afcent in this cafe, than in the cafe of a perpendicular afcent. Therefore if gravity varies in the reciprocal proportion of fome power of the diftance higher than unit, a body may run out to infinity in its orbit, if it be projected with a certain force.

22. If this force is the fame which it would acquire by falling from an infinite height, it will go off in a curve of the parabolic kind. But if it is projected with a greater force than that which would be acquired even by an infinite defcent, the curve will be of the hyperbolic kind. If it is projected with the fame velocity which it would acquire by falling from an infinite height (affuming different laws of gravity, but other circumftances fimilar) it will go off to infinity after a greater or lefs part of a revolution, or after a greater or fmaller number of revolutions, according as the power of the distance, which is reciprocally proportional to the gravity, is greater or lefs. The limit here is a quarter of a revolution from the apfis, or the place where the direction of the body's motion is perpendicular to the line drawn to the centre; for it must always take more than that to get off from the apfis to an infinite distance. If gravity obferve the reciprocal fefquiplicate proportion of the distance, then the body will go off in of a revolution. If it obferve the reciprocal duplicate, it will go off for ever in a parabola, in half a revolution. If it obferves the reciprocal power of the diftance, it will go off in a complete revolution. But if gravity obferve the reciprocal triplicate proportion of the distance, and the body be projected oblique to the radius, it will go off in an infinite number of revolutions *.

2

23.

In general, if gravity vary as the power of, the distance. reciprocally, and the body is projected obliquely upwards with Ꮓ

a

23. If gravity decrease in a lefs proportion than the reciprocal fimple proportion of the distances, and a body is projected from the apfis with any finite force whatsoever, it cannot rife for ever; but will have the fame velocity at any diftance, as it would have had at the fame diftance, fuppofing it had been projected at A directly upwards with the fame force of projection and fince any finite force would have been destroyed in the perpendicular, if the body move in a curve it muft return again, and after paffing the higher apfis, defcend again to the lower apfis, tho' that apfis be not in the fame place as before. If gravity increase as the distance increases, a fortiori the body will never be able to afcend to an infinite diftance, Thefe obfervations fhew the limits of the various forts of motions, that can proceed from various laws of gravity.

I.

WE

CHAP. IV.

Of the motion of the moon.

E have explained the motions of the bodies in the folar system, from gravity, and have taken notice of fome inequalities or errors in their motions, that arife from the fame principle.

a force that is to that which would carry it in a circle as I to

I

2

612m

it will rife for ever from the centre, and go off in the

part of a revolution, or in the n part of the revolution. Suppofing to be the excess of 3 above the number m. If the gravity follow the reciprocal proportion of the 28 power of the distance, the body will go off in 50 revolutions. Fluxions, 416, & feq.

99

See

But

But the manifold irregularities that are produced by it in the motion of the moon deferve particularly to be confidered, as fhe is the nearest to us in the fyftem, and as great advantages might be deduced. from her motions, if they could be fubjected to exact computation. Formerly, they who built fyftems had great difficulties to reconcile their principles with the phænomena: our author anticipates obfervations, and the more perfect our knowledge of the motions in the fyftem fhall become, the more will this philosophy be esteemed. Pofterity will fee its excellence yet more fully than we do, when the celeftial motions fhall be determined more accurately, by a feries of long-continued exact obfervations.

2. To give the principles of our author's computations on this perplexed fubject, in as plain a manner as poffible, we must recollect what has been already obferved; that if the fun acted equally on the earth and moon, and always in parallel lines, this action would serve only to restrain them in their annual motions round the fun, and no way affect their actions on each other, or their motions about their common centre of gravity. In that cafe, if they were both allowed to fall directly towards the fun, they would fall equally, and their respective fituations would be no way affected by their descending equally towards it. We might then conceive them as in a plane, every part of which being equally acted on by the fun, the whole plane would defcend towards the fun, but the refpective motions of the earth and moon would be the fame in this plane as if it was quiescent. Suppofing then this plane, and all in it, to have the annual motion imprinted on it, it would move regularly round the fun, while the earth and moon would move in it, with refpect to each other, as if the plane was at reft, without any

Book IV. irregularities. But because the moon is nearer the fun in one half of her orbit than the earth is, and in the other half of her orbit is at a greater diftance than the earth from the fun, and the power of gravity is always greater at a lefs diftance; it follows, that in one half of her orbit the moon is more attracted than the earth towards the fun, and in the other half lefs attracted than the earth; and hence irregularities neceffarily arife in the motions of the moon, the excefs, in the firft cafe, and the defect, in the fecond, of the attraction, becoming a force that difturbs her motion: add to this, that the action of the fun on the earth and moon is not directed in parallel lines, but in lines that meet in the centre of the fun.

3. To fee the effects of thefe powers, let us fuppofe that the projectile motions of the earth and moon were deftroyed, and that they were allowed to fall freely towards the fun. If the moon was in conjunction with the fun, or in that part of her orbit which is nearest to him, the moon would be more attracted than the earth, and fall with greater velocity towards the fun; fo that the diftance of the moon from the earth would be increafed in the fall. If the moon was in oppofition, or in the part of her orbit which is fartheft from the fun, fhe would be lefs attracted than the earth by the fun, and would fall with a lefs velocity towards the fun than the earth, and the moon would be left behind by the earth; fo that the diftance of the moon from the earth would be increafed, in this cafe alfo. If the moon was in one of the quarters, then the earth and moon being both attracted towards the centre of the fun, they would both directly defcend towards that centre, and by approaching to the fame centre, they would neceffarily approach at the fame time to

each

each other, and their distance from one another would be diminished, in this cafe. Now, whereever the action of the fun would increase their dif tance, if they were allowed to fall towards the fun, there we may be fure the fun's action, by endeavouring to feparate them, diminishes their gravity to each other; wherever the action of the fun would diminish their diftance, there the fun's action, by endeavouring to make them approach to one another, increases their gravity to each other: that is, in the conjunction and oppofition, their gravity towards each other is diminished by the action of the fun; but in the two quarters it is increased by the action of the fun. To prevent miftaking this matter, it must be remembred, it is not the total action of the fun on them that difturbs their motions, it is only that part of its action by which it tends to feparate them, in the firft cafe, to a greater diftance from each other; and that part of its action by which it tends to bring them nearer to each other, in the fecond cafe, that has any effect on their motions with refpect to each other. The other, and the far more confiderable, part has no other effect but to retain them in their annual courfe, which they perform together about the fun.

4. In confidering, therefore, the effects of the fun's action on the motions of the earth and moon with refpect to each other, we need only attend to the excefs of its action on the moon above its action on the earth, in their conjunction; and we must confider this excefs as drawing the moon from the earth towards the fun in that place. In the oppofition, we need only confider the excefs of the action of the fun on the earth above its action on the moon, and we must confider this excels as drawing the moon from the earth, in this place, in a direction

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