292

ASTRONOMY.

is more conformable lo astronomical observations; and thesecomputations afford an additional proof that the action of theEarth observes the same proportion to the distance which is herecontended for. Hence it follows, that the power which retainsthe Moon in her orbit is the same as that which causes bodies nearthe surface of the Earth to gravitate; for since the power by whichthe Earth acts on the Moon will cause bodies near the surface ofit to descend with all the velocity they are found to do, it is cer-tain no other power can act upon them besides ; because, if it did,they must of necessity descend swifter; and it is at length veryevident, that the power in the Earth which we call Gravity, ex-tends up to the Moon , and decreases in the 'duplicate proportionof the increase of the distance from the Earth . If to thc'motionof the satellite whereby it would be carried round its primary atrest, we superadd the same motion, both in regard to the velo-city and direction, as the primary itself lias, it will describe aboutthe primary the same orbit with as great regularity as if the primaryhad been indeed at rest. This proceeds from the law of motion,which makes a body near the surface of the Earth descend per-pendicularly, though the Earth he in a swift motion, that if thefalling'body did not partake of it, its descent would be remark-ably oblique. From this we learn, that if the satellite moved a-bout its primary with perfect regularity, besides its motion aboutthe primary, it would have the same progressive velocity withwhich the primary is carried about the Sun , in a direction paral-lel to that impulse of its primary. And, on the contrary, thewant of either of these, in particular of the impulse towards theSun , will occasion great inequalities in the motion of the secondaryplanet. The inequalities which would arise from the absence of.this impulse towards the Sun are so great, that by the regularitywhich appears in the motion of the secondary planets, it is pro-ved, that the Sun communicates to them the same velocity by itsaction as it gives to their primary at the same distance. Newtoninforms us, that upon examination lie found, that if any' of thesatellites of Jupiter were attracted by the Sun more or less than Ju-piter himself at the same distance, the orbit of that satellite, in-stead of being concentrical to Jupiter , would have its centre at agreater or less distance than the centre of Jupiter from the Sun ,nearly in the subduplicate proportion of the difference betweenthe Sun’s action upon the satellite and upon Jupiter . Therefore,if any satellite were attracted by the Sun hilt lOOdth part more orless Ilian Jupiter is at the same distance, the orbit of that satellitewould be distant from the centre of Jupiter no less than a lifthpart of the outermost satellite from Jupiter ; which is almost thewhole distance of the innermost satellite. I!v the like argument,the satellites of Saturn gravitate towards the Sun as much as Sa-turn himself at the same distance, and the Moon as much as theEarth . Tint same may be said of the satellites of the Georgium Sidus.

(106.) The Sun acts upon the secondary planets as much as uponthe primaries, at the same distance : but the action of the Sim up-on bodies is reciprocally in the duplicate proportion of the dis-tance ; therefore the secondary planets being sometimes nearerto the Sun than to tiie primary, and sometimes more remote, theyare not always acted upon in the same degree with their primary,but when nearer lo the Sun are attracted more, and when fartheroff are attracted less. Hence arise various inequalities in the mo-tions of the secondary planets. Some of these inequalities, how-ever, would take place though the Moon , if undisturbed by theSun, had moved in a circle concentrical to the Earth , and in theplane of the Earth ’s motion ; others depend on the elliptical figureand oblique situation of the Moon ’s orbit. One of the former is,that the Moon does not describe equal spaces in equal limes, butis continually accelerated as she passes from the quarter lo thenew or full, and is retarded again by the like degrees in returningfrom the new and full to the next quarter : but here we considernot so much the absolute as the apparent motions of the Moon with respect to us.

(107.) Let S, (Plate XXIII. Fig. 1,) represent the Sun , E theEarth, and M the Moon , moving in their orbit-. Because the meanaction of the Sun is the same both on the Earth and Moon , we mayconsider the M ion’s orbit as carried along with the Earth in its pro-gressive motion. Suppose the Mooir setting out from the quadra-ture at M, so that independent of the Sun ’s action it might move inthe circle MPNQ, draw 1)10 equal and parallel to ES, andjoin SO. Now the equal action ot the Sun at E and M may berepresented by S E and S M, of which SM is resolved into MOand O 3 : but M O does not affect the relative situations of the

Earth and Moon , it being equal and parallel to S E ; therefore thedisturbing force is SO, equal lo the radius of the Moon ’s orbit, andits effect lo bring the Moon nearer the Earth is as SO is toS E, or as the Moon’s distance to the Sun ’s distance from theEarth. Since this force brings the .Moon further below the tan-gent or line of its direction, it will make the orbit more curve at M ;froni.M to P and N the Sun’s action is greater on the Moon than onthe Earth , being inversely as the squares of the distances. From m,draw m c through S, of such a length that SE expressing the action ofthe Sun on the Earth, me may express the greater action of the Sun oil the Moon , draw mf equal and parallel to ES, join ef, andcomplete the parallelogram efm o,lhcn m c, the force on the Moon ,is resolved into mo, mf, of which w/'affects not their relative situ-ations ; hence m o is the disturbing force. Produce E m, and per-pendicular to Emdrawwir, audits parallel oq, and letro be paral-lel to E q, then the disturbing force m o is resolved into in q, whichtends to make the Moon recede directly from the Earth , and m rwhich tends to accelerate the Moon in the direction of its motion.At P the disturbing force is the excess ol the Sun ’s action on theMoon, the whole effect of which is to bring the Moon further fromthe Earth , so that it will make the orbit at P less curve. Since atM the disturbing force acts wholly to bring the Moon nearer theEarth, and at P to make it recede from the Earth ; it is evident thatat some point between M and P, as suppose at v, the disturbingforce will entirely act in the tangent to the Moon ’s orbit. At anypoint m, nearer to P, the perpendicular o q falls on E m ; producedbut nearer to M, it falls between E and m ; therefore from M to r,part of the disturbing force m q tends to make the Moon move di-rectly to the Earth; and from cto P, directly from the Earth ; in bothcases the part m r acts in direction of the Moon ’s motion, hence theMoon ’s velocity increases continually from M to P. And thus itmay be shewn that while the Moon moves from P to N, it is in likemanner retarded, for the distuibing force mr ads directly contraryto the Moon ’s motion, as from ta to x, being always directed to theline ES. At N the Moon is brought toward the Earth as at M.From N to Qand M the Sun’s action is k-ss on the Moon than on theEarth on account of the .Moon ’s greater distance. EclS E (Fig. 2,)be the force on the Earth , and mf taken, o.n m S, the smaller Ideeon the Moon . Draw me e'qual and parallel to S E, join J'e, complete the parallelogram femo, and revolve mo as before into mrand m q, and it will be evident from the above reasoning that theMoon will be accelerated from N to Q, and retarded from Q toM, that is, she will be retarded in the first and third, and accele-rated in the second and last quarters. At Q the whole effect is tomake the Moon recede directly from the Earth ; for the force aet-ing in the line QES, and being greater at E than at Q, the dis-tance between the Earth and Moon will be increased.

(108 .) The force, by which the tendency of the Moon to theEarth is diminished, is at P, as the difference of the squares of S Eand SP to the square of SP; at Q as the difference, of the squares ofS Qand S E to the square of S E, which differences are nearly equal,

..; these ratios

because S E is very great in respect to E P or E Q ;are nearly as QP : SE, for (S E -4- E Q) 2 — SE

(SE +

2 E Q x S E -f- E Q 2 : S E 2 is the same as

EQ

EQ + ~ : ES;

and because SE is very .great in respect ofEQ, the small quantity-g-jy may be neglected, and the ratio becomes that of 2 E Q : S E,

or, as twice the distance of the Moon to the distance of the Sun from tlie Earth : but it has been shewn that the tendency of theMoon to the Earth at M and N is augmented in the ratio of theMoon’s distance to that of the Sun , therefore the diminution at thesyzygies is nearly double the augmentation at the quadratures. Itis found that the Moon’s gravity to the Earth is increased at M andN about the 190th part of its whole mean gravity, and diminishedabout the l?5th part at Q and P ; lienee the iniluence of tile Sun on the whole is to increase the Moon’s distance from the Earth , andaugment its pcriodicaltime. The above analogies shew that all theseeffects must be greatest when the Earth is nearest the Sun : and itis so found by observation; the Moon ’s orbit lining dilated andits period protracted in- winter. Hence the Moon cannot de-scribe equal areas in equal times, except at the quadratures andsyzygies, and that it will differ most from this law about the oc-tants, and likewise that the figure of its orbit will be variable.

(109.) If we suppose the Sun ’s action to commence when theMoon is at M, (Fig. 3 ;) then since the Moon is brought nearer theB Earth