# The Mathematical Principles of Natural Philosophy (1729)/The Laws of the Moon's Motion according to Gravity

THE

LAWS

OF THE

MOON's

MOTION

According to

GRAVITY.

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The LAWS of the MOON's MOTION.

IN justice to the editor of this translation of Sir Isaac Newton's Principia, it is pro- per to acquaint the reader, that it was with my consent, he published an advertisement, at the end of a volume of miscellanies, concerning a small tract which I intended to add to his book by way of appendix; my design in which was to deliver some general ele- mentary propositions, serving, as thought, to explain and demonstrate the truth of the rules in Sir Isaac Newton's Theory of the Moon. ° T41 '* THE occasion of the undertaking-was merely accidental; for he shewing me a paper which I communicated to the author, in the year 1717, relating td the motion of the nodes of the Moonls orbit; I recollected, that the method made use 'of in settling the Equation for that” .motion, ' was equally applicable to any other motion of revolution. And therefore I thought' that it would not be unacceptable to a reader of the Principia, to fee the uses of the said method explained in the other Equations of 'fthe Moon's rhotion: Especially since the greatest part of the Theory of the Moon is laid down without any proof; and since those propositions, relating to the Moon's motion, which are demonilrated in the Principia, do generally depend upon calculations very intricate and abflrufe, the truth of which is not easily examined, even by those that are most skilful; and which however might be easily 'deduced from other principles. But in my progress in this design, happening' to find several general propositions relating to the Moon's motions, which serve to determine many things, which have hitherto been taken from the observations of” Aflzronomersz And I . having 5] having reason to think, that the Theory of the Moon might by these means, be made more perfeék 'and compleat than it is at present; I retarded the publication of the book, 'till I could procure due satisfaction by examining observations on places of the Moon. But finding this to be a work requiring a considerable time, not only in procuring such places as are proper, but also in performing calculations, upon a new method, not yet accommodated to practife by convenient rules, or ailifled by tables; I thought it therefore more convenient for the Book feller, not to Poop the publication of his impression any longet upon this accounts But that I' may in fomemeafure, satisfy those who are well oonverfant in Sir Mme Nefwton's Principia, (and I could Wifh that none but such would look over these papers,) that the said advertisement was not without some foundation; and that I may remove any fufpicion that the design is entirely laid aside, I have put together, altho' in no order, as being done upon a sudden refoluti. on, some of the Propositions, among many others, that I have by me, which seem chieiiy to be wanting in a Thef ory of the Moon, as it is a speculation A 3 foun<1@d 4 6 ] . founded on a physical cause; and those are what relate to the fearing of the m¢an motions. For altho' it be of lit? tie or no use in Astronomy to know the rules for ascertaining the mean motions of the Node or Apogee, since the faéi is all that is wanting, and that is othere wife known by comparing the observations of former ages with those of the present; yet in matter of peculation, this is the chief and rnofl nece£ fary thing required: since there is no other way to know that the cause is rightly assigned, but by 'showing that the motions are so much and no more than what they ought to be. i i But that it may not be altogether without its —use, I have added all the rules for the equation of the Moon's motion, except two; one of which is a monthly equation of the variation depending on the Moon's anomaly; and the other an equation arising from the Earth's being not in the focus of the Moon's orbit, as it has been supposed to be, in all the modern theories since Horrax. F or not having had time to examine over the observations which are necessary, but being oblig'd inflcad thereof; to take Sir .I/imr: Ne¢wz'on's theory for tny guide and direction, I cannot vsnwfs venture to depart from it too far, in eſtabliſhing equations entirely new; ſince I am well affured, upon the beſt au- thority, that it is never found to err more than feven or eight minutes. And therefore, hoping that the rea- der, who confiders the ſudden occafion and neceffity of my publiſhing theſe Pro- pofitions at this time, will make due allowance for the want of order and me- thod, and look upon them only as fo many diftinct Rules and Propofitions not con- nected: I ſhall begin, without any other preface, with fhewing the origine of that inequality, which is called the Va riation or Reflection of the Moon.

The Variation of the Moon. THE variation or refle- ction is that monthly in- equality in the Moon's mo- tion, wherein it more manifeftly differs from the laws of the motion of a pla- net in an elliptic orbit. Tycho Brabe makes this inequality to arife from a kind of libratory motion backwards and forwards, whereby the Moon is accelerat- ed and retarded by turns, moving fwifter in the firſt and third quarter, and flower in the fecond and fourth, which inequali-. ty is principally obſerved in the octants.

Sir *Isaac Newton* accounts for the 13
variation from the different force of grae
vity of the Moon and Earth to the
Sun, arising from different distances
of the Moon in its several ai'pe6l:s.
The mean gravity of the Moon to the
Sun, he supposes, is satisfied by the annual
motion of the ¢Moon round the
Sun; the gravity of the Moon to the
Earth, he supposes, is satisfied by a ref
volution of the Moon about the Earth.
But the difference of the Moon's gravity
to the Sun more or leii; than the
Earth's gravity, he supposeé, produces
two effects; for as this difference of
force may be resolved into two forces,
one acting in the Way, or contrary to
the Way, of the Moon about the Earth,
and the other acting in 'the line to or
from the Earth: the first: causes the
Moon to describe .a larger or smaller
area in the same time about the Earth,
according as it tends to accelerate or
retard it; the other changes the form of
the lunar orbit from what' it ought to
be merely from the Moonfs gravity to
the Earth, and both together make up
that inequality which is called the variation.
But since the real motion of the Moon,
Ii1D'a Iimaile motion, caused by a congnual
dew élzion from a itreight line, by
H the
l 19
the joint force of its gravity to the Sun
and Earth, thereby describing an orbit,
which inclofes not the Earth but the
Sun, is yet considered as a compound
motion, made from two motions, one
about the Sun, and the other about
the Earth; because two such motions
are requisite to answer the two forces of
its gravity, if separately considered: For
the very same reason, the Moon's motion
ought to be resolved into a third
motion of revolution, since there remains
a third force to be satisfied, and
that is the force arising from the altes
ration of the Moon's gravity to the
Sun. And this when considered, will
require a motion in a small ellipsis, in
the manner here described.
T H E circle ADFH represents t.he Fig;
orbit of the Moon about the Earth in the
center if, as it would be at a mean
distance, ' supposing the Moon had n'o
gravity to any, other body but the
Earth. The diameter AT F divides
that part of the orbit which is towards
the Sun, suppose ADF, from the part
opposite to the Sun, suppose AH E
The diameter at. right angles H ST D,
is the line of the Moon's conjunction
with or opposition to the Sun. The
figure PQLK is an Ellipsis, whose cen-f
ICI ll 1°
nr is carried round the Earth in the orbit
ABDEFIL hav' its longer axis PL
in length double Gfm§ lC shortcr aXi8 215,
and lying always parallel to TD, the line
joining the centers of the Earth and Sun.
Whllit the said figure is carried fromd
to B, the Moon revolves the contrary Way
from Qtr: N, so as to describe equal areas
in equal times about the centre of it;
and to perform its revolution in the
same time as the center of the said Elliptic
epicycle (if it may be so called,) performs
its revolution; the Moon being always
in the remoter extremity of its (hotter
axis in Qand K when it is in the quarters,
and in the nearest of its longer
axis at the time of the new and full Moon.
T H E shorter semiaxis of this Ellipsis
AQ is to the distance of its center from
the Earth AT; in the duplicate proportion
of the Moon's periodical time about
the Earth to the Sun's periodical time:
Which proportion, if there be 2 139 revolutions
of the Moon to the Stars in
160 fydercal years, is that of 4.7 to 8400.
T H E Fgure which is described by this
compound motion of the Moon in the Elliptic
epicycle, whilfi: the center of it is
carried round the Earth, very nearly repreiknts
the form of the Lunar orbit; supposing-
it without eccentricity, and tltgt
c l H
the plane was coincident with the plane of
the ecliptic, and that the Sun cQntinu'd
in the same place during the Whole reV0lution
of the Moon about the Earth.
F no M the above construction it:;}ppears,
that the proportion between e
mean distance of the Moon and its
greatest or least: distances, is easily all
iigned; being something larger than that
which is assigned by Sir War N efwtvfl in
the Sth proposition of his third book.
But as the cornputation there given, depends
upon the solution of abipuadratig
equation, affected with numera coeiiicif
ents; which renders it impossible to
compare the proportions with each other,
so as to fee their agreement or disagreement,
except in a particular application
to numbers; I (hall therefore fet down a
rule, in general terms, derived from his
method, which will be exact: enough,
unless the periods of the Sun and Moon
should be much nearer equal than they
are. LetL be the periodical time of the
Moon, S the period of the Sun, M the
synodical period of the Moon to the Sun,
and D be the difference of the periodsf
of the Sun and Moon; then, accordin
to Sir Muze Ne=wton's method, the difference
of the two axes of the Moon's
elliptic orbit, as it is contracted by the
aéliion
I. -I2
action of the Sun, is to the sum of the said
M--L
axes as 3L X, -7 to 4.DD-+SS. Bnt
according to the construéftion before laid
down; the iiid proportion is -as 3LL
I0 ZSS—L.L. ' e .
B Y Sir I/bac Newforfs rule, the difference
will he to the sum, nearly as
5 to 694; and consequently the. diameters
will be nearly as 689 to 699, or 69
to 7o: But by the latter mle, the difference
will be to the sum, nearly as I to
I 19; and the diameters or distances of
the Moon, in its conjun€tion and quadrature
with the Sun, will be as 59 to
60. Dr. Halley, (who in his remarks
upon the Lunar theory, at the end of
his catalogue of the Southern Ptars, first
took notice- of this contraction of the
Lunar orbit in the Syzygies from the phenomena
of the Moonis motion) makes
the difference of the diameters to the sum,
as 1 to QO; and consequently the greater
axis to the lesser, as 4 5-Q to 4435, ' °
B U T the difference, in these proportions
of the extream distances, tho' it
may appear considerable, is not, however,
to be diitinguiflfd by the observations
on the diameters of the Moon,
whilst the variations of the diameters,
from f
il13l
from' this cause, are interini'Xt the
other much greater variations, arising
from the eccentricity of the orbit..
T H E angle of the Moon's elongation H3
from the center, designed by B T N, is
properly the variation or reflection of the
Moon. The properties which are
evident from the description
F IR sT, It is as the line of the double
distance of the Moon from the quadrature
or conjunction with the Sun: For 'it
is the difference of the two angles BTA
-and N TA, whose tangents, by the con#
fl ruction, are in a given proportion.
SECONDL Y, A Tghe variation is, crateris
paribus, in the duplicate proportion
of the synodical time of the Moon's revolution
to the Sun. For the variation is
in proportion to themean diameter of
the epicycle, and that is' in:the duplicate
proportion of the synodicaltinie of revolution, ,
1 ' i A ' V
T H E greatest variation is an angle,
whose fine is to the radius, as the 'difference
of the greatest and least distances
fl' Q, and 'T L, that is 3./1€2, to their sum.
-According to the proportion of the lines
before described, this rule makes the elongation
near 29 minutes; which would
f p bc Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/478 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/479 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/480 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/481 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/482 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/483 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/484 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/485 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/486 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/487 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/488 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/489 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/490 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/491 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/492 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/493 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/494 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/495 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/496 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/497 le 34
portion nearly as the numbers 1 57 and
177. The duplicate of which proportion
is that of 107 to [56; which, according
to the rule, ought to be the proportion
of the least eccentricity to the
mean eccentricity.
S o that by this rule, the mean eccentricity,
(or half the sum of the greatest
and leal'c,) ought to be to the difference
of the mean from the least, (or half the
difference of the greatefr and the lea{'r,)
as 136 to 29.
How near this agrees with the Observations,
will appear from the numf
bers of Mr. Horrox or Mr. Flamped,
and of Sir Isaac Newton.
THE mean eccentricity according to
Mr. F/amfed or Mr. Horrox is»o.o5 5 2. 3 6,
half the difference between the greatest
and least is o o1 1 6 I 7; which numbers are
in the proportion of 1 5 5§ to 282 nearly.
Acconnmc to Sir Mac Newton,
the mean eccentricity is o.o 550 5, hall
the difference of the greatest and lean
is o.or 173; which numbers are in proportion
nearly as 13513; to 28%, each
of which proportions is very near that
above assigned.
BUT it is to be noted, that the rule,
which is here laid down, is true only upon
fuppohtion that the eccentricity exceeding Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/499 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/500 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/501 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/502 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/503 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/504 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/505 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/506 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/507 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/508 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/509 46] A
longer semi-axis CA, in the fubdupliezte
proportion of the longer axis to the sum of
the -two axes 5 as may be easily proved.
WHAT has been said, may be enough
to shew the form of the equant, and
the manner of' the motion about the
upper focus in general. But the precise
determination of the inequality of
the nrotiori, requires the knowledge of
the quadrature of the several sectors
of the equant, or at least, if any other
method be taken, of that which
is equivalent to lilch a quadrature.
There are divers methods for shewing
the relation between the mean and true
motion of a planet round the Sun, or
round the other focus, some more exact
than others. But the following seems
the molt proper for exhibiting in one
view, all the several hypotheses, and
rules, which are in common usein the
modern Astronomy, whereby it may
easily appear, how far they agree or differ from each other, and how much each
of them errs from the precise determination
of the motion, according to the
true law of an equal description of areas
about the Sun.
UPON the center F describe the el#
lipfis LN I, equal and similar to the clliptic
orbit /1 SD iP; but having its axes
FN 47]
FN and F L contrarily polited, that is,
the shorter axis LF lying in the longer
axis of the orbit A P, and the longer
axis FN parallel to the shorter C Q.
Let the focus of the laid ellipsis be in
And suppose two other ellipsis L B I and
L fl, to be drawn upon the common
axis Ll, one passing through the point
B, where the perpendicular FN interjects
the orbit, and the other through
the focus Let the line FR, revolving
with the planet in the orbit, be indefinitely
produced, till it interject the
f1r{'c ellipsis LN I (which was similar
to the orbit) in Q, the equant in p, and
the ellipsis LB I (drawn through the
intersection B,) in K. From the point K
let fall KH perpendicular to the line
of apsides QA P, and let it be produced
still it interject the Brit ellipsis LN/ in
O, and the ellipsis L fl (passing through
the focus f) in E. And lastly, in the
ellipsis LN/, let GM be an ordinate
equal and parallel to EH. In which
construction it is to be noted, that the
ellipsis L fl and LB/ are 'supposed as
drawn only to divide the line OKH
in given proportions, that KH may be
to UH, as the latus rectum of the orbit to
the transverse axis; and that E H or GM,
the bale of the elliptic .segment GLM,
ma? Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/512 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/513 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/514 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/515 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/516 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/517 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/518

III. THE third part of the inequali-

ty, answering to the trilinear space OKQ, being the difference of the elliptic sector OFQ and the triangle OFK.

```
THE sector OQF is proportional to
```

an angle, which is the difference of two angles, whose tangents are in the gi- ven proportion of the semi-latus rectum FB and the semi-transverse FN, or in the duplicate proportion of the lesser axis to the axis of the orbit. So that this sector, when at a maximum, is as an angle, whose sine is to the radius, as the difference of the latus rectum and transverse to their sum ; or as the diffe- rence of the squares of the semi-axes to their sum.

THE triangle OFK is proportional

to the rectangle of the co-ordinates OH and HF ; that is, as the rectan- gle of the line OH and its cosine, in the circle on the radius FN; or as the sine of the double of that angle, whose sine is OH ; that is, the double of the angle, whose tangent is to the tangent of the angle QFL, in the given ratio of the greater to the lesser axis ; or whose tangent is the tangent of the angle of mean motion answering to the elliptic sector QFL, in the duplicate of the said ratio. But this triangle OFK, when at a maximum, makes an angle of mean motion, which is to the angle called R, as BN, half the difference between the latus rectum and transverse axis, is to the double of the transverse axis.

```
So that the sector or triangle in orbits
```

nearly circular, is always nearly equal to the double of Bullialdus's equation.

```
THE triangle and sector being thus
```

determined, the equation for the tri- linear space is accordingly determined. From what has been said, it appears, that 1. THIS equation for the trilinear space OKQ, is to that for the triangle OKF, in a ratio compounded of BN, the difference between the semi-trans- verse and semi-latus rectum to the semi- latus rectum, and of the duplicate pro- portion of the sine OH to the radius ; or OKQ is to OKF, in a proportion compounded of the duplicate propor- tion of the distance of the foci to the square of the lesser axis, and the dupli- cate proportion of the line OH to the radius. For the trilinear figure OKQ and the triangle OKF, are nearly as OK and KH, which are in that pro- portion, and consequently it holds in this proportion to the double of Bulli- aldus's equation. 2. THIS equation, in different angles, is as the content under the sine comple- ment and the cube of the sine. For the triangle OKF, is as the rectangle of the sine and the sine complement. 3. IT is at a maximum, at an angle whole sine complement is to the radius, as the square of the greater axis is to the sum of the squares of the two axes; which in orbits nearly circular, is about 60 degrees of mean anomaly. 4. IN orbits of different eccentricities, it increases in the quadruplicate propor- tion of the eccentricity. 5. IT observes the contrary signs to that for the elliptic equant, called Bul- lialdus's equation ; subducting from the mean motion in the first and third qua- drants, and adding in the second and fourth, if the motion is reckoned from the aphelion. THE use of these equations, in find- ing the place of a planet from the upper focus, will appear from the following rules, which are easily proved from what has been said. LET t be equal to CA the semi- transverse, c equal to FC the distance of the center from the focus, b equal to CD the semi-conjugate, and R an angle subtended by an arch equal to the radius, viz, 57°. 17'. 44". 48"', or 57, 2957795 degrees. Take an angle T = cc/2tt R ; E = b/2t T; S = 4c/3b T .

```
The angle T be will the greatest e-
```

quation for the triangle OFK ; the an- gle S will be the greatest equation for the segment LMG ; and the angle E will be the greatest equation for the area OKFL. Which greatest equations be- ing found, the equations at any angle of mean anomaly, will be determined by the following rules.

LET M be the mean anomaly ;

and let τ be to T as the sine of the angle 2 M to the radius : In which pro- portion, as also in the following, there is no need of any great ex- actness, it being sufficient to take the proportions in round numbers.

```
TAKE e to E as the sine of 2M +/- 2 τ to
```

the radius; and s to S as the cube of the sine of M +/- τ to the cube of the radius.

THEN the angle QFL is equal to

M + e + s , in the first quadrant LN, or M - e + s , in the second quadrant Nl, or M + e - s in the third quadrant, or M — e — s in the fourth quadrant.

```
NOTE, That the small equation τ is al-
```

ways of the lame sign with the equation e; and in the case of the planets, always near the double of that equation.

THE angle RFA at the upper focus

F being known, the angle RSA at the Sun in the other focus, is found by the common rule of Bishop Ward ; viz. the tangent of half the angle RSA, is to be to the tangent of half the angle RFA, always in the given proportion of the perihelion distance SP to the aphelion di- stance SA. How these equations are in the several eccentricities of the Moon's orbit, will appear by the following Table. Eccentr, E. S. 0.040 1.23 09 0.045 1.45 13 0.050 2.09 17 0.055 2.36 23 0.060 3.06 30 0.065 3.38 38 0.070 4.14 47

To add one example; suppose the ec- centricity 0.060, the mean anomaly 30°. The sine of the double of the mean ano- maly, that is, the sine of 60 is to the ra- dius, nearly as 87 to 100; whence, if the equation E = 3'.06", be divided in that proportion, it will produce 2'.40" near- ly, for the equation e: the sine of M is, in this case, equal to 1/2 the radius, the cube is 1/8 of the cube of the radius; whence if the equation S=30" be divi- ded in the same proportion, it will produce near 4" for the equation s. There- fore the angle RFA, which is M + e + s, will be 3O°.2'.44"; and the half is 15°.l'.22" ; wherefore if the tangent of this angle be diminished, in the proportion of 1.06, the aphelion distance, to 94 the perihelion distance, it will produce the tan- gent of 13°.23'.13" ; the double of which 26°.46'.26", is the true anomaly or angle at the Sun RSA. And consequently, the equation of the center is 3°.13'.34" to be subducted, at 30 degrees mean anomaly.

WHEN the place of a planet is found

by this, or any other method; the place may be corrected to any degree of ex- actness by the common property of the equant, viz. that the rays are recipro- cally in the duplicate proportion of the velocity about the center. For in this case, if there be a difference between the mean motion belonging to the angle assumed at the upper focus, and the given mean motion, the error of the an- gle assumed is to the difference, as the rectangle of the semi-axes to the rect angle of the distances from the foci. But in orbits like those of the planets, the rules as they are delivered above are sufficient of themselves without further correction.

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[edit]POSTSCRIPT UPON reviewing these few sheets after they were printed off, which happened a little sooner than I expected, I fear the apology I have offered for delivering the propositions relating to the Moon's motion, in this rude manner, without giving any proof of them, or so much as mentioning the fundamental principles of their demonstration, will scarcely pass as a satisfactory one ; especially since there are among these propositions, some which, I am apt to think, cannot easily be proved to be either true or false, by any methods which are now in common use. WHEREFORE to render some satisfaction in this article, I shall add a few words concerning the principles from whence these propositions, and others of the like Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/526 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/527 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/528 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/529 Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 2.djvu/530 t [67] fimiiar to the orbit, whose lon er axis it the double of the eccentricity; glue center of the vibratory motion, that is the place Where it is swiftest, will be in the focus; the time of the lib ration, through the several f aces, is to be measured by feftors of? the said ellipsis, similar to those described by the body round the focus of the orbit; and the period of the vibratory motion will be the same with the period .of the revolution. In any other law of gravity, the equant for the vibratory motion, will either be of a form different from the orbit, or if it be of the same form, it must not be similarly divided. I may just mention, that the equant for the vibratory motion, in the case of the Moon, is a curve of the third kind, or whose equation is of four dimensions; but is to be described by an ellipsis, the center of the lib ration not being in the focus. - From this method of revolving the motion, it will not be difficult to shew the general causes of the alteration of the eccentricity and inequality in the motion of the apogee. For when the line of apsides is moving towards the Sun, it may be easily fhevvn, that since the external force in the apsides, is then centri-2 E .2 fugal, 1 [ § 3'] fugal, it will contribute to lengthen the f ace and time of the lib ration; by lengthening the space, it increases ' the eccentricity; and by lengthening the time of the lib ration, it prmlracts the time of the revolution to the apsis, and causes what is improperly called a motion of the apsis forward. But when the line of apsides is moving to the quadratures, the external force in the apsides, is at that time centripetal 5 which will contribute to shorten the space and time of lib ration; and by shortening the space will thereby leffen the eccentricity, and by shortening the time of lib ration, will thereby contract the time of the revolution to the apsis; and cause what is im# properly called a retrograde motion of the apsis. I shall only add a few remarks, which ought to have been made in their proper places. A As to the motion of the Moon in the elliptic epicycle (page 9.) it should have been mentioned, that there is no need of any accurate and perfect description of the curve called an ellipsis, it being only rg shew the elongation of the Moon; from the center of the epicycle; which doth not require any such accurate def@fiPIiQ11=

It should have been said, that whenf' S 1 the Moon is in any place of its orbit, suppose somewhere at N, in that half of the orbit which is next the Sun, it then being nearer the Sun than the Earth, has thereby a greater gravity to the Sun, than the Earth: which excess of gravity, according to Sir I/bac Nefwton's method, coniifts of two arts; one acting in the line N IC paraiiel to that which joins the Earth and Sun; and the other acting, in the line If B directed to the Earth, and these two forces, being compounded into one, make a force directed in the line NB; which is in proportion to the force of gravity, as that line NB is to TB nearly. Wherefore, as there is a force constantly impelling the. Moon, ” fpmewhere towards the point B, this force is supposed to iniiect the motion of the Moon into a curve line about that point; for the same reason as the gravity of it to the Earth, is supposed to infiect its motion into a curve line about the Earth: not that the Moon can actually have so many diftinct motions, but the one simple motion of the Moon round the Sun is fitp ofed to arise »frotn a composition of these several motions, In In the laik artiszlqzgvn lqlgeffmqli annual nations, (page t eruesou t Znlhavebeenadded. Let /E be the equation of the Sun's eenter; P the mean periodical time of thenodeora ee; Stherneanfynodical time of r£(;gSun's revolution tp the mae Of apogw Then win 3; /E be the annual equation of the node or apogee, according as S and P are can pounded. The like rule will serve for the annual e uation of the Moon's mean motion. if S be put for the Sun's period; P for the mean synodical period of the Moon to the Sun; and L for the Moon's period to the Stars: The annual equation of the Moods mean motion will be 3 LL 2 P S IE. Accordin to these rules when expounded, tlge equation for the node will e found to be always in proportion to the equation of the Sun's center, nearly as 1 to 13. The equation of the apogee to the equation of the Sun's center, as to to 53. And the equation of the Moon's mean motion to the same, as 8 to 77, It r