required. p. 113. l. 1. f. Theor. II. 1. Theer. IV. ibid. 1. penult. f. And when, r.
And whereas, p. 114. 1.4. f. EDG. 1. ADC. p. 116. l. 5. f. hBF. r. hBf. p. 117.1.
ult. f. 2: lf, r. 4 : lif. p. 122. 1.29. fubdulicate, r. fubuduplicate. p. 125. I, 20. £.
(of the Angle (BiViS). I. of the Angle (CBM). ibid. 1. ante penult t. (as
Fig. ) r. (es Fg. 29) p 126. 1. 1o. f. Fig. 35. r. Fig 35. p. 132. I. 27. f.
3 AB. r. 2 AB. p. 133. l. 28. f. Coneasic, & Corvex, ibid. 1. 29. f. Convet, f.
Concave. ibid. 1. 30. t. Fig. 36. r. F.g. 35, and 36. p. 141. 1 2. f. Bp. r. Bb.
p. 148. in Ma gin, f. Fig. 46. r. Fig. 47. p. 151. 1. 3. E. OF x. OE. P, $56.
1, 8, F. Chayſtalını, t. Chryſtaline.
In Aftronomy, Page 63. Line 15. Read, Three Diameters of the Moon.
In Chronology, Page 33 After thefe Words, (in the last two Lines
of the Text,) viz. and in the Calendar adjoining to this Treatife; add
as follows: Or else the Golden Numbers are call'd the Prime, as denoting
Luna Prima or the Firl Day of the New Moon; according to which way
of speaking the Full Moon is frequently fil'd Luna Quarto-decima, as
falling on the fourteenth Day after the New Moon inclufively.
JUL 7 1920
A
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1716
UNIVERSITY OF MICHIGAN
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