tor to the poles; but it has the same intensity in all points of the earth's surface in the same latitude.

Now the space through which a heavy body would fall during a second in the latitude the square of whose sine is, has been ascertained by experiments with the pendulum to be 16.0697 feet; but the effect of the centrifugal force makes this quantity less than it would otherwise be, since that force has a tendency to make bodies fly off from the earth. At the equator it is equal to the 288th part of gravity; but as it decreases from the equator to the poles as the square of the sine of the latitude, the force of gravity in that latitude the square whose sine is, is only diminished by two-thirds of or by its 432nd part. But the 432nd part of 16.0697 is 0.0372, and adding it to 16.0697, the whole effect of terrestrial gravity in the latitude in question is 16.1069 feet; and at the distance of the moon it is 16.1069. sin2 57' 4".17 nearly. But in order to have this quantity more exactly it must be multiplied by 35, because it is found by the theory of the moon's motion, that the action of the sun on the moon diminishes its gravity to the earth by a quantity, the constant part of which is equal to the 358th part of that gravity.

Again, it must be multiplied by, because the moon in her relative motion round the earth, is urged by a force equal to the sum of the masses of the earth and moon divided by the square of Cm, their mutual distance. It appears by the theory of the tides that the mass of the moon is only the of that of the earth which is taken as the unit of measure; hence the sum of the masses of the two bodies is

Then if the terrestrial attraction be really the force that retains the moon in her orbit, she must fall through

333. Let mS, fig. 68, be the small arc which the moon would describe in her orbit in a second, and let C be the centre of the earth. If the attraction of the earth were suddenly to cease, the moon would

fig. 68.

E

n

ገጽ

go off in the tangent mT; and at the end of the second she would be in T instead of S; hence the space that the attraction of the earth causes the moon to fall through in a second, is equal to mn the versed sine of the arc Sm.

The arc Sm is found by simple proportion, for the periodic time of the moon is 27 days.32166, or 2360591", and since the lunar orbit without sensible error may

be assumed equal to the circumference of a circle whose radius is the mean distance of the moon from the earth; it is

355 be put

355

113

113

113(2360591")

The arc Sm is so small that it may be taken for its chord, therefore Cm. mn; hence

(mS)

consequently

4(355) (Cm)3 =2Cm. mn; (113)2 (2360591")9

(113)* (2360591")2

Again, the radius CE of the earth in the latitude the square of whose sine is, is computed to be 20898700 feet from the mensuration of the degrees of the meridian: and since

consequently,

mn=

=0.00445983

2(355) (20898700)

(113) (2360591") sin 57' 4".17

of a foot, which is the measure of the deflecting force at the moon. But the space described by a body in one second from of the the distance at the earth's attraction

moon was

shown to be 0.00448474 of a foot in a second; the difference is therefore only the 0.00002491 of a foot, a quantity so small, that it may safely be ascribed to errors in observation.

334. Hence it appears, that the principal force that retains the moon in her orbit is terrestrial gravity, diminished in the ratio of the square of the distance. The same law then, which was proved to apply to a system of satellites, by a comparison of the squares of the times of their revolutions, with the cubes of their mean distances, has been demonstrated to apply equally to the moon, by comparing her motion with that of bodies falling at the surface of the earth.

335. In this demonstration, the distances were estimated from the centre of the earth, and since the attractive force of the earth is of the same nature with that of the other celestial bodies, it follows that the centre of gravity of the celestial bodies is the point from whence the distances must be estimated, in computing the effects of their attraction on substances at their surfaces, or on bodies in space.

336. Thus the sun possesses an attracting force, diminishing to infinity inversely as the squares of the distances, which includes all the bodies of the system in its action; and the planets which have satellites exact a similar influence over them.

Analogy would lead us to suppose that the same force exists in all the planets and comets; but that this is really the case will appear, by considering that it is a fixed law of nature that one body cannot act upon another without experiencing an equal and contrary reaction from that body: hence the planets and comets, being attracted towards the sun, must reciprocally attract the sun towards them according to the same law; for the same reason, satellites attract their planets. This property of attraction being common to planets, comets, and satellites, the gravitation of the heavenly bodies towards one another may be considered as a general principle of this universe; even the irregularities in the motions of these bodies are susceptible of being so well explained by this principle, that they concur in proving its existence.

337. Gravitation is proportional to the masses; for supposing the planets and comets to be at the same distance from the sun, and left to the action of gravity, they would fall through equal heights in equal times. The nearly circular orbits of the satellites prove that they gravitate like their planets towards the sun in the ratio of their

masses: the smallest deviation from that ratio would be sensible in their motions, but none depending on that cause has been detected by observation.

338. Thus the planets, comets, and satellites, when at the same distance from the sun, gravitate as their masses; and as reaction is equal and contrary to action, they attract the sun in the same ratio; therefore their action on the sun is proportional to their masses divided by the square of their distances from his centre.

339. The same law obtains on earth; for very correct observations with the pendulum prove, that were it not for the resistance of the air, all bodies would fall towards its centre with the same velocity. Terrestrial bodies then gravitate towards the earth in the ratio of their masses, as the planets gravitate towards the sun, and the satellites towards their planets. This conformity of nature with itself upon the earth, and in the immensity of the heavens, shows, in a striking manner, that the gravitation we observe here on earth is only a particular case of a general law, extending throughout the system.

340. The attraction of the celestial bodies does not belong to their mass alone taken in its totality, but exists in each of their atoms, for if the sun acted on the centre of gravity of the earth without acting on each of its particles separately, the tides would be incomparably greater, and very different from what they now are. Thus the gravitation of the earth towards the sun is the sum of the gravitation of each of its particles; which in their turn attract the sun as their respective masses; besides, everything on earth gravitates towards the centre of the earth proportionally to its mass; the particle then reacts on the earth, and attracts it in the same ratio; were that not the case, and were any part of the earth however small not to attract the other part as it is itself attracted, the centre of gravity of the earth would be moved in space in virtue of this gravitation, which is impossible.

341. It appears then, that the celestial phenomena when compared with the laws of motion, lead to this great principle of nature, that all the particles of matter mutually attract each other as their masses directly, and as the squares of their distances inversely.

342. From the universal principle of gravitation, it may be foreseen, that the comets and planets will disturb each other's motion, so that their orbits will deviate a little from perfect ellipses; and

the areas will no longer be exactly proportional to the time that the satellites, troubled in their paths by their mutual attraction, and by that of the sun, will sensibly deviate from elliptical motion: that the particles of each celestial body, united by their mutual attraction, must form a mass nearly spherical; and that the resultant of their action at the surface of the body, ought to produce there all the phenomena of gravitation. It appears also, that centrifugal force arising from the rotation of the celestial bodies must alter their spherical form a little by flattening them at their poles; and that the resulting force of their mutual attractions not passing through their centres of gravity, will produce those motions that are observed in their axes of rotation. Lastly, it is clear that the particles of the ocean being unequally attracted by the sun and moon, and with a different intensity from the nucleus of the earth, must produce the ebb and flow of the sea.

343. Having thus proved from Kepler's laws, that the celestial bodies attract each other directly as their masses, and inversely as the square of the distance, La Place inverts the problem, and assuming the law of gravitation to be that of nature, he determines the motions of the planets by the general theorem in article 144, and compares the results with observation.