| 


A New and moft Accurate 


THEORY 


Q-F- T-HE: «7 
1 


Moons Motion; 


| Whereby all her [regularities 
may be ſolved, and her Place 
45 # calculated to Two Mi- 
nutes. ; | 


. * by. 
That Incomparable Mathematician 


Mr. Is 4A NEwro, 
WET, 


And Publiſhed i in Latin by 


Mr. David GREGORY 
in his Excellent Aſtronomy. 


— 


—— — 5 . 3 K 


ö 'Z O N D 0 BE 
Printed, and fold by A. Baldwin in 
Warwick-lane. 1702. 


” : £ * 
DDr — = - —— 3 - = — _ : A => L "= — 
„5757FE P TTT — 


-— 


. 
— * 4 


- —_— a 


(ii 


TO THE 


READER 


| HE Irregularity of the 
Moon's Motion hath been 
all along the juſt Com- 
blaint of Aſtronomers ; and indeed 

have always look'd upon it as 4 
great Misfortune that a Planet 
o near us as the Moon is, and 


phich might be ſo wonderfully uſe⸗ 
A 2 ful 


iv To che Reader. 


ful to us by her Motion, as 
well as ber Light and Attracti- 


on ( by which our Tides are chitf-. 


ly eccaftoned) ſhould have ber Or. 
bit "fo unaccountably various, that 
it is in a manner Vain to depend 
on any Calculation of an Ecliple, 
a Tranſit, or an Appulſe of 


her, tho never ſo accurately made. 


Whereas could ber Place be but 


truly calculated, 
Places would be found every where 
at Land with great Facility, and 
might be nearly oneſsd at Sea 


without the help * 4 Teleſcope, 


which cannot there be uſed. 


This Alt of the Moon's 
Mation depends (as is now well 
known, ſince Mr. Newton hath 


demonſtrated the Law of Univer- 


1 * on the Attraction 


—M 


the Longitudes of 


, 4. Ft we 


Be, hs WY: 9 SS Ws 


To the Reader. v 
of the Sun, which perturbs the 
Motion of the Moon (and of all 
other Satellites or ſecondary Pla- 
nets) and makes her move ſome- 
times faſter and ſometimes ſlower 
in ber Orbit; and makes conſe- 
quently an Alteration in the Fi- 


gure of that Orbit, as well as of 
its Inclination to the Plain of 


the Eclipticck. But this being 


now to be accounted for, and 
reduced to a Rule; by this 
Theory ſuch Allowances are made 
for it, as that the Place of the 
ſhall be truly Equa- 
ted. 


This therefore being perfectly 
New, and what the Lovers of 
Aſtronomy have a long while been 
put m hopes to receive from the 
Great Hand that hath now fini- 

—_— (hed 


vi To the Reader. 

ſhed it; I thought it would be 
of good ſervice to our Nation to 
give it an Engliſh Dreſs, and 
publiſh it by : felf : For as 
Dr. Gregory's Aſtronomy is 4 
large and ſcarce Book, it is 
neither every ones Money that can 
purchaſe it, nor Acquaintance that 
can procure it; and beſides I 
tope we have a great many Per- 
ſons in England that have Skill 
and Patience enouzh to calculate 
a Planet's Place, who yet it may 
be don't well enough underſtand the 
Latin Tongue o make themſelves 
Maſters of this Theory in the Au- 
thor's own Words. At leaſt ] per- 
ſwade my ſelf, that a Theory! fo 
eaſy and plain as this, Which car- 
ries along with it ſuch a Pretence 
to Exattneſs, will encourage many 


Perſons to imploy themſelves in Aſtro- 


nomical 


— — = Ms — EC — I 


— — VB 


To the Reader. vii 


nomical ( alculation, which before 
poſſi bly they neglected, becauſe they | 
judged there was but little Exactneſs | 
to be attained in it. And this would 4 
be a very uſe ful way of ſpending ; 
their leiſure Fours; and if they | 
would oblige us with the Publication 
of good Ephemerides, Tables, cc. 
they would ſoon enflame others with a 


Deſae of purſuing theſe kind 7 


Studies. 


The 


—_— — — „ 


The Famous 
Mr. J ſaac Newton's 
THEORY 
9 1 
MOON. 


HIS Theory hath been 
long expected by the Lovers 


of Art, and is now pub- 

liſh'd in Dr. Gregory's A- 
ſtronomy, in Mr. Newton's own 
Words. 


B By 


E 


By this Theory, what by all Aſtro- 
nomers was thought moſt difficult and 
almoſt impoſſible to be done, the Ex. 
cellent Mr. Newton hath now effect. 
ed, viz, to determine the Moon's 
Place even in her Quadratures, and 
all other Parts of her Orbit, beſides 
the Syzygys, ſo accurately by Calcu- 
lation, that the Difference between 
that and her true Place in the Hea- 
vens ſhall ſcarce be two Minutes, 
and is uſually ſo ſmall, that it may 
well enough be reckon'd only as a 


Defect in the Obſervation, AndF 


this Mr. Newtos experienced by 

comparing it with very many Pla- 

ces of the Moon obſerv'd by Mr. 

— and communicated to 
im. | 


The Royal Obſervatory at Green- 
wich is to the Weſt of the Meridian of 


Paris 29, 19”. Of Uranibargh 12, 
517. 30”, And of Gedanum 18. 


48 /. 
The 


r 


1 


SA 2. 


(11) 


The mean Motions of the Sun and 
Moon, accounted from the Vernal E- 
quinox at the Meridian of Greenwich, I 
„make to be as followeth. The laſt 
Day of December 1680. at Noon 
| Þ (Old Stile) the mean Motion of the 
Sun was 9 Sign. 20, 347. 467%, Of 
the Sun's Apogæum was 3 Sen. 79. 
23 30". 


„ The mean Motion of the Moon at 
y that time was 6 Sign 19, 357, 45”. 
2þ And of her Apogee 8 Sign. 4. 28% 
'5”, Of the Aſcending Node of the 
Moon's Orbit 5 Sign. 24. 147. 35%. 


And on the laſt Day of December 
1700, at Noon, the mean Motion of 
the Sun was 9 Sign. 200, 437. 50%. 
Of the Sun's Apogee 3 Sign 7 . 44 7. 
1. 30 7, The mean Motion of rhe 
ol Moon was 105i 15. 197. 50 J. 
Of the Moon's Apogee 11 Sign 8, 
„18“ 20%, And of her aſcend» 

ing Node 4 Sign. 27. 247. 204, 
For in 20 Julian Years or 7305 Days, 
he | B 2 the 


" (12) 


the Sun's Motion is 20 Revolut. © Sign, 


o. 97. 4% Andthe Motion of the | 


Sun's Apogee 21. o/. 


The Motion of the Moon jn the 


fame Time 1s 247 Rev. 4 Sign. 13 5 
347. 5% And the Motion of the 


Lunar Apogee is 2 Revol- 3 Sign. 3 . 
50 . 15%, And the Motion of her Þ 


Node 1 Revol. o Sign, 26 ®, 507. 15 U. 


All which Motions are accounted 
from the Vernal Equinox : Wherefore 
if from them there be ſubtracted the 
Receſſion or Motion of the EquinoCtial 
Point in Antecedentia during that ſpace, 
which is 167. © J. there will remain 
the Motions in reference to the Fixt 
Stars in 20 Julian Years; viz, the 
Sun's 19 Revol. 1 1 Sign. 29. 52/7. 247, 
Of his Apogee 4. 207. And the 


Moon's 247 Revol. 4 Sign. 13 . 17 


25, Of her Apogee 2 Revol. 3 Sign. 
12 3310. 35. And of the Node of 
3 Moon 1 Revol. o Sign. 279. 67, 
55 


Accord- | 


han 7 


1 


According to this Computation the 
Tropical Year is 365 Days. 5 Hours. 48 7. 
| 57 J. And the Sydereal Year is 

| 365 Pays. 6 Hous. 9. 147. 


Theſe mean Motions of the Lu- 
minaries are affected with various 
| Inequalities : Of which, 


Ys „ 


y 

f 1. There are the Annual Equations 
* of the aforeſaid mean Motions of the 

Sun and Moon, and of the Apogee and 

e © Node of the Moon. | 

. 

| The Annual Equation of the mean 

„ Motion of the Sun depends on the 

1 Þ Eccentricity of the Earth's Orbit round 

t Þ the Sun, which is 16 +; of ſuch Parts, 

eas that the Earth's mean Diſtance from 

7, the Sun ſhall be 1000 : Whence ' tis 

e Þ called the Equation of the Centre; and 

is when greateſt 19, 567. 207. 

N. 

f The greateſt Annual Equation of 

„the Moon's mean Motion is 11 7. 49 J. 


of her Apogee 20 and of her Node 
9% 300. 
-| And 


614) 


And theſe four Annual Equations 
are always mutually proportional one 
to another: Wherefore when any of 
them is at the greateſt, the other 
three will alſo be greateſt ; and when 
any one leſſens, the other three 
will alſo be diminiſhed in the ſame 
Rat io. 


The Annual Equation of the Sun's 
Centre being given, the three other 
correſponding Annual Equations will 
be alſo given; and therefore a 
Table of that will ſerve for all. 
For if the Annual Equation of the 
Sun's Centre be taken from thence, 
for any Time, and be called P, and 
let = PSY Q, Q. QQ == R, :P= 
D, D + D E, and D-++ D= 
2 F; then ſhall the Annual Equation 
of the Moon's mean Motion for 
that time be R, that of the Apo- 
gee of the Moon will be E, and 
that of the Node F. 


Only obſerve here, that if the E- 
quation of the Sun's Centre be re- 
quired 


G as ©. am wo . ¼ m -wt ue oa as 


as mm ao 17 £m Sa a. 


N. po 
bs + Ara 


( 15) 
quired to be added; then the Equa- 
tion of the Moon's mean Motion 
muſt be ſabtracted, that of her A- 
pogee mult be added, and that of the 
Node ſabdutted. And on the contra- 
ry, if the Equation of the Sun's 
Centre were to be /ubdudted, the 
Moon's Equation muſt be added, 
the Equation of her Apogee ſub- 


ducted, and that of her Node 44. 
ded. 


There is alſo an Equation of the 
| Moon's mean Motion depending on 
the Situation of her Apogee in reſpect 
of the Sun; which is greateſt when 
the Moon's Apogee is in an Octant 
with the Sun, and is nothing at all 
when it is in the Quadratures or 
Svzygys. This Equation, when 
greateſt, and the Sun 7 Perigeo, 
is 37, 56%, But if the Sun be in 
| Apogao, it will never be above 3 7. 
34 ”, At other Diſtances of the 
Sun from the Earth, this Equation, 
| when greateſt, is reciprocally as the 
Cube of ſuch Diſtance, But when 


the 


(16) 
the Moon's Apogee is any where but 
in the OFants, this Equation grows 
leſs, and 1s moſtly at the ſame di- 
ſtance between the Earth and Sun, as 
the Sine of the double Diſtance 
of the Moon's Apogee from the 
next Quadrature or Syzygy, to the 
Radius. 


This is to be added to the Moon's 
Motion, while her Apogee paſſes 
from a Quadrature with the Sun to a 
Syzygy; but is to be ſubtratted 
from it, while the Apogee moves 
from the Syzygy to the Quadra- 
ture. - 


There is moreover another Equs- 
tion of the Moons Motion, which 
depends on the Aſpect of the Nodes 
of the Moon's Orbit with the Sun: 
and this is greateſt when her Nodes 


are in Octants to the Sun, and va- 


niſhes quite, when they come to 
their Quadratures or Syzygys. This 
Equation is proportional to the 
Sine of the double Diſtance of the 

Node 


; 
A 
- 
D 
* 
* 
7 


a 

Node from the next Syzygy ot 
Quadrature; and at greateſt is but 
47 /. This muſt be added to the 
Moon's mean Motion, while the 
Nodes are paſſing from their Sy- 
zygys with the Sun to their Quadra- 
tures with him; but ſabtracted while 
they paſs from the Quadratures to 
the Syzygys. 


From the Sun's true Place take 
the equated mean Motion of the Lu- 
nar Apogee, as was above ſhewed, 
the Remainder will be the Annual 
Argument of the ſaid Apogee. From 
whence the Eccentricity of the Moon. 
and the ſecond Equation of her Apo- 
gee may be compar*'d after the man- 
ner following (which rates place alſo 
in the Computation of any other inter- 
mediate Equations.) 


(18 ) 


Let T repreſent the Earth, TS a 
Right Line joining the Earth and 
Sun, TACB a Right Line drawn 
from the Earth to the middle or 
mean Place of the Moon's Apogee, 
equated, as above: Let the Angle 
STA be the Annual Argument 0 
the aforeſaid Apogee, IT A the leaſt 
Eccentricity of the Moon's Orbit, 
TB the greateſt, Biſſect AB in C 
and on theCentre C with the Diſtance 
AC deſcribe a Circle AFB, and 
make the Angle BCF = to the 

double 


(19) 
double of the Annual Argument. 
Draw the Right Line T E, that ſhall 
be the Eccentricity of the Moon's 
Orbit; and the Angle BT F is the 
ſecond Equation of the Moon's Apo- 
gee required. 


In order to whoſe Determina- 
J tion let the mean Diſtance of 
the Earth from the Moon, or the 
Semidiameter of the Moon's Orbit, 
be xt000000; then ſhall its greateſt 
Eccentricity T B be 66782 ſuch 
Parts; and the leaſt TA, 43319. 
So that the greateſt Equation of the 
Orbit, viz. when the Apogee is in 
the Syzygys, will be 79. 397. 30 J. 
or perhaps 7. 4o”. (for I ſuſpect 
there will be ſome Alteration ac- 
cording to the Poſition of the Apogee 
in S or in .) But when it is in 
Quadrature to the Sun, the preateſt 
Equation aforeſaid will be 4. 57 7. 
156%. and the greateſt Equation of 
the Apogee 129. 157. 4 “ 


C3 Having 


(20) 


Having from gheſe Principles made 

a Table of the Equation of the 
Moon's Apogee, and of the Eccen- 
tricitys of her Orbit to each degree 
of the Annual Argument, from 
whence the Eccentricity T F, and 
the Angle BTF (viz. the ſecond 
and principal Equation of the Apo- 
gee) may eaſily be had for any 
Time required ; let the Equation 
thus found be added, to the firſt E- 
quated Place of the Moon's Apogee, 
if the Annual Argument be leſs than 
go ?, or greatear than 180 ?, and 
leſs than 270; otherwiſe it muſt be 
ſubducted from it : and the Sum or 
Difference {hall be the Place of the 
Lunar Apogee ſecondarily equated ; 
which being taken from the Moon's 
Place equated a third time, ſhall leave 
the mean Anomaly of the Moon corre- 
ſponding to any given Time. More- 
over, from this mean Anomaly of 
the Moon, and the before-found 
Eccentricity of her Orbit, may be 
ſound (by means of a Table of E- 
qQuations 


EST FO gg > _JGyrammr —_— 


($173 


quations of the Moon's Centre made 
to every degree of the mean Ano- 
maly, and ſome Eccentricitys, vi 
4<090, .50000, 55000, 6ocoo, and 
65900) the Proſtaphereſis or Equa- 
tion of the Moon's Centre, as 1n the 
common way : and this being taken 
from the former Semicircle of the 
middle Anomaly, and added in the 
latter to the Moon's Place thus 
thrice equated, will produce the 
Place of the Moon a fourth time 
equated, 


The greateſt Variation of the 
Moon ( vis. that which happens 
when the Moon is in an Octant 
with the Sun) is, nearly, reciprocal- 
ly as the Cube of the Diſtance of 
the Sun from the Earth. Let that 
be taken 37 7. 25%, when the Sun 
is in Perizeo, and 33. 4o . when 
he is n Apog. eo: And let the Dit- 
ierences of this Variation in the Oc- 
tants be made reciprocally as the 
Cubes of the Diſtances of the Sun 
irom the Earth; and fo let a Tow 

c 


(22 ) 
be made of the aforeſaid Variation 
of the Moon in her Octants (or its 
Logarithms) to every Tenth, Sixth, 
or Fifth Degree of the mean Anoma- 
ly: And for the Variation out of 
x Octants, make, as Radius to the 
Sine of the double Diſtance of the 
Moon from the next Syzygy or 
Quadrature : : ſo let the aforefound 
Variation in the Octant be to the 
Variation congruous to any other 
Affpect ; and this added to the 
Moon's Place betore found in the 
firſt and third Quadrant ( account- 
ing from the Sun) or ſabducted from 
it in the ſecond and fourth, will 
give the Moon's Place equated a 
fiftli time. 


Again, as Radius to the Sine of 
the Sum of the Diſtances of the 
Moon from the Sun, and of her 
Apogee from the Sun's Apogee (or 
the Sine of the Exceſs of that Sum 
above 360.) : : ſo is 2. 107, to 
a fixth Equation of the Moon's 
Place, which muſt be ſubtrafed, if 
the 


623) 


the aforeſaid Sum or Exceſs be leſs 
than a Semicircle, but added, if it 
be greater. Let it be made alſo, 
as Radius to the Sine of the Moon's 
Diſtance from the Sun:: ſo 27. 
20 J. to a ſeventh Equation : which 
when the Moon's Light is encreaſ- 
ing, add, but when decreaſing, ſab- 
tract; and the Moon's Place will 
be equated a ſeventh time, and this 
is her Place in her proper Orbit. 


Note here, the Equation thus pro- 
duced by the mean Quantity 2. 
20 U. is not always of the ſame 
Magnitude, but is encreaſed and di- 
miniſhed according to the Poſition 
of the Lunar Apogee. For if the 
Moon's Apogee be in Conjunction 
with the Sun's, the aforeſaid Equa- 
tion is about 54”, greater: but 
when the Apogees are in oppoſitt- 
on, 'tis about as much leſs; and 
it librates between its greateſt Quan- 
tity 37. 14%, and its leaſt x 7. 26”. 
And this is when the Lunar Apogee 
is in Conjunction or Oppoſition with 

the 


the Sun's : But in the Quadratures 
the aforeſaid Equation is to be lef. 
ſen'd about 50 % or one Minute, 
when the Apogees of the Sun and 
Moon are in Conjunction; but if 
they are in Oppoſition, for want of 
a ſufficient number of Obſervations, 
I cannot determine whether it is to 
be leſſen'd or increas'd. And even 
as to the Argument or Decrement 


of the Equation 27. 207. above men. 


tioned, I dare determine nothing 
certain, for the ſame Reaſon, vir. 
the want of Obſervation accurately 


made. 


If the ſixth and ſeventh Equati- 
ons are augmented or diminiſh- 
ed in a reciprocal Ratio of the Di- 
ftance of the Moon from the Earth, 
i. e. in a direct Ratio of the Moon's 
Horizontal Parallax ; they will be- 
come more accurate: And this may 
readily be done, it Tables are firſt 
made to each Minute of the faid 
Parallax, and to every ſixth or fifth 


Degree of the Argumennt of the 
ſixth 


(25) 
ſixth Equation for the Sixth, as of 


the Diſtance of the Moon from the 
Sun, for the Seventh Equation. 


— 


From the Sun's Place take the 
mean Motion of the Moon's aſcen- 
ing Node, equated as above; the 
emainder ſhall be the Annual Ar- 
gument of the Node, whence its 
econd Equation may be computed 
after the following manner ia the pre- 
eding Figure: 


D Let 


Let T as before repreſent the 
Earth, 'I'S a Right Line conjoining 
the Earth and Sun: Let alſo the 
Line TACB be drawn to the Place 
of the Aſcending Node of the Moon, 
as above equated; and let STA 
be the Annual Argument of the 
Node. Take T A from a Scale, 
and let it be to AB:: as 56 toy 
or as 18 + to 1, Then biſſect BA 
in C, and on C as a Centre, with 
the Diſtance CA, deſcribe a Circle 
as AFB, and make the Angle BCF 
equal to double the Annual Argyu- 
ment of the Node before found: 80 
ſhall the Angle BTF be the ſe— 
cond Equation of the Aſcending 
Node : which muſt be added when 
the Node is paſſing from a Quadra- 
ture to a Syzygy with the Sun, and 
ſubducted when the Node moves 
from a Syzygy towards a Quadra- 
ture. By which means the true Place 
of the Node of the Lunar Orbit 
will be gained: whence from Table 
made after the common way, thc 
| Moon'; 


4 
0 
C 
c 
[ 
b 
8 
a 


22 — © — — 2 2 — — 2 


(27) 

Moon's Latitude, and the Reduction 
of her Orbit to the Ecliptick, may be 
compured, ſuppoſing the Inclination 
of the Moon's Orbit to the Eclip— 
tick to be 4®. 59 7. 35 J. when the 
Nodes are in Quadrature with the 
Sun; and 5 9. 17 7. 207, when they 
are in the Syzygys. 


And from the Longitude and La- 
titude thus found, and the given Ob- 
Iiquity of the Ecliptick 23 ©. 29 7. 
the Right Aſcenſion and Declination 
of the Moon will be tound. 


The Horizontal Parallax of the 
Moon, when ſhe is in the Syzygys 
at a mean Diſtance from the Earth, 
| make to be 577. 30 J. and her 
Horary Motion 337. 327. 3277. 
and her apparent Diameter 31 /. 
zo”, But in her Quadratures at a 
mean Diſtance from the Earth, I 
make the Horizontal Parallax of 
the Moon to be 567. 40”. her 
Horary Motion 32 J. 12 /. 2 ½. 
and her apparent Diameter 31 7. 

D 2 37, 


(28 ) 
3/7, The Moon in an Octant to the 
Sun, and at a mean Diſtance, hath 
her Centre diſtant from the Centre 
of the Earth about 60 3. of the 
Earth's Semi-Diameters. 


The Sun's Horizontal Parallax I 
make to be 10 % and its apparent 


Diameter at a mean Diſtance from 
the Earth, T make 327. 157. 


The Atmoſphere of the Earth, 
by diſperſing and refracting the 
Sun's Light, caſts a Shadow as 
if it were an Opake Body, at 
leaſt to the height of 40 or 50 
Geographical Miles (by a Geogra- 
phical Mile I mean the fixtieth Part 
of a Degree of a great Circle, on 
the Earth's Surface) This Shadow 
falling upon the Moon in a Lunar 
Eclipſe, makes the Earth's Shadow 
be the larger or broader. And to 
each Mile of the Earth's Atmoſphere 


is correſpondent a Second in the | 


Moon's Disk, ſo that the Semidiame- 
ter of the Earth's Shadow projeQed 
| upon 


e 808 © 


222 


_ a. ___—___—_. = 1 TY 


—  "W0 -.- DO 


, 
| 
F 
7 
) 


( 29) 
upon the Disk of the Moon is to be 
encreaſed about 5o Seconds : or 
which 1s all one, in a Lunar Eclipſe, 
the Horizontal Parallax of the Moon 
is to be encreaſed in the Ratio 
of about 70 to 69. 


Thus far the Theory of this In- 
comparable Mathematician. And if 
we had many Places of the Moon 
accurately obſerved, eſpecially about 
her Quadratures, and theſe well com- 
pared with her Places at the ſame 
time calculated according to this 
Theory; it would then appear whe- 
ther there yet remain any other 
ſenſible Equations, which when ac- 
counted for, might ſerve to improve 
and enlarge this Theory. 


Dr. Greg. Aſtr. Elem. Phyſ. & 
Geom. p. 336. 


FINIS.