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A COMPLEAT
SYSTEM
O F
ASTRONOMY.
In Two Volumes
CONTAINING
The Deſcription and Ufe of the SECTOR
the Laws of Spheric Geometry; the Projection of
the Sphere Orthographically and Stereographically
upon the Planes of the Meridian, Ecliptic and
Horizon; the Doctrine of the Sphere; and the
Eclipfes of the Sun and Moon for Thirty feven
Years.
Together with
All the PRECEPTS of Calculation.
ALSO
NEW TABLESs of the Motions of the Planets,
Fix'd Stars, and the Firft Satellite of Jupiter; of
Right and Oblique Afcenfions, and of Logiſtical
Logarithmns.
To the whole are Prefix'd,
ASTRONOMICAL DEFINITIONS
for the Benefit of Young Students,
By
241728
CHARLES LEADBETTER,Teacher of
the Mathematicks.
LONDON,
Printed for J. WILCOX,
Britain.
at the Green Dragon in Little
M DCCXXV
XXVIII.
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TO ALL
LOVERS
O F
Mathematical Learning;
This COMPLEAT
SYSTEM
O F
ASTRONOMY
Is, with all Humility, Dedicated
BY
Their moſt Faithful and Obedient Servant,
Charles Leadbetter.
*
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YMONOR
!
THE
PREFACE
W
A
•
STRONOMY is a Science, which
teaches the Method of Examining
and Calculating the Motions, Magni-
tudes, Diſtances, Conjunctions, Eclip-
fes, Apogeons, &c. of the Heavenly
Bodies, by the Aid of Numbers, Geo-
metry, Telleſcopes, Quadrants, and Micrometers.
By this we may walk through the Air and Ether,
and converfe familiarly with the moſt wonderful Parts
of the Creation. Atlas, the Lybian, and King of Mau-
ritania, forfook the Society of Men, and retir'd to the
higheſt Mountain in Africa (which therefore bore
his Name) that he might contemplate upon the Mo-
tions of the Fix'd and Erratick Stars; and is for that
reafon faid to bear up the Heavens with his Shoulders.
He was the Inventer of the Spheres, 1590 Years be-
fore Chrift.!
+
The Poets have feign'd the Moon to be in love
with Endymion, becauſe he fpent his time upon Rocks
rand Mountains (chiefly on Mount Latmos, in ftudying
the Nature of the Moon and Stars, 1445 Years before
Chriſt.
1
ૐ.
We
1
The PREFACE,
1
We are not at all furpriz'd to find fo many great
Men affect this Study (the Names of fome of whom
that have arriv'd to a very great Proficiency, you may
fee in the Preface to my System of the Planets demon-
ftrated, and endeavour the Knowledge of fuch things
as raife the greateſt Admiration in all who are igno-
rant of it.
To ſee a regular Succeffion of Day and Night,
a conſtant Return of Seafons, and fuch an harmoni-
ous Difpofition and Order of Nature, muſt neceffari-
ly be a Noble Contemplation, and agreeable not only
to the Nature of Man, but alſo to the Pofture of
his Body, which is Erect; when other Creatures
(wanting that Muſcle) are made to look downward
upon the Earth.
There have been great Contentions among the
Learned of different Nations about the Origin of
this Study, every one claiming an Intereft in it; as,
the Babylonians, Egyptians, Grecians, Scythians, &c.
But be that as it will, we now enjoy it in a very clear
Light, to the immortal Honour of thofe two
Great Geometricians, the late Sir Ifaac Newton, and
Dr. Edmund Halley, our prefent Aftronomer-Royal.
Upon this Science depend Navigation, Geography
and Dialling; without which 'tis impoffible they
fhould be maintain'd: For, first, the Mariner cannot
conduct a Ship thro' the unbeaten Paths of the
Ocean, without the help of it; but being well
skill'd in Aftronomy, he may, by the Knowledge of
Eclipfes, the Immerfions and Emerfions of Jupiter's
Satellites, and the Times of the Tranfits of the
Moon by the Fixed Stars and Planets, determine the
true Difference of Meridians between London, and the
Meri-
The PREFACE.
A
Meridian where the Ship then is; which reduc'd into
Degrees and Minutes of the Equator, is the true
Longitude found at Sea. And for this reafon I would
adviſe every Man who has the Care of a Ship, or of a
School, that he both well inform himſelf, and alſo thoſe
under his Inftruction, of the following Work, left
he be like a certain Wappmeer, who lately publifh'd a
Book of Navigation, and did not underſtand what he
had tranfcrib'd into it.
Secondly, The Geographer is affifted hereby, in lay-
ing down the Cities, Towns and Countries in their
true Longitude and Latitude.
I aftly, It is by the help of this Science, that the
Dialift is inform'd how to trace out the true Hour of
the Day (in any part of the World,) by the Shadow
of a Gnomon plac'd on a Plane, tho' never fo irregular.
We read of many Perfons, who in this Study have
trod fo near upon the Heels of Nature her felf, and
have div'd into things fo far above the Apprehenfions
of the Vulgar, that they have been believ❜d to be
Necromancers, Magicians, &c. and what they have
done, judg'd to be unlawful, and perform'd by Con-
juration and Witchcraft, altho' the Miſtake lay in the
Peoples Ignorance, and not in the others Studies.
To undertake a Work of this nature, is to launch
into the Ocean of Criticks. However, fince no abler
Pen would undertake this Herculean Task, I have
ventur'd to beſtow this my Twenty Years Study and
Labour among my Country-men, wifhing they may
reap as much Profit, as I have had Pleature in Com-
piling this Work.
But whofoever they be that read Authors, and do
not by their own Senfe abftract true Repreſentations
of the things themſelves comprehended in thoſe Au-
a 2
thors
The PREFACE
Authors Expreffions, they do not reprefent true,
but deceitful Idea's and Phantafms; by which means
they form certain Shadows and Chimera's, and all
their Theory and Contemplation (which they count
Science) fhew nothing but Weakneſs.
?
In the following Work I have added nothing fu-
perfluous, nor omitted any thing that would be of
ufe to the young Aftronomer: For in the firſt place,
I have given you all the Terms of Art us'd in A-
ſtronomy; by which the young Tyro is taught to ſpeak
properly, without any other Guide or Tutor whate-
ver.
The Body of the Work I have divided into Five
Sections The firft contains the Defcription and Uſe
of the Sector; the fecond contains Spheric Geometry;
in the third you have the Projection of the Sphere Or-
thographically and Stereographically, on the Planes of
the Meridian, Ecliptic and Horizon: In the fourth
Section you have the Doctrine of the Sphere, according
to the Apparent Motion of the Sun; wherein I have
given the Problems in the fame Order in which they
ought to be learn'd; with an Appendix of fuch Ta-
bles as I thought neceffary for compleating this
Work. The fifth and laft Section contains Aftrono-
mical Precepts, which fhew the uſe of the Tables in
an eafie and practical Method, being all that is re-
quir'd to be known by them.
Next after the Precepts follow the TABLES them-
felves, which are grounded upon Sir Ifaac Newton's
Radixes, and the Obfervations of Mr. Flameed: For
by comparing as many Obfervations on Eclipfes as I
could procure, I found that all our Aftronomical Ta-
bles were faulty in the times of thofe Eclipfes, giving
the times in the Afcending part of their Orbs too foon,
and in the Deſcending too late; which put me upon
endeavouring to rectifie that Errour.
I
The PREFACE.
I alfo obferving (as Mr. Flamfteed and Dr. Halley
had done before me) that Saturn mov'd too faſt by
the Tables, Jupiter too flow, and Mars too flow in
Aphelion, and too faft in Perihelion; thefe Irregula-
rities you will find rectify'd in the following Tables,
and brought to agree with the Obfervations of this
prefent Age.
I have alfo rectify'd the Præceffion of the Equi-
nox to its true Place, and near 800 of the Fixed Stars,
by my Aftronomical Quadrant; the Limb of which
is divided into Minutes of a Degree. The Method of
making Obfervations I have fhewn in Problems 41,
42,&c. of the Doctrine of the Sphere.
In the 119th Page of my Syftem of the Planets De-
monftrated, I have fhewn you, how the Declinations
of the Fixed Stars increaſe and decreaſe; and that, by
reaſon they move upon the Poles of the Ecliptic,
they are found at different Diſtances from the Vertex
of any Place: As, for Inftance, that Star in the Tail
of the Little Bear, which we call the Polar Star, was
not the Polar Star at the Creation of the World, nei-
ther will it be the Polar Star 13337 Years hence; for
it will then be 8° 56' 49" to the South of the Vertex
of London. This may feem to thoſe who are unac-
quainted with this Study, to be a Falfity: But I af
fure you, there is not any Propofition in Euclid more
demonftrable than this is, as you will find appear, at
the End of the Precepts.
In the Advertiſement publifh'd in Mr. Parker's Ephe-
meris, and the Ladies Diary for the Year 1728, I
promis'd the Tables of the Satellites of Saturn; but
for two reafons I have omitted them. First, Becauſe
they cannot be ſeen but with a very long Tube, and
good Glaffes; and therefore are not to be purchas'd
but at a great Expence. Secondly, They would have
fwell'd the Book too much: So that I have contented
my felf with the First Satellite of Jupiter only, the
Im-
The PREFACE,
Immerſion and Emerfion of which frequently happen,
and may be ſeen with a fmall Telleſcope.
All the Tables I have digefted in a new, plain, and
eafie Method: And to make the Work Compleat, I
have added two ſeveral forts of Logistical Logarithms,
with their Conſtruction and Ufe, having continu'd the
latter to two Hours in Time.
My Rules and Precepts are all plain and eaſie :
For whereas other Authors fall immediately to work
in Calculating an Eclipfe, without telling how to find
it (which is a very improper way of Teaching,) you
will have it otherwife here: For firft, I fhew you how
to find the the Ecliptic Boundaries or Limits; then,
how to find what Number of Eclipfes there will be in
any Year; next, in what Months and Days they fall.
Thus having proceeded gradually, I laft of all, fhew
how to calculate the fame Eclipfe when found, for
any determinate Latitude or Longitude on the Globe,
with their Geometrical Conftruction, the Laws and
Methods of General Eclipfes, the Tranfits, Occulta-
tions, &c.
It may be expected, that I fhould have given a
Multitude of Obfervations (as is cuſtomary in Works
of this nature) of the Places of all the Planets, E-
clipfes, &c. But I have purpoſely omitted them, well
knowing, that Authors have often made the Obferva-
tions and their Tables to agree, on purpoſe to fet the
better Gloſs upon them. I chufe rather to leave the
Tryal to their own Obfervations, than to trouble
them with any doubtful or unneceffary Truths.
And as I have now given the World a Compleat
Body, or Sistem of Aftronomy, in Two Parts; (the firſt
by Inftrument, or the true Syftem of the Planets De-
monftrated, a Book in Quarto, publish'd the laſt Year ;)
fo in this Treatife you have all the moft nice and
exact Rules of Calculation: Which two Books and
Inftruments I advife every Student to purchaſe, and
which
The PREFACE.
which with due Application, will make him a com-
pleat Aftronomer.
And farther, I recommend to the Reader's Perufal,
my Sheet of Eclipfes, lately publifh'd; in which are the
Types of all the Eclipfes that will be Vifible for 35 Years;
only you are to underſtand, that the two Total Types
of the Moon for the Year 1754, will not be Vifible
at london, but in America only, as you may fee, if
you compare them with the Table, Page 339, of this
Book.
In the Appendix to the Doctrine of the Sphere I have
given you fome very uſeful Tables, viz. the Latitudes
and Difference of Meridians of fome of the moſt Emi-
nent Cities and Towns, with the true times of the
Sun's Rifing and Setting, in Hours, Minutes and Se-
conds, to every Degree of the Sun's Declination North
and South, for thirty fix of the moſt noted Cities in
the World. You have likewife at one View, all the
Eclipfes that will happen for thirty feven Years to
come, under the Meridian of London; which are of
ufe in determining the Difference of Meridians between
any other Place and that of London: As, fuppofe in a
Ship at Sea, &c. you fhould obferve the Middle Time
of the Moon's Eclipfe, Febr. 2, 1729, to be at 6 a-clock
at Night: Look into this Table, and you will find the
Middle Time to be at 8 h. 44'; the Difference of Me-
ridians being 2 h. 44', which in Degrees is 41°; fo
that the Ship is then fo much to the Weft of the Meri-
dian of London.
THE
CONTORT ST
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THE
CONTENTS
A
'Stronomical Definitions, from Pagė 1, to
Page 52
SECTION I
Defcription of the Sector,
Use of the Sector,
Of the Line of Chords,
53
54
55
SG
A
Of the Line of Sines,
To lay down an Angle by the Chords, Sines, Tangents and
Secants,
To find the Verfed Sine of an Arch,
To find the Secant of an Arch,
57
58
ibid.
58
A Demostration to prove that Radius is a Mean Proportional
between the Tangent of an Arch and the Tangent Complement
of the fame,
Spheric Geometry,
SECTION IL
To find the Pole of a Great Circle,
To find the Pole of an Oblique Circle,
To lay down any Angle required,
To measure any Spheric Angle when Projected,
61
62
63
64
65
67
Page 68
70
To measure the Quantity of any Arch of a Great Circle,
To draw a Parallel. Circle,
To draw a Great Circle thro' any Point, making with the Primtive
any given Angle,
To draw a Great Circle thro' any two given Points within the
Periphery of the Primtive,
71
To draw a Great Circle Perpendicular to a given Great
Circle,
:
b
72
To
The CONTENTS.
To draw an Oblique Circle perpendicular to a Right Circle
given,
..
73
To draw an Oblique Circle perpendicular to a given Oblique
Circle,
SECTION 3.
113
74
To project the Sphere Orthographically on the Plane of the Me-
ridian,
75
To project the Sphere Stereographically upon the Plane of the
Meridian,
...
76
The Stereographick Projection of the Sphere on the plane of
the Ecliptic.
80
The Stereographick Projection on the plane of the Hori-
zon,
Direction for making an Horizorital Dial,
182
*86
SECTION 4.
The Doctrine of the Sphere,
88
Of the Obliquity of the Ecliptic,
ibid.
89
90
91
To find the Sun's Declination,
To find the Sun's Longitude or Place in the Ecliptic,
To find the Sun's Right Afcenfion,
To find the Sun's Amplitude,
92
To find the Afcenfional Difference, the true time of the Sun's
Rifing and Setting,
93
To find the Oblique Afcenfion and Oblique Defcenfion," 95
To find the Oblique Afcenfion and Oblique Defcenfion of the
Sun, Moon or Star,
:96
To find the Beginning, Duration and End of the longest Day, and
Night in any Latitude,
To find the Apparent Time of the Sun's Setting,
To find the time when the Sun will be due East and West,
To find the Sun's Azimuth when due East and Weft,
To find the Sun's Azimuth at the Hour of Six,
To find the Sun's Altitude at the Hour of Six,
98
100
тоз
105
106
108
To find the Sun's Altitude at any Hour when he is in the Equi-
noctial,
The Sun's Azimuth given, to find the Altitude,
109
110
To find the Sun's Altitude at any Hour, and in any Latitude pro-
pofed,
By the Latitude of the Place, Sun's-Declination and Altitude, to
find the Hour of the Day,
To find the Sun's Azimuth at any Time and Place,
115
122
To
1
The
CONTENT´S.
}
To find the Beginning and Ending of the Twilight,
To find when the forteft Twilight happens,
To find the true Declination of a Star or Planet,
To find the Right Afcenfions of the Planets, &c
128
13 1
132
138
To find the Right Afcenfion of the Planets, by having only the
Longitude and Latitude given,
140
By the Declination and Right Afcenfion of a Star, to find its
Longitude and Latitude,
To find the Sun's Declination when he rises and fets at any Hour
and Place,
A Table of the Declinations and Right Afcenfions of 42 fixed
Stars,
141
144
146
By the Sun's Azimuth, to find the Declination.
147
A Table of the Sun's Declination to every 5th Day for infcribing
the Month in Gunter's Quadrant,
149
A Table of the Sun's Meridian Altitude at London
150
A Table of the Sun's Altitude at every Hour in the Equator 151
A Table of the Afcenfional Difference to every Degree of the
Sun's Declination,
To draw the Azimuth in the Quadrant,
153
154
For the Sun's Altitude on any Azimuth, from 155 to 160
To find the Right Afcenfion of the Mid-Heaven,
161
To find the Medium Coeli in the Ecliptic, commonly called the
Culp of the Tenth,
162
To find the Meridian Angle,
163
To find the Declination of the Culminating Point
164
To find Altitude of the Mid-Heaven,
To find the Altitude of the Nonagefime Degree,
165
166
#
To find the Nonagefime Degree,
To find the Cusp of the Afcendant,
167
168
To find the Parallactick Angle,
172
To find the Parallactic Angle another way,
174
To find the Sun's Parallax in Longitude and Latitude
Of the Parallax of the Sun, Moon and Stars,
175
176
to page 183
Horizontal
Parallaxes, from 184 to
Shewing the Several Methods fed by Aftronomers for obtain-
ing the
How to make Cœleftial Obfervations,
To obferve the true Place of the Sun,
188
ibid
189
How to cbferve the true Places of the Planets from 191 to 211
By the Declination of the Planets to find the Longitude, 212
How to Erect a Cœleftial Scheme by the Doctrine of
Oblique - Angled - Spherc - Triangles, from 214, to
227
b 2
A
The CONTENTS.
A more Expeditious Way of Èrecting a Cœleftial Scheme, ibid.
To calculate Hour-Lines upon all Sorts of Dials that have Cen-
ters,
234
To find the true and apparent Times of the Southing of Fixed
Stars and Planets,
To find the time of the Rifing of the Stars and Planets,
243
248
To find the true times of the Setting of the Stars and
Planets,
To find the Cofmical Rifing of the Stars,
258
266'
To find the time of the Cofmical Setting of the Heavenly
Bodies.
268
To find the times of the Achronical Rifing of the Stars and
Planets,
272
The times of the Achronical Setting,
273
To find the true time of the Heliacal Rifing of the Stars and
Planets.
276
278
To find the time of the Heliacal Setting,
To find when a given Star or Planet will be in the Nonagefime
Degree,
280
291
To find the Logarithm of a whole Number confifting of 5, 6,
or 7 Places, and of a Fraction, &c. from 283 to
A Logarithm given, to find the Abfolute Number,
To draw a Tangent to a given Circle,
The uſe of the Appendix to the Doctrine of the Sphere,
ibid.
293
297
The use of Shakerley's Logistical Logarithms,
298
The use of Street's Logistical Logarithms,
300
A Table of Golden Number and Epacts,
305
A Table of Dominical Letters,
306
A Table of the Number of Direction,
307
A Table of the Movable Feaft and Terms,
308, 309,
310
A Table of Week-Days and Months-Days,
311
A Table of Number of Days from any one Day to another,
312
A Table fhewing the Roman Eafter
317
A Catalogue of Cities and Towns
315
the World,
Tables of the Sun's Rifing and Sitting at all the chief Cities in
from 318 to 335
A Table of all the Eclipfes of the Sun and Moon for 37 years,336
A Table of Break of Day,
A Table of the End of Twilight,
341
343
SECTION
1
The CONTENT S.
}
SECTION 5.
Aftronomical Precepts
How to Reduce any Meridian to that of London
337
How to find the Equation of Time,
339
How to compute the true Longitude of the Fixed Stars,
How to calculate the true Place of the Sun,
341
ibid
344
349
How to calculate the Sun's Ingrefs into any of the 12 Signs,
How to calculate the true Place of the Moon,
How to find the true time of the Conjunction or Oppofition
of the Sun and Moon,
353
How to calculate the true Heliocentric, and Geocentric place
of the five primary Planets, with an Example in each
Planet,
from Page 357 to 364
365
To find the Aphelions and Perihelions of the Planets,
How to find the Apogeon, and Perigeon of the Sun and
Moon,
369
How to find the Retrogradations of the Planets,
How to find the Mutual and Lunar Afpects,
372
375
How to determine the Ecliptic Boundaries of the Sun and
Moon,
378
How to find, in any year, how many Eclipfes there will be, and in
what Month they happen,
381
How to calculate an Eclipfe of the Moon,
386
How to delineate the Eclipfe of the Moon,
394
How to construct an Eclipfe of the Moon Geometrically,
395
How to calculate an Eclipfe of the Sun, from Page 398 to 408
How to delineate a Solar Eclipfe,
ibid.
How to calculate the times of a General Eclipfe of the Sun,
from Page 410 to 418
419
How to construct the Sun's Eclipſe Geometrically,
How to calculate the Tranfits of Venus and Mercury over the
Sun,
425
The Calculation of the Immersion and Emerfions of the firſt
Satellite of Jupiter,
433
How to find the true Hour of the Night by the shadow of the
Moon on a Sun-Dial,
436
A Demonstration that our Pole, was not the Artick Pole at the
Creation, &c.
437
Tables of Sirius's Rifing, Southing, and Setting, to every Day in
the Year at London,
443
The
The CONTENT S.
The Aftronomical Tables.
A Table of the Equation of Time,
A Table of the Præceffion of the Equinox,
Table of the Sun's mean Motions
ī
Page 2,3
4
from 8 to 30
'from 31 to 62
64
The Hourly Motion of the Moon from the Sun
65
56
67
Tables of the Moon's mean Motions
Table of the Parallax and Refractions,
Table for converting Hours and Minutes into Dgrees,
A Table fhewing the Lunar Afpects by Inspection,
4 Table of the Inclination of the Moon's Orb with the
Equinoctial,
Table of Declination and Meridian Angle,
Table of the Nonagefime Degree
71
73
from 75 to 80
Table of the Angle of the true Motion of the Moon from
the Sun that it makes with the Ecliptic,
A Table fhewing when the Sun, Moon or
Nonagefime Degree
The mean Motion of Mercury,
The Tables of Venus,
The Tables of Mars,
The Tables of Jupiter,
The Tables of Saturn,
A Catalogue of Fixed Stars,
1
81
Star will be in thè
82 to 87
3
from 88 to III
from page 112 to 135
from page 136 to 159
•· from page 160 to 183
from page 184 to 207
from 209 to 243
A Table of the mean Conjunction of the first Satellite of
Jupiter,
A Table of Right Afcenfions in time
and South Latitude
A Table of Oblique Afcenfions and
time, to fix Degrees of North
Shakerley's Logistical Logarithms
Street's Logistical Logarithms
from page 244 to 261
to Six Degrees of North
from page 262 to 285
Oblique Defcenfions in
and South Latitude,
from page 286 to 309
from 310 to 351
from 352 to 376
from 372 to
the End.
Logiſtical Logarithms continu'd by the Author,`
!
ERRATA.
J
1
r.
ERRAT A.
r.
AGE 8, Line 23, Read 0.231655; P. 33, 1. 8, r. Coſmical;
PAGE
p. 34, 1. 14, r. 50"; p. 48, 1. 1Ì,
dele the; P. 55, 1. 10, f
this extant; p. 64, 1. 3, dele of ; . P. 79, 1. 15, г. 66° 31′; P.
80, 1. 37, for in r. is; p. 97, 1. 8, r. 1' 7"; p. 98, 1. 2, r. 56°;
p. 104, 1. 12, г. 690 48; p. 120, 1. 9, r. Sine; p. 123, 1. 9, r.
Latitude; p. 125, 1. 16, r. half = 320 30'; p. 130, 1.6, r. X;
p. 144, 1. 15, r. 2 h. 50 27"; P. 144, l. 17, r. 3′h. 31′ 20″;
p.145,1.5, under time r.H; p.146, 1.ult. r.23′; p. 147, againſt 11,
r.24; P.153, 1.29, r.23°; p. 164, 1.16, r. To Sine R. A; p. 188, 149,
r. through; p.207, 1. ult. r. Solftitial Colure; p.251, 1. 20, r. Eftimate
Time; p. 269, 1.5, r. 85° 32'; P. 315, 1. 39, r. 53° 22′; p. 360,
1. ult. r. when Anomaly Commutation is less than 6 Signs, they are Oc-
cident if more than 6 Signs, Orient; p. 364, 1. 6. r. Novemb. 6;
P: 367, 1. 12, r. 1728; p. 370, 1. 21, r. one hour; P. 371,
1. 2, г.
42′ 17″; p. 389, 1. 10, r. is to fix Digits; P. 420, I. 16, r. VS 710
5′ 22″; p. 421, in the Scheme for Ir. H; P. 432, 1. ult. for
New, r. PRECEPT 20.
In the TABLES.
PAGE 3, at the foor of the Table write 11, 10, 9, 8, 7, 6
Signs; p. 26. against 23 hours, r. 56′ in both Columns;
P. 42, againſt 18 Days, r. 8° 56′ 58" under Node; p. 50, againſt 33
under Anomaly, r. 17° 57′52″; p. 110, under Inclination, for 8"
p. 158, for Sign 1 S. r. N; p. 64, Refraction at 33° Alt.
is 114, and at 43° Refract. is o' 52", and
r.o
2″,
#
}
55
66
Alt
34
29
70Refraction is 19,
01 180 1
90
p. 286, 1. Fa F. 51° 34′.
9
In the Calculations of the Places of the three Superior Planets Saturn, Jupi-
ter, and Mars, the Mean Motions differ a few Seconds from the Tables, by
reaſon I had finish'd the Calculations, before the Tables, that the pre might
not wait,
4
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at
A
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Stronomy: Or, the true Syftem of the Planets demonftra-
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the times when Venus, and Mercury may be feen in the Sun's Disk,
&c. with proper Cuts to each Planet: By which any Perfon
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not only fhewn, but plainly and fuccinctly demonftrated to the
Meanest Capacity, by Short and Eafy Rules and New Aftro-
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2. Magnum in Parvo; or the Marrow of Architecture, fhewing
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A
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GEO
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at the Hercules and Globe, next Door to the Fountain-Tavern, in
the Strand, London. Alfo Mathematical Books fold.
1
TH
Aftronomical Definitions.
A
A.
Chronical, Rifing and Setting of the Stars,
is, when they rife in the Evening in the
Eaſtern Horizon, as the Sun fets; and ſet:
in the Evening in the Weſtern Horizon with
the Sun. [See the Doctrine of the Sphere]
Ara's, are certain Periods of Time whence
Chronologers begin to compute; and the
moft eminent Era's among them are,
The Era of the World's Creation, which reckons 3949
Years before the Birth of Chrift, and which, according
to the Julian Account, began the 24th Day of October.
The Jewish Ara begins in Autumn, about the Year of
Chrift 344.
The Era from the Deftruction of Troy begins June 16.
The Era of Nabonaſſar begins Feb. 26, before Chrift 147%
The Era of the Olympiads begins from the New Moon in
the Summer-Solftice, 777 Years before Chrift.
The Era of Iphitus is only a Collection of the Olympick
Years; theſe two are the Era's chiefly uſed by the Greek
Hiftorians.
The Roman Era from the Building of the City, begins April
21, and is 752 Years before Chriſt.
The Chriftian Era from the Birth of Chrift, begins Decem-
ber 25.
B
The
•
2
Aftronomical DEFINITIONS.
The Turkiſh Æra, or the Hagira, began the 16th Day of Ju-
ly, Anno Dom. 623.
The Era of the Death of Alexander the Great, is the 12th
of November, 324 Years before Chrift.
The Julian Era takes it name from Julius Cæfar's Reforma-
tion of the Calender, which was done 45 Years before
Chrift, in the 708th Year from the Building of Rome, and
in the 731st Olympiad.
The Ethiopick, Abyffyne, or, as fome call it, the Dioclefian Æra,
others the Era of the Martyrs, becauſe it bore date with
a very fevere Perfecution; this Era began Auguſt 29,
A. D. 284, and in the firft Year of the Emperor Dioclefian.
'Tis uſed by the Ægyptians, and Abyſſynes.
The Perfick, or Jefdegerdick Era, takes its Date, either
from the Coronation of the laft Perfian King Fefdegeris,
or from his being Conquered by Ottoman the Saracen,
which was June 16, A. D. 632.
The Gregorian Era takes its Name from Pope Gregory the
XIII. and began in October, Anno Domini 1582, and from
the Reformation of the Calender.
Almicantarabs, fo called by the Arabians, are Circles of
Altitude parallel to the Horizon, (in any of the three Po-
fitions of the Heavens,) and you may imagine as many as
there are Points between the Horizon and Zenith.
Altitude, is the height of the Sun, Moon or Stars a-
bove the Horizon, in an Azimuth-Circle, and is counted in
Degrees, Minutes, and Seconds.
Almanack, an Arabick Word, and fignifies Diftribution,
or Numeration; whence our Annual Books wherein the
Days of the Month, Feftivals, Lunations, Motions of the
Heavenly Bodies, Eclipfes, &c. being fet down, are ſo cal-
led.
Amplitude, is an Arch of the Horizon, contained between
the Rifing and Setting of the Sun, Moon, or any Star, and the
Eaft and Weft Points of the Horizon, and numbred in De-
grees and Minutes, and is always of the fame Name with
the Declination of the Sun, Moon, or Star, which how to
find ſhall be fhewed in the Doctrine of the Sphere.
Analemma, is a Projection of the Sphere on the Plane
of the Meridian, Orthographically made by ftreight Lines
and Ellipfes, the Eye being fuppofed at an infinite Di-
ftance, and in the Eaft and Weft Points of the Horizon.
Ana-
Aftronomical DEFINITIONS.
3
Analogy, is much the fame with Proportion, and is often
ufed for that Word.
Angle, is made by the meeting of two Lines in a Point, and
may be of any quantity lefs than 180°; it is frequently
made ufe of by Aftronomers in theſe particulars, viz. An-
gle of Incidence in a Solar Eclipfe is formed by a Line
drawn from the Center of the Penumbra at the beginning
of the Eclipfe to the Center of the Earth's Disk; and
this is called the firſt Angle of Incidence. The fecond is
formed by a Line drawn from the Center of the Penumbra
to the Center of the Disk at the beginning of the Central
Eclipfe; that is, when the Center of the Penumbra firſt
touches the Earth's Disk. And the third Angle of Incidence
happens when all the Penumbra falls within the Disk, and it
is formed by a Line drawn from the Center of the Disk to
the Perimeter of the Disk in that Point where the Peri-
meter of the Penumbra laft toucheth it in its total Obſcu-
rity within the Disk. This third Angle can only happen
when the true Latitude of the Moon at the true time
of the Conjunction of the Sun and Moon is less than
the difference of the Semidiameter of the Penumbra and
the Earth's Disk.
Angle, of Incidence in the Moon's Eclipfe, is formed
by a Line drawn from the Center of the Moon, touching
the Axis of the Moon's Orb in the Center of the Earth's
Shadow, at the times of the Beginning or Ending of the E-
clipfe.
Angle, of the Sun's Pofition, is made by an Azimuth-
Circle, and the Meridian in the Zenith, the fame Azi-
muth being continued or ſuppoſed to pafs thro' the Center of
the Sun.
Angle Parallactick, is made by the Interfection of a
Vertical Circle with the Ecliptic thro' the Body of the
Sun, Moon or Star. This is of fingular ufe in the Compu-
tation of Solar Eclipſes.
Angle, of Inclination of the Earth's Axis, to the Axis of
the Ecliptic is 23° 294, and remains inviolably in all Points
of the Earth's Annual Orbit. This Quantity is called
by the Copernicans, The Earth's Reflection; but by the
Ptolemaicks, The Sun's greateſt Declination, or Obliquity of
the Ecliptic.
Angle, of Evection, is a fecond Inequality in the Motion
of the Moon, by which at her Quarters fhe is not in
that Line which paffes thro' the Center of the Earth and
B 2
Sun
4
Aftronomical DEFINITIONS.
Sun, as the is at her Conjunction and Oppofition. This Angle
in the Quadratures is about 2 Deg. 37'
Angle, of Reflection, is a third Inequality of the Motion
of the Moon, and arifes from her Apogeon, being chan-
ged as her Syftem is carried round the Sun by the
Earth's Motion. [See my Aftronomy, or Syftem of the
Planets demonftrated.] This Angle is greateſt, when the
Moon is 45 Degrees diftant from the Conjunction, Square
or Oppofition of the Sun, before, and after him; and in
Quantity is then 37′ 33″.
Angle of Ecliptic and Horizon, is the fame with the Alti-
tude of the Nonagefime Degree, and is of great uſe in
the Calculation of Solar Eclipfes, &c.
Angle of Direction, in the New Aftronomy, is made by
the meeting of the Axis of the Moon's Orb with the
Axis of the Globe in a Point, and that Point is always
at the Center of the Earth's Disk. It is of great ufe in
the Geometrical Conftruction of Solar Eclipſes; and if the
Sun be in Cancer, Leo, Virgo, Libra, Scorpio, or Sagittary,
the Earth's Axis lies to the right hand of the Axis of the
Ecliptic in the Projection: But if the Sun be in the oppo-
fite Signs at the time of the Eclipfe, viz. in Capricorn, Aqua-
rius, Pifces, Aries, Taurus, Gemini, then the faid Axis lies to
the left hand.
Annus Magnus, Or the great Year, contains 25920 Years,
this being that ſpace of time the fixed Stars are in perfor-
ming one entire Revolution at 50" per Year.
Anomaly, in Aftronomy, is the Diftance of a Planet in
Signs, Degrees, Minutes, and Seconds from the Aphelial
Point.
Antarick Pole, is the South Pole of the World, being
the fuppofed Center of the Earth's Axis, and is diame-
trically oppofite to the Artic Pole.
Antartick Circle, is a leffer Circle of the Sphere, and diſtant
from its Pole 23º 29'.
Antipodes, are thofe People that walk Feet againſt Feet,
fo as a right Line being drawn from the one to the o-
ther, fhall pals directly thro' the Center of the Earth :
Hence it follows, that the quantity of their Seaſons are the
fame with ours; only when it is Summer to the one it's Win-
ter to the other. The Antipodes of London fall on the Globe
in the unknown Southern Parts of the World.
Aphe-
Aftronomical DEFINITIONS.
5
I
Aphelion is that Point in our Syftem, in which a Planet is
at its greateft diftance from the Sun, and moves flow-
eft.
Apogeon, is when a Planet is at its greateſt Diſtance
from the Earth; the Moon in this Pofition moves flow-
eft.
Apparent, Conjunction, or Place of a Planet, is when
the Right Line that is fuppofed to be drawn thro' the
Center of the Planet, doth not pafs thro' the Center of
the Earth, the Caufe of which is the Parallax.
Apfis, fignifies the two Ends of the Tranſverſe Diameter,
of the Ellipfis in which the Planets move, and denotes
as well the Apogeon as Perigeon.
Argument of Latitude, is the diſtance of the Moon from the
North Nöde, in Signs, Degrees, Minures and Seconds. It is
upon this, that the Latitude of the Moon and Eclipſes of the
Luminaries depend.
Aries, a Conftellation of 21 Stars, lying in the Zodiack
in the figure of a Ram, and is the firft Sign, marked thus
r.
Artick Pole, the North Pole of the World; taking its
Name from Artos, the Bear, a Conftellation in the Northern
Part of the Heavens.
Artick Circle, is drawn 23° 29' from the Pole, and parallel
to the Equator.
Afcenfional Difference, is the difference between the Right
and Oblique Afcenfion, or Defcenfion; or, it is the Space
of Time the Sun rifeth or ferreth before or after Six a-
Clock; which is an Arch of the Equinoctial, meaſured
between that Point of it which rifeth with the Sun, Moon.
or Star, and that Part of it which comes to the Meridian with
them.
Afcendant, is the Eaſtern part of the Horizon. When
the Sun, Moon or Stars, are rifing, they are ſaid to be on the
Cufp of the Afcendant; it is alfo called the Eaftern Fini-
tor.
Afpet, from the Latin Afpicio, to behold, is a Correſpondence
or Familiarity of two Planets mutually beholding each other
with fome Ray Harmonically confidered; or when they are
pofited at fuch a certain diftance in the Zodiack, wherein they
mutually help or afflict one another. Of thefe Afpects-pro-
perly there are but four old ones, and eight new ones; to
which is added a Conjunction, tho' improperly called an
Afpect. Kepler defines an Afpect thus; That it is an
B 3
Angle
6
Aftronomical DEFINITIONS.
Angle formed on the Earth by the luminous Rays of two
Planets, efficacious to the ftirring up of Nature; for when
two Planets are joyned with, or be held of each other, they
feminate or breed fomething in fublunary Bodies according
to their own nature.
Afterifm, from After a Star, is the fame with Conftellation,
or Parcel of fixt Stars, fuppoſed to reprefent fome one Image
or Figure defign'd on purpoſe to diftinguiſh one Star from
another.
Aftronomy, from After, a Star, and Nomos a Law, is a Science
by which we are taught the Motions, Magnitudes and Di-
ſtances, and whatever belongs to the Knowledge of the Hea-
venly Bodies.
Atmosphere, is the lowest part of the Region of the Air,
with which our Earth is Compaffed all round. Its Height is
47.12 Miles, as I have Demonftrnted in my System of the
Planets.
Auge, the fame with Aphelion, which fee.
Aurora, the Morning-Twilight which begins to appear
when the Sun approacheth within 180 of the Eaſtern Horizon;
and this is always equal to the Time between the time of
Sun-fetting and the end of Twilight in the Evening.
Auftral, of, or belonging to the South; alfo Libra, Scorpio,
Sagittary, Capricorn, Aquarius, Fifces, are Auftral Signs; becauſe
they lye on the South fide of the Equinoctial.
Autumn, the third Quarter of the Year, beginning when the
Sun enters the Sign Libra, which is in this Age on September
12, or 13, bringing Harveft or Fall of the Leaf, and making
equal Day and Night,
Aux, the fame with Apogeon.
Axiome, is a Principle in any Science, fo evident, that it
need nothing but the Light of Reaſon to Demonftrate it.
Axis, of the World, is an Imaginary Line, conceived to
pafs through the Center of the Earth, from one Pole to ano-
ther.
Azimuths, or Vertical Circles, are great Circles interfecting
each other in the Zenith and Nadir (as Meridians or Hour-
Circles do in the Poles) and cutting the Horizon at Right
Angles. The Sun's Azimuth is of great ufe in Dialling and
in Navigation, which how to find, I fhall teach in the
Doctrine of the Sphere, following.
Be
Aftronomical DEFINITIONS.
7
B.
Biquintile Afpect, is a new Afpect obferved by John Kepler;
it contains parts of the whole Circle 4s. 24°.
Biffextile, the fame as our Leap-Year; and the reaſon
of the Name, is, becauſe in every fourth Year, they accounted
the 6th Day of the Calends of March twice; For once in four
Years the odd fix Hours above 365 Days made up juft a
whole Day; and that they place next after the 24th Day of
February, which cauſeth that Month in Leap-Year to have 366
Days; which must be carefully obferved in the Calculation of
the Planets Places.
Boreal, of, or belonging to the North: So the Boreal Signs
are Aries, Taurus, Gemini, Cancer, Leo, Virgo; becauſe they
lye on the North Side the Equinoctial.
CA
C.
Ancer, the Crab: In Calculating the Planets Places it is in
Number 3;
3; or the beginning of it is the third Sign Com-
pleat, and is thus marked; It is a Cardinal and Tropical
Sign, unto which when the Sun comes, which is about the 10th
Day of June, he makes longeſt Day and ſhorteſt Night to all
the Northern Inhabitants; and he has then the greateft
Declination, Amplitude and Altitude.
Calender, is the fame with Almanack; which ſee.
Canis Major, and Canis Minor; Two Conftellations; one
in the North, and the other in the Southern Hemisphere,
which rife with the Sun from about July the 9th to Au-
guft the 29th, and gives occafion to that time which is ge-
nerally very hot and fultry, to be called the Canicular, or
Dog-Days. See the Catalogue of the Fixed Stars.
Capricorn, the Goat, marked thus, VS, is the 9th Com-.
pleat Sign in Aftronomical Calculations: It is a Car-
dinal, and the South Tropical Sign, unto which when
the Sun comes, which is about the 10th of December,
he makes the ſhorteſt Day and longeft Night, to all
that inhabit this fide the Equator. The Sun has then
the greateſt South Declination and Amplitude, but the
leaft Meridian Altitude.
B 4
Car-
Aftronomical DEFINITIONS.
Cardinal Signs, are thefe four; Aries V, Cancer S, Li-
bra, Capricorn VS.
Colures, are two great Circles which interfect one ano-
ther at right Angles in the Poles of the World, and di-
vide the Zodiack into four equal Parts, and denote the
four Seaſons of the Year; that paffing thro' Cancer and
Capricorn is called the Solftitial Colure; and the other
that paffes thro' Aries and Libra, is called the Equinoctial
Colure.
Combuft, is when a Planet is within 8° 30' of the Sun,
either before or after him.
Commutation, is the Angle at the Sun, made by two
Lines, one drawn from the Earth, and the other
from a Primary Planet meeting in the Sun's Cen-
ter.
Complement, a filling up of any Arch or Angle, is
what that Arch or Angle wants of 90 Degrees, or that Part
by which it exceeds 90 Degrees, to make it up 180 Degrees.
Complement Arithmetical, marked Co. Ar. is what any
Logarithm wants of 9, and the Unites place taken from
10; thus the Logarithm 9.768345 being given, its Co. Ar.
is 0.231655 and fo of any other Logarithm, Sine, Tan-
gent or Secant, &c. it is of good ufe when the Radius comes
not in the Analogy, as you will find often in the Doctrine
of the Sphere.
Conjunction, (true,) as when the Sun and Moon (or any
other Planet) are exactly in one Degree and Minute of
the fame Sign, fo as
as a Line fuppofed to be drawn
thro' their Center will alfo país thro' the Earth's Cen-
ter.
Apparent is when their Centers lye in a right
I ne with the Eye of the Obſerver.
Constellation, fee Afterifm.
Conftruction, is the drawing of Lines, and forming of
Figures, or preparing the Propofition for a Demonftra-
tion.
Corollary, is a Confequent Truth, gained from fome pre-
ceding Demonftration.
Cofmical. Stars are faid to rife Cofmically, when they
rife in the Morning with the Sun; and to fer Colmically,
when they fet as the Sun rifeth. In the Doctrine of the
Sphere, I have fhewed how to Calculate the Colmical' Ri-
fing and Setting of any of the Heavenly Bodies.
Cul-
Aftronomical DEFINITIONS.
9
Culminating, or Culmen Cali, the higheft Point in Heaven,'
and
that any Planet or Star can rife to in any Latitude
when a Star comes to the Meridian of any place, 'tis laid to
Culminate Alfo the Southing of the Moon and Stars
are taken for the fame thing.
Cycle of the Moon, is a Revolution of 19 Years, in which
Time the Conjunctions and Lunar Aſpects are nearly the
fame they were 19 Years before. It is alſo called the
Prime, or Golden Number.
Cycle of Indiction, is a Revolution of 15 Years, of uſe
only at Rome. This has nothing to do with the Heaven-
ly Motion, being eſtabliſhed by Conftantine, Anno Domini
312, September 24, who fubftituted them in the room of
the Olympiads. They are fo called, becauſe they deno-
ted the Year that Tribute was to be paid.
Cycle of Eafter, is a Revolution of 532 Years found by
the Multiplication of the Solar Cycle 28, by the Lunar 19:
For in that time do the Holy Feaft of Eafter, and all
things depending thereon, return to the fame ftate again.
D.
DAY Natural, determined by the Sun's Motion (according
to appearance) round the Earth in near 24 Hours, tho'
really it is the Earth round her own Axis from the Weft to
the Eaft in near that Time: It is alſo called Civil, becauſe
it is sy divers Nations reckoned divers ways. The Ba-
bylonians begin to account their Day from the Sun-rifing;
as likewife do the Inhabitants of Nuremburg in Germany; the
Athenians, Jews and Italians from Sun fetting; the Egyp
tians and English at Midnight: But Aftronomers begin the
Day at Noon; to which tine are the places of all the Pla-
nets fupputated in our Ephemerides.
Day Artificial, is the time between the Sun's rifing and
fetting; to which is oppofed Night, which is the Time that
the Sun is under the Horizon.
Declination of the Sun, Moon, and Stars, are, their di
ſtance from the Equinoctial, reckoned on a Circle of Longi
tude in Degrees and Minutes between the Star and Equino-
tial, and is either North or South: The Sun's greateſt De-
clination is 23° 29', the Moon's 28° 45' 20"; but the Moon's
is not always fo much when he is in Cancer and Capricorn;
but only when her Nodes are in Aries and Libra; for
which
10
Aftronomical DEFINITIONS.
which purpoſe I have given you a Table of the Inclination
of the Moon's Orb with the Equinoctial.
Definition, is the unfolding, or explicating of the Nature
and Affection of a rhing.
Degree, is the 360th Part of a Circle, or the 30th Part of
a Sign.
Demonftration, is the proving of a thing by Definition and
Axiome; and fo from feveral Arguments drawing a Con-
clufion, that it has that Affection the Propofition did Af-
fert.
Defcenfion, of the Heavenly Bodies, is their going down,
or ſetting in the Weſtern Horizon.
Dexter Afpect, is made contrary to the fucceffion of the
Sign, as from V to . . &c.
Z.
Digit, proper y a Fager's Breadth; but in Aftronomy,
it is the Twelth part of the 'un's Diameter, made uſe
of, in Eclipfes; but the Moon's Digits may amount to a-
bout 23.
All above 12 fhew how far the fhadow of the
Earth is over the ſhadow of the Moon.
Direct; a Planet is faid to move Direct, when it moves
from Aries to Taurus, &c. The Sun and Moon are always
fo; but the Primary Planets are fometimes Retrograde.
at the Earth; for a demonftr.tion of which fee my A-
Stronomy, or Syftem of the Planets demonftrated.
Disk of the Sun and Moon, are their round Phaſes or
Faces, which at their great diſtance from us appear to us
plain, flat, or like Diſhes.
Disk of the Earth, is the difference between the Horizon-
tal Parallax of the Sun 10', and the Horizontal Parallax of the
Moon (which is different at different Times) which is de-
monftrated by the Diagram of Hipparchus, and uſed in the
Geometrical Conftruction of Solar Eclipfes, as I fhall fhew
at large in the following Sheets.
Diurnal, of or belonging to the Day. The Diurnal Moti-
ons of the Planets in Longitude and Latitude are what they
move from the Noon of one Day, to the Noon of the next
Day, which Quantities you may fee in the following Aftrono-
mical Tables.
Dog-Days. See Canis Major.
Dominical Letter; one of the firft feven Letters of the
'Alphabet, wherewith the Sundays are marked in the Almanacks.
with a Red Letter throughout the Year. In my Syftem of the
Planets demonftrated I have given New Numbers which fupply
the uſe of the Dominical Letters,) for finding what Day of the
Week any Day of the Month is for ever.
Dra
Aftronomical DEFINITIONS.
IT
Dragons-Head, or Afcending Node, is the North Interfection
of the Moon's Orb with the Ecliptic; to which when the Moon
comes, ſhe has no Latitude. It is Character'd thus .
R.
Dragon's-Tayl, or, Defcending Node, is the South Interfection
of the Moon's Orb with the Ecliptic, to which when the Moon
comes fhe has again no Latitude. It is alfo called Catabibazon,
and is diametrically oppofite to the Dragon's Head; and their
mean Motion is Retrograde, as may be ſeen by the follow-
ing Tables. It is Character'd thus 8.
Dragon, is a Northern Conftellation, which fee in the
Catalogue of the Fixed Stars.
Duplicate Ratio, is no more then the Proportion of the firft to
the third, in three continual Proportionals.
Duration, of an Eclipfe, Occultation, &c. is the Time it
continues to be Eclipfed, or hid from our Sight.
E.
Earth. In my System of the Planets demonstrated, I have
proved, that the Earth has an Annual and Diurnal Mo-
tion, and that the Sun is at Reft in the Center of the Univerſe.
Dr. Gregory faith, that the Earth's Axis keeps nearly parallel
to it felf in its Annual Revolution round the Sun; and that by
reaſon of its ſwift Diurnal Motion, purs on the Figure of an
Oblate Spheroid, fwelling out towards the Equatorial Parts,
and contracted towards the Poles; fo that the Diameter of it
at the Equator, is longer than the Axis by 63 Miles: For
Sir Ifaac Newton proved, that the Polar Diameter or Axis, is to
the Equatorial one, as 689 to 692. Therefore according to
Norwood's Meaſure, I have Calculated the Earth's
Circumference
Diameter
25035.84) Engliſh
7969.16
Miles.
Height of the Earth's Atmoſphere 47.12
The Cone of the Atmoſphere and Shadow does not reach fo
far as the Orb of Mars.
Eccentricity, is the diſtance between the Center of the
Ellipfis and the Focus. Here Note, that the Sun is feated
upon the lower Focus of the Ellipfis in the Syftem of the fix
Primary Planets, and the Earth upon the lower Focus in the
Moon's Syftem.
Eccentrick Place of a Planet, is the fame with the Orbit-
Place,
Eclipse,
12
Aftronomical DEFINITION S.
Eclipfe, is a Deprivation of Light. The Eclipfe of the Sun
(or truly the Earth) is caufed by the Interpofition of the
Moon's dark Body between the Sun and our Sight, and can
never happen but at the New Moon, when the Sun and
Moon are less than 180 from the Moon's Nodes; and by rea-
fon of the Nearnefs of the Moon to the Earth, and fudden
Change in Parallax, the fame Solar Eclipfe fhall be Total to
one Part ofthe Earth, to another Partial, and to another Inha-
bitant no Eclipſe at all.
;
The Moon's Eclipfe is real, and univerfal; and is cauſed
by the Interpofition of the Earth between the Sun and Moon
and this can never happen but at the Full Moon; within
lefs than 120 of her Nodes, for the Moon being an Opake Body,
borrowing all her Light from the Sun, is then deprived of
that borrowed Light, and fo is Eclipfed. There can never
happen more than fix, nor leſs than two Eclipfes in one Year;
and when two, they are both of the Sun. See my Aftronomy,
printed 1727.
Ecliptic, is one of the Six great Circles of the Sphere, in-
terfecting the Equinoctial in two Oppofite Points, Aries and
Libra, making an Angle therewith of 23° 29', called, the
Obliquity of the Ecliptic, equal to the Sun's greateſt Declinati-
on: In this Circle (according to appearance) is the Sun
always found, and the Earth truly, in the Oppofite Sign
Degree and Minute: It is divided into 12 equal Parts call'd
Signs, and every Sign into 30°, every Degree into 60', and
every Minute into 60'. It alſo toucheth the two Tropicks in
the very beginning of Cancer and Capricorn.
Elevation, of the Pole, is an Arch of the Meridian compre-
hended between the Pole and the Horizon, which is always
equal to the Arch of the Meridian between the Zenith and
Equinoctial; theſe being the fame with the Latitude of the
Place of habitation.
Ember-Weeks, are thofe Weeks in which the Ember-Days
fall; they were of great Antiquity in the Church in the
Primitive Times, and are four in Number, and were therefore
called by the ancient Fathers, Quatuor anni Tempora. the
four Cardinal Seafons on which the Circles of the Year
turn: They are the Wednesdays, Fridays, and Saturdays next
after Quadragefima Sunday, after Whit-funday, after Holy Rood-
Day September 14, and after St. Lucy's Day December 13:
They were at firft ordained for Quarterly Seafons of Devo-
rion; wherein as the first Fruits of every Sealon, the ancient
Chriftians put up their Prayers and Supplications to Almighty
God,
Aftronomical DEFINITIONS.
13
God, that thereby the whole Year, and every of its four Parts
might be bleffed; and uſed to eat nothing till the Eventide,
and then only a Cake baked under the Embers, or Afhes, which
they called Ember-Bread; thefe Ember-Weeks are chiefly
taken notice of on the Account of the Ordination of Priefts
and Deacons; becauſe the Canon now appoints the Sundays
next fucceeding the Ember-Weeks for the folemn Times of
Ordination, tho' the Biſhops, if they pleaſe, may ordain on
any Sunday or Holiday.
Emerfion, is the Time when any Planet that is Eclipfed,
begins to recover its Light again: It is moſt uſed in the
Ecliples of Saturn and Jupiter's Satellites.
Epact, from Epiago, a going round; is a Number that is
the difference between the common Solar Year 365d. sh. 49′
23", and the mean Lunar Year 354 d 8 h. 49' 12" which is
10 d. 21 h. oo' II"; but to avoid Fractions, the Number 11
Days is made ufe of, which fhews that the Moon Changes
fooner in any Month this prefent Year than fhe did in the fame
Month the laft Year; therefore it is of good uſe to find the
Days of the New and Full Moons, Age, &c. as I fhall fhew
in the Doctrine of the Sphere.
Ephemeris, a Diary or Day-Book; amongft Aftronomers,
Ephemerides, the fame with Almanack, which fee,
Epocha. the fame with Era, which fee.
Equation, is the difference between the Planets mean and
true Place; for if the Mean Anomaly be leſs than 6 Signs, the
Equation fubtracted; or if the Mean Anomaly be more than 6
Signs, the Equation added to the mean Place, the Sum or
Difference is the Eccentric or Orbit-place of the Pla-
net.
Equation of Time, or of Natural Days, confifts of two Parts;
the firſt Part depends on the Sun's Place, and is the Difference
between that and his Right Afcenfion, which in the first and
third Quadrants of the Ec iptic is to be added; but in the
fecond and fourth to be fubtracted; the Sum or Difference is
the first part of the Equation of Time. The fecond Part de-
pends on the Earth's Anomaly, and is only the Sun's (or
Earth's) Equation Reduced into Time; which, if the Mean
Anomaly be less than fix Signs, it addeth; if more, it fub-
tracteth to, or from the Equal Time, to gain the Apparent, if
both theſe Parts, add, or both ſubſtract their Sum; otherwiſe
their Difference is the abfolute Equation of Time; which
applied to the Equal Time according to the greater Title,
gives the Apparent Time,
Equi-
14
Aftronomical DEFINITION S.
Equiculus, or Equus Minor, is a Conftellation in the Nor-
thern Hemisphere. See the Catalogue of fixed Stars.
Equator, See Equinoctial.
Equinoctial, in the Heavens, or Equator on the Earth, is
one of the fix Great Circles of the Sphere, whofe Poles are
the Poles of the World. It divides the Globe into two equal
Hemifpheres, viz. North and South, and paffeth thro' the
Eaft and Weft points of the Horizon; and at the Meridian
it is always raiſed fo much as is the Complement of the La-
titude of the Place where you are; which Arch is alſo e-
qual to the Arch of the Meridian between the Zenith of a-
ny Place and Pole. Every 150 of this Circle, that paffeth by
the Meridian by the Diurnal Motion, is equal to one Hour
in Time. Alſo when the Sun (apparently) comes to this Circle,
which is about the 9th of March, and 12th of September, he
makes the Days and Nights equal all the World over, except
under the Poles.
Equinoxes, are the precife times in which the Sun, or Earth
enters into the firſt Point of Aries and Libra; and this they do
twice a Year, about the 9th of March and 12th of September,
which times are called the Vernal, and Autumnal Equinoxes,
making then the Days and Nights equal: And Aftronomers
have found by Obfervation, that the fpace of Time from
the Vernal Equinox, to the Autumnal, is 7 d. 18 h. 52' lon-
ger than the time from the Autumnal to the Vernal: From
which they come to know, that the Earth did not move or
keep an equal Place in all parts of its Orbit.
Erratick Stars, are the feven Planets; becauſe they wan-
der up and down the Zodiack: They are alfo called Er-
rones.
Evection. See Angle of Evection.
F.
FAculo, are certain bright or fhining Parts, which the
Modern Aftronomers have fometimes obferved upon,
or about the ſurface of the Sun; but they are but very fel-
dom feen.
Falcated. The Moon, (or any Planet) appears falcated,
when the enlightened parts are in the Form of a Sickle; as
the Moon doth in the firft and laft Quarters.
Fafcia of Mars, are certain Rows of Spots, parallel to the
Equator of that Planet, which look like Swathes or Fillets
round about his Body.
Finitor,
Aftronomical DEFINITIONS.
Finitor, the fame with Horizon; becauſe the Horizon fini-
ſhes or terminates your Sight, View, or Proſpect.
Firmament, by fome Aftronomers is taken for the Orb of
the Fixed Stars, or an Eighth Heaven; but more properly
'tis that space which is expanded or arched over us above in
the Heavens.
First Mover. See Primum mobile.
Fixed Signs of the Zodiack, according to fome are, Tau-
rus, Leo, Scorpio, and aquarius; and they are fo called, be-
cauſe the Sun (apparentl) paffes them refpectively in the
middle of each Quarter, when tha particular Seafon is more
fettled and fixed than under the Sign that begins or ends
it.
Fixed Stars, are fuch as do not, like the Planets or Erratick
Stars, change their Potitions or Diſtances in reſpect of one
another; and becau´e their Annual Motion is nothing but the
Receffion of the Equinox = 50"; therefore they are faid to
be fixed. They move upon the Poles of the Ecliptic, and
therefore never alter their Latitudes. Their diftance from
us is fo very great, that no Parallax in them can be diſcover-
ed, as Dr. Halley affured me; tho' Mr. Flamsteed wrote to Dr.
Wallis in the Year 1698, and affured him, that he had difcover-
ed a fenfible Parallax in the Earth's Annual Orbit in refpect of
the fixt Stars. It has been a Queftion amongst the Ancients,
whether the Light of the fixed Stars was innate; given them
by Almighty God at their Creation, or borrowed from the
Sun: The firft feems to carry moſt of truth in it; and our
Modern Aftromomers do now conclude each fixt Star to be
the Head and Chief Part of a diſtinct Mundane Syſtem, ha-
ving their feveral Planets carried about them. If fo, what a
vaft Expanſion muft there be in the Interftellar, or withour
our Planetary Syftem, to contain fo many vaft Bodies? Their
Scintillation, or Sparkling (Gaffendus, and Hevelius) think to
be caufed by that Native and Primogenial Light they are en-
dowed with, coming to our Sight at fo immenfe a diftance,
and paffing thro' different Mediums, which by a conftant E-
vibration of Lucid Matter, appears to our Sight to twinkle;
which is not obſerved in any of the Planers. As for their
Number, is what I fhall not pretend to give; but as many as
are uſeful and of Note, you will find in the following Ca-
logue.
Fomahant, a fixed Star, which fee.
16
Aftronomical DEFINITION s.
GA
G.
Alaxy, or Via Lattea. See Milk y-Way.
Gemini, the Twins; the third Sign in the Zodiack,
thus Character'd ; but in all Aftronomical Calculations.
it is numbred with 2.
Geocentrick, Place of a Planet, is that which is ſeen from
the Earth.
Gibbous, is a Term ufed in reference to the enlighten'd
Parts of the Moon, while fhe is moving from the firſt Quar-
ter, to the laft Quarter; for all that Time the light part is
Convex or Gibbous.
Golden Number, is the fame with Cycle of the Moon;
which fee.
Gregorian Year, is the Reformation of the Calender by Pope
Gregory XIII. in the Year 1582, and is in this Age 11 Days be-
fore the Old Stile uſed by us in England. It is alſo called the
New Stile; and the Places that reckon by the Gregorian or
New Stile, are, France, Spain, Portugal, Italy; and in Ger
many, all the Popish Electors and Princes, and all Poland.
Heliacal Rifing, is when a Star having been under the
Sun's Beams, gets from the fame fo as to be feen again in
the Morning before the Sun.
Heliacal Setting, is when a Star by the near Approach of
the Sun, firſt becomes inconfpicuous. The Moon may be
ſeen nearer the Sun; that is, at a lefs diftance from him than
any other Planet or Star; becauſe ſhe is nearer to us than
any of the reft; and alſo becauſe her apparent Diameter is
greater; fo that fhe may be feen at about 17 Degrees from
the Sun, when the other Planets cannot be ſeen till they
are near a Sign diftant from him.
Heliocentrick Place, is, that it would appear to an Eye at
the Sun; for the Planets would always appear Direct there,
and the Heliocentric Latitude is the fame with the Inclinati-
on of the Orb with the Ecliptic: For the Quantity of each
Planet's Inclination, or the greateft Angles you will find in the
following Tables.
Hemifphere, is the half of a Globe or Sphere, when 'tis fup-
poſed to be cut thro' the Center in the Plain of one of its great
Circles. Thus, the Equator divides the Terreftrial Globe into
the North and South Hemifpheres; and the Equinoctial the
Heavens after the fame manner. The Horizon alfo divides
the Earth into two Hemiſpheres, the one Light, and the other
Dark
Aftronomical DEFINITIONS.
17
Z
Dark, according as the Sun is above or below that Cir-
cle.
Heniochus, one of the Northern Conftellations.
Hesperus, the Name of Venus, when ſhe is the Evening-
Star.
Horizon, is one of the fix great Circles of the Sphere, which
divides the Heavens and the Earth into two equal Parts, or
Hemiſpheres, diftinguiſhing the upper from the lower It
is either Senfible or Apparent; or Rational or True Horizon.
The Senfible or Vifible Horizon, is that Circle which limits our
Sight, and may be conceived to be made by fome great Plain,
on the Surface of the Sea. It determines the Rifing and Setting
of the Sun, Moon and Stars, in any particular Latitude.
The Rational, Real, or true Horizon, is a Circle which en-
compaffes the Earth exactly in the Middle, and whoſe Poles
are the Zenith and Nadir.
Hour is the 24th part of a Natural Day, containing 60'
Minutes, and each Minute 60 Seconds. Theſe are Aftro-
nomical Hours, which always begin at the Meridian, and are
reckoned from Noon to Noon.
Hydra, a Southern Conftellation, and imagined to repreſent a
Water-Serpent.
Hyemal, Solftice. See Solftice.
Hypothefis, the fame with System, which fee.
I.
Lluminative, Month, is that Space of Time that the Moon
is vifible, to be feen betwixt one Conjunction and ano-
ther.
<
Immerſion, of a Star, is when it approaches ſo near the Sun,
as to be hidden in its Beams. The beginning of an Eclipſe
is alfo fo called; as alfo the Satellites of Saturn and Jupiter,
when they enter into their Shadows, are called Immerſions.
Inclination, of the Plains of the Orbits of the Planets to
the Plain of the Ecliptic, are the fame with Heliocentrick
Latitudes; which fee.
=
Inclination, of the Axis of the Earth,
it makes with the Axis of the Ecliptic
Ingrefs, is the Sun's Entrance into any
of the Ecliptic.
is the Angle which
23° 29'.
Sign, or other part
C
Inter
18
Aftronomical DEFINITIONS.
Intercalary Day, the odd Day made up of the fix Hours
every fourth Year; is put in the next after the 24th Day of
February, and that occafions the Leap-Year.
Interlunium, when the Moon has no Face or Appearance,
as being in Conjunction with the Sun (i. e.) New Moon.
Interftellar, a Word uſed by fome Authors to expreſs thoſe
parts of the Univerſe that are without, and beyond our Soy
Jar Syftem, moving round each fixt Star, as the Center of
their Motion, as the Sun is of ours. And if it be true, (as
'tis not impoffible, but each fixt Star may thus be a Sun to
fome habitable Orbs that may move round it,) the Interftel-
lar World will be infinitely the greater part of the Uni-
verſe.
}
Julian Year, is the old Account of the Year, inftituted by
Julius Cæfar, which to this Day we ufe in England, and
moft Proteftant Countries, and call it the Old Stile, in con-
tradiftinction to the New Stile, or Gregorian Account, which
fee. This Fulian Year is 365 Days, 6 Hours long.
bur 'tis too much by 10'io", which in about 134 Years
will amount to one whole Day.
Julian Feriod, is a Cycle of 7980 Years, produced by the
Multiplication of the three Cycles, viz. that of the Sun 28,
of the Moon 19, and that of the Roman Indiction of 15
Years. This was the Invention of Julius Scaliger, who fixt
the beginning of it 764 Years before the Creation; ſo that
at the Birth of Chrift it was 4713; therefore if to the Cur-
rent Year of Chrift you add 4713, the Sum will be the Year
of the Julian Period; and from the Year of the Julian Pe-
riod fubtract 4713, there will remain the Year of the Chri-
ftian Æra; or the Year of the Julian Period may be found to
any Year of Chrift, by fixed Multiplicators; which may be
found thus, viz. There must be fuch a Number found that
being multiplied by the Product of 19 by 15, as that Product
when Divided by 28, the Remainder will be 1.
Number will be 17; then 19×15 285 × 17 = 4845, the
Common Multiplicator.
X
This
Secondly, We muft find a Number, which being multiply'd
by the Product of 28 by 15 and that Product divided by 19,
leaves for the remainder 1, and this Number is 10; for
28 × 15 = 420 x 10 4200 the Common Multiplicator.
Thirdly, We muſt find a Number that being Multiply'd by
the Product of 28 by 19, and that Product divided by 15,
leaves 1. This Number is 13.
Then 28 x 19
36916, the Multiplicator fought. Then to find the Year
532 X
of
Aftronomical DEFINITIONS.
ig
of the Julian Period for any Year of Chrift, this is the
Rule; Multiply
Nuñ
4845
26916S
Sun,
the fixt Number 4200 by the Cycle of the Moon,
Indiction?
the Sum of theſe Products divided by 7980, the remainder
is the Year of the Julian Period to the given Year of
Chrift.
Example. This Year 1727, you will find the Year of the
Julian Period to be 6440.
Jupiter, is the higheſt Planet in our Syftem, except Saturn;
and his Motion round the Sun is fo adjufted, that the Square
of the time of his periodical Revolutions is as the Cubés
of his mean Diſtance from the Sun. And the fame immu-
table Law is obſerved throughout all the Planetary Syftem;
which was firft difcovered by Kelper, and fince demonftrated
by the great Sir Ifaac Newton. Mr. Flamfteed and Dr. Halley
have found by obfervation, that moved too flow by all
our Aftronomical Tables; which Defect I have taken into
confideration, and adjusted the Motion in the following
Tables.
►
Latitude, in Aftronomy, is the diftance of a Star or Planet
from the Ecliptic, meafured upon an Arch of a Circle of
Longitude from the Ecliptic towards the Poles thereof; but
the Geocentrick Latitude is the Angle that the Planets Lati-
tude appears under to an Eye on the Farth.
Latitude, in Geography, or on the Earth, is the Height
of the Pole of the World above the Horizon, which is always
equal to the Arch of the Meridian between the Zenith and
Equinoctial.
Leap-Year. The fame with Biffextile, which fee.
Lemma, is the Demonftration of fomething premiſed, in
order to fhorten a following Demonftration..
Leo, the Lion, the fifth Sign in the Zodiack, Character'd
thus, unto which the Sun comes about the 12th Day of.
July; and in Aftronomical Calculations is numbred with the
Figure 4.
Leffer Circles, of the Sphere, are thofe whofe Plains do not
pafs thro' the Center of the Sphere; and which do not divide
the Globe into two equal parts; but are parallel to greater
Circles; as the Tropicks and Polar Circles, and all Paral-
lels of Declination and Altitude.
C 2
Letter
20
Aftronomical DEFINITIONS.
Letter Dominical; fee Dominical Letter.
}
Libra, one of the 12 Signs of the Zodiack, Character'd
thus, unto which the Sun apparently comes about the 12th
of September, making equal Day and Night; and in Aftro-
nomical Calculations is numbred with the Figure 6.
Libration, of the Moon, is of three kinds; Firſt, in Longitude,
which is a Motion arifing from the Plain of that Meridian of
the Moon, (which is always nearly turned towards us,) being
directed not to the Earth, but towards the other Focus of the
Moon's Ecliptical Orbit; and fo to an Eye at the Earth, fhe
feems to Librate to and again. Secondly, in Latitude, which
arifes from hence, that her Axis not being perpendicular to the
Plain of her Orbit, but inclin'd to it, fometimes one of
her Poles, and ſometimes the other will nod, or dip a little to-
wards the Earth. Thirdly, the Moon has a kind of a Librati-
on, by which it happens, that tho' one part of her is not
really obverted, or turn'd to our Earth, as in the former
Librations; yet another is illuminated by the Sun: For fince
her Axis is perpendicular nearly to the Plain of the Ecliptic,
when ſhe is at her greateft South Limit, fome parts adjacent
to her North Pole will be illuminated by the Sun, while on
the contrary the South Pole will be in darkneſs; and theſe
Librations will be compleated in her Synodical Month.
Limit, of a Planet, is the greateft Heliocentric Latitude;
which fee,
Limit, for Eclipfes of the Sun and Moon, are certain
Diſtances of the New and Full Moons, from the Nodes of the
Moon; in which the Eclipſes always happen, and the Limits of
the Moon's Eclipſe are 12° 2′ 9″ and her utmoft Latitude
62' 25"; that is, if her Diſtance from either Node at the
Full Moon be more than 12° 2′9″, her Latitude will exceed
62' 25"; therefore there can be no Eclipfe at that time. The
Limit, or Boundaries of the Sun's Eclipfe are, 18° 20' 8"
and the Latitude of the Moon then 1° 34′ 16″. Theſe
are the Greateſt Limits: But there are yet two other Ex-
treams, which I call the Leaft Limits; that is, if the Diſtance
from the Nodes be fuch, it is poffible there may at that time.
be no Eclipſe; and they are theſe ;
Leaft Limits of
Sun
Moon 10° 19'
17" Lat. 53' 41"
16 35 5 Lat. 85 32
The Cauſe of theſe two Extreams of the Limit, is, the dif
ferent Diſtances of the Sun and Moon from the Earth at dif-
ferent times.
Line,
Aftronomical DEFINITIONS.
21
-
Line, of mean Motion of a Planet, is drawn from the upper
Focus thro' the Planets, and continued amongſt the fixed
Stars.
Line, of true Motion, is drawn from the Sun on the lower
Focus to the Planet, and continued amongſt the Fixed
Stars.
Line, of Nodes, is drawn from one Node to the o-
ther.
Logarithms, (from Logos, Reafon; and Arithmoi, Numbers)
are a Series of Arithmetical Numbers, invented for the eaſe
and expedition of Calculation by the Lord Neper, but im-
proved by Mr. Briggs.
Logiſtical Logarithms, are artificial Numbers deduced from
the Logarithms of abfolute Numbers, of which there are two
forts; one invented by Jeremy Shakerly, and the other by
Tho. Street. The firſt I have continued to 1° 18′, and the latter
to 120 Minutes, or two Hours, and there fhewn the Conſtructi-
on of them both. Shakerly's are made thus: To the Co. Ar.
of the Logarithm of 3600 Seconds in an Hour, add the Ab-
folute Logarithm of any Number of Minutes reduced into Se-
conds, or any Minutes and Seconds joyntly taken; the Sum is
the Logiſtical Logarithm fought.
=
Example. What's the Logiſtical Logarithm of 1'?
1 h.-3600" Logar. 3.5563025 Co. Ar= 6.4436975
60" Logar. add
1'
Logiſtical Logarithm of 1' is
1.7781512
8.2218487
But we reject the two Figures to the right hand.
The Conftruction of Street's Logistical Logarithms.
To the Logarithin of 1º=60', which is 3.5563 (omitting the
three Places to the right hand) add the Co. Ar. of the Minutes
reduced into Seconds: The Sum is the Logiſtical Logarithm
of any Minntes and Seconds under 60'.
Example. What's the Logistical Logarithm of 43′ 17″?
OPERATION.
1° 60' its Logarithm in Seconds 3.600"
43′ 17″ 25.97" Logar. Co. Ar. add
I =
Logiſt. Logar. of 43′ 17″ rejecting Radius is
C 3
3.5563
65855
.1418
For
22
Aftronomical DEFINITIONS.
For more than 60', take the Co. Ar. of the Logar. of 1h=60
which is 3.55630, Co. Ar. 6.44369; and to this add the
Logarithm of the Degrees, Minutes and Seconds all reduced
into Seconds; this Sum is the Logistical Logarithm fought.
Example, what's the Logiſtical Logarithm of 81' 50'?
OPERATION.
60'3600" Logar. Co. Ar.=
81 59=4910" Logar, add
Logift. Logar. of 81' 50" is
L. L. fignifies Logiſtical Logarithm.
6.4437
3.6911
1348
Longitude, in Aftronomy, is the Diftance of a Star or Planet
counted in the Ecliptic from the beginning of Aries, according
to the Order of the Signs, to the Place where the Stars
Circle of Longitude croffes the Ecliptic; fo that 'tis much the
fame as the Star's Place; and this may be either Heliocentric,
or Geocentric; which fee.
Longitude, in Geography, is an Ark of the Equator, inter-
cepted between the firft Meridian and the Meridian of the
Place; 'tis the difference either Eaft or Weft between the
Meridians of any two places counted on the Equator.
Lucifer, the Morning-Star. Venus is fo called when ſhe is
Oriental, and rifeth before the Sun.
Lucida Corona, a fix'd Star of the fecond Magnitude in the
Northen Garland, which fee in the Catalogue of Stars.
Lucida Hydra, a fixed Star.
Lucida Lyra, a fixed Star of the firft Magnitude in the
Conftellation Lyra.
Luminaries; the Sun and Moon are fo call'd by way of
eminence, for their extraodinary Luftre, and the great Light
that they afford us.
?
Lunar Afpects, are thofe that the Moon makes with the other
fix Planets; as, when he comes in o, *, 0, 4, or 8, with
them, then the Time is fo marked in the Ephemeris. ·
Lunary Months, are periodical, fynodical, or illuminative ;
which fee under thoſe Words.
Lunar Cycle; fee Cycle of the Moon.
Lunations, of the Moon, are the Times between one New
Moon and another; and this is greater than the Periodical
Month by two Days and five Hours; and is called the Synodical
Month; but this Synodical Month is unequal in every Month
in the Year: For in December, when the Earth is in Perihelion,
the time between one Conjunction of the Sun and Moon, and
the
Aftronomical DEFINITIONS.
23
the next, is more by about 12 Hours than it is in June, when
the Earth is in Aphelion, which in the firft Cafe is about
29 d. 19 h. and in the latter, 29 d. 7 h; the reaſon of which
is very plain: For the Earth (or Sun apparently) moving fafter
in December than they do in June, of neceffity there muft be
more Time ſpent for the Moon to come up to ♂ with the
Sun in the former, than in the latter.
Luni-Solar Year, is a Period made by multiplying the Cycle
of the Moon 19, by that of the Sun 28; the Product 532
Years, is the ſpace of Time in which the Holy Feaft of Eafter
makes one perfect Revolution, and every thing depending
thereon returns to the fame again that they were 532 Years
before.
Lupus, a Southern Conſtellation, in form of a Wolf.
1
}
M.
Magnitudes; the Stars are divided into fix feveral Sizes
or Magnitudes for diftinction fake; of which the greateſt
are called Stars of the first Magnitude; as Sirius, Ar&turus, &c.
the next to them in Brightneſs are called Stars of the Second
Magnitude; next Inferiours to them are called Stars of the
third Magnitude, next to them are call'd Stars of the
fourth Magnitude; the next lefs are of the fifth Magnitude, and
the next lefs are of the fixth Magnitude. See the Catalogue.
Mars, is the Name of one of the Planets, which moves round
the Sun in an Orbit between the Earth and Jupiter, and per-
forms his Revolution in one Year 3 21 d. 23 h. 27′ 30″; his
mean Diurnal Motion is 31' 27", his Orbit makes an Angle
with the Ecliptic of 19 52', and he is 15 times less than our
Earth; is of a red Colour like the Star Aldebaran, and is Re-
trograde once in two Years. He is called by the Poets Aris,
Pyrois, Mavors, and Gradivus. See the Syftem.
Mathematical Horizon, is the fame with true Horizon.
Mean Motion of a Planet, is, fuppofing it to move in a per-
fect Circle and equally every Day; divide 360° by the Num-
ber of Days in a Revolution, the Quotient will be the Mean
Diurnal Motion; which fee in the Tables of every Planer.
Mercury, the Name of one of the Planets, whofe Orb is
next the Sun; he performs his Revolution in 87 Days, 23 h. 15"
53", and his Mean Heliocentric Diurnal Motion is 4º, 5′ 32″;
his Orbit makes an Angle with the Ecliptic of 6° 54'; he is
never elongated from the Sun more than 29; and therefore
CA
feldom
24
Aftronomical DEFINITIONS.
feldom feen: He appears to us at the Earth Retrograde four
or five times every Year, and is 27 times less than our Earth :
this Planet is called by the Poets, Archas, Cyllenius, Hermes,
and Stilbone.
Medium Cali, is that Degree of the Ecliptic that is upon
the Meridian at any time of Day or Night.
Meridian, is one of the fix great Circles of the Sphere
paffing through both the Poles of the World, and cutting
the Horizon at right Angles, being equally diſtant between the
Eaft and Weft; unto which when the Sun or any Star come,
it is the higheft, or has then the greateft Altitude that it can
have that Day in that Latitude. The Stars are then alſo ſaid
to Culminate or be South, when they are upon the Meridian.
Meridian Angle, is the Angle made by the Ecliptic and
Meridian at any given Time of the Day or Night, which
can never be more than 90 Degrees when or VS Culmi-
nate; nor leſs than 66 Degr. 31 Min. when Y and are on
the Meridian. It is of great uſe in the Calculation of Solar
Eclipfes. See the Table for this purpoſe.
Metonick Year, invented by Meton the Athenian, is the
Time of 19 Years, the fame with the Cycle of the
Moon.
Micrometer, is an Inftrument invented by our Country-
man Mr. Townly; which being fitted to a Teleſcope, is to
take the Diameter of the Stars and Planets. See Philof.
Tranf. N° 25 and 29.
Milky-Way, Via Lactea, or Galaxy, is a white broad Path,
or Tract encompaffing the whole Heavens, and extending it
ſelf in the Sign of Capricorn from the Equinoctial to the Tro-
pic of Cancer, with a double Path; and the reſt of it is a
fingle one. Some of the Ancients, as Ariftotle, imagin'd, that
this Path confifted only of a certain Exhalation hanging in
the Air, but by the Teleſcope-Obfervations of this Age it
has been diſcovered to confift of an innumerable quantity of
fixed Stars different in fituation and magnitude; from the con-
fuſed Mixture of whofe Light its white Colour is fuppofed
to be occafioned. This Milk-Way begins at the Equinoctial
at Ophiucus, or Serpentarius, and paffeth thro' the Conftella-
tions of Aquila, Cygnus, Caffiopeia, Perfus, Auriga, part of O-
rion, part of Scorpio, Sagittarius, Monoceros, Argo, Navis and
the Ara, Irs greateft Declination North is about 65 Degrees,
and South 69 Degrees ; it croffeth the Equinoctial from South
to North in 5 Degrees of Capricorn, from North to South in 5
Degrees of Cancer. Its breadth where broadeft, is about
25
Aftronomical DEFINITIONS.
25
25 Degrees near Aquila; but in other places it doth not ex-
ceed 10 Degrees in breadth. In the Months of February and
Auguſt you have a full View of it in the Evenings.
Minute, is the 60th part of an Hour in Time, or of a De-
gree in Motion; fo that every Hour, or Degree of any great
Circle is divided into 60 Minutes, every Minute into 60 Se-
conds, and each Second into 60 Thirds.
Month, properly ſpeaking is the Time the Moon is in run-
ning thro' the Zodiack; and this fhe performs in 27 Days,
7h. 43'7". This is called the Lunar or Periodical Month ;
but the time between one Conjunction and another, with the
Sun, is called her Synodical Month; this, according to heril 1.
middle Motion, fhe performs in 29 Days. There is alfo a 29:12:44:
Solar Month, which is the Time the Sun takes in running
through one of the Signs in the Zodiack, which is about
30 Days; but not of an equal length; (fee Lunations.)
The Vulgar Computation of four Weeks, or 28 Days to the
Month agree pretty near to the Moon's periodical Month
mentioned above.
Moon, is one of the ſeven Planets, and the loweſt of all in
the Syftem; fhe is an Opake Body, borrows all her light
from the Sun, and refpects our Earth for her Center; and
not only the Moon it felf, but alſo her whole Syftem is car-
ried round the Sun along with our Earth in a Year, (ſee my
Inftrument made by Tho. Heath at the Hercules and Globe in
the Strand.) This and her Vicinity to the Earth, is the cauſe
of the great Difficulty we have in obtaining her true Place.
Her Periodical Revolution, in reference to the fixed Stars, is
27 Days, 7 h. 43' 7"; her Orbit interfects the Ecliptic in
two oppofite Points, call'd Nodes, making an Angle therewith
of 4° 59′ 35″ in Conjunction, and in Oppofition to the Sun
but in the Quadratures of 5° 17' 20"; fhe is always Eclipfed
at the Full, and within lefs than 12 Degrees of her Nodes.
For a farther account of all the Inequalities of this Irregular
Planet, I fhall refer my Reader to her Theory written by Sir
Ifaac Newton, which I have kept cloſe to in Compiling of the
following Tables of her Motions. She always appears to us
at the Earth Direct, and is 50 times leſs than our Earth.
Nothing is more common amongst the vulgar Country-Peo-
ple in the time of Harveft, than for them to talk of the Har-
veſt-Moon; which, they ſuppoſe, is always at the Full at one
and the fame time in Harveft, and that the rifes and fets fe-
veral Days together at the fame time; and that God gave
her that Light and Stability at that time (above the reft of
Τ
the
26
Aftronomical DEFINITION S.
the Year) to ripen and bring forwards the Fruits of
the Earth; but theſe are grofs Abfurdities, as I thus
prove.
1. Becauſe ſhe always moves Direct according to the order
of the Signs Eaſtward; and this Motion in Longitude, when
floweft, can never be less than 11 Degrees in one Day,
and this 11 Degrees in a right Sphere is 44 Minutes in Time;
fo that 'tis impoffible fhe can rife or fet two Days together at
the fame Time; and this Delay in her rifing will be greatly
increaſed when the is in Perigeon, or in a Sign of Right or
Long Afcenfion with South Latitudes; I fay, when theſe
three Teftimonies concur, there will be more than an Hour
and a half difference between the time of her rifing this
Night, and the Time of her rifing the next Night to the Inha-
bitants of England.
2. But that the Moon doth rife in an Oblique Sphere with-
in 9 or 10 Min. two Nights together, is plain from this De-
monftration; that is, when fhe happens in Apogeon, North
Latitude, and in a Sign of Oblique, or fhort Afcen-
fion.
3. It is alfo poffible in the Month of August to the Nor-
thern Inhabitants, that the Moon doth fet two or more Nights
together within less than 10 or 12 Min. of the Time each
Night; and this is when fhe is in Apogeon, South Latitude,
and in a Sign of Oblique Defcenfion.
Lastly, The great difference of the Moon's ſetting any
two Nights together in an Oblique Sphere, is caufed by her
being in Perigeon, having North Latitude, or in a Sign of
Right or Long Defcenfion. These three Teftimonies concur-
ring together, will caufe her to fet more than 1 Hour la-
ter this Night, than fhe did the Night before. The Moon
by the Poets is called Cynthia, Diana, Latone, Lucina, Noctilu-
ca, Phabe, Proferpina.
I 2
Motion, is a continual and fucceffive Mutation or change.
of Place.
Mutual Aſpects, are fuch as the
among themselves; as the of
the 8 4 9, &c.
?,
Primary Planets make
♂ O, A ?
24,
N.
Aftronomical DEFINITIONS.
27
t
N.
Nadir, is the Point in the Heavens feemingly under the
Earth, which is Diametrically oppofite to the Point di-
rectly over our Head's.
Nebulous Stars, feen thro' the Teleſcope, appear to be
Clufters of fmall Stars, leffer than thofe of the fixth Mag-
nitude.
Nocturnal Arch, of the Sun, is that space in the Heaven,
which he (apparently) runs thro' from the Time of his fetting,
to the Time of his rifing; and this is always equal to the
double of the Time of his rifing; as when he rifeth at four
a-Clock in the Morning; that doubled, is eight Hours, the
length of the Nocturnal Ark.
Nodes, in Aftronomy, are the Points or Interfections of the
Orbs of the Planets with the Ecliptic; and in the Primary
Planets theſe as well as the Aphelions, have a flow Progreffive
Motion, as you may fee in the following Tables of each
Planet. For the Moon's Nodes, fee Dragons Head and
Tail.
Nonagefimal Degree, is the 90th, Degr. or higheſt Point of
the Ecliptic at any given. Time of the Day or Night; and its
Altitude is always equal to the Angle that the Ecliptic makes
with the Horizon. It is of great ufe in the Calculation of
Solar Eclipfes.
Northern Signs, of the Ecliptic or Zodíack, are thoſe fix
which constitute that Semicircle of the Ecliptic which inclines
to the Northward from the Equinoctial, as Aries, Taurus,
Gemini, Cancer, Leo, Virgo.
Number of Direction, is a Number not exceeding 35; which
Number is the Boundary, or Limit of Eafter-Day, which
always falls between March 21, and April 25, exclufive, being
35 Days. This Number changes every Year, but not in a
due order; but if may be found Arithmetically thus:
1. From 26, fubtract the Epact for the Year propoſed; but
when the Epact is 28 or 29, then fubtract it from 56, and
reſerve the Remainder.
1
2. Divide, the Epact by 7; its Remainder fubtract from 8;
this Remainder fub. from the Dominical Letter, numbering
them thus, A 1, B2, C3, D4, E5, F6, G7; what remains now,
add
1
J
28
Aftronomical DEFINITIONS.
add to the firſt reſerved Number, which gives the Number of
Direction for the Year propofed. Note, when you cannot
fubtract from the Number of the Latter, borrow 7; and
if nothing remains, it muſt be called 7; and when the Epact is
28, add 2 to the remainder of the Sub. from 8; and when
the Epact is 29, you muſt ſubtract 5 from the Remainder of
the Sub. from 8; the Sum or Difference will be the true
refery'd Number; and in Leap-Year you must take the Letter
that ferves from February to the Year's end.
Example. What's the Number of Direction for the Year
of Chrift 1736?
Epact 28, Dominical Letters D C3. Then 56--28-28
and 287 Remains o=7.8-7=1+2=3---3=0 and 28+
7=35 the Number of Direction fought. But by the Tables in
the Doctrine of the Sphere you have it without any manner of
trouble.
i
O.
Blique Afcenfion, is that Degree and Minute of the
Equinoctial, which riſes with the Center of the Sun, and
Moon, or Star, in an oblique Sphere.
Oblique Defcenfion, is that part of the Equinoctial which fets
with the Center of the Sun, Moon or Star, or with any Point
of the Heavens in an Oblique Sphere.
Obliquity of the Ecliptic, is the Angle that the Ecliptic makes
with the Equinoctial, which is at Aries and Libra, where it in-
terfects it, and is 23 Degr. 29 Min. equal to the Sun's
greateſt Declination.
8,
Oblique Signs, are fuch as Afcend obliquely; thoſe are
VS, ~, H, V, Ŏ, II; and they will Deſcend rightly: Their
oppofite, &,m,, m,, do Aſcend right, and Deſcend
obliquely to the Northern Inhabitants.
Oblique Sphere, is where either Pole is Elevated any Number
of Degrees less than 90, and confequently the Axis of the
World, the Equinoctial and Parallels of Declination will cut
the Horizon obliquely, from whence comes the Name.
Occident, is the Weſtern Part of the Horizon, or 'tis that
part where the Ecliptic or Sun therein Defcends into the lower
Hemifphere.
Occident,
Aftronomical DEFINITION S.
29
Occident Eftival, is that Point of the Horizon where the Sun
fets at his entrance into the Sign Cancer when the Days are
the longeſt to all the Northern Inhabitants.
Occident Equino&tial, is that Point of the Horizon where the
Sun fets when he enters Aries or Libra.
Occident Hybernal, is that Point of the Horizon where the
Sun fets when he enters into Capricorn; at which time the
Days with us are ſhorteſt.
Occidental, (i. e. Wefterly.) In Aftronomy, a Planer is faid
to be Occident when it fets after the Sun; and in Ephemerides,
on the top Columns of Lunar Aſpects, you find Occi: Which
fignifies Occidental, and which fhews that Planet to be an
Evening-Star.
Occultation, in Aftronomy, is the Time that a Star or Planet
is hid from our fight when Eclipſed by the interpofition of
the Body of the Moon, or fome other Planet between it and
us.
Octant, or Octile, in Aftronomy, fignifies a Planet, &c. being
in ſuch an Aſpect or Pofition to another, that their places
differ the Eighth part of the Zodiack, or 45 Degr.
Olor, or Cygnus, the Swan, a Conftellation in the Northern
Hemiſphere. See the Catalogue of fixed Stars.
Ophiucus, one of the Northern Conftellations, the fame with
Serpentarius.
Oppofition, is that Pofition or Afpect of the Stars or Planets,
when they are fix Signs, or 180 Deg. diſtant from one another,
and is marked thus 8,
Orb, is any hollow Sphere; but the Orbs of the Planets are
thoſe Circles (or rather Ellipses) in which they move, and the
Ecliptic is called the Sun's or Earth's Orbit: They are not at
all in the fame Plain with the Ecliptic; but variouſly inclin'd
to it, and to one another at different Angles; the Plain of the
Ecliptic interfects the Plain of every Planet's Orbit in two
oppofite Points, call'd Nodes; the places of which and the
Inclinations may be feen in the Tables of each Planet.
Orbis Magnus is, the Orbit of the Earth in its Annual Re-
volution round the Sun. This, in respect to the vaft diftance
of the Fixed Stars, is no more than a Point.
Orient, is the Eaft Part of the Horizon; or, it is that part
of the Horizon where the Ecliptic, or the Sun therein Afcends
into the upper Hemisphere.
Orient Eftival, is that Point of the Horizon wherein the
Sun rifes when he enters Cancer.
Orient
3.0
Aftronomical DEFINITIONS.
}
Orient Equinoctial, is that Point of the Horizon when the
Sun rifes when he enters Aries and Libra, making the Days
and Nights equal.
Orient Hybernal, is that Point of the Horizon where the
Sun rifes when he enters Capricorn.
Oriental; In Aftronomy, a Planet is faid to be Oriental
when he rifes in the Morning before the Sun; fo in an
Ephemeris you will meet with Ori, on the Head of the Lunar
Afpects, which tells you, that, that Planet is then Oriental
of the Sun, or a Morning-Star.
Orion, a Southern Conftellation.
Orthographick, Projection of the Sphere, is the drawing the
Superficies of the Sphere on a Plane, which cuts it in the
middle, the Eye being placed at an infinite diftance vertically
to one of the Hemifpheres; in which all the Hour-Circles
become Ellipfes. Tis the fame with Analemma ; which
fee.
PAnfelene, fignifies the Full
P.
Moon.
Paracentrick Motion, is when a Planet approaches nearer
to, or recedes farther from the Sun or Center of Attracti-
on.
Parallax, is that Arch of a great Circle paffing thro' the
Zenith and true Place of the Sun, Moon or Star, and in-
tercepted between the true and apparent place: Becauſe the
true Place is fuppofed to be beheld from the Earth's Center;
but the Apparent from the Superficies; and that difference is
the Angle of Parallax; of which there are five ſorts, viz. in
R. Afc. Declination, Altititude, Longitude and Latitude: For
the underſtanding of which obferve theſe following Con-
fectaries,
CONSECTART 1.
If the diftance of the Moon from the Point Afcending or
Point Deſcending be leſs than her Altitude, fhe has then no
Parallax of Latitude; but this can never happen, but in ſuch
Latitudes where the Moon's Orb, or Ecliptic become Vertical
Circles
}
2. If
Aftronomical DEFINITION S.
31
2. If the Diſtance of the Moon from the Point Afcending or
Deſcending be juſt 90 Deg. then doth a Vertical Circle interfect
the Ecliptic at right Angles, and there is then no Parallax of
Longitude, but only of Latitude.
3. If the Vertical Circle paffing thro' the Moon's Center,
fall upon the Ecliptic at oblique Angles, then there is Parallax
both in Longitude and Latitude.
4. All places on the Earth that have more than 28° 46′ 20!!
of North Latitude, to them the Moon's Parallax is South,
and fhe is depreffed below her true Place, according as
ſhe is Eaft or Weft of the Nonagefime Degree. Theſe
Parallaxes are of fingular ufe in the Calculation of Solar
Eclipfes, &c.
Paralla&tick Angle, fee Angle.
Parafelene, a Mock-Moon.
Parhelion, a Mock-Sun.
Pafcha, Eafter-Day.
Path of the Vertex, is a Circle defcribed by any Point of
the Earth's furface, as it turns round on its Axis. This
Point is confidered as vertical to the Earth's Center, and
is the fame with what is called Vertex, or the Zenith. The
Semidiameter of this Path of the Vertex is always equal
to the Complement of the Latitude of the Point or Place
that deſcribes it.
Pagafus, a Conftellation in the Northern Hemiſphere.
Penumbra, in Aftronomy, is a faint kind of ſhadow, or the
utmoft Edge of the perfect Shadow which happens at the
Eclipfe of the Moon; fo that it is very difficult to deter-
mine where the Shadow begins, and where the Light
ends; as I have often proved by my Obfervation of Eclip-
fes.
Penumbra, in the new Aftronomy; Its Semidiameter is
equal to the Sum of the apparent Semidiameters of the
Sun and Moon: For if at the time of the true Conjunction
of the Sun and Moon, none of the Penumbra fall within
the Earth's Disk, the Sun will then no where on the Earth
be Eclipſed.
Periæci, are thoſe Inhabitants of the Earth who live under
the fame Parallels, but under oppofite Semicircles of the
Meridian, when they have the fame Seafons of the Year, viz.
Spring, Summer, Autumn, and Winter, at the very fame
time as alfo the fame length of Days and Nights; for
'tis
1
32
Aftronomical DEFINITION S.
'tis in the fame Climate, and at an equal diftance from
the Equator: But when 'tis Noon to the one, 'tis Midnight
to the other.
Perigeon, or Perigaum, is a Point in the Heavens wherein a
Planet is at its neareſt diſtance from the Earth. When the
mean Anomaly of the Moon is fix Signs, fhe is then in Peri-
geon, and her Diurual Motion is about 15 Deg. This
Point is always Diametrically oppofite to the Apogeon, ex-
tended by the Tranfverfe Diameter of her Elliptical Or-
bit.
Perihelion, is the Point in the Heavens where the Earth or
any of the Primary Planets are neareſt to the Sun: Their
Heliocentric Motions are now the fwifteft, and their mean
Anomalies are fix Signs. This Point is Diametrically op-
pofite to the Aphelion.
Period of the Eclipfes, fee Saros-
Periodical Month, is the ſpace of Time the Moon finiſhes
her Revolution in.
Periſcii, are the Inhabitants of the Frozen Zones ; for as
the Sun goes round them for fix Months, ſo doth their ſha-
dows; whence the Name.
Perfeus, a Constellation in the Northern Hemiſphere.
Phafes, in Aftronomy, is uſed for the feveral appearances
of the Planets, eſpecially the Moon and Venus, who feem to
our fight, obfcure, horned, half illuminated or full of
Light; and by the Teleſcope the fame is obferved in
Mars.
Phanix, a Southern Conftellation.
Phenomena, are appearances in the Heavens.
Phenomenon, any fingle Appearance in the Heavens,
as of an Eclipſe, Comet, &c.
Phosphorus, the Bringer of Light; it is the Name of Venus
when the is the Morning-Star.
Pifces, the Name of two Conſtellations the one in
the Zodiack marked thus , unto which the Earth comes
about the 12th of Auguft; the other in the Southern He-
mifphere.
Place of the Sun or Star, it is the fame with Longitude of
the Sun, Moon, or Star; which fee.
Place, (true,) of a Planet, is that which is pointed at by
a Line drawn from the Earth's Center to the Star.
Place, (apparent,) is that which is beheld by the Obferver
from the Earth's Superficies.
Planets,
Aftronomical DEFINITIONS.
33
Planets, are the feven Erratick Stars, Saturn 5, Jupiter 4,
Mars, Earth, Venus 9, Mercury, Moon D; which fee
under thoſe Words. The Sun being now exempted from the
being one of that Number.
Pleiades, the ſeven Stars; which fee in the Catalogue,
Poetical Rifing, and fetting of the Stars, are of three forts,
viz. Achronical, Cofmial, and Heliacal; which fee.
Point, of Station in Aftronomy, are thofe Degrees in
the Zodiack in which a Planet feems to stand ftill;
which always happens juft before and after their Retro-
gradation.
Polar Circles, are two leffer Circles of the Spheres, pa-
rallel to the Equinoctial, and 23 Deg. 29 Min. diftant from
the Poles of the World That about the North Pole is called
the Artick Circle; and that about the South Pole, the Antartick
Circle.
Pole-Star, is a Star of the fecond Magnitude, in the Tail
of the Little Bear; the height of it above the Horizon is
nearly equal to the Latitude of the Place: For a further
account of it, fee my Syftem of the Planets Demonftra-
ted.
+
Poles of the World, are two Points in the Axis of the
Equator, each 90 Deg. diftance from its Plain; one poin-
ting to the North, which is therefore called the North or Artic-
Pole; and the other Southward, which therefore is called.
the South, or Antartick Pole.
Poles of the Ecliptic, are two Points in the Solftitial Colure
23° 29′ diftant from the Poles of the World, lying exactly
in the Polar Circles'; fo that when the Sign Capricorn is on
the Meridian above the Earth, the North Pole of the E-
cliptic is on the Meridian above the North Pole of the
World; but when Cancer is on the Meridian above the Ho-
rizon, then the faid Pole of the Ecliptic is on the Meridian
under the Pole of the World. The Axis of the Sun and Moon
do nearly point to the Poles of the Ecliptic.
Pollux, a fixed Star, fee the Catalogue.
Poftulata, is a grantable Requeft, or fuch a Demand as
reaſonably cannot be denied.
Primary Planets, are Saturn, Jupiter, Mars, Venus, and Mer-
cury..
Prime of the Moon, fignifies the New Moon at her firſt ap-
pearing.
D
Primum
1
34
Aftronomical DEFINITIONS.
1
Primum Mobile, the firft Mover, according to the Ptolemaick
Aftronomy, is fuppofed to be a vaft Sphere, whofe Center
is that of the Earth; this, they ſuppoſed turned round in 24
Hours; but it is now found to be falſe, and the whole Hypo-
thefis is Exploded.
·
}
འ་
Proceffion of the Equinoxes; in the New Aftronomy, the
Fixed Stars are fuppofed to be immoveable; and that
the Earth travels round the Sun by its Annual Motion;
fo that its Axis makes always an Angle of 66 Deg.: 31
Min. with the Plane of its Orbit; and by the Earth's
Diurnal Motion once round its Axis in 24 Hours to the
Eaft, the Equinoctial Points are moved the contrary way
about 5ő Min. a Year; and for this reafon the Fixed Stars
feem to be carried forward according to the order
of the Sign, about as much in the fame Time.
*
Projection of the Sphere in Plano, is a true, Geometrical Deli-
neation, of the Circles of the Sphere, or any affign'd part of
them upon the Plain of fome one Great Circle, as on the
Horizon, Meridian, Equinoctial, Ecliptic, Colures, or on the
Tropicks, &c. and this is either Stereographick, which fup-
pofes the Eye to be but 90 Degr. diftant from, and perpen-
dicular to the Plane of the Projection; or Orthographick,
when the Eye is at an infinite diftance, in the Center of the
Projection.
Prometheus, or Hercules, the Name of a Northern Conftel-
lation; it is called alfo Engonafis.
Problem, is when fomething is propoſed to be done.
Proportion. When two quantities are compar'd one with a-
nother, in respect of their greatnefs or (mallnefs, that Com-
pariſon is called Ratio, Reaſon, Rate, or Proportion: But when
more than two Quantities are compared, then the Compari-
fon is more ufually called, The Proportion that they have to
one another. The Words Ratio and Proportion are frequenly
uſed promifcuouſly.
r. To three Numbers given to find out a fourth in a Du-
plicate Ratio ; as, fuppofe 3, 4 and 5, 525 × 4 = too ÷
☐ 3 = 11.
2. To three Numbers given to find a fourth in a Triplicate
Proportion; as, fuppofe 3, 4 and 5: 5 Cubed is `125, × 4
500 Cube of 3184, the Anfwer.
3. To two Numbers given to find a third, fourth, fifth,
fixth, &c. Number in a continual Proportion, to the two gi
ven
Aftronomical DEFINITIONS.
35
}
ven Numbers; as, ſuppoſe 2 and 4, 4 × 4 = 16 ÷ 2 = 8×8
= 644 = 16 × 16 = 2568
= 32 × 32 = 1024 16
= 64 × 64 = 4096 32 128. So I find the fix Numbers
÷ =
in a continual Proportional are 2, 4, 8, 16, 32; and ſo on ad
infinitum.
4. Between two Numbers given to find a mean Arithme-
tical Proportion; as fuppofe to and 20: Thus 10 † 20 →
302 15 the Anſwer.
5. Between two Numbers given, to find a Geometrical
Mean Proportion; as fuppoſe 10, and 20: Thus, 10 × 20 ≈
200, V = 14. 14 the Anſwer.
6. Between two Numbers given, to find a Mean Muſical
Proportion:
Rule. Multiply the Difference of the Terms by the leffer
Term, and alſo add them together: This done, divide the
Product by the Sum of the Terms; and to the Quotient,
add the leffer Term: This Sum is the Mufical Mean defired.
Example. Let 9 and 18 be given; I demand the Mufical
Mean Proportional.
81÷¬
Operation. 189= 9x9= 81, 18 +9
81, 18+ 9 = 27 and 81
27=3+9= 12 the Mufical Mean fought. This Mufical
Proportion is of excellent uſe in Philofophical Experiments of
Colours For if you take feveral Colours and put them on
2 Wheel, and diſtant one from another in this Proportion ;
turn the Wheel faft round, and they will all appear White.
Propofition, is uſed promifcuouſly, (i. e.) either for a Theo-
rem, or a Problem.
Proftapherefis, in Aftronomy, is the fame with Equation of
the Planets Orbit, and is the Difference between the Mean and
True Place; fee the Tables.
Pfeudoftella, in Aftronomy, fignifles any kind of Comet or
Phænomenon newly appearing in the Heavens like a Star.
Ptolemaic Syftem, fuppofes the Earth fixed in the Center,
and all the Heavenly Bodies moving round: But this is
falfe, as I have proved in my Aftronomy, or System of the
Planets Demonftrated.
Pythagorean System, is the fame with the Copernican Syſtem,
which fuppofes the Sun fixed in the Center of the World, and
all the Planets moving round: This is what we embrace,
and have demonftrated in the fore-cited Book.
D 2
Q₂
36
Aftronomical DEFINITIONS.
Qr
Q.
Vadragefima, is the firft Sunday in Lent, and fo called,
because 'tis about the 40th Day before Eafter, and on the
like account the three preceding Sundays are called Quinqua-
gefima, Sexagefima, and Septuagefima.
Quadrant, is the Quarter, or fourth part of a Circle,
graduated on the Limb with 90 Degrees; its Furniture are
Teleſcope, and Micrometer, to take the Altitudes and Diameters
of the Planets and Stars; and ſuch a one there is now at the
Royal Obfervatory at Greenwich-Hill, of near Eight foot Radius,
which was framed by Mr. Hern, and graduated by Mr. Graham:
It is an Inftrument of exquifite Workmanship; and being now
under the care of that skilful Aftronomer Dr. Edmund Halley,
it is fixed upon a ftrong Mural Arch, and exactly on the
Meridian, to take the Meridian Altitude of the Moon and
other Planets, as they pafs by.
Quadiatures, or Quarters of the Moon, are the middle Points
of her Obit between the Conjunction and Oppofition; and
they are ſo called, becauſe a Line drawn from the Earth to
the Moon, is then at right Angles with a Line drawn from
the Earth to the Sun; the Luminaries are then a Quarter of
the Zodiack or 90 Degrees diftant from each other, equal to
three Signs, and in an Ephemeris is thus Character'd □.
Quarters of the Year, are four in Number; the firſt begins
when the Sun apparently enters the Equinoctial Sigu Aries,
inaking the Days and Nights equal all the World over, ex-
cept under the Poles, and continues while the Sun is run-
ing thro' T, Ŏ, II. This is called the Spring-Quarter. The
Summer-Quarter begins abour the 10th Day of, June, and
continues while the Sun runs thro',,, making the
longeft Days to all the Northern Inhabitants. The third is
called the Autumn, or Harveft-Quarter, and begins about the
12th Day of September, continues while the Sun is running
thro',,, the Days and Nights are again equal. The
fourth and laft is called the Winter-Quarter, making then
ſhorteſt Days and longeft Nights to all the Inhabitants on this
fide the Equator. This Quarter continues all the time the
Sun is paffing thro' VS, **, *.
Quartile, the fame with Quadrature, which ſee.
Quincunx, is one of Kepler's new Afpects of the Planets, and is
when they are diftant from each other 5 Signs, or 150 Degr.
maked thus, VC, or Q
Quin-
1
Aftronomical DEFINITIONS.
37
Quindecile, is one of Kepler's new Afpects marked thus Q. d.
and happens when Planets are 24 Degr. diftant from each
other.
Quinguagefima, ſee Quadragefima.
Quintile, is one of Kepler's new Afpects, marked thus, Q;
and is when Planets are 2 S. 12° affunder.
R.
Radius, is the whole Sine, or Semidiameter of any Cir-
cle.
Rational, real, or true Horizon; fee Horizon.
Rays, or Beams of the Sun, Rays of Light, are either ac-
cording to the Atomical Hypothefis, thofe very minute Parti-
cles or Corpuſcles of Matter, which continually iffuing out of
the Sun, do thruft on one another all around in Phyfically fhort
Lines; or elfe, as the Cartefians affert, they are made by the
Action of the Luminaries on the contiguous Ether and Air,
and fo are propagated every way in freight Lines thro' the
Pores of the Medium.
Rays, Convergent, are thoſe which going from divers Points
of the Object, incline towards one and the fame Point tending
to the Eye.
Rays, Divergent, are thoſe which going from a Point of the
vifible Object, are difperfed and continually depart one from
another, according as they are removed from the Ob-
ject.
Reciprical Proportions, are, when in four Numbers the fourth
is leis than the fecond, by fo much as the third is greater than
the firft, and vice verfa: On which is founded the Indirect,
or Inverſe Rule of Three.
Receffion, of the Equinox,is the going back of the Equinoctial
Points every Year about 50". The Reafon of which is the
Earth's being thrown into a Spheroidical Figure by its Diurnal
Motion.
Reduction, in Aftronomy, is the Angle that is made between
the Axis of the Ecliptic, and the Axis of the Planets Orbit,
which is equal to the Quantity of the Ecliptic intercepted be-
tween the two Axes.
Reflection, in the new Aftronomy, is the diſtance of
the Pole from the Horizon of the Disk; which is the fame
thing as the Sun's Declination.
D 3
Ro
38
Aftronomical DEFINITIONS.
Refraction, Aftronomical, is that which the Atmoſphere
produceth, whereby a Star appears more Elevated above the
Horizon than really it is.
Refraction, Horizontal, is that which caufes the Sun or
Moon to appear on the edge of the Horizon, when they are
as yet fomewhat below it. In the following Aftronomical
Tables I have inferted Mr. Flamsteed's Tables of Refractions:
But this is varied by the Weather; and in places more Nor-
therly than London it has been much greater than has been aſ-
ferred by Mr. Flamsteed: For in the Year 1695, at a Town
called Pello, in the Latitude of 65° 53', ten Miles to the
Northward of Forneo in the Weſtern Bothnia, on the 14th
of June, at 12 Hours P M. when the Center of the Sun was
depreffed 40 Min. below the Horizon, he was feen by the
means of the Refraction at the Altitude of two Diame-
ters, Hodgson, Vol. 2. Page 274. And the known Experi-
ment of putting a Shilling into a Bowl of clear Water, doth
very well explain the nature of Refractions: But that
this may be underſtood by every one that would be an
Aftronomer, I fhall explain its Laws; which are thefe: A
Ray of Light paffing out of a fine into a more denfe Me-
dium, is Refracted downwards to the perpendicular L G; but
paffing out of a denfer into a finer Medium, the Rays of Light
will be Refracted from the Perpendicular; fo ED will be
turn'd out of its ftreight Courſe to DA: For if the Re-
fraction be made out of Air into Water, then the Sine of
Incidence is to the Sine of Refraction as 4 to 3;
if out
of Air into Glafs, the Sines are as 17 to 11, & vice verfa. A
L
A
44
B
ས་འཐ
E
M
K
Ray of Light paffing from
A to D, will not go
ftreight on to M, but
will be turned our of its
way to E: Make the An-
gle CDL of Reflexion
to the Angle ADL of
Incidence, and draw the
Chord A C; then is A B
the Sine of the Angle of
Incidence, and B C the
Sine of Reflexion : Make
A B = 4, and B F = 3;
draw F E, and DE, fo
is HDE the Angle of
The Rays of Light
Paffing
Refraction, and HE the Sine thereof.
Aftronomical DEFINITIONS.
39
paffing thro' Oyl of Turpentine and thro' Water, the propor-
tion is as 25 to about 16, which proves Oil is denſer than
Water.
,
Retrograde, in Aftronomy, is only appropriated to the five
Primary Planets, when by their proper motion in the Zodiack
they ſeem to move backward, or contrary to the Succeffion of
Signs, as Saturn did this prefent Year 1727, go back from
15° 8' AW to 8° 28', that is 6° 40′, and Venus from 23°
24 m, to 7° 56' m, that is, 15° 28′ Retrograde; but this
Motion is not real in the Heliocentric, but only in the
Geocentric Motion, occafion'd by the Annual Motion of
the Earth, as I have proved by the Inftrument in my Syftem
of the Planets demonftrated.
Region, Etherial, in Cofmography, is the vaft Extent of
the Univerſe; wherein are comprized all the Heavens and
Celeftial Bodies..
Retroceffion, the fame with Receffion, which fee.
Revolution, in Aftronomy is the Circumvolution of any
Coeleftial Body, till it returns to the fame Point in which
it was when it first began. The Time of the Revo-
lutions of each Planet you may fee under their
Names.
:
Right Afcenfion, of the Sun or Star, is that Degree of
the Equinoctial accounted from the beginning of Aries,
which rifes with it in a Right Sphere; or it is that De-
gree and Minute of the Equinoctial (counted as before)
which comes to the Meridian with the Sun, Moon, or
Stars, or with any part of the Heavens in an Oblique
Sphere. The reafon of which referring it to the Meri-
dian, is becauſe that is always at Right Angles to the E-
quinoctial, which the Horizon only is in a Right
Sphere.
༞、
Right Signs, are Cancer, Leó, Virgo, Libra, Scorpio, and Sa-
gittary. They are called Signs of Right Afcenfion; becauſe in
an Olique Sphere that part of the Ecliptic they pals,
nearly cuts the Eaftern Horizon as they rife at right An-
gles.
Rifing, of the Sun, Moon, or Stars, is their appearing a-
bove the Eaftern Horizon.
2
Ring of Saturn, is an opacous, folid, circular Arch and
Plane, like the Horizon of a Globe, of Matter compaí
fing entirely round the Planet, and no where touching:
Its Plane is at this Time nearly parallel to the Plane of
our Earth's Equator; the Diameter of this Ring is 2
ہے
D 4
of
40
Aftronomical DEFINITION S.
of Saturn's Diameters; and the Diſtance of the Ring from
the Planet, is about the Breadth of the Ring it felf. See
Hugens his Syftema Saturniana, 1659.
Roman Indiation, fee Cycle.
S.
Agittarius, is the Ninth Compleat Sign of the Zo-
diack; but in Calculations, Number 8, and Chara-
ter'd thus. The Earth enters this Sign about the 10th
Day of May.
Saros, is a Period for Eclipfes, and called both by Mr.
Flamsteed, and Dr. Halley, the Chaldean Saros; it contains in
Leap-Year 18 Years, 11 Days, 7 Hours, 43′ 15″; in a
common Year 18 y. 10 d. 7 h. 43' 15": The mean Mori-
ons of the Sun and Moon in 18 y. 11 d. 7 h. 43' 15'
equally os. 10° 48' 6"; of the Moon's Apogeon o§. 13° 39′
34"; of her Retrograde Node 11S. 18° 43' 38", and of the
Moon from the Sun, nothing. This in the 74th Page of my
Syftem of the Planets Domonftrated, I call Mr. Whifton's Period;
but Dr. Halley affured me, that that Gentleman had it from
himſelf, and defired me to let the World know fo much. This
Period may ferve very well for common ufe to examine
Eclipfes by; but not to truft to for the precife time; Therefore
I refer you to the following Precepts; where you have the
exact methods of Computing them.
Satellites, by Aftronomers are taken for thofe Planets who
are continually waiting upon, or revolving about other Planets;
as the Moon may be called the Satellite of the Earth; and the
reft of the Planets Satellites of the Sun; but the Word is
chiefly used for the New-difcovered fmall Planets, which make
their Revolution about Saturn and Jupiter; of which there
are five about Saturn, and four about Jupiter, which were firſt
difcovered by Galilæus.
Saturn, is one of the Primary Planets, and the highest of all
in the Planetary Syftem: He performs his Revolution, round
the Sun in 29 Years, 174 Days, 6 Hours, 36', 26". For his
other Motions fee the following Tables: He is Retrograde
once every Year.
Scholium, is a fhort Critical Expofition, gained from a form-
ef Demonftration, or a Corollary wanting an Explication.
1
•
Scorpion
1
Aftronomical DEFINITIONS.
41
Scorpio, is the Eighth Sign in the Zodiack, marked thus
m; but in Calculation, Number 7.
Unto this Sign the
Earth comes about the 9th Day of April.
Seafon of the Year, fee Quarters of the Year.
Secondary Planets, are fuch as move round others, whom they
reſpect as the Center of their Motion, tho' they move alfo a-
long with the Primary Planets in the Annual Orbit round the
Sun ; and theſe are the Moon and the Satellites of Saturn
and Jupiter.
Second, the Sixtieth part of a Minute, either of Time or
Motion.
Secondary Circles, are all Circles which interfect one of the
fix Great Circles of the Sphere at Right Angles; fuch are the
Circles of Longitude, cutting the Ecliptic at Right Angles;
alſo the Azimuths, or Vertical Circles in reſpect of the Hori-
zon.
Semiquadrate, is one of Kepler's new Afpects, marked thus
S. q. and is when two Planets are diftant from each other
1 S. 15°. Octile, or Sefs. Quadrate.
Semifextile, is one of Kepler's new Afpects, and marked
thus SS; it is made by two Planets of the diftance of one
Sign from each other.
Semiquintile, is when Planets are diftant from one another
36°, marked thus, Dec. Decile.
Septuagefima, Sunday, ſee Quadragefima.
Serpens, a Northern Conftellation, called the Serpent of
Ophiuchus.
Septentrional, Northern.
Serpentarius, or Ophiuchus, the Serpent-bearer a Northern
Conftellation.
Sefquiquadrate, is a New Afpect of 4 S. 15°, marked thus
Ss. 4.
Setting, of the Heavenly Bodies, is when they go down in
the Weltern Horizon. This is either true, or apparent: In
the following Doctrine of the Sphere, I have fhewn how to
Calculate both for any Time and Place.
Sextile, is an Afpect of the Planets, when they are diftant
two Signs, 60 Deg. being a fixth Part of the Zodiack, and
marked thus *.
Siderial Tear, is the Space of Time the Earth is going
round the 12 Signs of the Zodiack in reſpect of the fixed
Stars, which is 305 Deg. 6h. 9′ 14″.
Sigat,
42
Aftronomical - DEFINITION´S.
Signs, are the 12 Signs of the Zodiack, Aries V, Taurus ŏ,
Gemini II, Cancer, Leo, Virgo, Libra, Scorpio M, Sa
gittary, Capricorn VS, Aquarius, Pifces H.
Sinifter Afpect, is made according to the order of the Signs
from Aries to Taurus, &c.
î
Sirius, one of the brighteſt Fixed Stars in the Heavens.
Slow in Motion: The Planets are always flow in Motion
when their Anomalies are o Signs.
Solar Year, is either Tropical, or Siderial; which fee under
thele Words.
Solſtice, is the Time when the Sun (apparently) enters the
Tropical Points Cancer and Capricorn; is got furtheft from the
Equinoctial, and before he returns back towards it, feeming
to be for fome Time at a ftand, viz. that part of the Ecliptic
before and after the Tropical Points, lies near parallel to the
Equinoct al, and confequently while the Sun moves thro' theſe
10 Deg. of the Ecliptic, his Declination is infenfibly altered.
The Summer-Solftice is called Estival; and the. Winter, Hye-
mal.
· Solfitial Colures, are two great Circles of the Sphere,
meeting in the Poles of the World, and cutting each other
at Right Angles, paffing thro' the four Cardinal Points;
That which paffeth thro' Aries and Libra, is called the Equi-
noctial Colure; and that which paffeth thro' Cancer and Ca-
pricorn, the Solftitial Colure.
Southing of the Stars, is the fame with the time of their
Culminating, or being upon the Meridian; for then they are
juſt got half way of their Journey betwixt their Rifing and Set-
ting.
Southern Signs, ſee Auſtral.
Spherick Geometry, or Projection, is the Art of defcribing on
a Plain the Circles of the Sphere, or any part of them, in
their juft Pofition, and Proportion; and of meaſuring their
Arks and Angles when Projected. The Circles of the Sphere,
as to their Projection on any Plane, are of four kinds.
1. The Primitive Circle, or Limb which bounds the Pro-
jection, and with which "tis always made.
2. A Direct Circle, whofe Plane is directly oppofite to the
Eye; or when the Eye is in the Axis of the Plain.
3. Of a Right Circle, whofe Plane is Coincident, (that is,
falling one upon another) with the Axis of the Eye, or with
the Viſual Ray.
4. An
Aftronomical DEFINITIONS.
43
4. An Oblique Circle whofe Plane lies Oblique to the
Axis of the Eye, fo that it makes unequal Angles with it.
Spots in the Sun, or Macula. 'Tis certain, thofe Opake
Maffes which we ſee thro' the Teleſcope at the Sun, are not
Planets revolving at any, even the leaft diftance from him;
but Spots, adhering to him, revolving but once in about
254 Days; by which we come to know the Sun's Rotation
round its Axis.
Stars, are thoſe Glorious ſparking Diamonds in the Canopy
of Heaven, moving in that wonderful Order which was
given them at the Creation by the Almighty Tetragrammaton,
who then gave them a Law that shall not be broken, Pfalm 148.
6. of which there are two forts, viz. Fixed and Earratic;
which Words fee.
Station, ir Aftronomy, fignifies certain Places in the
Zodiack, where a Planet being arrived, ſeems to ſtand ſtill
for fome time in the fame Degree and Minute, and is, their
being Stationary; which always happens just before and
after their being Retrograde. See Point of Station.
Succeffion, of the Signs, is that Order in the which they
are uſually reckoned; as, firſt, Aries, Taurus, Gemini, &c. See
Signs.
Summer, one of the four Quarters or Seafons of the Year;
which fee.
Summer-Solftice, fee Solstice.
Sun, was one of the feven Planets (but is now exempted)
and refteth fixed in the Center of the Planerary Syftem,
and gives Light, Heat, and Motion to all the feven Pla-
nets.
Sun's Beams. A Star or a Planet is faid to be under the
Sun's Beams until they be more than 17 Deg. Elongated
from his Body, either before or after him; for till then they
cannot be ſeen with the naked Eye.
Sunday Letter, fee Dominical Letter.
Superiour Planets, are Mars, Jupiter and Saturn; they are fe
called, becauſe they move in Orbits round the Sun, which are
larger than that of our Earth, and ſo are above us with re-
gard to the Sun, and can never come between our Earth and
him.
Swift in Motion. All the Planets are fwift in Motion when
their mean Anomaly are fix Signs,
¿
Symbols,
1
44
Aftronomical DEFINITION s.
Symbols, are Marks or Signs of things invented by an Artiſt,
and peculiar to ſeveral Sciences, by which the knowledge of
the things themſelves is always more expeditiously, and moſt
times, more clearly convey'd to the Learner; eſpecially after
a little he hath enured himſelf to them. What Symbols I
make uſe of in this Treatife, are thefe following:
Given.
• Required.
R. Radius.
+ Plus, more.
-Minus, Lefs.
× Multiplication.
- Divifion.
Equal to.
=
cr. Side.
crs. Sides.
< Angles.
<< Angles.
z Sum.
X Difference,
Square.
Cube.
✓ Root.
S. Sine.
C. S. Co. Sine.
Sec. Secant.
C. Sec. Co Secant.
T. Tangent.
C. t. Co. Tangent.
Degrees.
Minutes.
?
Seconds.
... As.
: To.
:: So is.
▲ Triangle.
Synodical Anomaly, in the Moon's Syftem, is, the Aggregate
of all her Anomalies in one, viz. her Mean, Equated, Correct,
and Laftly, her Synodical Anomaly.
Synodical Month, is the fpace of time, viz 29. Days, 12
Hours, 45' contain'd between the Moon's parting from the
Sun at a Conjunction, and returning to him again; during
which time ſhe puts on all her Phaſes.
Synodical Revolution, is that Motion whereby the Moon's
whole Syſtem is carried along with the Earth round the Sun.
Syftem, properly, is the regular, orderly Collection or Com.
putation of many things together. In Aftronomy, the Syftem
of the World is the Örder wherein the Planets, move round
the Sun, of which there are ſeveral forts; but are all Exploded
except the Copernican.
Syzygia, in Aftonomy, is the fame with Conjunction of any
two Planets, or Stars; or when they are referred to the fame.
Point in the Heavens; that is, being in the fame Sign, Degree,
and Minute of the Ecliptic, by a Circle of Longitude paffing
thro' them both.
T.
Aftronomical DEFINITION 5,
45
}
T.
Tables Aftronomical, are fuch as are annexed hereunto.
Taurus, is the ſecond Sign in the Zodiack, unto which the
Earth comes about the 12th Day of October; it is marked
thus Ŏ, and in Calculations numbred with the Figure 1.
Teleſcope, is an Inftrument by which we difcover Objects
at a diftance: The Reflecting Teleſcope is the beft; becauſe
it brings the Object you look at into the Tube: But here are
two Inconveniences which this fort are liable to; firſt, that
'tis not very eaſie by it to find the Object you would fee.
2ly, the Concave Meraline Object Speculum, is fubject to
tarniſh, ſo that it will not continue long good. Of this fort
I have lately feen at Mr. Hern's in Chancery-Lane, London,
which feem to be free from the above mention'd Inconveni-
encies.
Temperate Zone, are two ſpaces on the Earth contained be-
tween the two Tropicks and Polar Circles; fee Zone.
Terms, at Westminster; there be four every Year, during
which Time Matters of Juftice are diſpatched.
The firft is called Hilary-Term, which begins the 23d of
January, and ends the 12th. of February.
The Second is called Eafter-Term, which begins (always)
the Wednesday Fortnight after Eafter-Day, and ends the Monday
next after Afcenfion-Day.
The Third is called Trinity-Term ; it begins the Friday next
after Trinity-Sunday, and ends the Wednesday fortnight af
ter.
The Fourth is Michaelmas-Term, which begins the 23d
Day of October, and ends the 28th Day of November next föl-
lowing. Note, that Eafter and Trinity-Terms are moveable;
and how to find them Yearly, you will meet with in a Ta-
ble following.
Terms, begin three Days fooner at Doctors-Commons than at
Weſtminſter.
Oxford-Terms, are four, viz. Hillary, or Lent-Term begins
January the 14th, ends the Saturday before Palm-Sunday.
Eafter-Term begins the 10th Day after Eafter, exclufive;
that is, Wednesday Sennight following, ends the Thursday before
Whitfuntide.
Trinity-
46
Aftronomical DEFINITION´S.
Trinity-Term begins the Wedneſday after Trinity-Sunday, ends
after the Act fooner or later, as the Vice-Chancellor and Con-
vocation pleaſe.
Michaelmas-Term begins October 10, ends December 17.
Note, the Monday after the 6th of July the Act begins.
Cambridge-Terms; Lent-Term begins Fanuary 13, ends
the Friday before Palm-Sunday.
Eafter-Term begins the Wedneſday Sev'night after Easter,
ends the Thursday before Whit-Sunday.
Trinity-Term begins the Wedneſday after Trinity-Sunday, ends
the Friday after the Commencement.
*
Michaelmas-Term begins October 10, ends December 16.
Note, the first Tuesday in July the Commencement-Act be-
gins.
The Irish Terms are the fame as Westminster, except that
Michaelmas-Term, which begins October 13, adjourns to No-
vember 3, and from thence to the 6th; it hath feven Re-
turns.
The Scotch-Terms.
Candlemas-Term, begins January 23, ends February 12.
Whitfuntide-Term begins May 25, ends June 15.
Lammas-Term begins July 20, ends August 8.
Martinmas-Term begins November 3, ends November 29.
The two Learning-Vacations in the four Inns of Court, Lon
don, viz. the two Temples, Lincolns-Inn, and Gray's-Inn, begin
the firft Sunday in Lent, and the firft after Lammas-Day, and
continue three Weeks and three Days.
N. B. If the Beginning and End of any of theſe Terms
fall on Sunday, then the Beginning.or Ending of the fame is
on Monday next following.
1
J
Terraqueous Globe, fignifies the Terreftrial Globe, from
Terra and Aqua; that is, Earth and Water; as they both to-
gether conftitute one Spherical Body.
Theorem, is when fomething is propofed to be demonftra-
red.
Time, is a certain Meaſure depending on the Motions of
the Heavenly Bodies, by which the Diſtance and Duration of
things are Meaſured.
Time of Incidence, is the time from the Beginning to the
Middle of an Eclipfe, and in. the Moon's Eclipfe is always
equal to the Time of half Duration.
Time
Aftronomical DEFINITION S.
47
}
Time of Repletion, is the Time from the Middle of a Solar
Eclipfe, to the End thereof.
Torrid Zone, is the fpace on the Earth between the two Tro-
picks. See Zone.
Tranfit, in Aftronomy, fignifies the paffing of any Planet
juft by, or under any other fixed Star; or of the Moon co-
vering, or going clofe by any other Planet: Alfo the Tran-
fits of Venus and Mercury over the Sun's Disk are underſtood
in the fame fenſe; that is, when they pafs between us and
the Sun, ſo as to make a black Spot on his Body.
Tredecile, or Sefquiquintile, is a New Aſpect of 3 Signs, 18
Degr. and marked thus T. d.
Trine Afpe&t, is the diſtance of 120 Degr. or 4 Signs, and
is marked thus A.
Triplicate Ratio, in four Continual Proportionals, is the pro-
portion of the firft Term to the fourth. As, in theſe four
Numbers, which are proportional 2:4:6:12. See Proportion.
Tropicks, are two leffer Circles of the Sphere, parallel to
the Equinoctial, and 23 Degr. 29 Min. diftant therefrom,
being the Bounds or Limits of the Sun's greatest Declination,
North and South. That which lyeth between the Equinocti-
al' and North Pole, is called the Tropick of Cancer; and the
other between the Equinoctial and the South Pole, the Tro-
pick of Capricorn. When the Earth is arrived to the Tropick
of Capricorn, which is about the 10th Day of June, fhe ma-
keth longeft Days to all the Northern Inhabitants; and re-
turning towards the Equinoctial, when being arrived at the
Tropick of Cancer, which is about the 10th of Dec. fhe then
makes longeft Nights and ſhorteſt Days to all that dwell ou
the North fide of the Equinoctial; and to thofe that live in
South Latitudes, juft the contrary Appearances. You muſt
underſtand, that the Sun is always apparently Diametrically
oppofite to the Earth
Tropical Points, are the very Points where the Ecliptic
toucheth the two Tropicks, which is in the very be-
ginning of Cancer and Capricorn, where the Solftitial Co-
Ture cuts them.
True place. See Place.
Twilight, is that dubious half Light which we perceive be-
fore the Sun-rifing, and after Sun-fetting. 'Tis occafion'd by
the Earth's Atmosphere, and the Splendor of the Ether which
environs the Sun. The Ethereal accended Atmoſphere of
the Sun, not fetting fo foon as, and rifing before the Sun; and
the Sun's Rays alfo illuminating the Earth's Atmoſphere
before,
48
Aftronomical DEFINITION S
1
:
before the Body of the Sun can appear, occafions a Light al
ways preceding at the Rife, and fubfequent to the Setting of
that Glorious Body: Which, tho' becauſe of many accidental
Variations in both the Sun's and Earth's Atmoſphere, it cannot
be always of the fame Degree of Duration or Brightnefs; yet
it uſually holds in the Evenings, till the Sun is about 18 Degr.
below the Horizon, and appears fo long before his Rifing in
the Morning. But from a due Confideration of the Sphere it
felf, it will be eafie to determine in any Latitude where the
the Parallel of Declination interfects the Parallel of
18 Degrees For to the Complement of the Latitude, add
the Complement of the Sun's Declination, and the distance
of the Parallel of 18 Degr. from the Zenith (which always
is 108 Degr) if the half Sum of thefe three be equal to 108,
then that is the Day the Parallel of Declination cuts the
Parallel of 18 Degr. on which Day the Sun has fuch Declina-
tion as makes up that Sum above named, and is the Day that
there begins to be no Night, but Twilight which is about
May 11; and when the Sun has paffed the Tropick, there is
another Day of the fame Length with the former, which in
this our Latitude of 51° 32' North, will be found to be July 10,
on which Days the Sun has the fame Declination, and conſe-
quently muft rife and fet at the fame Hours: So that from
May 11, to July 10, there is no Night, but Twilight But when
the half Sum above mentioned is more than 108 Degr. then
there is perfect Darkneſs at Midnight; which how to work you
will find in the Doctrine of the Sphere.
Tychonian Syftem, is that Hypothefis framed by Tycho Brahe,
in which he puts the Earth at reft, as the Center of the Moon
and fixed Stars; but the Sun moving round the Earth, is the
Center of the Primary Planets. See my Syftem of the Planets
Demonftrated, in which I have Exploded this Syftem.
V.
Acuum, is by Phyfiologifts fuppofed to be a ſpace devoid of
all Body; and the Planetary Regions in which the Hea-
venly Bodies move, muft needs be fuch; for otherwife a
Refiſtance muft accrue to the Planets Motions, which tho'
never fo finall, would in time be fenfible, and have an effect
in retarding the Motion of the Heavenly Bodies: But no
fach thing hath yet ever been obferv'd, or difcovered, tho'
the contrary is certain. Befides, fuch a thin Vapour as the
Tail of a Comet, can move thro' the Ather, (as fome call it,)
with
}
Arftonomical DEFINITIONS.
49
with incredible fwiftnefs, without being diffipated or drawn
from its Natural Courfe; which is in it felf a Demonftration
that there must be a kind of Vacuum in thofe Celeſtial Regi-
ons.
Variation, in Aftronomy, fee Angle of Reflexion.
Venus, is the Name of one of the feven Planets, and is the
moft Splendid of all the Primary Planets: For when he is
Occidental, and at her greateft Elongation from the Sun, fhe
often fhines fo bright as to caft a fhadow on the Earth; fhe
has her Increaſe, and Decreaſe in Light as the Moon, and
moves in an Orb between the Earth and Mercury, making her
Revolution round the Sun in 224d. 46h. 19′ 24″, and is never
found further off the Sun than 48 Degr. fhe has the leaft
Eccentricity, but the greateſt Geocentrick Latitude: For when
Retrograde in, he will have more than 8 Degr. North
Latitude; and when Retrograde in R, her Latitude will be
9.Degr. South. Every Eight Years you may nearly find her
in the fame place of the Heavens.
}
W
Vertex, is that Point in the Heavens, juft over our Heads,'
and the fame with Zenith; which fee.
Vertical Circles, the fame with Azimuths, which fee.
Vefper, the Evening.
Vefpertine. In Aftronomy, when a Planet fets after the Sun,
it is laid to be Vefpertine.
7
Via Lastea, the fame with Milky-Way, which fee.
Vindemiatrix, a fixed Star of the third Magnitude in the
Conftellation Virgo.
Virgo, one of the 12 Signs of the Zodiack, being the Sixth
in order, and thus marked but in Aftronomical Calcu-
lations, number'd with the Figure 5. The Earth enters this
Sign about the 8th Day of February.
Vifible Conjunction, of the Sun and Moon (in Aftronomy) is
that which is feen by an Eye from the Earth's Superficies,
which always differs from the Time of the true Conjunction,'
except they be conjoyn'd in the Nonagefime Degree; and
then the true and viſible or apparent time is the fame: The
reaſon of which difference is, that the True Conjunction is
made by a Line fuppofed to be drawn thro' the Earth's Cen-
ter; and the Vifible, by a Line from its Superficies: From
hence it will follow, that if the true Conjunction fall in the
Oriental Quadrant, that is, between the Nonagefime Degree
and the Eaftern Horizon, the Moon's Place is put forward by
the Parallax of Longitude, and then will the Viñible Conjun-
tion be before the True. But if the True Conjunction fall
•
{
E
in
50
Aftronomical DEFINITION s.
in the Occidental, that is, between the Nonagefime Degree
and the Weſtern Horizon, the Moon's Place is then retarded,
or put back fo much as in the Parallax is Longitude; confe
quently the Visible or Apparent Conjunction will follow the
true Time. The knowledge of thefe are of very great im-
portance in the Calculation of Solar Eclipfes.
Under the Sun's Beams, is when a Star or Planet is within
17 Degr. of the Sun's Body, either before or after him ;
fo that then they cannot be ſeen with the naked Eye.
Univerfe; the whole Mafs of material Beings, as, Hea-
ven, Earth, Stars, &c. are called by this Name.
Vortex, according to the Cartesian Philofophy, is a Syftem
of Particles of Matter moving round like a Whirl-Pool: By
this they endeavoured to folve the Motions of the Heavenly
Bodies. But Sir Isaac Newton proved it falſe; and therefore
it is Exploded.
Uraniburg. Any place where you view or contemplate the
Heavens and Heavenly Motions, may be called by this
Name.
Urania, the Heavenly Mufe.
Uranofcopia, a View of the Heavens.
Ursa Major, the Great Bear, called alfo by the Greeks, Ar-
os, and Helice; being a Northern Conftellation confifting
of 27 Stars; and it is alfo called Charles's Wain.
;
Urfa Minor, the Leffer Bear, called by the Greeks Arcos
whereupon the North Pole is called the Pole Artick, or Helice
Minor; becauſe of the fmall Revolution which it maketh
round about the Pole; or rather of Elice, a Town in Ar-
cadia, wherein Califto the Great Bear, and Mother of the lef
fer, was bred. It is likewife called Cynofura, becauſe tho' it
carrieth the Name of the Bear, yet it hath the Tail of a
Dog. Theſe two laſt Conſtellations never riſe nor ſet in the
Horizon of London.
W.
WInter-Quarter, one of the four Seaſons of the Year.
See Quarter.
Winter-Solstice, is, when the Sun apparently, enters the
firſt Minute of the Tropical Sign vs, making longeft Nights and
fhortest
Aftronomical DEFINITION S.
51
fhorteft Days to all the Northern Inhabitants. It happens
about the 10th of December.
X.
XIphias, the Sword-Fiſh, a Southern Conftellation.
Y.
Ear, is the time the Sun apparently takes to go thro' the
Twelve Signs of the Zodiack. This is properly the Na
tural or Tropical Year, and contains 365d. 5 h. 48' 57";
during which space of time all the Variety of Seatons are
Celebrated.
Zenith, is the Point in the Heavens right over one's Head,
being Diametrically oppofite to the Nadir, and is always
90 Degr. diftant from the Horizon: And here Note, that the
Arch of the Meridian between the Zenith and the Equinoctial,
is always equal to the Arch of the Meridian contained be-
tween the Horizon and the Pole; which is the fame with the
Latitude of the Place.
Zodiack, is a Zone or Girdle, furrounding the Heavens,
and cutting the Equinoctial at oblique Angles at Aries and
Libra, to 23° 29', being equal to the Sun's greatest De-
clination; and in the middle of this Zodiack lies the.
Ecliptic, or Via Solis, the apparent way of the Sun, and
Earth. The Breadth of the Zodiack is 18° 30'; for that
will take in the Latitude of all the Planets; lefs Breadth
would do only for the Planet Venus, who has fometimes,
9 Degr.' of Latitude: The Zodiack is equally Divided into
twelve parts called Signs, and Eleven of theſe twelve
reprefent living Creatures, viz. all but Libra the Ballance ;
for the reft are the Ram, the Bull, the two Naked Boys, the
Crab-Fish, the Lion, the Virgin, the Scorpion, the Shooting-
Horfeman, the Goat, the Water-Bearer and the two Fiſhes.
Zone, in Geography, is a Space of Earth or Sea, contained
between two Parallels of Latitude; and there are five in
Number, viz. Two Frigid, or Frozen; Two Temperate, and
One Torrid, or burning Zone.
The Frigid, are thofe Parts of the Globe comprehended
between the Poles and the Polar Circles: Therefore one
must be toward the North, and the other toward the South.
1
E 2
In
52
Aftronomical DEFINITIONS.
1
In the North Frigid Zone lies Iceland, Lap-land, Finmark,
Samajeda, Nova-zembla, Green-land, and fome part of North
America.
The South Frozen is not yet known, whether it contains
Land or Water. They are in breadth each 46° 58'. Theſe
Inhabitants are called Perifcii, becauſe their fhadow goes
round them.
The Temperate Zones lie one on the North fide the
Equator, between the Artic-Circle and the Tropic of Cancer;
the other on the South fide between the Tropic of Capricorn
and the Antartick Circle. Each of theſe is 42 Degr. broad.
Theſe Inhabitants are called Heterofcii: They caft their fha-
dow but one way.
The Torrid or Burning Zone contains all that Space be-
tween the Two Tropicks: The Breadth of this is 46° 58′ e-
qual to the Breadth of each of the Frigid Zones. The Inha-
bitants of this Zone are called Amphifcii, becauſe they caft
their fhadow round them; that is, at Noon fometimes to-
wards the North, and fometimes towards the South.
THE
53
1
THE
DESCRIPTION
And USE of the
SECTOR·
ļ
}
B
SECTION I.
Ecauſe the Projection of the Sphere, and
Geometrical Conftruction of Solar Eclip-
fes, are beſt performed by a Sector; and it
being that which I fall all along in this
Treatife make uſe of, I think it not imper
tinent to give my Reader a Page or two in
the Deſcription and Uſe of that Univeríal
and moſt uſeful Inftrument.
Euclid in his 9th Definition of his Third Book, fays, That
a Sector is a Figure contained under two Semidiameters,
and the Arch which ferves them for a Bafe.
This Inftrument is commonly made of Silver, Brafs, Ivory,
or Box-Wood, in length 6, 8, 9, and 12 Inches, with a Joint
like a Carpenter's Rule; fo that the faid Legs, together with
certain right Lines, drawn from the Center of the Joynt, con-
tain Angles of different quantities. The Lines that are com-
monly drawn upon the Face of this Inftrument, to be uſed
Sector-wife, are the Lines of Lines or equal Parts, numbred
with 1, 2, 3, to 10, and marked with LL; the x may
fometimes ftand for 10, the 2 for 20, or 200, &c. accord-
ing as the matter in hand requireth. Next thefe lies a Line
of Chords iffuing from the Center, and marked with ca
E 3
at
54
The Defcription and Uſe
at the End, and numbred 10, 20, 30, &c. to 60°; which
Chord of 60 is equal to the Radius of a Circle, or whole
Sine of 90 by Prop. 15. Book 4. of Euclid.
On the other Face of the Sector is a Line of Natural
Sines, numbred with 10, 20, 30, &c. to 90, and marked at
the end with SS. By the fides of the Sines lye two Lines
of natural Tangents iffuing from the Center alſo, and num-
bred with 10, 20, 30, &c. to 45°; becauſe the Tangent of
45, Sine of 90, and Chord of 60, are all equal to the Radius
of a Circle: Theſe Tangents are marked at the End with
TT. Between the Sines and Tangents on each Leg is a
Line offer Tangents, iffuing from two little Brafs Cen-
ters, and there beginning to be numbred with 45, 50, 60,
75, and marked with t. t.
This Line fupplies the Line of greater Tangents when
your Angle exceeds 45°. And on the fame Face with the
Chords, and equal Parts or Line of Lines, lies the Lines of
natural Secants, iffuing from two little Brafs Centers lying
betwixt the Chords and Line of Lines, and numbred with
20, 30, 40, 50, 60, 70, 75, marked with S, S. Theſe Chords,
Sines, Tangents, and Secants are all projected from the fame
Circle to the Radius of the Sector they are placed up-
on.
There are other Lines arbitrarily placed upon the Sector;
but tending nothing to my prefent purpofe, I fhall not
therefore trouble the Reader with their Deſcription or Uſe at
this time,
USE.
1. The Uſe of this Inftrment is fo very great through all
the Branches of Practical Mathematicks, that it ought to be
written in Letters of Gold.
And Firſt, I muſt explain the meaning of two Words ge-
nerally made uſe of in the Uſe of the Sector, viz. Lateral, and
Parallel.
When we fay the Lateral Line of Lines, Sines Tangent, or
Secant, we mean, that Line which is found upon the Face,
or Side of the Sector. And to take off a Parallel, Sine, &c.
is to ſet one Foot on your Compaffes on the Sine of 40, &c.
on one Leg, and the other Foot on 40 on the other Leg.
A Parallel Radius is when one Foot is fet in the Sine of 90
Degrees, or Tangent of 45, or Chord of 60, on one Leg, and
the other Foot on 90, 45, and 60 on the other Leg.
The
1
of the SECTOR.
55
}
1
The Line of Lines are actually Divided into 100 equal
parts each ; but we have only 10 put to them, which may
fignifie either themſelves alone, or to times themſelves, or
100 times themfelves, or 1000 times themſelves, as occafion
fhall require; fo that when you lay down, or take off any
Number of equal Parts, fet 1o, 10 to the given Radius, and
fet one Foot of your Compaffes on one Number, as fup-
poſe 70, and the other Foot on 70 on the other Line of Lines,
and extant is either 7, 70, 700, 7000, &c. according as the
the nature of your Queftion requires.
Theſe Line of Lines are uſeful to increaſe, or diminiſh a
Line in a given Proportion, to divide a given Line into any
Number of equal Parts; to find the Proportion between two
or more given Lines; to find a third Proportion to two gi-
ven Lines; or three Lines being given, to find a fourth Line
proportional to them; to find a mean Proportional between
two given Lines; or, to divide a Line in fuch manner, as a-
nother Line is already divided.
2. To open the Sector, that the two Lines of equal Parts
may make a Right Angle, if the whole lateral Length be ap-
ply'd over between 8 and 4; becauſe 8+ 6 = 100,
by 47 of the firſt of Euclid; and the Line of Lines then
on the Sector will ftand at a Right Angle.
3. The Line of Lines may be opened to a right Angle,
if the Lateral Line of 90 be applyed over parallel between
45 Deg. and 45. Dege. in the Sines; or if the Lateral Line
of 45 be apply'd parallel over the Line 30, and 30.
4. Line of Chords may be opened to any particular
Angle, by taking out the Lateral Chord of the Angle re-
quired, and applying it over in the Parallel of, 60, 60; and
you will have thofe Lines feverally to ftand open at the
Angle propofed.
Example, I would open the Line of Chords to an Angle
of 20 Degrees.
Take the Lateral Chord of 20, and apply it over Pa-
rallel on 60, 60; and then thoſe Lines ftand open at an An.
gle of 20, as was required,
E 4
1. 0.
56
The Defcription and Ufe
5. On the contrary, if the Sector be opened to any An-
gle at venture, you may find the quantity of it thus, viz.
take the parallel Chord of 60 Deg. and meaſure it on the
lateral Chord, and that will give the Angle that the Line of
Chords then ftands at. And obferve the fame of the Lines
of Sines, by confidering that the Sine is half the Chord
of the double Ark. As, if it were required to open
the Sector in the Lines of Sines to an Angle of 40; take
out the lateral Chord of 40, and to it open the Sector to
the Chord of 60; fo fhall the Lines of Sines be opened
to the Angle required. Or if the Semi-radius be apply'd over
between the Line 30 and 30, it will open the Lines of Sines
to that Angle,
W
Example, I would open the Lines of Sines to an Angle
of 45 Degrees.
Divide the lateral Chord into two equal parts, and lay
that extant parallel over the Sines 30, 30, and that ſhall
open the Lines of Sines to an Angle of 45. Degrees.
·
Note, It is one thing to open the Edge of the Sector to
an Angle, and another thing to open the' Lines on the
Sector to the fame Angle: For when the Sector is clofe
fhut, the Edges of it make no Angle; but the Lines of
Lines, Sines, Tangents and Secants, make then an Angle
of near 6 Degrees.
6. If you would examine the Lines of Chords, Sines,
Tangents and Secants, whether they be truly made, Pro-
ject them from a Circle of the fame Radius; and if you
would prove if the Sines, and Chords are truly projected
from the fame Circle, then open the Sector-Lines ftreight out
at length, and take the Sine 10, 10; that is, fet one foot of
your Compaffes on 10, on one Leg, and extend the other to 10
on the other Leg in a ftreight Line; carry this extant to the
Line of Chords; fet one foot in the Center, and the other
foot will exactly reach to the Chord 20 Degr. if your Sector
is truly made. The fame obferve of any other Degrees on
the Sines.
7. To Lay down an Angle of any quantity of Degrees.
This
of the SECTOR.
57
[
This may be performed, either by the Lines of Chords,
Sinės, Tangents, or Secants, due regard being had to each
particular Line.
1. By the Chords. Take the defign'd Radius in your
Compaffes, and open the Sector in 60, 60, on the Line of
Chords; then take parallel-wife the given Angle in your
Compaffes, and lay it down, and 'tis done.
Do fo for the Sines, by applying the Radius over the Sine
90, 90 and the Tangent if lefs than 45, over 45, at the end
of the Line, but if your Angle exceed 45 Degr. then you muſt
fet the Radius over the leffer 45, where are two fmall Brafs
Centers nearer the Joynt of the Sector alfo your Secants
muſt be taken from thofe Centers, on the other face of the
Sector, where 'tis marked at the end with S. or fometimes
with Sec.
་་
Example. Let it be required to lay down an Angle of 40
Degrees by the Lines of Chords, Lines, Tangents, and Se-
cants on the Sector. Draw A B, equal to your propoſed
Radius, and ſtrike the Arch; which take in your Compaffes,
and fet one foot in 60 of the Line of Chords; open the Sector
till the other foot fall in 60 on the other Leg of the Sector.
Now it is fitted to the given Radius; fo that any quantity of
Degrees may be laid down or meaſured to anſwer that Ra-
dius. But in the prefent Example 'tis only 40 Degr. Take
therefore 40 from the Line of Cords parallel-wife and fet
ir on the Arch B C; Draw A C, and the Angle B A C is an
Angle of 40 Deg. as was required.
But if your Angle exceed 60 Deg. then you muſt take
half, and lay it down twice; as fuppofe 80 Deg. take the
Cord of 40, and turning it on the Arch twice, fhall give you
the true Chord of 80: And fo of any other above 60.
2. By the Sines. Take AB in your Compaffes, and open
the Sector on 90, 90, to that extant; then take off the parallel
Sine of 40 Deg. and fet one foot in C, the other will reach
to E, and that is the Sine of the Arch B C, which meaſures
Angle B A C.
3. By the Tangents. Take A B, in your Compaffes, and
open the Sector on the Tangents of 45 Deg. to that extant ;
then
1
$8
The Deſcription and Uſe
"
then take off the Tangent of 40, parallel and fet one foot in
B, the other will reach exactly to D; then is B D the Tan-
gent of the Arch B C 40 Degrees.
;
4. By the Secants. Take A B in your Compaffes, and open
the Sector on the fmall Brafs Centers which lye toward
the Joynt of the Sector to that extant
then take off the parallel Secant of 40,
and it will, reach from A to D ; fo is
AD the Secant of the Arch B C, and
that meaſures the Angle B A C 40 Deg.
as was required.
A
A40
EB
;
And thus may any right-lined Angle
be either meaſured or laid down, by the
Line of Chords, Sines, Tangents, and Secants. And if you
have occaſion for the Verfed Sine of any Ark under 90, it
may be had by fubtracting the Natural Sine of the fame
Ark from the Radius; the Remainder is the Complement
of the Verfed Sine required; thus,
From Radius Sine of 90°
is
Natural Sine of
40
fub.
Verſed Sine of
50
is
1
10.000000
6.427876
3.572124
And the verſed Sine to any Angle above 9º, is had by,
adding the Natural Sines Excefs above 90 to the Radius:
Thus,
To Radius
900
10.000000
Add Natural Sine of
Verfed Sine of
130
40 add 6.427876
is 16.427876
And the Secant of any Ark is had by fubftracting the
Co. Sine out of the Double Radius: Thus,
Double Radius
An Angle of 40, its Co. Sine is
The Secant of 40 is
20.000000
9.884254
10.115746
Note, That Radius is a Mean Proportional berween the
Tangent of an Arch, and the Tangent Complement of the
fame Arch.
?
Demon-
of the SECTOR.
f
59
Demonftration per Euclid 13, 6.
Take the Tangent c D, and
fet it from 4 to b, in the lower
Diagram; take the Co. Tan-
gent be, and ſet it from b to c:
Biffect a c in e, on e as a Cen-.
ter with the Radius e a ec;
ftrike the Semicircle a D c, at
b; erect a Perpendicular to
touch the Periphery at D'; fo
fhall b D be the Geometrical
Mean, and is equal to the Ra-
dius ab of the upper Dia-
gram.
Alfo Radius is a mean
Proportional between the
Sine of an Arch, and the
Secant Complement of
the fame Arch, as is
proved above.
D
e
b
G
SECT-
:
{
涕
​SECTION II..
Of Spherick Geometry.
:
2
WHoever would be an Aftronomer, it is very requifite
that he be well-grounded in Numbers; and thereby
acquainted with Euclid's Elements; and that he may the bet
ter go thro' the Doctrine of the Sphere, I fhall here fubjoyn a
few Propofitions introductory to the Projection of the Sphere
on any Circle.
1. The Angle
of Interfection
of any two Cir-
cles on a Plane,
is equal to their
Angle made by
1
1
e
their Radii,
drawn from
their Centers, to
the Point of In-
terfection.
Conftruction. To the Point of Interſection 4, draw a e, a
Tangent, to the Circle a cd, and d 4, a Tangent, to the
Circle b a e.
Demonftration. Becauſe the infinitely fmall Portions of
the Circles, do coincide with the Tangents, and confequent-
ly have the fame Direction; therefore the Curv'd Lin'd An-
gle bac, formed by the two Circles, is equal to the Right
Lin'd Angle d af = b a ¢ 60°: And becauſe the Angle
dac Angle bae is a Right Angle, take away from each
the interjacent Angle dab = cae; there will remain the
Angle bac = Angle fa d= Angle ga e, which was to be de-
monftrated.
=
2. All
Of Spherick Geometry.
61
2. All Angles made by Circles on the Superficies of the
Globe, are equal to thofe made by their Repreſentatives on
the Plane of the Projection.
+
Suppoſe the Eye at a; project the Spherical Angle gbb, and
draw the Tangent b k to the Circle bb; alfo draw the Tan-
gent b to the Circle gb;
+
from the Point b, draw
be paralled to D h, and
joyn a c; draw ki at
Right Angles to Dk, to
meet the Tangent b i in
i; and then Draw f i.
Now by the 32,3. Euclid,
the Angle k ba, Angle
a c b = abc per Euclid
21, 3; and bƒ k = An-
bf
gle kb f per 29, I:
Therefore bk = fk by
the 6, i. Then in the
Triangles i kf, ik b, ik,
kf=ik, kb, and the
Angle i kƒ¨¨Right.
1
a
CMK
Wherefore the Angle ifk upon the Plane of the Projecti-
on, is equal to the Angle bk, forming the Spheric Angle
gbh; which proves the Angle made by the Circles is equal
to the Angle made by their Tangents. For the Eye at a,
Projects the Tangent b k in fk, and the Tangent bi in fi;
the Angles ifk, i b. k being both bounded by ik.
Or, as in the following Scheme, Draw all the Line; then
will the Angle a dm, be equal to the Angle a qd, in (the
Oppofite. Segment) which is qad; becauſe the A is an
fofceles; but the Angle q da = Angle higd; becauſe q dis
parallel to e g; therefore the Angle hg d = Angle a d m =
d'm
vertical Angle bdg; wherefore the A bag is an Ifofceles,
bd
and confequently db ishg.
Hence
1
62
Of Spherick Geometry?
{
a
m
\h
2
Hence the Trian-
gle 'Od h, hath two
Sides, oh and hd,
and the Angle obd
to two Sides,
and one Angle in
theo bg. Where-
fore all things are e-
G qual, and Confè-
quently the Angle
ob g Angle odb
.i
•
to the Curvilineal
Angle, made by the
two Circles.
PROPOSITION 1.
To find the Pole of any Great Circle.
F it be the Primitive Circle, its Pole is at e, the Cen-
ter.
1
2. If the Pole of a Right (cd) or
Perpendicular (a b) be fought, 'tis
90 Degrees diftant upon the Limb
from the Point where the Circle
cuts it. So the Pole of abis c
ord; and the Poles of c d are 4,
and b.
The Poles of every fmall Circle
are the fame with the Poles of that
great Circle to which they are Pa-
rallel.
PROP.
Of Spherick Geometry.
63
PROP. 2.
ROP.
To find the Pole of an Oblique Circle.
First, Confider, that this Circle muft cut the Primitive
in two oppofite Points, as in the Cafe of all great Cir-
cle.
2. The Pole of this Circle muſt be in a Line perpendicular
to its Plane.
K
3. This Circle's Pole muſt always lye between the Center
of the Primitive, and its own.
Let a bd, and a cd,
be two oblique Circles,
whoſe Poles are requi
red. A Ruler laid from
a to b and c, gives g and
h; take the Chord of
90. Degrees and fet it
from g to k and from
tok
b to i. Then a Ruler
laid from kto i feverally,
gives m, and L the two
Poles fought: So m is
the Pole of the Oblique
Circle, a bd, and I
of the Circles
a cd.
Note, The Pole of every
1
Circle falls in the Diameter of that Circle.
I
m
مة
+
:
1
PROP.
Of Spherick Geometry.
1
PROP 3.
To lay down on the Projection of any Angle required.
alamin
འད་
Here are three Cafes.
1 The Angle at the Center of the Primitive..
2 In the Periphery.
3 Any where in the Circle; but neither in the Center nor
Periphery.
ry.
a
P
First, An Angle at the Center is
laid down by the Line of Chords on
the Periphery, thus: I would have an
Angle of 51032. Take the Chord
H. of 51° 32' and fet it from H to P,
draw Pa and 'tis done. For the
Angle PaH is 51° 32.
1
Secondly, That the Angle-Point may be in the Periphe
d
•
RULE.
?
Take the Secant of the given Angle in your Compaffes,
and ſet one Foot at b, where you defign the Angular Point,
and with the other make a Mark
in the Diameter at e, that fhall be
the Center of a Circle that fhall
make an Angle at the Circumfe-
rence, as was required. Example,
I would have an Angle of 40 De-
grees, and the Angular Points to
to be at b and c.
b
JIANRAI
d
F
A
Open the Sector to the Radius of the given Circle fa, on
the Line of Secants, and take off the Secant of the given
Angle 40 Deg. carry this extant, and fet one Foot in bor c,
and with the other make a Mark in the Diameter at e ; fet
one Foot in e, and fweep the Arch b d c, fo fhall the Angle
abd
Of Spherick Geometry.
65
f
a b d a c d be an Angle of 40 Degrees as was requi-
red.
Thirdly, To make a Spherick Angle, where the Angle-point
is in any Point given within the Periphery, but not in the
Center.
Example. I would make an An-
gle of 50 Degrees at the Point b.
Rule. Lay a Ruler from d to the
given Point b, and it gives e in the
Periphery. Draw ge to meet the
Tangent of; Take cf, and fet
from g to b; Draw hi Parallel
to dg Take the Co. Tangent of
the given Angle, viz. 40 Deg. and
fet it from h to i, fo is i the Cen-
ter; on which draw (with the Ra-
L
ah
"1
:b
I
dius i b,) the Arch b L fo is the Angle ab L 50 Deg. as was
required.
PROP. 4
To measure any Spherick Angle, when Projected.
Here are three Cafes.
1. When the Angle-Point is at the Center of the Primitive
Circle.
2. When the Angle-Point is at the Periphery.
3. When the Angle-Point is within the Primitive, but not
in the Center.
First, To Meaſure an Angle at the Center of the Primitive
Circle.
You have no more to do, than to take the Arch in your
Compaffes, that is terminated by the two Legs of the An-
gle, and apply it to the Line of Chords, and 'tis done;
which being fo plain, needs no Example.
F
2. To
1
}
66
Of Spherick Geometry.
2. To Meaſure a Spheric Angle, the Angle-Point being
at the Circumference.
RULE.
By Prop. 2. find the Pole of the given Oblique Circle;
then meature the Diſtance on the half Tangent between the
Center of the Primitive; and the Pole of the Oblique Cir-
cle, and that's the Quantity of the Angle fought.
a
b
e
C
Example. Let the An-
gle abd be required :
First, The Pole of the
Oblique Circle b dc is
at f, the diſtance ef on
the Semi-Tangents is 40,
the quantity of the Angle
a b d a c d.
But if the two Poles are not in the fame Diameter,
then lay a Ruler to the Angle-Point, and to thoſe Poles fe-
verally, and that will reduce them to the Primitive Cir-
cle; which meaſure on the Line of Chords, as was taught
in the first hereof.
b
Id
n
e
Example. Let the Angle
d be be required to be mea-
fured: First, The Pole of
the Right Circle bc is at
n. by Prop. 1. and the Pole
of the Oblique Circle bd c,
is at f; lav a Ruler from b,
the Angle-Point to f, gives g
in the Circumference; and
laid from b to n, gives n; the
Chord ng, is 50 Degr. the
quantity of the Angle
fought.
Thirdly,
Of Spherick Geometry.
619
1
Thirdly, To meaſure a Spheric Angle when the An-
gle-Point is within the Primitive, but not in the Cen-
ter.
A
Rule. Find the Poles of
the two Oblique Ctrcles
that limit the Angle to be
meaſured by Prop. 2; and
a Ruler laid to the Angular
Point, and to thofe Poles
feverally, will reduce the MK
Angle to be meaſured, to
the Primitive Circle; which
meaſure on the Line of
Chords, as by the first here-
of.
Z
P
B
e
d
H
N
Example. Let it be required to Meaſure the Angle
PB Z.
First, Draw the Diameter e a of the oblique Circle PBS
the Pole thereof falls at c. Secondly, Draw the Diameter M H
of the Oblique Circle Z B N, the Pole thereof falls at d; a
Ruler laid from B the Angular-Point to c and d feverally,
gives ƒ and g in the Primitive; the Arch ƒg measured on
the Line of Chords is 18 Degrees, the quantity of the An-
gle required.
PROP. 5.
To measure the quantity of Degrees of any Arch of a great
Circle.
1. If the Arch be part of the Primitive, 'tis meaſured on
the Line of Chords.
2. If the Arch be any Part of a Right Circle (that is, a
Diameter that paffes thro' the Center of the Primitive) then
lay a Ruler from the Pole of the Right Circle, to the two
Extremities of the Portion of the Right Circle that is to
be meaſured, and it will give you two Points in the Primi-
tive; which meafured on the Line of Chords, is the quan-
F 2
city
68
Of Spherick Geometry.
}
tity of the Right Circle in Degrees and Minutes, as was re-
quired.
Example. In the laft Scheme, if it were required to mea-
fure the Part e d, of the Right Circle K, H, I lay a Ruler
to z, its Pole, and to e and d, gives N and i in the Primitive;
then open your Sector to the Radius e N, &c. and take off
Ni, and applying it parallel on the Line of Chords, gives
30 Degrees, the quantity of e d, as was required.
Or, the Portion of any Right Circle may be meaſured by
the Scale of half Tangents; fuppofing the Center of the Pri-
mitive to be in the beginning of the Scale; fo that if the
Degrees are to be reckoned from the Center, you muft ac-
count according to the Order of the Line of half Tan-
gents.
But if the Degrees are to be accounted from the Periphe-
ry of the Primitive (as will often happen) then you muſt be-
gin to account from the end of the Scale of half Tangents,
calling 80, 10; 70, 20, &c.
3. To meaſure any part of an Oblique Circle.
First, Find its Pole; there lay the Ruler; Reduce the
two Extremities of the Ark required to the Primitive Circle,
and then the diſtance between theſe Points on the Chords,
is the Quantity fought.
a
Thus in the laft Figure, if the quantity B K of the Ob-
lique Circle PBS were required, à Ruler laid to its Pole at
c, and to B and K, will give the two Points L, m, in the Pri-
mitive, which diſtance L m, on the Line of Chords is 63
Degrees, which is the quantity of B K, as was requi-
red.
PROP. 6.
To draw a Parallel Circle.
1. If it be to be drawn parallel to the Primitive Circle, at
any given diſtance, draw it from the Center of the Pri-
mitive with the Complement of that diſtance taken from
the Line of half Tangents.
1
2. If
Of Spherick Geometry.
69
་
F
"
1
1
2. If it be to be Drawn
parallel to a Right Circle
as, fuppofe cd parallel to ab
were to be drawn at 23°
diftant from it; from
the Chords take 23 Deg.
, and ſet it on the Primi-
mitive from a to c, and
from b to d; or fet
Complement 660
the Pole of a b,
Points c and d.
its
from P
to the
1
P
Then take the Tangents of the Parallel-diftance from the
Pole of the Right Circle a b, which is here 661; fet one
Foot of the Compaffes in c and d feverally, and make two
occult Arches, whoſe Interſection ſhall be the Center of the
Circle cd; and thus are the Tropicks and Parallel of Decli
nation drawn in the Stereographick Projection.
3. If it be to be drawn Parallel to an Oblique Circle.
Rule. From the Line of half Tangents, lay off the Paral-
lel diſtance from the Pole of the Oblique Circle given, both
ways in that Diameter of the Primitive Circle; and Note
thofe Marks: Then theſe Points Binfected, give the Center
of the Parallel fought.
Example. Let it be requi-
red to draw a Circle Paral-
lel to the Oblique Circle
abc, at the diſtance of 40
Degrees. First, find f, the
Pole of the Oblique Cir-
cle; then Meaſured f, the
diſtance of the two Poles
on the half Tangents, which
you will find to be 34 De-
grees; to which add 50 (the
Complement of the defign'd
Parallels diftance from the
a
h
b
le
500
C
34
784
X 16
Ob-
F 3
70
Of Spherick Geometry.
Oblique Circle,) to the Sum of 84; fet off the half Tan-
gents from d to g; then take the difference between 34
and 50=16, and fet it from d to h; Biffect g h in i; fo
is i, the Center of the required Parallel Circle.
Note, By this Prop. is the Path of the Vertex of any Place
drawn in the Copernican Projection.
PROP. 7.
To draw a great Circle thro' any Point making with the
Primitive Circle any given Angle.
Rule 1.
With the Tan-
h
a
d
gent of the given Angle,
fet one Foot in the Center;
deſcribe an Arch.
2. With the Secant of
the fame Angle, ſet one
Foot in the given Point;
ftrike an Arch, croffing the
former; the Interfection of
theſe two Arches is the Cen-
ter of the Circle required.
Example. Let it be required to draw a great Circle thro'
the given Point f, and to make an Angle with the Primitive
of 30 Degrees.
Open the Sector to the Radius ce, and take off the quan-
tity of the given Angle of 30 Degrees; fet one Foot of
the Compaffes in the Center of the Primitive Circle at e,
and Sweep the Arch at g: Open the Sector to the Radius
as above, at the little Center to the Sector, and take off
the Secant of 30; fer one Foot of the Compaffes in the
given Point f and ftrike the other Arch at g; the Inter-
fection of the Arches at g is the Center of the oblique Cir-
cle bfb, which paffes thro' the given Point ƒ and makes
an Angle with the Primitive of 30 Degrees as was re-
quired,
PROF
1
Of Spherick Geometry.
7፤
PROP. 8.
To draw a great Circle thro' any two given Points with-
in the Periphery of the Primitive Circle.
RULE.
1. Thro' either of the given Points and the Primitive Cir-
cle's Center, draw a Diameter, produceth it beyond the Pe
rimeter.
2. Cross this Diameter at Right Angles.
3. Through the Point mentioned draw a Line to the Ex-
tremity of the fecond Diameter.
4. At the end of this Line in the Periphery of the Primi-
tive Circle erect a Perpendicular, cutting the firſt Diameter
in a third Point.
5. Thro' the two given Points, and this third Point ftrike
a Circle, and it fhall be the great Circle required.
Example. Let it
be required to draw
a great Circle thro'
the two Points ƒ and
and
g.
1. Thro'ƒ and e
draw the Diameter
of the Primitive a
b.
2. Croſs it at
Right Angles with
the Diameter c d.
3. Draw a Line
K
either from f to c, or from ƒ to d.
G
الحق
4. To either of which lines fc, or fd, at the Extremity
of c or d, erect the Perpendicular ci, interfecting the
Diamerer a b, produceth in i, the third Point.
5. Thro' these three Points fg i, defcribe the Circle ƒgi
by the known Problem of finding a loft Center, Euclid 25, 3.
and 'tis done. Draw the Line KL; and if it paſs thro'
the Center of the Primitive, your Work is right; elfe not:
Becauſe then the required Great Circle cuts the Primitive
in two oppofite Points.
F 4
PROP.
1
72
Of Spherick Geometry.
PROP. 9.
To draw a great Circle Perpendicular to a given Great
Circle.
GENERAL RULE.
Draw a great Circle thro' the Pole of the given great
Circle, and it will be Perpendicular to a great Circle gi
ven.
Here are four Cafes.
1 Perpendicular to the Primitive Circle.
t
2 A Right Circle perpendicular to a Right Circle.
3 An Oblique Circle perpendicular to a Right Circle.
4 An Oblique Circle perpendicular to an Oblique Circle,
CASE I
To draw a Circle Perpendicular to the Primitive Circle given.
RULE.
Through the Center of the Primitive Circle draw a Dia
meter, and 'tis done: For the Center of the Primitive Cir-
cle is its Pole.
CASE II.
To draw a Right Circle perpendicular to a Right Circle.
RULE.
This is done by drawing the Diameter perpendicular to
the Diameter, or Right Circle given,
1
CASE
Of Spherick Geometry.
73
CASE III.
To Draw an Oblique Circle perpendicular to a Right Circle gi
ven.
RULE.
Find the Poles of the given Right Circle by Prop. 1.
Cafe 2.
2. Draw a Circle thro' thoſe two Poles.
Example. Let it be re-
quired to draw an Oblique
Circle perpendicular to the
Right Circle cd.
Lay the Chord of 90
from c to a and b, and
draw the Diameter a b; for
a and b are the Poles of
the Right Circles c d. With
any diſtance of the Com-
paffes fet one Foot in a and
b, and ſtrike the two Ar-
ches at f; then is f the
Center of the Oblique Cir-
cle ag b, and is at Right
•
b
a
9
d
Angles with the given Right Circle e d, as was requi-
red.
Note, That if the Oblique Circle required to be drawn,
were fo limited as to make a given Angle with the Primi-
tive Circle;
•
Then take the Secant of that Angle (as in Cafe 2. Prop.
3.) in your Compaffes; and placing one Foot in a or b
make two Arches croffing each other as before at f;
then ſhall ƒ be the Center: Or, take the Tangent of that
Angle, and fet it off in the Right Circle c d from the Center
e to f, and 'tis done.
But if the Point g be given, thro' which this Oblique Cir-
cle fhould paſs without any relation to the Angle it ſhould
make with the Primitive Circle, (tho' that be naturally given)
a Circle ftruck thro' the three Points 4 g and b, will anfwer
the Demand.
a
CASE
74
Of Spherick Geometry.
1
CASE 4.
To draw an Oblique Circle perpendicular to a Given Oblique
Circle.
RULE.
1
2.
Firſt, Find the Pole of the given Oblique Circle by Prop.
2. Thro' that Pole draw a great Circle; or, which is the
fame, draw fuch an Arch as may pafs thro' the Pole fo
found, and may interfect the Primitive in Points Diame-
trically oppofite.
a
h
d
K
2
Example. Let it
be required to draw
an Oblique Circle
perpendicular to the
Oblique Circle a b è̟.
First, I find the Pole
of the Oblique Cir-
cle a b c to be at g
then I lay a Ruler o-
ver the Center d, of
the Primitive (any
wife, becauſe it muſt
cut the Primitive in
oppofite Points) cut-
ting it in h and i;
Points i gh, and it
find the Center that will ſweep the three
fhall be the Center of the Great Circle required, and cuts
the great Oblique Circle a be at Right Angles at K. Note,
If the Point K in the Circle a b c be given, then draw a great
Circle thro' the two Points g and K by the 8th Prop. And if
it be required that the Circle igb, fhould make any gi-
ven Angle, it may be done by the 7th Propofition.
·
•
1
}
SEC-
75
SECTION III.
The Projection of the Sphere Orthographically and Stereo-
graphically, on the Planes of the Meridian, Ecliptic, and
Horizon.
PRO B. I.
To Project the Sphere Orthographically on the Plane of the
Meridian.
TAKE any convenient Radius from the Chord of 60 Deg.
on the Sector, and fweep the primitive Circle, which
doth always repreſent the Meridian of the Place projected
upon the Solftitial Colure. Draw HH for the Horizon, and
N for the Eaft and Weft Azimuth, the Zenith, and N
11 10.
Z
હ
G
a
12
1 2
H
M
T
b
R
F
A
b
H
N
B
5
101)
N
321
A
the
!
76
The Projection of the Sphere.
the Nadir of the Place; take the Chord of the propofed La-
titude(as fuppofe 51° 32') and fet it from H to P, and from z to
Æ, draw PS for the Earth's Axis, and EE for the Equi-
noctial Take the Cord of, 230 29', the diftance of the Tro-
picks from the Equinoctial, and ſet it from Æ on the Meridi-
an each way, and draw 12, and VS 12, parallel to the E-
quinoctial: Alfo fer the fame Chord of 230 29' on the Me-
ridian from the Poles at P and S, each way, and draw the
Polar Circles parallel to the Equinoctial: Take the Chord of
189 and fet it from H to I and I under the Horizon, which
L.
fhall be the Parallel of Twilight. Draw, VS for the Ecliptic,
and on it from the Center of the primitive Circle fet the
Sines of 300 and 60° each way, and place the Signs of the
Zodiack, and to fall in the Center, and and V8 at the
V
VS
Circumference: A A and a a are Parallels of Altitude; the firft
being drawn by the Cord of 20 Deg. and the other of 40 ; the
Meridians or Hour-Circles I have only drawn from Tropick
to Tropick, which if they were continued; would meet in the
Poles, and are Ellipfes, and are drawn, as I fhall now fhew in
the Azimuths.
First, Draw as many Parallels of Altitude as you pleaſe; as
here I have drawn one at 20, and another at 40 Deg. (the more
the better; then take the Sine of what Azimuths you defign
to project, from the Radius of the primitive Circle, and fet it
from the Center on the Horizon; as here I would draw the
45th and 11th Azimuths from the Eaft and Weft. I fet them
off feverally to m and n; theſe are the Points in the Horizon
thro' which the faid Azimuths muft pafs: But to find the
Points in the Parallels of Altitude thro' which they muft pafs,
take half the Parallel of Altitude, and make it the Radius of
the Line of Sines on the Sector=q A; from it take the Sine
of 45° and ſet it from R to q on the Parallel; alſo make a v
the Radius of the Line of Siñes on the Sector, and take from
it the Sine of 45° the required Azimuth, and fet, it on the
Parallel from v to x; fo are the Points x q and m, the Points
thro' which the 45th Azimuth muft pafs. Thus you muft find
Points under the Horizon, and by an even hand draw the
Elliptical Azimuths, which will all meet in the Zenith and
Nadir. And after the fame manner are the Hour-Circles
or Meridian drawn, by firft drawing as many Parallels to the
Equinoctial as you pleaſe towards the Poles; and by finding
Points, as has been fhewn in the Azimuths, firft marking them
in the Equinoctial by the Sines of 15,30,45,60 and 75°. The
Pro.
The Projection of the Sphere.
:77
•
or
Projection being finiſhed, we will fuppofe the Sun in 10°
20: (For then the Declination is the fame, viz. 14º 50º
28" N.) Draw DL, by fetting off the Chord of the Declinati-
on, which ſhall repreſent the Parallel of the Sun's Declination
for thoſe Days: Then
Fr
I B
B E
Y
will
be the
ED=
ԴՐ
b
B
D
B
L
B;
Longitude
Rifing or Setting.
Afcen Difference.
Time
is due Eaft or Weft.
Altitude a little before 9 a-clock.
Declination.
Right Afcenfion.
Semidiurnal Ark.
Seminocturnal Ark.
Amplitude.
To him that underftands how to lay down an Ellipfis by
the Line of Sines, the method of projecting this will become
exceeding eafie. This Projection I have contrived on Braſs,
fo that the Body of the Work moves within the Graduated
Meridian, to any Latitude; and upon it I have a moveable
Horizon with two Sights to take the Sun's Altitude, by which
it becomes an Univerfal Dial, and is of excellent ufe for
Sea-faring Men to find their Latitude and Hour of the Day:
For having the Sun's Zenith-Diſtance, (as fuppofe 20 Deg.
to the Northward) on the 6th Day of July, the Sun's Declina-
tion 21 Deg. 19' N. with your Pen mark on the Meridian the
Sun's Declination, and move the Projection till that Point
, touch 20°, the Zenith-Diftance, and then doth the North-
Pole of the Projection cut the Meridian at 1 Deg. 19 Min.
above the Horizon, and fuch is the Latitude of that Place
of Obfervation North. Here the Work by the Pen is not
only faved, but alſo Demonftrated.
PROB.
78
The Projection of the Sphere.
PRO B. 2.
!
To Project the Sphere Stereographically upon the Plane of
the Meridian.
༢
With the Chord of 60 Degrees fweep the primitive Cir-
cle; draw H H for the Horizon, and z N for the Eaft
and Weft Azimuth, the Zenith of your Habitation, and
N the Nadir. Take 51 Degr. 32 Min. the Latitude from
the Line of Chords, and fet it from H to P, and from
Z to Æ, Draw PS for the Axis, and Æ Æ for the
!
Æ
i
H
H
Æ
И
Equinoctial; take the Chord of 23 Degr. 29 Min. and
fet off the Tropicks and Polar Circles as in the laft Pro-
jection. Note, The Tropicks and all Parallels of Declination,
&c. Hours, Circles, Azimuths and Almicanters are drawn by
the Tangents of the Angles that they form with the Peri-
phery
The Projection of the Sphere.
79
phery of the Plane of the Projection: For as in the Ortho-
graphick Projection they were Ellipfes, here they are Cir
cles: Then, to draw the Tropicks; becauſe they are 66
Degr. 31 Min. fer one Foot in the Axis (being fuppofed
to be continued) and draw the Tropicks. Or, if the Secants
of their feveral Diſtances from their neareſt Pole, be fet off
in the Axis from Y, the Center of the Projection on the
fame fide of the Equinoctial as they lye, you will have
their feveral Centers, and the Tangent of the fame Diftance
will be their Radii, or Semidiameter. Therefore the
10
20
880
70
23 29
60
31
30
60
Parallel of 40
is drawn by the
250
50
Tangent of
40
бо
30
66 31
23 29
70
80
And the Hour-Circles
20
LIO
of
2 &
15
are drawn by
the Secants of45
30
бо
75
75
and marked in the Equi-60
noctial by their half Tan-45
gent from , of
.30
15
The Hour-Line of 6 is the Earth's Axis; therefore a
freight Line paffes through the Center of the Proje-
ction.
All the great Circles paffing thro' the Center are divided,
by the Line of Semi-Tangents of their feveral Divifions
from V. So the Ecliptick is divided by opening the Sector
to the Radius of the Primitive Circle; and becauſe every
Sign is 30 Degrees, take the Tangents of 15 Degrees, 30
Degrees feverally, and they will mark out the Places of
O,
, m, II, I, on one fide, and m, X, 7,~, on the other
fide; and Vs do fall at the Peripher, 90 Degrees from
Tand, whofe Semi-Tangent is 45 Degrees to the Ra-
dius
ļ
80
The Projection of the Sphere.
Radius of the Projection. Almicanthers, or Parallels of Al-
titude in this Projection; as alfo Parallels of Celeſtial Lati-
tude, are drawn as the Parallels of Declination; only the
Centers of the firft fall in the prime Vertical; but of the
latter, in the Axis of the Ecliptick.
In the following Problems of the Sphere I fhall always
make uſe of the Stereographick Projection. It is the com-
mon Method uſed by moft Authors to Project this in a Paral-
lel Pofition; and by drawing the Horizon of any particular
Place, they fixt it for that Latitude: But I have chofen ra-
ther an oblique Poſition, and have adapted the Pole to the
Elevation of London. I fhall leave it to the Reader's choice to
do what way he fancies beft; for the diftance from St. Paul's
to the Royal Exchange, is the fame, as from the Royal Exchange
to St. Paul's.
PRO B. 3.
The Stereographick Projection of the Sphere upon the
Plane of the Ecliptick.
७
A
WS
ㅍ
​This is what I call the
Copernican Projection of the
Globe; becauſe by it we
folve the Phænomena of
the Heavens according to
the Earth's Motion. Ac
any convenient Radius
ſweep the Circle, which
fhall here repreſent the
Orbit of the Earth, and
quarter it; fo fhall
repreſent the Equino-
ctial Colure, and VS
the Solftitial; their Inter-
fection being the Pole of
the Ecliptick at E. Take the Semi-Tangent of 23° 29' the
conftant Diſtance of the two Poles, and fer on the Solftitial
Colure from E to P, and that in the North Pole of the
Globe. Divide each Quarter of the Ecliptic in three equal
parts, and place the reft of the Signs in their order, as you
fee done in the Figure.
ļ
Tak
The Projection of the Sphere.
8t
Take the Tangent Complement of the Diſtance of the two
Poles 66° 31', and fet one Foot in E, the other will give the
Center of the firft Meridian, viz. Y P in the Solftitial
YP.
Colure continued: Thro' that Center, and at Right An-
gles to the Colure, draw an occult Line, and fet off the Tan-
gents of 15, 30, 45, 60, and 75°, which are the Centers of
the other Meridian. Then to draw the Path of any Vertex,
obſerve for London, thus: Add the Co. Lat. 38° 28' to the
Diſtance of the two Poles 23° 29', their Sum 61° 57'; fet
by the Semi-Tangents from E to A, and 38° 28'23° 29'-
14°59'; fet the Semi-Tangents thereof from E to B; biffect
A Bin C; fo is C the Center of the Vertex of London.
Now, fuppofe the Earth in , then will the Sin appear in
a; draw & E, and continue it till it meet with the
former Occult Line, and that will be the Center of 6 P 6;
draw ŎEm at right Angles to the place of the Sun and
Earth, and that fhall be Horizon of the Disk; and continue
it till it meet the former Occult Line, and that Interfection
ſhall be the Center of P, which is the proper Meridian
to the Place of the Earth and Sun that Day.
All that part which lies between the Horizon of the Disk
Ŏ Em, and the Place of the Earth, is in Darkneſs, as is
repreſented by the fhaded Part: The North Pole is now il-
luminated, and the Day is more than 12 Hours long, which
is fhewed by the Path of the Vertex of London, cutting the 6
a-Clock Hour-Line before 6 in the Morning, and after 6 at
Night. You may projects feveral Paths in one Scheme, and
by a moveable Horizon of the Earth's Disk (which I have
contrived) you may fee at one View what Places are illu-
minated, and what are not: For the Horizon of the Disk
moving upon the Pole of the Ecliptic, determines the Quan-
tity of Light and Darkneſs. So when the Earth comes into
, the North Pole of the Globe is then in Darkness; but
coming to or, the North Pole lies in the Horizon of
the Disk, and confequently there is an equal fhare of Day
and Night all over the Globe. This may be better under-
ftood by viewing that curious ORRERY lately made by Mr.
Wright at the Orrery and Globe in Fleetftreet, London, which
by the turning of a Handle, carries the Earth round the Sun
in its Parallelifm. This is the neceffary refult of the two
Motions of the Earth; that is, round its Axis, and its An-
nual one; and there needs no third Motion be feigned to
explain it, or to account for it. For as the Earth moves
Annually round the Sun, without the diurnal Motion, it
G
moves
82
The Projection of the Sphere.
moves only according to its Center of Gravity; and each
Point and Line in it keeps always the fame Pofition. Let
its Axis be one of thofe Lines; the Diurnal Revolution of
the Earth round this, which as to that Motion is fuppofed im-
moveable, cannot change the Pofition of, and therefore it
will be always the fame, i. e. always Parallel to it ſelf.
PROB. 4.
The Stereographick Projection of the Sphere on the Plane
of the Horizon.
With the Chord of 60 Degrees fweep the primitive Cir-
cle, which in this Projection reprefents the Horizon of our
Habitation.
Croſs it with two Diameters at right Angles, fo fhall 12
12 be the Meridian, and 6 6 the Prime Vertical, or
Azimuth of Eaſt and Welt, and z, will be the Zenith of the
Place.
•10
11
F
1/21
Then becauſe the Zenith of any Place is diftant from
the Pole equal to the Complement of the Latitude, take
therefore the Semi-Tangent of 38 Deg. 28 Min. and fet it
on the Meridian from z to P; fo fhall P be the North Pole of
the World in this Projection: Becauſe the Latitude is equal
to the diſtance from the Zenith to the Equinoctial, take the
Tangent of half the Latitude 51 Degr. 32 Min. viz. 25 Degr.
46 Min. and ſet it from to Æ, and the Secant of 38 Degr.
28 Min. from A to O, or the Tangent of it from to O,
will give the Point O, the Center of the Equinoctial.
Take the Tangent of the
given Latitude 51 Degr. 32
Min. and ſet it on the Meri-
dian from P to C, and draw
the Six-a-Clock Hour-Circle
6 P 6. Draw A B at right
Angles to the Meridian thro'
C, open the Sector to the
Radius P C, and take off the
Tangent of 15, 30, 45, 60,
and 75 Degrees, and fet them
off from C towards A and B,
and they ſhall be the Centers
of the other Hour-Circles,
as you fee in the Projection A
are drawn.
6
A
3 2 1
LÆ
VIS
12 11 10.
1
4
B
60 45 30 15 C 15 30 45 60
In
و)
The Projection of the Sphere.
83
In this Projection, Almicanters are all parallel to the
Primitive Circle.
And Azimuths are all right Lines paffing thro', the Ze-
nith, to equal Divifions in the Horizon; but omitted to avoid
Confufion.
Parallels of Declination are all Leffer Circles, and Paral-
lel to the Equinoctial; and their Interfections with the Me-
ridian, are found by fetting the half Tangent of their Di-
ftance from the Zenith, Southward, and Northward: Their
Centers are found by biffecting the Diſtance between those
two Points.
Thus: For the Tropicks Latitude, London,
l
t
Height Equinoctial 38 28
Obliquity Ecliptic 23 29
or thus
5 I
32
23
29
Meridian Altitude 61 57
28
3
90
Zenith
Tropick
from Zenith 283 to the Southward.
Then becauſe the Tropick of is juft fc much depref-
fed below the Horizon on the Meridian to the North, as
much as the Tropick of vs is Elevated to the South, 14
Degr. 59 Minutes; to this 14 Degr. 59 Min. add the Qua-
drant, or diftance of the Zenith from the Horizon, and the
Sum is 104° 59: Take the half Tangent thereof viz.
52° 29' and fer it on the Meridian from Z to E; Biffect
E in C; fo fhall C be the Center of the Tropick of Can-
cer. Or, becauſe the greateft Amplitude in the given Lati-
tude is nearly 39 Degr. 52 Min. North and South, take the
Chord thereof, and fet it from 6, 6, either way towards 12,
12, upon the Horizon to A A for %, and 4, 4, for vs; find
the Centers that will fweep thefe Points feverally, and that
will draw the Tropicks.
Secondly, For the Tropick of Capricorn :
38 28
or thus
23 29
Z Sub.
from
14 59
90
VS from Z 75
I
9
51 32
23 29
75 I VS from Z South.
31 30 Tang, from Z.
G 2
Deg!
1
1
84
The Projection of the Sphere.
To a Quadrant
Add Vs Depreffion
Deg. Min.
90.
6I 57
Vs from Z to the North 151 57
Half
75 58
Tang. from Z.
Take the Tangent of 75, 58, and fet it on the Meridian
from Z, beyond E Northward: Take the Tangent of 37
Degr. 30 Min. and fet it from Z to VS Southward; di-
vide theſe two Points into two equal parts, and that ſhall
give the Center of the Tropick of Capricorn a VS a: Or,
draw it by its Amplitude, as directed in the other Tropick.
And if you would draw the Parallels of every 5 Degrees
of Declination, and the Parallels of the beginning of each
Sign North, they may be done by this Table. Latitude
London.
Q
"
#1
!
51 32
5 O
046 32
041 32
IO O
O
025 46 0 128 28
02316
Iz3 28
02046 O 118 28
20 I 1 13 116 58
1816 O 113 28
II 29 33 40 2 27
15 O
20 O
03632
031 32 15 46
20 II 1531 20 45 15 40 22
23. 29
28 3
108 28
0164 14 0
061 44
0159 14
27 58 29 I
015644
054 14
108 17 45 54 8 52
152 29 3
14 3 3/10459
The First Column to the Left Hand fhews the Degrees of
Declination including the Equinoctial and Tropick of Can-
cer. The Second Column, the diftance of each Parallel from
the Zenith of London. In the Third are the half Tangents
to be fet off in the Projection from Z on the Meridian,
to the South. In the Fourth Column is the Diſtance of De-
clination from the Zenith to the North. The Fifth and laft
Column contains the half Tangents to be fet on the Meridian
from Z Northward. The midway between the North and
South Interfection on each Parallel with the Meridian fhews.
the Centers in the Projection to draw each Parallel
by.
督
​The
The Projection of the Sphere.
85
The Parallels of South Declination must be drawn by this
Table for Lat. 51 Degr. 32 Min. N.
#1
"!
O
51 32
25 46
128 28
164 14
5
56 32
28 16.
433 28
66 44
10 O 6132
30 46
138 28
69 14
11 29 33 63 I 33
31 30 46
143 28
71 44
15
20 0-
6632
71 32
33 16
148 28
74 14
35 46
148 39 1575 19 37
20 II 15
71 43 15
35 51 37
151 57
75 58
23 29
75 I
37 30 2
The Title of each Column is the fame with thofe above
deſcribed; only theſe are for South Parallel of Declinations,
and the other were for North.
To draw the Ecliptic.
This, you fee, is is two Halfs, viz. North and South, and
marked with their proper Signs. Becauſe the greateſt Meri-
dian Altitude of the Northern part of the Ecliptic is 61 Deg.
57 Min. at London, take the Secant of 61 Degr. 57 Min. and
fet it on the Meridian from, where the Tropick inter-
fects it Northward beyond E, fhall give the Center of the
part marked with Y, 8, II,,, . And becauſe the
leaft Meridian Altitude of the Southern Part in the given
Latitude is 14 Degr. 59 Min. take the Secant thereof, and
fet one Foot in your Compaffes in the Interfection of the
Tropick of Capricorn with the Meridian at VS, and the other
Foot will give the Point D, the Center of the Ecliptic mar-
ked with, m, ×, vs, ~~~, X;
To lay the Signs down on the Ecliptic.
See what Declination the beginning of each Sign has,
which are as is here fet down.
Deg. Min. Sec.
O
8 m II 29 33
o
O
Deg. Min. Sec.
O
M X II
29 33
20 II
དབྱིབྷི ནྟི སྡོ
T
a
20
11
23 29
North. And)✔
G 3
VS
23 29
I South.
O
Whole
1
86
The Projection of the Sphere.
Whoſe half Tangents you have in the foregoing Tables
for drawing the Parallels; and according to thoſe Dire-
ctions, if you draw thefe Parallels of Declination of the
beginning of each Sign, where they interfect the Ecliptic,
they are the Places where you are to write the Signs as
you fee in the Projection.
Secondly, The Ecliptic may be divided by firft finding the
Poles of each part of the Ecliptic; then lay a Ruler to
each Pole ſeverally, and to 30 and 60 Degrees in the pri-
mitive Circle, and that will truly divide each Half of the
Ecliptic as before.
Note, In all Stereographick Projections, all Diameters are
meaſured on the Scale of Half Tangents; the reafon of
which you have in the Spherick Geometry, with which
be fure to acquaint your felf well before you proceed to
the Projection of the Sphere. And this is the Ground of all
Dialling; or the true Projection of the Hour-Circles of the
Sphere on any Given Plane.
And if to this Projection there be fitted a Label or Index to
move upon the Center, and its Edge divided by the Line of
half Tangents, and Number'd with 10, 20, 30, 40, 50, 60, 70,
80, 90, from the Circumference to the Center, it will then
be fitly accommodated to perform many Conclufions of the
Sphere. As for Inftance, in Dialling: Let ftreight Lines be
drawn from the Center of the Horizon where the Hour-
Circles interfect it, and they fhall be the true Hour-Lines.
of an Horizontal Dial for the Latitude the Projection was
made, The Degrees and Minutes anfwering each Hour and
Quarter in the Limb of the Horizon are as is here fet
down.
And if at any time you
have a mind to make an Hori-
zontal Dial for the Latitude
of London, take theſe Degrees
and Minutes from the Line
of Chords, and fet them on
the Horizon from the Meri-
dian each way, and they
will mark out the Hour-
Lines of an Horizontal Di-
al.
2 3
I
250
3/38
2 3
Hour
I 2
Q
1 Hour
I
01 3
56
34
28
3
52
8
51 I 41 45
III
45
34
51
49
30
14
52
2 17
3 21 6
57453
35
I
57 47
2 24
62
6
20
66
33
I 27 36
36
2
31
571
6
2. For
1 2 3
The Projection of the Sphere.
87
Hour.
•
I 75 45
80
25
13
2 3
85
2. For an erect direct North or South
Dial. Lay the Graduated Index before de-
fcribed upon the Line 6, 6, and the
Hour-Circles will cut the Index in the
Number of Degrees and Minutes of every
Hour and Quarter, as is fet down in the following Table.
Theſe Degrees and Min. may by help of
the Line of Chords be projected into an E-
rect Direct North and South Planes, fetting
them off from the Meridian each way.
3. For a Vertical declining Dial. Let
the declination be 30 to the Weft. Lay
the Index to the Plane's declination in
the Limb, or primitive Circle, and the
Hour-Lines in the Projection will cut the
Index in the Degrees and Minutes that they
will have upon that Plane. In thefe Places
you must begin to Number the Index at the
Center with 10, 20, &c.
J
690
Hor.
12
1 2 3
00
Q
O
0
2
20
4
41
7
3
9
28
I
I I
56
14
27
17
4
19
45
I
22
2
3
2
1 2 20
3
3
4. For direct inclining Planes. By Sphe-
rick Geometry, project the Oblique Circles
repreſenting the inclining Plane, and find its
Pole, a Ruler being laid to its Pole, and to
the ſeveral Points where the Hour-Circles in I
the Projection crofs the Plane; the Ruler
will cur the Degrees in the Horizon that
the Hour-Lines muft have upon fuch an in-
clining Plane.
Laftly, For declining inclining Planes.
This Plane being projected, and its Pole
found, a Ruler laid to its Pole, and the Inter-
fections of the Plane, with the Hour-Circles,
fhall give in the primitive the Degrees of the
diftance that the refpective Hour-Lines muft
have upon that Planë.
2
3
25
o on cn
52∞
35
32
58
38
54
28
3 I
22
39
3
35
on mat
42 58
4+7
9
I 51 36
2 56 20
61 23
5 66 42
-23
72
200
78
17
3
80
6
90 0
The Gnomon of all Dials muft ftand parallel to the Earth's
Axis; and in the Doctrine of the Sphere I fhall fhew the rea-
fon of the Analogies for calculating Hour Lines on all forts
of Planes, for any place of the World.
G 4
SECT.
88
BY
SECTION IV.
The Doctrine of the Sphere.
Y the Doctrine of the Sphere, is meant, the Solution of fuch
Problems as relate to the Heavens, or Concavity of the
vifible World: In meafuring the Circles thereof, and Angles
they make with each other, I fhall fhew in a method more
Concife, and Methodical than any has done hitherto. I have
told you in feveral places of the Aftronomical Definitions, that
the Obliquity of the Ecliptic is fixed at 23 Degr. 29 Min.
which you muſt carefully remember; and which was deter-
mined thus:
At the Tower of London the the Height of the
North Pole is exactly
Deg. Min.
51 32
When Sun enters, his Merid. Altit. is there
When Sun enters V3, his Meridian Altitude is
there: Sub.
6I 57
14 59
46 58
23 20
Difference of Meridian Altitudes.
Half, is the Obliquity of the Ecliptic, or Di-
ftance of the Pole of the Equinoctial from
the Pole of the Ecliptic to Sun's Greateſt
Declination.
Greatest S
And here I fhall annex the Names of the Ancient Aftrono-
mers, and the Times when they flouriſhed, who have obferved
the Obliquity of the Ecliptic, and it's Quantity.
Bef. Chr.
280 Aristarchus
"
270
Eratosthenes
140 Hipparchus
Aft. Chr.
140 Ptolemy
825 Benimula
827 Almamon
828 Fabia Ebn Abumanfar
Q
23
51 00
23 51
OO
23 51
00
23 51
20
23 35
Oo
23 35 00
23 35
00
889
The Doctrine of the Sphere.
89
O
880
Mahumed Eben Gaber
23 35
00
911
Ababet Eben Corra
23 33 30
992 Abu Mahumed Al Cagandi
23 22
21
1269 Cojah Nafiroddni
23 30
00
1363 Eben Shatir
23 31
CO
1437 Vleg Fieg
23 30
17
1460, Regiomontanus
1490 Dominicus Maria Novaras Ferrarienfis 23 29
23
29
00
1514
Vernerus
1572 Tycho Brahe
1670 Hevelius
1670 P. Mengoli
1673 Mr. Flamsteed
23
28
00
23
31 30
23 30 20
23 28 24
23 29
Oo
The Reader from this must not conclude that the Obli-
quity of the Ecliptic has altered, but that the different Deter-
minations of it have arifen from the badnefs of the Obfervati-
ons, and a want of a true Knowledge of the Parallaxes and
Refractions of the Heavenly Bodies. See Philoſophical
Tranfactions, Numb. 163. and Marcus Manilius.
1
PROBLEM I.
The Sun's Greatest Declination being 23 Deg. 29 Min.
and his Place given, to find his prefent Declination.
Example 1728, April 29, at Noon, Sun's true Place by our
Tables is 19 Deg. 55 Min. 58 Sec. I demand his true De-
clination.
H
B
H
In the Right-Angle Sphe-
rick Triangle Y BQ,
Right Angled at B, are gi-
ven, the Sun's di-
ftance from the next Equi-
noctial Point V 49, Deg.
55 Min. 58 Sec. with the
Angle BYO 23
Min. to find B
fent Declination.
Deg. 29
the pre-
N
रु
VS
As
1
90
The Doctrine of the Sphere.
As Radius
To Sine
Longitude
O
13
90 00 00-10,000000
49 55 58- 9.883825
So Sine Angle BYO, obliquity 23 29 00- 9.600409
To Sine BQ Declination N.
17 45 19- 9.484234
Note, if the Sun be entering, or, that is, 60° from
the Equinoctial Points reprefented in the Rect-Angled Sphe-
rical Triangle byce, the Declination ce will be found to
be 20° 11' 15 South.
PROB. 2.
The Sun's prefent and Greatest Declination given, to find
his Longitude or Place in the Ecliptic.
This is the Converſe of the laft Problem, but of fingular
ufe in Aftronomical Obfervations, as I fhall fhew in its proper
Place.
Example. In the laſt Diagram let Angle B TO, and B O
be given, to find Y O, the Analogy is,
As Sine Angle BY O, Ob-
liquity Ecliptic.
To Sine B O, prefent De-
clination N.
So Radius
To Sine Longitude
from Y.
Q ] !!
23 29
co- 9.600409
17 45
19- 9.484234
90
00
00- 10.000000
49 55 58-9.883825
That is in Ŏ 19° 55′ 58", becauſe Declination was N. and
from Y.
I hope I need not acquaint my Reader that having any two
things given in a Right Angled Spherical Triangle, the third
may eafily be found, pre-fuppofing him well acquainted with
Trigonometry before he meddles with this Section.
*
+
}
PROB.
The Doctrine of the Sphere.
91
3
PROB. 3.
Given the Sun's Place, and Greatest Declination to find his
Right Afcenfion.
Example, April 29th Day at Noon 1728, I demand the Sun's
R. A. his Longitude being as in the preceding Scheme.
ANALOGT.
As C. t, VO, Longitude
To Radius
So Co. Sine Angle B YO
So t. B. R. Afcenfion
As Radius
To t, YO
Deg. Min. Sec.
49 5558 9.924848
9000
23 29 00
00-10:000000
29 00-4 9.962453
47 28 50--19:037605.
Or, by Tranfpofition, Say,
So C. S. Angle BYO
To t, B. R. A.
Dog. Min. Sec.
90
0900--10.000000
49 55 58--10,075151
23 2900-÷ 9.962451
47
2850-10.037004
כי
Note, If the Sun be in the firſt Quadrant of the Ecliptic
r, o, II, as in the Example above, then the fourth propor-
tional Arch is the Sun's Right Aſcenſion from Aries; but if
the Sun be in the fecond Quadrant,, MR, then you muft
fubftract the fourth proportional Arch from 180 degrees,
and the remainder is the Right Afcenfion from T. When
the Sun is in the third Quadrant, m,, you muſt add the
fourth proportional Arch (found as above) to 180, and that
Sum is the Sun's Right Afcenfion from T. Lastly, When the
Sun is in the laſt Quadrant of the Ecliptic vs, ≈≈≈, X, then
fubftract the fourth proportional Arch from 360 degrees, and
the Remainder is the Sun's Right Afcenfion from . So if
the Sun be o° you will find his R. A. by the preceding Me-
thod in the Triangle cc, to be 237° 481 36. And when
he is in the very beginning of, his Right Afcenfion is 302°
II! 24".
PROB.
92
The Doctrine of the Sphere.
*
PROB. 4.
Given, the Latitude of the Place, and the Sun's De-
clination, to find his Amplitude.
Example. Anno 1728, April 29th Day at Noon, the Sun's
declination was found by Prob. 1. to be 17 Degr. 45 Min.
19 Sec. I demand his Amplitude of rifing and ſetting at
London.
H
N
H
In the adjacent Scheme,
and Right Angled Spheri-
cal Triangle PHC, are
given, H P, the Latitude
of the Place, and P C the
Complement of the Decli-
clination, or Sun's diftance
from the North Pole, to
find CH, the Amplitude.
from the North, whofe
Complement YC is the
Amplitude from the Eaſt
and WeftPoints of the Hori-
zon.
ANALO G Y.
As C. S. of H P the Latitude
To Radius ;
Deg. Min. Sec.
51 32
00-- 9.793832
90 00
00--10.000000
19-- 9.484231
21 21-- 9.690399
So Sine Declination North
Declination North 17 45
To Sine C, the Sun's Am-
plitude.
29
Or the fame may more rationally be found in the Tri-
angle DC, in which are given, the Angle D V C =
38 Deg. 28 Min. the Complement of the Latitude of London,
and DC the Sun's Declination North, to find C the Sun's
Amplitude, from the Eaft and Weft Points of the Hori-
zon.
ANA-
The Doctrine of the Sphere.
93
1
་
ANALOGY.
As C. S. Latitude = Angle
Dr C
To Sine Sun's Declination.
North
So Radius, or Sine of the
Angle
To Sine
DC
C, the Ampli-
tude, North
Deg. Min, Sec.
51 32 00- 9.793832
17 45 19--9.4842 3 1
90 00
0--10.000000
29 21 21-- 9.690399
PROB. 5.
Given, the Latitude of the Place, and the Sun's
Declination, to find the Afcenfional Difference, and
confequently the true Time of the Sun's Rifing
and Setting, with the length of the Days and
Nights.
or
Example, Ler the Sun be
in the firft Scruple of
VS, and Latitude of London,
what's the Afcenfional diffe-
rence?
In the Right Angled Sphe-
ric Triangle CHP, there H
are given H P, the Latitude
of London 51° 324 and CP,
the Complement of the Sun's
Declination 66° 31', to find
the Angle CP H, the Com-
plement of the Afcenfional
C
S
J
N
Z
वि
C
H
D
A
difference. But it is better folved in the Triangle V dc,
in which are given, the Angle de the Sun's Declination,
to find d, the Time in the Equinoctial from the Sun's ri-
fing or fetting, to Six a-Clock.
:
ANA-
+6
The Doctrine of the Sphere.
ANALOG Y.
Deg. Min. Sec.
As Radius
90
00
To T. Latitude
5 I 32
00
10.000000
10.099913
So T. Declination
23 29
00
To S. Y d, Afc. Diff.
33
9 04
9.637956
9.737869
This 33° 9' 4" converted into Time by the Table for
that purpoſe, will ftand thus:
Sum, fub. and add
Sun riſes at
Sun fets at
h.
30=2
00 00 00
3=0 12 00 00
9=0 00
00 36
00
4=0
00
00 16
2
12 36 16
ро Oo
3
47 23 44
8
12 36 16
Double the Time of the Sun-rifing, gives the Length of
of Night; and the time of the Sun's Setting double, gives
the Length of the Day. And as the time of Sun-rifing and
fetting are the Complement of each other to 12 Hours; ſo
are the Length of the Day and Night the Complement of
each other to 24 Hours. The time of the Sun's rifing in
Northern Signs, is the time of his fetting in Southern Signs;
and the time of his fetting in Northern Signs, is the time of
his rifing in Southern Signs, & Contra. For inftance ; the Sun
rifes truly at 3 h 47 23" 44" when he touches the Tropick
of, which is the Time of his fetting when, he is in the
Tropick of vs, Alfo the Sun fets at 8 h. 12′ 364 16!!! when
in Capricorn: as you may the better be informed by the
Tables of the Sun's rifing and fetting for all the moſt Emi-
nent Cities in the World, which you will find at the End of
this Section.
PROB.
Į
The Doctrine of the Sphere.
95
}
---
PROP. 6.
Given the Right Afcenfion, and Afcenfional Difference, to
find the Oblique Afcenfion and Oblique Defcenfion.
If the Declination be
North
RULE.
fub. Afc. Diff. from R. A. Gives Ob. Afc.
add
add
Afc. Diff. to R. A. Gives Ob. Defc.
"Afc. Diff, to R. A. Gives Ob. Afc.
South
fub. Afc. Diff. from R. A. Gives Ob. Defc.
Example. Let the Right Afcenfion of the Sun be 47 Degr.
28 Min. 50 Sec. and Afc. Difference 23 Degr. 46 Min. 5 Sec.
in the Latitude of London, with N. Declination. What's the
Ob. Afc. and Ob. Defcenfion?
Right Afcenfion
OPERATION.
Afc. Diff. Sub, and add
Rem. the Ob. Defcenfion.
Sum is Ob. Deſcenſion
Deg. Min. Sec.
47 28 50
23 46 5
23 42 45
71 14 55
In things of this nature we always fuppofe the Sun's De-
clination to be unalterable for one Day; and therefore in
the Projection of the Sphere it is called, a Parallel of the Sun's
Declination, and is always drawn fo in the Projection: But
this, ftrictly speaking, is not fo; for they are not Parallel,
but a Spiral Line, the Sun, (or rather the Earth) deſcribes
from Tropick to Tropick, and the Declination near the E-
quinoctial Points alter in an Hour confiderably, but near the
Tropick more Slow: For they are proportional to Radius,
ઘડ
1
96
Sphere.
The Doctrine of the i
as are the Natural Sines, to the Semidiameter of the fame
Circle Therefore in any Operation where Exactneſs is re-
quired, you must always be careful to find the Declination
of the Sun, Moon, or Star, to the precife Time of the
Queftion, if you defign to be exact in your Calculations.
Special regard must be had to the Moon's true Declination
(becauſe her Motion is fwift) to the time of her Rifing,
Southing, and Setting, as I fhall fhew when I come to that
Precept.: Otherwife her Right Afcenfion, her Oblique Af
cenfion and Oblique Defcenfion will not be had true.
PROB. 7.
Given, the Latitude of the Place, and the Declina-
tion of the Sun, Moon, or Star, to find their Obli-
que Afcenfions and Oblique Defcenfions.
Example. Anno 1728, July 10, at Noon by our Tables
the Sun's Place is 28 Degr. 45 Min. 5 Sec. his Decli-
nation 20 Deg. 26 Min. 52 Sec. North. I demand the
Oblique Afcenfion, and Oblique Defcenfion at London, the
time of rifing and fetting, &c. Firft, draw DD the pa-
rallel of Declination, and POS. Then
Ꭰ
榮​禮
​In the Right Angled
Spheric Triangle PH O,
are given HP the Lati
tude of London, OP
the Sun's diſtance from
the North Pole, equal to
the Complement of the
Declination 69° 33′ 8″,
to find the Angle at
the Pole from Mid-
night.
ANA-
The Doctrine of the Sphere.
97
.}
As Radius
}
ANALOGY.
To t, of the Latitude
So t. Declination North
To Co. Sine of the
Semi-nocturnal Ark
Deg. Min. Sec.
90 00
00-10:
00-10.099913
51 32
32
20 26
59-- 9.571430
62 01
07-- 9.671343
This 62 Degr. 8 Min. 7 Sec. reduced into Time is 4 Ho
8 Min. 28 Sec. the Time of Sun rifing. Noté, If the Latitude
of the Place, and Declination of the Sun be of different
Names, that is, one North and the other South, then the
Ark found by the Analogy above when reduced into Time is
the Semidiurnal Ark or Time of Sun-fetting.
Example. December 10, the Sun in the very beginning
of VS, I would know the true Time of his rifing and fetting
at London.
In the Triangle O
Z P, or rather, in the
Triangle HS, are
given, the Latitude
*
HS, the Comple-
ment of the Sun's
Declination OS 66°
31′, to find the Angle
HSQ, the Semidiur-
nal Ark.
A
H
H
As Radius
ANALOGY.
N
`Deg. Min. Sec.
90 00
05--10.000000
To t. Latitude
51 32
00--10.099913
So t. Declination South
23 29
00-- 9.637956
To C. f. of the Semidiurnal Ark 56
50
57-- 9.737869
.ส
H
This
9.8
The Boctrine of the Sphere
Theſe 56 Hours, 50 Min. 57 Sec. converted into Time, are
3 Hours, 47 Min. 23 Sec. 48 Thirds, the true Time of the
Sun's fetting; which fubftracted from 12 Hours, leave 8 H.
12 Min 36 Sec. 12 Thirds, the true Time of the Sun's rifing,
on the Day and Place aforefaid; which gives the length of
the Day and Night, as is fhewn in PROB. 5.
N. B. The Complement of the Arch thus found is the A-
fcen. Difference.
PROB. 8.
i
To find the Beginning, Duration and End of the lon
geft Day, and the longest Night in any Latitude,
whofe Complement exceeds the Sun's Declination North
or South.
་
When the Declination of the Sun is equal to the Com-
plement of Latitude, and of the fame Name; that is, both
North, or both South; that then the Sun never defcends
below the Horizon of that Place; but his Center touches
in the oppofite part of the Meridian. And on the contrary,
when the Sun's Declination is of a different Denomination,
that there the Sun never Afcends above their Horizon; but,
its Center juft touches it when upon the Meridian.
Example. Let it be required at the North Cape Latitude
71° 25' to find that Day that the Sun begins with them, not
to ſet for fome certain Time, and alfo the Day when he be-
gins to diſappear, &c.
A
C
G N
d
H
A
'The Latitude being
71° 25' North, its Com-
plement 18° 35′ = Æ
Dthe Sun's Declination
North, where at Mid-
night you ſee it touches
the Horizon at H, and con-
tinues above the Horizon
all the time the Sun is
going to the Tropick, and
until he returns back unto
the fame Parallel of Decli-
nation DH. So that here
is no more to do than to
find
E
The Doctrine of the Sphere.
99
find the Sun's Longitude anfwering the Declination, which
in this Cafe is always equal to the Complement of the
Latitude of the Place; as is fhewed in PROB. 2.
►
ANALOG Y.
7
Deg. Min. Sec.
As Sine Sun's greatest Declination 23 29 00-- 9.600409
To Radius
So Co. Sine Latitude of the Place
To Sine of Sun's Long. anſwering
one Sine, fub.
Sun's Place
直
​90 00
00--10.000000
71 25.
00-, 9.503360
20-- 9.90295L
53
53
30
6
00
00
6 20 Sub.
1
Sun's Place
23
6
00 0 o From
4
6
53 40 Remains.
The Day of the Month anfwering
•
23 Deg. 6 Min. 20
Sec. is May 3, and the Day anfwering the Sun's Place
6 Deg. 53 Min. 40 Sec, is July 19; fo that from May 3,
to July 19, the Sun never fets at that Place; which is 77
Days.
And when the Sun's Declination 18 Deg. 35 Min. is South
increafing, its Parallel cd touches the Horizon when the Sun
comes to m 23 Deg. 6 Min. 20 Sec. which happens on No-
vember 4th; and when the Sun has returned back again
from the Tropick of vs, and has 18° 35' of South Decli-
nation, the Parallel cd now touches the Horizon on the
Meridian at c, and he begins then to rife, his Longitude
is 6° 53' 40", which happens upon the 15th of January;
fo that from Nov. 4, to Jan. 15, is 72 Days; which time the
Sun never rifes to them in the Latitude of 71° 25′ North:
And this Night of 72 Natural Days is fhorter than their
longeft Day by 5 Natural Days. But in the Southern Parts
of the World thefe Appearances are juft contrary, viz. when
tis Day in the North 'tis Night in the South, and when 'tis
Night in the North 'tis Day in the Southern Hemisphere.
What has been faid here for the Latitude of the North
Cape, the fame is to be obferved of all other Parallels
of Latitude within the Polar Circles.
H2
Eut
7
100
The Doctrine of the Sphere.
But herein is to be confidered, that the Calculation above
is performed, fuppofing the Sun to be free from Refraction;
(See the word Refraction;) but fince it is not fo, but that he
is Refracted in the Horizon more than half a Degree in our
Latitude, therefore it follows that the Inhabitants will fee
the Sun fooner than May the 3d. which is the Day truly when
they might expect him; and he will continue above their
Horizon longer than July the 19th, which is the Day that tru-
ly he will begin to diſappear to them at Midnight; fo that if
the true quantity of Refractions were known in all Latitudes,
then by the above inveftigation may the apparent Days of the
Sun's first appearing, and the Day of his diſappearing be
found, otherwiſe not.
PROB. 9.
.
Given, the Latitude of the Place, the Sun's Declination,
and Horizontal Refraction; to find the Apparent Time
of the Sun's rifing and ſetting.
The Apparent Time of the Rifing and Setting of the Hea-
venly Bodies always differs from the True Time; and this
is by the Rays of Light paffing thro' different Mediums, which
cauſes them to be turn'd or bent out of that ftreight Line
in which they fhould directly paſs. This Vifible Time is
of very great Moment in the Eclipfes of the Luminaries,
when the Sun or Moon rifes or fets Eclipſed, to find how
much of their Diameters are then obfcured, at the Vifible
Time of their rifing or fetting.
Example. Let it be required to find the Apparent or Vifi-
ble Time of the Sun's rifing and fetting the 10th Day of June,
when he is in the Tropick of Cancer?
In the adjacent Diagram, let v H repreſent part of the
Horizon, a the Equinoctial, PS the Earth's Axis, 4
the Tropick the Sun truly rifes at 4, but is feen to rife at
b when he is 33'-b ✪ below the Horizon. In the Oblique-
Angled Spherick Triangle ZPO are given Z P, the Com-
plement of the Latitude 38 Degrees 28 Minutes, Z O, the
Diſtance of the Sun from the Vertex 90 Degrees 33 Mi-
nutes and PO, the Diftance of the Sun from the North
Pole,
The Doctrine of the Sphere.
101
Pole, equal to the Com-
plement of the Declina-
tion 66 Degrees 31 Mis
nutes; to find the Angle
ZPO, the visible Semi-
diurnal Ark, or Apparent
time of Sun ſetting. Then
by the 11th Cafe of Ob-
lique - Angled
Spherick
Triangles, I perform the
work thus: .
1
Z
A
H
To Zb
90
add b Refraction
33
N
38.
Z P
Pa 66
28
31
Sum is Z O
9° 33
X 28
3
Half
45
16 2
half
14
en m
I
1
45
16
Add, and Subftra&t
14
1
Z ...
... 59
18
X.
3 I
15
a
NANX
ZP, S.
38 28
Co. Ar.
0206168
PO, S. 66 31
Co. Ar.
0.037547
18
9.934424
9.714977
1
19.893116
S. 59
X, S. 31 15
Z, Logarithms
Sine of 62 9 .19″
29.945558
Doubled, 124 Degr. 18 Min. 38 Sec. Reduced into
Time is 8 H. 17 Min. 14 Sec. 32 Thirds, the Apparent Time
of the Sun's ſetting, equal to the Semidiurnal Ark, whofe
Complement to 12 Hours is 3 H. 42 Min. 45 Sec. 28 Thirds,
the Apparent Time of his rifing on the given Day at London.
But the True Time of the Sun's rifing and fetting is 3 Hours
47 Min. 23 Sec. 44 Thirds, and 8 Hours 12 Min. 30 Sec qu 4843.
12 Thirds, by which the Sun is feen to rife fooner and
fet later by 4 Min. 38 Sec. 16 Thirds, which makes the
length of the Apparent Day longer than the Aftronomical
Day by 9 Min. 16 Sec. 32 Thirds.
x
H 3
1
9. 16.38
Example.
1
102
The Doctrine of the Sphere.
•
Example 2. Let it be required to find the Apparent Time
of the rifing and fetting of the Sun December 10, when he is
the Tropick of VS.
in
།
CA
In the Obliqué Angled
Spheric Triangle Z P,
are given ZP the Com-
plement of the Latitude
38 Degr. 28 Minutes, OP,
the Sun's Diſtance from the
North Pole, 113 Degrees
29 Minutes; and Z O, the
Sun's Diſtance from the
Vertex 90 Degrees 33 Mi-
nutes, to find the Angle at
A the Pole OPZ?:
*
+
N
Zb
OPERATION..
A
.0
90 O CP 90 +CO 23 29 PO 113 29
¿Refraction o 33 ZP=
Sum is ZO 90
Half
90 33
45
16
Add and Sub. 37 30
HIN HIN
38 28
X 75
// 37 30 /2/2
30/
I
Jurm
Z 82 47
Diff X 7 46
P 113 29 Complement S,
Z P
-NX
Z Logarithms
Half is Sine of
Doubled is
S.
66 31 Co. Ar. 0.037547
38 28 Co, Ar. 20.2c6168
S.
S.
82 47
746
9.996546
9.130781
19.371042
9.685521
"
28 59 47
57 59 34 Reduced into Time
is 3 Hours 51 Min. 58 Sec. 16 Thirds, the Semidiurnal Ark,
or apparent Time of the Sun's fetting; whofe Complement
to 12 Hours is 8 Hours 8 Min. 1 Sec. 44 Thirds, the Apparent
Time of the Sun's Rifing: But the true time of the Sun's
Rifing and Setting at London on the fame Day, is 8 Hours
12 Min,
The Doctrine of the Sphere.
102
12 Min. 36 Sec. 12 Thirds, and 3 Hour's 47 Min. 23 Sec.
48 Thirds, by which you fee the Sun rife ſooner and fet
later by 4 Min. 34 Sec. 28 Thirds; which makes the length
of the Apparent Day longer than the Aftronomical, by 8 Min.
8 Sec. 56 Thirds. And thus by thefe Examples may you
find the Apparent Time of the Moon's rifing and fet-
ting.
And as this Refraction occafions an Error in the Time of
the Sun's rifing and ſetting; ſo it likewife doth in the Am-
plitude: For the True Amplitude is a; but the Vifible,
rb, which you fee in the little Right-Angled Spherick Tri-
angle cb in theſe Schemes; and as in the firft the Declina-
tion is increaſed by the Refraction 33 Minutes; fo the
Visible Amplitude will be 40° 51' N. but in the laft Scheme
it is diminiſhed 33'. So the Viſible Amplitude is 38° 47°
South less than the True. And this ought to be carefully
minded by the Mariner; otherwife he will never attain the
true Variation of the Compafs, if he does not mind, to take
the Viſible Amplitude inftead of the True.
PROB. 10.
IO.
Given, the Latitude of the Place, and the Sun's De-
clination, to find the Time when he will be due
East and Weft.
Example Let the Sun be in the beginning of
and VS,
and let it be required to find the Time he is due Eaſt and
Weft in the Latitude of London.
Where the Tropicks cut
the Eaft and Weft Azimuth,
viz. at A and B, there draw
two great Circles, as PAS
and PBS, by which are.
formed two Rect-Angled
Spherick Triangles AZ P,.
and BNS Rect - Angled
at Z and N; in which are
Given A P B S, the Com-
plement of the Sun's Decli-
nation 66 Degrees 31 Mi-
H 4
nutes,
N
104
The Doctrine of the Sphere.
F
nutes, and ZP SN, the Complement of the Sun's
Declination 66 Degr. 31 Minutes, and Z PSN, the
Complement of the Latitude 38 Degr. 28 Minutes, to find
the Angles A PZ and BSN, the Times from Noon and
Midnight,
J
ANALOGY.
As Radius
To T. SN=ZP
So C. t. SB = AP
Tọ C, f. B S N≈APZ
As Radius
Deg, Min!
90 00--10.000000
38 28-- 9.900086
66 31-- 9.637956
69 41- 9.538042
Or, by Tranfpofition..
Deg. Mini
90 00--10,000000
To C t. Latitude
So T. Declination
51
32-- 9.900000
23
29-- 9.637956
To S. Sun's diſtance from 6 a-Clock 20
12-- 9.538042
This 69 48 reduced into Time, is 4 H. 39 Min. 12 Sec.
the Time in the Afternoon when the Sun is over the Weſt
Point of the Compaſs in the Summer, or under it in Winter,
which taken from 12, leaves 7 Hours 20 Min. 48 Sec. in the
Morning: Or, its Complement 20 12, converted into
Time, makes 1 H. 20 Min. 40 Seconds; which added to
6 Hours, makes 7 H. 20 Min. 48 Seconds; and fub. from
6, leaves 4 H. 39 Min. 12 Seconds, the Time as before, when
the Sun is due Eaft and Weft, when in the Tropick.
N. B. The Sun is never upon the Prime Vertical at 6 a
Clock, but when he is in the Equinoctial.
PROB.
་་
του
The Doctrine of the Sphere.
·PRO B. II.
Given, the Latitude of the Place, and the Sun's De-
clination, to find the Sun's Altitude when he is
due Eaft or Weft.
Example. Let the Sun be in the Tropick of S and
V; and let it be required to find his Altitude at London
when he is upon the Prime Vertical Circle ?
Draw two great Circles as P
AS and PBS, to cut the Prime
Vertical in the Tropick, and
then there is formed the Rect-
Angled Spheric Triangles AZP
and BNS Right-Angled at Z
and N, in which are given,
A P, the Sun's diftance from
the Pole BS, and Z P =
SN the Complement of the
Latitude 38° 28, to find Z A
=
、
BN the Complement of
the Sun's Altitude, when he is
upon the Prime Vertical Cir-
cle.
N
A
As C. S. of ZP
To Radius
So C. S. A P
To C. S. Z A
ANALOGY.
Deg. Min.
38 28-- 9.893745
90 00--10.000000
66 31-- 9.600409
59 24-9.706664
Whofe Complement 30 36 is the Sun's Altitude fought,
ANA-
2006
The Doctrine of the Sphere.
Or, by Tranfpofition in the Triangle YbA.
ན
Det Min.
196
As the S. of the Latitude Angle B TA 51
32-- 9.893745
90
00--10.000000
b A
TA
66
29-- 9.600409
30
36-- 9.706664.
To Radius
So S. Declination
To S. of the Altitude
The Sun's Altitude when in Northern Signs, as before;
and when in Southern Signs, it is his Depreffion below the
Horizon.
•
PROB. 12. MAG
Given, the Latitude of the Place, and the Sun's De-
clination, to find the Sun's Azimuth at the Hour
of Six.
Example: Let the Sun be in the beginning of for VS
in the Latitude of London; I would know the Sun's Azimuth
at the Hour of Six?
Æ
A
Draw ZA N, and ZdN
to interfect the Earth's Axis
in the Tropicks.
In the Rect-Angled Sphe-
rical Triangle APZ are
Given, A P the Complement
of the Sun's Declination,
Z P, the Complement of
H the Latitude of London, to
A
find the Angle A ZP the
Sun's Azimuth from the
North BH.
N
!
ANA
The Doctrine of the Sphère.
407
{
|
As t. A P
To Radius
So S. Z P
1
ANALOG T
Deg: Min.5
66 §1-010-862389
90 00--10.000001
30 28-- 9.793 8 3 2
74 53-- 9.431443 ..
To C. t. Angle AZ P
the South in VS.
Which is the Sun's Azimuth from the North, in, and from
1
By Tranfpofition in the Triangle T A B.
As Radius
Tot, A, the Sun's Declination
So C. S. Angle V, the Latitude.
Tot. TB, Azimuth from the
Eaft and Weft.
}
Deg. Min.
90 00--10,000000
23
29--9.637956
51 32-- 9.793832
15 07- 9.431788
Note, When the Sun is in Northern Signs, the firſt Ana-
logy is the Azimuth from the North; but when in Southern
Signs, from the South; whofe Complement to a Quadrant
is the Azimuth from the Eaft or Weft.
Or, to find it from the North, or South, you may ſay,
As Radius
To t. Sun's Declination
So C., of the Latitude
To C. t. of Azimuth
Deg. Min.
90
23
00--10.000000
29-- 9.637956
51 32- 9.793832
74 53-- 9.431788
1
T
PROB
108
The Doctrine of the Sphere.
PROB. 13.
Given, the Latitude of the Place, and the Sun's De-
clination, to find the Sun's Altitude at the Hour
Six.
Example. Let the Sun be in the beginning of or VS and
the Latitude of London: I demand the Sun's Altitude at Six
in the Morning, or Depreffion under the Horizon at Six at
Night.
A
Z
N
'As Radius
A
B
i
Draw the two great Cir-
cles Z A N, and Z d N,
to cut the Tropick and Axis,
in A and e; Then in the Tri-
angle APZ = eS N, are
given ZP 380.284, the Com-
H plement of the Latitude; and
AP 66° 31′, to find A Z the
Complement of the Altitude:
Or in the Triangle ABY A
e d' are given YA 23° 29',
the Sun's Declination, and
Angle, BY A = 51 32, the
Latitude of the Place, to
find B A the Altitude, or de
the Depreffion at 6 a-Clock.
A
ANALOGY.
To S. Sun's Declination
So S. Latitude
To S. Altitude
Deg. Min.
90 00--10.000000
23 29-- 9.600409
32-- 9.893745
II-- 9.494154
SI
18
8 11:
1
1
PROB.
The Doctrine of the Sphere.
109
PROB. 14.
Given, the Latitude of the Place, and the Hour of
the Day, to find the Sun's Altitude when he is in
the Equinoctial.
Example. Let the Latitude be London, and the Sun in the
Equinoctial at 10 in the Morning, or at 2 in the Afternoon;
what is then his Altitude?
Take the Semi-Tangent of
four Hours, which is the Time
from 6, and fet it on the Equi-
noctial from T to A; and
with the Secant of 30 Deg.
draw PAS, and alfo draw
ZAN: Then in the Right-
Angled Spherical Triangle
VBA, are given, Y A, the
Time from 6=60°, and the
Angle AV B 38° 28 the
Complement of the Latitude,
to find B A the Altitude at
that Time.
Z
H
B
N
Æ
F
ANALOG Y.
Deg. Min.
As Radius
90 00--10,000000
To S. of Time from 6
So G. S. of the Latitude
To S. A B the Altitude
бо
00-- 9.937531
51
32-9.793832
32 35--.9.731363
P.R.O B
པ་
The Doctrine of the Sphere.
"
む
​·PROB. IS.
Given, the Latitude of the Place, and the Sun's Azi-
muth, to find the Altitude.
Example. Let the Latitude of the Place be London, and
the Sun's Azimuth from the Eaft or Weft 15° 7' Northward;
what is then the Sun's Altitude ?
S
N
As C. t. Latitude
To Radius
So Sine Azimuth
To t. Altitude
As- Radius
To t. Latitude
A
B
P
+
Take the Semi-Tangent of
the given Azimuth from the
Eaft and Weft, and fet it on
the Horizon from to B,
and draw the Great Circle
ZBN: Then in the Rect-
Angled Spherical Triangle T
BA are Given, V B, the
Azimuth from the Eaft or
Weft 15° 7', and ithe Angle
AB-and
ATB 51°32', the Lati-
tude, to find B A the Alti-
tude at that time.
ANALOGY.
Deg. Min.
51 32-- 9.900086
,00--10.000000
.90
· 15
04-- 9.416283r
18 10-- 9.516193
By Tranfpofition,
So S. Azimuth from Eaft
To t. Altitude
Deg. Min.
90 00--10,000000
51 32- 10.099913
15 07-- 9.416283
18
10-- 9.516196
PROB
The Doctrine of the Sphere.
~PROB. 16.
Given, the Latitude of the Place, the Sun's De-
clination, and Hour of the Day, to find the Sun's
Altitude.
Example. Admit the Latitude of the Place be 51° 32 N.
the Sun's Declination 23° 294 N. at 10 in the Morning, or 2
in the Afternoon (for here the Declination alters but little in
that Time) that is, 2 Hours diftance from the Meridian; I.
would know the, Sun's Altitude?
+
From the Center of the
Primitive Circle ſet off four
Hours or 60 Deg. by help of
the Semi-Tangents on the
Sector, upon the Equinocti-
al to B, and with the Secant 、
of 30 Deg. draw the great
Circle PBS, and alfo draw
ZAN to interfect each o-
ther in the Tropick, the
place of the Sun at 10 or 2
a-Clock; by which there is
formed the Oblique Angled
Spheric Triangle. A Z P, in
which are given ZP Com-
A
69
}
d
B
f
H
A
N
plement of Latitude 38° 28′, AP the Complement of the
Sun's Declination 66° 31, and the Angle ZPB 30°, to
find A Z the Complement of the Altitude or Zenith-Diſtance,
which by the 6th Cafe of Obliques is anfwered thus; by
letting fall the Perpendicular Z d.
Firſt, For the Segment d P, I fay,
As C. t. of ZP
To Radius
So C. f. Angle ZP A
To t. d P
From A P
Remains A d
Deg. Min.
38 28--10.099913
90
30
fub.
00--10.000000
00-- 9.937531
34 32-- 9.837618
66 31
31
59
Or,
1
112:
1
The Doctrine of the Sphere.
'As Radius
Or, by Tranfpofition, Lay,
To C. t. of the Latitude
So S. Sun's diſtance from 6,
To t. of the fourth Ark
Deg. Min.
90.00--10.000000.
51 32-- 9.900086.
60 00--'9.937531
34 32-- 9.837617 ·
A General RULE.
If the Time given be between 6 in the Morning, and 6
at Night, this fourth Ark muſt be fubftracted from the Sun's
Distance from the North Pole: But if the Time given be
before 6 in the Morning, or after 6 at Night, then add this
fourth Ark to the Sun's Distance from the North Pole; the
Sum or Difference is the fifth Ark.
}
OPERATION.
From a Quadrant
Take the Sun's Declination North
90 00
23 29
Reft Sun's diftance from the North Pole
66 31
Fourth Ark fub.
34 32
Remains the fifth Ark
31 59
But if the Sun have South Declination, then it must be
added to 90, which gives his diftance from the North
Pole.
Now fay,
As C. f. of the fourth Ark Co. Ar.
34 32 00--0.084179
To C... of the fifth
So S. Latitude
CA
To S. Altitude
31 59 00--9.928499
51 32 00--9.893745
53 44 38--9.906423
Example z. Let the Sun be in the Tropick of Capricorn,
Latitude and Time of the Day as in the laft Example:
What's the Altitude ?
,
OPE
The Dottrine of the Sphere.
દ
OPERATION.
As Radius
To C. t. Latitude
So S. Sun's Dift. from 6.
To t. of the 4th Ark
Sun's Dift. from N. Pole
Remains 5th Ark
90
00- 10.000000.
5.1 32- 9.900086
60
00- 9.937531
34 32.- 9.837617
113 29
78 57
N
Now lay,
?
H
A
!
i
f
As Cf. of 4 Ark Co. Ar.
To C. f. 5 Ark
So S. Latitude
To S. Altitude = CA
Deg. Min.
34 32--0.084179
78 57--9.282544
51 32--9.893745
10 30-9.2604608.
Example 3. Fuly 13, Let the Sun's Declination be 20 Deg,
North, and the Time 10 Minutes before 5 in the Morning
or 10 Minutes paft 7 at Night, and Latitude 5t Deg 32 Mi-
nutes North; I deinand the Sun's Altitude? The Time from
6 is 1 Hour 10 Minutes; which converted into Degrees, is
17 Degrees 30 Minutes.
ફ
Now,
1
114
>
"..
of the Sphere.
The Doctrine !
As Radius
Now fay,
Deg. Min.
90 00-10.000000
To C. t. Latitude
51 32-- 9.900086
So S. Sun from 6,
17
30-- 9.498722
To t. of 4 Ark
14
04-- 9.398808
Sun from N. Pole add
70
00
Sum is the 5th Ark
84 04
}
N
As C. f. of 4th Ark Co. Ar.
To C. f. of 5th
So S. Latitude
To S. Altitude CA
Deg. Min.
14
04--0.013222
84 04--9.014399
51 32-9.893745
04: 47--8.921366
And after the fame manner may the Altitude of the Sun,
Moon or Star be found: But in things that require Exactneſs,
you muſt be ſure to find the Declination to the Time propoſed,
as I fhall fhew in its proper place; but in the Example above,
I ſuppoſed the Declination unalterable for that Day, which is
not fo it felf, but will ferve the prefent purpoſe well enough.
See Prob. 6. By the fame Inveſtigation I have found at Lon-
don the Sun's Altitude as is here fet down. Declination 19°
N. 1725 July 17.
at
!
115
The Doctrine of the Sphere.
J
;
S
at№ 3
3
4.
Ale
49 51
Hours, Altitude is 43
08
37 49
33 18
1
2.
And 1725, Aug. 6, at 8 Morning, Declination 13 Degrees
27 Minutes N. Sun's Altitude is 29 Degrees fere. Aug. 19,
Declination 8 Degr. 59 Minutes N. Altitude 45 Degr. 42 Mi-
nutes at one a-Clock. Aug. 21, at 10 Hours, Altitude 41°
381.
Aug. 26, at 11 Hours 27 Min. 28 Sec. Altitude 44 Deg.
22 Minutes. Anno 1726, Jan, 4, at one a-Clock, Sun's Alti-
tude is 16 Degr. 8 Minutes. April 1, at 9 Morn. Altitude
33 Degr. 33 Minutes. Aug. 2 d. 20 h. Altitude, 29 Degr.
48 Minutes at 8 H. 30 Minutes Morning, Altitude 34 Degr.
14 Minutes. August 16 at 2 a-Clock, Altitude 41 Degr. 54
Minutes; but half an Hour fooner it is 44 Degr. 42 Minutes.
Anno 1727, May 27 at 10 Morning, Sun's Altitude was 53
Deg. 6 Minutes July 6, at io, Altitude 51 Deg. 54 Minutes,
and at 6 it was 16 Deg. 34 Minutes: Thefe Altitudes I ob-
ferved at London with my Aftronomical Quadrant; and cor-
recting them by Refractions and Parallax, I found them all
to agree exactly; by which I pronounce the Elevation of the
Pole to be truly Afferted.
PROB. 17.
Given, the Latitude of the Place, the Sun's Declination
and Altitude, to find the Hour of the Day?
Example. Let the Latitude of the Place be 51 Degr. 2
Minutes North; Sun's Declination 23 Deg. 29 Min. North,
and Altitude 53 Degr. 44 Min. 38 Seconds: What's the Hour
of the Day?
With the Chord of 60 from the Sector, opened to the
Radius YÆ, fweep the Primitive Circle; draw PS the Earth's
Axis, to the Latitude of London, and Æ Æ at right Angies for
the Equinoctial, and H H the Horizon.
}
}
I a
Take
1
IFб
The Doctrine of the Sphere.
A
H
B
N
Take the given Altitude
53 Degr. 44 Min. 38 Sec.
from the Line of Chords,
and fet it upon the Meri-
dian from H to b and o,
and with its Co. Tangent
draw b A o which is a Pa-
rallel of Altitude; and
H where it cuts the Tropick
or Parallel of Declination,
which is at A, draw PAS
and ZAN; fo is there
truly projected the oblique
Angle Spherick Triangle A
Z P, in which are given
Æ
ZP, the Complement of the Latitude 38 Degr. 28 Minutes,
AP the Complement of the Declination 66 Degr. 31 Min.
and A Z the Complement of the Altitude 36 Degr. 15 Min.
22 Seconds, to find the Ang'e Z PA, the Hour from Noon?
Which by the 11th Cafe of Oblique Angled Spherical Tri-
angles, I perform thus:
OPERATION.
Z P Complement Latitude
AP Complement Declination
AZ Complement Altitude
Sum of all three
Half Sum
Complement Latitude fub.
38 28 00
66 31
00
36 15
22
141 14
22
.70 37 ΙΣ
3.8 28
00
32 9 11
70 37 II.
Complement Declination fub.
66 31 00
Difference
04 06
II
Difference
Half Sum
Having
The Doctrine of the Sphere.
117
{
Having prepared the Work above, then proceed thus:
Sides {ZP Compl. Latit Sine
A Compl. Decl. Sine
Difference of Co. Latit. and
and
Deg. Min.
Co. Ar. 38 28 60--0.2c6168
Co. Ar. 66 31 00--0.0.37547
Z Sine 32 09 11--9.726062
Z Sine 04 06-11-8.8.54613
18.824390
14 58 18-9 412195
29 56 36 converted'
Sum of the Logarithms
Half, is the Sine of
Doubled, is
into Time, is 1 H. 59 Min. 46 Sec. 24 Thirds from Noon ;
that is, 10 H. o Min. 13 Sec. 36 Thirds in the Morning.
And ſuch was the Hour of the Day at the time of this Ob-
fervation.
Or the Angle at the Pole may be found as in Problem 9.
Example 2. Admit the
Latitude
Declination
Sun's Altitude
•
51 32 N. What's the Hour of
23 29 S.
10 30
1
Draw the Parallel of Alti-
tude bO, by help of the
Line of Chords on the Se-
ctor, and it will interfect
the Tropick of vs (which
is here the Parallel of the
Sun's Declination,) in the H
Point where the two Ob-
lique Circles P dS and ZAN
muſt paſs. Therefore, in
the Oblique Angled Spherical
Triangle A ZP, all the fides.
are given to find the Angle at
the Pole.
N
the Day.
13
N
H
OPE-
I 3
1
118
The Doctrine of the Sphere.
OPERATION.
To Pd
Add Ad Declination South
Z is AP Sun from N. Pole
ZP Complement Latitude
AZ Complement Altitude
90
23 29
113 29
38 28
79 30
1
Sum
231 27
Half
115 43
38 28
Complement Latitude fub.
Difference
í
77 15 20
Half Sum
115 43
43 2
Sun from N. Pole fub,
113
29
Difference
24
14
"
ZP Comp. Latitude S. Co. Ar. 38 28 00--0.206168
Sides AP Sun from N. Pole 113°
29' Complement
Difference Co. Latit. Z. Sine
Co. Decl. Z. Sine
Sum of the Logarithms
Half is the Sine of
Doubled is
I
66 31 00--0.037547
77 15 30--9.989171
2 14 30--8.592335
18.825221
14 59 II-- 9.4126105
29 58 22. Converted
into Time, is 1 H. 59 Min. 53 Sec. 28 Thirds from Noon:
Confequently the Time of the Day is 10 H. o Min. 6 Sec, 32
Thirds in the Morning.
Or, if it be wrought as I have fhewed in Prob. 9. the Time
will be the fame as is found above.
}
1
EXAM
The Doctrine of the Sphere.
119
EXAMPLE.
Q }
Side oppof. to the required Angle 79 30 Sides P113 29
SZP 38 28
Half
39 45
Half Z. 2 Sides including re-
quired Angle
37 30 1
X 75 I
Z 77 15 2
4
I
// 37 30 1
X 2 14
Thus you fee the firft part of the Work is the fame; there
fore the Angle at the Pole will be the fame as is found in the
laft Work; for 'tis needleſs to work one thing over
twice.
Example 3. Admit at London I obſerve the Sun in the E-
quinoctial, and his Altitude to be 32 Degr. 35 Minutes;
I demand then the Hour of the Day?
Æ
Take the Chord of 32
Degr. 35 Minutes, and fet
it from H to E and O;
with the Tangent of its
Complement draw the Pa-
rallél of Altitude Æ O, and
it will cut the Equinoctial in
A, thro' A (by the Doctrine
of Spherick Geometry.) Draw H
PAS and ZA N, by which
you have the Oblique Sphe-
ric Triangle AZP, in which
are all the Sides given, to
find the Angle at the Pole,
from
Time
Noon
when the Obférvation was
or
made.
C
4
Ż
N
The Operation ftands thus, as in Prob. 9.
H
Æ
Deg. Min.
Deg. Min.
1
AZ Compl. Altit. 57 25
A P Sun from N. Pole
90 00
Half
28 42
2
ZP Compl. Latit.
38 28
Half Difference
25 46
Difference
51 32
Z 54 28
Half
25 46
X 02 56 2/2
I 4
Now
120
1
The Doctrine of the Sphere.
Now proceed thus;
Compl. Latit.
Sine 38 28 Co. Ar.
Sun from N. Pole
Sine 90 00
Sun,
Sine
54 284
2
Difference
Sine
02 56 2/2
Sum of the Logarithms
Half is the Sum of
15 I 3"
00.206168
10.000000
9.910546
8.710278
18.826992
9.413496
Doubled is
30 2 6. Converted into Time,
is 2 Hours 8 Seconds 24 Thirds from Noon, that is, 59 Min.
51 Sec.36 Thirds paft 9 in the Morning. Note, When any of
the Sides of the Triangle are a Quadrant, as here, A P is fo;
then the Logarithm or Sine of 90 Degrees may be omitted in
the Work, as you fee above I have taken no Notice of it. And
thus 'tis Evident how the true Hour of the Day may be
gained on any part of the Globe, if you have but a good
Quadrant to take the Altitude to Minutes of a Degree;
which is of excellent ufe in the Obfervation of Solar E-
clipfes, Regard being had to the Parallaxes and Refraction;
and alſo in finding the true Hour of the Night by the Moon
and Stars, as I fhall demonftrate in the next Section.
Secondly, I fhall fubjoyn another Method to find the true
Hour of the Day, by having the Latitude of the Place, Sun's
Declination and Altitude; which is that published by John
Collins. But becauſe he delivered it very abftruſely, I fhall
here Explain it, by way of Example. At London, on February
25, I obſerved the Sun to have 25 Degrees Altitude, and
4 Degrees 47 Minutes Declination South: What's the
Hour?
First, by Prob. 13. I find the Sun's Depreffion at the Hour
of 6 to be 3. 44: This remaining fixed for all that Day,
the Sun's Declination being fuppofed not to vary.
Now fay, As Co. Sine of the Sun's Declination,
To the Secant of the Latitude;
So in Summer is the Difference, in Winter the Sum of the
Sines of the Sun's Altitude, obferved, and of his Aititude or
Depreffion at the Hour of Six,
To the Sine of the Hour from 6, towards Noon in Win-
ter, and in Summer alfo, when the given Altitude is greater
than the Altitude at 6; but when it is lefs, then towards
Midnight,
}
OPE
!
The Doctrine of the Sphere.
OPERATION.
Deg. Min.
C. f. off Latitude
Declination
51
32 Co. Ar. 0.206168
4 47 Co. Ar. 0.001515
Sum, is the fixed Logarithm
0.207683
Given Altitude Natural Sine
4.226183
Sun's Depreff. at 6, Nat. Sine add
.651129
Sum, is Nat. Sine of
29 ri
4.877312
Logarithm Sine
29 II
9.688069
Fixed Logarithm add
0.207683
Z is the Logar. Sine of
SI 52
9.895752
which Converted into Time, is 3 h. 27' 28"+6 = 9 h. 27′
28" the Time in the Forenoon when
made.
the Obfervation was
51° 32 North, Sun's
Example 2. Latitude of the Place
Declination 23° 29′ N. and Altitude 53° 44′ 38″. What's
the Hour ?
First by Prob. 13 the Sun's Altitude at 6 is 18° 11.
Now the Work ftands thus:
Deg. Min.
C. f. of
S Latitude
Declination
51 32
2
Sum, is the fixed Logarithm
Co. Ar. 0.206168
23 29 Co. Ar. 0.037547
0.243715
Given Altitude, N. Sine
53 45
8.064446
Sun's Altitude at 6 N. S. fub.
18 II
3.120586
Rem. N. Sine of
29 36
4.943860
Logarithm Sine of
Fixed Logarithm add
Sum, is Logarithm Sine of
Converted into Time is
9 Hours 59 Min, 52 Sec.
3
59 58
9.693676
0 243715
9.937391
52 Sec. +6=
Hours 59 Min.
the Time of the Day.
29 36
"
PROB.
1
122
The Doctrine of the Sphere.
10
1
PROB. 18,
Given, the Latitude of the Place, the Sun's Declina-
tion and Altitude, to find his Azimuth from the
North.
Example. Admit at London, I obferve the Sun's Alti-
tude 50 Degrees, and his Declination 23 Degr. 29 Minutes
North; what is the Azimuth from the North?
H
NR
With the Chord of 60,
draw the Primitive Circle,
and fet off the Latitude of
London from H to P, and
draw the Axis P S, and to
it at right Angles the Equi-
noctial Æ Æ; fet off the
Altitude 50 Degrees by the
H Chord from H to B, and
and draw BP Parallel to
the Horizon, by the Tan-
gent of 40 Degrees; lay
off the Tropick by the
Chord 23 Degr. 29 Mi-
nutes, and draw it Parallel
to the Equinoctial, where
it cuts the Parallel. of Altitude B P, which is at A; there
draw the two Oblique Circles PÁS and ZAN; then
there is formed the Oblique Angled Spherical Triangle
AZP, and in it are all the fides given to find the An-
gle at the Zenith, which is the Sun's Azimuth from the
North,
N
1
$
*
OPE-
The Doctrine of the Sphere.
123
4
OPERATION.
ZP Complement Latitude
A Z Complement Altitude
A P Complement Declination
Complement Altitude fub.
Difference
Half Sum
Deg. Min.
38 28
40 00
31
66
Sum 144 59
Half 72 29 2
38 28
34 I
Complement Altitude fub.
Difference
72 29
40 00
32 29
22
Now Work thus:
Complement Latitude S.
Complement Altitude S.
Sine Differ. Co. Lat. and half Z
Sine Differ. Co. Alt. and half Z
Sum of the Logarithms
Half is Sine of
Double is
38 28 Co. Ar. 0.206168
40 00 Co. Ar. 0.191933
34 I
32 29 2/10
9.747842
9.730117
19.876060
9.938030
1
60 1
120 14 The Sun's Azi-
muth from the N. whofe Complement to a Semi-Circle is 60
Degrees, the Sun's Azimuth from the South.
Example 2. At London. I obferved the Sun's Altitude the
9th Day of January at 8 in the Morning to be 1 Degr. 14
Min. and Declination South 20 Degr. 11 Minutes; what's
the Sun's Azimuth from the North Note, when the Sun's
Declination is South, you must add it to 99, to get its diftance
from the North Pole,
1
t
PE-
&
124.
The Doctrine of the Sphere.
1
S
OPERATION.
Complement Latitude
Altitude
Deg. Min.
38 28
88 46
Sun's Diſtance from the N. Pole 110
II
}
Sum 237
25
Half 118
42
X{Co.
Co. Lat. and half Z
Co. Alt. and half
80 14
HINKIN
Z
29 56
>
حب
Deg. Min.
Sine Co. Latitude
38 28
Co. Ar.
0.206168
Sine Co. Altitude
88 46
Co. Ar.
0.000100
Sine Difference
80
→
14
9.993670
Sine Difference
9.698203
1
Sum of Logarithms
19.898141
·62 47
9.9490705
Half is the Sine of
Doubled, is
29 56
..125 34, the Sun's Azimuth from the
North, and its Complement to 180, is the Azimuth from the
South 54 Degr. 26 Minutes.
--
In the next Place, I fhall lay down Mr. John Collins his
Method of finding the Sun's Azimuth from the Eaft or Weft,
that fo the Reader may take which he likes beft.
ANALOGY.
As Tangent of half Complement of the Altitude,
To Tangent of half the Sum of Sun or Stars diftance from
the Elevated Pole, and of the Co. Latitude;
So Tangent of half their Difference,
To Tangent of a fourth Ark.
Then if this 'fourth Ark be less than half the Co. Altitude,
the Azimuth is acute, or lefs than go; if more obtufe, in.
both Caſes get the Difference of the two Arks; but if there
be no Difference, the Azimuth is 90 Degrees from the Me-
ridian. Then,
As R.
To t. of the faid Ark of Difference;
So t. Latitude.
1
Tọ
The Doctrine of the Sphere.
3125
To S. of the Azimuth from the Prime Vertical or Eaft and
7
Weſt.
EXAMPLE.
Latitude of the Place
Given Altitude of the Sun
Declination South
51 32
I 32 North.
25
04 47
Required the Sun's Azimuth from the North ?.
OPERATION.
90 Decl. S. O dift. à N.Pole 94 47
Complement Latitude
1
3828
I
2
1
Z 133 15 1 66 37 to 10.364121
56 19 28 9 t. 9.728412
X
Sum, fixed for that Declination
I
2
I
20.092533
Altit. 25 Compl. 65° 1 =
32 30 t.
fub.
9.804187
Tangent of
62 46 t.
10.288346
Difference
Latitude
30 16 t.
9.766095
51 32 t.
10:099913
Azimuth from Eaft Sine of
47 16 S.
9.866008
90 00
Sum 137 16 is the Sun's Azimuth
from the North as was required.
ས
The Sun's Azimuth from any of the four Cardinal Points
Eaft, Weft, North, or South, (for if you have it from any
one Point, you have it from the others alfo, by adding 90,
or fubftracting from 180 Degrees, as the Nature of your
Queſtion requires ;) is of very great ufe to the Mariner,
and Diallift: To the firft, in affifting him to find the
Variation of the Compafs; and to the other, in get-
ting the Declination of Planes whereon to draw Hour-
Lines to fhew the Hour of the Day. In order to the
obtaining
-126
The Doctrine of the Sphere.
obtaining of which, you must get the Horizontal Diſtance
of the Sun from the Pole of the Plane, and at the fame
Moment of Time (if poffible) take the Sun's Altitude with
a large Quadrant accurately divided; both which Inftru-
ments may be had of the beft fort, and at the loweſt Prices, of
Mr. John Fowler, Mathematical Inftrument-Maker at the
Sign of the Globe in Sweetings-Alley by the Royal Exchange,
London. Having gained the Sun's Azimuth, and the Di-
ftance of the Sun from the Pole of the Plane, obferve theſe
Rules.
1. When you make your Obfervation of the Horizontal
Diſtance, mark whether the Shadow of the Thread do fall
between the South, and that fide of the Quadrant which
was Perpendicular to the Plane; for then, add the Sun's
Azimuth from the South to the Horizontal Diſtance, and
that will give you the Declination of the Plane; and the
Declination of the Plane is then to the fame Point Eaft or
Weft as the Sun is.
2. If the Shadow fall not between them, then the Dif
ference between the Sun's Azimuth, and Horizontal Di-
ftance, is the Declination of the Plane: And here, if the
Azimuth be the greater of the two, then the Plane declines
to the fame Coaft whereon the Sun is; but if the Horizon-
tal Diftance be the greater, then the Plane declines to the
contrary Coaſt whereon the Sun is:
Note, The Declination thus found, is always accounted
from the South; and that all Declinations are accounted
from North or South, towards either Eaft or Weft; and
can never exceed 90 Degrees.
Example. Anno 1724, May 21, in London I obſerved the
Sun's Altitude in the Afternoon with my Aftronomical Qua-
drant, to be 14 Degr. 40 Minutes, and the Horizontal Di-
ftance of the fhadow from the of Pole of the Plane to be 22
Degr, 10 Minutes, between the North and that fide of the
Quadrant which was Perpendicular to the Plane. What is
the Plane's Declination, and to what Coaſt?`
I
O`P E-
The Doctrine of the Sphere.
127
OPERATION.
Sun's Alsitude gives Azimüth
from North
from South
Shadow fubftract
Plane's Deck from South Weftward
-
7,2 40
107 20
་ ་ 22 10
10
Example 2. At London I obſerved, the Sun's Altitude Fune
1, at 8 Morning, to be 369 26, and at the fame time the
fhadow of the Horizontal Diſtance between the South
and the Perpendicular 18 Degr. 30 Minutes. What's the
Plane's Declination?
OPERATION.
Q.
Sun's Azimuth. from North
98 14
<
South
81 46
Shadow, add
18 30
100 16-
Decl; from the S. by the E. Northerly
The Names of all forts of Planes are theſe following.
The Horizontal
The North or South Erect Direct
The Erect Decliner
The Recliner or Incliner
The Reclining, Declining
The Gonvex
The Concave
>Plane.
1
PROB.
T28.
The Doctrine of the Sphere
A T
PROB. 19.
C
Given, the Latitude of the Place, Sun's Declination
and Diftance from the Zenith, to find the Time of
Day-break in the Morning, and Twilight ending in
the Evening.
I have told you in the Definitions, that Day-break in the
Morning, and alſo the end of the Evening-Twilight is when
the Sun is 18 Degrees below the Horizon.
2-
!
་
Example. Let it be required in the Parallel of London on
the 5th Day of April, or on the 17th Day of August,
on which Days the Sun has ro Degrees North Declination,
and the Diſtance from the Zenith is always 18 +90 108°.
I demand the true Time of Day-break in the Morning, and
the end of the Evening-Twilight?
H
1
A
Z
S
N
Æ
=
Draw the Primitive Circle
repreſenting the Meridian of
the Place, HH the Horizon,
...be the Parallel of 18 Degr.
under the Horizon, dd, the
Parallel of, the Sun's De-
•
clination, and A where the
Parallel of Declination in-
H terfects the Parallel of 18
Degrees; there draw the
two Oblique Circles ZA N,
and PA S, by which is for-
med the Oblique Angled
Spherick Triangle A Z P,
in which are given Z P, the
Complement of the Latitude
38° 28', Z A the Sun's Di-
ſtance from the Zenith 108, and P A the Complement of
the Declination, or Sun's Diftance from the North Pole,
80 Degrees, to find the Angle at P, the Time from Noon of
the end of the Evening-Twilight ?
1
OPE
The Doctrine of the Sphere.
129
OPERATION, by the 11th Cafe of Oblique Triangles.
Deg. Min.
ZA =
108
AP
80
Half
54 o
Z P
38 28
X
Half and 20 46
Z74, 46
X 33 14
41 32
half 20 46
Deg: Min.
1
Sum Sine
Complement Latit. S.
Complement Decl. S.
Co. Ar. 38
28-- 0.206168
Co. Ar.
80
00-- 0.066648
74
46-- 9.984466
33
14-- 9.738820
19.936102
Difference Sine
Sum of the Logarithms
Half Sum is the Sine of
Doubled is
68 18- 9.968051
136 36 the quan
tity of the Angle at the Pole, which converted into Time,
is is 9 h. 6' 24' the End of Evening-Twilight; whoſe Com-
plement to 12 Hours is 2 h. 53' 36 in which is the true
Time of the Break of Day.
Example 2. What's the Time of Day-break in the Morn-
ing, and the End of Twilight in the Evening Dec. 10, at Lon-
don?
A
In the adjacent Diagram
HP is the Latitude of Lon-
´don 51° 32′, Æ VS the Sun's
Declination South 23° 29′
+90= 113° 29' A P the
Sun's Diſtance from the
North Pole Z A 108°, the
Sun's Diſtance from the Ze-
nith: And where the Tro-
pick of VS interfects the Pa-
rallel of 18°, which is at
A, there draw the two Ob-
lique Circles PAS.. and
ZAN, by which there is
formed the Oblique-Angled
H
Ъ
Æ
N
Ves
Æ
H
Spherical Triangle PAZ,
and in it the three Sides are given to find the Angle at P,
the Time from Noon.
με
2,01
K
OP E-
130
The Doctrine of the Sphere.
AP 113 29
ᏃᏢ 38 28
OPERTIÓN.
A Z
108
half
54
1
H 75 1
37 302/20
add and fub. 37 30 //
Z 91 30 Compl. 880 29' 2
X 16 29
"
Compl. Latit. S.
38 28
Co. Ar.
0.206168
Compl.
from N. Pole S. 66 29
Co. Ar.
0.037655
Sum, Sine
88 29
9.999849
9.453 128
19.696800
9.848400
Difference, Sine
16 29
Sum of the Logarithms
Half is the Sine of
44 51 2611
Double is
89 42 52 the quantity of the
Angle A PZ, which converted into Time, is 5 h. 58′51″
28", the End of Twilight, whofe Complement to 12 Hours
6 h. 1'8″ 32″", the Time of Day. And if, from the End of
Twilight you take the true Time of the Sun's feting, 3 h.
47' 24", there will remain 2 h. 11' 27" 28", the Dura-
tion of Twilight. Or, fubtract the true Time of the
Break of Day from the Sun's Rifing, and that will give the
Duration of Twilight as before.
There is a fecond Method of finding the Beginning and
Ending of Twilight, which I ſhall Exemplifie in the laſt Que-
ftion.
Compl. Latitude
OPERATION.
Sun's Diſtance from the North Pole
Sun's Diſtance from the Zenith
38 28
113 28
108
Sum
259 56
Half
129 58 Compl: 50ª 21
Sun's Dift. from the Zenith ſub.
108 O
X 21 58
Now
The Doctrine of the Sphere.
131
<
Now fay,
As Radius
To C. f. Latitude
So G. f. Declination
To S. fourth Ark
Say again,
Deg. Min. Sec.
Deg. Min.
90
00--10.000000
51
32-- 9.793832
23 29-- 9.962453
34
47-- 9.756 28 5
As S. fourth Ark
34 47
Co. Ar.
0.243764
To S. half Z
50 2
9.884466
So S. X
2 I 58
9.572949
Half is C. f. of
Double
44 51
Sum of the Logarithms
Angle at P as before; and confequently the Time is the
19.701179
26--
9.8508895
89 42 52 the
Quantity of the
fame.
PRO B. 20.
Given, the Latitude of the Place, and the Sun's De
preffion under the Horizon, to find when the Short-
eft Twilight happens in all the Year.
When the Declination of the Sun becomes equal to the
Difference between the Complement of the Latitude of the
Place, and the Depreffion 18 Degrees, and both North or
both South; then there is no Night, but Twilight. Thus,
in North Latitude 51 Degr. 32 Minutes its Complement
is 38 Degr., 28 Minutes; from which take 18 Degrees
the Depreffion, and there will remain 20 Degr. 28 Mi-
nures the Sun's Declination North, when the total Dark-
neſs ceaſes in that Latitude; and the two Days that
the Sun has that Declination North, are May 11,
and
July 10. See the Word Twilight in the Definitions.
K 2.
And
}
132
The Doctrine of the Sphere.
}
And by the Inveſtigation of the Problem, I find, that when
the Sun has 20 Degr. 28 Minutes South, which he hath on
January 10, and on November 10, that then in the Latitude of
51 Degr. 32 Minutes North, the Day will break at 5 Hours
46 Min. 8 Seconds, and End at 6 Hours 13 Min. 52 Se-
conds; and the Sun fets at 4 Hours 7 Min. 56 Seconds; there-
fore the duration of Twilight is 2 Hours 5 Min. 56 Seconds;
which is fhorter than when the Sun was in the Tropick of
Capricorn by 5 Min. 31 Seconds. And there is yet a fhorter
Time of the Duration of Twilight than this, as is plain if
you project the Sphere, drawing feveral Parallels of Decli-
nation: And where they interfect the Parallel of 18 De-
grees Depreffion, draw Great Circles to país thro' the Poles:
and then if you obſerve the ſeveral Arks of the Equinoctial,
intercepted you may there plainly fee them to be of an
unequal Length, and the fhorteft in all the Year at Lon-
don will happen Feb. 19 and October 1, when the Sun has
7 Degr. 2 Minutes, Declination South, which is found by
this Univerfal Canon ;
Min. Sec.
;
As Radius
90
00--10 000000
To S. Latitude
51
32-- 9.893745
So t. half Depreffion
9
00 9.194332
To S. Declination Sun South
PROB. 21.
2-- 9.088077
Given, the Longitude and Latitude, of a Planet or
Star, to find its Declination.
L
་
Example. Let it be required to find the Declination of
the Star called Arcturus, whofe Longitude the ft of F4-
nuary this Year 1727, is 20 Degr. 24 Min. 40 Seconds.
and Latitude 30 Degr. 57 Minutes North: Draw the Cir
cle ÆS vs P to reprefent the Solftitial Colure; E E the
Equinoctial P, and S its Poles; fet off the Chord of 23
Degr. 29 Minutes from A to and draw, vs, for
the Ecliptic, E and e its Poles: Then becauſe the Star
is in 20 Degr. 24 Min. 40 Seconds; that is, 69 Degr
35 Min. 20 Seconds from the Solftitial Colure, take
-
the
f
The Doctrine of the Sphere.
133
↑
the Secant of 69 degr. 35
min. 20 feconds, and draw
the Oblique Circle E A e
from Pole to Pole. Then
by Prop. 5. of Spherick Geo-
metry, lay down the Star's
Latitude. from B to A. and
thro' that Point; and the
Poles of the Equinoctial,
draw the Circle of Right
Afcenfion PAS; fo is
thère formed the Oblique :
Angled Spherick Triangle
APE, in which are gi-
ven, A E the Complement
of the Star's Latitude, 59
deg. 3 minutes, and PE
23 degr. 29 minutes, the
?
-
R
in
Z
N
E
Conftant Diſtance of the two Poles, with the included An-
gle equal to the Longitude of the Star, 20 deg. 24 min.
40 ſeconds, to find AP the Complement of the Star's De-
clination But for Conveniency of the Solution, I folve it
in the Triangle A Se, by letting fall the Perpen-
dicular SR: Then the Work will ftand thus:
As C.t. Se
To Radius
So C. f. Angle Se R
To te R
From e A
Subtract e R
There remains R A
Deg. Min. See.
23 29--
19.362044
10.000000
69 35-- 20 9.542519
8 36-- 57 9.180475
120
57
8 37
112 20 Compl. 67°
407
.
As C. f. e R
To C. f. RA
To C. f. e S
To S, D A Decl.
1
Then,
Q
8 37 Co. Ar. 0.004929
67 40
23 29
9.579777
9.962453
!
20 38 25
9.547159
K 3
f
Exam-
1
134
The Dottrine of the Sphere.
Example 2. Let the Declination of the bright Star, cal-
led The Virgin's Spike, be fought, whofe Longitude, is
20 degr. 2 min. 10 feconds, and Latitude 2 degr. 2 minutes.
South.
Z
B
P
E
Draw the Solftitial Colure
PESS with the Chord
of 60 degrees to any conve-
nient Radius, Æ Æ the Equi-
noctial, P and S its Poles;
fet off the Chord. of 29 deg.
29 minutes from E to,
and draw, VS, for the E-
cliptic E and e its Poles.
Then becauſe the Star is 69
deg. 57 min. 50 feconds, take
the Secant of 69° 57' 50",
and draw the Circle of Lon-
gitude E›A e, on the Circle
of Longitude; fet off the
Star's Latitude South from
B to A, and draw the Circle of Right Afcenfion PA-S;
then in the Oblique Angled Spheric Triangle A e S,
are given, e S, the Diſtance of the two Poles 23º 29', e A the
Complement of the Stars Laritude 87 degr. 58 min. with the
included Angle Se A, the Longitude of the Star from the
Solftitial Colure, 69 degr. 57 min. 50 feconds, to find SA,
the Complement of the Star's Declination.
S
As C. t. Se
To Radius
N
A
OPERATION.
So Co. f. Angle SeR
To t. e R fubt.
From e A
Remains RA
Deg. Min. Sec.
23 29
90
69 57
8 27
0--10.362544
0--10.000000
50-9.534803
59-- 9.172759
87 58 00
79 30
น
1
斗
​They
The Doctrine of the Sphere.
135
.
As C. f. eR
To C. f. RA
So C. f. e S
Deg
Then,
Deg. Min. Scc.
8 27
59 Co. Ar. 0.005339
79
30
9.260622
23 29-
9.962473
To S. A C Decl. South
9 44 30
9.228414
The fame by Tranfpofition, it will always bold.
As Radius
To S. Star's Longitude from y
So t. of the Obliquity
To t. of the firſt Árk
If the Declination fought be in the
Northern
Signs and
Southern
Signs and
Now obferve,
Deg. Min. Sec.
90
O
0--10.000000
20
2
I
10- 9.534803
23 29
& 27
59-- 9.172759
0-- 9.637956
North Latitude, Sub. the firft Arch from
Complement of the Star's Latitude, and
there remains the fecond Arch.
South Latitude, Add the firft Ark found as
above to the Complement of the Star's Lati-
tude, the Sum is the fecond Arch,
South Latitude, Subtract the firft Arch
from the Complement of the Star's Latitude,
and there remains the fecond Arch.
North Latitude, Add the firft Arch found
to the Complement of the Star's Latitude,
Land the Sum is the fecond Arch.
K 4
Exam-
}
1
136
The Doctrine of the Sphere.
EXAMPLE.
From a Quadrant
Sub. the Star's Latitude
Refts the Complement
Deg. Min.
90
0
2
2.
87 58. Then be-
cauſe the Star is in a Southern Sign and South Latitude,
(according to the third Canon above) fubtract the first Arch
8 degr. 27 min. 59 fec. from the Complement of the Star's
Latitude 87 degr. 58 minutes, and there remains 79 degr.
30 min. 1 fec. the fecond Arch. Then the fecond Analogy
IS,
As C. f. firft Arch
To C. f. of the ſecond
"
2
8 27
59
Co. Ar.
0.005339
79 30
I
}
9.260522
9.962453
? 44 30
9.228414
So C. f. Obliq. Ecliptic 23 29
To S. Decl. South
Example 3, Admit the Moon in II 11 degr. 28 minutes
with 5 degr. 2 minutes North Latitude: What's her De-
clination?
As Radius
OPERATION.
To S. of her Long. from T
So t. Obliquity Ecliptic
To t. firft Ark fub.
Complement's Latitude
Remains fecond Arch
As C. f. firft Arch
Now fay,
Deg. Min.
90 0--10.000000
71 28-- 9.976872
23
29-- 9.637956
22-9.614828
22
84 58 By the 1ft. Rule.
62 35
Deg Min.
22
23 Co. Ar 0.034019
62 35
9,663190
9.962453
9,659662
Exam:
To C. f. fecond
So C. f. Obliquity Ecliptic
23 29
To S. Declination North
27 10
The Doctrine of the Sphere.
837
6
{
Example 4. Let the Moon be in II 11 degr. 28° min.
as before, and 5 degr. 2 min. South Latitude: Then what's
her Declination?
&
As Radius
OPERATION.
To S. of her Longitude
So t. Obliquity Ecliptic
To t. firft Ark
Compl. D's Latit. add
Z is ſecond Ark
Deg. Min.
90
00--10.000000
7E 28-- 9.976872
23 29-- 9.637956
23-- 9.614828
22
84
58-- By the 2d. Rule.
107
25-- Compl. 72° 39′
Now fay,
3
As C. f. firft Arch
"To C. f. fecond
Deg. Min.
22 23 Co. Ar. 0.034019
72 39
9.4745 19
9.962453
9.47099 L
So C. f. Obliquity Ecliptic 23
To Sine Decl. North
29
17 12
By these two laft Operations you may fee what special
regard ought to be had to the Latitudes of the Planets and
Stars: For altho' their Longitudes be the fame, yet by reaſon
of their different Latitudes, they will Rife, South, and Sec
at different Times; but always get their true Declination,
and then you cannot mifs of the true Time. To know what
Stars Declination increaſe, and what decreaſe, ſee my Syftem
of the Planets Demonftrated, Page 119.
.
1
1
1
PROB.
1
73B
The Doctrine of the Sphere.
PROB. 22.
:
Given, the Longitude, Latitude, and Declination of
a Planet or Star, to find their Right Afcenfion.
Example. Let the Right Afcenfion of the Star Ar&turus
be required, whoſe
Longitude
Latitude
Declination
} is {*
Deg. Min, Sec.
20 24 40
30 57 oo North.
20 38 25 North.
In the first Scheme of the laft Problem, and in the Oblique
Angled Spherick Triangle APE are given, A EP, the
Longitude from ; AE the Complement of the Star's
Latitude 59 degr, 3 minutes, and AP, the Complement
of the Declination 69 deg, 21 min. 35 feconds, to find the
Angle at PD from ~.
OPERATION.
As Cf. Declination,
To C. f. Longitude
So C. f.
To C. f. Right Afcenfion
Add
Right Afcenfion is
20 24 49
Deg. Min. Sec.
+
20 38 25 Co. Ar. 0.0288 ne
9.971839
9.933.293
30 48 23
180
00
9.933944
30 57 00
210 48 23 Jan. 1, 1727.
all
The reaſon why you add a Semicircle, you have in Prob. 3.
the fame may alfo be found by the 11th Cafe of Oblique An-
gled Spherick Triangles; for in the fame Triangle A´É P,
the Sides are known, and required to find the Angle A PE,
the quantity of which is the Star's Right Afcenfion from
Capricorn.
!
Given,
The Doctrine of the Sphere.
139
Given,
Compl. Decl. A P
Compl. Latit. A E
69 21 35
59 3
Required Angle APE,
Dift. of the two Poles PE 23 29
OPERATION.
Sides including the required Angle EP 23 29
}
Side Oppofite to required 2
Angle is A E 59° 3'5
29 - 31. 30!
Half
Half Z Sides 22 56 17
}
Z 52 27 47
X 6 35
Deg. Min. Sec.
P 69 21 35
X
I = 22 56
45 52 35
17
S. Comp.Ded. AP 69 123mcolar.
21 35
-0.028812
S. Dift. Poles PE 23
29
Co. Ar.
0.399252
S. Z
52 27 47
9.399591
S. X
6 35 13
9.059604
Sum of the Logarithms
Half is the Sine of 29 36 5
Doubled is
47′ 50″.
19.387259
- 9.6936295
59 12 10 whofe Compl. to 270 is 21c9
Example 2. What's the Right Afcenfion of the Star call'd
the Virgin's Spike; its
Longitude
Latitude
Declination
{being {
4 20 2 ΙΩ
2
2 • South?
9 44 3 South?
O.PERATION.
As C. f. Declination
Deg. Min. St?.
9 44 30 Co. Ar.
Co. Ar.
0.036309
To C. f. Longitude
So C. f. Latitude
To C. f. Right Afcenfion
And
Right Afcenfion is
17 42
180 00 00
20
2 10
3
28
9.97 2885
9.999726
9.978920
197 42
28 Jan. 1, 1727.
PROB.
14:0
The Doctrine of the Sphere.
}
PROB. 23.
Given, the Longitude
the Longitude and Latitude of a Star or
Planet, to find the Right Afcenfion.
Example. Let the Right Afcenfion of the Star
Virgin's Spike, be required, whofe Longitude is
2 min. 10 fec. and Latitude 2 degr. 2 min. South.
ANALOG Y.
called the
20 degr.
As Radius
To S. Longitude from A
So C. t. Latitude of Spica
To t. of the first Ark
If the Lon-
git. of the
Star be
90
Deg. Min. Set.
20
2
0--10•@@doo.
10-- 9.534803
2
2
c-211.449732
--84 5 5.
2--10.984535
Now this General Rule is to be obferved;
PSES
~mvswX
And Latit.
viz.
North fubt.
South add
23 29, to, or from
the firft Angle, the
Z or X is the ſecond
North add
South fubt.Arch.
EXAMPLE.
Here the Star is in, and Latitude South; therefore
}
From the first Arch
Subt. the Obliquity Ecliptic
84 5
23 29
2 2
Remains the fecond Angle
60 36
2
Now
;
*The Doctrine of the Spheres
141
1
1
As S. of the firſt Arch
To S. Second
So t. of the Longitude
Tò t. R. A. from
Add a Semi-circle
!
Z. R. A. from T
Now fay,
84
5
2
2
60 36
20 2 10
17 42 52
180 Oo
Co. Ar. 0.002319
197 42 52 as before.
9.940127
9.561917
9:504363
२
And after this manner are the Tables of Right Afcenfions in
Time in this Treatife Calculated. In which may be obſerved.
that a Planet or Star having Latitude
SNorth in vs m x r ŏ II
amm
South in ∞ ( R
2
The Right Afcenfion is di-
miniſh'd, and confequently the Star comes fooner to the Me-
ridian than if it were in the Ecliptic.
But when the Latitude of the Star is
North in a ™
≈ a ❤ ~ m
South in VS ™ X PX I
The Right Afcenfion is in-
creaſed, by which a Planet comes later to the Meridian than
if it were fimply in the Ecliptic.
PROB. 24.
:
Given, the Right Afcenfion and Declination of a
Star or Planet, to find its Longitude and Lati-
tude.
This Problem is only a Converfion of the two laft; for in
the fame Triangle A EP there are given PE, the conſtant
Diſtance of the two Poles 23 degr. 29 minutes, and A P, the
Complement of the Declination, and the included Angle APE
the Right Afcenfion from Capricorn, to find A E, the Com-
plement of the Latitude of the Star, and the Angle A EP, its
Longitude.
Example. The Right Afcenfion and Declination of Ardu-
rus, is 210° 48′ 23″ and 209 38' 25" What's its Longitude
and Latitude?
}
•
First,
142
The Doctrine of the Sphere.
1.
Firſt, For the Latitude, or its Complement A E,
Let fall the perpendicular E T; then in the Right Angled
Spheric Triangle ET P.
Deg. Min.
As C. t. E P
To Radius
90
So C. f. Angle TPE, R. A. from vs
Tot, TP
59
12
23 29--10.362044
--10.000000
12-- 9.709306
33-- 9.347262
'As Radius
To t, E P
So C. f. Angle TPE
Or, by Tranfpofition, fay,
Deg. Min.
99 0--10.000000
-23 29-- 9.637956
59 12-- 9.709306
Tot, TP
fubt.
12 33-- 9.347262
From A P
69
22
Remains T A
56 39
Now fay,
Deg. Min.
12
33 Co. Ar. 0.010403
9.738241
9.962453
9.711097
As C. f. firft Arch P T
To C. f. fecond TA
So C.f. PE = Obliquity
To S. Latitude B A
56 49
23 29
30 57
Secondly, For the Longitude or Angle A EP. Now all
the Sides are known and the Angle at P; therefore by the
first Cafe of Oblique Angled Spherical Triangles, it will
hold.
As S. A E Co. Latitude
Deg. Min.
59 3 Co. Ar. 0.066707
To S. Angle P, R. A. from VS. 59 12
So S. A P, Co. Declination
9.933973
9.971208
9.971888
69
22
To S. Angle E
fubt.
69 36
From
180
'Angle A EP
110 24
Sub.
90
Remains Longitude in
20
24
Or
The Dott
·743
Doctrine of the Sphere.
Or, if I had fubtracted 69 deg. 22 min. from 90 degrees,
it would have given me the fame thing.
Theſe three laft Problems are of excellent uſe in making
Aftronomical Obfervation, as the young Student will preſent-
ly perceive, when he is a little acquainted with this fublime
Study. For by the 21st and 22d you may find the Right
Afcenfions and Declinations of all, or any of the Fixed Stars
in the following Catalogue, which R: A. being reduced into
Time; will be of excellent ufe to find the Hour of the Night
by the Stars; as I fhall fhew in its proper Place.
1
$
FAC
t
1
י
ATABLE
}
144
The Doctrine of the Sphere.
ATABLE of the Right Afcenfions, reduced into
Time, and Declinations of 42 Eminent Fixed Stars
for the Year 1727, being of ufe to find the Hour of
the Night.
I'
Stars NAME S.
Declina-
tion.
"
0
N the Breaft of Caffiopeia, Scheder, 55 2 N20
The Bright Star in the Tail of
the Whale,
Pole Star,
The Bright Star of Aries,
In the Jaw of the Whale, Mandibula,
Head of Medufa, Algol,
The Bright fide of Perfeus,
Brightest of the 7 Stars Pleiades,
R. A. in R. A in
Motion Time
"H.'
618300 25 14
19 29 S 30
7 26500 2941
87 54 No
22.9
Noo
259 32
+
910 20 0 36 41
275649 151 47
42 co 00
2 48 0
3953
00
48 51
2314
The South Eye of the Bull, Aldebaran, 1555
In the Goat, Capella,
The bright Star in Left Foot of
Orion, Rigel,
North Horn of the Bull,
The Left Shoulder of Orion,
South Horn of the Bull,
The middle Star in Orion's Belt,
The laſt in Orion's Belt,
Right Shoulder of Orion,
In the Great Dog's Mouth, Syrius,
Caftor, or the Head Northern Twin,
Procyon, the Little Dog,
Pollux, or the Southern Twin,
The Heart of Hydra,
The Lyon's Heart, Regulus,
The Southermoſt of the two pre-
ced. in Great Bear,
The Northernmost of them,
423645 350 27
40 46 14 30 3 458
001 52.50
525000 3 10 20
46
65 3
4 2012
45 45 50
74 73045630
833 S 12
752100 5 I 24
28 20 N51
77 1400 5 856
6 4
2056 45
125 S
28
7737 25 5 10 29
80 192
20 52118
343152218
27 I 2
719N26
16 20 S 58
81444 526 59
85 438
540 18
98 1640
633 1
7 17 3
7 2457
80
3227 N 0109 15 50
554
28 39
15111 14 12
14 112 739 72830
729
S 013831 20 9 14 Í
1317 No148 26 7 9 5344
57 50 36 161 16-26 10456
6313
The Tail of the Lyon, Deneb.
16 5
40 161 37 5010 46 ||
30 17346 10 11 35 5
The North of the 2 following in
the Great Bear,
58 34
28 180 27 40 12 147
2740
The first in the Tail of the Great
Bear,
519 30
190 23 00 12 41 32
The
The Doctrine of the Sphere.
145
•
The TABLE continued.
Stars NAME S.
In the North Wing of the Virgin,
Vindematrix,
The Virgin's Spike,
The Middle of the three in the Tail
of the Great Bear, ›
The laft but one in the Tail of Hydra,
The last of the three in the Tail of
Great Bear,
In the following Shoulder of the
Çentaur,
Arcturus,
The Scorpion's Heart, Antares,
The Brighteft in the Dragon's Head,
The brighteft Star in the Harp,
The Brightest in the Eagle,
The Mouth of the Southern Fish,
Fomaḥaut,
In the Flying Horfe, Scheat,
The Head of Andromeda,
Declina- R. A. in 'R.A. in
tion. Motion, | Time.
"
1225 40/19230521246 3
9445 30197 42 28 13 IC 50
56 22 N36 197 44 131056
21 43S 0196 227 13 410.
5042 N 0204 11 50 13 36 47
3437
20
50207 39 32 135038
25210482314 3
313
038
25 47 S 30243 10 20 1642 11
15132 N 0207 35 45 175 23
13833
Ic276 54 2018 27 37
Is 294 20 0 019 37 20
1937
810
313 S 30 340 34 59 22 42 20
26 35 N52342 37 20122 50 29
27 34 22358 3 3 3 0 2 3 5 4 1 +
34
+
I
PROB.
{
146
The Doctrine of the Sphere.
PROB. 25.
Given, the Latitude of the Place, and the Hour of the
Sun's fetting, to find its Declination..
Example. At London when the Sun apparently riſes at
5, and fets at 7 a-Clock, I then demand its Declination ?
D
A
H
A
Draw PHS B, to re-
preſent the Solftitial Colure,
H B the Horizon, EE the
Equinoctial; by help of the
Lines of Chords on the Se-
ctor fet of the Pole's Eleva-
tion from B to P 51 degr.
B 32 minutes; then becauſe
the given Hour is between
Six a-Clock and Midnight,
viz. 5 Hours = 75 degrees,
take the Secant of 75, aud
draw P, A, S, the given
Hour-Circle, and where it
cuts the Horizon which is at A, there the Parallel of the
Sun's Declination DD for that Day muft alfo interfect it:
Then in the Right Angled Spherical Triangle A BP, there
are given B P, the Pole's Elevation 51 degr. 32 minutes,
and the Angle APB 75 degrees, to find A P, the Comple-
ment of the Declination.
2
1
Æ
OPERATION. ..
Deg. Min.
As t. B P the Lat. 51 32--10.099913
To Radius
90
OC--I0.000000
73-- 9.313083 whofe Comp, is 119 37
So C. f. Angle APB 75 00-- 9.412996
To C. t. A P 78
Or,
The Doctrine of the Sphere.
147
÷
As Radius
Or, by Tranfpofition
Deg. Min.
go 0-10.00cono
To C. t. B. P. the Lat. 51 32- 99.00026
So C. 1. Angle APB 75 00- 9.412996
To t. CA the Decl. N 11 37- 9.313085
O Sets Declination.
Hours. o
21
4
40 South
$
1 I
37
6
O
၀၁
7
II
37 North.
8
21
40
And after the fame manner have I found the Declination as
in this Table on the Right Head.
- PRO B. 26.
Given, the Latitude of the Place, and the Sun's Azi-
muth from the South, to find his Declination when
be rifes and fets upon that Azimuth.
Example.
At London, when the Sun rifes and fers
upon the 100th Azimuth from the South, I demand then
his Declination?
Draw the Solftitial Colure
ZHNA, fet off the Latitude
from H to P, and from
ZE, draw EE for the
Equinoctial; and becauſe
the given Azimuth is a roo
from the South, that is 80
from the North, take the Se- H
cant of 80, and draw the
Azimuth Ź c N, where it
interfects the Horizon, which
is at c; thro' that Interfecti-
on draw the Hour-Circles
PcS; then in the little
Triangle Bc are given
B
И
P
H
Æ
Tc, the Azimuth from the Eaft or Weft 10 degrees, and
the Angle B = the Complement of the Latitude 38 deg.
28 minutes, to find the Declination B c.
c=
L2
ANA-
1
148
1
The Doctrine of the Sphere.
*
•
J
As Radius
ANALOGY.
To S. Y c Azimuth from E. or W.
So S. Angle BV c, Co. Latitude
To S. Bc Declination North
Deg. Min.
90 00--10.000000
00-- 9.239670
10
38 28-- 9.793832
6,12-9.033502
Example 2. What Declination has the Sun when he fets
at London upon the 50 degr. 10 min. Azimuth from South?
ANALOGY.
I
As Radius
To C. f. Azimuth from the South
So C. f. Latitude
To S. Declination South
Deg. Min.
90 00--10.000000
50 10- 9.816557
51 32-- 9.793832
23 29,- 9.600389
By which Calculations it appears that when the Sun is in
the Tropick of Capricorn, his Azimuth from the South when
he rifeth and fetteth, is 50 deg. 10 min. and its Complement
to a Quadrant is the Azimuth from the Eaft and Weft Points,
equal to the Amplitude 39 deg. 50 min. becauſe this Arch
of the Horizon meaſures the Angle at the Zenith, it being at
the Diſtance of 90 deg. from it.
Hence, becauſe theſe two Problems are very uſeful to
delineate the Hour-Lines upon Gunter's Quadrant, I fhall
here incert all the Requifites thereunto belonging for the La-
ritude of 51 deg. 32 min. North.
A TABLE
The Doctrine of the Sphere.
149
'
>
A TABLE of the Sun's Declination to every 5th Day
of the Month for the Year 1727, for Infcribing the
Months.
ļ
April
May
18
4 19
June
Months.
I
$
ΙΟ
15
Q
Q
1 O
21
219 58 18 47
January 21 S 42
February 13 46 12
March
3 25 I
8N 3510
48
2510 38 8
5000N 8 2 7
46 13 26
821
2 II
3 20
5
23
10 23
22 23
2923
2923
25
July
22 4 21
29/20
Auguſt
36 19
34
IS
8 13
54 12
1710 34
Septemb.
4 20 2
47 00
51 IS 7
October
7 S 18
8
Novemb. 17 40 18
17 40 18
48 10
3712 22
43 19
December 23 6 23 21 23
5420 57
29/23 25
Months.
20
25
30
O
O
January
February
March
4N. 4 5
April
May
15
June
July
Auguft
Septemb.
17S.27 16
6 54 4 58
21
2 22
23
5 59 77 SI
016 28 17 49
5422 32 23
II 22
18 24 17
·8 48 6
47/22
I
12
715
43
58
58 5
5+
5
O
6 55
38 17
7
9 22
3123
42/22
2
3
3 S. 4
October 14 315
Novemb. 21 49 22
December 23
14. 25
1
·
霉
​L3
A Ta-
150
The Doctrine of the Sphere.
}
5
A TABLE of the Sun's Meridian Altitude to every
5th Day for the Latitude of 51° 321 North.
I
Months.
5
10
15
。
January 16 46 17
4617
2618
30 19 41
February 24
March 135
24
42 26
3 27
50 29
40
336
38 38
3838
36 40
35
April 47
3:48
3050
1451
$4
May
56
32 57
31 58
31 58
36 59
33
June 61
3861
50 61
57,61
53
July 60
Auguft 153 36
3259
57 59
4,58
2
52
22 50
45 49
2
Septemb. 42 48 41
!
15 39.
19 3.7
21
October 31 10 29
40 27
51 26
6
Novemb. 20 48 19
45 18
3417
34'17
31
December 15 22 15
71459.15
3
Months.
20
25 | 30
O
January
February
March
~ 34
21
122
2824
3
31
34 33
30
42
32 44
27 46
19
April
53 28 54
56 56
17
May
60 2261
061 2.9
June
61
39 61
15 60 40,
July
56 5255
35 54
11
Auguft
47
16 45
26
43
33
Septemb: 35
35
24 33
28 31
33
October
24
25 22
25 22
50 21
21
Novemb. 16
16
39 15
5715
26
Decemb. 15 19 15
46 16:25
A
North Declinations added, and South fubtracted, to, or from
the Elevation of the Equinoctial, give the Meridian Alti-
tude.
The Doctrine of the Sphere.
ISI
{
}
A TABLE of the Sun's Altitude at every Hour
when he is in the Equator and Tropicks, Lati-
tude 51 Degr. 32 Minutes North, for drawing
the Hour-Linés. -
Equinoctial
Hours.
t..
Tropicks.
VS
12
0138
2861
5714 59
Il 115
036
36 56 59
5659
5013 · 54
ΙΟ
230
0132
32 3653
45|10
30
9∞
3/45
26
645
45
415
17
8
4
60
18
736 40
7
575
019
1627
22
6
690
018
IO
5
19
27
8
I
31
See the TABLE of the Sun's Rifing and Setting at London,
L 4
A TA-
152
The Doctrine of the Sphere.
ATABLE of Right Afcenfion to every 5th Deg. of Lon-
gitude for dividing the Equinoctial in the Quadrant.
ԴՐ
II
રી
m
0
27 34
57
$7 4890
0122 12 152
I
6
5
4 35
32 42
63 3 97 27 127 22
156 51
10
911
37 34
15 13 49
42 21
68 21 100 53 132 28
73 43 106 17137 29
161 33
166 12
20 18 27 47 32
79
7111 39 142 26
25 23 9 52 3884 33 116 57147 18 175 25
301 27 541 57 48 90 0122 12152 6 180
170 49
M
VS
ww
*
o 180
1
5184 35
10 189 11
0 207 54 237 48 270 0302 12332 6
212 42,243 3 275 27 307 22336 51
217 34,248 21 280 53 312 28341 33
15 193 43 222 21 253 43 286 17 317 29 146 12
20 198 27 227 32 259 7 291 39 322
25203 9 232 38 264 33 296 57 327
130207 54 1237 48/270 0322 12 332
i
26 350 49
18 355 25
6360 00
A TA-
The Doctrine of the Sphere.
153
A TABLE fhewing the Aftenfional Difference to every
Degree of the Sun's Declination for the Latitude of
51 deg. 32 min. North. Calculated by Problem 5.
d
Decl.
Sun.
Afcenfional
Difference.
;
-
I
2
3
گر
4
1 16
2,31
3 47
5 3
6 19
P
36
7
53
8
TO* II
}
9
II
29
ΙΟ
12
49
It
14
و
12
15
30
13
16 53
14
18
17
15
19 4.2
1-6
21
8
រ ។
18
19
20
2 2
2 I
22
22
24
25 40
27 15
28 52
30 32
32 16
37
7
23 29
33
9
?
To
154
The Doctrine of the Sphere.
To Draw the Azimuth in the Quadrant.
For this purpoſe you must firft calculate a Table fhewing
the Sun's Altitude above the Horizon when he is in the Equi-
noctial, Tropicks,, and fome other intermediate Parallels
of Declinations at every 5th or 10th Azimuth.
Thus, fuppofe the Latitude 51° 32! North and the Sun
in the Equinoctial on the 80th Azimuth from the South;
What's the Altitude ?
A NALOGY.
Deg. Min.
As Radius
90
00--10.000000
To C. t. of the Latitude
51
32-- 9.900086
So C. f. of Azimuth from Merid. 80
00-- 9.239670
Tot. Altit. in Equinoctial
7
51 - 9.139756
And after the fame manner is the fifth Column of the fol-
lowing Table calculated under and, which muſt be fi-
niſhed e're the other can be done.
Then if the Sun have Declination, the Meridian Altitudes.
are given in the foregoing Table; but when he is not on the
Meridian, but on fome other Azimuth, then ſay,
As the Sine of the Latitude, !
To the Sine of the Declination ;
So is the Co. Sine of the Altitude at the Equinoctial,
To the Sine of the fourth Arch.
Now obferve thefe Rules:
1. If the Latitude and Declination be both of one Deno-
mination, that is, both North, or both South, on all Azi-
muths from the Prime Vertical upon the Meridian, or leſs
than 90 Degrees, then add the fourth Arch found by the Pro-
portion above, to the Altitude at the Equinoctial;, that Sum
is the Sun's Altitude on the given Azimuth.
2. If the Latitude and Declination are both alike, and the
Azimuth more than 90 Degrees diftant from the South, take
the Altitude at the Equinoctial out of the fourth Arch, the
the remainder is the Altitude of the Sun on the given Azi-
muth.
3. When
▼
155.
The Doctrine of the Sphere.
3. When the Latitude and Declination are unlike, or of
different Names, then take the fourth Arch out of the Sun's
Altitude at the Equinoctial, and the remainder will give
you the Sun's Altitude on the given Azimuth.
Example. What's the Sun's Altitude on the 80th Azimuth
from the South, Declination 23 degr. 29 min. North, and
Latitude of the Place 51 degr. 32 min. North? The Alti-
rude in the Equinoctial was found before to be 7 degrees
51 Minutes.
OPERATION.
As the Sine of the Latit.
To S. Decl. North
Deg. Min.
52
32 Co. Ar. 0.166255
23 29
9.600409
So C. f. Altit. in Equino&t.
To S. fourth Ark
7 51
9.99591 I
30 17
9.702575
Now according to the firſt Rule, becauſe the Latitude and
Declination are both North, I add the fourth Arck 30 degr.
17 min. to the Sun's Altitude in the Equinoctial 7 deg. 51
min, and the Sum 38 deg. 8 min. is the Sun's Altitude upon
the given Azimuth, as was required.
And to make all yet plainer, I fhall add more Examples
in the Tropicks, and fhew how one Analogy ferves for
both Tropicks.
Firft, for Altitude Sun on the Meridian.
Deg. Min.
Height Equinoctial at Lendon
Declination add and fubtract
38
28
23 29
Z 61
57 M. Alt. S.
X 14
59 M. Alt. VS.
For Sun's Altitude on 10th Azimuth in the Tropicks.
Deg. Min.
As S. Latitude
51
32 Co. Ar. 0.106255
To S. Decl.
23 29
*9.600409
So C, f. Alt. Equino&.
38 2
9.896335
To S. of the Arch
23 38
9.602999
Z is the Altit. in
6I
40
is the Altit. in VS
14 24
For
156
The Doctrine of the Sphere.
For the Sun's Altitude on the 20th Azimuth from the
As S. Lat.
To S. Decl.
So C. f. the Equator
To S. of the Arch
Z is Alt. in
X is Alt. in VS
South..
Deg, Min.
5 I
32 Co. Ar. c.106255
23 29
9.600409
36 44
24
60
4
48
9.903864
9.610528
I2
40
For Altitude Sun on the 30th Azimuth.
Min. Sec.
As S. Latitude
51 32.Co. Ar. 0.106255
To S. Decl.
23 29
9.600409
So C. f. Altit. in Equino&t.
34 32
9.915820
To S. of the Arch
24 47
9.622484
Z is Alt. in go
59 19
X is Alt. in VS
9 45
And after this manner is the Sun's Altitude obtained in
the Tropick, when the Azimuth is lefs than 90 from the
South ; but when it is more, viz. 100, 110, 120 degrees
from the South, then obferve, that the Sun in the, Equi-
noctial has the fame Depreffion under the Horizon on the
100th Azimuth, that he has Altitude on the 80th Azimuth;
therefore fubtract the Altitude in the Equinoctial from the
fourth Arch, gives the Altitude on the given Azimuth.
Example. What's the Sun's Altitude in the Tropick of
Cancer on the 100th Azimuth from the South, Latitude as be-
fore?
OPERATION.
1
The fourth Arch for the 80th Azimuth is
Sun's Altit. in Equinoct. on 80th Azimuth ſub.
Sun's Altit. in on 100th Azimuth
Deg. Miu.
30 17
7 51
22 26
For
The Doctrine of the Sphere.
3157
}
For Sun's Altitude in on the 11oth Azimuth from South.
The fourth Arch for 70th Azmuth is
Sums Altit. in Equator on 70th Azimuth fub.
Sun's' Altit. in on 110th Azimuth
Deg. Min.
29 25
15. II
I [..
14 14
For Sun's Altitude in on the 120th Azimuth from the
South.
The fourth Arch for the 60th Azimuth is
Sun's Altit. in Equinoct. on the 60th Azimuth
Sun's Altit. in on the 120th Azimuth
Deg. Min.
28 14
21 38:
6 36
To find the Sun's Altitude in the beginning of a on
the 120th Azimuth from the South?
You must first find the fourth Arch. to the 60 Azimuth
thus:
As S. Latitude
To S. Deck.
So C. f. Alt. Equi. on 60th Azim.
To S. of the Arch
Deg. Min.
51
82 Cq. Ar. 0.106255
20
I I 15-* -9.537937
21
39
O--
24
I I
9.968228
9.612420
32
Sun's Altit. in on 12th Azi. 2
II
To find the Sun's Altitude in the beginning of my on
the 80th and 100th Azimuth from South.
As S. Lat.
To S. Decl.
So C. f. Alt. in Equino&.
To S. of the Arch
Z in Alt. on 80th Azimuth
Deg. Min.
SI 32 o Co. Ari e.106255
II
29 33
7 51 Oo
14 36. 00
22
27
X in Alt. on 100th Azimuth 6 45
9.299376
9.995911
9.401542
To
158
The Doctrine of the Sphere.
To find Sun's Altitude in the beginning of m on the 70th Azi.
muth from the South.
Deg. Min. Sec..
51 32 oo Co. Ar. 0.106255
As S. Latit.
To S. Decl.
II 29.
39
So C. f. Alt. in Equinoct.
15 - 11
Ӧ
To S. of the Arch
14 13
9.299376
9.984569
9.390200
X is Alt. in * M on 70 Azi.
0 58
To find Sun's Altitude in beginning of on the 50th Azimuth
from the South.
Z
Deg. Min. Sec.
As S. Lat.
51 32
。 Co. Ar. 0.106255
To S. Decl.
20 ΙΣ 15
9.537937
To C. f. Alt. in Equinoct.
To. S. of the Arch
27 3
9.949687
23 7
2.593879
X is the Altit. in on
the 50th Azimuth.
"} 3 56
O
And after this manner have I Calculated the following
Table of the Sun's Altitude upon every Tenth Azimuth in
the beginning of every one of the 12 Signs; (by help of
which and the following Table the Azimuths may be laid
down on a Quadrant) which may be done to every Degree
of Azimuth and to any particular Latitude at pleaſure; the
Degrees anſwering o, Azimuth are the Meridian Altitude of
the Sun, and the reft of the Table, is found by Galculation,
as I have fhewed above.
}
រ
1
A
The Doctrine of the Sphere.
139
[
>
{
A TABLE of the Altitude of the Sun in the be-
ginning of each Sign, for every ro Degrees of
Azimuth, in the Latitude of 51 Degr. 32 Min.
North.
Azi-
muth
from
South o
-
69
Q
O
✔
•
VS
૧૪
TMM ԴՐ
*
Mw
061
10 61
57 58 3949
3949
58 38
28/26 5818
1714 59
4058 22 49
3638
226
2817
4314 24
20 60
4957
26/48
31 36
44 25
716
312 40
30159
19 55
55
5046
50 46
5046
38 34
34
32/22
25/13
14 9 45
40 57 18 53
2743
5313x
19 18
46 9
9
15 5 33
5054
150
10 40
927
313
56 3
560
5
60 49. 54 45
5.135
2121
39 7 58
70 44
38 40
40
23/29
23 29
2512 II 0 581
8038 9133
44 22
277
51
9030 36 26
814
23
100 22 26 18
06.45
11016
14 9
56
120 6 36 2 321
t
After this manner did I Calculate all the Requifites for
Delineating the Hours and Azimuths upon a Quadrant for
Madrid in Spain, which was made by that Curious Work-
man Mr. Benjamin Scott, at the Mariner and Globe in the
Strand, London, who maketh this and all other Mathema-
tical Inftruments in Silver, Brafs, Ivory or Wood, either for
Sea or Land, according to the beft Improvement, and at the
lowest Prices.
By Prob. 18, the following Table of the Sun's Azimuth is
Calculated.
A TA-
160
The Doctrine of the Sphere.
A TABLE of the Sun's Azimuth from the South
at his Entrance into the 12 Signs, and at each
Hour and Quarter of the Day, for the Latitude
of 51 Degr. 32 Min. North.
Hours.
20
m ԴՐ
* M!
O
VS
12
II
40
O
O 0.
O
7 10 5 50
14 22 13
21 27 19
1 27 54 25
34 14 32
O
O
O
5
354
53 3
52 3
50 3 50
22 II
89
387
55 7
38 7
8
50 16
4014
1012
81 I
810
35
58!
22
1 118
5016
2014
4114
10
27
3023
2020 12/18
15/17
36.
40
12 37
381 32
32
3927
3927
40 24
40/24 221
5020
57
45 39 43
0 37
30/32
629 4026
40/26 2824
23
IO
250
51 47
55 42
036.2131 48 28
48/28
5027
46
55
31 52
41 47
040 2835 1232
1631
O
60
857
9 50
5044
2439 835 3734 IS
64
9161
16 54
3048
1042
3238 4837 34
9 3 68
8 65
14 58
4551
5446
0142
04040
71
50 69
54 62
78 46 76 269
75 24 72 35 66
3055
3049 2345
1443 43
858
5752
5752 4048
1746.44
3652
2055
51/52
950 រុក
8
81
5279 18 72
5565
3858
5754
24
85
6182
82
40-76
1068
5062
5
88
85 36 79
2072
265
I 2
91
87 48 82
2675
068
IO
7
593 5891
30 85
30 85
2878
471
8
96 49 94
23 88
88
2581
7
99 38 97
102
14 91
23
84
33100
5 94
2087
6 6105
6102 5297
107 50104 56100.
1090
I10 34108
24 103
113
16111
12 195
52
5 7116
0114
118
50116 52
is
4
121
40119 45
18122
36
22
124
8127
1129 50
Note the 129 sol is the Sun's Azimuth at the time of
the rifing and fetting of the Sun in the Tropick of Cancer,
and 50° 10' is the Azimuth when he rifes and fets in the
Tropick of Capricorn.
PROB
The Doctrine of the Sphere.
161
!
PRO B. 27.
Given, the Sun's Place, and Time of the Day or Night
(under any known Meridian) to find the Right Afcen-
fion of the Mid-Heaven:
To the Time propofed, find the Sun's Right Afcenfion by
Prob. 3; then reduce the Apparent Time of the Day or
Night into Degrees and Minutes, and add it to the Sun's
Right Afcenfion before found; that Sum is the Right Afcen-
fion of the Medinm Cali, or Mid-Heaven. If the Sum exceed
360°, reject 360° and the Remainder is the Right Afcenfion
of the Mid-Heaven.
Example. Anno 1728, March 13, at 22 Minutes paft 8 in
the Morning the Sun's Place is 3° 55'47" and his Right
Afcenfion 3° 36' 6"; I demand the Right Afcenfion of the
Mid-Heaven at London?
OPERATION.
Apparent Time Hours
Deg. Min. Sec.
20 = 300 00 об
in the Meridi- Minutes 22 =
an of London.
Sun's Right Afcenfion add
Sum, is R. A. Medium Cali
Complement fhort of Y
5 30
3 26
OO
об
308 56 6
360 00 оо
51 03 54
་
1
Note, When the Sum is less than 90 Degr. it is the R. Afc.
from ; if it be more than 90 Deg. and lefs than 180 Deg.
fub. from 180; if more than 180 Deg. and less than
270 Deg. fub. 180 Deg. from it: But if it fall in the laft
Quadrant, as in the Example above, fubtract it from 360,
and you have the Quantity of Degrees and Minutes that you
are to make uſe of in Trigonometrical Calculations.
}
M
This
162
The Doctrine of the Sphere.
This Problem is of fingular uſe in the Calculations of Solar
Eclipfes, as I fhall fhew in the Precepts for that purpoſe.
PROB. 28.
Given, the Obliquity of the Ecliptic 23 deg. 29 min. and
the Right Afcenfion of the Mid-Heaven, to find the
Culminating Point, or Medium Cœli in the Ecliptic.
Example. Let the Right Afcenfion of the Medium` Cali
be 308 deg. 26 min. 6 fec. what Point of the Ecliptic is then
upon the Meridian ?
VS
Draw the Primitive Circle,
which here repreſents the Sol-
ftitial Colure, by what has been
taught in the Projection of the
Sphere; draw the Equinoctial
a a, and the Ecliptic c;
then becauſe the given Right
Afcenfion of the Mid-Heaven
is 308 deg. 56 min. 6 ſec. that
is, 38 deg. 56 min. 6 ſec. from
the Solftitial Colure, take the
Secant of 38 deg. 56 min.
6 fec. and draw the Meridian
or Hour-Circle PbS, by which
ՈՐ
there is formed the Right Angled Spheric Triangle bc, in
which are given, 360°-308 deg. 56 min. 6. fec. b
51 deg. 3 min. 54 fec. and the Angle cb 23 deg. 29 min.
to find c, the diſtance in the Ecliptic from Y to the Meri-
dian.
ANALOG Y.
*
Deg. Min. Sec.
As t. Vb the R. A. M. Cali
51
3 54.--10.092639
To Radius
So C. f. Angle c Y b Obliquity
To Go r c
53 27 42-- 9.869814
90
00.--10.000000
23. 29
00--9.962453
Or,
The Doctrine of the Sphère.
163
As Radius
Or, by Tranfpofition,
Deg. Min. Sec.
90 00 00--10.000000
29 00- 9.962453
To C. f. Obliquity
23
To C. t. R. A. M. C.
51
To C. t. of its Dift. from
S. 53
27
354-9.907362
42-- 9.869819
This 53 deg. 27 min. 42 fec. is≈ 1
23
24
27
42
From Yor
12
00 00 OÒ
Rem. Culminat. Point,
io
6 32 18
Note, That the diftance found by Trigonometry is always
from the fame Equinoctial Point, or, that the Right,
Afcenfion of the Mid-Heaven was taken from.
PROB. 29.
Given, the Obliquity of the Ecliptic, and the Right
Afcenfion of the Mid-Heaven, to find the Meridian
Angle.
Example. In the Scheme of the laft Problem, there are
Given b, the Right Afcenfion Mid-Heaven 51 deg. 3 min.
54 fec. and the Angle cb 23 deg. 29 min. to find the An-
gle rcb.
As Radius
To f. Obliquity
ANALOG Ÿ
Th G. f. R. A. Med, Celi
To C, f. Meridian Angle
༡༠ 00 00--10.000ÒÒÒ
23
öö-- 9.600409
29 00-
51 3
54-- 9.798263
75 29 51- 9.391672
1
M 2
PROB
164
The Doctrine of the Sphere..
PROB. 30.
Given, the Obliquity of the Ecliptic 23 deg. 29 min. and
Right Afcenfion of the Mid-Heaven, to find the Decli-
nation of the Culiminating Point.
Example. Let the Right Afcenfion of the Mid-Heaven be
308 deg. 56 min. 6 fec. what's the Declination of the Culimi-
nating Point?
In the, Triangle Y bc of the laft Scheme, are given b
i deg. 2 min. 54 fec. Complement of 308 deg. 56 min. 6 ſec.
and the Angle cb, the Obliquity of the Ecliptic, to find c b
the Declination.
ANALOG Y.
Deg. Min. Sec. ...
As Radius
To t. Obliquity
To S. R. M. Heaven
90
00
00--10.000Q00
23 29
00-- 9.637956.
51 3
To t. Declination Cul. Point S. 18 40
54-- 9.89090I
22-- 9.528857
Note, When the Right Afcenfion of the Mid-Heaven is
more than 180 deg. as in the Example above, then the De-
clination of the Culminating Point is South; for then the
Culminating Point it felf is in a Southern Sign: But if the
Right Afcenfion be less than 180 deg. the Longitude of the
Culminating Point is in a Northern Sign, and Declination
North.
1
PROB.
The Doctrine of the Sphere.
165
1
PROB. 31.
Given, the Latitude of the Place, and the Declination
of the Culminating Point, to find the Altitude of the
Mid-Heaven.
Obferve, in North Latitudes,
If the Declination of the Culminating Point be North,
add it to the Complement of the Latitude, or Elevation of
the Equinoctial above the Horizon; the Sum is the Altitude.
of the Mid-Heaven.
But if the Declination be South, fubtract it from the
Complement of the Latitude, and the Remainder is the
Altitude of the Mid-Heaven.
In South Latitude, you muft fubtract the North Declina-
tion, and add the South to the Complement of the Latitude.
of the Place; the Sum or Difference, is the Altitude of
Mid-Heaven.
Example. At London,
The Elevation of the Equinoctial is
Declin. Culminating Point South is
Remains Altitude Mid-Heaven
Deg. Min. Sec.
38 28 Oo
18 40 22.
19 47 33
Example 2. At London, let the Declination of the Cul-
minating Point be 15 deg. 17 min. 46 fec. North; What's
the Altitude of the Mid-Heaven?
OPERATION.
Height of the Equinoctial
Decl. Culmin. Point North add
Altitude of the Mid-Heaven
1
Deg. Min. Sec.
38 28
1
15. 17 46
56 56 45
M
3
}
+
PROB.
166
The Doctrine of the Sphere.
1
•
PRO B. 32.
里​看
​Given, the Altitude of the Mid-Heaven, and the Me-
ridian angle, to find the Altitude of the Nonage-
fime Degree, or the Angle that the "Ecliptic makes
with the Horizon at any given Time.
Example. Anno 1728, March 13, at 22' paft 8 in the
Morning, I demand the Altitude of the Nonagefime Degree
at London ?
A
IT
B.
H
AQ
With any convenient Ra-
dius, with the Chord of
60 Degrees ſweep the Pri-
mitive Circle, which here
repreſents the Meridian of
London; draw H H for
the Horizon; then by the
foregoing Problem I find
the Altitude of the Mid-
Heaven 19 degr. 47 min.
38 fec. and the Meridian
Angle 75 deg. 29 min. 51
feconds: Take the Chord
of 19 degr. 47 min. 38 fe-
conds, and fet it from H
to A, which is the Altitude of the Mid-Heaven. Then be-
cauſe the Meridian Angle is 75 degr. 29 min. 51 feconds,
take the Secant thereof, and fweep the Ecliptic A B C; then
by having the Side AZ, the Complement of the Altitude
of the Mid-Heaven, yo deg. 12 min. 22 feconds, and the
Angle Z A N, the Meridian Angle, I find the Angle A Z
N to be 37 degr. 21 minutes; take the Secant of 37 degr.
21 min. and draw the Vertical Circle ZND, and it will
cut the Ecliptic at N in the Nonagefime Degree at right
Angles. Now in the Right Angled Spheric Triangle AZN
(right Angled at N) there are Given A Z, the Comple-
ment of the Altitude of the Mid-Heaven 70 deg. 12 min.
22 fec. and the Angle NA Z, the Meridian Angle 75 deg.
29 min. 51 feconds, to find NZ, the Complement of the
Altitude of the Nonagefime Degree.
ANA-
The Doctrine of the Sphere.
167
f
As Radius
ANALOGT
To S. Meridian Angle
So C. f. Altit. Mid-Heaven
To C. f. Alt. Nonagefime
Deg. Min. Sec.
90 00
00--10.000000
75 29 51-- 9.985937
19 47 48-- 9.973552
23 21 53-- 9,959489
PROB. 33.
Given, the Meridian Angle, and the Altitude of
Mid-Heaven, to find the Nonagefime De-
the
gree.
In the laft Scheme, there are the fame things given, to
find the Side AN, the Diftance of the Mid-Heaven from
the Place of the Nonagefime Degree.
ANALOG Y.
$
¦
Dcg. Min. Sec.
As Radius
To C. t. Alt. Mid-Heaven
So C. f. Meridian Angle
90 Oo 00--10.000000
47 38--10.443817
75 29 51-- 9.398651
19
Tot, Dift. Mid-Heaven from Nonag. 34 49 45- 9.842468
Now you are to obſerve, that if the Place of the Mid-
Heaven (as found by Prob. 28)
VS
r I II
add
be in vv
o al mm x tub.
the Diſtance of the Mid-
Heaven from the Nonagefime Degree to, or from the Place
of the Mid-Heaven, the Sum or Difference, is the Place
of the Nonagefime Degree.
1
M 4
Sa
168
The Doctrine of the Sphere.
1
So in the Example before us, the Place of the Mid-Hea-
ven is in, that is,
Dift. Mid-Heaven add
Place of the Nonagefime Degree,
S.
}}
10 6
32 18
Ι
4
49 45
II II 22 3
ļ
Theſe ſeven laft Problems, are of great ufe in the Cal-
culation of Solar Eclipfes.
PROB. 34.
Given, the Latitude of the Place, and the Time of
the Day or Night, to find the Cusp of the Afcen-
dant.
Example. Anno 1727, September 14, at 15' paſt 5 at
Night equal time, I would know the Degree and Minure
of the Ecliptic that is Afcending the Eaftern, Horizon at
London ?
{
*
Sun's place
OPERATION.
Deg. Min. Sec.
2 Į 51
Sun's Right Afcenfion
Time from Moon in Degrees and Minutes
Right Afcenfion Mid-Heaven
Add
Sum is Oblique Afc. Afcendant
181
52
78
45
260 37
90 OO 00
350 37
Complement next to Y
9:23
Now fay,
As Radius
To C. f. Oblique Afc. Afcendant
So C. t. Latitude of the Place
To C. t. of the Arch
Deg. Min.
90
0--10.000000
9 23-- 9.994150
51 32-- 9.900086
51
35-- 9.894236
Now
The Doctrine of the Sphere.
169
T
Obliquity Ecliptic add
Now Note, When the Oblique Afcenfion of the Afcen-
dant is nearer than, (as in this Example) then you muſt
add the Obliquity of the Ecliptic 23° 29′, to the firſt Arch,
the Sum is the Second; but if it be nearer than T, then
fubtract the Obliquity 239 29′ from the firft Arch, gives the
Second:
The first Arch is
The Second Arch
1
Deg. Min.
51 55
23 29
75 24
Now fay,
Deg. Min.
As C. f. Second Arch
75
24 Co. Ar. 0.598479
To C. f. firft
51 55
9.790149
So t. Oblique Afc. Afcend.
9 23
9.218142
To t. of its Dift. from S.
From
22
I
9.606776
12
OO 00
F
Cufp Afcendant
།
II 7 59
Further Note, that if the Second Angle be leſs than 90°,
the Diſtance in the Ecliptic found by the fecond Operation
above muſt be accounted from the fame Equinoctial Point
that the Oblique Afcenfion was reckoned from.
But if the fecond Angle be more than 90°, then the Di-
ftance in the Ecliptic muſt be accounted from the contrary
Equinoctial Point that the Oblique Afcenfion was reckoned
from.
If the Cufp of the Afcendant (or the Arch contained be-
tween the Meridian and Horizon) were required when ei
ther no Deg. of or is on the Meridian, then the young
Student will be at a lofs; becauſe the foregoing Analogy
will not hold; the Oblique Afcenfion of the Afcendant will
be 90 when no Degrees of Culminates, and 279° when
is there, that is, in both Cafes equally Diftant from T
and fo that the Obliquity of the Ecliptic cannot be ap-
ply'd as has been Directed. Then to remedy this Defect, there
muſt be a new Rect-Angled Spheric Triangle formed (and
when no Degrees of is on the Meridian) under the
;
Nor..
1
(70
The Doctrine of the Sphere.
A
A
Northern Horizon: Thus,
draw the Primitive Circle to
repreſent the Meridian of
London; and becauſe the
Meridian Angle is 66 degr.
31 min. when and
Culminates, take the Secant
of 66 degr. 31 min. and
draw Y A Æ for one half
of the Ecliptic, and
E,
a right Circle the Equino-
cial, to the Elevation of
London, HB the Horizon;
then in the right Angled Spheric Triangle EA B there are
given, Æ B the Complement of the Latitude 38 deg. 28 min.
and the Angle B E A, the Meridan Angle 66 degr. 31 min.
to find A Æ, the Arch of the Ecliptic between the Horizon
and North Meridian from Libra.
ANALOGY.
'As t. BÆ
Deg. Min.
38 28-- 9.900086
To Radius
90
00--10.000000
So C. f. Angle BE A
66
31-- 9.600409
To C. t. A E
63
22-- 9.700323
'A Semicircle
180
Oo
Remains A
116 38
That is the Arch of the Ecliptic from V on the South
Meridian to A above the Horizon; that is, 26° 38'
for the Cufp of the Afcendant in the Latitude of 519 32
North, when no Degrees of Y. Culminate.
1
Example 2. Let it be required to find the Cup of
the Afcendant at London when no Degrees of Libra Cul-
minatę.
+
This
The Dottrine of the Sphere.
This is Projected as the laft
was, by taking the Secant
of the Meridian Angle
66 deg. 31 min. and draw-
ing A for half the E-
Æ
cliptic, HH the Horizon,
and HHÆ the Meri-
dian; then in the Rect
Angled Spheric Triangle
AH, there are given
H, the Altitude of the
Medium Cæli, equal to the
Height of the Equinoctial
38° 281, and the Meri-
dian Angle HA 66°
31', to find
FO
H
A
A
A the Arch of the Ecliptic between the
Meridian and Horizon.
Tot. Latitude
As Radius
So C. f. Meridian
To C. t. of A
Analogy by Tranfpofition,
Deg. Min.
90 00--10.000000
SI 32--10.099913
66.31-- 9.600499
03 22-- 9.700322
which is equal to AA in the laft Scheme. Then 63° 22′
=2S. 3° 221 + 6 S. = 8S. 3° 22′ the Culp of the Afcen-
dant, when no Degrees of Culminate. Note, When no
Degrees of or vs Culminate, three of the five Parts in
the Triangle are Quadrants, and confequently the Arch of
the Ecliptic between the Meridian and Horizon is known to
be a Quadrant. And after the fame manner have I calcu-
lated this Table, fhewing the Culp of the Afcendant when
no Degrees of every Sign Culminate at London.
Mid-
172
The Doctrine of the Sphere.
Mid-Hea-
X # 8 > # # = ~ 9 ¤ ¤ < ||Mid.
→
ven.
i
Gufpof
the Afcen-
dant.
Arch E-
cliptic be-
tweenMe-
The fame
Arch in
Time.
lace
ridian and
0
26
Horizon.
h.
"
38
116 38
7 46 32
16
30
106 30
7
6
O
7 21
97
21
6
00 00
90
6
Oo
29
24
O
I
22
38
82
38
5
30 32
13
30
1
73
30.
5
54
O
3
22
63 22
4
13
28
25
15.
55 15
4
41
O
27
10
57 IO
3
48
48 40
00
00
90 Oo
6
2
50
122
50
8
II
20
;
4 45
124 45
8
I 19
1
'
PROB. 35.
Given, the Latitude of the Place, the Hour of the
Day, the Sun's Altitude and Distance from the Afcen-
dant, or Defcendant, to find the Parallactick An-
gle.
Example At London Anno 1733, May 2 d. 6 h. 35′ 39″
(by a former Inveſtigation of mine) is the apparent Time
of the vifible Conjunction of the Sun and Moon at which
Time I demand the Parallactick Angle?
A
*
1
OPE-
The Dottrine of the Sphere,
173.
OPERATION.
Sun's Place to the Equal Time
Sun's Declination North
Altitude of the Sun
Right Afcenfion
Apparent Time from Noon
Sum, R. A. M.Cœli
Add
Oblique Afc. Afcendant
Complement
Culp of the Afcendant
Cufp of the Defcendant
Sun's Place
Dift. O from Defcendant
Medium Celi in Ecliptic
Declination Culminat. Point North
Deg. Min. Sec.
52 57
୪
22
18
2 I
00
8
58
00
50
28
98
54
45
149
22
1
45
90
00
00
239
22
45
·59 22 45
11:37
00
II 37
00
22
52 57
ΙΙ
15 57
રી
27
ΙΟ
EXP
12 36 00
51 4
69 57
888888
Altitude Mid-Heaven
Meridian Angle
Amplitude North
30
42
Sun's Azimuth from the North
71
24
H
**
N
7
BKC
H
To Project this Problem';
with the Chord of 60 degr.
draw the Frimitive Circle
to repreſent the Meridian
of the Place, HH the Ho-
rizon, take the Altitude of
the Mid-Heaven 51 degr.
4 min. from the Line of
Chords, and fet it from
H to E, and the Amplitude
by the Semi-Tangents from
the Center to C; take the
Secant of the Meridian An-
gle 69 degr. 57 Min. and
draw E AC for the Eclip-
tic: Then becauſe the Sun's Azimuth at the given Time is
71 degr. 24 min. from the North, take the Secant of 71 deg.
24 min. and draw Z A N, which cuts the Ecliptic in A the
Place of the Sun: And now by theſe three great Circles, viz.
the Horizon, Ecliptic and Azimuth we have the Rect Angled
13.0
N
Sphe
}
•
The Dottrine of the Sphere.
#74
Spheric Triangle ABC, right Angled at B, in which are given
BA, the Sun's Altitude & deg. 58 minutes, and AC the Sun's
Diſtance from the Defcendant 11 Degr. 15 Min. 57 Se-
conds, to find the Angle B A C the Angle formed by the
Vertical Circle and Ecliptic.
ANALOGY.
Deg. Min.
'As Radius
90 00--10.000000
To t, B A O Altitude
8
58-- 9.198674
So C. t. A C,
à Defcendant
II
16-- 10.70678
To C. f. Angle B A C Parallactick 37
31-- 9,899352
PROB. 36.
Given, the Sun's Altitude and Distance from the No-
nagefime Degree, to find the Parallactick An
gle.
Example. Anno 1733, May 2 D. 6 h. 35' 39" at London,
I would know the Parallactick Angle?
OPERATI O N.
Cufp of the Defcendant
'Add
Nonagefime Degree
Sun's Place fub.
Diſtance
S.
Q
I II
3 00
37
00
4
II
37
I 22 $3
2
18 44
QEP
The Doctrine of the Sphere.
175
í
ANALOGY.
As Radius
To t. Sun's Altitude
So t. Sun's Dift. à Nonag.
To C.f. Parallactick Angle
90 00--10.000000
8 58--9-198674
78 44--10.700678
37 31- 9.899352
PROB. 37.
Given, the Sun's Parallax in Altitude, and the Paral
lattick Angle, to find his Parallax in Longitude and
Latitude.
Note, The Sun's Parallax in Altitude you will find in a
Table at the end of the Lunar Tables; in which the Paral-
lax in Altitude anſwers to the Sun's Altitude, the greateſt
Horizontal Parallax being 10".
First, For the Parallax in Longitude.
With the Sun's Altitude, take out of the Table the Parallax
in Altitude 10", and then fay,
As Radins
To t. Sun's Parallax in Altitude;
So C.. of Parallactick Angle,
Text of Parallax in Longitude.
'As Radius,
2. For the Parallax in Latitude.
To S. Parallax in Altitude;
So S. Parallactick Angle,
To S. Parallax in Latitude
..
To
176
The Doctrine of the Sphere...
The three laft Problems have refpect to the Ecliptic
only; but in regard the Moon and other Planets and Stars
are very feldom found there, therefore we must have re-
ſpect to the Orb of the Planet, and find the Angle that a Ver-
tical Circle forms with it.
PROB. 38.
Of the Parallax of the Sun, Moon, and Stars.
In the Definitions, I have told you what is meant by the
Word Parallax in general: I fhall in this Problem demon-
ftrate the Parallaxes in Altitude, Longitude, Latitude and
Horizontal: The ufe of them is fo very great that the
Knowledge thereof is the very Foundation of Aftronomy.
Becauſe from thence, the Diſtance of the Sun, Moon,
and Stars from the Earth may moft eafily be had; for in
the Triangle AB t, A B the Earth's Semidiameter, B the
Right Angle, and t the Angle of the Parallax being known,
'tis eafie to find any Side or Angle fought, and confequently
A t, the Diſtance of the Moon from the Earth's Center.
In the adjacent Figure,
let A repreſent the Earth's
Center ef the true Hori.
zon, GH the vifible, L
Bm, half the Earth's Su-
perficies, n co the Moon's
Orb; an Obferver at B,
views the Moon at i; but
an Eye from the Earth's
Center at A would fee her
at K: So likewife if we
put the Semi-Circle PD q
to repreſent the Orb of
e
Vá
PHIAGO 9
A
Mars (or any other Planet) an Obſerver ſtanding at B on.
the Earth's Superficies, will behold Mars at R; but from
the Earth's Center it would be feen at S, and this Parallax
vanishes in the Vertex or Zenith of your Habitation: For
viewing a Star at V, both from the Earth's Center at A, and
alfo from the Superficies at B, it will appear in one and the
fame
$
The Doctrine of the Sphere.
177
fame place of the Heavens; and the nearer the Horizon
the Stars are, the greater is the Parallax of Altitude, and con-
fequently the Horizontal is the greateſt of all. Therefore,
becauſe the Places of all the Heavenly Bodies are fupputa-
ted to the Earth's Center, (to which place an Obferver can-
not come) and we being upon its Superficies, fhews how
needful the Knowledge of theſe Parallaxes are to him
that would be an Aftronomer: For as the Stars are raiſed
by Refraction's, fo they are depreffed by Parallaxes; and
they depreſs the fame way that the Planets appear; that
is, if they appear to the Southward of the Zenith, the
Parallax depreffes them to the Southward; and fo dimi-
nifhes their Latitude if it be North; but increaſes it if
it be South And on the contrary, if the Planets appear
to the North of the Zenith, the Parallax increaſes their
North Latitude, and diminishes the South Latitude; all
which will be very plain and and eafie, if you feriouſly
confider the Diagram before you.
And further, is to be confidered the Remotenefs or Near-
neſs of a Planet to the Earth: For Saturn being far-
theft from it, has the leaft Parallax of all; and the
Moon being the neareft to it, has the greateft; and fo
of the others, according to their Diftance from the Earth;
whofe Horizontal Parallaxes at a middle Diſtance from
the Earth I have ftated thus:
"
Ђ
0
I
34
2
2
49
8
14
10
0
2
୨
→
13 59
8 57 50 00
When a Circle of Longitude paffing thro' the Poles of
the Ecliptic, paffes thro' the Nonagefime Degree, it then
cuts the Ecliptic at right Angles; but in all other places
it cuts it at Oblique Angles. Therefore, if a Planer appear
neither in the Ecliptic, nor in the Nonagefime Degree, the
Parallax in Altitude will caufe a Parallax both in Longitude
and Latitude.
N
Example.
178
The Doctrine of the Sphere.
Example. Anno 1727, September 15, at 8 at Night, at
London, I would know the Parallax of the Moon in Alti-
tude, Longitude, and Latitude?
First, For the Angle that the Vertical Circle forms with
the Moon's Orb, proceed thus, viz. either find the Moon's
Distance from the Afcendant, or elſe from the Nonage-
fime Degree; which you pleaſe will do.
Moon's true-
true<
Longitude
Latitude South
Declination South
Given Hour
Q
24 56
I 14.
14. 24
8
Moon South at
Moon from Meridian
Altitude
Sun's Place
Right Afcenfion
9
44
I
44
21
15
2
57 34
182 43
Time from Noon Degr
Right A. M. Cœli
Add
Oblique A. Afcendant
Complement
Cufp Afcendant I
Moon's Place fub.
Moon from the Afcendants
Complement
S.
Angle formed by Vertical Circle?
3
·M 2
120
302 45
90
00
392 43
32 43
3 29
24 56
8 33
21 27
86 39
The Requifites above I have hd by the foregoing
Problems; and now for the given me 8 at Night, find the
Moon's Mean Anomaly S. 11° 14′ 20", and with that
take out of the Table her Horizontal Parallax 55 min. 12
feconds; and then for her Parallel in Altitude, the Analogy
is
As
}
The Doctrine of the Sphere.
179
As Radius,
To C. f. Moon's Altitude
So S. Horizontal. Parallax,
To S. Parallax in Altitude,
;
Deg. Min. Sec.
90 00. 00--10.00ôõõ¯
00 - 9.969321
21 17
00 55
12-- 8.205639
51
22-- 8.174956
21 17
True Altit. of the Moon
Vifible Altitude
'As Radius,
00
20 25 38
2. For the Parallax in Longitude.
ANALOGY.
To t. Moon's Parallax in Altitude;
So C. f. Angle of her Orb and
Vertical Circle,
To t. Parallax Longitude
Deg. Min, Sec.
00
1
90 00 00--10.000000
51 22-- 8.174421
86 3900-- 8.766675
} 86
00 3 00-- 6.941096
Note, If the Moon, &c. be between the Afcendant and
the Nonagefime Degree, the Parallax of Longitude muſt be
added to her true Longitude; but if fhe be between the No-
nagefime Degree and Defcendant, the Parallax of Longitude
must be fubtracted from the Moon's true Place in Longi-
tude ; the Sum or Difference is the Moon's Visible Lon-
gitude.
EXAMPLE.
6.
Afcendant
Sub.
Nonagefime Degrees
Moon's true place
Parallax Longitude ſub.
Moon's Visible Longitude
23
10
3 20
0 00
3 29
24 56 to the Weft.
00 3 00
1Q
21 56
N 2
Note,
i
180
The Doctrine of the Sphere.
}
Note, If the Moon's Longitude be lefs than the Place of
the Nonagefime Degree, fhe is then in the Occidental Quad-
rant; but if more, in the Oriental.
3. For the Moon's Parallax in Latitude.
As Radius,
ANALOGY.
To S. Parallax in Altitude';
Deg. Min. Soc.
90 00
00 51
So S. Ang, of her Orb and Vert. Cir. 86 39
To S. Parallax in Latitude,
True Latitude Moon South, add
Vifible Latitude Moon South
00
51
I 14
2
00--10.000000
22- 8.174956
22--
00 9.999257
16-- 8.174213
00
5 16
DEMONSTRATION.
B/K
G
H
DL
A
0
Let HO, be the visible
Horizon, whoſe Pole is V,
E A an Arch of the Moon's
Orb, whofe Pole is P, VL,
a Vertical Circle, paffing
thro' the Moon's true Place,
in K, and the apparent
Place in G; fo will K G
be the Parallax in Altitude:
Thro' K and G, draw two
Circles of Longitude as PK,
and PG, cutting the Moon's
Orb in K and C; then will
KC be the Parallax in
Longitude, and GC in La-
titude; V BD is a Vertical Circle paffing thro' the Nonage-
fime Degree, BK the Moon's Diſtance from the Nona-
gefime.
When the Sun, Moon or Planets are in the Nonagefime
Degree, then there is no Parallax in Longitude, and then the
Parallax in Latitude and Altitude are equal. And if they be
in the Vertex, there is no Parallax of Latitude nor Altitude,
but only in Longitude: If they be neither in the Vertex nor
Nonagefime Degree, they have Parallax in Longitude, Lati-
tude and Altitude, as has been above Calculated and Demon-
ftrated.
PROB.
*
The Doctrine of the Sphere.
181
PROB. 39.
Given, the Horizontal Parallax of a Planet, the
Altitude of the Nonagefime Degree, and its Di-
Stance from the Nonagefime Degree, to find the Pa-
rallax in Longitude and Latitude.
Rule. To the Logiſtical Logarithm of the Horizontal
Parallaxes of the Planet, add the Sine of the Altitude of the
Nonagefime Degree, and the Sine of the Diſtance of the
Planet from the Nonagefime Degree; the Sum of theſe
three Logarithms is the Logiſtical Logarithm of the Pa-
rallax in Longitude.
Example. Anno 1727, September 15. at 8 a-Clock at
Night at London, I would know the Parallax of the Moon
in Longitude and Latitude?
You must first find theſe requifites by the foregoing Prob-
blem, and fet them down in order thus:
D. H.
Given Time
1727 Sept. 15
8 00 00
Moon's Place
24
56 00
Sun's Place
2
Sun's Right Afcenfion
Time from Noon
Sum, Right Afcenfion M. Cali
Complement
Medium Cali in Ecliptic
Meridian Angle
Declination Gul. Point South
57 34
182 43 00
120 00 00
302 43 00
57 17 00
00
30 00
00
38
28
OÛ
77 34.00
20
Altitude Equator at London
Altitude Mid-Heaven
Altitude Nonagefime Degree
18 23 00
22 4
Dift. Mid-Heav. from Nonag. Degr. 32
sa 00
Nonagefime Degree
*
3
29 00
Moon's Place fub.
24 56 ao
Dift. Moon from Nonag. Degree
8 33
00
Mean Anomaly Moon
7 II
14 20
Horizontal Parallax Moon
00
55 12
N 3
Now,
182
The Doctrine of the Sphere.
Now, for the Parallax in Longitude of the Moon, the
Work ftands thus:
Horizontal Parallax Moon
Altitude of the Nonag. Degree
Dift. Moon from Nonag. Degree
Parallax in Longitude of Moon
Deg. Min. Sec.
00 · 55
12 LL 9.96379
22
4
00 S. 9.57482
8 33
00 S. 9.17223
3
5 LL 8.71085
00
2. For the Parallax in Latitude.
To the Logiſtical Logarithm of the Horizontal Parallax of
the Planet, add the Co. Sine of the Altitude of the No-
nagefime Degree; the Sum of theſe two Logarithms is
the Logiſtical Logarithm of the Parallax in Latitude.
OPERATION.
Deg. Min. Sec.
Horizontal Parallax D
OO 55
12 L L 9.96379
1
Altitude Nonagefime Degree
22 4
00 CS 9.96696
Parallax in Latitude D
00
51
9 LL 9.93075
And thus, have I given my Reader two feveral ways of
finding the Parallaxes; and fhall leave to his Choice to take
which he likes beft.
The greateſt Parallax of the Longitude may be found by ad-
ding the Logiſtical Logarithm of the Planets greateſt Hori-
zontal Parallax, to the Sine of the Angle of its Orb with
the Horizon (which is the fame with the Altitude of the No-
nagefime Degree in the Planet's Orb,) and the greateſt Di-
tance of the Nonagefime Degree, which is 90°; the Sum
of theſe three Logarithms is the Logiſtical Logarithm of the
greateſt Parallax of Longitude of the Planet that can happen.
Example at London in the D, fuppofing her North Node
to be in no Degrees of Aries, and the upon the Meridian in
no Deg. ; the Angle formed with the Horizon and Orb,
will be 67° 14' 20",
ОРЕ
The Doctrine of the Sphere.
183
"
OPERATION.
Altitude Nonagefime in her Orb
Deg. Min. Sec.
Greateſt Horizontal Parallax D
67
00 61
If
24 L L 10.01001
20 S. 9.26479
56
37 LL 9.97480.
Her greateſt Dift. from Non. Deg. 90 00 00 S. 10.00000
Her greateſt Parallax in Longitude
Alfo at London her greateſt Parallax in Latitude, may be
found by the following Operation.
Greateſt Horizontal Parallax
Leaſt Altit. Nonag. in her Orb
Greateſt Parallax of Latitude
Deg. Min. Sec.
0 61
61
24 LL 10.01001
9 41
40 CS 9.99375
I 00
31 LL 10.00376
And leaft Parallax in Latitude may be found at London
by the following Operation.
Deg. Min. Sec.
Leaft Horizontal Parallax
00
54
59 LL 9.96208
00 21
20 CS 9.58759
16 L L 9.54967
Greatest Alt. Nonag. in her Orb 67 14
Leaft Parallax in Latitude
And thus by obferving the Premifes, you may find the grea-
teſt Parallax in Longitude, the greateſt and leaft Parallax in
Latitude of a Planet, in any Latitude; which may be of good
uſe to give you a right Idea of the Parallaxes; and will allo
fhew you how they increaſe and decreaſe, and confirm your
Calculation, by knowing between what two Numbers your
Parallaxes at fuch a time and Place muft fall.
1
PROB
N 4
∙184
The Doctrine of the Sphere.
P. R. O B. 40.
Shewing the feveral Methods made use of by Aftronomers for obtain
ing the Horizontal Parallaxes of the Heavenly Bodies.
B
A O
M
EHI
D
K
ET
C
I
The firft that I fhall fhew,
is that famous Diagram of
Hipparchus, and made ufe
of by all Aftronomers to
this Day; exemplified in
finding the Sun's Hori-
zontal Parallax.
Let A be the Center of
the Sun,
M that of the New Moon.
E the Center of the Earth.
F the Center of the full
Moon in Perigeon.
L the Center of the full
Moon in Apogeon.
Let all thefe Centers fall
in the right Line A,
譬
​M, E, F, L, D.
A B the Sun's true Se-
midiameter.
The Angle A E C, is the
Apparent Semidiame-
ter Sun.
The Angle MEG is the
Apparent Semidiame-
ter Moon.
The Angle FEK, the
Apparent Semidiame
ter of the Earth's Sha-
dow when the Moon
is in Perigeon.
The Angle LEN, the
Apparent Semidiameter
of the Earth's Shadow
when the Moon is
in Apogeon.
And the Angle EA I, is
the Sun's Horizontal
Parallax.
The
1
|
The Doctrine of the Sphere.
185
亨
​1:
1
The Angle E M I, is the Moon's Horizontal Parallax.
The Angle EFI, is the Moon's Horizontál Parallax when
fhe is in Shadow in the Ferigeon
And the Angle EL I, the Moon's Horizontal Parallax when
in the Earth's fhadow in Apogeon.
The Angle EDI is half the Angle of the Cone of the
Earth's Shadow, equal to the Apparent Semidiameter of
the Sun view'd from the top of the Shadow.
A E is the Diſtance of the Sun from the Earth; and ME
the Diſtance of the New Moon: EF the Diftance of the
Full Moon in Perigeon, and EL the Diſtance of the A-
pogeon, full Moon: ED the Axis of the Earth's Shadow.
Draw OI parallel to A M, and HK to E F
Having thus prepared the Work, the Sun's Horizontal Pa-
rallax will be difcovered thus, A E CADCEA I the
AEC
Sun's Horizontal Parallax.
1. The Semidiameter of the Sun, lefs by his Horizontal
Parallax, is equal to the Semi-Angle of the Cone of the
Earth's Shadow: A E CEAI or AIO EDI: For
becauſe A E and OI are parallel, the Angle A IO=
Angle EAI by 29th of the firft of Euclid.
2. The Horizontal Parallaxes of the Moon, lefs by the
Semi-Angle of the Cone of the Earth's Shadow, is equal to
the Apparent Semidiameter of the Shadow.
EMI EDI FEK.
3. The Sum of the Horizontal Parallaxes of the Lumina-
ries is equal to the Sum of the Apparent Semidiameters of
the Sun and Shadow of the Earth.
Angle EAI EMI AEC+FEK.
=
Therefore,
if from the Sum of the Horizontal Parallaxes of the Sun
and Moon, you fubtract the Sun's Semidiameter, there will
remain the Apparent Semidiameter of the Earth's Shadow in
the lace where the Moon paffes through. E AI+EMI,
AECFEK
Thus far the Method of Hipparchus for finding the Sun's
Horizontal Parallax, which is now determin'd to be no
more than 10 Seconds: Therefore this Angle being.fo ve-
ry fmall, it is a very difficult Point to come at the true
Diftance of the Sun from the Earth, (I may fay, almoft
impoffible.) For an Eye at the Sun would behold the Earth's
Semidiameter under that Angle; confequently the Diſtance
of that glorious Body from us must be exceeding great.
We
186
The Doctrine of the Sphere.
We can easily by Trigonometry find the Length of the
Earth's Shadow E D; for in the Right-Angled plain Tri-
angle EDI, there are given, E I, the Earth's Semi-
diameter 3984.58 English Miles, and the Angle EDI=
to the Sun's apparent Semidiameter ſeen from the Ver-
tex of the Cone, which at a middle diftance of the Sun
from the Earth, is 16 min. 5 feconds, to find E D, the
Length of the Earth's Shadow.
As t. Angle EDI,
To E I,
Semidiamer;
So Radius,
To E D, Miles.
Therefore I fay,
Deg. Min. Sec.
Oo 16 5
7.650043
3983.58
3.600382
90
00 00
10.000000
851802
5.930339
Divide the Length of the fhadow 851802, by the Earth's
Semidiameter, 3984.58, and the Quotient 254 ferè is the
Length of the Shadow in Earth's Semidiameters. Al-
fo, if from ED you fubtract E L, the Moon's Apogeon-
Diſtance, there will remain LD, the Length of the Sha-
dow beyond the Moon: And fince the Diameter of the
Earth 7969.16, is to the Diameter of the Moon 2151, as
100 to 27; therefore the Altitude of the Earth's Shadow,
will be to the Altitude of the Moon's in the fame proportion;
becauſe the Conical Shadows are fimilar Figures; and there-
fore the Height of the Moon's Shadow will be 229986.54
Miles.
For, as 100: 27:: 851802: 229986.54;
Which divided by 3984.58, the Earth's Semidiameter,
the Quotient will be 57 388 Semidiameters of the
Earth.
Secondly, The fecond way of finding the Horizontal Pa-
rallax, is by obferving the exact Time that the Moon is
in the Quadratures, which fhe is twice every Month: And
by obferving this Moment, of time when he is Biffected,
that in the very fame Moment in which the Plane of
that Circle of Illuminaiion is found in the Eye of the
Spectator, or in the Center of the Earth, the Center of
the Sun is in that right Line, which is perpendicular to
the fame Plane, and paffes thro' the Center of the Moon.
And thus you have a right-Angled Plane Triangle formed
by a Line ſuppoſed to be drawn from the Earth's Center to
the Sun, from the Sun to the Moon, and by the Line of
t
Illumi-
The Doctrine of the Sphere.
187
Illumination or Biffection of the Moon, to the Earth;
and the Angle at the Sun, is that which Aftronomers
call, for Diftinction, the Menftrual Parallax, or, the Dif
ference of Pofition that there is in the Sun, as feen from
the Earth, and as feen from the Moon. The manner of
obferving the exact Moment of time when the Moon is
Dechotomized, must be done with a very large Teleſcope,
that the whole Difcus of the Moon may be taken in,
and her Spots repreſented to the Eye diftinctly at one
View. This being gain'd, they fearch out by actual Ob-
ſervation, or by Aftronomical Tables, the true Places of the
Sun and Moon for that Moment, and their Difference of Pla-
ces will be the Angle in the former Triangle formed
at the Earth: And thus in the Triangle all the Angles are
known, with the Diſtance of the Moon from the Earth, and
confeqnently the Diſtance of the Sun from the Earth is
eafily gained.
But notwithſtanding that great Subtilty of Wit and
Reaſon in both thefe Methods, yet many Defects there
are in them, which forbid us to expect an acurate In-
veſtigation of this Parallax by means of either of them.
For,
As to that of the Diagram of Hipparchus, there are a
great many things neceffary to be prefuppofed; which
are each of them fo difficult to be obferved, that we can
never come to that Exactneſs as the Cafe requires. As 1.
The Sun's apparent Magnitude. 2. The Horizontal Paral-
lax of the Moon; and 3. The Semidiameter of the Sha-
dow in the Place of the Moon's Tranfit. See Mr. Whifton's
Lecture, Page 70.
Thirdly, The third Method to find the Sun's Horizon-
tal´ Parallax, is, by Inveſtigation of the Parallax of Mars,
Venus, or Mercury; by Mars, when in oppofition to the
Sun; and by the other two, when in Conjunction Retro-
grade, and feen in the Sun's Disk: For in theſe Pofi-
tions they are nearer the Earth, than at other times
therefore moft proper for this purpoſe. See the first of
thefe handled by Mr. Whifton in his Aftronomical Lecture 7.
and that of Venus in the Sun, by Dr. Halley, Phil. Tranſ.
No. 348; and alfo at the end of his Obfervations, and
Catalogue of the Southern Stars, he gives feveral ways to
find the Parallaxes of the Sun and Moon.
Mr. Auque
188
The Doctrine of the Sphere.
Mr. Auzut alſo gives a Method to find the Moon's Parallax,
on a Day when he is in her Perigee or Apogee, and in
the moſt Northern Signs. Thus, by taking her Diameter
near the Horizon, and alfo in her greatest Altitudes, the
Difference of them will fhew the proportion of her Di-
ftance with the Semidiameter of the Earth; but this way
cannot be practis'd in England, becauſe the Moon is never
in our Zenith.
Dr. Gregory in Vol. 1. of his Elements of Aftronomy, gives
in the feventh Section, no less than 19 Propofitions for
this purpoſe.
PROB. 41.
How to make Cœleftial Obfervations.
To obſerve the true Places of the Heavenly Bodies, as of the
Sun, Moon and Stars, is a Work of the greateft importance in
Aftronomy; becauſe it requires large Inftruments, a good
Obfervatory, where there is a perfect clear Horizon, a thro-
rough Knowledge in Geometry, and withal, due Care in
making your Obfervations.
At the beginning of this Section, I fhew'd how the´Obli-
quity of the Ecliptic was obtained; and here I ſhall inform
how you may take the Sun's Altitude tho' you be not provi-
ded with an Aftronomical Quadrant: The Method is this;
Take your walking-Cane, or Stick of any convenient length,
and divide it into any Number of equal Parts, 10, 100,
1000, &c. Let it be ftreight, and fet it perpendicular to
the Horizon, when the Sun fhines on a plain level Place.
A
BC
Then ſuppoſe the Stick be divi-
ded into 100 equal Parts, and I
find the length of the Shadow
be to contain 52.6 Parts ; then
in the Right-Angled plain Tri-
angle ABC, Right-Angled at B,
there are given, AB, the Height
of the Stick 100 parts, and BC,
52.6, to find the Angle A CB the
the Length of the Shadow
apparent Altitude of the Sun.
!
ANA-
The Doctrine of the Sphere:
189
ANALOGY, AB made Radius.
As AB the Staff height,
To Radius ;
So BC the Shadow,
To C.t. Angle A C B;
Refraction fub.
Remains
Sun's Semidiameter
Remain's
Parallax add
True Alritude of Sun
100
2.000000
90 00
10.000000
52.2
1.721395
62 14 9.721395
0027
00
62 13 33
00
16
61
57 33
00 00 4
61 57 27
Hence, becauſe the Altitude was taken by the Shadow of
the Staff, and not by the Croſs-Hairs in a Teleſcope, there-
fore I fubtract 16 for the Sun's Semidiameter, becauſe the
Rays come from the upper Edge of the Sun, and not from
the Center: But when you obſerve by Teleſcope-Sights, with
two Croſs-Hairs, then you need not uſe any fuch Dedu-
ction of the Sun's Semidiameter; becauſe then you take the
Sun's Center at once.
And Secondly, becauſe it was the Sun's apparent Altitude
that was obſerved; therefore the Refraction is fubtracted,
and the Parallax added; for they are always of contrary
Effects. And when it is a true Altitude found by Calcula-
tion; then to that true Altitude you muft add, the Refra-
ction, and deduct the Parallax, and by that means you will
gain the apparent Altitude.
"
1
Secondly, To Obferve the true Place of the
Sun, &c.
Firft, In a known Latitude, fix a large Aftronomi-
cal Quadrant of 6, 8, or 10 Foot Radius (the larger
the better) truly upon the Meridian, and let its Limb be
truly divided into Degrees, Minutes, and Seconds, or any
other Divifions, as you fhall think fit; let there be a Te-
leſcope with two Crofs-Hairs on the Object-Glafs to take the
Center of the Sun, Moon, or Star when they come upon the
Meridian. Then, if it is the Sun, find its true Altitude,
as
190
The Doctrine of the Sphere.
as above has been fhewn, by correcting the Apparent by
Refraction and Parallax; which true Altitude, if it be lefs
than the Elevation of the Equinoctial in the Place of Obfer-
vation, then fubtract the true Altitude found from the
Height of the Equinoctial, and the Remainder will be the
true Declination South, of the Sun, Moon, or Star, ob-
ſerved. But if the true Altitude exceed the Complement of
your Latitude, then fubtract the Complement of your Lati-
tude from the true Altitude, and you will gain the
true Declination of the Sun, Moon, or Star, obferved
North.
Then by Prob. 2. you may find the Place of the Sun,
by having given the prefent Declinations, and the Ob-
liquity of the Ecliptic, as in that Problem I have given an
Example. And as now we have a perfect Catalogue of Fix-
ed Stars, there is no Method more certain for determining
the Places of the Planets, than by obſerving their near Ap-
pulfes to the Fixed Stars. See Phil. Tranf. Nº. 369.
And by obferving their Diſtances from the Fixed Stars,
we curiouſly gain their Places in Longitude and Latitude, as
I fhall fhew in the next Problem.
!
PRO B. 42.
The Longitudes and Latitudes of the two known fix-
ed Stars, with their Distance from a Planet, &c.
to find the Longitude and Latitude of a Planet, Co-
met, or new Star.
Example. Admit the Moon be obferved diftant from
the Fixed Star called Mirach, or the bright Star in the
Girdle of Andromeda 27 degr. 13 minutes, the Longitude
of Mirach then being Y 26 degr. 33 min. 34 feconds;
and Lat. 25° 56′ 19 N. and at the fame time fhe be obfer-
ved diftant from Aldebaran 41º 13', the Longitude of Alde-
baran was II 5° 57' 50" with Latitude 5° 29' 50" South. I
demand the Longitude and Latitude of the Moon at the
Time of the Obfervation?
Q
1
}
¦ Prom
1
[
The Doctrine of the Sphered
19F
Projection,
M
A
E
1
With the Chord of 60°
fweep ZH NO, to repreſent
the Solftitial Colure, HO
the Horizon, VS the Eclip-
tic. Then becauſe Mirach
is 63° 58' 56' from the faid
Colure, take the Secant of
63° 58' 56", and draw EA F; H
lay off the Latitude 25 56 19
North from the Ecliptic to
A; fo fhall A repreſent this
Star in the Projection: Alfo
becauſe Aldebaran is diftant
24° 34' 40" from the fame
Colure, take the Secant
thereof, and draw EBF;
lay off the Latitude 5º 29' 50' from the Ecliptic South at
B; fo fhall B reprefent Aldebaran in the Projection; from A
and B draw two Circles at their Diſtance from the Moon,
obſerved feverally, and they will interfect at ; then draw
EDF, and compleat the Triangle A B ; ſo ſhall A be
the Place of Mirach, B of Aldebaran obferved,
of Moon required.
N
rs
The Trigonometrical Calculation.
the Place
In the Oblique-Angled-Spherical-Triangle, A BE, are
given AE 64° 3′ 41" the Complement of the Latitude of
Mirach, BE 95° 29' 50" the Diſtance of Aldebaran; from the
North Pole of the Ecliptic, and the Angle BE A 39° 24′ 16″
the Difference of Longitude of the two Stars, to find
A B, the Diſtance of the two known Stars. By the 10th
Cafe of Oblique Angled Spherical Triangles, by firſt
letting fall a Perpendicular from A, upon E B.
}
>
OPE-
192
The Doctrine of the Sphere.
4
OPERA T‍I O N.
As C. t. A E,
To Radius ;
So C.. Angle BE A;
To t. of the 4th Arch
From E B
Remains 5th Arch
Dég. Min.
64
90
4-- 9.686898
qp-- 10,000000
39 24-- 9.888030
57
40--10.201132
95. 30
37 50
Or, by Tranfpofition, Say,
{
Deg. Min.
As Radius,
To t. A E;
90
00--10.000000
64
4--10:313102
So C. f. Angle BE A;
To t. 4th Arch
.39 24-- 9.888030
57
40--10.201 132
1
Now Say,
Deg. Min.
As C. f. 4th Arch,
57 40 Co. Ar. 0.271773
To C. f. 5;
So C. f. A F
37 50
9.897516
64 4
9.640804
To C. f. A B.'
49 47
-2.810093
1
}
1
1
Secondly, In the Triangle A BE, are given, all the Sides,
Deg. Min. Sec.
viz.
BE 95 29 50
1
41 to find the Angle A BE
A E 64 3 41
47 00
A B 49
F
}
BY
The Doctrine of the Sphere.
193
•
Deg. Min. Sec.
29
47
By the 11th Cafe of Oblique Spherical Triangles the Work
ſtands thus:
Sides includ. the SB E 95
required Angle B A 49
Deg. Min. Sec.
50 AE 64 3 41
I
00
32
32 I
50
X 45
42
50
22 22
51 25
AE
A E
32
I 50
Z 54 53 15
X 9 ΙΟ
25
S.BE 950 29' 50" Com. 84
30
10
Co. Ar.
0.001002
S. A B
S. Z
S. X
49
47
00
Co. Ar.
-0.117129
54 40
15
9.911606
9 10 25
9.202560
Sum of the Logarithms,
Half is the Sine of 24 26 3
Doubled, is
19.233297
9.616648
4852 6 the Angle A BE
Or, the fame Angle may be formed thus:
OPERATION.
Deg. Min. Sec.
BE 95 29 50 Complement 84° 20′ 10″
A.E. 64 3 41
BA 49 47
Oo
Z209 20 31
1 104 40 15
15
3 41
X 40 36 34
>
Complement 75° 19' 45"
Side oppofite to the required Angle.
AE Sub 64
S. AE
84 30
10 Co. Ar.
0.002002.
S. BA 49 47
oo Co. Ar.
0.117129
9.985613
S. Z 75 19 45
S. X
40 36 34
Z Logarithms
Half is C. f. of 24 26 3
Doubled is 48 52 6 the
9.813578
19.918322
9.959161
Angle
Angle A B E as before.
Third-
194
The Doctrine of the Sphere.
1
Thirdly, In the Triangle A B>, are given all theſe Sides,
Deg. Min. Sec.
25 to find the Angle A B .
A B
49 47 00
viz.B
41 13
A
27 13
118
13 28
こ
​Z
half 59 6 44
A fub.
27 13 3 Side oppoſite to required Angle.
X 31 53
Deg. Min. Sec.
41
S. A. B
49 47
∞o Co. Ar: 0.117129
S. B
41
13
25 Co. Ar. 0.181115
T
S. half Z 59
6 44
S.
X 34
53 41
9.933576
9.722930
19.954750
C. f. of
Z Logarithms
Doubled 36 40
LABE ad 48
Z is
}
18 20
03
9.977375
6 the Angle AB
52
6
32
12
EB {85
Fourthly, In the Triangle EB are known; BE 95
29′ 50″, B 41° 13′ 25″, and the Angle E B 85° 32′
12", to find E, the Moon's diftance from the North
Pole of the Ecliptic, and the Angle BE >, the Moon's
Longitude.
FIRST,
The Doctrine of the Sphere.
195
FIRST, For the Side E>; by fuppofing à Perpendi-
cular let fall from > upon the Side B E.
ANALOG T
As C. t. B
To Radius
So C. f. Angle BE
To t. of 4th Arch
Deg. Min. Sec.
41 13
25--10.057416
90
Oo
00--10.000000
85 32
12-- 8.891137
3 54
4-- 8,833721
Or, by Tranfpofition,
Deg. Min. Min.
90 Oo 00--10.000000
As Radius,
To t. B;
41
13
་
32
.25-- 9.942584
12-- 8.891137
{
54
4-- 8.833721
95 29
50
So C.f.Ang. D BE, 85
Tot.of the 4thAr. 3
From BE
Rem. 5th Arch 91 35 46 Complement 88° 24′ 14″:
Now fay,
Deg. Min. Sec.
As C.f.of 4th Arch,
3 54
4 Co. Ar. 0.001007
To C. f. of 5.
88
24 14
8.444880
So C. f. B),
41
13
13 25
9.076301
To C. f. E > Comp. 88
47 49
From
90
Rem. Lat. South
I
Oo 00.
12 II
+
8.322188
Or, if you fay, to the Sine
of the Latitude, it will fave
the trouble of fub. from 90,
2
A
Laftly,
196
The Doctrine of the Sphere.
Lastly, For the Angle BE
Say, By the firft Cafe of Spherical Triangles.
As S. DE,
Deg. Min. Sec.
To S. Angle » BE;85
So S. BD,
88 47
49 Co. Ar. 0.000095
32 I2
9.999205
41 13 25
8 34
9:818885
9.818185
To S. Angle BED 41
Hence, becauſe the Moon was in Antecedence of Aldebe-
ran at B, therefore fubtract the Angle BE D from the Place
of Aldebaran 15° 57' 50", and the Remainder will be the
true Place of the Moon in Longitude.
Longitude of Aldebaran
Angle BE > fubtract
t
Rem. the Longit. of the Moon
Note, If the Moon had
baran, then the Angle B E
own Reafon will direct..
S.
=1
Q
2 5
57 50
I II 8 34
24 49 16
been in Confequence of Alde-
muſt have been added, as your
Example 2. Anno 1680, April 14th, at 8 h. 15' P. M, Mer-
cury was obſerved diftant from Procyon 55° 47′30″, Procyon
at that time was in 21° 22′ 1″ having 15° 57′ 55″ South
Latitude; and at the fame time Mercury was found by In-
ftrument to be 22° 11' 55" diftant from the North Horn of
Taurus, or Southern Foot of Auriga; this Star being then
in I 18° 5′ 36″ with 5° 21′ 34″ North Latitude, I demand
the Longitude and Latitude of Mercury at the time of the
Obfervation? Mr. Hogdfon's Syftem. Math. Page 452.
PRO-
1
1
197
The Doctrine of the Sphere.
PROJECTION.
;
Becauſe the two fixed
Stars lye on each Side of
the Solftitial Colure, there-
fore I project it on the Plane
of the Equinoctial Colure
ԴՐ is the Ecliptic, E and
Fits Poles: Then becauſe
Procyon is 68 degr. 37 min.
59 feconds from, take
the Secant thereof, and
draw E A F, on which
fet off the Complement
of its Latitude 74 degr.
2 min. 5 fec. from F to A;
fo is A the Place of Procy-
on. Secondly, Becauſe the
Bull's Horn is diftant from
1
P
F
D
8
+xc
Aries 78 degr. 5 min. 36 feconds, take the Secant thereof,
and draw EB F; fet off the Complement of its Latitude
84 degr. 38 min. 26 fec. from E to B; fo is B the Place
of this Star in the Projection. Draw two occult Circles.
at the Diſtance of Mercury, obferved from A and B, fe-
verally, and they will interfect at C; thro' E C and F,
draw the Oblique Circle and then is C the Place of
Mercury in the Projection at the Time of the Obferva-
tion. Lastly, Draw A B, BC and CA; fo is the Projecti-
on finiſhed. Now for the Trigonometrical Calculation, ob-
ferve the following Steps.
First, In the Oblique Angled Spherical Triangle BEA,
there are given. (1.) A E, the Diftance of Procyon from the
North Pole of the Ecliptic 105 degr. 57 min. 55 feconds.
(2) BE the Complement of the Latitude of the Horn of
Taurus 84 deg. 38 min. 26 feconds. (3.) The Angle BEA
33 degr. 16 min. 25 fec. the Difference of Longitude of the
two known Stars, to find A B their Diſtance.
0 3
Let
!
198
1
The Doctrine of the Sphere.
Let fall the Perpendicular BD; then in the Rect-Angled
Spherical Triangle ED B.
Deg. Min. Sec.
As C. t. BE,
To Radius;
84 38
90 Oo
26-- 8.972266
00--10.000000
So C. f. Angle BED,
33 16
25-- 9.922235
To t, DE.
83 35 49--10.949969
As Radius,
To t. BE;
Or, by Tranfpofition,
So C. f. Angle BED,
To t, DE fub.
From A E,
Rem. A D,
As C. f. DE,
To Cf. DA,
So C. f. BE,
To C. f. BA,
Now fay,
Deg. Min. Sec.
90
O☺--10.000000
84 38 26--11.027734
33 16 25-- 9.922215
83 35 49--1-0.949969
105 57 55
22 6
22
Deg. Min. Sec.
83 35 49 Co. Ar. 0.952639
22 22 6
9.966927
84 38 26
8.970364
39 14 19
9.889030
1
Secondly,
The Doctrine of the Sphere.
199
1
Secondly, In the Oblique-Angled-Spherical-Triangle A BE
are given all the Sides,
Deg. Min. Sec.
A E 105 57
57 55
viz B E 84
38
26
to find the Angle B A E.
A B 39
14
19
Z
229 50
40
half
BE Sub.
:
114 55
84 38
20 Complement 65° 4′ 40″
26 Side oppofite to required Angle.
X 30
16
54
Deg. Min. Sec.
!
S. AE
74 2 5
Co. Ar.
0.017c84
S. A B
S. half Z
S. X
Z of the Logarithms
Half is C. f.
Doubled is
39 14 19 Co. Ar. 0.198904
65
4 40
30 16 54
29 52 16
9.957550
9.702647
19.876185
9.938092
59 44 32 is the Angle B A É.
The Angle BAE may be found by this Analogy:
As S. of the sth Arch A D,
To S. of the 4, DE
Deg. Min. Sec.
6 Co. Ar. 0.419578
22' 22
83 35 49
So t. Angle B E A of X Long. 33
To t. Angle B AE,
16 25
59 44 CO
9997282
9.817048
10.233908
Thirdly,
04
200
The Doctrine of the Sphere.
Thirdly, In the Oblique Spherical Triangle ABC, are gi-
ven all the Sides,
viz.
{
Deg. Min. Sec.
A B 39
14 19
ired
A C 55 47 3 required the Angle B A C.
B C 22
II 55
Z
I 17
13
44
Half 58
36
B C. fub.
22
I I
52
55 Side oppofite to Angle fought.
X
36 24 57
Deg. Min. Sec.
39 14 19 Co. Ar.
0.198904
55 47 30 Co. Ar.
0.082495
58 36 52
9.931296
36
24 57
9.773524
19.986219
Half is C. f.
IO
ΙΟ
47
9.993109
20
21
34 the Angle BAC.
39 44 32
S. A B
S. A C
S. half Z
S. X
Z Logarithm
Doubled is
Add Angle B´A E
Z Angle CAE
80 боб
Fourthly, In the Oblique-Angled-Spherical Triangle ACE,
there are known, AE, the Diſtance of Procyon from the
North Pole of the Ecliptic 105 deg. 57 min. 55 feconds,
AC, the obferv'd Diftance of Mercury from Procyon 55
deg. 47 min. 30 feconds, and the included Angle CAE juſt
now found, 80 degr. 6 min. 6 feconds, to find C E, the Com-
plement of Mercury's Latitude, and Angle CE A the Lon-
gitude of Mercury.
}
First,
The Doctrine of the Sphere.
201
First, For CE, by fuppofing a Perpendicular let fall from
C, upon A E.
OPERATION.
As C. t. CA,
So C. f. Angle CA E,
To Radius;
To t. of 4th Arch,
From A E
Remains 5th Arch
Deg. Min. Sec.
55 47 30 9.832389
OO--IQ. 000000
90 00
80
6
14 II
6-- 9.235277
26-- 9.402888
105 57
55
91 46 29 Comp. 88° 13′ 31″.
Or, by Tranfpofition.
As Radius
To t. CA;
So C. f. Angle C A E,
To t. of 4th Arch,
A
Deg. Min. Sec.
90 00 00--10.00000
55 47
30--10.167611
80 6
06-- 9.238905
14
II
26-- 9.402888
As C. f. of 4th Arch,
To C. f. 5;
So C. f. CA,
To S. Latitude Nor.
Now fay,
Deg. Min. Sec.
14 II 26 Co. Ar. 0.013459
88 13 31
8.490934
9.749894
8.254287
55 47 30
I OI 44
A
For,
2
20%
The Doctrine of the Sphere.
For the Angle CEA.
Deg. Min. Sec.
As S. CE.
88 58
16 Co. Ar. 0.000069
To S. Angle
C AE;
:80 6
6.
9.993486
So S. CA,
55 47
30
9.917505
To S. Angle CEA,
54 34 9
9.911060
Or thus:
Deg. Min. Sec.
As S. of 5th Arch,
88 13 31 Co. Ar. o.co0208
To S. of
4 ;
14 II
26
9.389428
S. t. Angle C A E,
80 6
6
10.758212
To t. Angle C E A,
5434
ΙΟ
10.137846
One Sign
30 00
00
S.
Sub.
I
24 34
ΙΟ
罾
​Procyon
3
21.22
I
1
Place
୪
I 26 47 51
Example 3 Anno 1686, February 11, at 6 h. 16 min.
P. M. the Diſtance of Venus from the Head of Androme-
da was 24 degr. 18 min. 20 ſeconds. The Head of An-
dromeda at that Time was in T 9 degr. 55 min. 33 fe-
conds, and Latitude 25 degr. 41 min. 1 fecond North.
And at the fame time fhe was diftant from Aldebaran
46 degr. 54 min. 40 feconds. The Place of Aldebaran
was then 5 degr. 23 min. 40 feconds, with 5 degr.
29 min. 49 seconds South Latitude. I demand the
Longitude and Latitude of Venus at the time of the Ob-
fervation.
This Figure is Projected upon the Plane of the Solftitial
Colure; becauſe the Longitude of all thefe Stars falls be-
tween and . So that ve is the Ecliptic, E and
F its Poles; the Oblique Circles E AF and E BF, are
drawn by the Secants of the Diſtance of the Stars from
the Solftitial Colure, and ECF as has been taught a-
bove.
The Doctrine of the Sphere.
203
bove: A is the Place of An-
dromeda, B of Aldebaran, and
C of Venus. First, In the Ob-
lique-Angled-Spherical-Trian-
gle AB E, are given, (1.)
AE the Complement of the
Latitude of the Head of An-
dromeda 64° 18′ 59″. (2.) B E
the Diſtance of Aldebaran from
the North Pole of the Ecliptic
95 degr. 29 min. 49 feconds.
(3.) The Angle A E B, the
the Difference of Longitude
of the Head of Andromeda
and Aldebaran 55 deg. 28 min.
7 feconds, to find AB the
diſtance of the two Stars.
By which I find AB to be
62 deg. 9 minutes, omitting
Seconds.
B
Aq
와
​Secondly, In the Triangle A B E, all the Sides are gi-
Deg. Min. Sec.
ven.
A E 64 18 59 By which I find the Angle A B E
viz. A B 62
BE 95
9 00
9 49
59 deg. 4 min.
Or as S. 5th Arch, to S. 4; So t. Long. to t, Angle
ABE as in Page 199.
Thirdly,
viz.
In the Triangle ABC, all the Sides are
known.
Deg. Min. Sec.
A B 62 09 оо
A C 24 18
LB C 46
20
54
40
S
By which I find the Angle A B C
23 degr. 28 min.
To
}
204
The Doctrine of the Sphere.
I
To the Angle ABE
Add the Angle ABC
Sum, is the Angle CBE
59 04
23 28
82
32
Fourthly, In the Triangle CB E, are known the
Deg. Min. Sec.
#
BE 95 29 49
Sides B C 46 54 40
29 49
Angle C BE 82 32
And Angle CE B
By which I find CE
Deg. Min.
88 21
46 26
S. Deg. Min. Sec.
From Longitude of Aldebaran
Sub. Angle C E B
Rem. Longitude Q
With Latitude North,
2 5
23. 40
I
16 26 00
0
18
57 40
I
39
[
Example 4. Anno 1687 September 29, at 6 h. 14 min.
P. M. Mars was obſerved from the following Star of
the three in the Head of Sagittary, 35 deg. 41 min.
15 ſeconds, the Star being then in VS II degr. 54 min.
54 feconds, having 1 degr. 28 min. 59 fecond's North
Latitude. And at the fame time the distance of Mars
was obferved from the bright Star in the Eagle 37 deg.
53 min. 30 minutes, this Star being then in VS 27 degr.
21 min. 34 feconds, with 29 degr. 19 min. 11 ſeconds
North Latitude: I demand the Longitude, and Latitude
of Mars at the time of the Obfervation,
•
In this Figure Vs, is the Ecliptic E and F its Poles, and
A reprefents the Star in the Head of Sagittary, B the bright
Star in the Eagle; their Circles of Longitude EAF and
EB F, are drawn by the Secants of their diftance from
Vs, and C the required place of Mars, by the Inter-
fections of two_Circles projected by their diftance from
A and B, by Problem 6, of Spheric Geometry. In the
Oblique-Angled-Spherical-Triangle, AB E, there are given
A E, the Complement of the Latitude of the firft
Star
เ
The Doctrine of the Sphere.
205
1
I
Star A, 88 deg. 31 min. 1
fecond, B E, the Comple-
of the fecond Star B, go deg.
40 min. 49 feconds, and the
Angle A E B being the dif-
ference of Longitude 15 deg.
26 min. 40 feconds between
the two known Stars, to find
A B, their diſtance, which I
find to be 31 deg. 30 mi-
nutes.
V$
મ
E
J
Secondly, In the Triangle A EB, all the Sides are
Dég. Min. Sec..
A E 88 3 1
I
viz.
A B
30
C
1
known,
12 By which I find the Angle BAE
26 deg. 28 min.
BE 60 40 49
Or,' As S. 5th Arch, to S. 4th; So t. Angle AEB,
Tot. Angle BAE 26 deg. 28 min. as in Page 199.
Thirdly, In the Triangle B AE, are known the
Deg. Min. Sec.
A B I
31 30
Sides AC 35 41 15
ВС 37 53 30
To the Angle BAE
Add the Angle BAC
Sum, is Angle CAE
By which I find the Angle
BAC 71 deg. 36 min.
26 28
71 36
98 04 Comp. 81 deg. 56 min.
Lastly,
200
The Dottrine of the Sphere,
t
Laftly, In the Triangle CA E, are known, the
Sides A £ 38
C A
CA 35
Deg. Min. Sec.
35 41 15
Deg. Min
88
Angle C AE 98
4
31 I
00
By which I find CE
And the Angle A EC 35 21
93 29
Hence becauſe the Perpendicular G D, falls without the
Triangle, it may feem more difficult to the Young Tyro;
therefore I fhall put down the Operation.
And firſt, for the Segment AD.
As C. t. AC,
To Radius ;
So C. f. Angle DA C
To t: AD,
E A, add
Z=ED
Deg. Min.
35 41--10.143796
90
00--10.000000
81 56-9.147136
$ 45-- 9.003440
88 31
94 16 Compl. 85° 44′
$
Now Lay,
As Č. f, DA, thể 4th Árch,
To C.f. D E, the 5th;
Deg. Min.
546 Co. Ar. 0.002191
85 44
8.871565
So C. f. A C,
35 41
9.909692
To S. C G Latitude South,
3 29
8.783441
1
Lastly
The Doctrine of the Sphere.
207.
As S. CE,
1
Lastly, For the Angle A E C
To S. Angle CAE
So S. A C,
To S. Angle A EC;
Deg. Min.
số 31 Có. Ar. 0.000003
81
59
9.99568z
35
41
9.765896
35
21:
9.76238₤
S. Deg. Min. Sec.
{{{
To the place of the Star in the Head
of Sagittary
Add the Angle A E C,
Sum, is Longitude of Mars
9 II 54 54
I 5 21
ΙΟ 17 15 54
Example 5. Anno 1688, Auguft 28, at 8 h. 10 min. P.M.
at the Royal Obfervatory at Greenwich, the diſtance of Jupi-
ter from the preceding Shoulder of Aquarius, was mea-
fured, and found to be 32 degr. 48 min. 40 fec. the Star in A-
quarius's Shoulder in 19° 3'3", and having 8° 38′ 43″
North Latitude; and at the fame time the Diſtance of Jupiter
from the following of the two Stars in the Eagle, was
found to be 36 deg. 45 min. 15 feconds; this Star was
then in VS 15 degr. 27 min. 14 feconds, with 36 degr.
13 min. 48 feconds North Latitude. I demand the Longi-
tude and Latitude of Jupiter at the time of the Obfer-
vation; A is the place of the firft Star, B of the fecond,
and C of Jupiter, VS the Ecliptic, and E and F its
Poles, E VS F is the Equinoctial Colure.
1
}
Firſt,
1
208
The Doctrine of the Sphere.
First, In the Oblique Angled Spheric Triangle AB
E, there are known the
Deg. Min. Sec.
SA SI
A E 81 21 17
Sides BE 53 46 12
{{
Angle BE A 33 35 49
49,
By which I find the Side A B
41 deg. 7 min.
:
B
नै.
Secondly, In the Triangle AB E, are known all
Deg, Min. Sec.
A E 81 21 17
the Sides,
viz.B E 53 46
I 2
A B 41
7
00
By which I find the Angle BA E
42 deg. 40 fec.
'
Thirdly, In the Triangle ABC all the Sides are
Deg. Min. Sec.
A B 41 7 17
viz.ZA C_32: 48 40
B C 36 45 15.
known,
By which I find the Angle BAC
61 deg. 54 min.
To
The Doctrine of the Sphere.
209
To the Angle
Deg. Min.
BAE 42 40
Add the Angle
BAC 61
54
Sum is the Angle CAE
104 34 Compl. 75° 26'
Lastly, In the Triangle ACE, are known the
Deg. Min. Sec.
A E 81
Sides AC 31 48
32
Angle CA È 104
17
40
By which I find CE
Deg. Min.
90 28
34
00
And the Angle AEC 31 38
Hence, becauſe Jupiter is in Antecedence of the Star A,
therefore the Angle A EC 31 degr. 38 minutes fubtracted
from the Place of the Star A, will give the Place of Ju-
piter in Longitude.
Place of the Star A is
Angle A EC Sub.
Place of Jupiter
And Latitude South
}
S. Deg. Min. Sec.
10
19 3
3
I
I 38
00
9
17 25
3
00
00 28
Example 6. Anno 1688, March 30 D. 11H. 40 Minutes
P. M. at the Royal Obfervatory at Greenwich, the Distance
of Saturn from the Lyon's Heart, was found by Obferva-
tion 56 degr. 18 min. 15 feconds; this Star (noted by the
Letter A in the Scheme) at that time was in 25 degr.
29 min. 40 feconds; and having 26 min. 38 feconds North
Latitude, and at the fame time the Diftance of Saturn
from Arcturus was found to be 28 degr. 31 minutes, Ar-
&turus at that time being in 19 min. 52 min.
12 fe-
conds, and having 30 degr. 57 minutes North Latitude;
this Star in the Scheme is reprefented by B, and Saturn
by C; I demand the Longitude and Latitude of Saturn at
the time of the Obfervation.
P
E
210
The Doctrine of the Sphere.
69
Z
f
E
F
Tr
SA
EF vs repreſents the
VS
. Solftitial Colures, VS the
Ecliptic E and F its Poles.
A the Lyon's Heart, B Ardu-
rus, C the place of Saturn
required.
1
In the Oblique-Angled-Spheric-Triangle A B E, there
are given, the
Deg. Min. Sec.
22
}
Sides {AE 89 33
Angle AEB 54
22
32
BE 59 03
00
By which I find the Side A B
59 degr. 46 min.
Secondly, In the Triangle ABE, are known all
the
Deg. Min. Sec.
A B 59 46 007 By which I find the Angle
Sides, viz B E
A E
59 03
89 33
00
22BAE 53 deg. 50 min.
Thirdly, In the Triangle A B C, are given all the
Deg. Min. Sec.
59 46 00
A B
viz. AC 56 18
56 18 15
BC 28 31
00
Sides,
By which I find the Angle
BAD 33 degr. 28 min.
To
The Doctrine of the Sphere.
211
:
Deg. Min:
To the Angle
BAE
53 50
Add the Angle
BAC
33 28
Sum is the Angle
CAE
87 18
Lastly, In the Triangle A E C, are known the
1
Deg. Min. Sec.
Deg. Min.
Sides AC 56 18
Angle CAE 87. 18
SAE 89 33
15
227 By which I find CE
Compl. is the Lat.
87 23
2 37
oo
And the Angle A E C
56 17
1
To the Longitude of the Lyon's Heart
Add the Angle A E C
Sum, is the Longitude of Saturn
With Latitude North
S. Deg. Min. Sec.
4 25 29 40
کر
Ι 26 17 00
6 21
46 40
2
37 OQ
And thus have I given my Reader a full Explanation of
the Method for finding the true Places of the Planers, by
knowing their Diſtance from the fixed Stars; and by which,
if he is but furnished with a good Aftronomical Quadrant,
and is careful to take the Diſtances true, he cannot mifs
of the true Places of the Planets; becauſe the Method
is grounded upon undeniable Principles: Which Method
I have followed in Compiling the following Tables; and
I doubt not but you will find the Places of the Primary Pla-
nets to agree with Obfervation in all Parts of their Orbits,
as I have often proved: But the Moon I dare not fo much
boaſt of, for want of more Oblervations; for it requires no
lefs than 194400 Obfervations to Compleat her Theory; that
is, in every Minute of the Zodiack, and throughout one Re-
volution of her Apogeon. And I dare boldly affirm, that
there is not any perfect Theory of the Moon extant; but
in time I hope it will be compleated.
P 2
PROB.
212
The Doctrine of the Sphere.
1
1
F
PROB. 43.
Given, the Latitude and Declination of a Star, or
Comet, to find its Longitude.
Example. Ann. ante Chriftum 294, Timocharis obferved (as
related by Sherbone, Fol. 12. V. Wing, Inftan. Fol. 56, and
Street, Page 16.) the Pleiades to have 14 degr. 30 minutes
North Declination, with 4 degr. North Latitude. I de-
mand then their true Place in Longitude?
7
А
اسم
Projection, Let PVS.
repreſent the Solftitial Colure,
Y the Equinoctial, P and
Sits Poles; VS the E-
cliptic, E and F its Poles;
draw PAS, and E AF, to
interfect each other at A in
the given Declination and
Latitude. In the Oblique
Angled Spherical Triangle
APE there are known PE,
the conftant Diftance of the
two Poles, 23 degr. 29 mi-
nutes, AP the Complement
of the Declination 75 degr
30 minutes, and AE the
Complement of the Latitude
86 degrees, to find the Angle 4 E P, the Longitude of the
Pleiades from the Solftitial Colure. By the 11th Cafe of
Oblique-Angle-Spheric-Triangles, the Work ftands thus:
Deg.
The Doctrine of the Sphere.
213
Deg. Min.
AE 86 00
AP 75 30
PE 23 29
AES.
PE S.
Z S.
X S.
29 Compl. 87 degr. 31 min.
Z 184
Half· 92
AP fub- 75
59
30
X 16 59
Z of the Logarithms
Half is C. 1. of
Doubled, is
Deg. Min.
86
Oo
23 29
87 31
16
1
59
Co. Ar. 0.001059
Co. Ar. 0.399591
9.999592
9.465522
19.865764
31
3
9.932882
%
62 6 the Angle A EP;
which
fubtracted from the Colure leaves T 27 deg. 5 4 min. the Lon-
gitude of the Pleiades at the time of the Obfervation.
S. D. M. S.
Long. of the Pleiades after Chrift 1727 Years 1 26 10 58
Long. of the Pleiades before Chrift
294 Years
be
1
Sum 2021
0 27 54 00
0 28 16 58
60
1696
60
2021)101818(50"
By which I prove the Annual Receffion of the Equinox to
50 ſeconds, as I have incerted in the following new Tables.
PROB.
P 3
214
The Doctrine of the Sphere.
1
PRO B. 44.
Given, the Latitude of the Place, and the Time of
the Day or Night, to Frect a Coeleftial Scheme,
according to Regiomontanus,
The principal Authors which have given their Opinions.
concerning the Dividing of the Heavens into Twelve Parts,
which they call Houſes, are, (1.) Ptolemy; (2.) Alcabitius ;
(3.) Campanus; (4) Regiomontanus; which laft is generally
received, and called, the Rational Way of Regiomantanus.
1. Ptolemy adviſes, that the Heavens fhould be Divided
into Twelve Houles by Domifying Circles of Pofition drawn
thro' the Poles of the Ecliptic, and thro' every 30 degrees.
thereof, beginning to reckon at the Afcendant, and counting
every 30 degrees of the Ecliptic for the Space of one
Houfe.
2. Alcabitius would have the Houfes of Heaven to be Di-
vided by Domifying Circles, or Circles of Pofition drawn
from the Poles of the World thro' every 30 degr. of the
Equinoctial, beginning at the Point of the Ecliptic Afcen-
ding; and counting 30 degrees upon the Equinoctial from
thence, to be the Cufps of the feveral Houſes.
3. Campanus, divides the 12 Houſes by the Circles of Po-
fitions paffing thro' each 30 degrees of the Prime Vertical
Circle, or Azimuth of Eaft and Weft; and where they
then cut the Ecliptic, are the Cufps of the feveral Hou-
fes.
4. Regiomontanus, divides the Houſes of Heaven by Cir-
cles of Pofition paffing thro' the Interfection of the Me-
ridian and Horizon, and cutting the Equinoctial in every
30 degrees from the Afcendant, and the Point where they
then cut the Ecliptic are the Cufps of the feveral Houſes:
And to find thefe Points of the Ecliptic, is what falls di-
rectly under the Denomination of the Doctrine of the
Sphere; which, when you are acquainted readily how
to
1
The Doctrine of the Sphere.
215
1
to perform for any time of the Day or Night, will
be the only help to Learn you to know the Conftellations
of Heaven, and thereby readily to know any Star or Pla-
net when you fee them in the Heavens; which is the
main End and Defign of this and the following Problem.
Example. Anno 1728, August 28 D.
Auguſt 28 D. 10 H.
10 H. 41 Min.
Apparent Time, at London; I would know the Points of
the Ecliptic where the Circles of Pofition interfects it;
and alſo what Conftellations and Stars are above the Ho
rizon, and what are below it?
N. B.
The Horizon is a Circle of Pofition for the Afcen-
dant and Deſcendant, and the Meridian for the Medium Cali,
and Imum Cæli.
J
Projection.
With any Convenient
Radius of the Chord of
60 degrees, draw the Pri-
mitive Circle NWS E,
which here repreſenteth
the Horizon of the Place:
NZS the Meridian, W
ZE the Prime Vertical,
or Eaft and Weſt Azi-
muth. Then becauſe the
Equinoctial at London
makes an Angle with the
Horizon of 38 degrees
28 minutes, take the Se-
cant thereof, and draw
WAE for one half of
the Equinoctial under the
Horizon, and W BE for
W
N
413
T
P
K Z G
B
S
E
for the other half above the Horizon: Then by Prob. 2
of Spheric Geometry find the Pole of the Equinoctial
WBE, which is at P; take the Chords of 30, and ou
Degrees feverally and let them from S to C and b, a
Ruler laid from P to C and d, will give the Places in
the Equinoctial where the Circles of Pofition maft cut it,
and interfect the North and South Points of the Horizon at
N and S.
P 4
Lafts,
216
The Doctrine of the Sphere.
Lastly, To draw the Ecliptic, you muſt by Prob. 34.
find the Cusp of the Afcendant, and by Prob. 4. its Am-
plitude, and by Prob. 32. the Angle of the Ecliptic and
Horizon. By help of the Chords fet off the Amplitude
from E to, and from W to VS, take the Secant of
70 degr. 19 minutes, the Angle that Ecliptic makes with
the Horizon and Draw VS and VS V for the Ecliptic:
And where the Circles of Pofition N f S and NGS cut the
Ecliptic on one Side the Meridian, and NKS, NLS, on
the other, are what we are ſeeking, and are vulgarly
called the Cufps of the Houses. Now to find
to find the faid
Cufps by Trigonometry, you muft obferve the following
Steps:
Given Time Apparent 1728, Aug.
Equation of Time fub
Equal Time
I
Sun's Place then
Sun's Right Afcenfion
Apparent Time from Noon add
Sum, is Right A. Mid-Heaven
Add
Oblique Afcenfion 11th Houſe
Add
Oblique Afcenfion 12th
Add
Oblique Afcen. Afcendant
Add
Oblique Afcen. 2
Add
Oblique Afcenfion 3
Complement
{
D. H. M. S.
28 ΙΟ
45.
00
2
28
10
38
d co
52
8
m
16
21
54
167 28 00
160
15 00
327 43
30 OO Oo
357
43 00
3༠ 00 00
27 43 00
30 00 00
$7
43 00
30 00 00
87
43
00
30 00 .00
T17 43 00
62
17 fhort of
The Work being thus prepared, the next thing to be done,
is to find the Elevation of the Pole above each Čircle of Po-
ſition which in the Projection are equal to the Complement of
the Angles fb E and GiE; that is, the Interfection of
the Equinoctial and Circles of Pofition.
1
And
The Doctrine of the Sphere.
217.
And firft, in the Right-Angled-Spherical Triangle Efb
there are given, the Angle ƒ Eh, the Latitude of the Place
51 degr. 32 minutes, and Eh 30 degrees, to find the Angle
fb E. But here I muft give you to underſtand, that this
Triangle hf E, is not the Triangle it felf in which the
things given and required, lie; but the oppofite, and
confequently the Complement of the things given and re-
quired: Therefore, becauſe a Complement falls upon a Com-
plement (according to Lord Neper's Catholick Propofition,) I
take the Sine of hE, and the Tangent of the Angle fb E;
and by Tranfpofition, that the Radius may come firſt in the
Analogy, I take the Tangent of the Angle f Eh, and ſa
for the 3d, 5th, 9th and 11th Houfes thus:
1
Deg. Min.
As Radius,
9.0 00--10.000000
00-- 9.698970
32--10.099913
11- 9.798883
51
To S. Circle Pofition from the Meridian; 30
So t. Latitude Given,
To t. height Pole above Circle Pofition, 32
32
Its Complement is the Angle Gi E 57 degr. 49 min.
Secondly, For the Elevation of the Pole above the Cir-
cle of Pofition of the 2d, 6th, 8th and 12th Houſes, in the
Triangle iGE.
ANALOGY.
Deg. Min.
As Radius,
90
00--10.00000 €
To S. Circle Pofition from Meridian;
So t. given Latitude,
32--10.099913
60 00-- 9.937503
51
To t. Height Pole above that Cir. Pofition 47 28--10.037444
Its Complement is the Angle fhE 42 deg. 32 min.
1. For the Cusp of the 10th Houſe.
In the Right-Angled-Spherical-Triangle T B 4, are given,
BY the Right Afcenfion of the Mid-Heaven from 32° 17'
and the Angle B q the Obliquity of the Ecliptic, to find
7 the Diſtance in the Ecliptic of the Culp of the 10th from
T.
ANA-
F
窄
​218
The Doctrine of the Sphere,
་
,
ANALOGY.
As t, BT, R. A.
To Radius;
So C. f. Angle BYq,
To C. t. qr,
Deg. Min.
32
17-- 9.800557
90
00--10.000000
23
29-- 9.962453
34
34--10.161896
Or, by Tranfpofition, fay,
Deg. Min.
As Radius,
90
00--10.000000
To C. t. R. A. M. C.
32
17--10.199443
So C. f. Obliquity,
23
To C. t. of dift. from T
From
S.
12
29-- 9.962453-
34 34--10.161896
oo oo
Sub. the Diſtance =
I 4 34
Cufp 10th Houſe
ΙΟ
25 26
2. For the Culp of the 11th Houfe.
In the Oblique-Angled-Spherical Triangle 11, Yi, are
given, i 2 degr. 17 minutes the Complement Oblique
Afcenfion from Aries, the Angle 11, 122 degr. 11 mi-
nutes, and the Angle 11, the Obliquity of the Ecliptic,
to find 11, that is, the Diſtance of the Cufp of the Ele-
venth in the Ecliptic from Aries.
By the third Axiom of Oblique-Angled-Spherical Trian-
gles the Work ftands thus:
Deg: Min.
From a Semi-circle
180 00
Take the Angle GiE
57 40
Rem. Angle Vill
122
II
Angle 11
add and ſub.
23 29
Sum
145 40
72 50
Difference
98 42
49
21
Oblique Aſcenſion Houſe
Half
357 43
178 51 Compl.
1
19
Now
The Doctrine of the Sphere.
219
As S. half Z Angles,
To S. half their X;
Now fay,
So t. half Ob. Afc. Houfe,
To t. half X of 11 Y and rii,
As C. f. half Angles,
To C. f. half their X;
So t. half Ob. Afc. Houſe,
To t. half Z Sides,
Half X Sides add
Deg. Min.
72
49
Say again,
50 Co. Ari 0.019792
21
I 9
0 55
J
9 880072
8.302634
8.202508
Deg. Min.
72
50 Co. Ar. 0.539554
49
21
9.813872
I
9
8.302634
2
36
8.656060
55
S:
3 31.
00-00
Sum, fub.
From
Remains
12, 00
I I 26 29 Cufp 11th Houfe;
or Point in the Ecliptic where the Circle of Poſition cuts
it.
3. For the Cufp of the 12th Houfe.
In the Oblique Angled Spheric Triangle T bf are known
b, the Oblique Afcenfion of the 12th Houſe 27 degr.
43 minutes, the Angle fb, the Obliquity 23 degr.
29 minutes, and the Angle bf, the Angle formed by
the Circle of Pofition and Equinoctial 137 degr. 28 mi-
nutes, to find the Distance in the Ecliptic f, the Cup
of the 12th Houſe.
OPE-
?
1
.
220
i
The Dottrine of the Sphere.
OPERATION.
From a Semi-circle
Deg. Min.
180 00
Sub. Angle fhE
42 32
Rem. Angle
b f
137
28
Add and Sub. Angle fh
23 29
Sum™
160
57
80° 28'
Difference
113
59 2/
56 59
Oblique Afcen. Houſe
27 43
Half
13 51
!
Now Say,
Deg. Min.
As S. half Z Angles
To S. half X
80 28 Co. Ar. 0.06039
56 59%
9.923509
So t. half Oblique Afc. Houſe
13
51
9.391907
To t. half X of the Sides
ΙΙ
5 1
9.321655
Say again,
Deg. Min.
80 28 Co. Ar. 0.780884
56 59
9.736303
13
51
9.391907
39 2
9.909094
11 51
S. 50 53
As C. fi half Z Angles
}
To C. f. half their X;
So t. half Ob. Afc. Houſe,
To t. half Z Sides,
Add half X Sides,
Z= Side Y f
That is,
I 20 53 the Cufp of the
12th Houfe, or the Point of the Ecliptic where the Circle of
Pofition cuts it.
4. For
The Doctrine of the Sphere..
221.
4. For the Cufp of the Afcendant.
In the Oblique-Angled-Spherical Triangle E, there
are given E, the Oblique Afcenfion of the Afcendant
57 degr. 43 minutes, and the Angle VEF to the
Latitude of the Place 51 degr. 32 minutes, with the
Angle E¯ = to the Obliquity of the Ecliptic
23 degr. 29 minutes, to find the Diftance of the
Cufp of the Afcendant from Aries.
OPERATION.
Deg. Min.
180 00
38 28
From a Semi-circle
Sub. Angle S Eb
Rem. Angle TES
141 32
Add and Sub. Angle ETƒ 23 29
Sum
165 I
Difference
118
3
|| ||
82° 30′
59 I
Oblique Afc. Houſe
57 43
Half
28 52
Now Say,
Deg. Min.
As S. half Z Angles,
To S. half their X;
82 30 Co. Ar. 0.003731
59 I
So t. half Oblique Afc. Houfe
28 SI
To t. half X of the Sides
25 28
·9.933141
9.741066
9.677938
Say
222
The Doctrine of the Sphere.
As C. f. half Z Angles,
To C. f. half their X;
So t. half Ob. Afc. Houfe,
To t. half Z Sides,
Add half X of the Sides
X = Side TS
That is,
Say again,
Deg. Min.
82
30 Co. Ar. 6.884302
59
I
9.711629
28 51
9.741069
65
17
10.336997
25 28
90 45
00 45 the Cufp of the
Afcendant, or the Point where the Horizon cuts the Ecliptic
at the given Time and Place.
1
(
5. For the Cufp of the Second Houfe.
the 2 R, are
In the Oblique Angled Spherical Triangle 2 R,
known, R the Oblique Afcenfion of the Houfe 87
degr. 43 minutes, the Angle of the Circle of Pofition
with the Equinoctial 2 R 42 degr. 32 minutes,
the Angle 2~ R 23 degr. 29 minutes,
in the Ecliptic, the Diſtance of the Cufp from
bra.
OPERATION.
From a Semi-circle
Sub. Angle 2 R
Deg. Min.
180 OO
42 32
to find
and
2
Li-
Rem. Obtufe Angle at R
Add and Sub. Angle 2R 23
Sum
Difference
Oblique Afc. Houſe
137 28
29
160 57
113 59
87 43
Half
43 51
A II
!
80° 28'
56 59
1
Now
The Doctrine of the Sphere.
223
As S. half Z Angles,
To S. half X,
Now fay,
So t. half Oblique Afc. Houfe,
To t. half X of the Sides,
As C. f. half Z Angles,
To C. f. half their X
So t. half Ob. A. Houſe,
To t..half Z Sides
1
Add half X of the Sides
Sum, from
Take away
Deg. Min.
80 28 Co. Ar. 0.006039
56.59
4351
39
15
Say again,
Deg. Min.
51
7226
9.923509
9.982562
9.912110
80 28 Co. Ar. 0.780884
56 59
9.736303
43
9.982562
10.499749
39 15
ԴՐ
III 41
69.69
90 00
Remains
21 41 the Cufp of
the Second, or the Point of the Ecliptic where the Cir-
cle of Pofition cuts it.
Laftly, For the Cup of the Third Houfe.
In the Oblique Angled Spheric Triangle 3 T, there are
known, the Oblique Afcenfion of the Houſe 1172 43' in the
Equinoctial TE, the Angle T3 made by the Circle
of Pofition and Equinoctial 57° 49', and the Obliquity of
the Ecliptic to the Angle T3 23° 29', to find the Di-
ftance in the Ecliptic.
3
=
Į
:
1
OPE-
224
The Doctrine of the Sphere.
OPERATION.
From a Semi-circle
Take the Angle ~ T 3
Reft Obtufe Angle at T
Add and fub. Obliquity
Sum
Difference
Deg. Min.
180 Oo
57 49
122 II
23 29
་
4072° 50′
49 21..
1
145 40
98 42
Oblique Afc. Houſe
117 43
Half
58 51
Now fay,
Deg. Min.
As S. half Z Angles,
To S. half their X;
72 50 Co. Ar. 0.019792
49
21
9.880072
So t. half Ob. Aſc. Houſe,
58 51
10.218654
To t. half X Sides,
52 43
10.118518
Say again,
Deg. Min.
As C. f. half Z Angles,
To C. f. half their X;
72
50 Co.Ar. 0.559554
49 2.1
9.813872
So t. half Ob. Afc. Houſe,
58 51
10.218654
75 00
10.572080
52
43
127 43
To t. half Z Sides,
Add half X Sides,
Sum
Sub. 4 Signs
That is,
=
રી.
120 00
7 43 the Cufp of
the third Houſe, or the Point of the Ecliptic where the
Circle of Pofition Curs it.
Thus have I given you a Method of Erecting a Figure (as
they call it) by the Doctrine of Triangles; in which, you
are to obferve, that if you keep in the Latitude of Lon-
don, the half Sum of the Angles, and alfo the half Dif-
fence is unalterable, and therefore being once Collected,
and
་
The Doctrine of the Sphere.
225
and to them their Sines and Co. Sines, as is here fet down,
it will greatly fhorten the Work when you have occafion to
fet a Scheme for the fame Latitude.
Houfes
کرو
72
$249
50 S. Co.Ar. 0.019792
49 21 S.
9.880072
30 S. Co.Ar. 0.00373i
II
Latit. 51° 32′ North Afcend.82
259
80
12
56
59 S.
I S.
9.933141
28 S. Co.Ar. 0.006039
9.923509
And thus by the above Calculation I have found the
Culp of the Twelve Cœleftial Houſes to be as here follows:
The Cufp of
10 Houfe is
I I
12
Ι
2
And the Cufp of
1 $ 32 2P269 690 X
Deg. Min.
25 26
29
26
20
53
00 45
21 41
43
3
7
4 Houfe is 25
a
26
5
m 26 29
678
20
53
00
45
VŠ 21
4Í
7 43
Note, You need only to Calculate for the Cufps of the
Six Houſes mentioned firft above; for the Cufps of the o-
ther Six are always the fame Degree and Minute of the oppo-
fite Sign.
And thus have I given you the Face of the Heavens at
the Time and Place propofed; where, if it be a clear
Night, and you will but take the Pains to caft your Eyes
up to the Heavens, you may fee Ar&urus near fetting in
the Weft, and above him you may fee Hercules, Lyra,
the Eagle, and Swan all-North-Weft; and on the Meridian.
Q
is
226
The Doctrine of the Sphere.
1
!
is Pegaſus, the Water-Bearer, and the Planet Saturn; all the
other Planets are under the Horizon, but Jupiter is near Ri-
fing. Look South-Eaft, and you'll fee the Whale, and a-
bove him the Ram, and above the Ram, Andromeda; look
a little more Northerly, and you may fee the Bull and
Pleiades; above them is Perfeus, and above Perfeus, is Caf
fiopeia in her Chair; on the Meridian between the Ze-
nith and North Pole is Cephas; look a little more to
the North, and you may fee Hercules and the Goat, with
that glittering Star Capella about 26 Degrees high; and
between the Pole and the Horizon you have the Great
Bear. Thus is the Spangled Canopy of Heaven re
prefented to your View at the Time and Place above
mentioned; which will be nearly fo every 29th Day of
Auguft for this Age. But by reafon of the Sun's appa-
rent Annual Motion, which is about 60'a Day, the Hea-
venly Bodies feem to Rife, Culminate, and Set about 4
fooner every Day; becauſe one Degree is 4' in time,
which in 15 Days makes one Hour; and thus you may
reckon all the Year round 15 Days to an Hour: Which
Method will ferve well enough for common uſe to learn
you to know the Conftellation and Fixed Stars; for by
knowing at any time what Sign is Afcending, Culmina-
ting and Defcending, and then by looking into the Ca-
talogue of Fixed Stars, you will there fee what Stars
are at that time in that part of the Heaven,
a little
practice in which will make you as well acquainted with
them, as you are with your familiar Friends, or with any
one you know paffing along the Street.
By the Tables of Houſes in Mr. Parker's Ephemeris, you
may readily find what Signs are in any part of the Hea-
vens at any Time: Thus, with the Sun's Place enter the
Column under 10, and right againft in the next Column
on the left hand is the Sun's Right Afcenfion in Time,
which add to the time of the Day or Night propoſed;
which Sum, if less than 24 Hours, feek in the Column
under the Sun's R. A. in Time; but if the Sum exceed
24 Hours, take the Overplus, and right againſt that Num-
ber towards the right Hand are the Cufps of the 10th,
11th, 12th, 1st, 2d, and 3d Houfes.
1
Example.
The Doctrine of the Sphere.
224
1
· Example. Anno 1728, August 28 D. 10h. 41' P.M. I would
know the Cufps of the 12 Cœleftial Houſes perform'd by the
Table of Houſes ?
Suu in my 16° 21' 54" gives R. A. in time
Apparent Time from Noon add
Sum
H. M. S:
ii
9 52
10
41 00
21
50 52
:
Seek this 21 H. 50' 52" in th Column under A. R. O in
Time; and right againſt it on the right hand are the Cufps
of
Deg. Min.
10th Houfe W
25 261
11th
*
26 29
12th
the<
53
If
2d
3d
21
41
20
ஞ00
45
And the Cufps of the 4th,
5th, 6th, 7th, 8th and 9th Hou-
fes,are the fame Degree and
Minute of the oppofite Sign:
7
43
PROB. 45:
Given, the Latitude of the Place, and the Time of
the Day or Night, to Erect a Cœleftial Scheme
by the Doctrine of Triangles, according to Regio-
more Expediously than was hewn in
montanus,
the laft Problem.
First, Obferve, that if the Oblique Afcenfion of the
Houſe be less than 90, or more than 270 degrees; then add
the Obliquity of the Ecliptic 23 deg. 29 min. to the firft Arch,
gives the Second.
Q i
But
228
The Doctrine of the Sphere.
But if the Oblique Afcenfion of the Houſe be more than
90, and less than 270 degrees, then fubtract the Ob-
liquity of the Ecliptic 23 deg. 29 min. from the firſt Arch,
gives the fecond.
And if the ſecond Angle be leſs than 90 degrees, the
Diſtance in the Ecliptic muft be accounted from the fame
Equinoctial Point, that the Oblique Afcenfion was reckoned
from.
But if the ſecond Angle be more than 90 degrees, then the
Diſtance in the Ecliptic muſt be reckoned from the contrary
Equinoctial Point that the Oblique Afcenfion of the Houſe
was reckoned from. See Page 169.
Example. Anno 1728, August 28 d. 10 h. 41′ P. M. at
London, I would know the Cufps of the Twelve Cœleftial
Houſes ?
OPERATION
D. H. M. S.
Anno 1728, Auguſt
Equation of Time, ſub.
28
ΙΟ
41 Appar. Time.
2 52
Equal Time
Sun's Place
Sun's R. A.
28 10 38
08
m 16 21 54
167 28
00
Appar. Time from Noon add 160
15
R. A. Med. Cali
327
43
00 Compl. 32° 17′
Add
30 00
00
Ob. Afc. 11th Houſe ·
357 43
00 Compl. 2 17
Add
30 00 00
Ob. Afc. 12th Houſe
27 43 OO
Add
Ob. Afc. Aſcendant
Add
Ob. Afc. 2d Houſe
Add
!
Ob. Afc. 3d House
30 00 00
57 43 00
30 00 00
87 43
00
30 90 00
117 43 oo Compl. 62 17.
1
Firſt,
The Doctrine of the Sphere.
+
229
Firit, For the Elevation of the Pole above each Circle
of Pofition.
In this Diagram PH
SB, repreſents the Me-
ridian of the Place, Æ Æ
the Equinoctial, P and
S its Poles. 10, 4, the
Ecliptic, HA B a Cir-
cle of Pofition 30 degr.
diftant from the Meri-
dian, Hb B a Circle of H
Pofition 60° from the Me-
ridian above the Horizon,
HDB and HCB are
Circles of Pofition un-
der the Horizon, cut-
ting the Equinoctial in
30 and 60 degrees from
the Meridian at G and
=
10
Z
I
72
b
1
D
B
4
h
C
S
Æ
re-
b, as the other two do at e and ƒ above the Horizon,
where they all meet at H and B, the South and North In-
terfections of the Horizon and Meridian, and confequently
are nothing elſe but moveable Horizon; in which the Ele-
vation of the Pole above thofe Circles of Pofition are
preſented by PA SC, of the 11th, 3d, 5th, and 9th
Houfes; to which the Angle He E is equal: Alfo the
Height of the Pole above the Circle of Pofition of the 12th,
2d, 6th, and 8th Houfes is reprefented by Pb SD, to
which the Angle at the Equinoctial HĮ is equal
Therefore, to find the Elevation of the Pole above the
Circle of Pofition of the 11th and 3d Houfes in the Rect-
Angled-Spheric Triangle He, there are given HÆ, the
Complement of the Latitude 38° 28', and Ee the Distance
of the Circle of Pofition in the Equinoctial from the Meri
dian 30°, to find the Angle He Æ,
}
Q 3:
ANA-
230
The Doctrine of the Sphere.
ANALOG F
Deg. Min.
As t. HÆ Co. Latitude,
38
28-- 9.900086 ·
To Radius ĝ
90
00--10.000000
So S. Ae, Circle from Meridian
30
00-- 9:598970
To C. t Angle He E,
$749-- 9.798884 whofe
Complement 32 deg. 11 minutes, is the Elevation of the Pole
above that Circle of Poſition.
Secondly, For the Height of the Pole above the Circle of Po-
fition of the 12th and 2d Houſes.
In the Rect-Angled Spherical Triangle H E f, are given
HA, the Height of the Equinoctial, or Complement of the
Latitude 38 degr. 28 minutes, and Ef 60 degrees, the
Diftance of the Circle of Pofition from the Meridian, to
find the Angle HĮ, made by the Equinoctial and Circle of
Pofition.
ANALOG Y.
Deg. Min.
30 28-- 9.900086
As t. HÆ, Co. Latitude
To Radius;
90
So S. Ef, Circle from Meridian,
To C. t. Angle Hf E,
бо
42
00--10.000000
00-- 9.93753 1
32--10.037445 whofe
Complement is 47 degr. 28 minutes, the Height of the Pole a
bove that Circle of Pofition.
Note, Theſe being once found for any one Latitude, are
always the fame in that Latitude. And by Tranfpofition,
thefe Analogies will be the fame as is fhewed in Page
217.
Thirdly, For the Cufp of the 10th Houſe.
In the Rect-Angled Spheric Triangle V Æ 10, there are
known Æ the Complement of the Right Afcenfion se deg.
17 minutes, and the Angle 10, the Obliquity 23° 28',
to find
10 in the Ecliptic.
ANA-
The Doctrine of the Sphere.
231
}
ANALOGT
As t. Æ. R. A. M. Cali.
To Radius;
So C. f. Angle Æ Y 10,
To C. t. V 10,
From
Deg. Min.
32
90
17--9.800567
00--10.000000
23 29-- 9.962453
34 34--10.161896
Or, by Tranfpofition, as in Page 218.
Sub. the Diſtance
Deg. Min. Sec.
12 Oo OO
I 04.
Rem. Cufp 10 in 10
34
25 26 the fame as before.
Fourthly, For the Cufp of the 11th Houſe.
As Radius,
ANALOGY.
To C. f. Oblique Afc. Houfe ;
So C. t. Elevat. Pole above,
To C. t. of the firſt Angle,
Obliquity Ecliptic add
Sum, is Second Angle
Dig. Min.
90
2 2
32
OC--10.000000
17- 9.999655
II--10.231132
12--10.20077&
23 29
32
55 41
Now fay,
Deg. Min.
As C. f. Second Angle,
To C. f. firft;
So t. Oblique Aſc. Houſe,
To t. of Dift. from Y,
From
Rem. Cufp 11th Houfe
12
26
26 34
26
55 41 Co. Ar. 0.248901
32
[2
9.927469
2 3
17
8.600669
8-777037
i
Fiftbly,
Q 4
1
232
The Doctrine of the Sphere.
t
As Radius,
1
Fifthly, For the Cufp of the 12th Houſe.
ANALOGY.
To C. f. Oblique Afc. Houfe;
So C. t. Pole above Circle,
Sum, is Second Angle,
Deg. Min.
90 oc--10.000000
27 43-- 9947070
47 28-- 9.962560
50 55″- 9.909630
23 29
74 24
To C. t. first Angle,
Obliquity Ecliptic add
Now fay,
Deg. Min.
As C. f. fecond Angle,
74 24 Co. Ar. 0.570377
To C. f. firft
So t. Ob. Aſc. Houſe,
To t. paſt Y
That is,
50 55
9.79965 L
27 43
9.720476
50
56
10.090630
୪
20
56 the Cufp of the
Twelfth Houfe.
As Radius,
Sixthly, For the Cufp of the Afcendant,
ANALOG Y.
GY.
To C. f. Ob. Afc. Houſe
So C. t. Pole above Circle Pofit.
To C. t. firft Angle,
Obliquity add
Second Angle
Deg. Min.
90
00--10.000000
57 43-- 9.727628
51
32-- 9.900086
67 00-- 9.627714
23
29
90 29 Compl. 89° 31'.
Now Say,
Deg. Min.
As C. f. Second Angle,
89 31 Co. Ar, 2.073781
To C. f. firft;
67
00
9.591878
So t. Ob. Afc. Houſe,
To t. fhort of,
57 43
89 13
10.199443
11.865102
Hence,
1
The Doctrine of the Sphere.
233
Hence, becauſe the fecond Angle was more than 90 de-
grees, this Diſtance 89 degr. 13 min. is from, and not from
Y, contrary to the Oblique Afcenfion's Diſtance.
Therefore from
Sub.
Rem. Cufp Afcend.
As Radius,
S. Deg. Min.
6
00 OO
2 29 13
3 Co 47
Seventhly, For the Cufp of the 2d Houſe.
ANALOG T
Deg. Min.
90 00--10.000000`
87 · 43-- 8.600332
So C. t. Pole above Circle Poſition 47 28-- 9.962560
87 55--8-562892
23 29
To C. f. Ob. Afc. Houfe;
To C. t. of the firſt Angle
Obliquity add
Sum, is Second Angle
111
24 Compl. 689 36.
Now fay,
1
Deg. Min.
As C. f. fecond Angle
To C. f. firft
68 36 Co. Ar. 0.437854
87 5.5
8.560540
11.399323
10.397717
So t. Ob. Aſc. Houſe
Tot fhort of
From Libra
Sub.
Cufp 2d House
87 43
68
II
6 00 00
2
2 3
8 II
2I
49
As Radius,
Lastly, For the Cuſp of the 3d Houſe.
ANALOG T.
Deg. Min.
To C. f. Ob. Afc. Houſe ;
So C. t. Pole above Circle Pofition,
To C. t. firſt Angle,
Obliquity Sub.
Rem. Second Angle
9000--10.oooooa
62 - 17-- 9.667545
32 II.-10.2011 27
53 32-- 9.868668
23 29
30
3
1
Nom
}
}
234
'C
The Doctrine of the Sphere.
!
As C. f. fecond Angle,
To C. f. firſt
So t. Ob. Afc. Houfe;
To t. from
From Libra.
Sub.
Rem.
Houſe.
Now fay,
Deg. Min.
53 32
30
3 Co. Ar. 0.062687
9.774046
62 17
10.269524
52
34
10,116257
6 0
I
22 24
ટી. 7 26
26 Cufp of the 3d
And thus you may expeditiously and exactly know at any
Time and Place the true Face of the Heavens.
PRO B. 46.
Given, the Latitude of the Place, and the Distan
ces in the Equinoctial, to Calculate Hour-Lines
upon all forts of Planes that have Centers.
I do not intend in this Place to learn you the whole Art
of Dialling, (for that would take up a Volume of it
felf,) but only to fhew the reaſon of fuch Analogies as relate
to Central Dials, that falling directly under the Doctrine of
the Sphere.
First, For the Horizontal Hour-Lines.
In the Projection of the Sphere Prob. 4. Page 82, in the
Rect-Angled-Spheric Triangle C 12, 1, are given, C 12,
the Elevation of the Pole above the Horizon, equal to the
Latitude of the Place 51 degr. 32 minutes, and the Angle
12 C1 the Diſtance of one Hour in the Equinoctial 15°,
to find the Side 12, 1, the Diſtance of one Hour-Line upon
the Plane of the Horizon.
ANA-
The Doctrine of the Sphere.
235
As C. t. Angle P
ANALOGY.
Deg. Min.
15 00--10.571947
To Radius
So S. P 12 the Latitude,
To t. 12, I upon the Plane,
Or, by Tranfpofition,
As Radius,
To S. Latitude;
So t. Dift. in Equino&ial,
To t. Dift. upon the Plane,
90
00--10.000000
SI
II
32-- 9.893745
51-- 9.321798
Deg. Min.
90
00 10.000000
SI
32
9.893745
35 00
9.328052
II 51
9.321797
And after this manner is the Table in Page 86, calculated,
being Hours, Halves, and Quarters on the Horizontal Plane
for the Latitude of 518 32'.
2. For an Erect Direct Dial.
This Plane is reprefented by the Line 6, 6, and in the
Rect-Angled Spheric Triangle ZP1, are given Z P, the
Zenith diſtance or Complement of the Latitude of the Place,
and the Angle ZP 1 to the Equinoctial Diftances of the
Hour of One from the Meridian 15°, to find the Distance
of the One a-Clock Hour-Line from in the Plane 6, 6,
As C. t. Angle at P,
To Radius
So S. P Z,
;
ANALOGY.
To t. of Hour from Z,
As Radius,
To C. f. Latitude ;
Or, by Tranfpofition.
So t. Angle at the role,
To t. one the Plane,
Deg. Min.
15
00 10.571947
90
38
00 10.000000
28 9.793132
9 28 9.221885
Deg. Min.
90 00 10.000000
51 32 9.793832
IS
00 9.428052
9 28 9.221884
And
236
!
The Doctrine of the Sphere.
And after the fame manner are the Hour-Diſtances in the
Table, Page 87, calculated.
3. For Erect Decliners.
In Page 126, I have fhewn you how to take the Declina-
tion of a Plane; and when that is found, there are three
other Requisites to be known, before you can draw the Hour-
Lines.
1. The Inclination of the Meridian of the Plane, with
the Meridian of the Place.
2. The Height of the Pole, or Stile above the Place.
3. The Diſtance of the Subftile from the Meridian
Line:
And when thefe Requifities are found, then the next thing
to find, is the Diſtance of each Hour-Line from the Subſti-
ler-Line.
-
Example. Latitude 51 degr. 32 min. North, Declination
of the Plane 22 degr. 8 minutes Eaft. I demand all the
Requifites for Drawing Hour-Lines upon fuch a Plane?
Proje&ion. With the Chord of 60 fweep the Primitive Cir-
cle, which fhall reprefent the Dial's Plane.
G N
1
F
I
P
A
D
K
B
Take 29 deg. 8 min.
the Plane's Declination,
and ſet it from N to D;
lay a Ruler from Z to
D, and it will cut the
Horizon in A, thro'
which the Meridian of
the Place muſt paſs. Or
take the Secant of the
Complement of the
Plane's Declination 60
degr. 52 minutes, and
that will draw, the Me-
dian ZA N. Then by
Prob. 2, of Spheric Geo-
metry, find the Pole of
this Oblique Circle of
Meridian Z AN, which
is at E. Take the Chord
of 51 degr. 32 minutes
the Latitude of the Place,
and
The ་
237
Doctrine of the Sphere.
and fet from B to L, and from N to K; draw E L, and it
will cut the Meridian ZA N, in P; a Ruler laid from E
to K gives E; find a Center in G F. To draw E E the E-
quinoctial, lay a Ruler from P to C, and draw FG for
the Subftile or Axis of the World; and now there is the
Right Angled Spheric Triangle, Z PF Right-Angled at F,
in which,
ZF, is the Subftile's Diſtance from the Meridian.
Z P, the Co, Latitude of the Place.
FP, the Height of the Pole above the Plane; or Stile's
Height.
And the Angle
P F Z, is the Right Angle
PZF, is the Complement Plane's Declination.
FPZ, is the Plane's Difference of Longitude.
First, For the Inclination of the Meridians, or the Angle
FPZ.
ANALOGY.
As C. t. Angle P Z F,
To Radius:
So C. f. Z P,
To C. t. Angle ZP F
!
Or, by Tranfpofition,
Deg. Min.
52 9.746132
60 52
90 00 10.000000
38 28 9.893745
35 27 10.147613
Deg. Min.
As Radius,
90
00 10.000000
To C. t. Decl.
29
8 10.253868
So S. Latitude,
SI 32 9.893745
To C. t. Inclinat.
35
27 10.147613
Or fay.
As S. Latitude,
Deg. Min.
SI
32 9.893745
To Radius;
So t. Declination,
To t, Inclination
90
00 10.000000
29
8 9.746132
35 27 9.852387
1
1
2. For
1
238
The Doctrine of the Sphere.
!
1
¡
2. For the Height of the Stile P F.
ANALOG T
As Radius
So S. PZ
So S. Angle PZF
To S. PF
Deg. Min.
90
00 10.000000
38 28 9:793832
60 52 9.941 257
32 54 9.735089
Or by having found the Angle at P, you may make it
an adjacent Extream, and ſay,
As C. t. Z P
Deg. Min.
38 28--10.099913
90
00--10.000000
So C. f. Angle ZP F
1
35
27-- 9.910956
To Radius
To t. P F
32 54-- 9.811043
3. For the Diſtance of the Subftile from the Meridian Z F
ANALOGY.
As C. t. Z P
To Radius
So C. f. Angle PZ F
To t. Z F
Deg. Min.
38 28-10.099913
90
00--10.000000
60 92-- 9.6873 89
21 9-- 9.587476
Or, by having the Inclination of Meridians, or Diffe-
rence of Longitude,
you may make it an
'As Radius
To S. Z P
Angle Z PF; by the firft hereof,
Oppofite Extream, and fay,
Deg. Min.
90
00--10.000000
38 28-- 9.793832
So S. Angle Z PF
To S. Z F
35
27-- 9.763422
2 I
9-- 9.557254
Now the Requifites being found, and the Plane's Diffe-
rence of Longitude FPZ 35 degr. 27 min. being more
than 30 degr. or two Hours in the Equinoctial, fhews, that
the Subftiler-Line will fall between the Hour of 9
and
10
1
The Doctrine of the Sphere.
239
10 in the Morning; becauſe the Plane declines to the
Eaft.
Now before we can calculate the Hour-Diſtances from
the Subftiler-Line, we muſt prepare a Table of the Equi-
noctial Hour-Diſtances, as follows; in the ſecond Column
of which Table; and in the firft Column put down the
Hours and Quarters, fo many as will fall on the Plane
and then to make the fecond Column, proceed thus
Inclination Meridians
Deg. Min.
35 27
Sub. 2 Hours =
30 00
Dift. Subftiler from 10
5 27
9 a Clock from the Meridian
45
00
Inclination Meridian Sub.
35
27
Hour of 2 from Subftile
9 33 on the other Side it.
Then you are to Note in the Equinoctial, one Hour is 15
Three quarters is
Half an Hour is
One quarter is
00
11 15
7 30
3 45
Then by adding, and fubtracting 3 degr. 45 min. conti-
nually, I Compleat the fecond Column of this Table, which
are the Degrees and Minutes in the Equinoctial anfwering
to every quarter of an Hour, fetting the Plane's Difference
of Longitude 35 degr. 27 min. in the fecond Column againſt
12 a-Clock in the firſt Column.
Having finiſhed the firft and fecond Column, the third
Column is made by Calculation thus, for a quarter before
10 a-Clock in the Forenoon.
As C. t. Angle at Polę
To Radius
So S. Stile's Height
To t. dift, on the Plane
Deg. Min.
I 42--11.527546
90
00--10,000000
32 54-- 9,734939
55- 8.207393
00
1
Or,
1
1
240
The Doctrine of the Sphere.
Or, by Tranfpofition, fay,
As Radius
To S. Stile's height
So T. Angle at Pole
To t. on the Plane.
Deg. Min.
90 00--10.000000
32 54-- 9.734939
* 42-- 8.472454
00 55-- 8.207393
And after this manner are the Hour-Diſtances on the
Plane in the third Column found, which may be fet upon
the Dial's Plane, by help of the Line of Chords from the
Subftiler-Line, as has been fhewn in the 87th Page.
The TABLE.
Equi- Hour
Equi-, Hours
noctial on the
Hours. Diftan. Plane.
38 12 90 0190
Hours.
no&tial on the
Diſtan Plane.
Q
Merid. Subft.
3 I
420
55
3
88 18186
52
ΙΟ
05
272
58
4
O
84 33 80
2
I 80 48 73
24
19 125
2 I2 577 7
277
367
3
316 429
14
373
18 61
5
I I
020 27 II
26
069
69
3355
32
I 24 12 13
43
165
48 50
24
262
345
39
2 27 57
5716
3 31 4218
33
358
58
18 41
20
I 2
0135
27 21
9
6
054
3337
20
150
I 50 48 33
40
247
3 30
16
139 1223
2/42 57/26
346
54
49
42 29
58
343
18 27
6
I
о
50
2733
20
0139 33124
1.35 48 21
2132 3 18
IO
I 54
1236 59
23
257
$7 57 40
57
47
61
42 45
338
18
24 33 13
I 20
48 Ir
9 28-
16 19
56
40
2
065
27 49 56
169
1255
2
272 5760 33
376
8
1
a
-2
17
76 42 66
29
313
IS 7
19 3
80
27172
48
019
22 5
13
184
1279 25
1 S
2
2
4813
3
9
287
57 86 14
I
7 || 38'.
༡
12
90
0190
1
0
4. If
The Doctrine of the Sphere.
241
4. If your Declining Plane, recline from the Zenith,
then before you can draw Hour-Lines thereon, you must find
the new Latitude, and new Declination.
Example. In the Latitude of 51 degr. 32 min. N. a Plane
declines to the Eaft, or Weft 24 degrees, and reclines from
the Zenith 54 degrees; what is the new Latitude and De-
clination.
ANALOGY.
As Radius
To C. f. Declination
So C. t. Reclination
Deg. Min.
90
00--10.000000
24
00-- 9.960730
54
00-- 9.861261
33 34-- 9.821991
To t. of the Arch
Now obſerve thefe Rules, in South Recliners.
1. This fourth Tangent muſt be compared with the given
Latitude, and the Complement of their Difference is the
new Latitude.
Given Latitude
Fourth Tangent ſub.
Difference
Complement
1
Deg. Min.
51 32
33 34
17 58
72
2 is the new Latitude.
2. If the fourth Tangent be equal to the given Lati-
tude, then the Difference will be nothing; and fo the Plane
will be a Polar Declining Plane; and the Hour-Lines are
parallel, and the Stile Parallel to the Plane.
3. If the fourth Tangent be greater than the given Lati-
tude, then the North Pole is Elevated in South Decliners.
But if the fourth Tangent be leffer than the given Latitude,
then the South Pole is Elevated in North Decliners.
2. In North Recliners.
Rule 1.
The fourth Tangent found as before, is to be
compared with the Complement of the given Latitude,
and their Difference is the new Latitude.
Rule 2.
If the fourth Tangent be equal to the Com-
plement of the given Latitude, that Declining Reclining
Plane will be an Equinoctial Plane Declining.
R
2. To
1
242
The Doctrine of the Sphere.
2. To find the New Declination.
ANALOGY.
As Radius
To S. of the Reclination
So S. old Declination
To S. new Declination
Dcg. Min.
90
00 10.000000
54 00
9.769219
24-00
9.609313
13 50 9.378532
3. To find the Angle made between the Meridian and Ho
rizon.
ANALOG T
1
Deg. Min.
As Radius
To S. Reclination
So t. of old Declination
To C. t. of the Angle
90
0@ 10:000000
54 .00
9.907958
24 00
9.6-8585
70 II
9-556543
the Angle that the Hour-Line of 12 muft make with the Ho-
rizon. So that a Dial made (according to the Directions a
bove,) for the Latitude of 72 deg. 2 min. North, and De-
clination 13 deg. 50 minutes, will be the true Hour-Lines
upon a Plane in the Latitude of 51 deg. 32 min. North,
Reclination 54 deg. and Declination 24 degrees.
5. Of the Direct South Recliner.
1. If the Plane on which you are to draw Hour-Lines be
a Direct South Recliner,
Take the Difference between the Plane's Reclination and
the Complement of the Latitude of your Habitation, and that
will give you a new Latitude, where that Direct Reclining
Plane will become an Horizontal Plane. If the Reclination
be equal to the Complement of the Latitude, then the Pole
has no Elevation, and thofe Hour-Lines muſt be drawn
as under the Equinoctial, viz. all Parallel by their Natural
Tangents.
2. If the Plane be a Direct North Recliner, and that Re-
clination be equal to the Latitude of the Place, add it to the
Complement of the Latitude, and that Sum will be 90,
for the Latitude under the Poles of the World; where
you have no more to do, than to divide a Circle (the
Equi-
The Doctrine of the Sphere.
243
J
}
Equinoctial) into 24 equal Parts, and the Limbs drawn to
the Center fhall be the true Hour-Lines on fuch a North Re-
clining Plane But, Note this by the way, that this Dial is
of no uſe in North Latitude when the Sun is in Southern
Signs, nor in South Latitude when he is in Northern
Signs.
And whatever the Reclination of this North Plane be,
add it to the Complement of the Latitude of your Habi-
tation, and that ſhall give you a new Latitude, where it
will become an Horizon-Plane, the Hour-Lines upon which
are drawn as has been fhewn in the Horizontal Dial; to
which I refer you.
PROB. 47.
To find the true and apparent Times of the Southing of
the fixed Stars and Planets.
For this purpoſe I have calculated Tables of Right Aſcen-
fions in Time to fix Degrees of North and South Latitude,
which are chiefly intended for the Planets, or thole Stars
whofe Latitudes exceed not Six Degrees: And to take out-
the Right Afcenfion of the Sun, enter the Table with the
Place of the Sun, the Sign on the Head and Degree in the
firſt Column on the left Hand; and under no Degrees of La-
titude, (for the Sun is always apparently in the Ecliptic) and
in the Angle, or Place of meeting, you have the Hour and
Minute of the Sun's Right Afcenfion, remembring to make
proportion for the Minutes of the Sun's Place; becauſe the
Tables give the Right Aſcenſion only to even Degrees of the
Places of the Planets and Stars: Alfo enter the Tables with
the Place of the Planet, and in the Column of the Degree of
its Latitude (if it has any at that Time) you will have the
Hour and Minute of the Planet's Right Afcenfion, minding
to make proportion both for the Planet's Longitude, and La-
titude, if its Place be not even Degrees. In the 144th
and 145th Pages, I have given you a Table of 42 Emi-
nent Fixed Stars with their Right Afcenfions in Time; but
if the Star whofe Time of Southing you want to know, be
not in this Table, nor its Right Afcenfion to be had in the
Tables of Right Afcenfions, then you must find its Right
Afcenfion by Problem 22 and 23.
R 20
Ha-
1
244
The Doctrine of the Sphere.
1
That's hand
Having gained the Right Afcenfion of the Sun, and alſo
of the Star or Planet, fubtract the Sun's Right Afcenfion
from that of the Star or Planet, and the remainder is the
true time of the Star's being upon the Meridian: And if
the Hours are less than 12, the Time is in the Afternoon of
that Day; but if more than 12, 'tis in the Morning of the
following Day; becauſe, as I told in the Definitions under
the Word Day, that Aftronomical Time begins at Noon:
And further Note, that if the Star's Right Afcenfion be lefs
than the Sun's, fo that Subtraction cannot be made, then add
24 Hours to the Star's Right Afcenfion, and out of that Sum
take the Sun's Right Afcenfion, and the remainder will be
the true Time of the Star's Southing, Culminating, or
that it will be then upon the Méridian of your Habita-
tion.
But you must be fure to get the Place of the Sun and Pla-
net as near to the true Time of Southing as poffible, other-
wife you will err 2 or 3', more or lefs in the true Time of
the Planet's Southing; and therefore before we can obtain
the true Time, it is neceffary to have the Eftimate Time of
their Southing, which to know, fubtract the Sun's Place from
the Planet's Place in Longitude, the remainder reduced into
Time by the Table, Page 66, will give you the Eſtimate
Time near enough for this purpoſe. To know this Time, find
the Longitude of the Planets, and their Right Afcenfions to
their Places at the Eftimate Time will produce the true
Times of their Southing.
Example. Anno 1727, October 19, I would know the true
Time of the Pleiades coming to the South?
1. For the Eſtimate Time.
S.
。 1
"
Longit. of the Pleiades
Sun's Place at Noon
Remains
is Eftim. Time of South.
I 26 II 37
627 33 3
6 28 38 34 This, Red. into Time,
h.
"
13 54 34 16
Sun's Place then is 6 28 7 44 R.A.13 44 28 Sub.
Right Afcenfion Pleiades 24 Hours added
• "
h.
27 21 20 From
13 46 52
That is, 46' 52'! paft one a-Clock on the 11th Day in the
Remains the true Time of Southing
Morning.
Exam-
The Doctrine of the Sphere.
· 245
Example 2. Let it be required to find the true Time of
the Southing of the Head of Medufa, on the 5th Day of No-
vember 1727 ?
OPERATION.
"
S. 2
I 22 16 42
Long. of
{Meduſa, Algol
O thatDay
that Day at Noon. 7 23 39
3
Remains
5 28
37 39
H.
Reduced
H.
10
7 24 9 7 R.A.
15 27 4
2650 27
11 23 23
into time is Eftim. time South. II 54 30 36
Right Afcenfion of Algol 24 Hours added
Sun's Place then is
True Time of Southing
Example 3. Let it be required to find the true time of
the Southing of Jupiter, December 6, 1727?¸
OPERATION.
་་
R. Lat. 。°
2 R. Lat. 0° 59′ S. D.
at Noon.
S.
Long. of {Jupiter
Sun
1 22 55 2
8 25 7 14
Remains
4 27 47 48
H.
of{
Reduced into Time is 9 51 II 12
Sun's Place then
H.
"1
8 25 32 21 R.A. 17 40 32 Sub.
Right Afcenfion Jupiter 24 H. added
True time Southing
27 22 40 From
9 42 8 P. M.
And after this manner are the Eſtimate and True Times
of the other Primary Planets and found; but be-
cauſe the Moon is fwift in Motion, her Place to the Efti-
mate Time is practically found by the Logiſtical Loga-
rithms.
Example. Anno 1727, October 10, I would know the true
time of the Moon's Southing?
R 3
1
OPE-
f
246
The Doctrine of the Sphere.
!
1
Eftimate Time
OPERATION.
Moon's Age 7
Multiply by .8 Tenths.
5.6
H. .6
5 36 of Southing.
at Noon<
II
D's place10Day
Diurnal Motion
!
SVS 23 45 Lat. 4 22 S.D.
AW 600
3 39 S. D.
12 15
0 43
Now fay,
H. M.
If
24
oo L.L. Co. Ar. 602 1
Give
12
15
6900
What
5 36
10300
Anfw. 2 51
13221
VS
VS
23 45 D's place 10th at Noon
26 36 D's place at Eftim. South R.A.
27 47 O's place at Eftim. time R.A,
's Southing
True Time of
Add Moon's Mean Motion à O
Eftimate time of Southing 11th Day
H. M.
19 58
13 43
6 15 P.M.
0 48
7 3
Place
I I
12
Day at Noon
D. M. D. M.
~ 6 o Lat. 3 39
Diurnal Motion Moon
18 6
12 6
2 48
ཨ་
54
1
5
Now
:
The Doctrine of the Sphere.
247
H. M.
Now Jay,
If
24
00 LL 6021
Give
I 2
06
6954
What Eſtimate
7
03
9300
Anſwer
3 33
12275
D's place add
6
6
00
H. M.
> at Eftimate * 9
ΑΥ
43
R. A.
20
52
at Eftim.
28
51
R. A.
13
47
South at
Difference in one Day add
Eſtimate time
Place 12 Day at
> [13] Noon |
Diurnal Motion
1
7
00
05
50 the 11th Day.
7 55 the 12th Day.
Now fay,
D. M. D. M.
18 6 Lat. 2 48 S. D.
29 52
II 46
I 51
0 57
D. M.
If
24 00 LL
602 1
Give
II 46
7075
What
7
55
8796
Answer
3 53
11892
12 Day 18
06
H. M.
> Eſtimate 21
29
59 R. A.
21
41
53 R. A.
13
52
> South at
7
49
Difference in one Day add o
Eftimate time 13 Day
44
8
33
D. M.
Place
14
Day at Noon
Diurnal Motion >
29
52 Lat. 1
51 S.D.
* 11 40
O
50
II 48
I
I
*
Now
R 4
1
!
248
The Doctrine of the Sphere.
Now fay,
D. M.
If
24 00 LL
6021
Give
What
Anſwer
I 1
48
7063
8
33
A
12
8462
1.1546
Add
29
52
H. M.
*
4
04 R. A. 22
26
53 R. A. 13
54
m
True Time's Southing 8 32
And after this manner, if you pleaſe, you may proceed for
the whole Year, always taking care to get the Places of the
Planets as near to the Time of Southing as poffible, as is Ex-
emplified above.
PRO B. 48.
Given, the Latitude of the Place, and the Places of
the Stars and Planets, to find the true Times of
their Rifing.
This may be performed two ways.
1. By Subtracting their Semidiurnal Arch from their
Time of Southing, you will have the Time of their Ri-
fing.
2. By the following Tables of Oblique Afcenfion; for if
you fubtract the Oblique Afcenfion of the Sun, from the
Oblique Afcenfion of the Planet, and to the remainder add
the Time of Sun-rifing, you will gain the true Time of the
Rifing of that Star or Planet.
Example. Anno 1727, October 10, I would know the true
time of the Rifing of the Pleiades at London ?
1
OPE-
i
The Doctrine of the Sphere.
249
OPERATION.
Their Declination North
Their Afcenfional Difference in Time
Add
Their Semidiurnal Arch fub.
Their Time of Southing from
Deg. Min. Sec.
23 14 00
2 10 48
60
8 10
48
13 46 52
True Time of their Riſing in the Evening 5
36
4
Note, To find the Semidiurnal Arch of a Star or Planer,
if their Declinations be North, then add the Afcenfion Dif-
ference in Time to Six Hours; but if the Declination be
South, fubtract the Sum or Difference, is the Semidiurnal
Arch. Or by Prob. 7. you may find the Semidiurnal Arch
without the Afcenfional Difference.
f
2. By the following Tables of Oblique Afcenfions.
t
Deg. Min. Sec.
off
Sun
Long. of Pleiades 26
8
11 37 Ọb. A.
28 7 44 Ob. A.
1
20
0
14
40
Q
Remains
10 40
Sun rifes that Morning at London add
Sum, is the true Time of Riſing
Evening as before.
6
56 4
17
36 4 in the
Sub.
12
§ 35
4
Example 2. Let it be required to find the true Time
of the Rifing of Sirius, on the 5th Day of November
1727.
Declination South
Afcen. Difference
In Time, fubt.
From
Semidiurnal Arch,
Time of Southing
Time of Rifing
OPERATION.
Deg: Min. Sec.
16 20
21
58
40 O
1h. 26
40
600
4
IS
6
3 found by the laft Problem.
IQ
32
43
Note,
33 20
250
The Doctrine of the Sphere.
Note, That if the Declination of a Star exceed the Com-
plement of the Latitude of your Habitation, and be of the
fame Name, viz. both North, or both South, then that
Star doth not rife nor fet in that Latitude: As for inftance,
the Head of Medufa's Declination is 39 degr. 53 min. North,
exceeding the Complement of the Latitude of London
38 degr. 28 minutes, proves that Star never rifeth nor fetteth
at London, but has 1 degr. 25 min. Altitude when on the
Meridian under the North Pole.
To find the Rifing of the Planets by the Tables of Ob-
lique Afcenfions, you muft firft find the Eftimate Time of
Rifing, and to that Time the Places of the Sun and Planet,
or as near to the Time as poffible.
Having by the foregoing Problem, found the true time of
Southing, enter the Table of Semidiurnal Archs in the Ap-
pendix, with the Longitude of the Planet; and if it has at
that time little or no Latitude, you will have the Semidiur-
nal Arch pretty near the true; but if the Planet (whofe ri-
fing is required) have confiderable Latitude, as Venus and the
Moon often have, then by Problem 21, find its Declination,
obſerve what Sign and Degree the Sun is in when he has
the fame Declination with the Planet; with which take out
the Semidiurnal Arch, and fubtract it from the Time of
Southing; and that is the Eftimate Time of the Planet's
Rifing; to which Time compute the Places of the Sun and
Planet, and with their Places, take out their Oblique Afcen-
fion, and then proceed as has been taught above.
*
Example. Let the true Time of the rifing of Jupiter
enquired, December 6, 1727 at London'?
OPERATION.
be
H.
M: S.
42 8 P.M.
40 O
2
2 8 P,M.
True Time of Southing 9
Semidiurnal Arch ſub. 7
Eftimate Rifing
Place of <
24
୪
Remains
Deg. Min. Deg. Min.
22 56 Lat, 0 59 South.
OX 25 12
Sun riſes that Morning at
True Time 4 Rifing
H. M.
Ob. Afc.
1 49
Ob. Afc. 19 51
5 58
8 12 add.
2 10 P. M.
N. B.
The Doctrine of the Sphere.
.
251r
|
t
N. B. You muſt borrow 24 Hours to the Oblique Afcen-
fion of the Planet, if it be leſs than the Sun's, and reject 12,
as in the Example above.
Example. Let it be required to find the true Times of
the Rifing of the Moon at London, on October 19, 20, 21,
22, 23, 24, and 25th Days? The Work ftands thus:
Full Moon 19th Day at One in the Morning.
Sun fets that Night at 4 h. 48 minutes; the time from the
time of the full Moon in the Morning, to the Sun's fetting
that Evening is 15 h. 48. Then if 24h: 48′ :: 15 h. 48′:
31'36".
If
Give
What
Answer.
By the Logiſtical Logarithms.
H. M.
24
O
48
15 48
o LL Co.Ar.6021
O 31 36″
969
5795
2786. Theſe 31 min. 36 fec.
added to the Time of the Sun fetting 4h. 48 min. gives 5
hours 19 min. 36 feconds, the Eftimate of the Moon's Ri-
fing.
Now for her Place at that Time.
Deg. Min.
Ŏ 11
Deg. Min.
11 48 Lat. 4
»
N.'A.
4
36
36
Place>19 Day at Noon 24
20
the
Diurnal Motion >
22
12
34
Say again,
H. M.
If
24
0 L L 6021
Give
12
34
6789
What
Answer
>'s place
) ୪
8
Om
Remains
Sun-rifing add
5 20
1 2 48
10512
13322
II
48
H. M.
14
36 Ob. Aſc.
O
53
6
46 Ob. Afc.
16 29
9 24
7 12
True
}
2$2
The Doctrine of the Sphere.
1
True Time Moon-rifing
H. M.
4.36 P.M. the 19 Day.
Diurnal Motion
à O add
0 48
Eftimate Rising 20 Day
5 24
D.
M. D. M.
Plac
[20] Day at Noon
the 21
Diurnal Motion D
Now fay,
Ŏ 24
I 7
12
7
45
22 Lat. 4. 36 N.
4 58
0 22
If
Give
What
Answer
Dadd
}
୪
H. M:
24
0 LL
6021
12
45
6726
5
24
1045 L
2 52
13198
24
22
H. M.
27
m 17
Remains
Sun's Rifing add
True Time
14 Ob.Afc. I 18
42 Qb.Afc.
Rifing
Difference of Rifing add
Eftimate Time > Rifing
15
35
9 43
7 14
4 57 P.M. the 20 Day.
0 21
5 18 the next Day.
For the Moon's Place at that Time.
D. M.
Place
the
|21|
21 Days at Noon
I 7
II 20
7 Lat. 4
6
58 N.
5
7
Diurnal Motion >
I 2
59
9
H. M.
If
24 00 LL 6021
Give 12 59
5 18
What
6648
10539
Answer 2 52 13208
21 Day 7
7
H. M.
I
SI
4 L
II 9 59 Ob. Afc.
Om 8 46 Ob. Afc. 15
Remains
Sun-rifing add
True time > Rifing
Difference of Rifing add
Eftimate Rifing
10 10
7 15
5 25 P.M. the 21 Day.
O
28
5
53 the next Day?
1
FOR
The Doctrine of the Sphere.
253
For the Moon's Place at that Time:
Place the 22 Day at Noon
23
Deg. Min. Deg. Min.
6 Lat. 5
TI 20
3
17
7 N.
4
59
Diurnal Motion>
13 · 11
Ho. Min.
Tf
24
00 LL
6021
Give
13
II
6581
What
5 53
10085
Anſwer
3
14
12687
22 Day 20
6
H. M.
> II 23
20 Ob. Afc.
O m 9
2 38
47 Ob.Afc. 15 47
Remains
10 51
Sun-Rifing add
7
17
True Time > Rifing
6
8 P.M. the 22 Day.
Difference of Rifing add
O
43
Eftimate time
Rifing
6 51 the next Day:
For the Moon's Place at that Time.
Deg. Min.
Place the 23 Day at
22 Day at Noon
3
Deg. Min.
17 Lat. 4 59 N.
16 41
4
35
Diurnal Motion >
13 24
24
Ho. Min.
If
24
00 LL
6021
Give
13 24
6510
What
6 51
9428
Answer 3
49
11956
→ 23 Day 3
)
17
H. M.
6 Ob. Afc. 3
42
II
7
7
m 10
50 Ob.Afc. 15
Remains
Sun-rifing add
True Time Rifing
Difference of Rifing add
Eftimate Time ↓ Rifing
8 P.M. the 23 Day.
I 00
808 the next Day.
For
3
49
19
!
1
1
254
The Doctrine of the Sphere.
}
1
For the Moon's Place at that Time.
Place the 24 Day at Noon 16
>
25
Diurnal Motion >
6
Deg. Min. Deg. Min.
16 41 Lat. 4
35 N.
16
13 35
3 54
0 41
Ho. Min.
If
24
O
。 LL
6021
Give
13
35
645 I
What
8 8
8679
Answer
4
36
11151
》 24 Day 16
41
H.
M.
1
in
21 17
Ob. Afc.
5
3
O in m
11 55 Ob.Aſc. 16
O
Remains
13
Sun-rifing add
True Time Rifing
3
7 21
8 24 P.M. the 24th Day.
Differ. of Rifing add
I
16
Eftimate Time
Rifing
9 40 the next Day.
For the Moon's Place at that Time.
V
Place the 25
|26| Days at Noon
Diurnal Motion >
H. M.
Deg. Min. Deg. Min.
。16 Lat. 3
14 6
54
54 Nor.
2
59
13
50
O
55
If
24 o LL Co.Ar. 6021
Give
13 50
6372
What
9
40
7929
Answer
5 34
10322
→ 25th Day o
16
H.
M.
> in a
5
50. Ob. Afc.
6
31
O in m
12
58
Ob. Afc.
16
6.
14 25
Remains
Sun Rifing add
True Time > Riſing
7 23
9 48 P.M. the 25th Day.
In the Examples above I have all along omitted Seconds,
which is the practical Method of finding the Rifing of the
Moon; but if you would be more curious, then you may
by the foregoing Problems find the true Oblique Afcenfions
of
The Doctrine of the Sphere.
255
of the Sun and Moon to the given Latitude, and from thence
her true Time of Rifing in Hours, Minutes, and Seconds.
As, for inftance, that we may have the true Time more
exact, we muſt calculate the true Places of the Sun and Moon
to the Time above found, with the other Requifites, as is
here fet down.
Deg. Min. Sec.
Sun's Place m
Declination South
12 48 42
15 42 00
Aſcen. Differ.
20
45
Right Afcen.
220
22
00
Ob. Afcen.
241
07
00
Time Sun's Rifing
4
37
00
Setting
7
23
00
Moon's Longitude a
6 00
00.
Latitude North
3
34
Oo
Declination North
22
16
00
Afcen. Differ.
31
01
ΟΙ 00
Right Afcen.
128
23
23 00
Oblique Afcen.
097 22
Moon's Right Afc.
128 23
o + 360
Sun's Right Afc.
220
22
O
Moon Southing
268
I
S.
In Time
Sun's Place then m
Right Afcen.
Moon's Place a
Latitude North
Right Afcenfion
Sun's R.A. Sub.
Moon's Southing
In Time
Oblique Afcen. >
Oblique Afcen. O
Remains
In Time
the Tables of Oblique Afcenfions.
H. M.
17 52 4
13 19
220 51
ΙΟ
34
3 13
133 46 360°
+
220 51
272 55
18
II 40 P.M.
97
241
22 + 360º.
216 15
7
14 25 The fame as by
Bur
256
The Doctrine of the Sphere.
5
But, to a Quadrant or
H. M. S.
6 00 00
Add Afc. Differ.
2 04 04
Semidiurnal Arch >
08
04 04
Southing
18
I I
40
True Time
Rifing
ΙΟ 07 36
Thirdly, The true Time of the Rifing of the Planets may
be obtained, if you fubtract the Sun's Right Afcenfion,
from the Oblique Afcenfion of the Planet; and if the re-
mainder exceed fix Hours, take the Overplus; but if the re-
mainder be lefs than fix Hours, add fix Hours thereto;
the Sum of Difference is the Hour and Minute of their
Rifing.
Example. In the laft Work of the Moon,
> Oblique Afcen.
Right Afcen.
Rem. Rifing
H.
M.
6 31
14 42
9
49 as before.
What I have fhewn above concerning the rifing of
the Heavenly Bodies, has been in refpect of true Time;
but by reaſon of Refractions and Parallaxes, that the true
Time fo found is not the apparent Time, or the Time
that you fee; therefore to obtain the apparent Time of
their Rifing, regard muſt be had both to Refraction and
Parallax; and as the Stars are raiſed by Refraction, and
depreffed by Parallax, their Effects are always contrary;
fo that the Apparent Time will always differ from the True,
except when the Refraction and Parallax are equal.
Example. In the Moon to the Time laft wrought.
Anno 1727, October
Mean Anomaly >
Horizontal Parallax
Horizontal Refraction fub.
Moon's true Altitude
D. H. M. S.
00
25 9 48
4 20 49 6
60 31
33 00
27 31 when
her Center begins to appear in the Horizon. Now to find
the Difference of Right Afcenfions of the Moon and Mid-
Heaven, at the time of her Apparent Rifing, we have gi
ven the Latitude of the Place 51 degr. 32 min. North,
the
The Doctrine of the Sphere.
257
the Moon's Declination 22 degr. 16 min. North, and the true
Altitude of the Moon 27 min. 31 feconds, to find the Horary
Arch of the Equinoctial. That is, in the Oblique Angled
Spherical Triangle B PZ, there are given,
ZP the Complement of
the Latitude 38 deg. 28 min.
BP the Complement of
A
the Declination 67 deg. 44
min.
BZ the Complement of.
the Altitude 89 deg. 32 min:
29 fec. to find the Angle H
BPZ, the difference of the
right Afcenfion of the Moon,
and Mid-Heaven.
D
N
B
OPERATION.
Deg. Min. Sec.
ZP 38 28 Oo
BP 67 44
00
BZ 89 32 29
Z 195 44 29
I
2 97 52
52 14 Compl. 82° 7′ 45″ ½
BZ 89 32 29
X 8 19 45.
Deg. Min.
S. Z P
S. BP
38 28 Co.Ar. e.206168
67 44 Co.Ar. 0.033656
S. Z
82 8
9.995894
S. X
8 20
Z of the Logarithms
Z is C. f. of
Doubled is
60° 3°
9.161164
19.396882
9.698441
120 6 the Angle BP Z.
S
Moon's
1
258
The Doctrine of the Sphere
·
Moon's R. A.
Sub. Angle B PZ
Rem. R. A. M. Cœli
Sun's Right Afc. ſub.
Rem.
Deg. Min.
128
23
120
6
17+ 360 3680 17'.
8
220
22
147 55
Timé when the Moon's Center afcends the Vifible Horizon
9 h. 51′ 40″; that is, 3' 40" later than the Time of her
real Afcent above the true Horizon.
Note, Whenever the Refraction is more than the Horizon-
tal Parallax, then the Excess is the Depreffion of the Moon
below the true Horizon. The neceffity of knowing the ap-
parent Times of the Rifing and Setting of the Luminaries,
is in order to pronounce whether an Eclipfe, or Occultation
will be vifible at a given Place, which by this and the Ninth
Problem are performed.
"
If you would have the time when the Moon's lower Limb
afcends the Horizon, then add the Moon's Semidiameter,
(which at that time is 16' 26") to her Altitude 27 min. 31 fe-
conds, and you will have for her Altitude 43 min. 57 fec. to
A B. Then work as above has been taught.
PRO B. 49.
{
Given, the Latitude of the Place of your Habitation,
and the Places of the Stars and Planets, to find
the True and Apparent Times of their Set-
ting.
This (as in their Rifing) may be performed two fe-
veral ways; either by their Semidiurnal Archs, or by the
Tables of Right and Oblique Afcenfions hereunto an-
nex'd.
1. By their Semidiurnal Archs, find the true Times of their
Southing by Prob. 47, and then their Semidiurnal Arch as is
Thewn in Page 249; add theſe two together, and that will
give the true Time of the Star's fetting.
2. By
The Doctrine of the Sphere.
259
2. By the Tables of Right and Oblique Afcenfion, with
the true Places of the Planets (as near the time as poffible,)
take out their Oblique Defcenfion, which is done by entering
with the oppofite Sign and Degree of the Planet's Places,
and with Latitude of a contrary Name in the Tables of Ob-
lique Afcenfion; remembring to enter with the oppofite Sign
and Degree of the Sun's Place under no Degrees of Lati-
tude; which done, fubtract the Oblique Defcenfion of the
Sun, from the Oblique Defcenfion of the Planet; and to
the remainder add the time of the Sun's fetting that Day,'
that Sum is the time of the Planet's ſetting.
Or from the Oblique Defcenfion of the Planet, fubtract
the Right Afcenfion of the Sun; and if the remainder ex-
ceed fix Hours, fubtract fix Hours from it; but if it be leſs
than fix Hours, add fix Hours; the Sum or Difference is the
time of the Planet's ſetting.
Examples in all the Cafes follows:.
Anno 1727, October 10, I would know the true time of
the fetting of the Pleiades at London ?
OPERATION.
To their time of Southing
Add their Semidiurnal Arch
Their time of Setting
H. M. S.
13 46 52
8 10 48
2I 57 40 That is 57" 40"
paft 9 in the Morning,
2. By the Tables of Oblique Afcenfion.
Ŏ
Deg.Min.Sec.
Long. of Pleiades 26 11 37 But their oppofite Places are
1
Sun
28 28 O
H. M.
Pleiades m 26° 11′ 37" Lat. 40 South Ob.Defc. 17 42
Sun
Remains
V 28 28 o
Sun fets that Day at
Time of their fetting
Ob.Defc, o 49
16 53
24
5 4
57 as before.
Sa
3. By
260.
{
The Doctrine of the Sphere.
3. By the Tables of Right Afcenfions.
OPERATION.
Pleiades Ob. Defcéntion
Sun's Right Afcenfion
Remains.
Sub.
Time of their Setting
H. M.
17.42
1, I. 46
15 52
6
9 56 in the Forenoon of
the 11th Day.
Example 2. What time doth Sirius fet at London the 5th
Day of November ?
OPERATION.:
Ho. Min. Sec.
IS 6 3
Semidiurnal Arch, add 4 33
Time of Southing
Time of Setting
20
19 39 23
Example 3. Let the time of Saturn's ſetting be required at
London, October 14, 1727?
Saturn 10 8
Sun 7 I 32
S. D. M.
Longit. of S
33 Lat. 0° 59' S.
Remains
37
this Reduced into Time, is
6 h. 28′ 4″ the Eſtimate time of Southing,
True time of Southing is
Declination South
Afc. Difference
In Time
Semidiurnal Arch add
Time of Setting
6 47
19
6
25
50
I 43 20
Į
4 16 40
II
3 49
2. By
•
The Doctrine of the Sphere.
261
2. By the Tables of Right and Oblique Afcenfion.
Oppoſite Places of
{
<5
Sh
Deg. Min. Deg. Min.
8 34 Latit. o
59 North.
H. Min
5 Ob. Defc.
7
I
Ob. Defc.
56
Remain
6
S
Sun fetting add
4
58
Saturn fets at
II
3 as before.
1
Or thus.
H. Min.
5 Ób.Afc.
OR. Afc.
7
I 59
Remain
5
2
Add
60
Time of fetting II
2 as before.
-
¡
\
Example 4. Let the Time of the Moon's fetting be fought
November 3d, 1727, at London ?
New Moon the 2d Day at 28′ paſt 4 Morning.
Sun fets that Night at 4 h. 23 minutes; the time from
the New Moon is 35 h. 55 minutes.
Then if 24h. 48′:: 35 h. : 55': : 1 h. 11'50".
Sun's Setting add
Sum Eftimate time Moon ſetting
4 23 00
5 34 50
For the Moon's Place then.
Deg. Min.、 Ho. Min.
Place the Day at Noon<?
Diurnal Motion Moon
4 Lat. 4 58 South,
Z 9
22 51
13 47
5.
2
{
S 3
Now
262
The Doctrine of the Sphere
Now Say,
}
H.
M.
If
24
oo Co.Ar. 6021
Give
13 47
6388
What
5 35
10313
Answer
3
12
12722
> in
9
4
in
12
16
O
in m 2I
52
Oppofite Places are
Deg. Min.
Deg. Min.
Ho. Min.
) E 12
16 Latit.
4 59 N.
Ob. Defc.
1
58
୦ ୪ 2 !
52
Ob. Defc.
I
40
Remain
Sun fets at
Time of the Moon's fetting
0
4 23
4 41
18
3
Deg. Min.
'I 12 16 Ob.D.
ŏ 21 52 R.A.
Remain
Subtract
Moon fetting
Or thus:
Ho. Min.
1 58
3 17
10 4I
6
4 41 as before.
Or the Eſtimate time of the Moon's fetting may be found
by taking the R.A. of the Sun, and the Ob.A. of the Moon
to their Places on the 3d Day at Noon; and their Difference,
rejecting 6 Hours, will be 4 hours 30 minutes; to which
time compute the true Places of the Sun and Moon, and
to thoſe Places, take out of the Tables the Right Afcen. of
the Sun, and the Oblique Defcen. of the Moon, and their
Difference (adding or fubtracting fix Hours) is the Time of
the Moon's fetting. But if your Cafe require more Exact-
nefs, you muſt to this time laft found compute the Places of
the Sun and Moon. Then working as before is taught,
you will obtain the true and correct time of the Moon's fet-
ting.
2. To
The Dottrine of the Sphere.
263
2. To find the Time of the Moon's fetting by her Semi-
diurnal Arch.
True Time of Southing
Eftimate Time of fetting
Moon's Place then ✰ II
Moon's Declination South 27
Afc. Difference
In time Sub.
Semidiurnal Arch add
Time of Moon's ſetting
H. M.
1
12
4
30
39 Lat. 4° 59′ S.
1 I
40
17
2 41
3
18
52 to her Southing.
4
30
30 52
8 from 6 Hours.
Obferve, by reafon of the Moon's fwift Motion, her
Semidiurnal Arch is always Changing, which caufes a diffe-
rence between the time of her Rifing and Setting from that
Time, found by the Oblique Afcenfion or Oblique Defcen-
fion. But in the fixed Stars and other Planets, whofe Motions
are flow, their Rifing and Setting, found by the Semidiur-,
nal Arch, will agree with the times found by the Oblique Af-
cenfions, &c.
Laftly I fhall fhew the Investigation of the Apparent
Times of the Moon's Setting.
Anno 1727, Nov. 3, Moon ſets at
Mean Anomaly Moon then.
Horizontal Parallax
Horizontal Refraction fub.
Moon's true Altitude
H. M. S.
4
4I
oo at London.
8 S.15
37
03
58 41
33
25
41 when her
Center begins to defcend the Western Horizon.
Therefore, in the Oblique
Angled Spherical Triangle
BZP are given B Z, the
Complement of the Altitude.
89 degr. 34 min. 19 feconds,
BP the Diſtance of the
Moon from the North Pole
of the Equinoctial 117 deg.
II minutes, and ZP the
Complement of the Latitude
38 degr. 28 minutes, to find
the Angle BPZ, the Dif-
ference
$ 4
D
H
A
B
N U
Æ
جالية
2
264
The Doctrine of the Sphere
ference of the Right Afcenfion of the Moon and Mid-Hea-
ven.
OPERATION.
Deg. Min. Sec.
Z P 38 28
O.
o Complement 62 deg. 49 minutes
BP 117 II
BZ 89 35 19
Z 1 245 13 19
Half 122
36 39 Complement 57 deg. 23 min. 21 fec.
B. Z 89 34
19
X
33 2
20
Deg. Min.
38 28 Co. Ar. 0.206168
62 49 Co.Ar. 0.050826
S. Z P
S. B P
S.
Z
57 23
S. X
33
2
Z of the Logarithms
C.f. of
24 22
9.925464
9.736497
19.918955
9.9594775
Doubled 48 44 is the Angle BPZ.
R.A. add 249
21
Z.R.A. M.C. 298 5
O'sR.A.fub. 229 26
Remains 68 39, which reduced into Time, is 4h. 34 mina
36 fec. the Apparent Time of the Moon's defcending the We-
ftern Horizon.
If you would have the Time when the Moon's upper Limb
defcends the Horizon, fubtract her Horizontal Semidiameter
15 min. 55 fec. from her Altitude 25 min. 41 fec. and you
will have for the Altitude of the upper Limb 9 min. 46 fec.
when it leaves our Hemiſphere; which, by working accor-
ding to the preceding Method, I find it to fet at 4 h. 37 min.
$ feconds.
OPE-
1
The Dottrine of the Sphere.
265
1
OPERATIQ N.
Moon's Altitude when her Center2
defcends the Western Horizon.
Horizontal Semidiameter fub.
Min. Sec.
25 41
L
15 5.5
Alt. of her upper Limb when fetting 9 46
From
90
Zen. Dift. upper Limb to BZ 80 50
BP
Ꮓ Ꮲ
Z
half
B Z fubt.
1
О
}
1
14
}
。 Compl: 62° 49'
117 II
38 28
O
245 29 14
122
44 37 Compl. 579 15′ 23″
89 50 14
X
32 54 23
S. ZP
Ꮓ Ꮲ
Deg. Min.
38 28 Co.Ar. 0.206168
S. B P
62
49 Co. Ar. 0.050826
S. half Z
57
13
9.924816
S. X
32 54
9.734939
1
Z Logarithms
half is C.f. of 24 41
Doubled is 49 22
22
'R.A. add 249 21
R.A. M.C. 298 53
229 26
✪ R.A.
19.916749
9.9583745
Angle BP Z, fuppofing that Scheme.
to ferve for this Work.
Remains 69 17 which in time is 4 h. 37 min. 8 feconds,
the time that the upper Limb of the Moon defcends our Ho-
rizon. And thus have I given you all the Methods of finding
the Rifing, Southing and Setting of the Sun, Moon and Stars,
both True and Apparent Times, which was never before ſo
Methodically, and fully handied by any.
1
•
PROB.
1
266
The Doctrine of the Sphere:
PROB. 50.
Given, the Latitude of the Place, and the Oblique Af
cenfion of the Star or Planet, to find the Time when it
will rife Cofmically.
Every Star rifes with that Point of the Ecliptic, that has
the fame Oblique Afcenfion with it: And confequently at
the fame time with the Sun, when he poffeffes that Degree
of the Ecliptic.
Therefore, by Problems 5th and 6th, having found the
Oblique Afcenfion of the Star, fubtract 90 degrees from it,
and the remainder will be the Right Afcenfion of the Mid-
Heaven at the time of the Star's Rifing: Then by the 34th
Problem find the Gufp of the Afcendant; which done, fee
what Day of the Month the Sun is in that Degree of the
Ecliptic that is then Afcending; for that is the Day, that
that Star rifeth Coſmically.
Example. Let it be required, at London to find the time
when the Pleiades rife Cofmically, Anno 1727 ?
See the Work.
Deg. Min. Sec.
261058
Longitude:
୪
Latitude North
4 00 37
Declination North
23 14
00
Pleiades Afcen. Difference fub.
32 42
Right Afcenfion
52 50
00
Oblique Afcenfion
20
8 00
Right Aſc. M. Cœli
290
80.00
Degree of the Ecliptic Afcending Ŏ 13 degr. 46 min. the
Sun is in this Place of the Ecliptic about the 23d Day
of April; on which Day the Sun and Pleiades rife together,
Example 2. At London, I would know the Day when F-
mahaunt rifes Cofmically ?
Longi
The Doctrine of the Sphere,
257
Deg. Min, Sec.
Longitude of Fomahaunt m 20
Latitude South
59. 50
21
4. $4
Declination South
31
3 30
Afc. Difference add
49 19
Right Afcenfion
Oblique Afcenfion
340
34
•
29
·53
59.
59
31
Degr. of the Ecliptic Afc. & 29
。 The Sun is in this
Place of the Ecliptic about May 10, which is the Day this Star
rifes Cofmically at London.
Example 3. I demand the Day that the bright Star in the
Eagle will rife Cofmically at London?
I fhall put down all the Work as follows.
Deg. Min. Sec.
Longitude of the Star
MAS 27 54 54
Latitude
29 19 II
Declination North
8
10 1,5
Right Afcenfion
294
20 00
Afcenfional Difference. fub.
10 24 09.
Oblique Afcenfion
Now by Problem 34, find the Point of the Ecliptic Aſcending.
283 56
00 from T 763-4'.
Deg. Min.
As Radius
90
00--10.000000
To C. f. Ob. Afc. of the Star 76
04-- 9.381643
So C. t. Latitude London
51
32. - 9.900086
To C. t. of
7.9
10-- 9.281729
Obliquity add
23 29
Z, is the ſecond Angle
102 39 Complement 770 21!
}
Now fay,
Deg. Min,
As C.f. fec. Angle
77
21 Co.Ar. 0.659566
To C. f. firſt
79
ΙΟ
Sot. Ob. Afc. *
76
4
9.274049
10.605386
Tot. from A
1
74
3
10.543991
!
That
268
be Doctrine of the Sphere.
;
That is 14 deg. 3 minutes, to which Place of the Eclip
tic the Sun comes about the 25th Day of November, on which
Day this Star rifes cofmically. This Method is more Expe-
ditious, than any ever publiſhed that I know of.
PROB. 51.
Given, the Latitude of the Place, and the Oblique
Defcenfion of a Star, to find the Time of its Cofmical
Setting.
Every Star fets with that Point of the Ecliptic, that has
the fame Oblique Defcenfion with it, and confequently at
the fame time as the Sun rifes, when he poffeffes that oppo-
fite Point of the Ecliptic.
By Problem the oth find the Oblique Defcenfion of the Star,
and to it add 180 degrees; that Sum is the Oblique Af-
cenfion of the Afcendant; to which find by Problem 34, the
Point of the Ecliptic then Afcending, and that is the Place
of the Sun, at the Time that the given Star féts cofmi-
cally.
>
Example. What time
What time at London do the Pleiades fet
coſmically?
See the Work.
Deg. Min. Sec.
Longitude Pleiades
୪
26 10 58
Latitude
Declination North
Right Afcenfion.
Afcen. Difference add
Sum, Oblique Defcenfion
Add
Oblique Afcenfion
4. 00 37
23 14 00
52 50
32
42
85 32 00
180 00
00
265 32 00
Complement paſt
85
32 od
>
Now
The Doctrine of the Sphere
26g
As Radius
Now fay, by Prob. 34
To C.f. Ob. Afcen.
Deg. Min.
90 00--10.000000
85
00-- 8.891421
To C.t. Latitude
To C.t. of
51
32- 9.900086
86
28- 8.791507,
Obliquity Sub.
23:29
Deg. Min.
As C.f. of ſec. Angle 62 · 59 Co.Ar. 0.342705
To Cf. of firft Angle 86 · 28 ·
So t. Ob. Afc.
To t. from
t
85 32
60.4
7
0
That is,
8.789787
11.107258
10.239750
4, to which place the Sun comes
November the 12th Day, and that is the Day that the Pleiades
fet coſmically.
Example 2. Let the Day that Fomahaunt fets cofmically
at London be required?
OPERATION.
Deg. Min. Sec.
Longitude of Fomahaunt 29
Latitude
Declination South
Right Afcenfion
29 59 50
21 4 54
31 03 30
340 34 59
49. 19
00
Afcen. Difference
Oblique Defcenfion
'Add
Oblique Afcenfion
Complement fhort of
Degree Afcending
a
291
180
III
15 59
oo oo
15 59
69
44 Ι
II
5 I oo to which Place of
the Ecliptic the Sun comes the 24th of July, and that is the
Day fought.
Example 3. What Day doth the middle Star in Orion's
Belt fet cofmically at London ?
1
A Syno-
The Doctrine of the Sphere.
A`Synopfis of the Work.
Longitude of the *
Latitude South
Declination South
Right Afcenfion
Afcenfional Difference fub.
Rem. Ob. Defcenfion
Add
Oblique Afcenfion
Compl. paft
Degree Afcending m
Deg. Min. Sec.
I 19 38: 34
24 33 30
I
24 49
80 .34 23
I
47 00
78 47 23
180
258 47 23
78 47 23
J
}
}
25 16 the Sun comes to this
Degree of the Ecliptic about the 6th of November, which is
the Day fought. What has been faid of the Fixed Stars,
the fame is to be obferved of the Planets.
}
7
}
1
4 TABLE
The Doctrine of the Sphere.
271
1
ATABLE of 42 Fixed Stars, with the Days when
they Rife and Set cofmically at London.
Cofmical CofmicaD
}
STARS Names.
Rifing. Setting.
F
Irft in Pegafus's Wing, Marchab
Right Shoulder of Aquarius
Extream Star in the Wing of Pegaſus,
Laft in the Goat's Tail
Brighteft Star in the Ram's Head
1
Fan..
1 Sept 13
9 August 18
28 Sept.
24
Feb.
7 July
31.
March
4 October 24
That in the former Horn called the firft
T
ΙΟ
19
In the Tail of the Whale
April
12 Sept.
4'
Brighteft of the Pleiades
23 Nov. 12
In the Whale's Mouth, Mencar
May
20 October 15
North Horn of the Bull, Foot of Auriga
15 Nov .IO
Fomabaunt
May
2
10 July
24
North Eye of the Bull
20 Nov.
13
In the Belly of the Whale
23 Sept.
12
South Eye in the Bull, Aldebaran
28 Nov.
13
South Horn of the Bull
June
5
30
Caftori
9 Feb.
3
Pollux
22 Jan
19
Middle Star in Orion's Belt
July
3 Nov
6
Hare's Thigh
Leffer Dog, Procyon
19 Decemb. T
22 October 18
Great Dog's Mouth, Syrius.
31 Novemb. 5
Lyon's Heart
Auguft 9 Feb:
13
Lyon's Back
Hydra's Heart
Lyon's Tail
Deneb.
10 April 16
20 Dec.
17
22 April 15
Vindemiatrix
Sept. 10 May
7
1
Ar&turus
15 June I2
Virgin's Girdle
19 April 19
Bright Star of the Crown
Virgin's Spike
Right Shoulder of Hercules
Left Shoulder of Hercules
Head of Hercules
28 July
9
October
4 March 27
6 July
12
TO
25
21
It
30 August 21
7
Swan's Bill
Right Shoulder of Ophiucus,or Serpent-bearer. Novemb. 5 July
Lower Wing of the Swan
Vulture's Tail
Right Knee of Ophiucus, or Serpent-bearer Nov.
Scorpion's Heart
Brighteft
in the Eagle
In the Thigh of Pegafus, Scheat
In the Head of Andromeda
5 Sept. 15
11 Auguft. £
16 Fane
22 May
Nov.
Decemb.11 Sept.
PROB.
7
25 August 2
26
25 October 11
}
2.72
The Doctrine of the Sphere.
PROB. 52.
Given, the Latitude of the Place, and the Oblique
Afcenfion of a Star, to find when it will rife
Achronically.
This Problem is folv'd in the preceding for inafmuch as
the Point of the Ecliptic anfwering to the Oblique Afcenfion
rifes with it; therefore its oppofite Point must be the Place
of the Sun, when the Star rifes Achronically: Confequently
the Cofmical Rifing and Setting being known, the Achronical
Rifing and Setting of the fame Star is known alfo: As for
inftance, if I would know the Degree of the Ecliptic the
Sun is in when the Pleiades rife Achronically at London, ha-
ving found that 13 degr. 46 min. of Taurus rifes with them,'
therefore it tells me that Scorpio 13 deg. 46 min. (being the
oppofite Point) will fet as the Pleiades rife; and the Sun
paffes that Place about the 26th Day of October; and that is
the Day at London when the Pleiades rife Achronically. But
to make it more plain, I fhall give the Trigonometrical Inve-
ftigation, by Prob. 34.
Longitude of the Pleiades
ŏ 26
Deg. Min. Sec.
10 58
Latitude North
4 00 37
23
14
00
52
50 00
Declination North
Right Afcenfion
Afcenfion Difference fub.
Oblique Afcenfion paft Y
'As Radius
To C. f. Ob. Afc.
So C. t. Latitude
To C. t. of
Obliquity add
As C. f. of ſecond Angle
To C. f. of the firft
So t. Ob. Afcen.
To t. paft Y
1
32 22 00
20 08 00
Deg. Min.
90
00--10.000000
20 8-9.972617
51 32-- 9.900086
53 17- 9.872703
23 29
76 46 Co.Ar. 0.640321
53
17
20 8
43 46
9.776598
9.564202
9.891121
Thar
The Doctrine of the Sphere.
273
That is, Taurus 13°46', and the oppofite Point of the E-
cliptic thereto, is Scorpio 13º 46'; which Point the Sun
comes about the 26th Day of October, the Day on which the
Pleiades rife Achronically at London.
Example 2. I would know the Day at London when Foma-
haunt will rife Achronically at London ?
SOLUTION
In Page 267. I found the Degree of the Ecliptic the Sun
is in to be Taurus 29° 31' when that Star rifes Cofmically;
therefore the Sun muft poffefs Scorpio 29° 31 min. when the
fame Star rifes Achronically, and to this Place the Sun comes
about the 10th of November.
PROB. 53.
Given, the Latitude of a Place, and the Oblique Defcen
fion of a Star, to find when it will fet Achronically.
The fame Degree of the Ecliptic that defcends with the
Oblique Defcenfion of the Star, is that Place that the Sun.
muft poffefs when the given Star fets Achronically. There-
fore, as the oppofite Point of the Ecliptic that the Sun pof-
feffes when a Star rifes cofmically, makes its Achronical Ri-
fing; fo the Sun muſt be in the oppofite Point of the Ecliptic
when a Star fets Cofmically, to cauſe the Star's Achronical
Letting.
Example. At London the Day of the Achronical ſetting
of the Pleiades is required.
This is folved in Page 269; for there I have found that
the Sun muſt be in Sagittary 09 4 to caufe their Cofmical
fetting; therefore the oppofite Degree Gemini 0° 4′ muft
be the Place of the Sun to cauſe their fetting Achronically;
and to that Place of the Ecliptic the Sun comes the 10th Day
of May. Hence, the Pleiades then fet Achronically at Lon-
dan. More Examples in things fo plain were needlefs
becauſe the Work of thefe two Problems, is performed in
the Coſmical Rifing and Setting.
}
T
か
​274
The Doctrine of the Sphere.
'
A TABLE of 42 Fixed Stars, with the Days when they rife
and fet Achronically at London.
STARS Names.
Fin
If in Pegaſus's Wing, Marchab
Right Shoulder in Aquarius
Extream Star in the Wing of Pegafus
Laft in the Goat's Tail
Bright Star in the Ram's Head
That in the former Horn. called firft * r
The Brightest of the Pleiades
In the Tail of the Whale
In the Whale's Mouth, Mencar
North Horn of the Bull, foot of Auriga
Fomabaunt
North Eye of the Bull
In the Belly of the Whale
Achron., Achron.
Rifing. Setting.
Fuly
6 March 11
Feb.
13
13
August 5
March 22
13
Fan. 27
Sept.
7 April 21
16
I 2
October 16 March 2
26 May 10
Nov.
15
19 April 12
May 9
20
Io Jan.
21 May 12
March 10
24
29 May
Decem. 6
12
30
Decem. 10 August 6
21 July 23
South Eye of the Bull, Aldebaran
South Horn of the Bull
Caftor
Pollux
Middle Star in Orion's Belt
30 May
5
Leffer Dog, Procyon
In the Hare's Thigh
Jan. 14
Fune
6
In the Great Dog's Mouth, Syrius
Lyon's Heart
Feb.
Lyon's Back
Hydra's Heart
S
In the Tail of the Lyon
15
18 April 15
26 May 3
August 18
October 19
I Fune 17
Deneb.
18 October 18
Vindemiatrix
March
Ar&turus
7
Nov.
9
Virgin's Girdle
12
Dec.
[I
1600ober 22
Bright Star of the Crown
Virgin's Spike
17
Far.
6
April
Right Shoulder of Hercules
I Sept.
30
3 Fan,
9
Left Shoulder of Hercules
Head of Hercules
9
Fan.
21
Swan's Bill
19 Feb.
6
16
Right Shoulder of Ophiucus, Serpentarius
27
May
Lower Wing of the Swan
3
Fan.
S
March 5
Vulture's Tail
4
Right Knee of Ophiucus
9
Fan.
28
16 Decem. 7
Scorpion's Heart
Brightest Star in the Eagle
In the Thigh of Pegafus, Scheat
In the Head of Andromeda
Fune
22 Novem. 6
25 Jan.
29
II March 23
26 Decem. 25
From
The Doctrine of the Sphere.
275
1
From the four laft Problems it is manifeft, that from the
times of the Stars Cofmical Setting, to the times of their
Achronical Rifing, they are visible above the Horizon,
from the time of their Rifing, to the time of their Set-
ting, in North Latitudes, if the Stars have South Decli-
nation.
And on the contrary, from the time of their Achronical
Setting, to the time of their Coſmical Rifing, they are alto-
gether invifible, and never appear above the Horizon from
the Setting of the Sun, to his Rifing.
As for inftance; Fomahaunt fets Cofmically on July
the 24th, and rifes Achronically the 10th of Nobember
all which time (being 109 Days) this Star riſes after
Sun-fetting, and fets before Sun-rifing; confequently vi-
fible from the time of its Rifing, to the time of its Set-
ting. But from January 20, the time of its Achronical
Setting, to May 10, the time of its Cofmical Rifing, it
never appears in our Hemiſphere, but when the Sun is
there, and therefore inviſible.
2. And from the time of its Cofmical Rifing May 10,
to the time of its Achronical Setting January 20, it conſtantly
appears above the Horizon at fome part of the Night or
other.
But if the Stars have North Declination, (as ſuppoſe the
Pleiades) they are Vifible from the time of their Achro-
nical Rifing October 26, to the time of their Cofmical Ri-
fing April 23, or till they approach fo near the Sun as to
become Combuft, which fhall be the Bufinefs of the next
Problem.
T 2
PROB.
}
276
1
The Doctrine of the Sphere.
PRO B. 54.
+
Given, the Latitude of the Place, and the Depreffion
of a Star below the Horizon, and the Time of its
Cofmical Rifing, to find the Time of its Heliacal
Rifing.
}
It is known by Obfervation that the fmalleft fixed Stars are
not vifible' till the Hemifphere is wholly free from the Sun's
Rays; that is, till after the End of the Evening, and before
the Beginning of the Morning- Twilight, which is, when the
Sun is 18 degrees below the Horizon; and that Stars of fe-
veral Magnitudes may be feen when the Sun is Depreffed
below the Horizon, as is here fet down.
Degrees
12
2
Stars of
the
13
Magnitude, may be feen 14 below the Ho-
when the Sun is
rizon.
16
:
1
Example. Let the time of the Heliacal Rifing, of the
Pleiades be required at London ?
į
You muft firft by Problem 32, find the Altitude of
the Nonagefime Degree in the given Latitude, to the
time of the Gofmical Rifing of the Star, and the Steps
of the Calculation you muft obferve as is here fet
down.
Lati-
The Doctrine of the Sphere.
277
Deg. Min.
Latitude of the given Place 51 32 N.
Pleiades rife Cofmically April 23
Sun riſes that Morning at
4
35 20"
13
46 found in Page 266.
41 18
248 50
+
290 8
69 520
VS
18
35
82
7
I :
28
8
16
17
3
Complement ſhort of Y
Sun's Place then
Sun's Right Afcenfion
Time from Noon
Sum, R. A. M. Cœli
Medium Cali is
Meridian Angle
Compleat Latiudé
Declination Cul. Point South 22
Altitude Mid-Heaven
Altitude Nonagefime Degr. 18
Z
E
A
B
D
C
D
F
The requifites above being
found, I fhall now explain
what is required in the adja-
cent Figure, in which, ZH
NO repreſent the Meridian
of the Place, HO is the Ho-
zizon, E AF the Ecliptic,
Z BV, a Vertical Circle, H
DCD the Parallel of De-
preffion of the Pleiades 14
degrees when they become
vifible, after their Conjun-
ction with the Sun, inter-
fecting the Vertical Circle
and Ecliptic at C: Now
in the Right Angled Spherical Triangle ABC, right
Angled at B, there are given, B C the Stars Depreffion 14
degrees, and the Angle B A C Angle H A E, the Altitude
of the Nonagefime Degree, or Angle that the Ecliptic makes
with the Horizon, to find A C, the Diſtance in the Ecliptic,
between the Cofmical Point at A, and the Heliacal Point at
C.
= =
N
T3
ANA-
278
The Doctrine of the Sphere.
A NALOGY.
Deg. Min.
As S. Angle BAC Alt. Nonag. 18
3-- 9.491147
To BC the Depreffion
14
00-- 9.3836.75
90
00--10.000000
To S. of the Arch A C
51
20-- 9.892528
So Radius ;
1
Which added to the Place of the Sun Taurus 13° 46', at
the time of the Coſmical Rifing, gives Cancer 5° 6' for the
Place of the Sun at the time of the Heliacal Rifing: To
this place of the Ecliptic the Sun comes the 16th Day of
of June, on which Day the Pleiades will begin to appear
after their Conjunction with the Sun, and will be feen in the
Morning before the Sun rifes.
PROB. 55.
Given, the Latitude of the Place, and the Depreffion
of a Star below the Horizon, and the time of the
Achronical fetting, to find the time of its Heliacal
fetting.
Example. Let the time of the Heliacal fetting of the
Pleiades be required at London ?
The time of the Achronical fetting is May 10.
Sun fets that Evening at
Sun's Place then
Time from Noon add
Sun's Right Afcenfion
Right Afcenfion M. Cali
Complement fhort of
Medium Cœli in Ecliptic
Meridian Angle
Hou. Min. Sec.
7
50
20
I
00
4 found in Page275.
57
52
117
35
175 27
4 33
m
25
24
66 36
Declinat. Cul. Point North
Complement Latitude add
Altitude Mid-Heaven
Altitude Nonagefime Degree
I 58
38 28
40 26
45
41
Now
The Doctrine of the Sphere.
279
}
H
E
Now in the adjacent Fi-
gure, ZHNO reprefents
the Meridian of London,
ZN the Prime Vertical,
HO the Horizon, E AF the
Ecliptic, making an Angle
with the Horizon of 45 deg.
41 minutes, D CD is the
Parallel of Depreffion of the
Pleiades at the time of their
fetting Heliacally 14°. There- D
fore, in the Right-Angled
Spherical Triangle ABC,
there are given B C 14°, and
the Angle BAC 45° 41′,
to find the Arch A C, the Di-
ſtance between the Achroni-
cal Point and the Heliacal
Point.
I A
ANALOG Y.
As S. Angle B A C, Alt. Nonag. 45
To S. B Č the Depreffion
To S. of the Arch AC
So Radius ;
AB
P
N
Deg. Min.
14
41-- 9.854603
00 - 9.383675
90
00--10.000000
19
45-- 9.529072
००
This 192 45' fubtracted from the Place of the Sun Gemini
。° 4' at the time of the Achronical fetting, leaves Taurus 10º
23', for the Place of the Sun at the time of the Heliacal
fetting of the Pleiades; and to this Place of the Ecliptic the
Sun comes the 20th of April, which is the laft Day of
their appearing until the Day of their Heliacal Rifing, June
16: So that from April 20, to June 16, the Pleiades cannot
be feen; but all the other part of the Year they may by
thoſe who inhabit the North Parallel of 51° 32'. The
fame Method is to be obſerved in calculating the times
of the Heliacal rifing and fetting of any other Fixed Star:
But for the Planers you muft obferve the Depreffion of the
Sun, when they rife and fer Heliacally, as is here ſet
down.
T 4
Deg.
1
1
280-
"
The Doctrine of the Sphere.
Ђ
26
Kata
Deg.
I I
10
I L
ΙΟ
5 may be feen in the Day.
5
may be ſeen in the Day.
The Knowledge of theſe Poetical Rifings and Settings of
the Stars were of great Efteem among the Ancient, and
were very uſeful to them in adjufting the times fet apart
for their Religious and Civil Ufes; but now they ſerve
no other end to us, than to inform us of the time when we
may look out for a Star or Planet to make our Obſervations
upon it as occafion ſhall require.
PRO B. 56.
of
Given, the Latitude of the Place, with the Day
the Month, and the Planet's Place at the time it's
on the Meridian, to find the time it will be in
the Nonagefime Degree.
Rule. If it is the Sun, add three Signs to its Place at
Noon; but if any other Planet, add three Signs to its
Place at the time it is South; then with the Place of the
Sun at Noon, or with the Place of the other Planet at
the time of its Southing, enter the Table, fhewing when
the Sun, Moon, or Star, will be in the Nonagefime De-
gree, Page 82, &c. under R. A. and in the next Column
on the left hand under Time, is the Right Afcenfion in Hours
and Minutes; which write out, and referve. Then with
the Place of the Planet, and the Sum of three Signs, en-
ter the ſame Table in the Column under O. A. and against it
on the left hand under Time, is the Hour and Minutes anſwer-
ing, which write out alfo: Then if it is the Sun, the Diffe-
rence between theſe two Quantities of time thus taken out
of the Table is the Time that the Sun will be in the Nona-
gefime Degree on the Day propoſed.
1
[
M
But
1
The Doctrine of the Sphere.
But if it be a Planet or Star, then this Difference of Hours
and Minutes added to the Time of its Southing will give you
the Time that Day or Night that it will be in the Nonage-
fime Degree.
Example. Anno 1727, July 5, at London, I would know
the Time that the Sun will be in the Nonagefime De-
gree?
OPERATIO N.
Deg. Min.
4
Ho. Min.
Sun's Place at Noon is
23 15 gives R.A.
7
7
40
Add
3 · 00 00
Sum
6 23
15 gívés O.A. 8 12
Difference in the Time paft Noon
32
the
Sun is in the Nonagefime Degree.
Example 2. September 8, What time is the Sun in the
Nonagefime Degree?
OPERATION.
Deg. Min.
Ho. Min.
Sun's Place at Noon is 25 56 gives R.A. 11 45
Add
Sum
3
oo oo
8
25 56 gives O.A. 13 55
2 10 the
Difference in Time paft Noon
Sun is in the Nonagefime Degree at London.
Example 3
Anno 1727, July 15, what time will the
Moon be in the Nonagefime Degree at London ?
*
:
J
OPE-
282
The Doctrine of the Sphere.
OPERATION.
Ha. Min.
6 47
P.M. H.
Moon South at
Her Longitude then
7
21
10 gives R.A. 15 15
Add
3
00
Sum
10
Oo
21 10 gives O.A. 16 51
Difference in time
Southing add
> in Nonagefime
I
37
6 47
8
24 P.M.
Example 4. Anno 1727,
October 18, I would know
the time the Moon will be in the Nonagefime De-
gree.
OPERATION.
H. M.
H. M.
Moon South at
II 53 P.M.
Longitude then Ŏ
5 34 gives R.A. 2
13 fub.
Add
3
00
00
}
Sum
4 5 34 gives O.A. o 50 from
Remain
22
37
Time of Southing add
11 53
10
30 Night.
Moon in the Nonagefime Degree
Example <. I would know what
time the Lyon's Heart
will be in the Nonagefime Degree the firſt Day of March in
this Age.
H. M.
53 -
}
2 gives R.A. 9
H. M.
South at
Longitude Cor. 26
10 22
Add
3 0
O
Sum
7 26
Remains
Time of Southing add
2 gives O.A. 11 20
I 27
10 22
The Star is in the Nonagefime Degree at 11 49 atNight. Ex-
The 283
:
Doctrine of the Sphere.
H. M.
51
Longit.
21 gives R.A. 6 45
Example 6. What time will the Star Syrius be in the No-
nagefime Degree, February 1, in this Age?
Syrius South at 8
3 10
H. M.
Add
3
O
ら
​Sum
6 10
21 gives O.A. 7
Remains
0, 15
Time of Southing add
8
5x
Syrius is in the Nonag. Degree
96 at Night at Lon-
don.
PRO B. 57.
Of the General ufe of Logarithms; fhewing how to
find the Logarithm of a whole Number confifting of
5, 6, or ⋆ Places, &c. or of a Mixt, or of a Decimal
Fraction.
The Tables of Logarithms of abfolute Numbers we find
Printed in moft Books of the Mathematicks; and they perform
that by Addition and Subtraction, which Common Num-
bers do by Multiplication and Divifion: But becauſe the
Tables run no further than 10000, and in Aftronomy having
frequent occafion for a Logarithm to 5, 6, or 7 Places, I fhall
make it my Bufinefs in this Problem to explain what is need-
ful to be underſtood in theſe Logarithms.
Every Logarithm is noted with its proper Index, or Cha-
racteriſtick; and thefe Indices are feparated from the rest of
the Logarithm, to the Left-hand by a Dot (); as appears
here below.
Cha-
284
The Doctrine of the Sphere.
;
I and
10 and
TheCharacteriſtick of all
Numbers between
100 and
1000 and
10000 and
100000 and
1000000 and
10000000 and
100
I
1000
2
10000
100000
}
>is
1000000
10000000
บ
100000000
100000000 and 1000000000
1000000000 and 10000000000
7
From hence it is evident, that the Characteriſtick is al-
ways lefs by one, than the Number of Places of its abfo-
lute Number unto which it doth belong. And the Loga-
rithm of the abfolute number Unity with ten Cyphers an-
nexed is as follows.
Numbers. Logarithms.
110.0000000
10 1.0000ONO
100/2.0000000
1000 3.0000000
10000 4.0000000
10000015.0000000
1000000 6.0000000
10000000 7.0000000
100000000 8.0000000
1000000000 9.0000000,
And here alfo, it is to be Noted, that the Logarithm of 2,
of 20, of 200, of 2000, &c. is the fame, having regard to
the Characteriftick, as in this Table.
'
Num-
1
The Doctrine of the Sphere.
285
i
Logarithms.
Numbers.
2
0.3010300
20
1.3010300
200
2.3010100
2000
3.3010300
20000
4.3010300
200000
5.3010300
7
0.8450980
70
1.8450980
700
2.8450980
7000
3.8450980
70000
4.8450980
700000
5.8450980
7000000
6.8450980
7.8450980
70000000
The like of any other Numbers.
In the Tables, all Numbers under 100, have their Loga-
rithms anſwering; but obferve to prefix the proper Chara-
cteriſtick 1. thereto.
2. From 100, to 1000 in the Column under o, is the Lo-
garithm anſwering.
But if you want any Number from 1000 to 10000, then
find the three firft Figures in the firft Column under the
Number, and the Figure that ftands in the Place of U-
nits at the top of the Table, and in the Angle, or Place of
meeting, is the Logarithm fought, being mindful to put the
proper Characteriſtick 3. So the Logarithms of 1681, you
will find to be 3.2255677; any other whole Number under
10000 you have in the Tables in Mr. Hodgson's. Syftem of Ma-
thematicks; which I recommend to the young Student's per-
ufal. In like manner the Logarithm
Num-
286
The Doctrine of the Sphere.
Numb
Logarithm.
54567.
3.6596310
456.7
2.6596310
45.67
1.6596310
4.567
0.6596310
•4567
1.6596310
.04567
2.6596310
of
.004567 is 3.6596310
.09876
2.9945811
.9876
1.9945811
9.876
0.9945811
98.76
1.9945811
987.6
2.9945811
9876.
3.9945811
By this it appears how the Logarithm of a whole Number,
a Mixt, or Decimal Fraction is found, only by changing the
Characteriſtick; or thoſe with this Mark on the top of the
Figure fhew they are leſs than Unity, or defficient Logarithms.
Thus, if you would find the Logarithm of a Vulgar Fra-
ction, fubtract the Logarithm of the Numerator, from, the
Logarithm of the Denominator, taken fimply as a whole
Number, the Remainder fhall be the Logarithm of the gi
ven Fraction.
Example. Let the Logarithm of be required?
Logar. of{4}is
Logar. of is
0.6020600
0.4771213
0.1249387
So you will find the Logarithm of to be 0.3010300, and
of 4 to be -0.06020600, which are the Logarithm of 2 and
4, only fignified by the Sign Minus, before them to fhew
they are Fractions.
2. To find the Logarithm of a Vulgar Mixt Number.
譬
​个
​RULE.
The Doctrine of the Sphere.
287
RUL
Reduce the given Mixt Number into an im-
proper Fraction, and fubtract the Logarithm of the Deno-
minator, taken as a Whole Number; the remainder is the
Logarithm fought.
Example. What's the Logarithm of 40 †?
Being Reduced, is this improper Fraction 24
S2.3096302.
Logar. of {204} is {3.3989700
Logar. of 40 is
-1.6106602
3. To find the Logarithm of a Decimal Fraction.
RULE. To the given Decimal, put its proper Deno-
minator, then (as in the Vulgar) fubtract the Logarithm of
the Numerator, from the Logarithm of the Denominator,
and the remainder is the Logarithm of the Decimal
fought.
Example. What's the Logarithm of 25? With its De-
nomination it will ftand thus:
S2.0000000
Logar. of{100} is {2.3999400
25
1
Logar. of .25 is --0.6020600 the fame with 4 of the
Vulgar Fraction found above.
So likewife will you find the Logarithm of .5 to be
---0.3010300, and of 75 to be ---0.1249387.
Note, that the Logarithm of a Fraction is always defective,
that is, the Value thereof is always lefs than Nothing; for
the Logarithm of being 0.0000000, the Logarithm of 2,
c. which is less than 1, muft needs be less than nothing;
and by how much a Fraction approaches nearer to 1, by
fo much leſs is the Quantity of its Logarithm ;
as in
the Logarithm of the Fractions above, you fee that the Lo-
garithm of is less than the Logarithm of, and the Lo-
garithm of is lefs than the Logarithm of 4, &c.
4. To find the Logarithm of a Decimal Fraction another
way.
RULE. Find the Logarithm of the given Decimal
(without the Characteriftick) as if it were a Whole Num-
ber; that done, take the Complement Arithmetical of that
Logarithm, and place before it, its proper Characteriſtick,
which must confit of fo many Units as there are Cyphers
before the Decimal Fraction, and that is the Logarithm
fought.
Exam-
288
The Doctrine of the Sphere.
Example: „What's the Logarithm of this Decimal 75 ?
The Logarithm of 75 as whole is
Subtract it from
Logarithm of .75 is -
1.8750613
1.0000000
0.1249387
5. How to find the Logarithm of a Decimal Mixt Num
ber.
RULE. Seek the Logarithm of the Number given, as
if it were whole, without the Characteriſtick, and place be-
fore it the proper Characteriſtick belonging to the whole
Part thereof, and that fhall be the Logarithm of the given
mixt Number.
Example. What's the Logarithm of this Mixt Num-
ber 40.8?
The Logarithm of 408 taken as a whole Number is
6106602; before which prefix 1. the proper Characteri
ſtick to the whole Number 40, gives for the Logarithm of
40.8---1.6106602. After the fame manner will you find
the Logarithm of this Mixt Decimal 9.876 to be
0.9945811.
6. To find the Logarithm of any abfolute Number confiſt-
ing of 5 Places.
For this purpofe Sir Jonas Moor in his Math. Comp. and
Harris in his Lexicon, have a Table of proportional Parts
with the Difference of each Logarithm, whoſe uſe is this;
ſuppoſe the Logarithm of 35786 were required; feek the
Logarithm of the four firft Figures towards the Left, viz.
3578 which is 3.553640 and the common Difference is 121;
with this Difference enter the Tables of Parts proportional
(in either the fore-cited Books) in the Column under Diff.
and then Lineally againſt that Number, and under 6 (the Fi-
gure in the Unit's place of the given Number 35786) you
will find 72, the proportional Part; this being added to the
Logarithm of 3578 viz. 3.553640, makes 4.553712.
But becauſe theſe proportional Parts are not always
Printed with the Tables of Logarithms, and confequent-
ly do not fall into the hands of every Buyer of Mathema-
tical Books, therefore for this reafon I fhell fhew how to
find the Logarithm of an abfolute Number confifting of
5, 6, or 7 Places without the help of thoſe Tables of propor
tional Parts.
Exam-
The Doctrine of the Sphere.
289
Example. Let the Logarithm of 35786 as before be requi-
red ?
Taking away the 6 from the Unit's Place, the Logarithm
of 3578 is 3.6536403
and of 3579 is 3.5537617
Difference
Multiply by
Product
1214
6
728.4
Log.of 3578 add.5536403
Log.of 35786--4.553713 1
1
Note, Ever mind to cut off from the Product fo many Fi-
gures to the right Hand as you multiply the common Diffe-
rence by.
7. To find the Logarithm of any Abfolute Number confift-
ing of 6 Places.
RULE. Take the Logarithm out of the Canon to four
Places to the left Hand of the Abfolute Number; and alfo
the next greater Logarithm; and take the Difference of theſe
two Logarithms, and Multiply it by the two Figures that wère
taken away from the Right Hand of the Abfolute Number;
from the Product cut off two Figures to the Right Hand,
and add the other part of the Product to the Logarithm of
the four Figures firft taken out of the Canon; that Sum is the
Logarithm fought.
Example. Let the Logarithm of 101265 be fought ?
OPERATION.
Logar. of
Logar. of
1012 is 3.0051805
1013 is 3.0056094
Difference
4289
Multiply by
Product
65
21445
25734
2787.85
3.0051805
5.0054593 Ever remember
Logar. of 1012 add
Logar, of 101265 is
1
i
V
[Q
290
The Doctrine of the Sphere.
to prefix its proper Characteriſtick, which here is 5, be-
cauſe the number of Places of the Abfolute Number were
Six.
8. To find the Logarithm of any Abfolute Number con-
fifting of feven Places.
Example. Let it be required to find the Logarithm of
1012659 ?
OPERATIO N.
Logar. of<
1012 is 3.0051805
1013 is 3.0056094.
Difference
4289
Multiply by
659
38601
41445
25734
Product add
3026.451
ToLogar. of 1012 .0051805
1
Anſwer
was required.
6.0054831 is the Logarithm of 1012659 as
And likewife you will find the Logarithm of 1367631 to be
6.1359699.
Now to prove that 6.1359699 is the true Logarithm of
1367631, take of 6.1359699, and it is 2.0453233, and
the Cube Root of 1367631 is 111. This done, I look into
the Canon, or Table of Logarithms, and I find that 2.0453233
is the Logarithm of 111.
Which proves that 6.1359699 is the true Logarithm of
1367631. Note, That taking of a Logar. Extracts the Cube
Root of its Abfolute Number; and take the of any Lo-
gar. and you will have the Square Root of its Abfolute
Number.
PROB.
The Doctrine of the Sphere.
291
3
PROB. 58.
A Perfect Logarithm being given, to find the Ab-
folute Number thereunto belonging...
}
If the Characteriſtick of a Logarithm be under 4, then its
Abfolute Number is under 10000, and is eafily found in the
Tables of Logarithms: But if the Characteriftick be 4, 5,
6,7, &c. then the Abfolute Number will exceed the Verge
of the Tables; and obferve this Rule: By what has been
faid in Page 284. you may fee, that when the Characteri-
ftick is 4, that then the Abfolute Number will confift of $
places; make the Characteriftick 3, and look in the Ta-
bles for the given Logarithm, or the neareſt thereunto,
and take the Difference between the given Logarithm, and
the neareſt in the Tables: Alfo take the Difference between
the greater and leffer Logarithm, and ſay,
As the whole Difference of the two Logarithms in the
Table, which is greater and leffer than your given Lóga-
rithm,
Is to the Difference between the next leffer Logarithm
found in the Table, and your Logarithm given;
So is 10,
To the Figure that is to ſupply the Unit's Place of the Ab-
folute Number.
But if the Characteriſtick be 5,
Then fo is ICO,
To the two Figures that are to fupply the Places of the
Units and Tens in the Abfolute Number
But if the Characteriſtick be 6, then fo is 1000,
To the three Figures, that are to fupply the Units, Tens,
and Hundreds Places of the Abfolute Number.
Example. Let the given Logarithm be 4.5537131, and its
Abfolute Number required?
1
V 2
OPE
292
The Doctrine of the Sphere.
OPERATION.
By Changing its Index to 3, it will then be 3.5 5 371 31
Neareſt lefs in the Tables is 3578
Difference
Logarithm of 3578 is 3.5536403
Difference
3579 is 3.5537617
1214
3.5536403
728
.
Now Say,
As 1214: 728 :: Io:6, which put in the Unit's place of
3578 it makes 35786 for the Abfolute Number fought,"
Example 2. Let the Logarithm be 5.0054592; I demand
the Abfoute Number anſwering thereunto 254
OPERATION.
By changing the Characteriſtick to 3, the neareſt Number
in the Tables is 1012.
Logarithm of 1032 is 3.0051805
Difference
of{ 1033 13.0056094
Given Logarithm is
Logarithm of 1012 is
Difference
4289
11
11
1
3.0054592
3.0051805
2787
Now Say,
As 4289 2787: 100: 65, which put in the Unit's and Tens
Places of the Number 1012, it makes it 101265, which is the
Abfolute Number anſwering the given Logarithm.
Example 3. Let the Logarithm given be 6.0054631, and
the Abfolute Number required?
ļ
PE
!
The Doctrine of the Sphere.
293
}
OPERATION.
By changing the Index to 3, the Number in the Table an-
fwering the neareſt leſs, is 1012.
Given Logarithm is
Logar. of 1012 is
Difference
Logar. of 1012 is 3.0051805
of{ 1013-is 3.0056094
Difference
4289
1
3.0054631
3.0051805
2826
Now Say,
1
As 4289 2826: 1000: 659 which put to the right Hand
of the Number 1012, makes it 1012659, which is the Abfo-
lute Number anſwering to the given Logarithm.
1
1
PRO B. 59.
To Draw a Tangent-Line to a given Circle.
This Problem being mislaid when the Spheric Geometry
was Printed off, is the reafon it is placed here. Open the
Compaffes to any
convenient Extent,
and on the Center
A, draw the given
Circle; fet one
Foot in D (in any
part of the Circum-
ference) and ſweep
the Semicircle A
BC; Draw A B,
the Radius of the
given Circle from
its Center, to the
Circumference.
where the Semi-
D
FA
A
1
1
circle
V 3
294
The Doctrine of the Sphere.
;..
circle interfects it; then Draw B C to meet the Diameter of
the Semicircle, at C, and 'tis done; fo fhall B C be a true
Tangent to the given Circle, as was required: Becauſe the
Angle A B C is a right Angle, as being made in a Semi-
circle; as per Euclid 31, 3.
PROB. 60.
Shewing the Ufes of the Tables, in the Appendix to
the Doctrine of the Sphere.
The firft Table gives you by Infpection the Golden Num-
ber and Epacts in both Accounts for any Year of our Lord
from 1700, to 1799, incluſive.
To find them Arithmetically.
For the Golden Number, add 1 to the preſent Year, and
divide the Sum by 19; the Remainder is the Golden Number,
and the Quotient is the Revolution that the Sun and Moon
have made fince the Birth of Chrift.
•
1
For the English Epact, multiply the Golden Number by
11, and divide the Product 30; what remains, is the E-
pact.
W!
}
For the Roman Epact, fubtract 11 from the English Epact
(until the Year 1800) and the Remainder is the Roman E-
ract.
2. The fecond Table gives you the Dominical Letters in
both Accounts till the Year 1800.
To find them Arithmetically..
For the English Sunday- Letter, divide the Year, its 4th Part,
and 4 by 7; the Remainder fubtract from 7, gives you the
Number of the Letter, as is here fet down.
·
I. 2. 3. 4. 5. 6. 7.
A. B. C. D. E. F. G.
For the Roman Letter, divide the Year, and its 4th Part
by 7, the Remainder fubtract from 7, gives you the Num-
ber of the Roman Letter reckoned as above. Alfo by the firſt
part of the Work you will difcover whether it be Leap-year
or what Year paft; for if 1 remain when you divide the
Year by 4, then 'tis the firft paft Leap-year; if 2 remain, 'tis
the 2d paft; if 3 remain, 'tis the 34 paft; but if nothing re-
main, 'tis Leap-year.
3. The
;
The Doctrine of the Sphere.
295
3. The next Table is a perpetual Table of the Number
of Direction, whofe ufe is to find out the Moveab e
Feafts and Westminster-Terms Yearly. How to find it Arith-
metically I have fhewed in the Definitions, under the Words
Number of Direction.
4. Enter the 4th, 5th, and 6th Tables with the Number
of Direction in the firft Column on the left hand for the given
Year, and right againſt it you have all the Moveable Feaſts
and Terms for the faid Year, in the English Account.
Arithmetically.
Seek the Epact for the Year propoſed; and if it is lefs
than 28, or 29, ſubtract it from 47; but if it be 28 or 29,
fubtract it from 77, the remainder is the Day of the Month
in March or April, or Eafter Limit for that Year; which if
it be less than 31, look in the Month of March, and count on
from that Day or Limit, till you come to the Sunday-Letter
for that Year; for that is Eafter-Day. But if the Limit ex-
ceed 31, fubtract 31 from it, and count in April from the
Day or Limit, until your reckoning end at the Dominical
Letter for the given Year, and that gives you Eafter-Day in
April. Or having the Dominical Letter for any given Year,
number it as above fet down, and add 4 to it always; this
Sum take from the Limit, and what remains, is Eafter-Day
in March, if the remainder was less than 32; but in April, if
it was more than 31.
Secondly, In our Common-Prayer-Book we have the Prime
or Golden Number in a Column to the left Hand in every
Month, whofe uſe at firſt was to find the New Moons, and
Eafter-Day; but Time has worn out the firſt, and now made
it uſeleſs; but its other uſe ſtands good now, and will direct
you to Eafter-Day in any Year in the English Account, if you
carefully obferve this Rule,
In March after the firft C,
Look the Prime where ever it be,
The third Sunday after that, Eafter-Day fhall be;
And if the Prime on Sunday be
Then reckon that for one of the three.
1
V 4
For
286
The Doctrine of the Sphere.,
!
For the Roman-Eafter fee the following Table, which fhews
it by Infpection for this Century; and the Difference in Days
every year from the English Eafter.
5. The Table fhewing what Day of the Week begins any
Month is very plain; for having the Dominical Letter for the
given Year, find that from the Head, and guide your Eye
down from it till you come right against the Month, and
there is the Name of the Day of the Week that begins
that Month.
6. The next Table fhews you the Day of the Week any
Day of the Month falls on in both Accounts; for in the firſt
Column you have the Julian or English Months, ftanding a-
gainst the Letter (in the firſt Column of Letters) that begins
the Month it ftands againft: And in the laſt Column to the
Right Hand are the Gregorian or Roman Months, ftanding a-
gainst the Letter that begins the Month.
31.
Under the Dominical Letters are the Day of the Month to
Then fuppofe I would know what Day of the Week
the 25th Day of October falls on in both Accounts
1727 ?
For the English, firft, find the Sunday-Letter A, in the firſt
Column October; and becauſe I find it ftands againſt the Sun-
day-Letter, that informs me that the 1, 8, 15, 22, and 29
Days are all Sundays, and that the 2, 9, 16, 23, 30, are all
Mondays, and the 3, 10, 17, 24, 31 are all Tuesdays in October,
and January, &c. as the Figures underneath fhew. And the
4, 11, 18, 25 are all Wednesdays, &c.
For the Roman, their Sunday- Letter is E which I feek in
the fame Line right against October on the Right Hand, and
call E Sunday, F Monday, G Tuesday, A Wednesday, which
is the firſt Day of the Month: Then I go to the Fi-
gures; and becauſe the 1ft is Wednesday, the 8, 15, 22,
29 are Wednesdays, 2, 9, 16, 23, 30 are Thursdays, the 3, 10,
13, 24, 31 are Fridays, and 4, 11, 18, 25 Days Saturdays
in October in the Roman Account. But for this purpoſe (in
the English Account) I have given you Expeditious Ta-
bles in my System of the Planets Demonftrated, Prin-
ted for Mr. Wilcox in Little-Britain, and for Mr. Heath
in the Strand.
The
7.
The Doctrine of the Sphere.
Line a
Month is
Head
abc
7. The Table for the Number of Da, s is obvious co
meaneft Capacity; for find the Month in the
the Head, and under it in the Came Column in a
the Number of Days from any Day of the Month
to the fame Day of the Month in any other Moths: As,
from August 29, to January 29, 15 153 Days; the like of any.
other. And this is ufeful in computing the man Place of
the Moon's Nodes; for knowing the Place of the No th
Node any one Year, (as fuppofe January 1, 1727 he in
o S. 4° 27' 25'', and I would have its mean place October 25
next following,) I look into this Table, and find om Ja-
nuary 1, to October 1, 273 Days; to which add 24 Days,
make 297 Days from January 1, to October 25 Inclufive: Then
becauſe the mean Diurnal Motion of the Node Retrograde is
3'11" 191" ×297 15° 45' 27" for the Mean Motion of
the Node in that Time, which fubtracted from the Place of
the Node January 1, 4° 27' 25" leaves 18° 41′ 58″ for
the Mean Place of the Node October 25, as was required.
All the other Tables in the Appendix are ſo obvious to the
meaneft Capacity, that nothing needs be faid by way of Ex-
planation.
8. For the Moon's Age, add to the Epact for the given
Year the Day of the Month, and the Number of the Months,
as is here fet down; and if the Sum is under 30, that is
the Moon's Age; but if it exceed 30, caft away 30, and
the remainder is the Age of the Moon. The Months muft
be Numbred thus: 1
2
I
2 3 4
5 6.
8
January February March April May June July August September
ΙΟ
ΙΟ
October November December.
9. For the Day of the New Moon, add the Number of
Months, and the Epact together, and fubtract the Sum from
30; but if the Sum exceed 30, fubtract it from 59, and the
remainder is the Day of the New Moon according to her
Middle Morion.
The Day of the Full Moon is gained by fubtracting the a-
bove mentioned Sum from 15; but when Subtraction cannot
be made, borrow 30 Days, and the remainder will give you
the Day of the Full Moon according to her Mean Motion.
PROB.
1
208
The Doctrine of the Sphere.
PROB. 61.
Shewing the Use of Shakerley's Logistical Loga
rithms.
Theſe Logarithms are of excellent uſe in all forts of A-
ftronomical Calculations, either in Time or Motion, the Fi-
gures running along the Head of the Table beginning with
。 under Motion, and fo on with 1, 2, 3, 4, to 6o, are
either Degrees, Minutes, or Seconds, &c. as the Cafes
require.
If they be Degrees, then the firft Column of Figures
are Minutes; if they be used as Minutes, then thoſe in the
first Column are Seconds, as they are entitled; and the Logi-
ftical Logarithm anſwering thoſe Degrees, Minutes, or Se-
conds, may be uſed in reſpect of Motion; and thoſe Fi-
gures in the Column under H, are the Hours and Minutes
of Time, and run to 24 Hours. Lastly, the Line of Figures
on the Head of the Table beginning with o, 60, 120, 180,
C c. to 3540, are the Minutes in fo many Hours as the Fi-
gure above reprefents; or they are the Seconds in fo many
Minutes as is contained in the Figure over them: Theſe are
uſeful to find the Curtated Diftance of a Planet from the Sun,
or Earth:
I fhall give in each an Example.
1. If the Sun Apparently move 2' 27" in an Hour, or 60'
Time, what will it move in 57′ 30″?
OPERATION.
H.
Min. Sec.
If
Give
What
Answer
I or
60
on L.L. 10.00000
2 27
57 30
2 21
8.61101
9.98152
8.59253 reject Radius.
Se.
The Dottrine of the Sphere.
299
Secondly, the fame Proportion wrought in reſpect of Time.
}
If one Hour
"
Co.Ar. 1.38021
Give
2 27
8.61101
What
57 30
8.60131
Answer
2 21
8.59253 reject Radius.
Lastly, Suppoſe the Mean Anomaly of Mercury be 8 S. 39
16' 50" what's the true Logarithm of his Diſtance from the
Sun ?
S. Deg.
is 4.563022
S8 4 is 4.564704
8 3
Logar. of
of{
Difference
1682
Now to find the Logiftical Logarithm of this Difference
1682, you muſt feek on the Head of the Table, and there un-
der 28 you will find 1680; then take the 2 in the firſt Co-
lumn, and the Common Angle is 966952, the Logiſtical Lo-
garithm of 1682.
If one Degree or
Give that Difference
What will
Answer 420 + 53
1
Now fay,
Min. Sec.
бо o LL 10.00000
1682
}
9.66952
9.44802
472
9.11754 reje& Radius.
16 50
That is, you find 420 at the top of the Table, and 52 in
the firft Column to the left Hand; which added together,
make 472. Now becaufe the Planet is in the Aſcending
Part of the Orbit, the Logarithm of its Diſtance from the
Sun increaſes; therefore you muſt add the proportional Part
472 to the Logarithm of the Anomaly 8 S. 30= 4.563022,
and the Sum is 4.563494, the Logarithm of Mercury's Di-
ftance from the Sun, anſwering to his Mean Anomaly 8 S.
3° 16′ 50″:. And thus you must proceed at all times in
finding the Logarithms of the Planets Diftances from the
Sun; becauſe the Logarithms and Equations of the Planets
are
300
The Doctrine of the Sphere.
are Calculated only to even Degrees of their Mean Anoma-
lies; but it generally falls out in finding the Planets Places
from Aftronomical Tables, that their Mean Anomalies at the
time fought, contain odd Minutes and Seconds, which muſt be
accounted for as above.
The further and general Uſe of theſe Logiſtical Loga-
rithms I have fhewn in Problem 39,, in finding the Paral-
laxes.
PROB. 62.
Shewing the Use of Street's Logistical Logarithms.
Theſe Logiſtical Logarithms were firſt Printed in Street's
Aftronomia Carolina; and run only to 60; but I have conti-
nued them to twice that Number, viz. to 120' or 2 Hours in
Time; they ferve expeditiously to find the proportionalPart in
any Aftronomical Calculation. In which, the Top-line of
large Figures which run from o to 119, may be taken as De-
grees, Minutes, or Seconds, either in Time or Motion, as
the Cafe requires; and the firft Column of every Page may
be either Minutes, Seconds, or Thirds; that is, if the Fi-
gures on the Head be taken as Degrees, then theſe in the
firft Column are Minutes; but if the other be Minutes, then
theſe are Seconds, &c. And the Figures in the fecond Line
beginning with o, and running to 3540, are the Minutes
in the Degree that ftand above them: Or they are the Se-
conds in thoſe Minures. And theſe are of uſe in finding the
proportional Part of large Differences in the Logarithms of
the Planet's Diftances from the Sun.
To find a Logiſtical Logarithm to any Degrees and Mi-
nutes, look for the Degree on the Head; and the Minutes
in the firft Column on the left Hand, and in the Common
Angle or Place of Meeting is the Logarithm fought.
As, fuppofe, you want the Logarithm of 200 15' or of
20' 15", you'll find it to be 4717; and the Logarithm of 24
Hours you'll find after the fame manner 3979; always re-
membring when you work by theſe L. L. to reject Radius
10000. And I generally uſe theſe for finding the Proportional
Parts, rather than Shakerley's; becauſe there are fewer Fi-
gures in them: But in my Practice I generally ufe a Sli-
ding-
The Doctrine of the Sphere..
30I
}
ding-Rule, which I recommend to the ingenious Student,
and fuch a one as is lately improv'd by my ſelf and Mr.
Fowler, Inftrument-Maker, and Sold by him at the Globe in
Sweeting's Alley by the Royal Exchange, London, as Advertiſed
in the Ninth Edition of Everard's Stereometry.
Example. Let the Mean Anomaly of Mars be 2S. 17° 25′
38", I demand the true Equation, and Logarithm of his Di-
ftance from the Sun at that time?
S.
0
OPERATION.
To{21}is Equation<
Differences
Q
11
10 4 00
10 7 29
29 Logar.
329)
{
5.195244
5.194606
636
Now for the Equation, fay, by the L.L.
1
"
If one Degree or
60
00 LL
Give the X
3 29
12341
What Anomaly
25 38
3693
16034.
•
Anfw. Propor, Part I 30
Now. becauſe the
the Equation is increafing, this Proportional Part 1 min. 30
muft be added to the Equation anſwering 2 Signs 17 degrees,
and it makes 10 deg. 5 min. 30 fec. the true Equation to be
fubtracted.
For the Logarithm, fay, by the L.L.
-y
"
If one Degree or
60
00 LL
Give X
7528
What Anomaly
3693
Anfw Propor. Part
11221.
636.
25-38
272
Here becauſe the
Planet is going towards his Perihelion; the Logarithm of his
Diſtance from the Sun decreafes; therefore the Proportional
Part 272 muft be fubtracted from the logarithm anfwering
to 2 S. 170, and there will remain 5 194972, the Logarithm
of the Diſtance of Mars from the Sun
And after the fame manner you muſt alwas find the Pla-
nets Equation and Logarithm-Distance from the Sun or Earth
anſwering to their Mean Anomalies; at the time when you
feek their Places.
What
..
The Doctrine of the Sphere.
૩૦૮
What other Varieties may fall in your way in ufing thefe
Logiftical Logarithms, may be known by the following Ex-
amples; by which you may fee when to add, and when to
fubtract the Logarithms, according as they are more or leſs
than 60 Minutes.
}add
14075
All Varieties of working by Street's Logistical Logarithms.
"
60 。 LL
If
Give 2 27
What 57.3°
Anſw. 2 21
ܐ
13890 add.
185
//
26 17 LL
60 co
3558 fub.
I 49
15.563 from
3 48
11978
If 60 0 LL
O
32 8 LL
2712 from
Give
2 24
What 75 9
13979 fub.
fub.
977
60 00
O
57 20
197 fub.
Anſw. 3 0
13002
47 4
2515
60 0 LL
O
36 21
104 53
63.34
2176 fub.
2426 from
34 19 LL 2426 from
60:00
79 1739 38:1801 fub.
138 34 59 17 625
}
250
60 0 LL
33 19 LL
2555
63 41
258 from
бо
O
58 48
88 fub.
60 8
Sada
add
62 24
170
108 17
2564
60 0 LL
88 21 LL
1680
40.45
1680
88-21
1680S
fub.
·60·00
2
O fub.
40 45
1688
60 60
60 0
88 22 LL
69 0 LL
60 о
70
670 fub.
69 17
1680 from
O
625 ſub.
46 0
1154 from
47 3
1055
53.40
484
16
j
1
The Doctrine of the Sphere.
+393
fub.
IOCCO
52
2
27
7421
11717
occo }
"}
O 16 2 LL
Six Digits
4 15
1° 35 25
#
5731 fub.
24
"61 0
o L.L. 3979 add
7782 from
1000
}à
11498}
15767
© 16 8 LL 5704
6º 00
4
IO
}à
IO O
Sum 4050 fub.
3732
24 0 L.L.C.A.6021
25 25
h.
14 38
D
16 40 LL
5563 add
10000 à
9 27
62 0
142 to
5 46
6128 Sadd
8027)
10176
Sum 5705 fub.
Omit an Unit to the left Hand.
58 26LL.CoAr.885
220 19
4295
23
O
O
3979 Sadd
-ad
dd
9 27
2494
> 16 40 LL
60 O O
38 20
13 48 0
Or thus:
5563 fub.
10000
1946 } à
6383
D14 52 L.L.C.A. 3941
S
13 53
6358
6°
O O
50 14
20
24
16 15
o L.L.
48 0
3979 from
969
2838
add
31 13
Sum 3807 fub.
+
21 24
"
1392 fub.
4477
Or thus:
92 42 L.L.
24 O
18907
3979
add
dd
82 41 Co.Ar. 8608
24 0
92 42 L.L. 18932
3979
3}add
Sum
5869 from
10000
add
82 41
772
4713
21 24
4477.
62 26
172
Or thus:
24 0 L.L.
3979
48 o Co. Ar.
03:
add
31 г3 Co.Ar
7162
f
19 40 L.L. 4844 fub.
24 O
7 3
add
13279 from
39792
9300
62 22
172
8 36
8435
Or
9304
The Doctrine of the Sphere.
1
#
3979 add
9300
"
Or thus:
19 40 Co.Ar. 5156
24 O
7 3
90.0
。 L.L. 1761 add
661
}add
1499 fub.
5.1.31
84 44
Sum
2422 from
836
8 36
8435
48 31
923
19 49 L.L.
4211
ASA
રે
47 51
26 22
3571
1982 add
90
Or thus:
0.L.L.
1761
51 31
661 Sadd
4554 fub.
84 44 Co.Ar.8500)
63 40
257
48 41
922
Or thus:
19 49 L.L. 4811
26 22 Co. Ar.6429
60 o L.L.
O
87 46
1651 fub.
47 51 Co.Ar. 017 Sadd
Sadd
32 44
2632 from
47 52
981
63 40
257
60 o L.L.
87 46
1651 from
84 57 L.L.
1910
6º 43 22
66 45 0
9506 Sadd
397 fub.
42 58
1450 fub.
6251
201
118 OLL.C.A.7063
11016from Z
100 20°
84 52
5º 12 9
10619
1..
71 12 L.L.
7 12
2 2
7457
9208 Sadd
14699)
nici
24652
?
0 12
40 5
}add
80 10 L.L. 1258
1752 add
4664)
7,674 Sum
I L.L. 537 Sub. às.
72 9
115 27 L.L. 2842 from
60 о
81 12
+
1315 ſub.
1527
111 48 L.L. 2702
53 40
60
0
28 49
120
484 add
O
3186
0 L.L. 30107
1197 Sadd
1107
20 30-
10 15
46 30
53
3
2
12952
add
46 30
2.52
13208
18 I
5224
.26170 Sum.
0 10
25633
2233 add
1506.
802
42 13
AN
AN
APPENDIX, &c.
A Table fhewing the Golden Number
and Epacts in both Accounts, to the Year
1800.
1
1
Rom.
Epact.
Gold. Numb.
Engl. Epact.
Anno Domini.
10/20 9 1700 1719 1738 | 1757 1776 1795
II
I 20 1701 1720 1739 1758 1777 1796
12 12 I 1 1702 1721 1740 1759 1778 1797
13 23 12 1703 1722 1741 1760 1779 1798
14 423|| 1704 1723 17421761 1780 1799
1515 4 1705 1724 1743 1762 1781
16 26 15 1706 1725 1744 1763 1782
17 7 26
1818
7
19 29 18
1707 1726 1745 1764 1783
|
1708 1727 1746 1765 | 1784
1709 1728 1747 1766 1785
1728|1747
111 29 1710 1729 1748 1767 1786
222 11 1711 1730 1749 1768 1787
1
II
33 22 1712 1731 1750 1769 1788
414 31713 1732 1751 1770 1789
5 25 14 1714 1733 1752 1771 1790
6 625 1715 1734 1753 1772 1791
717 6 1716 1735 1754 1773 1792
828171717 | 1736 1755 | 1774 1793
9| 9|28||1718|17371756|1775 | 1794
Seek the Year of our Lord, and right againſt it on the
left Hand you have the Roman and English Epacts, with
the Golden Number under their proper Titles for the Year
propofed.
+
X
306
An APPENDIX to the
i
་
A Table fhewing the Cycle of
the Sun, and Dominical Let-
ters in both Accounts for
100 Years.
Letter Rom.
Letter Engl.
Cycle Sun.
Anno Dom.
F|D C|| 1702 1728 1756| 1784
BAFE
I
2 E
3
4
C
BAG
5
6
G
F
C
E
B
a
DCAG
10
B
F
D
II A E
AGE
1701 1729 1757 1785
1702 1730 1758 1786
1703 1731 1759| 1787
1704 17321760 1788
1705 1733 1761 1789
17061734 1762 || 1790
1707 1735 1763 | 1791
1708 1736 1764 | 1792
1709 1737 1765 1793
1710 1738 1766| 1794
1711 1739 1767 1795
1712 1740 1768 1796
1713 1741 176 1797
12
G
D
13 F EC B
14
D
15
C
16
B
F
1714 17421770 1798
1715 17431771 1799
17
A GE D
18
F
C
19
E
20 D
BA
1716 17441772
1717 1745 1773
1718 1746 1774
1719 1747 1775
21 C BG F 1720 1748 1776
22
2222
23
24
AGF
E
1721 1749 1777
D 1722 1750 1778
25 E DBA
C
1723 1751 1779
2222
1724 1752 1780
26
C
G
1725 1753 1781
27
B
F
1726 1754 1782
172717551783
28 A E
Find the Year of our Lord, and againſt
it on the left Hand is the Cycle of the
Sun, and Sunday Letter in both Ac-
counts.
1
1
1
1
Doctrine of the SPHERE.
307
A Table fhewing the
Number of Direction
Gold. No.3 +/NO N∞
for ever.
A
BCDEFG
20 | 21 | 22
2∞
16 17 18
I
19/20
2
5
6
7
8
9
10 11
4
26 27 28 29
19 13 14 15
| |
30
24 25
16
17
5
6 7
8
2
34
26 27 21 22
23
24 25
12 13 14 15
16
10 11
33 34 35 29
30
31 32
9
19/20
19 20 21 22 23
24 18
ΙΟ 12 13 7 8 9
26 27 28 29
19 20 21 15
30 31 32
10 11
II
Year to the
1582 October 5
Add
10
1600
IO
1700
II
24 Febr. 1800
12
1900
13
2000 B 13
2100
14
2200
15
2300
16
2400 B 16
2500
17
2600
18
2700
19
2800 B 19
2900
20
3000
21
3100
22
3200 B 22
3300
23
3400
24.
13500
25
A Table to reduce the Julian
Gregorian.
II
12
16 17 1
13
5
6
7
8
9; 10
14
25 27
28 29
23 24 25
15
16 17
34
17
12 13 14 15
5 6 7 I 2
26 20 21 22 23 24 25
18 12 13 14 15 9 10 11
19 33 34 28/29 | 30 | 31 | 32
Enter this Table with Gol-
den Number on the Left-hand,
and Dominical Letter on the
Head for the given Year, and
in the place of Meeting is the
Number of Direction for the
faid Year.
J
X 2
308
An APPENDIX to the
after
geffima quagef.
123 +
2 2 2 2
aulaw
78
22mm
a
ΙΟ
II
A Table fhewing the moveable Feaſts in both
Accounts for ever.
NoDi.
Sundays Septua- Quin- First
Epiph, Sunday. | Sunday.
1 Jan. 18 Feb. 1 Feb. 4
Middle
Eaſter
Day of
Lent.
Lent Day,
Sunday.
March 1 Mar. 22
I
19
!
I
20
2
21
22
23
24
25
234
5
2.
23
6
3
24
4
7100
4
25
56 78
8
5
26
9
ΙΟ
I Į
27
78
28
29
26
12
9
30
27
10
13
ΙΟ
31
3
28
II
14
11 April 1
12
3
29
12
15
12
2
13
14
17
MMM m', m
M. H. NI2 2. 2 2 2 2 2 ala amm
19
20
21
22
23
ao
3
3 Feb.
3
3
4
4
4
33
30
13
16
13
31
14
17
14
34
I
4
4
4
9
24
4
10
1 2
3456
2 2 2 2
78
15
18
15
.5
16
19
16
6
17
18
78
20
17
21
18
19
22
19
789
20
23
20
IO
21
22
23
24
AWN I
24
25
26
27
2222
21
II
22
12
23
13
24
14
25
5
II
26
5
12
27
5
13
28
5
14
2222
25
28
26 Mar.
I
27/
28
2222
25
15
26
16
27
17
3
28
18
29
5
15 Mar.
30
/www
5
16
5
17
б
18
333
33
19
34
20
+234,56
I
4
29
19
5
30
20
31
21
7 April 1
22
୨
35
21
7
ΙΟ
234
23
24
41
25
Doctrine of the SPHERE.
309
Sunday.
I
Apr. 261 Apr. 30 May 10 May 17
27 May I
3
28
2
A Table fhewing the moveable Feafts in both
Accounts for ever:
|N.Di. Năm
Rogat. Afcent. White Trinity
Sundays
after
Advent
Day. Sunday. Sunday. Trinity. Sunday.
27
Nov. 29
II
12
1 2
18
27
30
19
27
Dec.
I
4
29
3
13
20
27
2
ا
30
4
14
21
27
3
May 1
1 2 3
15
22
26 Nov. 27
6
16
23
26
28
7
17
24
26
29
4
8
18
IO
9
19
II
12
ΙΟ
20
13
14
15
раз
II
21
2-2 2 2
25
26
30
26
26 Dec.
27
26
28
26
1 2 3
I
3
1:00
12
22
29
25
Nov. 27
13
23
30
25
28
10
14
24
31
25
29
「5.
II
15
25 June I
25
30
17
18
78
12
16
26
2
25
Dec.
I
13
17
27
3
25
2
19
14
18
28
4
25
3
22
w/www/
K N N N N N N N N
20
15
19
29
24
Nov. 27
16
20
30
24
28
17
21
31
7
24
29
18
22 June I
24
19
23
2
བ བ
8
24
30
24 Dec.
I
20
24
3
10
124
2
26
21
22
22
25
4
II
[24
3
26
12
23 Nov. 27
23
27
13
23
28
24
28
7
14
23
29
25
29
15
23
30
26
30
9
16
23 Dec.
I
32
27
3 I
ΙΟ
17
23
2
33
28 June
34
29
35
30
123
II
18
23
3
I 2
19
22 Nov. 27
13
20
22
28
310
An. APPENDIX to the
A Table fhewing the moveable
Terms for ever.
No.Di
Eafter Easter Trinity
Term Term Term
I rinity
Term
Begins. Ends. Begins. Ends.
April 8 May 41 May 22, June 10
1 2 3 4
78
. 9
5
23.
II
10
24
12
II
7
25
13
12
8
26
14
13
27
15:
14
ΙΟ
28
16
IS
ΙΣ
29
17
9
16
12
IO
17
13
mm
30
18
31
19
ΙΙ
1.18
14 June
1
20
12
19
15
2
21
?
1
13
20
16
3
22
14
21
17
4
23
15
16
56
1.22
18
5
24
23
19
125
1.7
24
20
26
7
·18
25
21
.∞
8
27
19
26
22
28
9
20
27
23
ΙΟ
29
21
28
24
I I
30
22
29
25
12 July
I
23
30
26
13
24||Máy I
27
14
2 3
"
2 2 2 A
25
2
28
IS
4
26
27
28
1
3
29
16
5
4
30
17
31.
18
www/wwwNI
29
6 June 1
30
8
1 2 3 til
19
7100
8
20-
9
21
IO
22
II
33
34
35
X10
III
0812
ΊΟ
23
12
6
24.
13
25
14
เ
ནས་པཱའི
•
Doctrine of the SPHERE.
311
1
A pril
1
A perpetual Table, fhewing what Day of
the Week begins any Month for ever.
Months A | B | C | D | E | F | G
January |Sunday/Saturd. Friday Thurf. Wedn. [Tuefd. (Mond.
Februar Wedn. Tuefd. Mond. Sunday Saturd Friday Thurf.
March Wedn. Tuefd. Mond. Sunday Saturd, Friday Thurf.
Saturd. Friday Thurf. Wedn. Tuefd. Mond. Sunday
May
Mond. Sunday Saturd. Friday Thurl. Wedn. Tuefd.
June Thurf Wedn. Tuefd. Mond. Sunday Saturd. Friday
July Saturd Friday Thurf. Wedn. Tueſd. Mond. Sunday
Auguft Tuefd. Mond. Sunday Saturd. Friday Thurf. Wedn.
Septem. Friday Thurf. Wedn. Tuefd. Mond. Sunday Saturd.
October Sunday Saturd. Friday Thurf. Wedn. Tueſd. Mond.
Novem. Wedn. Tuefd. Mond. Sunday Saturd. Friday Thurf.
Decemb. Friday Thurf. Wedn. Tuefd, Mond. Sunday Saturd.
A Table to find what Day of the Week
Day of the Month falls on for ever, in
both Accounts.
Julian M.I
Jan.October_A |_ B C D
any
Days of the Week.
Gregor.M
E
F
|
Fe.Mar.No. DE
F G
A B
G |April July
C Auguft
April July
GA
B C
D E
B C
May
D
E
FIG
June
E
F
Ꮐ A
B
C
Auguſt
CD
E F
GIA
F Sep. Decem.
A October
Ban. May
D Fe.Mar.No.
Sep. Decem. F
G
A B
C
D
E June
Days of the
Month.
I
2
3 14
5
6
7
8
9
10
ΙΟ
II
ΙΙ 12
13
14
15
16
17
18
19 20
21
22 23
24 25 26 | 27 | 28
29 30
31
312
A Table of the Number of Days, from any one Day in one Month, to the fame Day
in any other Month in a common Year. In Leap-Year after February, add a Day
more to the Numbers here fet down; it is of ufe when you find the
Arithmetically.
From January Februa. | March | April
Nodes
May June
une July August Septent. [October | Novem. [Decemb.
Feb. 31 Mar. 28 April 31 May 30 June 31 July 30 Aug. 31 Sep.
35 Jan. 31
To
3100. 30 Nov. 31 Dec.
Mar. 59 April 59 May 61 June 61 July 61 Au. 61 Sep. 62 Oct. 61 Nóv. 61 Dec. 61 Jan. 61 Feb. 62
April 90 May 89 June 92 July 91 Aug. 92 Sep. 92,O&. 92 Nov. 92 Dec. 91 Jan. 92 Feb. 92 Mar. 90
May 120 une 120 July 122 Aug.122 Sep. 123 Oct. 122 Nov. 123 Dec., 123 Jan. 122 Feb. 123 Mar.120 Apr. 121
June 151 July 150 Aug.153 Sep. 153 Oct. 153 Nov.153 Dec. 153 Jan. 153 Feb. 153 Mar.151 Apr. 151 May 151
July 181 Aug.181 Sep. 184 Oct. 183 Nov.184 Dec. 183 Jan. 184 Feb. 184 Mar.181 Apr. 182 May 181 June 182
Aug.212 Sep. 212 Oct. 214 Nov.214 Dec. 214 Jan. 214 Feb. 215 Mar.212 Apr. 212 May 212 June 212 July 212
Sep. 243 O&. 242 Nov.245 Dec. 244 Jan. 245 Feb. 245 Mar.243 Apr. 243 May 243 June 243 July 242 Aug.243
O&. 273 Nov.273 Dec. 275, Jan. 275 Feb. 276 Mar.273 Apr. 274 May 273 June 273 July 273 Aug.273 Sep. 274
Nov.304 Dec. 303 Jan. 306' Feb. 306 Mar.304 Apr. 304 May 304 June 304 July 303 Aug.304 Sep. 304 Oct. 304
Dec. 334 Jan. 334 Feb. 337 Mar.334 Apr. 335 May 334 June 335 July 334 Aug-334 Sep. 335 O&. 334 Nov.335
Jan. 365 Feb. 365| Mar.365 Apr.365| May 365| June 365| July 365| Aug.365|Sep. 365|O&. 365 Nov.365 Dec. 3651
Doctrine of the SPHERE.
313
A Table of the femi-diurnal Ark,
to every Degree of the fix firft
Signs of the Ecliptic Lat. Lon-
don.
II
H
H
H
5917
511
16
27
I 7
7
26
417
37
36
67
87
Н
69
6
61660
10 7
616
୪
127
1417
816
167
9 6
106
523
53 8
5 7 55 8
77 568
97
7
578
107 588
127 59 8
14
18 7 158
2017 1778
8
H∞∞∞∞∞∞ ∞o ∞o los ∞o ∞ ∞ ∞ l:o ∞s ∞o ∞o ∞o loo ∞ ∞ ∞ ∞
8
4
8
68
618
28
--
H
Ω mm
H
исо
1317
7 5015 59
137 496
58
12 7
47 6
127
46 6
54
II 7 45 6
52
117
43 6
50
10 7
42 6
48
10 7
416 46
97
40 6 44
97
7 39 6
42
116 2217 19
126 247
218
136 267
23
8
14 6 287 24
·8
15
6 307 268
166 32 7
28 8
2016
17 6 3417 30
18 6 36 7
19/6 387 33
4017 35
318
216
22 6
23 6
42 7 37
44 7 39
467 40
246 487 42
428
25 6 501.7 44
2616
2716
28 6 56 7 48 8
587498
5917 51
296
3016
5217 45
5417 47
8
.co oo co l∞o ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ 100 ∞ ∞ ∞ ∞ 100,00 00 ∞ ∞
8
8
aars W
7 8
7 8
88
88
8
8
27
8
∞∞
8 9 8 I7
ΙΟ
8
8
10 7
597
17
II7
8
II 7
87 386
7 38 6 40
87 37
77
38
366 36
34
32
30
7 7 35
67 33
67 31 6
ww
57 306 28
47 286 25
47 266 24
37 23
6 22
2016
20
17 6 18
07
17 15 6 16
13 6
14
5717 II 6 12
5617 9 6
16
8
127 557 7 6
8
127
5317 5
13 7 5217 3
137
1317
517
5016 59
314
An APPENDIX to the
H
6
015
/
15
584
594
25
564
57 4
35 5 5414 5514
415
5214 53
A Table of the femi-diurnal Ark,
to every Degree of the laft fix
Signs of the Ecliptic Lat. Lon-
don.
О
વા
H
m
H 1H H
474
10/5
474 115
484 135
484 145
*
I
1 2
4
600
↑ W
H
14 103
8
3
713
53
43
49 4
155 8
55 504 524
5 484 50. 4
75 464 48 4
8 5 444 46 4
95
33
494 175
ΙΟ
213
504
1815
I
3 504 1915
3
514
205 16
42 4 453
3
514
215
18
2468
5
40 4
43 3
58 3
52 4
225 20
11 5
384
413
57 3
524
23 5 22
12
5 36 4
393
563
3
534
245
24
13 5
34 4
37 3
55 3
534 255 26
14 5 324
363
54 3
54 4 275 28
15 5 304
34 3
543
544
295 30
16
5 284 323
53 13
5514
30
305 32
17 5
5 264
30 3
533
564
3215 34
18 5 244
293
52
523
5614
345 36
19 5 22 4
27 3
5 3
574 375 38
20
5
20 4
25|3
25 3
513
584 405 4:0
215
18 4
23 3
503
594 435 42
225 16 4
21
213
504
04 455 44
235 144
203
494
14 475 46
245 12 4
18 3
49 4
34 495 48
255 10 4
16 3
3 48
484
44 51 5 50
265
84 153
48
54 53 5 52
275
64 133
484
74 555 54
28 5 414
123
47484 575 56
295.
24
II3
474
914 59 5 58
3015 114
103 47 4
το 15
I I 6
1
Doctrine of the SPHERE.
315
A Table of the Latitudes, and Difference of Meridians from)
London, of fome of the most eminent Cities in the World.
Lat.
Dif. M
PLACES NAME S.
O
h
Aberdeen
57 N 60 S 7
Amſterdam
52 29
o A 21
Archangel
64
30
2 A 42
Arles in France
43 40
o A 27
1
Aix la Chapelle in Germany
50
48
。 A 44
Barbados, middle
13
243
S 53
Berlin
52
33
。 A 54
Boſton in New England
42 O
4 S 45
Cambridge
52
17
o A o
Conftantinople
43
о
2 A 7
Copenhagen
55
43 o A 50
Cracovia
50 10 1
A 18
Danzick
54 13
1
A 16
Douglaſs in Man-Iſle
54
40 S 18
Dublin
53
20
Edenburg
56
7
0 S28
O S 12
Elfinour
56
ט
0 A 50
Exeter in England
50
44 0
S 14
Ferrara in Italy
44
Fez in Barbary
33
10 О
540 A 47
0 S 24
Fort St. George
13
8
5
A 24
Gibraltar
36 30
0525
Glafcow
55 50
O S 17
Greenwich Obfervatory
51
281/0 A
Hamburg
53
57
o A 41
Hanover
52
35
0 A 40
Hoaignam in China
Jamaica, middle
ava Eaſt-end
Terufalem
Kelmar in Denmark
Kebeck, or Quebeck
Kendal in England
Leverpool
Lisbon
33 357 A 56
18 255 S 4
65 20 7 A 34
32N 302 A 22
56 40 I A 6
46 554 S 39
15 O SIO
54
53
20
38
42
O SIO
O S 37
London
51
32
O
Madrid
40
ΙΟ
0 S 13
Mancheſter
53
22
O S 10
Marfeilles
4.3
20
o A 22
Mofcow.
155 25
2 A 38
Y 2
3
316
An APPENDIX to the
1
A Table of the Latitude, and Difference of the Meridians from
London, of fome of the moft eminent Cities in the World.
Lat.
Dif. M
PLACES NAMES.
Naples
New York
Noremberg
North Cape
h
41N 8 1 A o
41
49
404 S 48
29 o A 49
71 25
1 A 28
Oftend in Flanders
51
Oxford
51
II o A 12
460 S 5
Ozala in Japan
35
5
7
Paris
48
51
o
A 32
A io
Petersburgh
бо 4
2
A 36
Port Mahon
39 45
O A 16
Quaqu in Guinea
4 16
O S 16
Revel in Finland
59 13
I A 36
Rome
41
Roterdam
$2
8
500 A 52
O A 16
Scanderoon
36
30
2 A 27
St. Chriftophers
17
Stockholm
59 30
304 S 6
I A 10
Suratt in India
21 30
4 A 48
Syracufa in Sicily
37
I
Á I
Tamefwaer in Hungary
47 30
i A 21
Tangier, Africa
35 55
0 S 25
Trent in Germany
47 20
。 Á 54
Toledo in Spain
39 54
54 0 Sis
Turin in Italy
Valentia in Spain
Venice in Italy
Vienna
Virginia Cape Charles
Warfaw in Poland
Uraniberg, Tycho Brahe's Obfervatory
Waterford in Ireland
44 500 A 29
39 45 OSI
360 A 52
45
36
48 14 I A I
37 47 4 S 57
55
540 A 52
52
14 I A 27
52
7 0 S31
Wiggan in England
53
340 SII
Wincheſter
!
SI 3
Woodſtock
51
53
Worceſter
52
14
9
Yarmouth in Wight
50 41
OS 7
York
54
o S 4
Zamora in Spain
41 45
о
o S 19
1
Doctrine of the SPHERE.
317
A TABLE of the true Time of the
Roman Eaſter, with the Difference in Days
from the Julian Account, until the Year
1800.
Difference.
Roman
Year
Eafter.
Difference.
Roman
Year
Eafter.
Difference.
Roman
Year
Eafter.
Difference.
Year
Roman
Eafter.
1700 Ap. 11 01728 Mar.2835,1756 Apr. 1871784 Apr. 11 0
1701 Mar.27351729 Apr. 17
1702 Apr. 16 01730 Apr. 9
1703 Apr. 801731 Mar.25
1704 Mar.23,35 1732 Apr. 13
1705 Apr. 12 71733 Apr. 5
1706 Apr. 4 01734 Apr.25
1707 Apr. 24 1735 Apr. 10
1708 Apr. 8 71736 Apr.
01757 Apr. 10 01785 Apr. 328
01758 Mar.26 35 1786 Apr. 23 o
351759 Apr. 15 71787 Apr. 8 o
71760 Apr. 601788 Mar.3028
01761 Mar.2235 1789 Apr. 19 o
01762 Apr. 11
1709 Mar.3135 1737 Apr.21
1710 Apr. 20 01738 Apr. 6
71739 Mar.29
1711 Apr. 5
1712 Mar. 27351740 Apr. 17
01741 Apr. 2
71763 Apr. 3
71790 Apr. 4 o
01791 Mar.27 28
1351764 Apr. 22
01792 Apr. 15 o
C1765 Apr. 7 71793 Mar.3135
71766 Mar. 3035 1794 Apr. 20 0
Mar.3035
1767 AP01795 Apr. 120
71769 Mar.2635 1797 Apr. 16
35
C1768 A
71796 Mar.2735
1713 Apr. 16
1714 Apr. 1
71742 Mar. 2535
1770 Apr. 15
01798 Apr. 8 o
1715 Apr. 21
71743 Apr. 14
C1771 Mar.31
71799 Mar.2435
1716Apr.
1716 Apr. 12
1717 Mar.28
01744 Apr. 5
1
01772 Apr. 19
7
35 1745 Apr. 18
71773 Apr. 11 o
1718 Apr. 17
71745 Apr. 10
01774 Apr. 328
1719 Apr 9
1720 Mar.3
1721 Apr. 13
01747 Apr. 2281775 Apr. 16 7
J
281748 Apr. 14
71776 Apr. 7 7
71749 Apr. 6
01777 Mar.3028
1722 Apr. 5 01750 Mar. 29
28 1778 Apr. 19 0
1723 Mar.28 28 1751 Apr. 11
71779 Apr. 4 7
1724 Apr. 16
01752 Apr. 2
71780 Mar.2635)
1725 Apr. 1
1726 Apr. 21
1727 Apr. 13
71753 Apr. 22
01754 Apr. 14
01781Apr. 15 of
01782 Apr. 70
1755 Mar.3035 1783 Mar.23 35
318
An APPENDIX to the
Sun's Declination North.
.. ·
A perpetual Table of the Sun's Rifing and
Setting for thefe Places. True Time.
Aberdeen
Amfterdam
Latitude 57° 6' Latitude 52° 29'
Sun Rifes Sun Sets. Sun Rifes. Sun Sets.
O
h
#h
#h
О
6
о
06
О 0.6 о 06
O
HOME TO N∞ alo bound
15
5.3 486
6 125
54 486
5 12 I
2
5 47 366
12 245
49 366
10 24 2
3
5 41 246
18.365
18.365
44 206
15 40 3
4
5 35 126
24 485
39 86
39 86
20 52 4
5
28 566
31
45 33 526
26 85
6
5
22 366
37 245
37 245
28 326
31 38
6
5
16 166
43 445
43 445
23 126
36 48 7
15
9 526
50 8/5
17 486
42 12 8
5
3. 2016
56 405
12 246
47 369
4 56 48,7
3 125
6 526
53 810
6
955
I 206
58 40|1 1
ΙΟ
II
12
13
14 50
50 4
43
14 36
14
4 29 207
15
4 22 8
7
4
14
7 127
16
17.
16 444
55 4417 4 1612
23 494 50 c7 ΙΟ 013
30 404 44
44 27 15 58/14
37 5214
14 487 45 124
18 3 59 248
19 3 51 248
20
222 22
38 167
21 4415
32 167 27 4416
-52 484 28 87 33 5217
0 364 19 5217 40 818
8 364 13 247 46 3619
6 487
25 324 о 8
21
22
3 43 48
3 34 288
25 248
16 56/4
53 12/20
О 021
34 363
53 018
7 022
23
3 15 568
14 1623
23
293
17 52/23 29
44 43 45 448
II 168 48 443 42 88
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
Sun's Declination South.
Doctrine of the SPHERE.
319
A perpetual Table of the Sun's Rifing and
Setting in thefe Places. True Time.
Archangel
Barbados
Latitude 64° 30 Latitude 130 24
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
Sun's
Declination North.
h
h
#h
,
OHNM+ no no ao m
6 O 06
О 6
О
16
о O O
I
5 51 486
8 245
58 486
I 12 I
2
5
43 126
16 485
58 86
3
34 486
25 125
57
57
86
I 52 2
2523
4
5 26 166
33 445
56
816
3 52 4
5
17 44 6
42 165
55 126
4 48 5
5
9 4
16
50 505
54 166
5 44
0 2016
59 495 53 166
6 44 7
8
4 51 287
8 3.25
52 206
7 40
9
4 42 247
17 365
51 206
8 409
ΙΟ
II
12
13
14
4
33 127
26 485
50 2016
9 40/10
4- 23 487
36 125
49 206
10 4011
4 14 87
45 525
48 246
1.1 3612
4 127 55 485
47 40,6
12 2013
53 568.
6 45
46 246
13 3614
15
3 43 168
16 445
45 206
14 40 15
16
3 32 128
27 485
44 206
15 4016
17
3
20 328
39 285
43 16,6
16 44 17
18
3 8 358
51 245
42 126
17 48
4818
19
2
55 89
4 525 41
86
18 4219
20
21
22
23
2 2
41 49
18 565
40 46
19 5620
25 409
34 205
39 46
20 56 21
12
8 249
51 365
51 365
37 566
22
4,22
I
49 32 10
10 285
36 48
36 48,5
23 12 23
23 291
37 28 10
22 325
36 166
23 4423 29
Sun. Sets. Sun Rifes. Sun Sets. Sun Rifes.
Sun's Declination South.
1
320
An APPENDIX to the
A perpetual Table of the Sun's Rifing and
Setting for thefe Places. True Time.
О
Boſton New-Eng-
Cambridge
land Lat. 420 39 Latitude 520 17/
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
h
a
16
#11: 1 "h
O 016
I 5 56 2016
о
་
2
3
Sun's Declination North.
IO
I I
12
Z13
14
15
OHN Mtro 7 ∞ a
0.6
О 0.6
3 405
55 206
5 40 I
5
52 246
7 365
49 246
10 36 2
5
48 566
11 45
47 366
12 24 3
4
5
45 126
14 485
39 166
20 44 4
5
5
41 286
18 325
34
34 06
26 05
5
37 486
22 125
28 486
31 12 6
5 34 06
26
016
26 0/5
05
23
23 286
36 327
5
30 166
29 445
18 86
41 528
9
5 26 286
33 325
12 446
47 16 9
5
22 366
37 245
37-245
8 206
52 4010
5
18 446
41 165
I 406
58 2011
16
17
19
62 auf 3 3
5
14 486
42 124
56 127
3 4812
10 52
52/6
49 81
14
51 327
92813
5
6 566
53
44
44 48 7
15 1214
5
2 2 16
a
57
8
2I
4 38 567
4
58 447
I 164 33 07
27
415
0/16
4
53 367
6 244
26 527
33 817
18
4 50 207
9 404
20 367
39 241 18
4 46 07
07
14
14 04
15 167
45 44 19
20
4 41 407
18 204
9 127
50 48 20
21
4 37 87
22 524
I 017
59
02 1
22
4 32 367
27 243
54 08
6 022
43
4 27 567 32 43
46 528
|
13 8/23
23 294 25 407
34 203
43 168
16 4423 29
Sun Sets. Sun Rifes. Sun Sets. SunRifes.
1
1
Sun's Declination South.
Doctrine of the SPHERE.
321
A perpetual Table of the Sun's Rifing and
Setting for theſe Places. True Time.
Conftantinople
Copenhagen
Latitude 430 Latitude 55° 43′
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
Sun's Declination North.
0
h
1
7.h
Wh
1
042344o no ao
6
о 016 о об O
016
O
O
I
5 56 166
3 445
54 86
5 32 I
52 326
7 285
48 166
II 44 2
5 48 486
II 125
42 246
17 36 3
5 45 46
14 565
36 286
23 32 4
324
5
5 41 2016
18 405
30 326
29 28 5
5 37 326 22 285
24 326
35 28
34 446 25 165
18 326
41 28 7
5
29 526 30 85
12 246
47 36 8
9
5 26 06 34 05
6 166
53 54 9
IO
I I
12
13
14
15
15
22 126 37 485
O 46
59 5610
5
18 126 41 484
53 447
6 1611
14 166 45 444
47 207
12 40 12
15
II 166 49 444
40 48 7
19 2013
5
6 126 53 484
34 527
25 814
15
2 86
57 524
27 247
32 36 15
16
4 57 56,7
2 44
20 287
39 3216
17
4 53 487
6 124
13 247
13 247
46 3617
18
4 49 287
10 324
6 87 53 52|18.
19
4 45 47
14 563
59
59 40 8 I 2019
20
4 41 407
19 203
50 568
9 420
21 4 36 87 23 523
7
23 523
42 568
17 421
22
4 31 287 28 323
34 368
34 368
25 24 22
23.
4 26
437 33 163
25 568
34
3.4 423
23 294 24 247.35 363 21 40 8
38 2023 29
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
:
Z
Sun's Declination South.
322
An APPENDIX to the
A perpetual Table of the Sun's Rifing and
Setting in thefe Places. True Time.
Cracovia
Dantzick
Latitude
500 10 Latitude 54° 13′
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
h
Wh ·
I
Sun's Declination North.
OH2Mtno too a
6 O 16
O 016 0 016
O о
5
55 126
4 485
54 286
5 32 I
5
50 246
9 365
48 526
II 82
3
5 45 366
14 245
43 206
16 40 3
4
5 40 446
19 165
37 446
22 16 4
5 35 566
24 45
32 86
36
27 52 5
6
5 31 46
28 565
28 565
26 2816
26 286
33 32
15
26 816
33 52
20 44/6
39 16
5
21 126
38 485
15 06
768
45 O
5 16 126
43 485
9 566
50 44
9
10
ΤΙ
12
511 126 48 485
3 2016
56 40/10
15
6 46 53 564
57 247
2 3611
5 0 566
59 44
51 247
8 36 12
13
4 5.5 447
4 164
54
45 167 14 44 13
14
4 51 247
8 364
39 07 21
97_21
014
15
16
4 45 47
4 39 367
1
14 564
32 447 27 1615
20 244
2
17
4 34 07
26 04
26 167 33 4416
19 167
19 167 40 24 17
18
4 28 167
31 444
12 527 47 818
19 4
22 2817
37 324
5 527 54 8/19
20
4
16 327
43 283
58 408
I 2020
21 14
10 247
49 363 51 168
8 44/21
22
14
4 87
55 523
43 368
16 2422
23 3
57 368
2 243 35.448
24 1623
23 293
54-288
5 323
31 448
28 16 23 29
Sun Sets.'Sun Rifes. Sun Sets. Sun Rifes.
L
Sun's Declination South.
Doctrine of the SPHERE.
323
i
A perpetual Table of the Sun's Rifing and
Setting for theſe Places. True Time.
Douglafs
Dublin
Latitude 54° 4' Latitude 53° 20′
Sun Rifes Sun Sets. Sun Rifes. Sun Sets
་
O
h
Wh
1
Wh
6
О 016
O 016 о 016
О
O
Sun's Declination North.
7
H234ino No a
I
5 54 286
5 325
54 4016
5 20 I
5 49 06
ΙΙ 95
49 166
10 44 2
5 43 246
16 365
43 566
16 4 3
5 37 52 26
22 8
5
38 326
21 28 4
5 32 166
27 445
33 126
26 48 5
5 26 406
33 2015
27 326
32 28 6
5
21 06
39 05
21 486
38 12 7
5 15 166
44 445
44 445
16 326
43 28 8
5
9.326
50 285
10 526
49 89
ΙΟ
II
12
13
5
3 406
56 20 5
5 126
4 57 487
2 125
0.286
54 48 10
59 3211
48/10
14 51 487
8 124 53 407
6 2012
4 45 447
14 164 47 207
12 4013
14
4 19 32,7
20 284 41 167
18 4414
15
4 33 127
26 484
4
35 397
35 367
24 2415
16
4 26 48,7
33 124
29 87
30 5216
17
4 20 117
39 484
22 527
37 817
18
4 13 287
46 324
16 187
43 3218
19
14 6 327
53 28 4 9 127
50 48/19
1
20
3 59 2818
0 324 2 567
57 420
21
3
52 48
7 503 56 128
3 4821
22 13
44 288
15 323
50 48
95622
23
36 358
23 243
43 32
16 2823
23 293
32 408
27 203
37 128
22 4823 291
Sun Sets. Sun Rifes. Sun Sets Sun Rifes.
Z 2
Sun's Declination South.
An APPENDIX to the
324
A perpetual Table of the Sun's Rifing and
Setting for thefe Places. True Time.
Edenburg
Latitude
T
560 7 Latitude 130.81|
Fort St. George
Sun's Declination North.
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
Sun's Declination South.
h
h
h
h
O
5 О
O
O
О
IO
II
12
13
14
15
16
17
18
Ham & no noo ao H&M tron∞
I
5 54 416
5 565
59 46
O 56 I
2
5 48 46
11 565
58
816
I 52 2
5 42 46
17 565
57 126
2 48 3
4
5
36
86
23 525
56 166
3 44 4
5
30 46
29 565
29 565
55 206
55 206
4 40 5
5
24 06
36 05
54 246
536 6
5
18 486
41 125
53 246
6 367
II 406 48 205
52 286
732 8
9
S
5 286
54 325
51 326
8 289
4
59 87
0 525
50 366
9 24 10
4 52 407
7 205
49 366
10 2411
4 46 127
13 4815
48 366
II 2412
4 39 167
20 245
47 406
12 2013
4 32 487
27 125
46 406
13 2014
4
25 567 34 45
45 406
14 2015
4
18 527 41 85
44 406
15 2016
14
II 407 48 205
43 406
16 2017
4
4 167
55 445
42 366
17 2418
19
3 56 408
3 2015
41 366
18 2419
20
3 48 448
11 165
40 325
19 2820
21
3 41 368
19 245
39 286
20 3221
22
123
2 33
3
32 48
27 565
38 206
21 40 22
4022
3 23 128
36 485
36 485
37 165
37 165
22 4423
23 293
18 488
18 488
41 125
36 446
23 16 23 29
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
1
:
Doctrine of the SPHERE.
325
A perpetual Table of the Sun's Rifing and
Setting for theſe Places. True Time.
Gibraltar
Hamburg
Latitude 360 30' Latitude 53° 57'
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
06
#h
0 016 O 016
Sun's Declination North.
o. I h
#h
"h 1 Wo
6
ㅎㅎ
​9
IO
II
12
13
14
15
16
17
18
19
OHNM &nonoo alo 1 2 3 4 5 6 7∞ a
O O O
4
ཡས ་ ་
5 5
46
2 565
54 3216
5 28 I
5 54 46
5 565 49 016
II O 2
5 51 46
8 565
43 286
16 32 3
5 48 86
11 525
37 566
22 4 4
5
45 126
14 485
32 246
27 36 5
5
42 126
17 485
26 486
33 12 6
7
5
39 86
20 5215
21
S6
38 52 7
5
36
36 86 23 525
15 286
44 32 8
5 33
86 26 52/5
9 446
50 16 9
5 30 06 30 015 3 566
5 26 566 33 44 58 4/7
56 410
1 5011
5 23 486 36 124
52 47
7 5612
5
20 406 39 204 46 07
14 013
17 326 42 284
39 527
20 8/14
5
14 166 45 444
33 367
26 2415
15
II 06 49 04
27 127
32 4816
15
7 446 52 164
20 407
39 2017
15
4 206 55 404
13 567
46 418
5
0 566 59 44 7 47
52 56 19
20
4 57 327
2 284
O 018
о
20
21 4 54 017
6 03
52 408
7 2021
22
4 50 247
9 363
45 88
14 5222
23
4 46 487
13 123
37 208
22 4023
23 294 45 07 15 03
33 248
26 36 23 29
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
Sun's Declination South.
1
326
An APPENDIX to the
A perpetual Table of the Sun's Rifing and
Setting for thefe Places. True Time.
Hanover
Jamaica
Latitude 52° 35'Latitude 180 25'
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets
Sun's Declination North.
Q
h
Wh
h
h
J
о
I
2
3
4
7
9
IO
II
12
13
14
15
16
17
18
19
O mot ~ Mtro t∞ al OH Nm+ no no a
6
O
о 016
O O
5
54 486
5 125
58 40/6
I 20 I
5
49 326
10 285
57 246
2 36 2
5
44 206
15 405
56 86
3 52 3
5 39
39
46
20 565 54 526
5 84
5
34 246
25 365.53 286
6 32 5
5 28 246
31 365 52 06
8
06
5
23 46
36 505
50 44/6
9 16
5
17 406
42 205
49 206
10 40 8
5
12 166
47 445
48 06
12 09
78 9
5
6 446
53 165
46 406
13 2010
5
I 86 58 525
45 206
15 4011
4
55 287
4 325
43 486
16 1212
4 49 447 10 165
42 326
17 2813
4 43 567 16 45
41 56
19 O 14
4 31
MM 2
4 38
38 97 22 05
39 496
20 2015
567 28 45
38 86
21 5216
4 25 487 34 125
36 446
23 16
17
4 19 287 40 325
7
40 325
35 166
24 4418
4 13
07 47 05
33 486
26 12 19
20 4 6 247
53 365
32 126
27 48 20
21 3 59 328
0 285
30 406
29 2021
22 3 52 288
7 325
29 126
30 48 22
23 3 45 128
14 485
14 485
27 286
27 286
32 32 23
23 293 41 368
18 245
26 446
33 16 23 29
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
Sun's Declination South.
Doctrine of the SPHERE.
327
1
A perpetual Table of the Sun's Riſing and
Setting for theſe Places. True Time.
Jerufalem
Kelmar
Latitude 32° 30' Latitude 569 40
Sun Rifes. Sun Sets.
Sun Sets.
Sun's Declination South.
O
I h
!
"h
Wh
О
16
O 016
O
a
O 16
O O
Ι
5
57 286
2 325
53 56
64 I
2
5
54 526.
5
85
47 48 6
12 12 2
3
5
52 206
7 405
41 446
18 16 3
4
5
49 486
IO 125
35 366
24 24 4
5 47 126
12 485
29 286
30 32 5
5
44 406
15 205
23 4
6
36 56 6
5 42 06
18 05
17 0 06
43 이 ​7
5 39 386
20 225
10
40 6
49 20 S
5 36 526
23 85
4 1616
55 44 9
ΙΟ
5
34 126
25 484
25 484
57 487
2 1210
I I
5
31 356
28 244
28 244
51 167
8 4411
12
13
14
15
16
5
28 526
31
84
4 44 367
15 2412
26 06 34 04
37 487
22 12 13
23 246 36 364
30 567
29 414
15
20 406 39 204
39 204
23 5217
36 815
5
17 566
42 44
16 367
43 24 16
17
5
15 46
44 564
9 127
50 48 17
18
15
12 126
47 48 4
1 367
58 24 18
19
5
9 2016
50 403
53 448
6 1619
222 22
20
5
6 246
53 363
45 368
45 368
14 24 20
21
5
3 286
58 323
323
37 128
22 4821
22
5
I 206
59 493
28 248
31 36 22
23
4 57 127 2 483
19 168
19 168 40 44 23
23 294
55 447
4 15
3
14 368
45 24 23 29
Sun's Declination North.
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
}
I
A
An APPENDIX to the
328
A perpetual Table of the Sun's Rifing and
Setting for thefe Places. True 'Time.
O
Leverpool
Lisbon
Latitude 53° 22' Latitude 38° 45'
Sun Rifes. Sun Sets.
h
Sun's Declination North.
Sun Rife
'h
h
2
3
4
9
IO
ΙΙ
12
13
14
16
17
OHNm+ mo noo ao HNM+MON
6 0° 06
о об
О
ㅎㅎ
​이 ​0
I
5
54 366
5 245
56 486
3 12 I
5 .49 166
10 445
53 366
6 24 2
5 43 526
16 85
50 206
940 3
5 38 246
21 365
47 486
12 52 4
5.
5 33 06
06
27 05
43 566
16 4
5 27 286 32 325
41 366
19 24 6
15
22 06 38 05
37 246
22 36
7
5
16 246 43 365
34 86.25 52 8
15
10 486 49 125
31 486 29 12 9
5 5 86
54 525
27 286
32 32 10
4 59 247
0 365
24 246
35 3611
4 53 327
6 285
20 446
39 1612
4 47 407
12 205
17 166
42 44 13
4
41 407
18 205
13 386
46 22 14
15 4
35 327
24 285
IO 206 49 40 15
4 29 167
30 445
6486.
6 486 53 12 16
4 22 527
37 85
3 125
46 4817
18 4 16 207
43 404
59 327
0 28 18
19 4
9.407
50 204.
55 487
4 12 19
20
14 2 487
57 124 52 47
7 5620
222
21 3. 55 408
3 48 248
22
4 204 48 4/7
11 5621
11 364 44 167
15 4422
23
3 40 438
23 293 37 08
19 164 40 207
19 40 23
23 04
38 447
21
16 23 29
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
Sun's Declination South.
Doctrine of the SPHERE.
$29
A perpetual Table of the Sun's Rifing and
Setting in thefe Places. True Time.
London
Madrid
Latitude 510 32' Latitude 400 10
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
#h
О 06 О 016
יד
Ꮒ
01234567∞ ao H&M 4 SO D∞ O!! O H2MM
12
222
0 06
O
О
I
5 55 166
444/5
56 366
3 24 I
5 49 566 10 45
53 166
6 44 2
5 44 526 15 85
49 526
ΙΟ 83
39 486
20 125
46 286
13 32 4
1
5
34 446
25 165
43 46
16 56 5
5 29 3616
30 245
39 406
20 20 6
5 24 246
35 365
36 86
23 52 7
8
19 166
40 445.
32 486
27 12 8
9
14 06 46 of
015
29 166
30 44.9.
8 446 6
51 165
25: 446
34 1610
3 206 56 405
II 5
12
13
14
15
16
17
18
19
•
4
57 507 2 45
52 287 7325
46 527 13 8
4 41 87 18 525
4 35 247 24 365
4 29 287 30 325
4 23 287 36 324
7 446
4 016
Ο
52 1615
56 016
0 126 59 48 17
22 166
37 4411
1840
18. 406
41 2012
15 06
45 013
11 246
II 246 48 36/14
Sun's Declination, South..
$20
21
22
123
23 293 47 248 12 364 33 567
Sun Sets Sun Rifes Sun Sets. Sun Rifes.
56 2017
3 40/18
4 17 167 42 441
10 567 49 44
52 247 7.3619
48 247
11 3620
4 287 55 324
57 443 2 164
44 247
15 3621
40 167
19 44 22
350 528 9 84
36 407
23 56,23
26 423. 29
Sun's Declination North.
A T
}
330
An APPENDIX to the
A perpetual Table of the Sun's Rifing and
Setting for theſe Places. True Time.
Mofcow
New-York
Latitude 55° 25' Latitude 419 40
Sun Rifes. Sun Sets Sun Rifes. Sun Sets.
•
Th
Wh
"h
•
01234 ~o no a o
0 06
0
016
О 016
O
O
I
15 54 126
5 485
56 246
3 36 I
1
5
48 246
11 365
52 526
7 8 2
5
42 16,6
17 445
49 2016
10 40 3
5 36 446
23 165
45 446
14 16
4
5
8
30 526
29
5
42
816
5
24 566 35
566 35 4/5
5
38
38 326
17 525
21 28
7
5
19 06
5
13 06 47
96
41
34 566
25 4
47
31 166
28 44 8
9
15
6526
526 53
527 366
32 24 9
0 446
59 16
4 48 87
41 487
11 525
18 12
456 7∞ a
4
35 27
24 485
4 28 327
31 28 15
14
21 447
4 14 447
38 16
45 16 4
14
7 327
52 28
4
O
8
7
59 524
IO
II
12
13
14
IS
16
17
18
19
4 54 327 5 285
6 36 4,10
20 126 39 48 11
16 24 6 43 3612
12 326 47
47 28
8 446 51 16
13
14
6 06 54 015
226 59 38 16
3 28
17
5
23 56
5610
125
Sun's Declination South.
56 327
4
52 587 7
218
48 367
II 24 19
20
3 52 328
7 28
4
46 47
13 56 20
21
22
3
44 498
15 2014
40 87
19 5621
23
2 33
3 36 288
23 324
35 447
24 1622
3 28 08
32 04
31 24
7
28 36 23
36/23
23 293 23 448 36 164
29
O 7. 31 023 29
Sun's Declination North.
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
ין
Doctrine of the SPHERE.
331
A perpetual Table of the Sun's Rifing and
Setting for theſe Places. True Time,
North-Cape
Oxford
Latitude 71º 25' | Latitude 510 46′
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
Sun's Declination South.
h
O
6
// h
06 0 06
Wh
ยา
1
0 016
4
7
9
II
12
13
14
15
16
1.7
18
~~~+ no no ao SST SEA
2 21
H H
5 48 86
11 525
54 566
5'
I
2 5 36 86
23 525
49 526
10 82
3 5
24 86 35 525
44 44
446
15 16 3
5
12 06
48 05
39 406
20 20 4
1
4 59 40:7
205
34 326
25 22 5
4 47 87
87
12 525
29 206
30 40
6
4 34 207
25 405
24 86
35 52 7
4 21 127
38 485
18 566
41 4
14 7 367
52 24/5
13 36/6
46 249.
3 53 288
6 325
7 406
52 2010
3 38 448
21 165
2 526
57 811
3
23 88
36 524
57 247
2 3612
3 6 328
53 284
51 527
8 813
2 48 329
II 284
46 127
13 4814
28 369 31 244
40 287
19 3215
5 569 54 44
34 407
25 2016
I
38 2010 21 404
28 407
31 2017
O
3611
59 36 11 0 244
0 244
22 447
37 1618
19 S. Se.
not
S. Ri. not 4
16 207
43 40 19
20
21
22
123
no
All
Night.
Day and
All Night
and no Day.
4
9 567
50 420
4
3 327
56 2821
23 29
3 46 88
10 2423
13 5223 29
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
3 56
56 368
49 368
3 24 22
Sun's Declination North.
F
A a 2
332
An APPENDIX to the
A perpetual Table of the Sun's Rifing and
Setting for thefe Places. True Time.
Paris
Rome
Latitude 489 51 Latitude 41° 50
Sun Rifes. Sun Sets Sun Rifes. Sun Sets.
6 O 06
2
Sun's Declination North..
ΙΟ
II
12
Z13
14
is
16
17
18
OHNM+iro 7∞
alo-23 tronco
отоб
0 06
Ο
I
5 55 286
4 325
56 246
3 36 I
5 50 326
9. 28
5
52 406
7 20 2
3:
4
55
5
46 166 13 445 46 486
13 32 3
41 405
18 205
45 366
14 24 4
5
37 05
23 05
42 46
17 56 5
5 32 2016
27 405
38 246
21 36 6
7.
5
27 406
32 205
34 486
25 12 7.
15
22 566 37 45
31 86
28 52 8
15
18 126
41 485
27 246
32 36 9
15
13 286
46 325
23 406
36 2010
8 366 51 245
51 245
19 566
40 411
5 3 401
406
56 205
16 86
43 5212.
4 58 447
1 165
12 206
47 4013
14 53 407
6 2015
8 246
51 3614
Sun's Declination South.
4
48 367
11 445
4 286
55 32 15
4 43 247
16 365
0 326
59 2816
4 38 47
21 564
56 287
3 3217
4 32 407
27 204
52 247
7 318
19
4 27 127
32 484
48 127
11 48119
།
{
20
4 21 327 38 284
43 567
16 420
21
4 15 487.44 124
39 367
20 2421
[22
4
9 527 50 84
35 127
24 48 22.
23
14
3 447 56 164
30 407
29: 2023
23 294
0 487 59 124
28 287 3 3223.29
7
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes
1
ر کیا
Doctrine of the SPHERE.
333
1
A perpetual Table of the Sun's Rifing and
Setting for thefe Places. True Time.
St. Christophers
Stockholm
Latitude 179 30' Latitude 599 26'
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets
Sun's Declination North.
oh
4h
Wh
་་
Wh
О 16
о 016
,0 016 O
О
I 5
58 446
1 165
53 126
6 48 I
2
5
57 286
2 325
46 286
13 32 2
3
5
55 126
3 485
39 496
20 20 3
4
5 55 06
5 05
32 386
27 12 4
46 7∞ alo
5
53 406
6 205
25 566
34
34 45
5 52 246
7 365
19 06
41 06
5
51 86
8 525
13 86
46 527
5
49 526 10 55
4 566
55 4 8
5 48 326
11 284
57 487
2 12 9
2 mthno 700 alo
222 2
ΙΟ
II
12
13
14
15
16
17
18
5 47 166
12 444
50 327
9 28 10
5
45 566
14 44
43 167
16 4411
5 44 356
15 244
35 407
24 2012
5 43 166
16 444
27 567
32 413
5
41 566
18 44
20 87
39 5214
5 40 366 19 244
12 47
47 515
5 39 196 20 444
3 527
56 816
5
37 26 22
22 83
55 168
4 44 17
5 36 286 23 323
46 288
13 3218
19:
5 35 46 24 563
37 208
22 4019
20
5 33 356
26 243
27 488
32 1220
21
5 32 126 27 483
17 528
42 821
22
5 31 446 29 163
7 208
52 4022
23
5 29 166 30 442
56 129
3 48 23
23
295 28 3:26 31 282
50 4.89
9 24 23 29
1
Sun Sets. Sun Rifes-Sun Sets. Sun Rifes.
Sun's Declination South.
334
A APPENDIX to the
་
A perpetual Table of the Sun's Riſing and
Setting for theſe Places. True Time.
Vienna
Virginia
Latitude 48° 14' Latitude 37° 47′
Sun Rifes. Sun Sets. Sun Rifes. Sun Sets.
SunRifes.
h
"h
#h
Wh
3
Sun's Declination North.
9
10
II
12
Z13
OMNMt no to alO-
6 O 06
O
6
O
c16
O
I
5
$5 326
4 285
56 456
3 12
2.
5
S1 46
8 565
53 4016
6 20 2
46 486
13 125
50 246
4
5 42 86
17 525
47 166
9363
12 44 4
5
37 326
22 285
44 3 616
15 245
5
33 46
26 565 41
816
18 52 6.
7
5
28 366
31 245
38 06
22
5
24 46
35 565 35
35 016
5 19 286
40 325
31 566
22
25
28
MÓ ACO a
9
5
14 286
45 325
27 526
32
810
5
II
06 49 05
24 366
35 2411
5
5 46
54 565
21 126
38 48 12
5
0 46
59 565
17 486
42 1213
14
4 55 167
4 445
14 246
45 3614
IS
4 50 127
9 485
10 566
49 415
16
4 45 247
14 365
7 326
52 2816
17
4
40 167
19 445
3 526
56 817
18
4
34 407
25 205.
0 166
59 44 18
19
4 28 127
31 484
57 07
3 019
2 2 2 2 2
120
4 23 487
36 124
52 527
7 820
21
4 18 127 41 484
49 247
10 36 21
22
23
14
12 247 47 364
46 127
13 48 22
4
6 327 53 284
42 527
17
823
23
294
3 367
56 244
39 247
20
36 23 29
Sun Sets.'Sun Rifes. Sun Sets. Sun Rifes.
Sun's Declination South.
}
Doctrine of the SPHERE.
335
f
A perpetual Table of the Sun's Rifing and
Setting for theſe Places.
Warfaw
True Time.
York
Latitude 52° 14' Latitude 54° o
Sun Rifes Sun Sets. Sun Rifes. Sun Sets
Sun's Declination North.
に
​· Wh
Wh
0
O O O
HAM + Noo ao = ~ MILO O 2 2 2 2 3
I
5
54 526
5 85
54.326
5 28 1
2
5 49 406
10 205
49 Có
II 이 ​2
3
5
44 286
15 325
43 286
16 32 3
4
5 39 166
20 445
37 566
22
44
5 34 46
25 565
32 2016
27 40 5
5 28 486
31 1215
26 446
33 16 6
7
5 23 286
36 325
21 86
38 52 7
8
5 18 126
41 485
15 246
44 36 8
5 12 4816
47 125
9 406
50 209
14
ΙΟ 15
II
12
13
14
15
16
7 2416
15 1 526
14 56 2017
52 305.
S..
3486
56 12 10
I
58 84
57 567
2
411
3 404 51 567
50 407
9 204 45 567
8 412
14 413
4 44 207
15 404
39 447
20 1014
4 39 47
20 564
33 287
26 32 15
4 33 87
26 524
27 07
33 016
17
4 27 47 32 564
20 287
39 3217
18
14
20 487 39 124
13 447
46 16 18
19
4 14 287 45 324
6 527
53 819
20
14
7 567
52 43
59 448
O 16/20
21
4 1 127
58 483
52 248
7 3021
22
3 54 16,8
5 443
44 528
15
822
23.
3 47 88
12 523
37 98
23 023
23
293
43 368
16 243
33 88 26 5223 29
Sun Sets. Sun Rifes. Sun Sets. Sun Rifes.
Sun's Declination South.
336
An APPENDIX to the
A
TABLE
Of all the Eclipfes, both Viſible and
Invifible of the Sun and Moon,
Moon, that will
happen 'from the Year 1728, to the Year
1764, under the Meridian of London.
Year
}
1
Long
A
Lat. Digits
Months and Days Long. Lat.
Dh
1728 Febru. 13 19 14 m
Lumin.
Digits Vifible
19
Ꮕ
or
Invifible
6
00 36
N D9
Aug.
Febrù. 28 8 0
85
*
20 370 45
26 390 35
N A
NA
S D
Aug.
J
1729 Jan.
17 18 39 m
13 Set. Ecl.
Invifible O
Invifible
Invifible O
Invifible O
Febru. 2 8 44
Febru 16 933
July,
14: 13 58
July
28 13 20
1730 Jan.
7 637 v
Jan.
22 15 34
17 Viſible
Invifible O
47
Vifible)
Invifible O
53 Vifible
Invifible
t
July
July
Dec.
1731 June
23 13 om 11 290 46 S A
31526
18 4.26 m
27 22 27 1
8 13 557
22 17:57
1 22 45
2 12
I I Ο ㅇ ​이
​9 271 20 SD
25 150 6 S A19
9 221 23-NA
2 511 5 N'D
16 150 8 N A18
1508
28 330 39 SD
14 100 46 S A2
22 150 15 S A6
6 70 53 N A
17 450 2 NA
28 40-55 SDI
11 350 19 S A
6 190 43 ND
6 550 44-N A
A)….
7 Rif.Ecli. O
Invifible>
Invisible O
52 Visible
Inviſible
Invifible
Invifible. O
Invifible >
June
Dec.
I
Dec.
17 13 5
1732 May
June
Nov.
28
2 12
17 150 14 S D
6 4 53 m
25 381 21 S D
Nov.
Dec:
20 9
9 59.10
5 21 12
25
20. 3. N D 20 48 Vifible >
480 32 NAO
1
I Vifible
I 61 2 SAI
Invifible
Invifible O
Doctrine of the SPHERE.
337
The TABLE of Eclipfes continued.
Months and Days Long.
Dh!
Year
1733 May
May
2
17
Octob. 26
Lat.
Lat.
19
་
>Digits Visible
6 378 22 520 8 ND
6501
4 44m 14 170 43 S D
O
or
Invifible
20 Vifible
8 170 32 N A3
25 Rife Ecl.
Invifible
Invifible
Invifible
Nov. IO I 68 29 160 39 S A
1734 April 21 22 88 12 390 3 ND
1735
Octob. 15 6 15m
March 26 22 41
April 11 10
11 10 588
Sept. 20 13.33
O&ob. 4 14 40
I 2 36
March 15 11 52
March 30 19 2
Aug. 24 21 30m
Sept. 8 14 24
1736 March
1737
March
5
3 40 4 S D
17 80 42 S D
2 70 40 S A
8 180 39 ND
8 180 39 N D5
22 120 37
22 120 37 N A
22 211 17 ND
6 360 IN A 21
21 381 23 SA
13 01 17 S D
27 1901 S A 20
11 401 19 NA
Sept. 23 5 44
Febru. 18 3 4X II
11 90-39 N D10
О 0 35m 25 590 43 N A
Aug. 14 12 58/1m 2 390 33 SD
28 15 44 16 220 42 S A4
7 5 57 29 590 I SA
1738 Febru. 7 5 57
Aug.
Aug.
3 23 3
1739 Jan.
13 10 54
Jan.
27 16 4
July
July
Dec.
1740 Jan.
June
June
Dec.
Jan.
Dec.
1741 June
Nov.
MW
9 4 18v
24 4 23
18 20 49 v
2 10 25
17 8 7 M
7
12 14 20
27 21 23
7 10 557
21 11 49
11
1 21 46
26 17 44
22 140 21 S D4
4 380 37 S D6
19 20 42 S A
27 160 26 N A
11 390 12 N D7
8 190 27 N D2
23 100 1 N A 20
8 201 24 SA
2391 IS D
17 100 19 S A
A/20
27 270 39
27 270 39 ND
11 460 40 N A5
22 160 49 S D
16 160 1S A
B b
Lumin.
Invifible O
Invifible
Invifible
37 Vifible
Invifible
Invifible
45 Viſible
Invifible
Inviſible
32 Vifible
Invifible O
55 Viſible
Inviſible
Invifible O
34 Viſible
Invifible
8l Vifible
22 Vifible
Invifible
Invisible
6 Viſible
10 Viſible
29 Vifible
Invifible
Invifible
Invifible >
Invifible Q
49 Viſible
Inviſible
Invifible
338
An APPENDIX to the
Year
The TABLE of Eclipfes continued.
Months and Days. Long. Lat.
Digits Vifible
or
Dh!
O 10
10
Invifible
Lumin.
1742 May
7.23 38 m 28 110 49 N D
Invifible
May
22 12 42
12
80 28 N A
Invifible
Nov.
1 0 308
19 580 43 S D
Invifible
Nov.
15 18 157
4 530 39
SA
Invifible
1743 April
12 21 478
3 45 1.22 SD
Invifiblé
1746 Feb.
Aug. 19 12
April 27 3 21
May 12 5 54 II
Octob. 6 2 43
Octob. 21 15 358
Nov. 4 18 27 m
1744 April I 951
April 15 8 32m
Sept. 24 13 18
Octob. 10 0 48
1745 March 21 14 56
Sept. 14 5 2
24 3 44/10
2 40 36 NA
23 460 33 N
ND
9 17.0
9 170 1 S D21
23 301 17 S A
23 170 39 S D
6 510 35 S A8
13 30 44 N
28 250 40 NA
12 28,0 2 NA
16 590 37 N D8
March 10 14 54 I 220 42 N A
57 180 39 S D6
22 22 160 43 S A
52 NN
20 321 21 SD
Sept. 3 21
1747 Januar. 29 2
Feb. 13 17 2m 6 180
27 17 18
6 180 5 SA 19
Feb.
20 181 21 N A
July 25 20 508
Aug. 8 20 52
13 191 9 ND
26 470 4 N A
Inviſible
Invifible
42 Viſible
Invisible O
Invifible O
Invifible>
Invifible
17.320 7 N
Invifible
Invifible
Invifible
30 Vifible
Invifible
Invifible
Vifible
Invifible
Invifible
Invifible
2370 IND
Invifible
41 Invifible
Inviſible
O
6 Viſible
I
Aug. 24 9 28mm 11 481 26 SA
1748 Januar. 18 15 25
Feb.
2 23 49
July 13 22 30
July 28 11 34
januar. 7 7 17
June 18 21 24V
July 3 О 21
Dec. 12 8 8
27 21 12
1749 Januar.
Dec.
80
9 430 40 S D
25 160 46 S A
2 380 4 ND9
16 320 49 N A4
28 570 2 NA
8 310 59 S
21 590 15 SA
2 140 44 N D4
18 60 13 S D7
Inviſible
Invifible
53 Vifible
38 Vifible
Invifible
Invifible D
Invifible
36 Vifible
9 Vifible
Doctrine of the SPHERE.
339
The TABLE of Eclipfes continued.
Year
Months and Days. Long. Lat.
Digits Vifible
ΟΙ
Dh
Invifible
Lumin.
1
1750 June
June
8
9 9
226
6 51
28 160 15 S D16 Vifible
11 280 58¸S A
9
Invifible
Nov.
17 13 197
6 46
461 22
22 S D
Invisible
Dec..
1 18 32
21 1303 N D 21
6 Viſible
Dec.
17 6541
7
11 23 ND
Invifible
1751 May
13 2
132 5111
28 13 58
1752 May
Octo.
April
May
Nov. 6 12.43 m
21 9 47
Nov.
1753 April
April
1754
2 5 458 23 140 6 ND
25 13 59m 14 20 5 SD
6 6 20 27 520 45 S D
21 19 378 12 570 39 SA
Sept. 30 21 36 19 110 41 N D
Octo. 14 21 59 m 3 100 LI S A8)
18 141 20 N D
17 240
17 240 2 S D21
Mar. 12 5 52m
Mar. 26 19 47
10 22. 178
Sept. 5 I 13
2 941 19 SA
23 301 21
S D
17
3 220 51 ND
440 28 N A10
25 200 44 SD
10 350 39 S A3
14 Rif.Ecli.
Invifible O
Invifible O
D
8 Vifible
Invifible O
18 Viſible D
Invifible O
Invifible O
D
Invifible
3
Vifible
Invifible
41
Invifible D
Invifible O
Invifible
Sept.
19 22 28
7 550 1 ND 21
12 Invifible
Octo.
4 13 31
22
351 17 N A
Invifible
1755 Mar.
I 9 45
94522
22
20 40 N D
Invifible
Mar.
16 12 12 V
7
Aug.
Sept.
1756 Feb.
Aug.
2동 ​22
18 13 48
25 20 30 m
22 40
13
590 42 S A7
190 37 SD
13 Viſible
Invifible
27
X
10
40 34 SA
5808 ND
Invifible
Invifible
1757Jan.
14 7 12 m
23 19 6
Feb.
July
Aug.
3 10 45
Dec.
1758 Jan.
29 6 11
12 18 13
an.
27 16 37
une
23 20 55
7 1 2.7
2 520 7 NA
15 480 38 S D6
0 90 41 S Aj
19 11 53 ~ 7 510 30 ND 11
22 100 49 N A
19 331 20 N D
4 200 ON A 21
19 28 23 SA
13 81
S D
43 Set. Ecl. D
Invifible
32 Viſible D
Invifible
Invifible O
27 Set. Ecl.
Invifible
Invifible
Invifible
B b 2
340
An APPENDIX to the
•
The TABLE of Eclipfes continued.
Year
Months and Days. Long. Lat.
> Digits Vifible
D h
July
9 4 44
Dec.
1759 Jan.
June
Dec.
1760 May
18 19:29
I 19 46
13
5 23
6969 $ 5
SA
8
2 14
Το
27 450 15
8 400 40
22, 550 39
2460
27 290
N D
N A6
I SD
1
oz. SA
07.
41 Set. Ecl.
Invifible
invifible o
or
Invifible
Lumin.
Invifible
Invifible
O
18
93
9 357
8 510
52
N Do
47 Viſible
June
I
1 19 22
22 360 20
S.D4
35 Vifible
Nov.
II
9 18ПI
I 80 44
SD6
30 Vifible
1
Nov.
26 2 27
16 2038
38
SA
Invifible
1761 April
May 7 10
23 5 408 14 331 24
2m 28 30 II
S D
N D17
May
22 13 24 12 341 9
N A
Octo.
16 0 29m
4 141 27
ND
Octo. 31 23 438
20 210 2
SD
Nov. 15 2 157
4 381 16
SA
1762 April
12 17 28
3 580 42
SD
April
26 15 36, M
17 260 32
S Alto
Invifible
31 Viſible
Invifible
Inviſible O
Invifible)
Invifible O
Invifible O
o Viſible
Octo.
Octo.
1763 April
5 20 12
21 9 I
I 22 15
24 oo 16
N D6
10 Vifible O
9 118
9 250 39
NA 6
42 Vifible
23 120
1
S D
Sept.
25 13 5
25 135
13 280
13 280 2
N DI
Inviſible
Invifible O
1764 Mar.
II 47 15
27 500 3.9
27 560 39
ND8
15 Vifible
Mar. 20 22 9
12 100 4
SD10
7 Vifible O
Aug.
29 19 18
18
0042
SD
Sept. 14 5 5
3
30 4
SA
Invifible)
Invifible O
A TABLE
T
Doctrine of the SPHERE.
341
1
A TABLE of break of Day,
of break of Day, Latitude
51° 32′ N.
Days.
Janua. Feb. Mar.
April
May
June
H
H H
H
H
H
5215
7
13 4
1913
4 I
23
25 52 5
12 4
t
17 3
2
I
19
315
51 5
104
15/2
59 I
14
5
5015
84
13 2
57 I
9
4915
614
ΙΙ
2
54 I
· 4
5
485
414
812
510
58
4715
24
62
:49.10
53
46
014
4/2
4710
48
45 4
5914
2
2
4410
40
44 4
57 4
02
4010
24
1
11 5.
42 4
5513
1215
41 14
1
53 3
135
40 4
513
1415
3914
5013
572
55/2
532
5012
384
48 3
482
M32 2 2
350
14
32
29
26
23
16
5
3614
47 3
45 2
19
ab ml a
17 5
3514
45 3
432
16
185
34 4
43 3
40 2
'12
1915
334
41 3
38 2
9
2015
3214
3913
352
215
3014
3613
332
2
225
29 4
343
31 I 59
23 5 274
323
29 I
56
No
Night but Twi-light.
No Night.
24 5
25 4
30 3
26 I
53
25
24/4
28 3
23
50
5 22 4
25 3 20 I
47
2 2 2 MM
27
5 21 4
233
17 I
44
28
5
1914
21 3
14 I 39
29
5
17
3
II I
33
30
16
3
81
28
a mo
31
14
3
6
$
342
An APPENDIX to the
}
Break of Day, Latitude 51°
32 N.
Days.. |
July
Aug.
Sept. Octob. Nov. Dec.
H
H
H
H
H
/ H
}
123
2
Nō Night.
aal
~~~ 2222 222
613
3-54
4145
33 5
59
2
103
374
43 5
34 5
59
143
394
455
355
59
173
41 4
475
365
59
2 203
444
495
37
233
47 4
5.1.5
39
I
263
5014
535
40
I
293
52 4
55 5
416.
I
323
54 4
57 5
426
I
35 3
5614
595
4316
I
17.1.2
3.83
595.
0.1.5
45.
I
12 O
30 2
414
I5
25
46 6
Ι
13
40 2
1
444
315
47
I
14
15
O O
48
2
47 4
5
5
48
I
52 2
50.14
75
85
496.
I
161:0
582
5314
10 5
9.1.5.
5116
I
171
22
564
125
10 5
526
I
181
82
594
145
1215
53 6
I
19 1
143
24
1615
13 5546
I
20 I
20 3
514 185
1s།༨
546
I
21
22
1 2
I
2413
74 21.1.5
£75.
545
59
I
2813.
1014
23 5
185
55 5
59
23 I
3213
13 4 25 5
205
58
24 I
36 3
16
4
27 5
215
56
58
25 I
40 3
194'
295
225
5615
57
26 I
443
214
31
5
245
57 1.5
57
27 I
48 3
24
244 33 5
255
575
56
28 I
513
3
27 4
355
27 5
575
56
29 I
54 3
304
30 I
57 3
3:24
3715
395
285
585
5 55
3112
013
34
m m
305
58
5 54
32
53
Doctrine of the SPHERE.
343
A TABLE of the end of Twi-light,
Lat. 51° 32′ N.
Days.
Jan. Feb. March April May June
A
H
H
H
H
H
H
1234
16
86
4717
418
50 10 37
216
86
487
438
5810
IO 41
6
916
50 7
4519
I ΙΟ
10 46
1016
52 7
479
3 10 51
116
5417
499
6 10 56
66
126
5617
529
911
76
136
587
5419
11 II
27
816
147
07
56 9
13 11
12
157
ΙΟ
16
7 3
78
I 7 58
9
9
1611
20 II
20
36
116
187
5
8
319
126
1316
197
7
2017 9
519
79
14 6
1516
217 10 8
2217 12 8
ΙΟ
10
9
12
9
2 2 mm M
1616
247 13 8
1716
2517 15
8
186
2617
17
19/6 277
206 28 7
1918
218
∞ ∞ ∞ ∞ ∞
1519
25 11 46
28
31
34
37
41
22
179
209
9 51
44
48
259
54
216 30 7
24 8
279
58
226 317 268
246
2516 3617
4316 3317 28 8
3517 30 8
328
29 10
I
31 10
4
34 10
7
37 10
10
40 10
No Night all this Month,
but Twi-light.
No Night but Twi-light.
2616 38 7 358
27 6 39 7 378
41/7
286
29 6
3016
31 6
43
44
46
3918
∞∞
13
43,10 16
46 10 21
49 10
52 10
54
27
32
344
An APPENDIX to the
A TABLE of the end of Twi-light,
Lat. 51° 32′
51° 32′ N.
2
3
ΙΟ
II
Days.. |
No Night but Twi-light.
July
Aug.
Sep. Octo. Nov. Dec.
H
H
H
H
H
H
aa
548
257 19 6
27 6
I
508
237 17 6
2616
I
9
4618
217 15
256
I
9
43 8
197
13
9
40/8
16 7
II 6
a a
246
I
236
O
9
37 8
137
9
215
59
348
1017
7
י
205
59
9
318
87
9
288
67
9
2518
417
5MH
6
1915
59
3
6
1815
5
59
16
175
59
片名
​II 43 9
22 8 017
ol6
155
$9
12
13 11
II
3019
19 7.
5916
5816
14/5
59
II
2019
1617
5716
5616
135
59
14 11
1219
13.17
55
5516
54
6
125
59
15/11
89
1017
5316
52
6
115
59
16 II
219
77 506
51
6
17 10
58
47 486
5016
1810
529
17
46 6
486
a∞ 7
915
59
5
59
5
59
19 10 468
5817
4416
>
47
6
5
59
20 IO 4018
8
-55 7
4216
4516
615
59
21 10 368
5317 39 6.
4316
616
Ι
22 10 328
507
37 6
4216
23 10 288
5
6
I
24 10 248
4717
35 6
40 6
5
2
447
33 6
3916
2
2510
208
417
316
385
3
2610
168
397 2916
3616
3
27 10
128
36,7
2716
3516
3
4
28 10
9.8
3317 25 6
336
3
4
2910
6:8
3017
2316
32 6
216
3010
3
8
257
216
3016
216
31
10
08
26
6
28
6
7
A
337
**********************
SECT. V.
Aftronomical PRECEPTS, in the Ufe of the follow
ing New Tables; fhewing how to Calculate the Equa-
tion of Time, Planets Places, Ingreffes, Aphelions,
Retrogradations, Eclipfes, (both Particular and Ge-
neral) Occultations, Appulfes, Tranfits, Immerfions,
and Emerfions of the Satellites, &c. for any Time and
Place propofed.
PRECEPT 1.
To Reduce any other Meridian to that of London
& contra.
TH
"HE Epocha's or Radixes of the Middle Motions of the
Planets in the following Tables are accomodated to the
laft Day of the Julian Year for the Meridian of London.
In the Catalogue of Cities, feek the Place defired, and right
against it is the Elevation of the Pole, and Difference of
Meridians from London, either Eaft or Weft, as the Letters
4, or S, which fignifie Add, and Subtract, denote: That
Place with A againft it lies to the Eaft; and that with S, to
the Weft of London. ..
Then fuppofe I am at Rome, and there at 8 a-Clock in the
Morning it be required to Calculate the Places of the Pla-
nets from thefe Tables: Before I can begin the Work, I
muft reduce the Time at Rome, to the Time at London.
In the Table of Places I find Rome lies 52' to the Eaft of
London; therefore, contrary to the Title, fubt. 52' from the
Time at Rome, gives the Time at London ?
?
་ ་་
1
B b
OPE
338
Aftronomical Precept.
OPERATION.
H. M.
Time at Rome
800
Difference of Meridians fubr.
52
Remains, the time at London
7
08
Secondly, If it be 7 h. 8 in the Morning at London, what
time is it then at Rome?
Time at London
7
H. M.
8
Difference Meridians add
52
8 0
Time at Rome
Thirdly, Admit at Leverpool, when it is there Noon, what
time is it then at London?
Time at Leverpool
Difference of Meridians add
***
Time at Londòn
H. M.
12. CO
00
10
12 10.
1
Laftly, Suppoſe at London it be 12 h. 10'; what time is it
then at Leverpool?
H. M.
Time at London
12 10
Difference Meridians fub.
O
10
Time at Leverpool
12
Theſe are all the Varieties that can happen in this Precept.
:
..
*
PRE
Aftronomical Precepts.
339
PRECEPT 2.
To find the Equation of Time; and to reduce the Equal
Time to the Apparent, & contrà.
I have told you in the Definitions what the Equation of
Time is; and for this purpoſe I have Calculated two Tables,
which you will find in Pages 2 and 3, of thefe Tables: The
first part is gained by entring with the Sun's Place on the
Head, if the Sun be in the firft or third Quadrant of the E-
cliptic, and the Degree on the left Hand Defcending: But if
the Sun be in the fecond or fourth Quadrant, then find the
Sign he is in at the bottom, and the Degree on the Right
Hand Afcending; and in the Place of meeting is the firſt
part of the Equation of Time in Minutes and Se-
conds.
Secondly, With the Sun's Mean Anomaly, enter the Table.
in Page 3, finding the Sign on the Head, if he be in the
firft Semicircle of the Ellipfis; that is, if the Mean Anomaly
be 0, 1, 2, 3, 4, or 5 Signs and the Degree on the left
Hand Defcending; but if the Sun (or Earth) be in the
fecond part of the Ellipfis; that is, if the Mean Anomaly be
6, 7, 8, 9, 10, or 11 Signs, find the Sign at the bottom, and
the Degree on the Right Hand Afcending, and in the Place
of meeting you have the fecond part of the Equation of
Time in Minutes and Seconds; which is to be added or ſub- ·
tracted as the Table directs: Then if both Parts add, or
both fubtract, their Sum; otherwife their Difference (accor-
ding to their greater part) is the Abfolute Equation of
Time.
Example. Anno 1728, November 5th Day at Noon, I de-
mand the true Equation of Time.
Sun's Place
Anomaly
Deg. Min. Sec.
24 24 57
Gives
9
4 17 31 00
Gives
Min. Sec.
29 + to
5 19+ to
time.
time.
Sum 14 48 Add to the E.
qual, or Subtract from the Apparent Time.
}
Bb 2
Of
349
Aftronomical Precepts.
Or otherwife without the help of the Tables, you may ar
ny time find the true quation of Time thus:
L
By Prob. 3, find the Sun's Right Afcenfion to the time pro-
poſed, and take the Difference between that and the Longi
tude, which fhall be the first part of the Equation of Time,
and is to be added to the Apparent Time, when the Sun is in
the fecond or fourth Quadrants; but fubtracted, if he be in
the firft or third.
The ſecond part of the Equation of Time is the Elliptic E-
quation, taken out of the Table, Pages 28, 29, and 30,
which is to be added to the Apparent Time in the laft fix
Signs of Mean Anomaly, and to be fubtracted in the firſt fix
Signs.
When thefe two Parts are of the fame kind, that is, both
add, or both fubtract their Sum; otherwife their Difference
is the Abfolure Equation of Time, which according to the
greater part, added to, or fubtracted from the Apparent
Time, you will have the Equal; but to reduce the
Equal to the Apparent, uſe the contrary Titles.
Example. Anno 1728 November 5th Day at Noon, I would
know the true Equation of Time ?
Sun's Longitude in Degrees
Right Afcenfion
Difference, is the first part of }
the Equation
Elliptic Equation add
Sum
£}
Deg. Min. Sec.
234 24 57
*232 2 29
2
22
28 fub,
1 19 48 fub,
3 42 16. Which
reduced into Time by the Table in Page 66, is 14 min. 49
fec 4 thirds, the Abfolute Equation of Time; which is
to be added to the Equal, or fubtracted from the Apparent
Time.
PRE
Aftronomical Precepts.
341
PRECEPT 3.
To Compute the true Longitude of the Fixed Stars.
The Catalogue of Fixed Stars I have rectified to the be-
ginning of the Year of Human Redemption 1727; there-
fore you have no more to do, than to take the Præceffion of
the Equinox out of the Tables, in Pages 4 and 5, for any
interval of Years, and add it to the Place of the Star in the
Catalogue for Time, after 1727 ; but fubtract for Time be-
fore, and you will gain the true Longitude of the Star en-
quired after.
Example. Anno 1740, Fannary 1, I would know the true
Place of the Pleiades?
From 1727, to 1740, is 13 Years.
1727 Place of the Pleiades is
13 Year Motions add
Place of the Pleiades
Deg: Min. Sec
Ŏ 26 10 58
10 50
8 26 21 48 The like
you are to obſerve for any other Fixed Star in the Cata-
logue, And the Præceffion of the Equinox is given any Year
by Inſpection.
PRECEPT 4.
To Calculate the true Place of the Sun.
1. From the Tables of the Sun's Mean Motion, write ouz
the Longitude and Anomaly anſwering the Years, Months,
Days, Hours, Minutes, and Seconds, (if occafion be, which
added up feverally, are the Mean Motion of the Sun for the
time propofed; remembring in Leap-Year after February to
take the Days of the Month on the Right Hand under Biffex-
tile.
2. With the Sun's Mean Anomaly thus collected, enter the
Table of his Equation, with the Sign on the Head (if under
6 Signs) and the Degree in the firſt Column on the Left-
Hand, deſcending; but if the Anomaly be more than 6 Signs,
!
find
342
Aftronomical Precepts.
1
find the Sign at the Foot of the Table, and the Degree on
the Right-hand Afcending; and in the Common Angle, or
place of meeting is the Elliptic Equation, and Logarithm of
the Diſtance of the Sun from the Earth anfwering; which,
according to the Title, added to, or fubtracted from, the Mean
Longitude of the Sun before found, will give his true Place
to the time propoſed; ever obſerving to find the Equation
and Logarithm anſwering to the Mean Anomaly, as has been
fhewn in Page 299.
Example. Let the Sun's true place be required on April 7,
1728, at Noon?
OPERATION.
Equal Time: Long.
| Anom. O
១.
S O
17289 20 11 436 11 58 5² If 60 о
April 7, B3 6 35 37 3
32/13
6 15 2 Give 120 14771
Mean Moto 26 47 209 15
Equat. add 1 49 301
Sun's Place.lo 28 36 50L. 5.002443
34 12What 34 12 2441
Answer 68 17212 to be
added to the Log. of 9 S.
18º makes the trueLog.
à anſwering the Mean
Anomaly.
Example 2. Let the Place of the Sun be required on
Auguft 29 at 36' 46'' paft 8 a-Clock in the Morning at Low-
don, Equal Time?
OPERATION.
Equal Time. Long. O, Anom.
S. O
S.
1728 9 20 11 436 11 58 52 If
Aug. 28, Biff. 7 27 32 277 27 31 31 Give
Hours 20
Minutes 36
46
Seconds
Mean Mot.
Equat, Sub.
Sun's trae
Place
5 18
49 17
I 29
2
Min.Sec.
60 。 L.L. 。
0 42
49 17 What 21 11
19331
4522
1 29. 0 15 23852 This
I
2 propor. Part is fubtracted
I
34 582 10 21 from the Equation of 2 S.
48 15 Le 5.00257190, and there remain 19
48'16", the trueEquation.
5 16 46 42
1
Exam-
т
Aftronomical Precepts.
349
Example 3. Let the Place of the Sun be required 3949
Years before Chrift, on the 17th Day of April, under the Me-
ridian of Babylon at Noon, that being the fuppofed Time of
the World's Creation..
In order to Calculate the Places of the Planets to any given
Time before Chrift, fubtract the Motion anſwering to as
many Years as will bring in, or exceed the Number of Year's
intended, from the Mean Motion anſwering to the firft Year
of Chrift; and to the remainder add the Motion of ſo many
Years as will make up the Complement of the Number of
Years you fubtracted, to the given Number of Years before
Chrift. As in this Example, 40 +11=51 fub. from 4000,
leaves 3949.
Or fubtract the Mean Motion from the first Year of Chriſt,
which anſwers to the given Number of Years before Chrift,
having regard whether the Year be Biffextile or Common
and then to fubtract the Motion of a Day more from thẻ
reft, as you may fee by the Examples following.
Equal Time.
Long. O
Anom. O
S.
O
"1 S. Q
"
Anno Chrifti, 9 7 53 10 6
29 53
40
4000 Years.
-
00
13
2010
20 T༢ 20
AnteChr.4000 8
40
1111 29 20
Radix 3949| 8 7
April 17,Com. 5 15
XMer.3h.14'.
7
39
50 8
fub.
9
40 20
• 18
811
29
36
8
3811
29 9
S
36 8 8
25
33
1
27
52 3 15
27
34
7
7 59
Mean Mot.
11 22
54
27/12
24 OI
II
53
Equation add
Sun's Place. 1 23 6
20
By which it appears that the Sun did not enter Arses till
about the 24th Day of April, at the Creation,
Of
844
Aftronomical Precepts
Or you may work thus.
Years. S. Deg. Min. Sec S. Deg. Min. Sec.
22 40
0011 OO 10
6 48 0011 21 3
300000
900.00,
40 | 00
18
8/11
29
36
8
911
29 49
1811
29 39
51
3934 Motion
00
29
35
16/20
20 58
59 fub.
Year Xt.
9
7
53
10 6
29
53
40 from
Rem.
8 8
17
441 8
9
24
41
59 8
58
8
8
18 36 8 8
25
Sub. one Days Motion
Radix
So 'tis needleſs to proceed any further;
will be the fame as was juft now found,
33 as before:
for the Sun's Place
23° 6 min. 20 fec.
PRECEPT 5.
To Calculate the Sun's Ingrefs into any of the Twelve
Signs.
In order to make this Work as fhort and plain as poffible,
I have here underneath given the Elliptic Equation when the
Sun enters every one of the 12 Zodiacal Signs for the Year
1728; which are to be added or fubtracted to or from the
Sun's Mean Longitude of any given Year, the Sum or Diffe-
rence fubtracted from the Number of the Sign you are feek-
ing the Sun's entrance into ; and from that fubtract the near-
eft lefs Day, Hour, Minute, and Second, in the Tables of
Mean Motion, and you will fpeedily gain the true time of
the Sun's Ingrefs into the fame Sign; as the following Exam-
ples will make plain.
1
1
The
Aftronomical Precepts.
345
1
The Table of Elliptic Equation.
1728
XT DE LEIEV
"
O 42 39 add?
3º 41 add
48 2 6 ma
55 add
32 add
*
I
ԴՐ
I
54
I
48
I
12
41
add
O
16
50
add
43
40
fub.
I
32
6
fub. I
I
55
20
20
fub.
I
47
31
fub.
I
I I
17 fub.
VS
O
from the
Sun's Mean Longitude for
any given Year.
Add to, or
Subtract
16 25 fub.
Example. Let the true time of the Sun's Ingreſs into Aries
be required, Anno 1728?
The Work ſtands thus:
}
Anno 1728, Sun's Mean Longitude
$
S. Deg. Min. Sec.
9 20
II 43
Equation add
1
54 55
Sum Sub.
9
22
6 38
From Aries
12
O
O
Q
1
*
Rem.
March 8, Biffext. fub. 27
Rem.
2 7
53
22
•
1 26
SI 59
Hours 21 fub.
51 45
1
Rem.
Minutes 4 fub.
.11 Thirds.
9. 51
Rem.
1
Seconds 28 fub.
I
9
Се
By
346
Aftronomical Precepts.
By which it appears, that the Equal Time of the Sun's
Entrance into the Equinoctial Sign at London 1728, is
March 8 d. 21 h. 4 min. 28 ſec. At which time the Sun's
Mean Anomaly is 8 S. 19 deg. 52 min. 5 feconds; which
gives the Equation of Time 7 min. 40 feconds, to be fub-
tracted from the Equal Time 8 d. 21 h. 4 min. 28 feconds:
Therefore the Apparent Time of the Vernal Equinox is March
8 d. 20 h. 56 min. 48 feconds.
And if thoſe Gentlemen that are qualified, and fitted
with proper Inſtruments, will pleaſe to obſerve the fame,
I do not at all doubt but that the Event will very nearly
anfwer.
And after the fame manner have I found the Equal Times
of the Sun's Ingrefs at London to be as follows, with the
Mean Anomaly at the fame Time.
IO 20 34 II
Q's Anom.
D. H. M. S.
S
D. M. S.
Fan.
9 5 25 37 enters
Feb.
7
20
25 25
March
8
21 4
28
April
8
IO
9 O
May
9
I I
II 13
Fune
9
20 20
O
17285
July II
7 22 19
Aug. 1 I
13 30 49
Sept.
KUDHULZIENS
78
6 21 4 30
20 16 24
8 19 52
S
9
19 58 24
2 on
II 9 25
Octob. 11 16 44 IS
Nov. 10 12 21 26
Dec. 10
O 44 25
8 II 15 25 enters
Fan.
Feb.
7
2
March 9
April
·14 12
253 10
8 15 57
May 9 15 59 14
Fune ΙΟ 27 42
1729 July II 13 10 33
Aug. II 19 19 28
Sept. II 15
Octob. 11 23
ી.
11 21
29 58
O 22 30 24
I 23 18 46
2
3
23 41 56
23 34 3
4 22 57. 45
VS 5 22 2 49
K&DEGLEIENS
6 21 3 27
7 20 15 21
8
19 51 1
9 19 57 19
10 20 33 5
II
O
14
O
33
17
O
VS
S
Nov. 10 18 20
Dec. 10 6 33 43
21 28 52
22
29 18
23 17 41
I
2
23 40 52
3 23 33 0
4 22 56 45
22 I 47
1730
Aftronomical Precepts.
347
1
O Anom.
D.
H. M. S.
S.
(Fan.
8 17
5
170 in
Feb.
7
8
4
4
March
9
8
ԴՐ
42
00
April
8 21
45 SI
May
9
2 47
Fune
10
7 56 00
1730 July
II
18 59
Aug.
12
I 7 18
m
}
Sept.
II
21
3
00
Octob.
12
4
22
19
Nov.
II
00 10 00
IXS DHONIES
D. M. S.
6 21 2 26
*
7
20
14
19
8
19 49
58.
୪
9
1956
10 20 32 Oo
16
II
21 27
47
ી.
O
22
28 13
I
23
16
16
36
2
23
49
48
3 23
31 57
4
42 55
41
I
Dec. 10 12 23 25
vs 5 22 00 46
By reaſon of the Motion of the Earth's Aphelion and
Nodes, the Elliptic Equation is never the fame again
when the Sun enters the fame Sign, &c. for by Cal-
culation I have proved that in 72 Years time, the diffe-
rence of the Equation in the fame part of the Orbit, is, as
is here fet down.
When enters
XT CHOLEIENS
VS
Min. Sec.
2 22
I
O
37 fub.
25
53
2 ΟΙ
હ
2
2
34
add
24. Zlub.
I 35
35
I
2
22
577
58 add
30
to,
or from the Equations of
the Year 1728 in 72 Years,
and ſo proportionable for any
intermediate Years.
1728 =
Example. Idemand the time of the Sun's Ingrefs into the
Tropical Sign Capricorn, Anno 1740? Firft 1740
12 Years?
Then ſay, As 72 Y: 2 min. 30 fec:: 12 Y: 25 feconds.
This is to be added to the Elliptic Equation of the Year 1728,
16 min. 25 fec. in Capricorn, and it makes 16 min. 50 fec. for
the Elliptic Equation in Capricorn for the Year 1740.
Сc 2
Now
348
Aftronomical Precepts.
1
Now the Operation ftands thus:
Anno 1740, Sun's Longitude
Equation for 1740 fub.
Rem. fub.
S. Deg. Min. Sec.
9 20 17 ΙΟ
16 50
9 20 CO 20
From VS
9 0
Rem.
II
9
59 40
Dec. o Biffextile fub.
I I
9
3 45
1
Rem.
Hours 22 fub.
55 55
54 13
42 Thirds.
2
Rem.
I
Minutes 41 fub.
I 41
Rem.
Seconds 23 ſub.
Rem.
Thirds 32 fub.
58 Fourths.
56 40
I 20
I 19
I
I
1
Rem.
Fourths 25 fub.
Rem.
O
By this Calculation it appears that the Sun will enter Ca
pricorn Anno 1740, at London, December 9 d. 22 h. 41 min.
23 fec. 32 thirds 25 fourths, equal Time, the Anomaly at
that Time being 5S. 21 deg. 50 min. 44 feconds, which gives
the Equation of Time 1 min. 7 fec. to be added to the E-
qual Time, which makes the Apparent Time of this Solar
Ingrefs on December the 9th, 22 h. 42 min. 30 fec. 32 thirds
25 fourths.
1
PRE-
}
Aftronomical Precepts.
349
PRECEPT 6.
To Calculate the true Place of the Moon.
1. By Precept the 4th, find the Sun's true Place, to the gi-
ven Time.
2. In the Tables of the Mean Motions of the Moon, write
out her Longitude, Anomaly, and Node; to the Year, Month,
Day, Hour, Minutes, and Seconds given, add the Motions of
Longitude and of Anomaly into two feveral Sums; but the
Node (becauſe it is Retrograde) muft be fubtracted; that
is, fubtract the Motions of Days, Hours, Minutes, and
Seconds, from the Mean Motion anfwering the given
Year; and thus you will have the Middle Motions of the
Moon collected to the given Time.
3. With the Mean Anomaly enter the Table of the Moon's
Elliptic Equation, in Pages 51, 52, and 53; and take out
the Equation and Logarithm of her Diſtance from the Earth,
(as I have fhewn in that of the Sun) taking the propor-
tional Parts to the Minutes and Seconds of her Mean Ano-
maly; and (according to the Title of the Table,) the Equa-
tion added to, or fubtracted from the Mean Longitude and
Anomaly, gives her Place firft Equated.
4. From the firft Equated Place of the Moon, fubtract the
Sun's true Place; the Refidue is the Diſtance of the Moon
from the Sun; which double, and with the double Diſtance
enter the Table, Page 54, 55, and take out the Moon's Re-
flection, which (according to its Title) apply to the Equated
Anomaly, and it gives you her Anomaly corrected. Alfo
with the Diſtance of the Moon from the Sun take out of
the fame Table the Logarithm of the Chord of Evection: Or,
to the Logarithm of the Diameter of the Circle of Evection
3.640432; add the Sign of the Diſtance of the Moon from the
Sun, rejecting Radius, is the Logarithm of the Chord of E-
vection; which reſerve.
5. To find the Synodical Anomaly; the Moon paffing from
the Conjunction or Oppofition of the Sun to the Quadratures,
the Complement of the Diſtance of the Sun and Moon to a
Quadrant is to be added to the Correct Anomaly before found.
But from the Quadratures to the Conjunction or Oppofition,
the Excefs of the Diſtance of the Moon from the Sun is to
be
}
Aftronomical Precepts.
350
{
!
be fubtracted from the Correct Anomaly; and the Sum or
Difference is the Synodical Anomaly.
6. From the Logarithm of the Diſtance of the Moon from
the Earth, (found by the 3d hereof) fubtract the Logarithm
of the Chord of Evection; and to the Refidue add the Radius.
the Sum is the Tangent of an Arch; from which reject 45º.
Then ſay,
As Radius,
To the Tangent of the remaining Arch;
So is the Tangent of the half Synodical Anomaly,
To the Tangent of an Arch, whofe Difference from the
half Synodical Anomaly is the Angle of Evection; which, if
the Synodical Anomaly were lefs than fix Signs, it fubtracts;
if more, it adds.
7. If the Reflection, and Evection, both add, or both fub-
tract their Sum; otherwife their Difference according to the
greater part, is the fecond Equation; which added to, or
fubtracted from the Longitude of the Moon firſt Equated,
gives her Longitude in her Orb.
8. To find the Moon's Latitude, and Reduction from her
Orbit to the Ecliptic.
1. With the double Diſtance of the Moon from the Sun,
enter the Table, Page 56, and there take out the Equation of
the Mocn's Nodes; which, (according to its Title) added to,
or fubtracted from the equal Place of the Node, gives the
true Place; which fubtracted from the Moon's Longitude in
her Orb, leaves the Argument of Latitude.
2. With the Diſtance of the Moon from the Sun, take out
the Exceſs of the Moon's Latitude above 5 degrees, in Page
57; which added to 5 degrees (always) gives the Angle of
the Moon's Orb and Ecliptic at that time.
Then for her Latitude, by Trigonometry, it will always
hold.
As Radius,
To Sine Moon's Distance from the neareſt Node;
So Sine of the Angle of her Orb with Ecliptic at that
time,
To the Sine of her Latitude. Which is North, if Argu-
.ment of Latitude be leſs than 6 Signs; but South, if more.
3. With the Argument of Latitude enter the Table, Pages
58 and 59, and take out the Reduction; (according to its
Title) being apply'd to the Moon's Orbit-Place, gives her
Longitude reduced to the Ecliptic.
Ex-
Aftronomical Precepts.
351
Example. Anno 1728, Let the Place of the Moon in Lon-
gitude and Latitude be required April 7th Day at Noon?
Equal Time Longit.
、
Anno 1728
Apr. 7 Biff.
Anom.
Node>
?..
S.
9 17 46
7 I 17 I
19 20 50
6.2021
I
151
15 II
53
8fub. 5 11
22
Mean Mot.
Equat. fub.
>'s Pla. fub.
4 19
3 13 4 11 I 2
3 32 56
3 3
23111
sladd on
ΙΟ
00
3 x
<8
23
4 15 30 17 4
7 39
-7 I
10
-8
4
O's Pla. fub.
c 28
6
Reft fub.o
Dift.
à O
3
16 5;
7
3 46
1
་
!
Double Dift.
fec.Equ.fub.
Din her Orb
2 13
27, 4 7
4 fub 16
4 12 47 12
N. Node fub. 11
Argum. Lat. 5
Tr. Lat.N.D.
Reduct. add
19 26 Log. Di.Cir.Ev. 3.640432
53 278073°5′33″.980848
5 3 20 25 59 Log.Ch.Evect 3.6 1200
1 25 12 59 Log.) ab.
10 58 54
1 48 18
As Rad. to
29 17
Ecl. Place 4 12 52 39
So tSy. An.
Tot.of the Ar.ful.
Rem.Ev.lub.
Reflect. fub.
2d Equa.fub.
t 87°29′39′
1.45
4980219
11.358932
c fub.
t.42 299-- 9.961961
55 12 59--10.158 69
52 49 55--10.120230
2 23
20
}}add
2 43 5 Sum.
2 43
For the Moon's Latitude, and Ecliptic Place.
With the Diſtance of the Moon from the Sun 3 S. 16 deg.
53 min. 27, enter the Table, Page 57, and you'll find the
Angle of the Moon's Orb above 5 degrees to be 16 min. 10
feconds; which, added to 5 degrees, make 5 degr 16 min.
10 ſeconds the Obliquity of the Moon's Orb at that time, and
the Moon's Orbit Place 4 S. 12 degr. 47 min. 12 feconds fub-
tracted from the Place of the South Node 5 S. 10 deg. 58
min. 54 feconds leaves 28 deg. 11 min. 42 feconds, the
Moon's Distance from the neareft Node.
Theſe things being known, ſay,
As Radius
To S. of
from
Deg Min. Sec.
90 Co
00--10 000000
1
1
To S. Obliquity Orb
To S. Latitude N.D.
}
28
II
5 16
42-- 9.674378
10-- 8963029
2 29 13-- 8.637407
For
f
352
1
Aftronomical Precepts.
For the Ecliptic-Place.
With the Argument of Latitude 5 S. 1 deg. 48 min. 18
fec. enter the Table, Page 58, and take out the Reduction
5 min, 27 feconds; which (according to its Title) added
to the Moon's Orbit Place 4 S. 12 deg. 47 min. 12 feconds,
gives her Ecliptic Place 4 S. 12 deg. 52 min. 39 feconds.
Example 2. Let the Moon's Place be fought on August 29,
at 36 min. 46 lec. paſt 8 in the Morning.
Equ.Time Long.
S Q
Anno 1728
1
>
Anom. >
S. 9
Node>
"
S.
9 17 46 1
9 20 50 1511
15
11
53
Aug. 28 Bif
Hours 20
9 25 30 40
10 58 49
8
20 39 42
. I 2
45
44
10 53 15
39
Minute 36
Seconds 46
Equat. add
19 46
Mean Mot 7 24.35 447 043
19 36
5
25
25
} 12
48
28
0 43 13!I
2 23
25
2 22 3
2
's Pla. Eq 7
5
26 58 13 7
16 46 42
22 3: Sub. 1
II I
6 56
16 29
O's Pla.fub
Dift. D à G2
Double Dift.
22 Equa. add
Refl
3 5 45
add 23 56 Log.Di.Cir.Evect. 3.640432
10 11 31 7 3 29 41 S.àО70°11′31″ 9 973513
4 20 23
in her Orb 7
N.Node fub.11
2 27
A
add19 8 2 Log.Chor.Ev. 3.613945
7 23 18 ic Log.) from
29 25 19 3 26 39 5 t.87 29 46
16 29 3 20 55 45 о o fub.
1
Argu. Latit 8 28
Tr. Lat. S.A.
Reduct. fub.
2
4.973196
11.359251
859 As Rad. toft.42 29 46-- 9.962160
1528 So Sy.An. t.63 20 55--10.299396
25 To the Archt.61 17 45--10.261556
Eclip. Place 7 29 24 54
1
2
3 10 Evect. add
0 23 56 Refl. add
2 27 06 ad Equation.
For the Moon's Latitude and Ecliptic Place.
¡
With the Diſtance of the Moon from the Sun 2 S. 10 deg.
11 min. 31 feconds, Enter the Table, Page 57, and it gives
the Angle of the Moon's Orb with the Ecliptic above s
deg. 15 min 38 feconds; which added to degr. makes
5 deg. 15 min. 38 feconds, the Obliquity of the Moon's
Orb at that Time. And the Place of the South Node
5 S. 1 deg. 16 min. 29 fẹc. fubtracted from the Moon's
Orbit
;
Aftronomical Precepts.
353
Orbit-Place 7 S. 29 deg. 25 min. 19 feconds, leaves 2 S.
88 deg. 8 min. 5o fec. the Moon's
28 deg. 8 min. 50 fec.
Diſtance from the neareft Node.
Now for the Latitude, ſay,
50
Deg. Min. Sec.
As Radius
90 00
00--10.000000
To: S. of
from 2
88 8
50-- 9.99977.3
So S. Obliquity Orb
5
15
38-- 8.962298
To S. Latitude South A
5
15
28-- 8.96207 £
Lastly, For the Ecliptic-Place.
With the Argument of Latitude 8 S. 28 deg. 8 min. 50
fec. enter the Table, Page 59, and take out the Reduction
25 ſeconds, which (according to its Title) fubtracted from
the Moon's Place in her Orbit 7 S. 29 deg. 25 min. 19 fec.
leaves 7 S. 29 degr. 24 min. 54 fec. the Moon's Place redu-
ced to the Ecliptic.
PRECEPT 7.
To find the true Time of the Conjunction or Oppofition of
the Sun and Moon.
1
This may be performed three ſeveral ways.
1. By the Logiſtical Logarithms.
2. By the Table of Lunar Afpects in Page 67.
3. By the Table of the Mean Hourly Motion of the Moon
from the Sun in Page 65; which Method is this.
With the Epact for the given Year, find the Day of the
New or Full Moon, as has been fhewn in Page 297; to
which Day at Noon compute the True Place of the Sun,
and the firft Equated Place of the Moon. If theſe two Pla-
ces be the fame Sign, Degree, Minute, and Second, then have
you the true Time of the Full Moon: Or if their Places dif
fer juft x Signs, then have you the true Time of the Full
Moon: But if their Places differ at Noon (as most common-
ly they do) fubtract the leffer Place from the greater, and
with this Difference enter the Table, Page 65, and ſee how
many
Dd
7
*
354
Aftronomical Precepts.
many Hours and Minutes, or Minutes and Seconds of Time
the Diſtance of the Sun and Moon anſwers to; which, if the
Sun's Place at Noon was more than the Moon's firft Equated
Place, then this Difference in Time muſt be added to the
Day at Noon above found by the Epact; but if the Moon's
Place exceeds the Sun's, then are the Luminaries paft the
Conjunction, or Oppofition: Therefore you muſt fubtract
the Time found in the Table from the Noon of that Day;
and this Sum or Difference, is the fuppofed Time of the
New or Full Moon; to which Time compute again the
Sun's true Place, and the firft Equated Place of the Moon;
and if their Places now agree, then you may conclude
you have the true Time of the New or Full Moon; but
if you find a difference in their Places, you must enter
the aforefaid Table, and take out the Time anſwering.
to that Difference, and add or fubtract it, to, or from the
time laft found, according as the Moon's Place was lefs
or more than the Sun's: And thus you muſt proceed un-
til you find the Sun's Place, and the firft Equated Place of
the Moon to agree in Signs, Degrees, Minutes, and Seconds;
for then you may be affured that you have the true equal
Time of the New or Full Moon: And ever remember to
make a Repetition of your Work until you find a Concur-
rence in the Places of the Sun and Moon: Here you are
to Note, that the time of the New and Full Moons are more
eafily obtained than the Times of the Sextile, Square or
Trine; by reafon that feveral Inequalities of the Moon va-
nifh: An Example will make all plain to the diligent Rea-
der.
Example. Let it be required to find the time of the Full
Moon in January, Anno 1730?
}
OPE
1
Aftronomical Precepts.
355
I
OPERATION.
Epact for the Year is 22, fub. from 45.
Refts
at Noon{
Jan. 23, at Noon
Difference paft 8
ត
૧.
In the Table give 7 hor.
Remain
Minutes 20
fub.
Rem.
Seconds 40
ſub.
2'3d Day.
Deg. Min. Sec.
14 30 31
18 14
21
3
43
50
3 33 20
10
30
10
ΙΟ
20
20
}
From the 23d.
H. Min, Sec.
00
Sub.
7 40
40 40
Remain
22
16
39 30 then
30 then
O
14º II' 51"
20 14.
14 21 08
Difference paft &
Minutes 18 fub.
9 17
9
9
Remain
Seconds 16 fnb.
Rem..
8
8
Days H. Min. Sec.
From January 22 16 39 20
Sub.
18
16
1
Remain
22 16 21 4
Dd 2
At
F
356
Aftronomical Precepts
Deg. Min. Sec.
At which time the 14
AW 14 II об
ટી 12
Difference paft &
17
Minutes 2 fub.
Rem.
Seconds 20 fub.
;
Days H. Min Sec.
From January 22
16 21 4
Sub.
2 20
Remain
22
16 18 44
I
II
I OI
10
LO
Deg. Min. Sec.
At which time the
Differ. paft &
Seconds 6 fub.
Rem.
14
II
0
4.
14
II
3
3
3
o
So that the preciſe time of this Full Moon is January 22 D.
16 h. 18 min. 38 feconds; at which time the Sun's true place
is 14 deg. 11 min. and the Moon in 14 deg. 11 minutes.
After this manner muft you find the equal Times of the New
and Full Moons. This is the moft expeditious of all other
Methods made ufe of by Aftronomical Writers; which Me-
thod is my own, and will become eafie if you will but work
upon your Slate, and find the proportional Parts of the
Elliptic Equations by a Sliding-Rule, as mentioned in
Page 301.
PRE-
Aftronomical Precepts.
357
PRECEPT 8.
To Calculate the true Heliocentrick, and Geocentrick.
Places of the five Primary Planets 5, 4, d, f,
and. §.
t
1. By Precept 4, find the Sun's true Place, and the Loga-
rithm of his Diftance from the Earth to the given Time.
2. Out of the Tables of the middle Motions of the Pla-
net, write out the Longitude, Anomaly, and Node, to the
Year, Month, Day, Hour, Minute, and Second; if need
be, add them up feverally; fo have you the Mean Motions
of the Planet to the Time propoſed.
3. With the Mean Anomaly take out the Elliptic Equation
of the Planet and its Logarithm, referve the Logarithm and
apply the Equation to the Mean Longitude, and (according
to its Title) either add or fubtract, and the Sum or Difference
will give you the Heliocentric Place of the Planet in its Orbit
from the Vernal Eqninox.
4 From the Heliocentric Orbit-Place, fubtract the North
Node of the Planet, and the refidue is the Argument of La-
titude; with which take out the Reduction of its proper
Table, and (according to its Title) added to, or fubtracted
from the Heliocentric Orbit-Place you will have the fame
Place reduced to the Ecliptic.
5. From the Longitude of the Sun, fubtract the Heliocen-
tric Ecliptic Longitude of Saturn, Jupiter, Mars; but from
the Ecliptic Heliocentric Longitude of Venus or Mercury fub-
tract the Longitude of the Sun; the refidue is the Angle at
the Sun, or Anomaly of Commutation; of which take the
half; and if the half be more than three Signs, take its
Complement to fix Signs or 180°.
6. From the Logarithm of Saturn's, Jupiter's, Mars's Di-
ftance from the Sun, fubtract the Logarithm of the Sun's
Distance from the Earth; to this remainder add the Ra
dius, and you will have the Tangent of an Arch; from
which reject 45º.
But
F
358
Aftronomical Precepts.
But in the two Inferiours Venus and Mercury, take the Lo-
garithm of their Diftance from the Sun, out of the Loga-
rithm of the Sun's Diſtance from the Earth, and to the remain-
der add Radius, and it is the Tangent of an Arch, from which
reject 45 Degrees.
Then, As Radius,
To Tangent of the remaining Arch,
So is the Tangent of half Anomaly of Commutation, or
its Complement, To the Tangent of an Arch. Whofe Sum
and Difference to the half Commutation is the Elongation
and Parallax of the Earth's Orb. Or otherwiſe, the Paral-
lax of the Earth's Orb may be found.
Thus, In Saturn, Jupiter, Mars, fubtract the Loga-
rithm of their Diſtance from the Sun, from the Loga-
rithm of the Sun's Diſtance from the Earth (ie. the grea-
ter Logarithm from the leffer, the Radius being firft ad-
ded,) and the remainder will be the Tangent of an Arch;
to which you must add 45°.
But in Venus and Mercury fubtract the Logarithm of the
Sun's Diſtance from the Earth, from the Logarithm of the
Planet's Diſtance from the Sun, (the Radius being firft ad-
ded) and the remainder is the Tangent of an Arch; to which
always add 45°. Then,
As Radius,
To Co. Tangent of that Sum,
So is the Tangent of half Anomaly of Commutation, to
the Tangent of an Arch.
This Paragraph is no more than the fecond Axiom (or
Norwood's 3d) of Plain Trigonometry; for here are always
two Sides, and the Angle included given, to find the other
two Angles; that is, the Diſtance from the Earth to the Sun,
and the Diſtance of the Planet from the Sun by their Lo-
garithms with the Angle at the Sun always given, to find
the Angle at the Earth, being the Elongation, and the An-
gle at the Planet, being the Parallax of the Orbit. This I
have made plain in my System of the Planets Demonstrated,
Printed Anno 1727, for Mr. Wilcox in Little Britain, and for
Mr. Heath, Mathematical Inftrument-maker in the Strand,
London.
7. In Saturn, Jupiter, Mars, the fourth proportional Tan-
gent added to the Anomaly of half the Angle at the Sun, or
Commutation, gives the Elongation; but fubtracted, gives the
Parallax of the Earth's Orb.
But
Aftronomical Precepts.
359
But in Venus and Mercury the Sum of the fourth propor-
portional Tangent added to half the Angle at the Sun, or
Commutation, gives the Angle at the Planet or Parallax of
the Orb; but fubtracted, gives the Angle at the Earth, or E-
longation from the Sun.
8. If the Anomaly of Commutation be less than fix Signs,
the Parallax of the Earth's Orb is to be added to the Helio-
centric Longitude of Saturn, Jupiter, Mars; but in Venus
and Mercury to be fubtracted: If the Anomaly of Commu-
tation be more than fix Signs, the Parallax of the Earth's Orb
(or Angle at the Planet) is to be fubtracted from the Helio-
centric Longitude of Saturn, Jupiter, Mars; but in Ve-
nus and Mercury to be added; the Sum or Difference, is
the true Geocentric Longitude from the Vernal Equi-
nox.
Or, in Saturn, Jupiter, Mars, if the Anomaly of Com-
mutation be less than fix Signs, fubtract the Elongation;
but if more than fix Signs, add the Elongation to the Sun's
Place. In Venus and Mercury, if the Anomaly of Commu-
tation be less than fix Signs, add the Elongation; but if
it be more, fubtract, to or from the Sun's place; the Sum or
Difference is the true Geocentric Longitude of the Planet as
before.
9. For the Geocentric Latitude of the Planets.
With the Argument of Latitude take out of the pro-
per Table the Planets Inclination, or Heliocentric Lati-
tude; and then ſay,
}
As the Sine Commutation Co. Ar.
To S. of Elongation;
So is the Tangent of the Heliocentric Latitude,
To the Tangent of the Geocentric Latitude.
Or fay,
As the Sine of Elongation Co. Ar.
To the Sine of the Commutation;
So Co. Tangent of the Heliocentric Latitude,
To Co. Tangent of the Geocentric Latitude.
Example. Let the true Place of Mercury be enquired the
first Day of January at Noon, Anno 1728, under the Meri-
dian of our Table's equal Time?
See
360
Aftronomical Precepts.
Tempus datum Longit.
S.
A
See the Work..
Anom.
S. o
Anno 1728, 5 21 50 49
January 1,
4 5 32
S.
Node ?
8 42 45
I
15
10 40
4 5 32
1
Mean Mot.5 25 56 139
Equat. add 21 16 5
Hel Or. pla. 6
12, 48 17
12,48
Node fub.
17 12 18 Log. Qà
I 15 10 40 Log. ✪ à
น
Arg. Latit.
Reduc. add
5 2
1 38 T. 23°4′18″
10 32 4dd45 0 o
Hel.Ecl. pla. 6 17 22 50 C.T.68 4 18
Sun's pla.fub 9 21 37
Anom.Com.18 25 45 21
29 Ta.47 7 20
Ta.23 26 27
Sum70 33 47
Half
Complem.
I 17
4 12 52 40
17 7 20
Diff.23 40 53
Parallax add 2 10 33 47
Geo. place 8 27 56 37
Dir.Orient.l
4.622239
4.992785
9.629454
9.604843
10.032291
9.637044
Parallax add.
Elongat. fub, from the
Sun's Place will give the
Geocentric
Place the
fame, by adding the Pa
rallaxes to the Heliocen
tric Ecliptic Place.
1
For the Latitude.
With the Argument of Latitude 5 S. 2 deg. 1 min. 38 fe-
conds, take the Inclination out of its proper Table 3 deg. 14
min. 11 fec. N.D. and then ſay,
As S. Commutation Co. Ar.
To S. Elongation
So T. Inclination North Deſc.
To T. Geocentric Latitude N.D.
Deg. Min. Sec.
85 45
E5
21--0.001193
23. 40
53--9.603848
3
14 11--8.752399
1 18 17--8.357.449..
N. B. In three Superiours h, 4, d, if the Anomaly of
Commutation be less than fix Signs, they are Oriental;
if more, Occidental. But in the two Inferiours 2 and 9,
when
1
เ
Aftronomical Precepts.
361
Example 2. Anno 1728, Let the Place of Venus be rẹ-
quired April 16th Day at Noon, 1728 ?
Given Time Longit.
Anno 1728
[S
4 28 5557
55:57
Apr. 16 Biff. 5 21 25 56
Mean Mot.10 20 21 531
Equat. fub. O II 4
Anom. R
Node ?
"
S.
S.
6 21 586 2 14 12
5 21 25 41
013 22.47
Log. lào
Hel.Orb. Pl. 10 26 10 49 Log.à
21
4.862278
5.003469
Node fub.
2
14 12
Argum. Lat. 8
5 58
21 Ta. 35°50′ 461
28 Add45 0
9.858809
Reduct. fub.
2
12 C.T.80 50 46
9.207201
Hel.Ecl. Pla. 10 20
8 37
Ta. 38 36 36
9.902315
I
7 21 48
Tan. 7 19 57...
9.109516
Sum45 56 33 Parallax add
O's Pla. fub.
Anom.Com. 9 12 46 49 Diff 31 16 39 Elongation fub.
Half
4 21 23 24
1
8 36 36
Complem.
Parallax add I 15 56
Geo.Lon. ?
Dir. Orient.
O
6 5
For theLatit.
AsS.Co. C.A. 77 13 11--0.010895
10 To S. Elong.31 16 39--9.715321
10 SOT.Incl.SĂ. 3 5 34--3-732644
ToGe.La.SA. 1 38 50--8.45876 0
3
testatieateusutatatertiaintertoutuitectents etrctents
E e
}
Ex-
1
362
Aftronomical Precepts.
Example 3. Let the Place of Mars be fought June 11th,
Day at Noon 17282
Equ.Time. Long.
*
S. Q
Anom. &
Node d
:
S:
}
7
2
Argu. Latit.
I 17 42 26 Ta. 32°43 55
Anno 1728 4 16 18 21 15 12 21~17 42 26
June 11 Biff. 2 25 25 25 2 25 24 55
Mean Mot 7 11 43 46 210~37 16
Equat. fub
Hel. Orb.Pl.
Node fub.
:: 9:38 oc
4
Log Đào
5 46 Log ởàO
5.007229
"
5.199168
9.808061
5 14 23
20 Add45.00 cc
Redu&t. fub.
36.7.77 43 55
9.337362
3
2 m
I
5 3.7
Half
Dir.&Occid,
Hel. Ecl. Pl. 7
Sun's Place
Anom.Com 7 28 59 21 Diff. 39 28 38 Parallax fub.
3 29 29401
Complem. 2 0.30 20For the Lato
Parallax fub I 9 2838
Geo Long.5 22 37 38 So t.He.L.NDoo 29 52
6 16a. 60 30 20
10.247456
Ta. 21 1:42
9.584818
Sum81 32 2 Elongation add
As S. Com8 59 21 C. Ar. 0.066983
To S. Elong. 81 32 2
9.995242
Tot.Ge.L.NDoo 34 28
7.938894
8.001119
!
!
1
·
EXAM
Aftronomical Precepts.
363
S.
17
4
5 11 27
Example 4 Let the Place of Jupiter be required Auguft
26, 1728, at Noon?
EqualTime. Longit, 1s."
Anno 1728,
S.O
1 26 56
Aug. 26Biffex. o 19 52
Mean Mot.
Equat. add
2 16 48 21
#
Anom. 24
Node 24
"
S.
11.
7 16 48 533
7 57 30
30
0 19 51 16
86 40 9
Hel.Orb. Pl. 2 21.59 48.
Log. → à
5.002902
Node fub.
3 7 57
30
Log. 24 à
5.708603
Ta. 110 8' 20
9.294299
Argum. Lat. II 14 2 18
Add 45
O
Reduct. add
15
C.T. 56 8.20)
9.826714
}
5.13 59 16
Hel.Ecl.Pla. 2 22 O
Sun's Place
Anom.Com.2 21.59 13
'I' 10 59 36,
Half
Parallax add
Geo. Lon. 24
Dir.and Ori
3
1
For theLatit at the Earth.
IO 44 43 As S. Com C4.81°59′13″0.004261
To S. Elong
2 44 46 SOT.Incl.s.D
TòT.Geo.La.SD
71 14 29-9976296
: 00 21 47-7.801710
@0 20 49-7-782327
Tan. 40 59 36
9.939061
3 Tan. 30 14-53)
9.765775
Sum 71 14 29 Elongation fub.
Diff. 10 44 43 Parallax add
Ee 2
Ex-
2
364
Aftronomical Precepts.
t
1
Examples. Let the Place of Saturn be fought November
6, 1728, at Noon ?
Equal Time.
Long. H
Anom.
Node T
S. a
S.
"
S.
"
Anno 1728,
10 21 42
36
Novem. Biff
10 25
40 Ĩ
* 22 31 563 21
0 10 23 55
F3 30
M. Mot. 11
2
7 407
'd
Σ
2 55 51-
Equat. fub.
5.37 29
Hel. Or.Pla. 10 26 30 11 Log.
Node fub. 3 21 13
30
Log.
O
à O
་
- 4.994518
5.991720
Tan.5°44' 50'
Arg. Lätit. 7 5 16 41
Add 45 00
Reduct,fub.
I 33
CT. 50 44 50
Hel. Ecl. Pla.2 26 28
Tan.45 31 32
Sun's Place 7 25 25 35 Sum85 17 39 Elongation add
38 Tan.39 46' 7
9.002798
9.912283
10.007968
9.920251
Ano. C om.
8 28 56
Diff. 5 45 25 Parallax fub.
57
Half
4 14 28 28
Complem.
Í 15 31
32
For the Lat. at the Earth.
As S.Com.C.
A.88°56′ 57″0.000073
Parallax add
To S. Elong.
5 45
351
So T.Incl.S.A.
Geo.Longh 10 20 43
Dir.and Oc.
13
ToT:Ge=Lat.SA
85 17:39-9-998533
I 26 42-8.401831
I 26 25-8.400437
<
PRE-
1
Aftronomical Precepts.
365
!
1
PRECEPT 9.
Shewing how to find the Time when any of the Primary
Planets will be in their Aphelions, and Peribe-
lions.
First, You must underſtand, that when the Mean Anomaly
of a Planet is no Signs, Degrees, Minutes, nor Seconds,
then that Planet is in Aphelion; and if it be juft fix Signs,
then it is in Perihelion. Thefe things being known, fubtract
the Mean Anomaly for the given Year from 12 Signs, and
ſeek the remainder in the Months of that Planet, and what
Day you find it ftand againſt, is the Day that that Planet is
in its Aphelion,
2. Subtract the Mean Anomaly for the given Year from
'fix Signs, and feek the remainder as before, and you will have
the Day that it is in its Perihelion.
Example. Anno 1728, I would know the Days that Mer-
cury will be in Aphelion and Perihelion ?
From
OPERATION.
Aphelion.
S. Deg. Min. Sec.
I 2 00 DO 00
Sub.Anom.for 1728 9
Remains
1
8 42 44
2 21 17
and
Perihelion.
S. Deg. Min. Sec.
6 00 00
9 08 42 44
16 and 8 21 17 16
First, I feek in the Months of the Mean Motions of Mer-
and I find this Anomaly 2 S. 21° 17′ 16″ ſtand againſt
theſe Days, viz.
sury;
Fanuary 10
April 17(which are the Days that Mercury will be in
Fuly
14
October 10
Aphelion in the Year 1728.
Alfo
366
Aftronomical Precepts.
Alſo I ſeek the Anomaly 8 S. 21° 17' 16" as above, and I
find it againſt theſe Days,
4
March the Days on which Mercury will be in Peri-
SM
May 31
27 S
Aug. 27
Nov. 23
helion Anno 1728.
Example. In Venus, Anno 1728 ?
From
OPERATION.
At Aphelion
S. Deg, Min. Sec.
12
Sub. Anom. for 1728 6 21
Remains
Perihelion
S. Deg. Min. Sec.
6 oo .00 00
21 58 7
00 oo and
5807
6
5
8
53 and 11
8
I
53
April
Anse 1728 Aphelion.
Anno
Novem, 18
!
July 29, in Perihelion.
So Venus in the Year 1728, comes twice to her Aphelion,
and once to her Perihelion, as above.
Example in Mars, Anno 1728 ?
““
Perihelion.
S. Deg. Min. Sec.
00 00 and 6 00 00 00
Aphelion.
S. Deg. Min, Sec.
From
I 2
Sub. Anom: for 1728
11 1512 21
Remain
00 14 47 39
II
15
12
21
6 14 47 39
Aphelion Jan. 28. But doth not reach his Perihelion till Jan.
1729. For the Anomaly 6 S. 14° 47' 39" is not to be found
in the Months of the Mean Motions of this Planet.
Exam-
Aftronomical Precepts.
367
A
Example in Jupiter for 1728.
5 1
t
Aphelion
S. Deg. Min. eec.
From
12
O
and 6
O
Perihelion.
S. Deg. Min. Sec.
O
O
Sub. Anom. for 1728
7 16 48 53
7 16 48 53
/
Rcmains
4 13
II 7 and
10
13 II 7
Theſe Numbers cannot be found in the Months of the
Mean Motions of Jupiter; which proves he doth not come to
either of thoſe Points in the Year 1728.
Example in Saturn for the Year.
From
Aphelion.
'
S. Deg. Min. Sec.
12 00, 00 oo and
Sub.Anom. for 1728 1 22 31 56
Remains
Perihelion
S. Deg. Min. Sec.
6
00 Oo Oo
I
22 31 56
4
ΙΟ 7 28 4 and 4
7 28
Theſe Anomalies cannot be found in the Months of the
Mean Motions of Saturn; which fhews, he doth not come
to thoſe Points in the Year 1728. And thus I have given you
a New and Expeditious Method to find the Days when
the Planets will be in Aphelion and Perihelion. The Times
of the Earth's Aphelion and Perihelion are found the fame
way.
Example in the Earth for the Year 1728?
Aphelion.
S. Deg. Min. Sec.
From
12 00 Oo o and
Perihelon.
S. Deg. Min. Sec.
6 00 00
00
Sub. Anom. for 1728 6 11 58 52
Remains
5 18
6 II 58 52
1 8 and IL 18 I 8
The
$68
Aftronomical Precepts.
The Anomaly S. 18° 1' 8" anfwers to June 18, OR
which Day the Earth is in Aphelion: And the Anomaly 11S.
18° 18' I find the neareft unto it right against December 18,
the Day that the Earth is in Perihelion. But to find the pre-
cife Time of the Aphelion or Perihelion, you muft work as
in the Solar Ingreffes; thus, for the Time of the Earth's A-
phelion.
From
June 18, fub.
S. Deg. Min. Seo.
5:18 I 8
5 17 33 8
Rem.
Rem.
28
Ӧ
Hours II fub.
27
6
54 Thirds.
51 45
2 15
2
15
Minutes 21 ſub.
Rem.
Seconds 55, ſub.
Rem.
By which it appears that the Earth will be in Aphelion,
Anno 1728, June 18 d. 11 h. 21′ 55″ P.M.
For the Time of its Perihelion.
S. Deg. Min. Sec.
From
\II 18 I 8
Decembber 18, fub.
II 17 54 58
Remain
6 10
Hours 2, fub.
4 56
Remains
I
14 Thirds.
Minutes 30, fub.
1 13 $5
Rem.
5
Seconds 2, fub.
S
Remain
Anno
Aftronomical Precepts.
369
Anno 1728, December 18 d. 2 h. 302 the Earth is in Peri-
helion.
For proof of your Work, if to thoſe times above found
you collect the Anomaly of the Earth, you will find it in
the Aphelion to be nothing, and in the Perihelion fix Signs.
And after the fame manner you may find the precife
times of the Perihelions and Aphelions of the Primary Pla-
nets.
PRECEPT 10.
To find the Times of the Apogeon and Perigeon of the Sun
and Moon.
I
The Method for this is the very fame as has been ſhewn
in the laft Precept; but here you are to Note, that when
I mention the Earth's or Sun's Anomaly, it is all one and the
fame thing; for 6 S. 11 deg. 58 min. 52 feconds is the
Earth's Anomaly as well as the Sun's for the Year of
Chrift 1728, Current; and that the time of the Earth's
Aphelion is alfo the time of the Sun's Apogeon; and the
time of the Earth's Perihelion, is likewife the time of
the Sun's Perigeon, which were found in the laſt Precept,
and ſo needs not be repeated here: But the Places of the
Earth and Sun are ever Diametrically oppofite.
Secondly, Becauſe of the Moon's fwift Motion, fhe tran-
fits the Points of the Apogeon, and Perigeon feveral times
every Year; therefore let it fuffice to find the true time
of her tranfiting her Apogeon in January in the Year
1728.
~
A
A
F.f
OPE
370
Aftronomical Precepts.
OPERATION.
From
S. Deg. Min. Sēt.
12
Sub. Mean Anomaly 1728,
Rem.
Fan 5, fub.
2 2
9 20 50 15
9 9 45
$ 19 30
Rem.
3.
Hours 7 fub.
3
50 15
48 38
Rem.
I
3.7.
Minutes fub.
I
5
Rem.
32
Seconds 59 fub.
32
Rem..
So the Moon is in Apogeon 1728, Fan. 5th 7h. 2 min. 59 fe-
conds. And in 27 deg. 31 min. 9 feconds.
For the Time of her next Perigeon.
S. Deg. Min. Sec.
6.00
co 00
9 20 50 15
1
From
Sub. Anomaly for 1728
Rem.
Fan, 19, fub.
.8
∞ ∞
8
9
9 45
8 14 5
Rem.
55 40
Hours 22 fub.
32 39
Rem.
23 I
Minutes 42 fub.
22 52
Rem.
9
Seconds 17 fub.
9.
Rem
、;
}
Se
Aftronomical Precepts.
37x
So that the Moon is in Perigeon Jan. 19th 1 h. 46° 17!',
1728, in m 29 deg. 3 min. 15 feconds. And thus you may
find in every Month of the Year the equal time that the
Moon is in Apogeon, and in Perigeon: For when you
have fubtracted the Mean Anomaly for the Year from 12, .
and 6 Signs ſeverally, thoſe remainders may be found in e-
very Month of the Year, which has been taught above.
You will have the times of the Moon's Tranfiting thofe
two Points in her Syftem.
D. H. M. S.
5 7 2 59 in 27° 31 min. 9 fec.
20 21 31
!Fan.
Feb.
1
Feb.
28
9
40
8
March 27
22
58
42
Anno
1728
April 24
May 221
I 2
17 17
35
50
in
Fune
18
14
54 24
Apo-
Fuly
16 4
12 57
Aug. 12 17
31 32
geon
Sept.
9
6
50
6.
Octob.
6
Nov.
Nov.
30
Dec.
28
30∞
(Jan.
20 8 43
9 27 15
45 48
22
12 4 19 in Ŏ 7º 25 min. 33 fec.
D. H. M. S.
I 42
42
17 in m 29° 3 min. 15 fec.
19
Feb 1,5 15 Oo 52
Mar. 14 4
19 24
April 10 17
Anno 1 May
37
37 56
8 6 58 22
1728 Fune 4
> in Fuly 2
Peri- July
20° 15
92
29 22
33
8
40
52 15
geon Aug. 26 12 9 19
Sept. 23 3 19 35
o&. 20 14 47 58
Nov. 17 4
Dec. 14 17 25.
6 32
2 ) in m 5º 53 min. 28 fec.
And
!
Ff 2
372
Aftronomical Precepts.
1
And as I have given you here the Moon's Mean Place a
the firſt and laſt times that fhe is in Apogeon and Perigeon
in the Year 1728; fo you may find her Place at the other
times, as fet down, if you do but collect her Mean Motions
to thoſe times feverally.
t
PRECEPT II.
To find the Time of the Retrogradations of the Pla-
nets.
What I intend here, is to find out the Days when the Pla-
nets become Retrograde; and firft, you are to understand,
that Saturn and Jupiter are Retrograde every Year; Mars
once in two Years; Venus is fix times Retrograde in the ſpace
of Eight Years; Mercury three or four times every Year.
In order therefore to make the Work plain and eafie, we
muſt have the Angle at the Sun, or Anomaly of Commuta-
tion when the Planet becomes Retrograde; and altho 'tis
impoffible (by reaſon of the Different Pofitions of the Earth
at different times) to fix this Angle to each Planet fo as to be
Perpetual; yet that it may be of fervice herein, I have fta-
ted that Angle to each Planet, as is here fet down.
Ret. Limit.
Dir. Limit.
S. D. M.
路
​3 24 30
S. D. M.
8 ö
4
630
be more than <5
8
7 23 30
6 22
5 15
6 15
4 28
)
7
2
ده
the Planet is
Retrograde.
RULE. Subtract the Mean Longitude of a Superiour
Planet for any given Year from the Mean Longitude of the
Sun for the fame Year, and that is the Mean Anomaly of
Commutation; which, if it be not between the Retrograde
and Direct Limit, (as fpecified by the Table above) then
the Planet is Direct: And to find when it will become Re-
trograde, fubtract the Angle at the Sun fo found from the
Retrograde Limit, and feek the remainder in the Month
of the Solar Tables for all the Planets except Mercury, and
fee
Aftronomical Precepts.
373
fee what Day it anſwers to; on which Day compute the
Longitude of the Sun, and Heliocentric Place of the Pla-
net; find the Angle at the Sun now; and if it be ſhort of
the Retrograde Limit, fubtract it from it, and feek the remain-
der in the Solar Tables, and add this time to the time firſt
found in the Solar Tables; to this time compute the Places
of the Sun and Planet as before, and the Angle at the Sun;
and if it is yet fhort of the Retrograde Limit, fubtract it
from it, and work as before, till you find it agree with the
Retrograde Limit, and then you will have the time that the
Planet becomes Retrograde.
Example. In the Year 1727, I would know when Saturn
becomes Retrograde.
OPERATION.
1727 Mean Long. of{
Angle at the Sun
Retrograde Limit
In the Solar Tables
So
May 1575
Ş. Deg. Min. Sec.
9 20 26 3
ħ 13 9 29 14
II 10 56 49
3 24 30
4 13 33 11 is May 15.
'༡
4 35
3
10 9 39
44
3 24 55
19.
Angle at the Sun
This agreeing
with the Limit, fhews Saturn becomes Retrograde May
15.
And if to this Point you add a Year and 12 Days, it
will give May 27, 1728, the Day that Saturn becomes
Retrograde. See the Table in my Syftem of the Planets De-
monftrated, Page 102.
1
Exam-
374
Aftronomical Precepts.
Example in Mercury,
S. Deg. Min. Sec.
ŏ
1728 Mean Long.of
5 21 50 41
9 20 II 43
Angle at Sun
8 1
38 58. This being more.
than the Direct Limit, fhews the Planet to be Direct.
Retrograde Limit
Angle at Sun fub.
In Tables
S. Deg. Min.
4 28
8
Σ 39
26 21 is March 5.
S. Deg. Min. Sec.
2 20 26 57
March 5,{8
Angle at Sun
Limit Retrog.
11
26
9 37
24 17 20
2
4 28
Oo
00
t
Diſtance
2 3 42 40 give 15 Days in Mer-
cury's Tables; added to March 5, give 20 of March, the Days
that Mercury becomes Retrograde; to which add 125 Days
(ſee the above-cited Book, Page 66,) and that points out July
23, when Mercury becomes Retrograde again. He is now
in Virgo; add 112 Days, and that points out November 12,
when Mercury becomes Retrograde a third time in the Year
1728. And thus you may proceed in any other Planet; by
adding the Diſtance of Days from one Retrogradation to a-
nother, you will nearly have the Day of the next Retrograda-
tion of the fame Planet; which Diftances between each Re-
trogradation of all the Planets you have Tables of in my
Book above-mentioned,
PRE-
Aftronomical Precepts.
375
PRECEPT 12.
To find the Times of the Mutual and Lunar Af
1
pects.
To perform this, you must first have the Motion of all
the Planets computed to the Noon of feveral Days fuccef-
fively; as, for a Month, or for a Year, &c. And for the
Mutual Aspects, it doth fuffice to find the Day only, becauſe
of their Slow Motions; but becauſe of the Moon's fwift
Motion, her Aſpects with the other Planets are (or ought
to be) computed to the precife Time.
Then having the Motions of all the Planets in readineſs
for a Month, begin with any two of them, and guide
your Eye down their Columns, and fee if you can find
the fame Degree and Minute of an Aſpect-Sign (a Table
of the Aſpects you have in my Aftronomy before-mention-
ed, Page 17,) for that is the Day of the Afpect.
Example. This Year 1727, I look in the Month of No-
vember, and I compare the Places the Sun and Saturn toge-
ther, and find on the 22d Day the Sun's Place at Noon
10 deg. 52 minutes, Saturn 10 deg. 36 minutes; this be-
ing two Signs afunder, makes the Sextile-Afpect; but the true
Time was before Noon that Day; becauſe the Sun, being
the Swifter Planet, is a few Minutes lefs than two Signs
diſtant from Saturn. The true time of this Aſpect is found
by the Logiſtical Logarithms, thus,
Diur. Motion of
1 {{
Deg. Min.
I
00
O
-5 Diſtance 21 Day at Noon 44′
Diurnal Motion of à O.
55
Now
1
876
Aftronomical Precepts.
H. Min.
.32
Now fay,
Òì 55 Co.ÁN LiL. 622
If
Give 24 00
What o 44
Answer 19 12
3979
1347
4948
By which the true time of the Afpect is 21 d, 19 h. 12
minutes.
Alſo in the fame Month, I compare the Places of the Sun
and Jupiter together, and find on the 8th Day at Noon the
Sun in Scorpio 26 deg. 41 minutes, and Jupiter in Taurus 26
degr. 31 min. Retrograde; that is, a few Minutes in Mo-
tion paſt the Oppoſite
Diurnal Motion of {
J
à 2
Deg. Min.
I
Ο
6 8
Diurnal Motion
I 9 Diſtance at Noon
o deg. 59 minutes; therefore the time of this Oppofition
in November 7 d. 20 h. 30 minutes: And after this manner
I compare the Sun's Place with the other Planets; by which
I diſcover all the Aſpects that he makes with them. Then
I take Saturn's Place, and compare it with the Places of Jupi-
ter, Mars, Venus, and Mercury feverally; by which i fhall dif-
cover all the Aſpects that he makes with them. Then I
compare the Place of Jupiter with Mars, Venus and Mercury,
and his Afpects with them are difcovered. Next, I compare
the Place of Mars with the Places of Venus and Mercury; and
Laftly, I compare the Places of Venus and Mercury together;
and if they form any Afpects with each other, I fhall dif-
cover the Day whereon it falls; and the Hour and Minute
may be found by the Logiſtical Logarithms, as is fhewn a
bove; obferving, if the Planets are both Direct, or both
Retrograde, that you take the difference between their Diur-
nal Motion; but if one be Direct, and the other Retrograde,
the Sum, and this Sum or Difference, fhall be the Diurnal
Motion of the Swifter Planet from the Slower. And after
the like manner muft you examine each Month in the Year;
by which Method, not one Mutual Aſpect can eſcape your
ſpection.
In-
Se-
1
1
1
Aftronomical Precepts.
377
Secondly, For the Lunar Aſpects.
Compare the Longitude of the Moon with every Planet
feverally, as has been fhewn in the Primary Planets a-
above, and you will diſcover the Days of the Lunar A-
fpects; the Hour and Minute may be had by the Tables of
Lunar Afpects, Page 67, &c. by entring the Table with the
Diurnal Motion of the Moon from the Planet; and the firſt
Column on the left hand, with the Diſtance of the Moon and
Planet on the Day at Noon before the Afpect, and the An-
gle, or place of Meeting, is the Hour and Minute of the
Time of the Afpect.
Example. Anno 1727, in November, I would know the time
of the Conjunction of the Moon and Jupiter?
By comparing the Longitude of the Moon with that of
Jupiter in the faid Month, I find that fome time be-
tween the 16th and 17th Day at Noon they will be
Conjoyn'd.
OPERATION.
S. Q
3
at Noon is2
Į
Place of {17} Day
D
Diurnal Motion of
Add
1
Diurnal Morion ◄ à 2
I 20 19.
S...
1 25 19
12525
12 46 of 24
8
8
(Retrog
1254
S.
Deg. Min.
Place of{
16 Day at Noon 1
25 27
1 20 19
Their Dift. at Noor
5 8
Ey entering the Table of Lunar Aſpects, with the Diur.
nal Motion of the Moon from Jupiter, and their Diſtance at
Noon, as before directed, you will find the time of the Con
junction to be the 16 d. 9 h. 33 minutes. And in a Conjun-
ction of the Moon with the Planets, if you have regard to
their Latitudes, you may difcover whether there will be an
Occultation or not. In the Example before us, the Latitude
of the Moon is 4 degr. 36 min. North, and the Latitude of
Jupiter 1 degr. 4 minutes South; which added together,
G g
make
378
Aftronomical Precepts.
make 5 degr. 40 minutes, their Difference in Latitude; by
which I fee the Moon paffes far above Jupiter at the Con-
junction; and therefore free from Occultation. This Me-
thod is to be obferved in the Moon and other Primary Pla-
nets, whether it be a Conjunction, Sextile, Square, Trine, or
Oppofition.
PRECEPT
13.
Shewing how to Determine the Ecliptic Boundaries of the
Sun and Moon.
Firft for the Moon.
OPERATION.
The Moon's Perigeon Horizontal Parallax
Sun's Horizontal Parallax add
Sun's Apogeon, Apparent Semidiameter fub.
Min, Sec.
61 24
00 10
Sum
61 34
15 49
45 45
16 40
Sum
62 25
Greateſt Apparent Semidiameter Earth's Shadow
Moon's Perigeon Semidiameter add
i
In the Table of the Moon's Latitude, Page 58, the Argu-
ment of Latitude anſwering to this Latitude 62 min. 25 fe-
conds, is 12 degr. 1 min. 22 feconds; that is, before and
after 6 or 12 Signs. For if the Diſtance of the Moon from
either Node at the time of the true Oppofition to the Sun,
be less than
great. Limit.
S.
| G |
S.
12° 1′22″, or more than
12° 1′22", or more than 170 58′38″.
|,
II
The Moon at that Full will be Eclipfed; elfe not. And
if at the time of the Oppofition of the Sun and Moon, the
Latitude of the Moon exceed the Sum of the Semidiame-
ters of the Moon's and Earth's Shadow, the Moon at that
time
Aftronomical Precepts.
379
time will not be Eclipfed; but if lefs, fhe will. See the
Word Limit in the Definitions. For the leaft Limit work
thus.
Min. Sec.
Apog. Horizontal Parallax Moon
Sun's add
Sum
Sun's Perig. Semidiameter fub.
Appar. Semidiameter Earth's fhadow.
Moon's Apogeon Semidiameter add
Sum
54 59
о ΙΟ
55 59
16 22
38 47
14 54
53 41. The
Argument of Latitude anſwering this Latitude thus, 53 min.
41 fec. is o S. 10 degr. 19 min, 17 feconds: That is, before
and after Six and Twelve Signs.
S.
Thus :
S.
10 deg. 19 min. 17 fec. or || 19deg. 40 min. 43 fec.
And the Mean Limit is o S. 11 deg. 5 min. 4 fec.
4
S.
Thus :
S.
11 deg. 5 min. 4 fec. or 18 deg. 54min. 56 fec.
I I
Secondly, To determine the Ecliptic Boundaries of the Sun
Perigeon Horizontal Parallax of the Moon
Sun's fub.
Min. Sec.
61 24
O IO
Rem.
Perigeon Horizontal Semidiameter of
14
16 22
16 40
..
Sum
G g 2
94 16
In
380
Aftronomical Precepts.
1
In the Table of the Moon's Latitude, the Argument of
Latitude anſwering to this Latitude 94 min. 16 feconds, is
18 degr. 20 min. 8 feconds; that is, before and after Six
and Twelve Signs.
.་
Thus,
S.
S.
8} 180
{
-189 20′ 80″ or{5}11° 39′52″ Greateſt Limit.
For the leaft Limit.
Apogeon Horizontal Parallax of the D
Min. Sec.
54. 59
10
Sun's fub.
Difference
54 49
Apogeon Semidiameter of{
15 49
14 54
Sum, to > Latitude
85 32
The Argument of Latitude anſwering to this Latitude 85
min. 32 feconds, is o S. 16 deg. 35 min. 5 ſeconds the leaft
Limit; that is, before and after Six and Twelve Signs.
S.
Thus;
S.
fec. And
0 | 16 deg. 35 min. 5 fec.|5|13 deg. 24 min. 55 fec.
6
the Mean Limit, that is, when the Luminaries are at a mid-
dle diſtance from the Earth, is 17 deg. 21 min. 52 ſeç. before
Six and Twelve Signs.
S
8 | 17 deg, 21 min. 52 fec. or
}
Thus,
S.
|
12 deg. 38 min. 8 feconds. Sa
that when you are ſeeking an Eclipſe of the Sun, you muſt
make uſe of the Limit the Sun is neareſt ro; as, if he be
in Apogeon, take the leaft Limit, &c.
And
}
Aftronomical Precepts.
381
And if at the true time of the true Conjunction of the
Sun and Moon, the Moon's true Latitude be less than the
Sum of the Apparent Semidiameters of the Sun and Moon,
added to their Differences of the Horizontal Parallaxes, the
Sun will then be Eclipfed fomewhere on the Earth; elfe
not.
Otherwiſe, If at the Apparent Time of the Viſible Con-
junction of Sun and Moon, the Viſible Latitude of the Moon
be less than the Sum of their Apparent Semidiameters, then
the Sun will be Eclipfed at that Time and Place on the
Earth.
But if the Moon's Vifible Latitude exceed that Sum,
then will the People of that Place who behold the
Moon's Visible Latitude to be fuch, fee no Eclipſe at
all.
PRECEPT 14.
To find in any Year, how many Eclipfes there will
be, and in what Months they happen.
First, You are to observe, that the Sun enters the
Twelve Zodiacal Signs on thefe Days, as hereunder ſet
down.
January, February, March, April, May, June, July, Auguſt,
9
7
9
*
ԴՐ
9 10
୪ II
10
12
12
૧.
m
September, October, November, December.
12
12
II.
IO
M
Z
VS
2. Look into the Table of the Moon's Mean Motions for
the given Year, and fee what the Radical Place of the
Moon's North Node is; for in thoſe Months in which the
Sun enters the Signs that the Moon's Nodes are in, will the
Eclipfes of the Sun and Moon fall in that Year. And the
Moon's Nodes being always Diametrically Oppofite, if
there happen an Eclipfe in Fannary, there will alſo be one
in July; becauſe the Sun enters Aquarius in January, and Lea
in July, Aquarius and Leo being Oppofite Signs, &c. And
if the Nodes change their. Signs in that Year in which you
are
382
Aftronomical Precepts.
ľ
are ſeeking the Eclipfes, then in the Months preceding the
Months above found, will there alſo be an Eclipfe; and
thefe Months I call the Node-Months.
1
3. By Precept 7, find the Equal Time of the New and Full
Moons in the Node-Months, and alfo in the Months next before
and after the Node-Months, (by which means you will be
fure not to miſs the Eclipſes that Year.) Set down the true
Places of the Luminaries at the New Moon, and the Place
of the Moon at the Full; and from thefe Places feverally
fubtract the Place of the Moon's North Node for the time
given; and this Remainder is called the Argument of the
Moon's Latitude; which, if it be leſs than the Limits of E-
clipſes ſet down in the laſt Precept, there will be an Eclipſe at
that time, elſe not.
A
Example. Let it be required to find how many Eclipfes
there will be of the Sun and Moon in the Year of Chrift
1730, and alſo in what Months they happen?
The Method of your Examination for the whole Year will
ftand thus:
The Year 1730, is the fecond paft Leap-Year; the Epact
is 22; and the Radical Place of the Moon's North Node is
Aquarius 6 deg. 29 min. 16 feconds, but comes back into
Capricorn on May 1. Confequently, the Months in which the
Eclipfes will happen, are January, February, June, July, Auguft,
and December. Note, Altho' the Eclipfes cannot happen
in ail the Months above written, yet they are the Months
in which you muft examine the Lunations, and by
that means you will be fure not to miss any Eclipfe that
Year.
New D 1730, Jan. 7 d. 6 h. 57′ 44″ in 9
North Node fub.
S. Deg. Min. Sec.
28
33
I I
10
6 6
6 Sun's
11
22 27 5 Ecl.
Ecliptic Bounds Sun are from
11 11 39
52 Invif.
To ..
0 18
20
Argument Latitude
T
Full
Aftronomical Precepts.
383
S.Deg. Min.Sec!
Full) 1730 Jan. 22 d. 16 h, 18″ 38″ ) in 4 14 FI
North Node fub.
Argument Latitude
Ecliptic Bound. Moon are from
To
I
10
5 17 13
6 8 53 48
5 19 49 43
6 10 19 17
New 1730 Feb. 5 18 28 38 in 10 28 25 21
Argument Latitude
› Eclip.
viſible.
North Node fub.
10 4 32 26
0 23 52 55
Paſt the
Bo,noEcl.
Ecliptic Bounds Sun are from
∙11 12 38 8
To
O 17 21 52
North Node fub.
Argument Latitude
Full 1730, June 18 21. 319 in 9 8 8 47fhort of
9 29 29 47 thort of
27 29 31
II 10 39 16
Boun.
no E-
Ecliptic Bound. Moon are from
11 17 58 38
clipfe.
To
0 12 I 22
New D 17.30, July 3 17 14 39
14 39
in
in
3 22 17 9 Sun is E-
North Node fub.
Argument Latitude
Ecliptic Bound. Sun are from
To
3 22 17
9 26 42 22(clipfed
5 25 34 47 part vi-
5 13 24 55ible.
6 16 35 5
Full 1730, July 18 4 26 24 in 10 6 6 48
North Node fub.
Argument Latitude
Ecliptic Bound. Moon are from
To
Moon is
9 25 56 24 Eclipfed
0 10 10 24 Invifible.
11 17 58 38.
O 12 I 22
New 1730 Aug. 2
1730 Aug. 2 8 34 55 in 4 20 39 52Paft the
North Node fub.
Argument of Latitude
Ecliptic Bound. Sun are from
To
9 25 8 8 9 Boun. no
6 25 31 43 Eclipfe.
5 13 24 55.
6 16 35 5
1
L
L
Fall
$84
Aftronomical Precepts.
d. h. min. fec.
Full 1730, Dee, 12 20 26 5 >
North Node fub.
Argument of Latitude
Ecliptic Round. Moon are from
To
S.Deg.Min.Sec.
1
in 3 2 22 5hort of
9 18 7 13
5 14 15 38
5 19 49 43
6 10 19 17
New 1730, Dec. 27 22 9 45 in 9 17 44 50
North Nade fub.
Argument Latitude
Ecliptic Bound. Sun are from
To
+
9 17 19 18
theBoun.
no Eclip.
Sun isEcl.
invifib.by
0 0 25 32 reafon of
II 11 39 52her great
9 18 20 8 Parallax.
Si
By the Work above, I have examined all the New and
Full Moons in the Year 1730, that are poffible of produ-
cing an Eclipſe; and I find within the Circumference
thereof, there will be five Luminarian Eclipfes, viz. three
of the Sun, and two of the Moon: And for this purpoſe
alſo, I have Calculated the following Table; which, if
you enter with the Moon's Mean Anomaly at the time of
any Eclipſe, and take out the Argument of Latitude, and
add it to the Augument of Latitude at the rine of any
Eclipfe, that will fhew you whether the next Lunation
will produce an Eclipfe or not: For if the Sum, be within
the Limits of Eclipfing (as determin'd in the laft Pre-
cept) there will be an Eclipfe; elſe not.
•
ΑΤΑ
Aftronomical Precepts.
385
!
rian Eclipfes.
Mean Anom. D
A TABLE of the Mean Motion of the Argument of
Latitude of the Moon, for discovering of the Lumina-
Argument Latitude.
1
S.
Deg. S.
S.
Deg.
Min. See:
00
00
6
14
04
35
15
II
6
14
17
05
T
00
II
6
14
29
35
I
15
ΙΟ
6
14
42
2
10
6
14
54
35
2
15
9
6
15
07
05
3
00
9
6.
15
.19
35
3
15
8
6
IS
32
05
4
00
8
4
15
7
00
7
aa a
15
44
35
15
57
05
16
09
35
5
15
6
6
16
22
05
6
00
6 6
16
34
35
Example. I have found that the Sun is Eclipfed the 28th
of December 1730: I would know at one View whether
the next full Moon in January 1731, will be Eclipſed or
not?
OPERATION.
Moon's Mean Anomaly
Argum. Latitude December 28, is
Augument Latitude per Table
Argument Latitude
S. Deg. Min. Sec.
5 19 47 18
Oo
00
25 32
6 16 26 05
6 16 51 37
This Sum, is the Argument of Latitude at the Full Moon
in Fanuary 1731; which far exceeding the greateſt Limit
of the Moon's Eclipfe, proves that, that Full Moon will
paſs below the Earth's Shadow; and confequently free from
any Obſcurity.
Hh
Note,
386
Aftronomical Precepts. -
Note, When the Argument of Latitude falls near the Li-
mit, then you muſt carefully examine that Lunation as has
been taught in Precept 13; otherwife 'tis poffible you may
mifs of diſcovering à ſmall Eclipſe.
PRECEPT 15.
To Calculate an Eclipfe of the Moon.
First, In order hereunto, you must fet down the Calculati-
on of the Sun's and Moon's Place to the Equal Time of the
True Orbit-oppofition; and for' an Example, I fhall take the
Moon's Eclipfe which I have found in the laft Precept to hap-
pen January 23, in the Morning, Anno 1730, the time of the
true Oppofition found as has been fhewn in Precept 7, and
ftands thus:
Eq.Time & Longit.O
S.
Longit. Anom. O ,,
S. 0
Anno 1730
9 25 42 12
January 22
o 21 41, 4
6 12 27 14
o 21 41. O
Hours, 16
Minutes 18
39 25
44
39 25
44
Seconds 36
Mean Mot. 10 13
Equat. add
Sun's Place.'I® 14 II
2
2
3 26
7 4 48 25
I 7 34
1
Equal
Aftronomical Precepts.
387
Eq. Time 8 Longit, Anom,
>
>
Node >
S. o
S. O
S.
Anno 1730
6 19 42 42
4
I 20 35 10
6 29.16
Jan. 22,
9 19 52 50
9
17 25 47
I 954
Hours 16
8 47
3
8 42 36
1
Minutes 18
9
53
Seconds 36
20
9 48
27
20 !
Sub. 1 12 3
Mean Mot. 4 18
32 48
1 27 39 610 5 17 13
Equat. fub.
4
21 48
>
in her Orb 4 14 II 00
Node fub.
IO 5 17 13
8 53 47
Argu. Latit. 6
Tr.Lat.S.A.
Reduct. fub.
46 20
2
Eclip. Place 4 14 9
O
1. With the Mean Anomalies of the Sun and Moon, take
out of the Table, Page 62, their Hourly Motion, thus:
Hourly Motion of?
29
O
Min. Sec.
2 32
31
19
Hourly Motion of Dà 28 47
Now for time of Reduction, fay,
As Hourly Motion of Moon from Sun
To one Hour, or
So is Reduction
To Time of Reduction
Min. Sec.
28 47 L.L. 3190
60
2 Oo
4 ΙΟ
14771
11588
This time of Reduction thus found, (in any Eclipfe) ap-
ply'd to the equal Time firſt found, according to its firft Ti-
tle, gives the Equal Time of the Middle of the Eclipſe ;
and apply'd contrary to its firft Title, gives the Equal Time
of the true Ecliptic Oppofition. Thus, if the Latitude of
the Moon be Afcending (either North or South). then the
time of Reduction must be fubtracted from the Equal Time
of the true Orbit-Oppofition; the remainder is the Equal
Time of the Middle of the Eclipfe: Add the Time of Re-
Hh 2
ducti-
388
Aftronomical Precepts.
duction to the
Equal Time of the Orbit-Oppofition, and
the Sum is the Equal Time of the true Ecliptic Oppofi-
tion.
2. But when the Latitude of the Moon is Defcending (See
the Schemes, Page 61,) which is when the Argument of
Latitude is more than 5 or 11 Signs, and less than 6 or 12,
add the time of Reduction to the Equal time of the true
Orbit-Oppoſition, gives the middle of the Eclipſe; and ſub-
tracted, you will have the true time of the Ecliptic Op-
pofition. Thus in the Eclipfe before us;
D.
Eq. time of the true Orb. 8 at Lon.1730 Jan. 22
Time of Reduction fub. and add
Equal Time of the Middle of the Eclipſe
Equal Time of the Ecliptic Oppofition
Equation of time fubtract
S Middle
Ap. Time of the Ecliptic 8
16
H. M. ·S'
18 36
4 10
22
16
14
26
22
16
22
46
14 24
22
16
00 2
22 16
8 22
3. With the Mean Anomalies of the Sun and Moon, take
out their Horizontal Parallaxes (the Sun's being ever 10 fe-
conds) and Apparent Semidiameters, and from the Sum of
the Horizontal Parallaxes, fubtract the Apparent Semidia-
meter of the Sun; the remainder will be the Apparent
Semidiameter of the Earth's Shadow that the Moon at that
time paffeth through.
Horizon Parallax of{
Sum
Semidiameter Sun fubtract
Appar. Semidiameter Earth's Shadow-
Semidiameter Moon add
Sum
Moon's true Latitude fubtra&t
Remain the Parts deficient
Min. Sec.
10
56 14
56 24
16 18
40
6
IS 15
21
'55
46 20
9 !
I
Hence,
Aftronomical Precepts.
389
Hence, becauſe the Parts deficient are leſs than the
Moon's Diameter, it flrews, the Eclipfe will not be total;
but if they be equal, then the Eclipfe will be total without
continuance. But if the Parts deficient be more than the
Moon's Diameter, then the Eclipfe will be total with con-
tinuance.
Now for the Digits Eclifped, fay,
Min. Sec.
As the Semidiameter Moon
15
15 L.L. 5949
As to Six Digits
6
00
00
10000
So are the Parts deficient
9
I
To the Digits Eclipfed
3 32
51
8231
12282
4. To find the Scruples of Incidence, or Motion of half
Duration.
This may be performed Four ſeveral ways.
1. By the 47th of the firft of Euclid.
2. By Trigonometry.
3. Logarithmetically.
4. By Shakerley's Logiſtical Logarithms.
Firft, In the right-Angled plain Triangle AP M, right-
Angled at P, there are given in the following Scheme, A P,
the true Latitude of the Moon, at the time of the true Op-
pofition 46 min. 20 feconds, and A MPN the Sum of
the Moon's Semidiameter and Earth's Shadow 55 min. 21 fe-
conds, to find PM PN the Motion of half Duration.
OPE-
390
Aftronomical Precepts.
ན་་
OPERATION.
Min. Sec.
Min. Sec.
{
46 20
55
21
60
67
2780
3321
2780
3321
222400.
3321
19460
6642
5560
9963
9963.
7728400
Square of A M, 11029041
Square of A P fubtract 7728400
Extract the Square Root of 3300641(1817 fec. fquare Root
in feconds; which Divided by 60, gives 30 min. 17 fec. =
BC, the Motion of half Duration of the Eclipfe.
Secondly, By Trigonometry.
1. For the Angles at A and M.
As Sum Semidiameters A M
To Radius
So Moon's Latitude A P
To C. f. Angle BAM
Deg. Min.
3321
3.521169
3.444045
90 00--10.000000
2780
33 10-- 9.922776
As Radius
Again,
Deg. Min.
90
To Z Semid. Moon's and Earth's Shad. 3321
0--10.000000
3.521269
So S. Angle P A M
To P M Motion of half Durat.
fore.
Ľ
33
10-- 9.738048
1817
3.259317 as be-
J
Thirdly,
Aftronomical Precepts.
Thirdly, Logarithmetically.
RULE. The Rect-Angle made of the Sum, and Dif
ference of any two Numbers, is equal to the Difference
of the Squares of thofe Numbers. That is, take the Lo-
garithms of the Sum and Difference of the Semidiameters
of the Moon's and Earth's Shadow, and of the Latitude of
the Moon; the half Sum of the two Logarithms is the
Logarithm of the Scruples of Incidence or half Dura-
tion.
OPERATION.
Sum of Semid. D's and 's Shad. in Sec. 3321
Latitude Moon in Seconds
2780
Sum
6101--3.785401
Difference 541--2.733197
Sum of the Logarithms
6.518598
Half Sum of the Logarithms.
1817--3.259299 is the
Half Durations in Seconds, as found above.
Lastly, By Shakerley's Logiſtical Logarithms.
RULE. Subtract the Logiſtical Logarithm of the Sum
of the Semidiameters of the Moon's and Earth's Shadow,'
from the Logistical Logarithm of the Moon's Latitude,
the remainder is the Sine of an Arch; to the Co. Sine
of which Arch, add the Logiſtical Logarithm of the Sum
of the Moon's and Earth's Shadow; this Sum, fhall be the
Logiſtical Logarithm of the Scruples of Incidence of half
Duration,
3
OPE-
i
|
392
Aftronomical Precepts.
1
OPERATION.
Min. Sec.
Latitude of the Moon
46 20 LL
9.88774
Sum of the Semid.) and ✪ fhadow
fhadow 55 21 EL
9.96496
Remains the Sine of
56 50
9.92278
Co. Sine of
$6 50
9 73805
Sum Semid. Moon and Earth Shad.
55 21 LL
9.96496
Scruples of Incidence
.30 17 LL
9.70301
5. To find the Time of Incidence, or half Dura-
tion, and from thence the Beginning, and End of the E-
clipse.
By Street's Logiſtical Logarithm, ſay,
As true hourly Mor. Moon from Sun
To one Hour, or
Min. Sec.
28 47 LL.
3190
60
00
0000
So are the Scruples of Incidence
.30
17
2969
To Time of half Duration
63
8
221
For the Beginning and End of the Eclipfe.
Apparent Time of the Middle
Time of half Duration fubtract and add
Appar. Time of the Beginning
Ending
D.
H.
M. S.
22 1.6 00 2
I 3 8
22
14 56 54
22 17 3 10
6. To find the Latitude of the Moon at the Beginning and
End of the Eclipfe.
The moft exact way is to Calculate the Place of the
Moon in Longitude and Latitude by the 6th Precept. But
because that is fomething troubleſome, and the ufe of her
Latitude being for no other end than to ferve for Drawing
the Type, or a Repreſentation of the Eclipfe in Plano, there-
fore the following practical Method is fufficient for this pur-
poſe.
First,
Aftronomical Precepts.
-393
1
First, Find the Motion of the Sun in time of Incidence
and add it to the Scruples of Incidence,
As one hour, or
Is to Sun's hourly Motion
So the time of Incidence
To Motion Sun in that time
Scruples of Incidence add
Sum
Min. Sec.
60 00 LL
2 32
13745
63
8
2
40
221
13524
30
17
32 57 Subtract this Sum,
from the Argument of Latitude at the Middle, gives the
Argument of Latitude at the Beginning; and added, gives
the Argument of Latitude at the End; by which Arguments
of Latitudes find the Moon's true Latitude anſwering there-
unto by the Table, Page 58.
OPERTION.
S. D. M. S.
Argument of Latitude at the Middle
Sum, fubtract and add
6 8
53 47
32 57
Argument Latitude at End
Beginning
6 8
20 50
9 26 44
Min. Sec.
Beg. 43
307
Hence, the Latit. at
South Afcending.
End 49 10
1
Note, The Latitude is Afçending either North or South,
until the Moon be three Signs diftant from her Nodes; be-
cauſe it increaſes all that time; but if the diftance be more
than three Signs, then 'tis Defcending towards the Nodes,
and therefore the Latitude decreaſes.
From the foregoing Calculation I have found the
Appar. time at
London of
the
D. H. M. S.
Begin. of the Eclip. 1730, Jan. 22 14 56 541
Middle
Ecliptic 8
End
16
00
02
16
8
P.M.
22
17
3
10
Total Duration
Digits Eclipfed are
2
6
16
3 32
51
I i
To
.394
Aftronomical Precepts.
7. To delineate the Eclipfe of the Moon in Plano.
1. From the Line of Lines on the Sector opened to any
convenient Radius, (or from any Scale of equal Parts) take
the Semidiameter of the Earth's Shadow in your Compaffes,
and fet one Foot in A; defcribe the Circle B C DE, this
ſhall repreſent that part of the Cone of the Earth's Shadow
and Atmoſphere 40 min. 6 feconds, cut off in that place.
which the Moon paffeth thro' at that time.
2. With the Sum of the Semidiameters of the Moon and
Shadow 55 min. 21 feconds taken from the fame Scale,
decribe the Circle, F, G, H, I: Draw FI, to repreſent a
Horizontal Line.
3. At the time of the middle of the Eclipfe find the Al-
titude of the Nonagefime Degree, which in this Eclipfe at
London is 37 degr. 50 minutes; then by help of the Lines
of Chord on the Sector fet from F to G; draw G A thro'
the Center of the ſhadow and it ſhall repreſent the Ecliptic at
that Time and Place.
"
G
FLB
N
K
P
}
H
E
I
t
M
4. Take
Aftronomical Precepts.
395
·
4. Take the Latitude of the Moon at the beginning of the
Eclipſe 43 min. 30 feconds, from the Line of Lines on the
Sector (fer to the fame Radius as you drew the Circles by)
and fet it from A to K, becauſe the Latitude is South; (had
it been North you muſt have fet it from A towards D,) take
49 min. 10 ſeconds, the Latitude at the end, and ſet it from
A to L; then by help of your Parallel-Ruler draw K M,
and NL, parallel to GA, and draw M N, which fhall repre-
fent the Moon's Orb during the time of the Eclipfe, and
ſhall lye in a true Poſition at that Time in reſpect of the Ho-
rizon of London.
Laſtly, Divide M, N, into two equal Parts, at P, with the
Semidiameter of the Moon 15 min. 15 feconds, on M, P
and N; ſeverally fweep three Circles; fo fhall that at M,
repreſent the Moon at the beginning of the Eclipfe, that
at P at the Middle or greateſt Obfcuration, and that at N, the
Moon when he begins to Emerge out of the Earth's Shadow
and Atmoſphere, or the final End of the Eclipſe. A P, is
the Axis of the Moon's Way, to which fhe always comes at
the middle of the Eclipfe; A L is the Axis of the Ecliptic,
to which ſhe comes at the time of the true Ecliptic Oppofi-
tion. And the Angle L, A, P, is the double quantity of the
time of Reduction, as is manifeft if you compare the Scheme
with the Calculation.
8. To Construct an Eclipfe of the Moon Geometri-
cally.
The greateſt part of this Work is performed in the 7th
Paragraph of this Precept, fo that here is nothing to be done,
but only to divide the Moon's Orb, into Hours and Minutes
of Time; which being performed, you may preſently ſee
at any time during the Eclipfe, how many Digits are dark-
ned at that Time.
To Divide the Moon's Orb.
Confider what Hour is nearest to the middle of the Eclipfe,
which in this Example is Four a Clock in the Morning,
whofe Difference from 4 is only 2"; and then I fay,
*
Ii 2
wh
If
396
Aftronomical Precepts
If one Hour, or
Give Hourly Motion Moon from Sun
What dift. from 4 a-Clock give ›
Anfwer, Motion Moon from Sun
Min. Sec.
бо
o LL
12847*
3190
2
32553
I
35743
Take this 1 in your Compaffes from the fame Scale the
Diagram was laid down by, and fet it from the middle of
the Eclipſe on the Moon's Orb at P towards M, and that
Point fhall be the Place of the Moon at Four a-Clock,
(but in this Cafe it is fo fmall, that 'tis fcarce difcerni-
ble ;) but had the Middle of the Eclipfe been before 4 -
Clock, then the Diſtance in the Moon's Orb must have been
laid down towards N, as your own Reafon will direct
you better than a Multitude of Words. Take 28 min. 47
fec. in your Compaffes, the Hourly Motion of the Moon from
the Sun, and fet one Foot in the Moon's Orb at the Hour of
Four (just now found,) and turn the other Foot towards
M; that fhall give the Hour of Three in the Morning, and
turn'd towards N, fhall give the Place of the Moon at the
Hour of Five: And thus you may mark out the Orb of the
Moon in Hours and Minutes during the time of the Eclipfe,
which you will find to agree exactly with your Calculation,
diftinguishing the Minutes by fmall Dots along the Moon's
Orb.w
Take the Semidiameter of the Moon 15' 15/' in your Com-
paffes from the fame Scale, and upon ftrong Paper, or fine
Card; fweep a Circle as per Figure, and draw the Diameter
AB, which Divide into 12 equal Parts,
•
> >
or Digits, and with the Semidiameter
of the Earth's Shadow and Atmoſphere
40 min. 6 feconds, draw Eleven Arch-
Lines; fo fhall you have the Body of
Moon divided into 12 Digits: Cut this
Moon out, and put a Pin thro' the Cen-
ter; carry the Point of the Pin gently a-
long the Moon's Orb, always keeping the Point A truly to the
Center of the Shadow; and by this Method, you will fee at
every Hour and Minute of time how many Digits and Parts
of Digits of the Moon's Body are. Eclipfed during the whole
time of the Deliquium And by this Method Projected upon
a long Sheet of Paper, I always fhew Gentlemen the nature
of a Lunar Eclipſe.
4
I
9. Ta
Aftronomical Precepts.
39.9
To find the Scruples of Half Total Darkneſs in a To-
tal Eclipfe of the Moon, and thence the Continuance, Be-
ginning, and End of Total Darkneſs.
RULE. From the Semidiameter of the Earth's Sha-
dow, fubtract the Semidiameter of the Moon; the remainder
reduce into Seconds, and alfo reduce the Moon's Latitude in-
to Seconds; the Half Sum of the Logarithms of the Sum
and Difference in Seconds fhall be the Motion of half Con-
tinuance in the Total Darkneſs, as has been fhewn in finding
the Scruple of Incidence in the Partial Eclipfe.
Example. In the Eclipfe of the Moon, March 15, 1736, ac-
cording to a former Inveſtigation of mine, the
Min Sec.
S's Shadow 44 18 Lat. 》 0º 51'
Semidiam. of's
16 33
Difference
27 45
60
Dierence in Seconds
1665.
Latit. Din Seconds
SI
Sum
1716, Logar.
3.234517
Differenee
1614 Logar.
3.207904
Sum of the Logarithms
6.442421
Motion of half continuance 1664 Half
3.221210
7
7
Which Divided by 60, gives 27 min. 44 feconds; thes
fay by the Logiſtical Logarithms,,
་་
As true Hourly Motion, à Q
To one Hour, or
So Mot. of half Cont. to t. Dark.
To the time of half Darkneſs
Min. Stc.
.'.34 5.0 L.L. 2362,
60 0
27 44
47 41
3350
998
This half Continuance of Total Darkneſs, fubtracted from
the middle of the Eclipſe, gives the time of the Beginning
off the Toral Darkneſs; and added to the time of the Mid-
le, gives the Time of the End thereof.
r
Second-
198
Aftronomical Precepts.
Secondly, By Shakerley's Logiſtical Logarithms, work as has
been fhewn in the Partial Eclipſe.
OPERATION.
Latitude Moon
Difference
Sine of
Deg. Min. Sec.
O 5 I
LL. 8415127
27 45 LL. 9.66511
Co. Sine
1- 45
I
45 O
8:48616
9.99979
Difference
27 45 LL. 9.66511
Mot. of half Total Darkneſs
27 44 LL. 9.66490 as be-
fore.
PRECEPT 16.
To Calculate an Eclipfe of the Sun, to any particular
Place on the Globe.
First, By Precept 14, I have found that in the Year 1737,
there will be four Eclipfes of the Luminaries, viz. two of
each Light: (See my Treatife of Eclipfes, for 26 Years, and
alfo my Sheet for 35 Years ending with the Year 1761.
Which is fold by my Self, and by Mr. Wright, Mathema-
rical-Inftrument-Maker to his Majefty at the Globe and Orrery
in Fleetftreet, London,) and on February 18, there will be a
great and visible Eclipfe of the Sun, whofe Calculation fol-
lows for the Meridian and Latitude of London.
By the foregoing Problems of the Doctrine of the Sphere,
you muſt carefully find the Requifites, and fet them down
thus:
Middle Time true & 1737, February
Place of Sun and Moon in her Orb
Mean Anomaly Sun
Mean Anomaly' Möón
North Node fub.
Argument Latit.
7
1
D.
H. M. S.
18
I 1.
*
8
51 48
1 oo
29
22
47
5·18 19-7
5 22 536 I´S
True
Aftronomical Precepts.
199
[
True Lat. Moon N.D.
Reduction add
Ecliptic Place Moon
Hourly Mor. of{
Hourly Mot. of > à O
Time of Reduct. fub.
Equal Time true Eclipt. & February.
Equation of, Time fub.
Appar. Time true Eclipt. o
Appar. Time from Noon in Degrees
Sun's Place
Sun's Right Afcenfion
1
Right Aft. Med. Cæli paft Y
Medium Cæli in Ecliptic T
Meridian Angle
Declin. Cul. Point North
Alt. Equat. London add
Altitude Mid-Heaven
Altitude Nonagefime Degree
Nonagefime Degree o
D. H. M. S.
38 34
*
2
12 43
I
40
11
7
2.
30
30
4
27
34
?
3
38
18
I
48
8 10
18
1
35
27.
* II
S 13
342
33 16
23
51
51 46
6
25
I
6
59 00
66
41
2 47
00
38 28
41 15
46 20
00
24 17
I
16
I 20
10
47
55 10
Diſtance Mid-Heaven from from Nonag. add
Diſtance Sun from Nonagefime
Horizontal Parallax Moon from Sun
Parallax Longitude Moon from Sun
Ecliptic Place Moon
Visible Place Moon
30 39
*
II
7 2
* 10
36 23
Read Page 179. And from thence you will gather, that
becauſe the Luminaries are between the Defcendant and the
Nonagefime Degree, the Vifible Conjunction will follow the
True.
Parallax in Latitude Moon from Sun
True Latitude Moon N.D.
Visible Latitude North
}
Min, Sec.
38 06
38 34
00 28
And becauſe the Eclipfe falls in the Occidental Quadrant,
you muſt ſeek the Requifites juft now found. by Problems 27,
28, 29, 30, 31, 32, 33, and 39; and to an Hour, (to 50, 40,
or to 30 Minutes, more or lefs, as you ſhall find moft conve
nient for you prefent purpoſe) after the time of the true
Con-
Aftronomical Precepts.
1
7
·Conjunction and fet them down as you fee in the following
Order.
:
2. To 1 Hor. after true & February
I
Sun's Place
'Sun's Right Afcenfion
Appar. Time from Noon add
Right Afcenfion Med. Cali
Declination Culminating Point North
Med. Celi in Ecliptic Y
Meridian Angle
Altitude Equi. London add
Altitude Nonagefime
Altitude Mid-Heaven
Dift. Mid-Heaven from Nonagefimé
Nonagefime Degree
Diſtant Sun from Nonagefime
Mean Anomaly >
Horizontal Parallax
à O
Parallax Longitude 》 à O
Parallax Latitude> à O
ད་
D. H. M. S.
18 2 35 27
* II 7 43
342 36 0
38 51
45
21 27 45
23
13
0
68
14
9
38
28
47
30
51 9
O
18 46
ว
I I 59 00
60
51
17
11
I 0
0
13
29
55 9
37 31
34 56
α
3. To find the Vifible Motion of the Moon from the Sun in
- any Time propoſed.
1. If the Eclipfe happen in the Oriental Quadrant, and
the Parallax of Longitude of the Moon from the Sun
-Increaſe add
the Difference of the Parallax of the
Decreaſe fubtract
Longitude the Moon from the Sun in an Hour, or in any o-
ther time, to, or from the true Motion of the Moon, from
Sun and you will have the Visible Motion of Moon from
Sun in the fame Time.
2. But if the Eclipfe fall in the Occidental Quadrant (as
the preſent Eclipfe doth) and the Parallax of Longitude,
Increaſe fubftract 3 the Difference of the Parallax of Longi-
Decreaſe add
tude Moon from Sun in an Hour, or in any other Time, to,
or from the true Motion of the Moon from Sun, and you
will gain the Visible Hourly Motion of Moon from Sun in
the fame Time,
1
Ac
ļ
Aftronomical Precepts.
401
H. Min. Sec.
Min. Sec.
I
At-
35 27 Paral. Longit.
30
.. 3
à O
37 3
Difference
I
00
Increaſe fub.
6 52
27 33
20 41
2 35 27
True Hourly Motion Moon from the Sun
Viſible Hourly Motion Moon.from the Sun
Now fay, for the Time of the Vifible Conjunction,
As Vifi. Hor. Mot. Moon from Sun 26
To one Hour, or
So Par. Long. true d
To Interval of time add
Min. Sec.
41 LL
60
Q.
30 39
-88 55
4625
α
2917
1708
r
Becauſe the Eclipfe falls in the Occidental Quadrant, this
Interval or Diſtance, between the True and Visible Conjun-
ction of the Sun and Moon must be added to the Time of
the True Conjunction, and the Sum is the Time of the Vifi-
ble Conjunction. But when the Eclipfe is in the Oriental
Quadrant, you muſt fubtract that Diſtance.
Apparent time true February
Interval add
Vifible ♂ is
Sun's Place then
Sun's Right Afcenfion
Appar. time from Noon add
Right Afcen. Med. Cali
Medium Cali in Ecliptic
Meridian Angle
Decl. Cul. Point North.
'
Alt. Equin, London add,
Altitude Mid-Heaven
Altitude Nonagefime Degree
Dift. Mid-Heav. à Nonagefime add
Nonagefime Degree Ŏ
Dift. Sun from Nonag. Degree
Kk
A
D. H. M.
18 I 35
I 28
S.
27
28
35
Feb. 18
3 4 22
,40
* 11 8
342 37
51 30
28
42.
5 I
69 33
II 47
38 28
5
૬૦
15
53
I 2
16 12
17
3
65 50
I 2
Hori-
402
Aftronomical Precepts.
Horizontal Parallax Moon from Sun
Parallax Longitude Moon from Sun
Diftance of Sun and Moon
Motion Sun in 88′ 55″ is
Sum, add
Arg, Lat. at true ♂
Arg, Lat. at Viſible ♂
True Lat.
at Vifible & N.D.: ...
•
*
Vifible Latit.
at Vifible
-N.A.
D. H. M. S.
55 9
t
40 18
40
18
3 42
00
93 44
44
123 136 15
5 23
१
20
IS
Parallax Latitude Moon from Sun
Vifible Latit. at true ♂ N. "
I
34 46
33 2
44
428
By which the middle of the Eclipfe happens after the, Vi-
fible d. See Page 61, in the Tables..
To the Parallax in Longitude Moon from the Sun at the
Vifible, add the Motion of the Sun in the time between
the True and Visible o; that Sum fubtract in the Oriental
Quadrant; but in the Occidental add to, or from the Ar-
gument of Latitude at the time of the True os the Sum
or Difference is the Argument of Latitude at the time of the
Vifible Conjunction; to which find the Moon's true Latitude
out of the Table, Page 58, and by her Parallax her Vifible
Latitude as you fee it wrought above.
Min. Sec.
As one Hour, or
60
o LL.
0
To the Sun's Hourly Motion.
2 30
13802
So is Dift. from Viſible ♂ to true
88 55
1708
To Mot. Sun in that time
3 42
12090
Min. Sec.
To Hor. Mot. Moon from Sun
As one Hour, or
To Dift. à Vifible & to True
To Mot. Moon from Sun in that time 40 49
60
o LL.
0
27 33
3380
88 55
1708
1672
Semidiameter of
Sun
S
Moon
Sum
Visible Latitude fub.
Rem. Parts deficient
Just
16 13
15 I
}
31 14
I 44
29 30
1
For
1
403
Aftronomical Precepts.
For the Digits, fay,
As Semidiameter Sun
To Six Digits
So are Parts Deficient
Min. Sec.
16 13 LL
5682
10000.
29 30
·3083
100
55 OO
To Digits Eclipſed
{
7401
4. To find the Scruples of Incidence, or Motion of half
Duration.
This may be done all the four ways, as I have ſhewn in
the Moon's Eclipfe; but need not repeat them here; there-
fore I fhall, work this Example by Shakerley's Logiſtical Loga-.
rithms.
OPERATION.
Deg. Min. Sec.
1 44 LL.8.46073.
LL. 9.71647
3114
Visible Lat. Moon Visible d
Z Semid, of the Sun and Moon
1
Rem, the Sine
3 II
8.74426
To the Co. Sine
3 II
9.99933
Add Z Semid. Sun and Moon
31 14 LL. 9.71647
31 II
9.71580
Scruples of Incidence
5. To find the Vifible Hourly Motion of Moon from the
Sun to an Hour before the time of the Visible Conjunction,
you must repeat the Work again, as you may fee here fer
down.
1
One Hor. before Vifible & February
Sun's Place
Sun's Right Afcenfion
Appar. time from Noon add
Right Afcenfion, Med. Cali
Medinm Cali in Ecliptic T
Kk 2
D. H M. S.
18 2 4 22.
*
11 6 1
342 34
18
31 5 30
13.39 30
14 51 00
Meri-
1
404
Aftronomical Precepts.
}
D. H. M. S
Meridian Angle
Declinat. Culmin. Point N.
Altitude Equinox London
Altitude Mid-Heaven
67
13
5 52
38
28
00
44 20
48-44
00
Altitude Nonagefime
Dift. Mid-Heav. from the Nonagefime add
Nonagefime Degree &
Distance Sun from the Nonagefime
Mean Anomaly Moon
Horizontal Parallax Moon from the Sud
Parallax Longitude Moon from the Sun
Parallax Latitude Moon from the Sun
True Hor. Motion Moon from the Sun
Diff. Par. Long. in this Hor. Increasing fub.
Vifible Horiz. Motion Moon from the Sun
I
6. At 1 Hour after Vifible & February
Sun's Place is
Sun's Right Afcenfion
Apparent Time from Noon
Right Afcenfion Medium Cali
Medium Cæli in Ecliptic
Meridian Angle
Decl. Culmin. Point N. add
21 37 00
6
28
00
55 21 42
IO 29 56 34
55
34
7
36
23
27
33
· 6
11
21
22
D. H. M. S.
18 4 4 22
* II II 18
୮
342 39 00
6I 5
30
ŏ 16 45
43 44
30
00
73 16
16 43
00
00
Altitude Equin. London
Altitude Mid-Heaven
Altitude Nonagefime
Dift. Mid-Heav. Nonagefime add
Nonagefime Degree &
Dift. Sun from the Nonagefime Degr.
Mean Anomaly Moon
Horizontal Parallax Moon from the Sun
Parallax Longitude Moon from the Sun
Parallax Latitude Moon from the Sun
True Hor. Mot. Moon from the Sun
Diff. Par. Long. in the Hor. Increaſing ſub.
Viable Motion Moon from the Sun
38 28 00
55 11 00
56 51 00
19.00
4 00
I I
1 2
28
96 52 42
I 1 53
55 9
44 58
10
30
27 31
20
4
51
22
I
1
1
7. To
Aftronomical Precepts.
1
1
7. To find the Middle, Beginning, and End of the E-
clipfe.
By the Visible Latitude of the Moon at the time of the
True and Vifible Conjunction; you may fee the Visible La-
titude is NA; enter therefore the Table, Page 60, with the
Visible Latitude at the time of the Vifible Conjunction r
min. 44 feconds; and take out the Motion of the Moon
from the Sun, 9 feconds; which, becauſe the Latitude is
afçending, is to be divided by the Vifible Hourly Motion
of the Moon from the Sun to one Hour before the Vifible
Conjunction 21 min. 22 feconds, and the Operation ſtands
thus by the Logiſtical Logarithms.
As Vifible Hor. Motion D à @
To one Hour, or
So is Motion
To the time à Vifible & to Mid.
P
Min, Sec.
·
21
22 LL
4484
60 OO
Oo 09
oooo
00 25
26021
21537
D. H. M. S.
Vifible d 1737 February
Sub.
18
3 4 22
25
*Middle of the Eclipfe
18
3 3 57
8. For the Time of Incidence, and the Beginning of the
Eclipſe.
Here you must take the Vifible Hourly Motion of the
Moon from the Sun to an Hour before the Viable Conjun-
ction 21 min. 22 feconds, and fay,
As Vifible Hourly Motion à
To one Hour, or
So are the Scruples of Incidence
To the Time, fub.
Middle of the Eclipfe
Time of Incidence fub.
Min. Sec.
21
22 LL
4484
60 Ơo
0000
31 II
87.34
2842
1264
D. H.
M. S.
18
3
3 57
I 27 34
I 36 23
9. For
Beginning is February 19
t
1
}
!
406
Aftronomical Precepts.
7
9. For the Time of Repletion, and End of the Eclipfe.
Here you must take the Vifible Hourly Motion of the
Moon from the Sun to an Hour following the Viſible Con-
junction, 22 min. 51 feconds, and fay,
As Viſible Hourly Motion à O
To one Hour, or
So Scruple of Incidence
Min. Sec.
22
51 LL
4193
60
:
O
O
31
II
2842
81
54
1351
D.
H. M. S.
To Time Repletion add
Middle of Eclipfe
Repletion add
February 18 3 3 57
I 21
54
End of the Eclipfe is
Feb. 18
4 25 51
10. In order to delineate a Solar Eclipfe, we muſt have.
the Latitude of the Moon feen at the time of the Beginning
and End of the Eclipfe, which is found at the Beginning by
repeating the former Work, as is here fet down.
D. H. M. S.
Beginning of the Eclipfe February
18 I 36 23
Sun's Place
*
II 5 14
Sun's Right Afcenfion
342 33 00
Apparent Time from Noon add
Right Afcenfion Medium Cælî
Medium Cæli in Ecliptic
Meridian Angle
Declination Culminating Point North
Altitude Equin. Zondon add
Altitude Mid-Heaven
Altitude Nonagefime
Diſtance Mid-Heaven from the Nonagefime
Nonagefime Degree
Distance Sun from the Nonagefime
Mean Anomaly Moon
Horizontal Parallax Moon from the Sun
Parallax Longitude Moon from the Sun
x
୪
24 5 45
6 38 45
7 15
66 41
2
38
41
53
28
2 I
46 25
24 13.
O
I 28 .0
50 22 46
10 29 41 18
55 10
30 47
Scru-
Aftronomical Precepts.
L
407
1
Scruples of Incidence
Difference
Diſtance of the Sun and Moon
Mot. of the Sun in 87'34" is
Scruples of Incidence add
Sum, Sub.
Argument Latitude at Vifible o
Argument Latitude at the Beginning
True Latitude Moon N. D.
Parallax Latitude Moon from Sun
Vifible Latitude Moon's South Descending
D. H. M. S.
31 ΙΕ
00
24
24
3
39
31
II
34 50,
1
5 23
20 15
5 22
45 25
37
7
38
2
15
11. For the Latitude of the Moon feen at the End of the
Eclipfe, you muſt again make a Repetition of your former
Work, as in the following Order.
End of the Eclipfe February
Sun's Place
Sun's Right Afcenfion
Apparent Time from Noon add
Right Afcenfion Med. Celi
Medium Cæli in Ecliptic o
Meridian Angle
Declination Culminating Point North
Altitude Equ. London
Altitude Mid-Heaven
Altitude Nonagefime Degree
Diſtance Mid-Heaven from the Nonagefime
Nonagefime Degree ш
Diſtance Sun from the Nonagefime
Mean Anomaly Moon
Horizontal Parallax Moon from the Sun
Parallax Longitude Moon from the Sun
Scruples of Incidence add
Sum
Diſtance of the Sun and Moon
Motion Sun in 81′ 54″ is
Scruples of Incidence add
Şum, add
1
D. H. M. S.
18 4 25
51
* II 12 18
342
40 00
66 27 45
21
49
7 45
34 Oo
74
53 00
18
ΙΙ
00
38 28
56 39
00
743
57 57
9 44 00
I
18
8a
5 42
II 1 13 35
55 9
46
3
35 II
77 14
77 14
3 25
31 II
34 36
Argu-
408
Aftronomical Precepts.
Argument Latitude at Vifible d
Argument Latitude at End
True Latitude Moon N.D.
Parallax Latitude fub.
Vifible Latitude Moon North afcending
the
D. H.
M. S:
5 23 20 15
S 23 54 51
3-1
46
29
16
2 30
And thus from the foregoing Calculation I have found
London of the
Appar. Time at
Begin. 1737 February
Middle
Vifible o
End
Total Duration
Digits Eclipfed
D. H. M. S.
18. 1 36.23
3 3 57 P.M.
3 4 22
4 25.51
2 44 28
10°55 00
SP.M.
12. To Delineate the particular Eclipfe of the Sun in Plane.
· Open the Sector to any convenient Radius, and from the
Line of Lines take the Sun's Semidiameter 16 min. 13 fec.
I
O
E
A
J
id
Aftronomical Precepts.
409
in your Compaffes, and fweep the Circle D G D I, to repre-
the Sun; through its Center draw the Line HO to repre-
ſent an horizontal Line; with the Sum of the Semidiameters
of the Sun and Moon, (which in this Example is 31 min.
14 fec.) defcribe the Circle (on the fame Center) AEXC:
Take the Altitude of the Nonagefime Degree at the time of
the Visible Conjunction 53 degr. 12 minutes, and fet the
Chord thereof from H to E; draw E C for the Eclipfe at
that time, and A X at right-Angles for its Axis Take the
Visible Latitude of the Moon 15 fec. South at the beginning
of the Eclipſe, and fet it from the Center of the Sun to e;
draw e f parallel to E C, the Ecliptic: Then take the Visible
Latitude of the Moon at the end, 2 min. 30 fec. North, and
fet it from the Sun's Center to c, and draw be parallel to
the Ecliptic EC; draw bf which fhall here repreſent the
Moon's Visible Way. Lastly, Take the Semidiameter of the
Moon 15 min, 1 fec. in your Compaffes from the fame Scale
of equal Parts; and fetting one Foot in f, defcribe a Circle
which fhall reprefent the Moon at the beginning of the Eclipfe;
with the fame Extent of the Compaffes fet one Foot in the
middle between band f, and defcribe a Circle: Thiş
reprefents the Moon at the time of the greateſt Obſcura-
tion of the Eclipfe, and will fhew you likewife the Digits
of the Sun then Obfcured: Carry the fame Extent of your
Compaffes, and fer one Foot at b; draw a Circle which fhall
repreſent the Moon at the End of the Eclipfe; and thus you
may repreſent, or Typifie any Solar Eclipfe to any particular
Place on the Earth, in its true Pofition at that Time; which
was first publish'd by me, in my Treatife of Eclipfes, and
which is performed by having only regard to the Altitude of
the Nonagefime Degree at the time of the middle of the E-
cliple, as is fhewn. Thus have I finiſhed the practical Me-
thod of Calculating the Sun's Eclipfe for a particular place
on the Globe; in which you are to obferve, that whatever
City or Town you would do it for, that you take the Com-
plement of the Latitude of that Place, which is always equal
to the Elevation of the Equinoctial, and apply it to the De-
clination of the Culminating Point, (as you may fee I have
done, and as I have taught in Prob. 31 ;) and by duly obſer-
ying the Premiffes, you will truly gain the Appearance of the
Sun's Eclipfe at that Place, whofe Complement of the Lati-
tude you made ute of in your Work.
LI
PRE-
}
1
!
↓
410
Aftronomical Precepts.
PRECEPT 17.
To Calculate the Times of the Principal Appear-
ances of a Solar Eclipfe under any known Me-
ridian.
"
And for an Example, I fhall take the Eclipſe of the Sun,
which will happen February 18, 1737 ?
1
D. H. M. S.
Equal Time of the true o is
February
18
}
I SI 48
Moon's Place in her Orbit
Ecliptic Place of the Moon
Equation of Time fub.
Apparent Time in the Moon's Orb
Sun's Place from the Earth
!
18
.12
I 39
43
S
*
II
5
22
*
II 5
22
* II
༡
2
Argument of Latitude
True Latitude of the Moon N.D.
True Hourly Motion of the Moon from the Sun
5 22
36 15
38
34
27 34
Declination of the Sun South
7 25
12
Horizontal Parallax of the Moon
55
20
Horizontal Parallax of the Sun
fub.
ΤΟ
Remains the Semidiameter of the Earth's Disk
55
ΙΟ
Min. Sec.
Sun
Semidiameter of
Moon
16 13
15 I
Sum is Semidiameter Penumbra
Angle of the Moon's Way
Semidiameter Earth's Disk
Semidiameter of the Penumbra
Sum
5
31. 14
43 0. This is taken
out of the Table, in Page 81.
Min. Sec.
1
55 10
31 14
86 24
?
1
་
Hence,
Aftronomical Precepts.
411
Hence, becauſe the Semidiameter of the Disk and Penum-
bra is greater than the Moon's true Latitude at the Equal
Time of the true Conjunction, it fhews the Sun (Vulgarly
ſpeaking) will be Eclipfed; or rather, that fome part of the
Earth's Inhabitants will be deprived of the Sun's glorious
Light And becauſe the Moon's true Latitude is less than
the Semidiameter of the Disk, it fhews, the Sun will be cen-
trally Eclipfed to fome part of the Earth.
Min. Sec.
1
Earth's Disk
55 ΙΟ
Semidiameter of
Penumbra
31
14
Difference
23 56
Becauſe the Difference is less than the Moon's true Lati-
tude, it proves, the Penumbra will not all fall within the
Disk, and that there will be but two Angles of Inci-
dence.
Projection.
From the Line of Lines on the Sector, take the Semi-
diameter of the Earth's Disk 55 min. 10 fec. in you Com-
paffes, and fet one Foot in the Center of the Sun; fweep the
Circle A B C D, which fhall here repreſent the Horizon of
the Disk of the Earth; from the fame Scale of equal
Parts take the Sum of the Semidiameters of the Disk and Pe-
numbra 86 min. 24 fec. in your Compaffes, and fet one foot
in the Sun as before, and draw the Circle E F GH; then
draw, EG to reprefent the Ecliptic, and FH its Axis.
Now becauſe the true Latitude of the Moon is North Def
cending, the Axis of the Moon's Orb will lye to the Left-
Hand of the Axis of the Ecliptic: Take therefore 5 degr.
43 minutes the Angle of the Moon's Way, from the
Line of Chords, and fet it from F to I, and draw IK
for the Axis of the Moon's Orb. From the Scale of equal
Parts take the Moon's Latitude 38' 34" North; and becauſe
'tis North, fet it on the Moon's Axis from the Sun to L,
through Land at Right Angles to IK, draw M N, which
fhall repreſent the Moon's Orb, or Line of her Way over the
Disk, or Path of the Penumbra.
LI 2
}
Take
412
Aftronomical Precepts.
1
Take the Semidiameter of the Penumbra 31 min. 14 ſec
in your Compaffes, and fet.one foot in M, O, L, P, --and N
प
IF
N
I
B
M
SLT
D
C
G
i
HK
R
feverally; then draw Circles which fhall repreſent the Center
of the Penumbra at thoſe Places in its paffage over the Earth's
Disk Draw Lines from the Center, to M, O, P, and
N; fo fhall there be feveral Triangles formed, viz. the Tri-
angle ML, OOL, OLP, and OL N, in which are
given, firft in the Triangle OL M, M the Sum of the Se-
midiameters of the Disk and Penumbra 86' 24", and OL
the Latitude of the Moon 38' 34", to find the Angle MOL
the first Angle of Incidence, and L 'M, the Motion of the Penum-
bra from M to L, which is the half Motion of the general E-
clipfe.
First,
Aftronomical Precepts.
413
* Firſt, for the Angle of Incidence LM O.
As Z Semidiameters in Second M
To Radius.
So Moon's Latitnde in Seconds
O
To C.f. Ang. M O L 1ft ▲ of Incidence
Sec.
5184
Deg. Min.
၄၁ ၄ဝ
2314
3.714665
10.000000
3.364363
63 30 9.649698
Secondly, for the Motion of half Duration L M, »
fay,
Deg. Min.
As Radius
To O M in Seconds
90 o
IC.000000
5184
3.714665
So S. Angle LOM
To L M, the Motion of half Duration
63 30
4639
9.951791
3.666456
That is 77 min. 19 feconds,
For the Time that the Penumbra is moving from M,
to L, fay,
As true Hourly Motion of > à O
To one Hour, or
So Motion of half Duration M L
To the Time.
Min. Scc.
27
34 LL.
3378
60 00
77 19
ΙΙΟΙ
Hence, becauſe the Proportion above will exceed the Ta-
bles of Logistical Logarithms, you must take half the Num-
bers, and what comes out muſt be doubled, and that will
be the true Anſwer.
}
X.
OP E-
414
Aftronomical Precepts:
OPERATION.
Min. Sec.
As half Hourly Motion à @ 13 47 LL. 6388 from.
To half an Hour
So half the Motion ML
30 00
38 39
•
84
8
30102
1919+
Z 4920 fub.
168 16
16
To half the time
Doubled is
That is 2 h. 48 min. 16 feconds
1468
Secondly, In the Triangle OLO, there are given OO,
the Semidiameter of the Disk 55 min. 10 feconds, and O
L the Moon's Latitude, to find LOM, the fecond Angle
of Incidence, and OL, the half motion of the Central E-
clipse.
Say, for the Angle L © M.
As Semid. Disk ≈.☺ O feconds
Sec.
3310
3.519828
Min. Sec.
To Radius
90 O
10.000000
So Moon's Latitude in fec. LO, 2314
To C.f. Angle LO M
3.364363
45 39
9.844531
For the Motion L O, fay,
}
As Radius
To,
O
So S. Angle LO
To LO
Which Divided by 60, are 36 min. 15 ec.
Deg. Min. :
90 0 10.000000
3310
?
·3.519828
45 39. 9.854356
2175 3.374184
*
For
Aftronomical Precepts,
419
}
For the time fay,
Min. Sec.
As true Hour, Mot. Moon from the Sun 27 34 LL 3378
To one Hour, or
So Motion LO
To the Time
60 00
36 15
78 55
2188
1190
3. We are to find the Inclination of the Axis of the Globe
with the Axis of the Ecliptic.
ANALOGY.
As Radius
To C. f. of the Sun's Longitude
So T. of the greateſt Reflection
To T. of the Inclination
Deg. Min.
90
00--10.0000oo
18 55-- 9.975887
23
22
29-- 9.637956
20-- 9.613843
Now you are to obferve, that if the Sun be
the Axis of the Globe Right-Hand
DIMM
in{a}
Left-Hand
of the Axis of the Ecliptic.
Open the Sector to the Radius F on the Lines of
Chords, and take of the Chord of the Inclination 22 degr.
20 minutes, and fet it from F to q; draw q R, and that
fhall be the Axis of the Globe projected in the Earth's
Disk.
Now becauſe the Axis of the Globe, and the Axis of the
Moon's Orb lie both on the fame fide of the Axis of the E-
cliptic, therefore the Angle q OF is Negative; then from
the Angle 40 F 22° 20' fubtract the Angle I O F 5° 43',
there Remains the Angle qOI 16° 37' the Angle of Dire-
ction. Now to find the Motion L S.
>
In the Triangle LOS, right-Angled at L, are given, the
Angle of Direction LOS 16° 37', and the Moon's Lati-
tude L 38' 34", to find LS.
1
ANA-
Aftronomical Precepts.
}
As Radius,
To L,
ANALOGY.
Deg. Min.
90
Sec.
00 10.000000
Latitude in Seconds
1
2314 3.364363
So t. Angle LOS Direction
To LS the Motion
16 37 9.47+842
690.6
*
2.839205
This 690 fec. is Ir min. 30 feconds, to Reduce it
into Time, fay,
Min. Sec.
As true Hourly Mot. Moon from Sun 27
34 LL.
3378
To one Hour, or
60 00
So is the Motion L S
II 30
7175
To the Time
25 2
3797
Theſe 25 min. 2 fec. added to the Apparent Time of the
Middle, give the Apparent Time when the Meridional Sun
will be Centrally Eclipfed.
And by the foregoing Calculation of this general Eclipfe
I have found,
The firft Angle of Incidence
The Second
The Mot. of half Durat. of this general Eclipfe I
The Motion of Semiduration of this
Central Eclipfe in the Disk.
From the Axis to the nearest approach
Hence, half the time of the general Eclipfe
Half Duration of the Central Eclipfe
The time from the Axis to the Middle fub.
D. M. S.
16.3 30 30
45 39 00
Ι
19
O
36 10
I I
30
2 48 16
I
18 55
O
0 2
25 2
4.
For
Aftronomical Precepts.
417
4.
For the apparent Time of the neareſt approach of
the Moon to the Center of the Disk.
D.
H. M. S.
's Orb. Feb. 18
I 39 5
3 38
18
I 35 27
18
I
Apparent time true d in the
Time Reduction fubtract and add
Ecliptic &
Middle
42 43
Having found the middle of the Eclipfe, or the apparent
Time when the Center of the Penumbra comes to L, if to
and from that you ſubtract and add the Semidurations feve-
rally, you will have the Beginning and End of the general
Eclipfe.
EXAMPLE.
D. H. M. S.
Middle of the Eclipfe 1737
Feb.
18
1 42 43
Semiduration fubtract and add
2 48 16
Beginning of the Eclipſe
17
22 54 27
End
18
4 30 59
1 18 55
18
00 23 48
End
18
3 1 38
Half Durat. of the Cen. Eclip. fub. and add
Beginning of the Central Eclipſe
From the foregoing Calculation I have found at London the
times when
The Penumbra firſt touches the Disk, and the Eclipſe firſt
of all begins in the Earth; its Center is then at M 17 d.
22h. 54 min. 27 feconds.
1
ì
M m
The
418
Aftronomical Precepts.
.
D. H. M. S.
18
00
23 48
The Center of the Penumbra enters the
Earth's Disk and the Central Eclipfe
firft begins, and is then at O,
The Nonag. Sun Centrally Eclifed at T
Middle, or Center of the Penumbra is now at L
The Meridional Sun Centrally Eclipfed at S
Center of the Penumbra paffes off the Disk,
and the Central Eclipfe ends at P
The Penumbra paffes off the Disk, and
the Eclipfe ends in all places of the
Earth at N
2 35 27
42 43
2
7 45
3.
1- 38
4 30 59
0-}
5 36 32
After they have continued in paffing o-
ver the Earth,
•
Laftly, To determine the Latitudes of thofe Places on
the Globe, and their Longitude from London, where any
of thofe Appearances happen.
Theſe Places are all eafily found by any who are well
skill'd in Sphericks; but becaufe, the Book fwells too much,
I must defire the Reader to be content only with the
Latitudes and Longitudes of the Places themſelves, and defer
the Calculation till another Opportunity.
Hence, the Latitude and Longitude from London where
Lat.
Sun beg. Ecl. as he rifes at M 9 48
9 48 | 107
Sun rifes Centrally Eclip. at O 27 29
the Sun Cen. Eclip. in the Mer. S 39 26
Sun fets Cent. Eclipfed at P 60
Eclip. Ends at Sun-fétting N 42
7
40
Long.
J
40 15 W
92 4 00 W
31 56 15 W
31 30 30 E
15 25 15 E
1
!
PRE-
1
4.19
Aftronomical Precepts.
PRECEPT 18.
To Conftruct the Sun's Eclipfe Geometrically.
The Projection that I fhall here defcribe, is that mentioned
by Mr. Flamsteed in the 27th Page of his Doctrine of the
Sphere; and is, if a Plane be conceived to touch the Moon's
Orbit in that Point, wherein a Line connecting the Centers
of the Earth and Sun, interfects the faid Orbit, and ftands
at right Angles to the aforefaid Line: and if an infinite
Number of ftreight Lines be fuppofed to paſs from the Cen-
ter of the Sun, thro' this Plane of the Periphery of the Earth,
to its Axis, as likewife to the Axis of the Ecliptic, and
the Path of any Vertex; the faid Lines will Othographically
project the Earth's Disk, its Axis, the Axis of the Ecliptic,
and the Path of the Vertex, on the aforefaid Plane. And
this is the Projection we are to delineate.
+
In Problem 3, of the Projection of the Sphere, I have fhewn
how the Path of any Vertex may be drawn; and that when
the Sun's apparent Place is either in Aries, Taurus, Gemini,
Cancer, Leo, or Virgo, the North Pole of the Globe lies
to the right Hand from the Axis of the Ecliptic, and the North
Pole of the Globe lies in the illuminated part of the Disk:
But if the Sun be in Libra, Scorpio, Sagittary, Capricorn, Aqua-
rius, or Pifces, then the North Pole lies in the Obfcure.
If the Sun be in the Equinoctial, the Paths of the Verti-
ces, will be projected in right L'nes upon the faid Plane;
but if the Sun be not in the Equinoctial, then the Path will
be Ellipfes upon the faid Plane.
The Tranſverſe Diameter of the Ellipfis reprefenting any
Path is equal to double the right Line of the Diſtance of the
faid Vertex from the Pole; that is, equal to twice the Co-
Sine of the Latitude of the Place or Vertex ; but the Con-
jugate, to the Difference of the right Sines of the Sum and
Difference of the Diſtances of the Path and Sun from the
Pole; that is, equal to the Sine-Complement of the Sun's
Declination added to the Co-Latitude of the Place, lefs the
right Sine of the Difference of the Complement of the Sun's
Declination and the Co-Latitude of the Place.
M m 2
The
1
420
Aftronomical Precepts.
The Tranfverfe Diameter lies at right Angles to the
Earth's Axis, and the Conjugate coincides with it. For an
Example I fhall conftruct the Eclipfe of the Sun which will
happen February 18, 1737, for London.
Open the Sector to any convenient Diftance, as OA, and
draw the Semicircle ABC; this fhall reprefent the Sou-
thern half of the Earth's illuminated Disk Projected on the
Plane of the Ecliptic A O C.
Take the Chord of 23 deg. 29 minutes, the conftant Di-
ftance of the Pole of the Ecliptic and the Pole of the Equino-
ctial, and fet it from B, to dand e; draw de, in which the
Pole of the Globe will be always found.
Make de ge the Radius of a Line of Sines, and fet
off the Sine of the Sun's Distance from the Solftitial Co-
lure VS 7° 5' 22"; (becauſe the Sun at the time of the true
dis in 11° 5' 22") and if the Sun be in ,,,, m,
or, then the Axis of the Globe muft lie to the right Hand
of the Axis of the Ecliptic; but if the Sun be in VS, *, X,
T, 8, or I, then to the Left. So in our Example, the
Sun being in X, I fet the Sine of 71° 5' from G to P, and
draw P for the Axis of the Globe. Or by Calculation it
will always hold.
As Radius
To C. f. Sun's Longitude
To t. Diſtance of the two Poles
To t. Inclination two Axis PO B
Deg. Min.
90 00-101000000
18. 55-- 9.975887
23 29-- 9:637956
22: 20-- 9.613843
being fet to the
Take the Chord of 22° 20' (the Sector
Radius B) in your Compaffes, and fer one Foot in B,
the other Foot will reach almoſt to d; draw dP O, and it
will repreſent the Axis of the Globe, as was found juft be-
fóre by Projection.
The next thing to be done, is to draw the Path of the
Vertex of London; for the doing of which you must always
have in readineſs the Sun's Declination, which in this Exam-
ple is 7º 25' 21' South; which being known, work thus:
Sun's
Aftronomical Precepts.
421
Sun's Declin.
Complement
Co. Latitude
Sum
Complement
Difference
Deg. Min. Séc.
7 25 21 South.
82 34 39 Diſtance from the South Pole.
-38 28 O
121
2
39.
58 57
21 Vertex à O at Midnight.
44 6
39 Vertex à at Noon.
1
Make B the Radius of a Line of Sines on the Sector,
and take the Sine of 58° 57421" in your Compaffes, and
fet it from O to I; alfo take 44 degrees 6 minutes 39
ſeconds and fet it from O to H; fo is I the Meridional
Interfection of the Nocturnal Arch of the Path with the Axis,
and H the Interfection of the Diurnal Arch of the Path of
the Vertex of London with the Meridian.
MET
V
:6
a
TB
G
2
H7
R
10
T
TIX
a
T
A
Nь
Biffect HI in K, thro' K, at right Angles to P; draw
an occult Line; then from the fame Radius of the Line of
Sines take the Sine of 38 degrees 28 minutes, the Comple-
plement of the given Latitude, and fet one Foot of your
Compaffes in K; turn the other each way to 6, 6; draw
the Line 6, 6, and it fhall repreſent the Tranfverfe Diameter
of the Earth's Ellipfis to the Vertex of London, and HI the
Conjugate.
Make
422
Aftronomical Precepts.
事
​Make half the Tranſverſe, viz. 6 K the Radius of a Line
of Sines, and take the Sines of 15, 30, 45, 60, 75° feverally
in your Compaffes, and fet them feverally in the Tranfverfe
Diameter from K each way towards 6, 6; thro' theſe Points,
draw Occult Lines parallel to P O'; make HK the Radius
of a Circle on the Line of Sines, and take the Sines of 75,
60, 45, 30, 15°, and fet one Foot in the tranſverſe Diame-
ter feverally on each fide K, at 15, 30°, &e. and let the o-
ther Foot fall in the Occult Line; fo will you have Points
thro' which with an even Hand if you draw a Curve, it will
be an Ellipfis, and repreſent the Path of the Vertex of Lon-
don, to which fet the Figures 12, 1, 2, 3, &c. as in the Dia-
gram: Note, You need only draw the Diurnal Path. And
this is no more than laying down an Ellipfis by the Line
of Sines, which I prefume every one of my Readers is well
skill'd in doing.
Take the Sun's Declination 7 degrees 25 minutes from the
Line of Chords in your Compaffes, and fet it from A to L;
draw L, take the Co. Latitude of London 38 degrees 28
minutes, fet it from A to M, let fall M N, perpendicular
to the Ecliptic A C, and it will cut LO in O'; transfer
O in the Axis from O, and it will reach (in this Ex-
ample almoft to K; thro' this Point if you draw a Line
parallel to 6, 6, it will give you the Amplitude of the Path
of the Vertex of London, and does fhew you that the Sun
that Day riſes after Six in the Morning, and fers before Six
at Night. Otherways, by Calculation thus,
As C. f. Sun's Declination c
Deg. Min.
7
25-- 9996351
90
0--10.000000
To Radius
So S. of the Latitude of London
To C. f. of the Arch & S
As by the Projection above defcribed.
51 32-- 9.893745
37
52 9.897394.
How to place the Moon's Orb in the Projection.
With the Argument of Latitude 5 S. 22 degr. 36 min.
15 feconds at time of the True Conjunction, and true
Hourly Motion of the Moon from the Sun 27 minutes 34
feconds; take out of the Table, Page 81, the Angle of
the Moon's Vifible Way 5 degrees 43 minutes, and becauſe
the Moon's Latitude is N.D. fet it by help of the Line of
Chords from B to q, and draw Og for the Axis of the
Moon's
Aftronomical Precepts.
42 3.--
Moon's Orb. Set the Line of Lines on the Sector to the
Radius of the Earth's Disk B 55 min. 10 feconds,'
O
and, as the Sector now ftands take off the Moon's true
Latitude 38 min. 34 feconds at the time of the true Con-
junction, and fet in the Axis of the Moon's Orb from O
to R; thro' R, at right Angles to q draw TV, which
fhall reprefent the Way of the Moon over the Earth's Disk
during the time of the Eclipſe.
To Divide the Moon's Orb.
The Middle Time of the true Conjun&t. is Feb. 18
Equation of Time fub.
D. H. M. S.
I
51 48
12 43
Apparent Time of the Orbit-Conjunction
Time of Reduction add
18
I 39 S
3
38
Apparent Time Middle of the Eclipfe
18 I 42
43
That is, when the Moon's Center paffes the Axis of her
Orb, being 42 min. 43 feconds paft One a-Clock.
As one Hour, or
Now fay,
Min. Sec.
50
。 LL.
27 34 3378
To true Hourly Motion Moon from Sun
So Time more than I a-Clock
To Mot, Moon from the Sun in that time
1
1476
42 43
19 37 4854
Take this 19 min. 37 feconds in your Compaffes from the
Line of Lines on the Sector open'd to the Radius of the
Disk 55 minutes 10 feconds, and fet one Foot in the Inter-
fection of the Moon's Orb with its Axis; turn the other
Foot towards the Right-Hand, and where it falls, is the
Hour of One: Then take 27 minutes' 34 feconds in your
Compaffes from the fame Scale of equal Parts, fet one
Foot in the Moon's Orb at I, and turn the other Foot each
way, and it will mark out the Hours of 12 and 2 a-Clock,
or the Places in the Orb where the Center of the Penumbra
will be at thofe Hours.
ገፈ
1
Laft-
1
*
424
Aftronomical Precepts.
Lastly, Divide each Hour into 60 equal Parts, and then
you will have given the Place of the Moon's Center
in the Line of her Way to every fingle Minute in
Time.
To find the Time of the Visible Conjunction.
Having divided the Path of the Vertex, and the Line of
the Moon's Way into their proper Hours, &c. take a Ru-
ler and lay on the Moon's Way, and move it at right-
Angles therewith from the Right-Hand to the Left, until
the Edge thereof cut the fame Hour and Minute in the
Line of the Moon's Way, that it doth in the Path of
the Vertex; for that is the true Time of the Visible
Conjunction at that Place for which the Path was drawn.
So in this Example I find the Vifible Conjunction at
London to be at 4 minutes 22 feconds paft Three in the
Afternoon.
From the fame Scale of Minutes take the Semidiameter
of the Sun 16 minutes 13 feconds in your Compaffes and
fet one Foot in the Path of the Vertex of London at the Hour
and Minute of the Time of the Vifible Conjunction, and
there deſcribe a Circle which ſhall repreſent the Body of the
Sun, at that Time and Place.
Alfo, from the fame Scale take the Semidiameter of the
Moon 15 minutes 1 fecond in your Compaffes, and fet one
Foot in the Line of the Moon's Way at the time of the
Viſible Conjunction, and there deſcribe a Circle; this
fhall reprefent, the Body of the Moon at that Time and
Place.
Divide the Sun's Diameter into 12 equal Parts, by fet-
ting the Sector to the Radius of 12 upon the Line of Lines ;
from which take the Sun's darkned Space, and apply that
Extent of the Compaffes to the Sector open'd as now dire-
cted, and you will have the Digits of the Sun's Diameter
then Eclipfed; which in the Eclipſe before us is 10 Dig.
55 min.
To find the Beginning of the Eclipfe.
Take the Sum of the Semidiameters 31 min. 14 feconds
in your Compaffes from the Line of Lines on the Sector
open'd to the Radius of the Earth's Disk 55 minutes 10
feconds; and carry this Extent of the Compaffes one Foot
along the Moon's Way, and the other along the Path of the
Vertex
>
t
The Doctrine of the Sphere.
425
!
い
​Vertex until both Points fall in the fame Hour and Minute,
and that is the beginning of the Eclipfe: Which in this Ex-
ample you will find to be Feb. 18 d. i h. 36 minutes 23 ſe-
conds.
For the End,
Carry the former Extent of the Compaffes on towards the
left hand, one Foot in the Path of the Vertex, and the other in
the Moon's Way, till each Point fall in the fame Hour and
Minute of Time, and that is the Apparent Time of the End
of the Eclipfe: Which in this Example you will find to
be Feb. 18 d. 4 h. 25 min. 51 feconds, the End of this Solar
Eclipfe at London.
In which Conſtruction,
Long.
of Latit.
3
b 3
O 3
the Parallax
the Parallax of Latit
î
Altit.
Min.Sec.
40 18
Moon à Sun which
meafur. on the Scale 33
of
Sun
equal Parts is
533
And, as this Method is entirely free from all Parallaxes; fo
by it you may readily Conftruct any Solar Eclipfe for any
Latitude, or Occultation of the Fixed Stars or Planets, as
has been taught in this Solar Eclipfe; only minding to pro-
ject them by a Sector of a Foor Radius, and let the Proje-
ction be as large as poffible.
PRECEPT 19.
To Calculate the Tranfit of Venus and Mercury over
the Sun.
F
For an Example I fhall here fhew how to inveſtigate the
Paffage of Mercury over the Sun October 31, Anno 1736.
?
First, You must find the Time of the True Conjunction
by Precept, having regard to the Motion of Mercury from
the Sun inftead of the Table, in Page 65.
Na
:
Equal
426
Aftronomical Precepts.
D. H. M. S.
Equal Time of the True o at London 1736 0&.
Equation of Time add
Apparent Time in Mercury's Orb
30
22 39 26
15 38
30
22 55
Mean Anomaly of Mercury
of{Sun
4
12 27
22
5
14 27
4
Heliocentric Place of Mercury
Anomaly of Commutation
Hourly Motion of
Hourly Motion of Mercury à O
I
19 22
20
Geocentric Place of Sun and Mercury R
7
19 22 20
6
0 O
}
2
}
35
5
57
3 26
True Diftance of
Sun from the Earth
Mercury from the Sun
98874 4.995082
31197
4.494110
Remains Diſtance
Mercury from the Earth
67677
4.830441
D.
H.
M. S.
North Node of Mercury
I IS
17
20
Argument Latitude
4
5
0
Reduction fub.
I
49
Inclination, or Heliocentric Latitude N.A.
29 29
With the Mean Anomaly take out of their Tables, the
Logarithms of their Diſtance from the Sun and Earth, and
to thofe Logarithms (by Problem 58) find the Abfolute
Numbers to them; then fubtract the Abfolute Number
of Mercury, from that of the Sun, and the Remainder
will be the Abfolute Number of Mercury from the Earth in
Parts; to which Parts, find by Problem 57, the Logarithm
thereunto, as you fee are inferted above.]
Now for the Latitude of Mercury feen from the Earth at
the Time of his True Conjunction with the Sun, fay
as is taught in my Syſtem of the Planets Demonſtrated, Page 24.
•
?
OPE-
Aftronomical Precepts.
427
OPERATION.
4.494110
As the Dift. of Mercury from the Earth 67677 Co.Ar. 5.169559
To Diſtance of Mercury from the Sun 31197
So T. Heliocentric Latitude N.A.
To T. of the Geocentric Latitude N.A.
29′ 29″--7.933291
13 36 --7.596920
Now becauſe the Latitude of Mercury is Afcending, you
muft find the Geocentric Places of the Sun and Mercury
to Six Hours before the Time of the True Conjunction.
To Six Hours before the true & October
J Sun
True Geocentric place of Mercury
Sun
Mean Anomaly of Mercury
Heliocentric Latitude
Geocentric Latitude Six Hours before
Geocentric Latitude at the true o
Difference in Latitude
D. H. M. S.
30 16 39 26
7 19 7 II
7 19 42
41
4 12 12
34
5 13 25
41
18 23
8 29
13 36
$
7
60
307 fec..
Now for the Angle of Mercury's Visible Way over the
Sun, fay,
As the Elongation in Seconds
To Radius,
Sec.
2146
Deg. Min.
'90 do
}
3.331630
10.000000
So Difference in Latitude
3༠༡
2.487138.
To T. of the Visible Way
88 28"-9.155508
Open the Sector to any convenient Radius, and, take the
Semidiameter of the Sun 16 minutes 16 feconds (found
in the Table, Page 62,) in your Compaffes, and draw the
Nn 2
Semi-
428
Aftronomcal Precepts.
H
E B
NE
ས
I
K
I
D
ОС
M
Semicircle A B D, which fhall reprefent half of the Sun's
upper Vifible Periphery A D, a portion of the Ecliptic:
Open the Chord of 60' on the Sector to the Radius C B,
and take the Chord of 8 degr. 8 min. 28 fec. the Angle
of Mercury's Vifible Way; and becauſe Mercury is Re-
trograde, and Latitude afcending, fet the Chord of 8
degrees, 8 minutes, 28 feconds from B to E, and draw
CE for, the Axis of Mercury's Visible Way; take the
Latitude 13 minutes 36 feconds at the Time of the True-
Conjunction, and fet it on the Axis of the Ecliptic from
C to F; draw HI through the Point F at right Angles to
C-E; fo fhall, HI be the Vifible Way of Mercury over
the Sun Draw HL, NO, and I M parallel to B C
the Axis of the Ecliptic, and H K, and NG parallel to
the Ecliptic A D. Then is C F the Latitude at the time
of the True Conjunction, CN the neareft Diftance of the
Centers of the Sun and Mercury, NO the Latitude of Mer-
cury at the Middle of the Ecliple, HL his Latitude at the
Beginning, I M his Latitude at the Central Egrefs or End:
For becaule Mercury is always Retrograde at the time he
is feen in the Sun, therefore he touches the Sun's Disk at
H, and goes off at I.
From this Demonftration it is plain, that HK = NG,'
HN to NI, and NK to GI,
Like-
Aftronomical Precepts.
I
Likewife the Angles KHN, GNI, CNO; and BCE
or FCN are equal to the Angle of the Vifible Way of
Mercury with the Ecliptic, which was found above to be 820
8' 28"
NF, is the Motion of Mercury from the Sun in Longitude
in the interval between the True Conjunction and middle of
this Mercurial Eclipfe, HK LO=NG=O M thé Mo-
tion of half Duration. And laftly, A H and DI are the
Arches of the Sun's Visible Periphery intercepted between
Mercury and the Ecliptic in his Ingreſs and Egrefs.
Now having explain'd the Nature of the Calculation, I
fhall proceed to find the Requifites by plain Trigonometry
in the following Order.
"
First, In the Triangle CNF, right Angled at N, are
given C F the Latitude of Mercury 13 minutes 36 feconds
at the True Conjunction, and the Angle FCN 8 degr. 8-
min. 28 feconds, to find C N, the neareſt approach of their
Centers; which I find to be 13 min. 28 feconds.
Secondly, In the Triangle NOC, right-Angled at O, are
known C N 807"8, and the Angles as before, to find OC
the Motion from the Middle to the True Conjunction, which
I find to be 114.3 1 min. 54 feconds.
Thirdly, In the famè right-Angled plain Triangle NO C,
with the fame things given to find NO, the Latitude
of Mercury at the Middle, which I find to be 799.7 fec.
= 15 min. 19 feconds.
Fourthly, In the Triangle CHNACIN, there are
given CN 808 fec. ferè, the neareft approach of Mercury
to the Center of the Sun, and CHCI the Sun's Semi-
diameter in Seconds 976 fec. to find H NNI the Mo-
tion of half Duration, which by the 47th of the firſt of Euclid
I find to be 547/fec: 9 min. 7. feconds.
+
+
=
Fifthly, In the Triangle H N K, are given HN = NI
547 fec. and the Angle HNK 8 degr. 8 min. 28 feconds, to
find H K=N G; which I find 541 fec. 9 min. 21 fec. 4.
Sixthly; In the fame Triangle, there are given as before,
to find NK FG, which I find to be 78 feconds, and
ON 799:7 KN78= OK 721.7 min. LH the La-
≈
titude of Mercury when he first touches the Sun's Periphery.
Alfo O N 799.7 fec. + GI 78 fec. MI 877.7 fec. the
Latitude of Mercury when he goes off the Sun's Disk.
•
4
Seventh-
450!
Aftronomical Precepts..
Seventhly, Having found the Motions of all the Arches of
Mercury's Paffage over the Sun, I fhall fhew how to find the
Times that he takes to move over thoſe ſpaces of his
Orb, thus.
1
First, For the time that he moves from O to C, fay,
As the part of the Elong. 357 fec.
To one Hour, or
So is the Motion OC
To the time
fub.
Min. Sec.
5 57 LL. 10036
60
00
114=
I 54
14994
· 19 9
4958
8. For the Time of half Duration.
As part of Elongation
To one Hour, or
So is the Motion HK
To to Time of half Duration
9. For the Arch A H.
Min. Sec.
5 57 LL 10036
60
фо
9
Ι
90 56
8231
1805
*Sec.
As Semidiameter O, CH
976
2.989450
Deg. Min.
To Radius,
90 00
So CH Latitude at beginning
To S. A H
721.7
47 41
10.000000
·2.858357
9.868907
Lastly, For the Arch D I. In the Triangle CIM, there
are given CL, the Sun's Semidiameter, and the Latitude
of Mercury when he goes off the Disk = M I, to find the
Angle M CI, which is meaſured by the Arch D I.
As CI
Sec.
976
Deg. Min.
90
+
877-7
To S. Angle MCI = Arch DI
64
To Radius,
Se MI
1*
12.989450
10.000000
*** 2.943346
9.953896
Now,
Aftronomical Precepts
Now, from the foregoing Calculations I have found,
D. H. M. S.
Apparent Time True Conjunction 1736 08. 30
Time from O, to C, fub.
Middle of the Eclipfe
Half Duration ſub. and add
Central Ingreſs, or Beginning
Central Egrefs, or End
Total Duration
22 55 4
19 9
30
22
35 55
I
30 56
30
21
4 59
3 I
Oo
6 SI
3
$2
The TYPE.
I
H
1
At the Middle time of this Tranfit of Mercury, he may
be ſeen in the Sun not much unlike a Patch on a Lady's
Face, and the Sun is then Vertical to the South-Weft
Parts of Africa, in Latitude 17 degrees, 36 minutes South,
and Longitude 21 degrees, 31 minutes, 15 feconds Eaft
from the Meridian of London; confequently Viſible to all
Europe, Africa, and Part of Afia.
After the fame manner you may Calculate the Time of
Venus's being feen in the Sun by the Inhabitants of our
Earth, which happens but twice in this Century, and
that of Mercury fixteen Times; whofe Calculations all at
large I have now by me in M. S. [See my Treatise of
Eclipfes] If the Type of Venus or Mercury be projected
by a large Scale, and their Orb, or Viſible Way be divi-
ded into Hours and Minutes of Time, as has been fhewn in
dividing the Moon's Orb in Page 395, it will pleaſantly
repre-
༔
1
143
afronomical Precepts.
reprefent to your View at every Moment of Time the
Place of the Planet during the Time of its Tranfiting the
Sun's Disk. So that he, who is able, and fitted with proper
• Inftruments ta obſerve the Times, I do not at all doubt, but
will find them to agree, with my Calculations.
1
Fortisakontiratie afisatisats
:
1
New
A
Aftronomical Precepts.
433
PRECEPT 20.
Shewing how to compute the true Times of the · Im=
merfion and Emerfion of the first Satellite of Ju-
piter.
Theſe Tables of this Satellite were firſt publiſh'd in the
Philofophical Tranfactions by M. Caffini, and reduced to the
following Method by Mr. Pound in the faid Tranfactions:
* But becauſe the Eclipſe of the first Satellite of Jupiter affords
the beſt Mean of determining the Longitude of Places on
'the Land where Teleſcopes of a convenient length may be
uſed, thirteen of theſe Eclipfes happening every twenty
three Days, it is requifite that the Obferver know near
when theſe Opportunites offer themſelves, left on the one
hand he let them flip, or elſe grow weary by a long At-
tendance on them, Phil. Tranf. N. 361.
Out of the firft Table take the Epoche for the Year with
its correfponding Number A and B ; and then add, out
of the Tables of Months, the Day, Hour, Minutes and
Seconds, neareſt leſs than the time of the Eclipſe you ſeek,
together with its Number A and B ; the Sum of the
Times, is the Mean Time of the Conjunction or middle of
the Eclipfe.
2. With Number A, thus collected take out the firft
Equation of the Conjunction; as also the Equation of
Number B, is always to be added to Number B before
found.
3. With Number B fo Equated, take out the fecond E-
quation of the Conjunctions; and in the laft Table, the
third Equation, as alfo the Semiduration of the Eclipſe an-
fwering to Number A.
4. To the Mean Time of the middle of the Eclipſe, add
all thoſe three Equations; the Sum fhall be the true Equa-
ted Time of the middle of the Eclipfe fought.
5. If Number B Equated be less than 500, fubtract the
Semiduration, and you will have the time of the Immer-
fion; or if it be more than 500, adding the fame, will
give the time of the Emerfion. (See the 28th Page of my
Syftem of the Planets Demonftrated.
O o
N B.
434
Aftronomical Precepts.
N. B. The times thus found are Equal Time, which muft
be reduced to the Apparent, as has been taught in the 2d
Precept. And in Leap-Year after February you muſt ſubtract
one Day from the Day of the Month.
Example. Let it be required to find the Emerſion of Ju-
piter's firſt Satellite January 20, 1728 ?
OPERATION.
D. H. M. S. Nu. A.
Nu. B.
1728
Fanuary 19
21 20 8
630
636
II
14 36
4
51
Mean Conj. 20
8
34
44
634
687
Equated
{
Ι
2
3
I am
I 9 19
4 Equat. B.
5
6
38
691 B. Equat.
Middle
20
9
49 47
Semid. add
I
4 44
January
20
ΙΟ
54 31 Emerfion Equal Time,
Equat. fub.
'Appar.Time 20
14 4
10 40 27
Example 2. Let it be required to find the Immerfion
'of Jupiter's firft Satellite Auguſt 26, 1728?
OPERATION.
D. H. M.
M. S.
1728
0 21 20
8
630
635
Auguſt Biff. 25 22 20
54
55
593
26 19 41
2
685
229
I
15
53
I Equat. B.
Equation2
8 33
3
I
I
230
Middle
26 21
21 6 36
Semid!
Aftronomical Precepts.
435
Semid. Sub.
D.
H.
M. S.
I
6 2
Auguft
Equat. add
26
20
O
34 Equal time of the Immersion,
2 18
Apparent
26.20 2
52 Immerfion.
At which time, the true place of the Sun is 14° 47′
58", and Mean Anomaly 2 S. 8° 139" which gives the
Equation of Time 2' 18" to be added as above; therefore
the Apparent Time is at 2' 5244 paſt 8 a-Clock in the Morn-
ing of the 27th Day.
Example 3. Let the time of the Emerfion of this Satel-
lite be required December 25, 1728.
OPERATION.
D. H. M. S.
A B
1728
O 21 20 8
630
636
December Biff 24
6 45 40
83 900
December
25
4
5 48
713 536
I
17 40
o Equ. B.
Equation
2
0 13
3
I
25
Equat.
25
5
25
6 Middle
Semidur. add
I
6
51
December
25
6
31 57
Eq. time fub.
4 14
25
6 27 43
'Apparent
1 Revol. add 1
T. of the next 27.
18 28 36
。 56 19 Emerfion.
002
After
436
Aftronomical Precepts.
After this manner may you easily compute the times of
the Immerſions and Emerfions of this Satellite; and if you
are fitted with a good Teleſcope and Pendulum-Clock, you
may compare your Obfervations with your Calculations, and
I doubt not but you will find them agree, as I have often ex-
perienced. When thefe Tables of the firft Satellite of Fu-
piter were publifh'd by Mr. Pound, there were feveral Ty-
pographical Errors, which I have taken care to correct, by
the Directions of the Reverend Author; from whom I
receiv'd the Corrections done by his own Hand, which
I have apply'd; and now thefe Tables appear free from any
Error, I hope, to the fatisfaction of the moft Curious in
Aftronomy.
PRECEPT 21.
Shewing how to find the Hour of the Night by the
Shadow of the Moon upon a Sun-Dial.
First, By Problem 47, find the true time of the Moon's
Southing Then obſerve, if the Shadow of the Moon fall
among the Morning-Hours upon a Sun-Dial, whatever the
Shadow wants of 12 a-Clock by the Dial, fubtract from
the time of the Moon's Southing: But if the Shadow of
the Moon fall among the Afternoon-Hours, fo many as it
is paft 12 by the Dial, add to the true time of the Moon's
Southing; the Sum, or Difference, is the true Hour and Mi-
nute of the Night.
Example, Anno 1728, January 16, at London, the true time
of the Moon's Southing will be at 13' paft 12 at Night;
and I obferving the Shadow on a Sun- Dial to fall upon the
Eleven a-Clock Hour-Line, then what will be the Hour
of the Night ?
OPE
Aftronomical Precepts.
437
OPERATION.
True time of the Moon's Southing
Shadow fhort of 12 fub.
Remains the true Hour of the Night
H. M.
12
13
I
O
71 13
Example 2. Admit the Moon is South at 7 h. 30 Af-
ternoon; and obferving the Shadow upon the Dial to fall
on the Hour of 1 h. 30; I defire to know the true Hour
of the Night?
OPERATION.
H. M.
True time of the Moon's Southing
Shadow paft 12 add
7 39
I 30
Sum, is the true Hour of the Night
9 0
Theſe Rules being fe plain, it is needleſs to give any
more Examples.
PRECEPT 22.
The Calculation, and Demonftration of our Pole-Star,
that it was not the Pole at the Creation of the
World, &c.
In the 19th Page of my System of the Planets Demonftra-
ted, I have fhewn you in what Signs the Fixed Stars increaſe,
and in which their Declinations decreaſe, and alſo hinted that
the prefent Polar Star would in Proceſs of time be to the South
of the Zenith of London; I fhall in this place clear up that
Point, and make it plainly appear to the meaneft Capacity,
that the Star of the fecond Magnitude in the End of the
Tail of the Leffer Bear was not the Polar Star at the Creation
of the World.
}
First,
438
Aftronomical Precepts.
९
First, then, You are to underſtand that this proceeds from
two Cauſes; the one by the Receffion of the Equinox, which
makes all the Fixed Stars feem to move in Confequentià 5011
a Year; the other, is their moving upon the Poles of the E-
cliptic, and therefore always keeping at the fame diſtance from
the Ecliptic; but the Declinations or Diftance from the
Equinoctial being always altering, (as I have fhewn in my
fore-cited Book) hence it is, that the Fixed Stars do not.
always keep the fame Places in the Heavens, but are found
fometimes on this fide, fometimes on that, fometimes to the
North, at other times to the South of your Zenith: And
this may ſeem a Paradox to the unskilful in Aftronomy; but
I can affure you, there is not any one Propofition in Euclid
more Demonftrable in it felf, than is now the Cafe before
us.
I fhall illuftrate this, in giving the Work of the preſent
Polar Star's neareft approach to the Pole, and alfo fhewing
when it will be to the South of the Zenith of London.
OPERATION.
S.
D. M. S.
D. M. S.
1727,Long. of the Pole of 2
Sub.its Longitude from 3
24
45 31 Lat. 66
4
II N.
0
O
O
Pole is now fhort of
5
14 29
бо
314
Nom Say,
60
Year.
if 50″ :
1 :
18869 Seconds.
I
5|0)188619 (377 Years.
1727 add.
38 2104 Sum,
36
Rem. 19
Lat.
Aftronomical Precepts.
439
}
}
Latitude of the Pole Star is
Obliquity of the Ecliptic add
Pole's Declination
D. M. S.
66 4 II North.
23 29 O
89 33 II North.
By the Work above it appears, that 377 Years hence,
which will be in the Year of Chrift 2104 the Pole Star's Lon-
gitude will be in the firſt Minute of Cancer, and then its
Declination will be 89° 33' 11" North, whofe Comple-
ment to 90° is o° 26' 49", which is the neareſt Approach
of the Polar Star to the Pole it felf that can poffibly
be.
As, when the Polar Star's nearest Approach to the
Pole is when its Longitude is in the firſt Minute of
Cancer; fo its greateft Distance from the Pole will be
when its Longitude is in the firft Minute of Capricorn. The
next thing therefore to be done, is to find how long time it
will be e're its Longitude will be in the firſt Minute of Ca-
pricorn, that is, in what ſpace of time it will by its Annual
Motion move a Semicircle, or from Cancer to Capricorn;
which is found by this Proportion ;
Years
If 50" : I :: 180
60
10800
60
510)64800/0 (12960 Years.
Remains
By the Work above, I have proved, that in 12960 Years.
the North Polar Star (and alfo every fixed Star) will move
half round the Heavens, and in that time will alter their
Declination 46 degr. 58 minutes, equal to the diftance of
the two Tropicks; therefore if you fubtract the diſtance
of the two Tropicks for the Declination of the Polar Star
when in Cancer, there will remain the Declination of the
Polar Star when in Capricorn.
OPE-
1
1
Aftronomical Precepts.
OPERATION.
Pole's Declination when in Cancer is
Diſtance of the two Tropicks fub.
Pole's Declination when in Capricorn
Declination of the Zenith of London
Diſtance of the Polar Star
D.
M. S.
89 33
II North.
46 58
0
42 35
II North.
51 32 O
8 56 49 to the
South of the Zenith, or Vertex of London: Then to find
when this will be, if to the Year of Chrift 2104, which is
the time the Polar Star will be in Cancer, you add 12960
the Years it is in moving from Cancer to Capricorn, the Sum
15064 is the Year of Chrift that the Polar Star that we now
obferve will then be 8° 56' 49 to the South of the Zenith
of London.
And if you would know what Star will be the Polar Star
in the Year of Chrift 15064 when our Pole that now is,
will be to the South of the Zenith of London 8° 56′ 49";
it will be a Star of the third Magnitude in the Calf of the
left Leg of Hercules, whofe Longitude in the Catalogue you
will find 16° 2', and Latitude 69° 22' North. And the
Star that was the Polar Star at the Creation of the
World, was a Star of the fecond Magnitude in the
Tail of Draco, which I have alfo Noted in the Cata-
logue. For the better clearing up of this Point, and for
your own Satisfaction, I fhall acquaint you how you may
prove, what has been faid by the Celestial Globe. Thus,
take in your Compaffes from any great Circle of the Globe,
(as from the Equinoctial or Ecliptick) the Latitude of the
preſent Polar Star 66° 4' 11"; carry this extant and fer one
Foot in the very beginning of Copricorn, viz. in the Point
where the Solftitial Colure, the Ecliptick and Tropick meet,
or touch each other, and the other Foot will fall in the fame
Colure 8° 56'49" fhort, or South of the Zenith of London,
which is the place on the Globe where the Pole will be in
the Year of Chrift 15064.
'Tis
Aftronomical Precepts.
441
}
'Tis certain that all the fixed Stars, do appear every Day
to rife and fet, and to move with a Circular Motion from
Eaft to Weft; the Plains alfo of theſe Diurnal Circular Re-
volutions being at right Angles to the Earth's Axis, or paral-
lel to the Equator.
All which is fairly and eafily accounted for, by fuppofing
our Earth to revolve round its own Axis in 24 Hours from
Weft to Eaft, (as I have proved in my Syftem of the Planets De-
monftrated:) But the Eye of the Spectator moving round to-
gether with the Earth, that muft appear to him immoveable
as a Ship doth to thoſe that are in it, till by Obſervation and
Judgment they find it otherwiſe.
There are above a 1000 Stars which appear, or that are
Vifible to the naked Eye; but the Teleſcope hath diſcover-
ed above 20 times as many more; and the larger and bet
ter theſe Glaffes are, the more are ftill diſcovered. With
my 13 Feet Tube I have ſeen above 20 in the Conftella-
tion Pleiades where the naked Eye can fee no more than
fix.
Σ
That the Fixed Stars by reafon of their immenſe Diſtance,
are to be looked upon as Points (unleſs ſo far as their Light
is dilated by Refraction) is plain from hence, That when
by the Appulfes to them they are Eclipfed or covered by her
Body, their Light doth, like that of the Planets in the like
cafe, vanifh or difappear gradually, but at once and all to-
gether; and when they emerge again out of the Eclipfe,
they do not become Visible by Degrees, but as it were in-
ftanteouſly, or at leaſt, in the ſpace of one or two Seconds.
The Diſtance therefore of the Fixed Stars, feems hardly
within the reach of any of our Methods to determin; but
from what has been laid down, we may draw fome Con-
clufions that will much illuftrate the Immenfity of it.
1. That the Earth's Annual Orbit is but as a Point in
comparison of the Diſtance of the Earth and Fixed Stars.
2. That could we advance towards the Stars 99 parts of
the whole Diſtance, and have only Part remaining, the
Stars would feem no bigger to us than they do here; for they
would fhew no otherwife than they do through a Teleſcope
which magnifies an hundred-fold.
I
3. That at leaft nine Parts in ten, of the ſpace between us
and the Fixed Stars, can receive no greater Light from the
Sun or any of the Stars, than what we have from the Stars
in a clear Night.
P P
4. That
442
Aftronomical Precepts.
4. That Light takes up more time in travelling from the
Stars to us, than we in making a Weft-India Voyage; That
found would not arrive to us from thence in 50000 Years,
nor a Cannon-Ball in a much longer time; this is eafily compu-
ted by allowing (according to Sir Ifaac Newton and Mr. Pound)
7 Minutes for the Journey of Light from the Sun hither;
and that Sound moves about 968 Feet in a Second of time,
as they found by the Eclipfes of Jupiter's Satellites.
!
Sirius
1
>
Sirius's Rifing, Lat. 519 32′ N.
Days.
January Febr.
March
April
May
June
H.
H.
H.
H. ¿ H.
H.
bood a on
I
*6A 25 | 4A 18
2
6
21 4 14
3
6
17
4 10
4.
6
13
4
5
6
10
4
6
6
6
78
6
3 58
2 3 54
06 200
5
59 3 50
9
5 55
3 46
ΙΟ
5
5
3 43
I
II
5
47
3 39
I
32 2
2 2 2 2 2 2 2 2 2 = food
2A 31
OA 3910M45 8M40
27
35 10
41 8 36
23
O
31 10
37 8
8 32
2120
O
27 IO
33
8
28
16
23 10
30
8
24
12
O
20 ΙΟ
25
8
20
8
0
4
Ι
9
16 ΙΟ 22 8.
12 IC 18 8
IO 14 8
15
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7
58
5
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10
8
2
55
O
I 10
6
12
5
43
3
35
I
52 11M58 10
2
77
58
54
13
5
39 3
31
I
48 11 54 9
58 7
50
14 5 35 3
27
I
44 II
50 9
54 7
46
15
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23
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46
9
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20
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9
467
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I
3311
-39
9
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18
5 19
3
12
I
30 II
35
9
38
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7
30
19
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8
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26
11
31
9
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26
20
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3
5
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22
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28
9
297
21
2 I
·5
6
3
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24
9
25 7
17
22
4
2
2
57
I
14 II
20
9
21
7 13
23
4
59
2
53
I
10 II
16
9 17
7
9
24
4
54 2
49
I
711
12
25 4 50 2
46
Ι
311
28
9
13
7
5
9
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7
26
4
45 2
42
I
0
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4 9
5
6 57
27
4
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28
4.36
2 2
39
O
5611
35
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52 10
56
29
4
32
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49 10
52
30
28
4
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4510
48
31
4
23
O
42
8∞∞∞∞
I
6
53
57 6
536
8 49 6 41
49
45
8
45
PP 2
443
1
444
Sirius's Rifing, Lat. 51° 32' N.
Days.
July August Septem. O&tober Novem. Decem.
H.
I
H.
6M36 4M34
H.
H.
H.
H.
2M40 12M51 10A 51
2
6 32
4 30
2 36 12
47 10
47
6
28
4
26
2
33 12
43 10
43
4
6
24
4
22 2
2912
40/10
49
8
8
5
6
20
4
18
2
2612
3610
35
8
6
6
16
4
14
2
22 12
3210
7
6
12
4
I I
8
6
8
4
7
9
6
4
4
10
5
59 4
mo
3
2 2 2
2 18 12
28 10
3 2
1512
II
2510
30
26
22
8
8
11 12
21
ΙΟ
18
8
2
812
12
18
10
14
8
ΙΙ
5 55
3
56
2
4 12
14 10
10
ते
∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ t
.8A 43
8 38
34
30
26
21
17
12
8
3
7 59
12
5
5
3
52
2
0
I 2
10 10
6
7 54
13
5 47
3
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6,10
2
7
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9
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37
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54
9
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9
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7
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5
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30
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46
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42
7
19
5
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27
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42
9
37
7
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23
20 5 20 3 24
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7
19
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3
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II
35 9 27
7
15
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3
16
I
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31 9 23
7
II
23
5
83
13
I
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27 9 18
24
5
43
9
I
17 ΙΙ
24 ୨
14
7
3
2 2 2
6 7∞ a
25 5 O 3
26
27
28
29
4 56 3
4 52 2
4
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30 4 40
5
I
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6
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2
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15 9
6
6
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58
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12
6
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7
8
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3
2
47
54 10
59
8
8 54 6 4.1
49 6 33
31
4 35
2 43
10
55
6
31
1
Sirius's Southing
445
Days.
H.
H. 1
H.
11
3
Ham+n6200 a
I
IOA 57
8A 50
January Febr. March
H. 'H. 1
5 A11 3A 17
April May
Fune
H. 1
7 A 3
IO 53
8 46
7
5
7
IO 47
8
42
6 56
5
4 IO
44 8
38
6
52
43
O
5
10
40 8
34
6
48
4
56
10
36 8 30
6
45 4
52
2
7
10
31
8
26
6
41
4
48
8 10 27
8
22
6
37
4
44
442
9
IO 23
8
18
6
34
4
40
ΙΟ
ΙΟ 19
8
15
6
31
4.37
I I
10
15
8
I I
6
27
4 3 33
4
12
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I I
8
7
6 23
4
30
13.
IO
68
3
6
20
4
26
14
I a
2
8
6
16
4
22
15
9 58
7
56
6
12
4
18
16
9 54 7
52
6
17 9 497
48
6
18
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19 9 41
7
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5 58
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4
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них
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°M 1
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20 9 37
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21 9 33
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25 9 16 7 18
m on cn2 2
37
5 54 4
2
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3
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36
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5 21 3 24
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12
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IIỐ
1
446
Sirius's Southing
Days.
Fuly
Auguft
Septem. Octob.
Novem. Decem.
Ι
2 3
=
H.
IIM 8
ΙΙ
I I
H.
}
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3 M23 IMIS
8
5
19
3
18
1
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5 15
3 14
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7
5
10 52 8
50
6
57
6
ΙΟ
48
8
8
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6
53
7
ΙΟ
44
8
43
6 50
555 5
12
5 4
∞ 4.0
8
8 ΙΟ 40 8
39
6
46
4
57
9
10 36
8 35
6
434
53
10
10
II
ΙΟ
12
10
on a d
31
8
13
271 8
23
ΙΟ 19
8
14
10
15 8
15
ΙΟ I I
16
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7 8
17
18 IO
IO
mo
3
O
19
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∞∞∞∞∞∞∞∞T
20 2 2
32
6 40 4
Mmm 222N
3
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6
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30
I
2011
31
9
9
7
16
3
26
ΙΣ
∞ 3
8
The Time of Sirius's Setting, Lat. 51° 32 N. 447
Days.
January
Febr. March
April May Fune
H.
H.
H.
H.
H. TH. 1
#2 on +
I
3 M29
1M22
11 A35 9A 43
7 A49 5A 44
3
4
6
78
9
10
1 I
12
13
14
15
16
17
18
2
19
20
21
22
mmm on on on m 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3
25
I
18
11
II
319 9 39
7
45 5 40
3 20
I
14 II
279
35
357
7
41
41 5
5
36
3
16
Ι
IO II
23 9
31 7
37 5
32
3
12
I
611
20 9
28
7
33
335
5 28
3
8
I
2
II'
169
9 24
7
305
3
3
12
2
4 12 A58
5512
51 12
4712
I I
139
20 7
26 5
54 11
10 9
16
7 22
50
50111
II
69
12 7
18
ИИИИ
24
20
5
16
5
10
47 II
39
97
14 5
6
43 Io
59 9
5 7
10 5
2
2
4312
39 10
559
7
6
4 58
2 39 12
3510
51 8
57
2
7
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2
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3512
31 10
478
53
6
584
50
2
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2612
22 12
1812
14 12
9/12
1610
2710 43 8
2310 40 8
50
506
54 4
46
46 6
50 4
42
2010
3618 43
6
46
46 4
33 8
396
12
ΙΟ
298
36
366
424
38 4
38
34
4 30
910
268
32 6
33 4
25
2 5
I 2
5
10
22
8 28 6
29
4 21
I I 2
I ΙΟ
188
246
25
4 I 17
23
1
5711
57
10
20
148
6 21 4 13
24
1
5311
53 10
ΙΟ
8
16
166
17 4 9
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50 10
78
12
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318
86
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SH
5
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41
41 II
4210
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I 30
9 52
9
56 5
56 3 49
9 49
7
52 5
523
45
31
I
26
9
46
5
48
448 The Time of Sirius's Setting, Lat. 519 32′ N.
}
Days.
July August Septem. October Novem. Decem.
H.
.1
H:
H.
H. TH. 1
H.
"}
I
7
8
9
10
II
12
13 2
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15
on on on m m m m m m od da 2 2
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3
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369 47 7 47
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5
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10 7
6
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4 54
2 47
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2 40 O 40
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498
586
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2
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∞ 4
34 in 6
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461 8
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6. 35.
4 23
0
24 10 328 39 6
18
910
5.10
55
0
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10
68
2 8
58
∞∞∞∞ ∞ ∞ ∞ ♪
J4 8 19 6 10 3 57
8
II
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6
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8
45
53
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58 3
7
59
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3
53
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3
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258 316
12 JO 22 8 27 6 18.
3 2
31
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27
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23
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6 14 4
I
The End of the First Volume,
JUL 10 1920
!
1
鉴
​ļ
!
.
1
}
A
1
>
i
A 426968
DUPL
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