390 THE MATHEMATICAL PRINCIPLES [BOOK III PROPOSITIONSPROPOSITION I. THEOREM I. That the forces by which the circumjovial planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to Jupiter's centre ; and are reciprocally as the squares of the distances of the places of those planets/ro?/i that centre. The former part of this Proposition appears from Pham. I, and Prop. II or III, Book I : the latter from Phaen. I, and Cor. 6, Prop. IV, of the same Book. The same thing we are to understand of the planets which encompass Saturn, by Phaon. II. PROPOSITION II. THEOREM II. That the forces by which the primary planets are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the sun. ; and are reciprocally as the squares of the distances of the places of those planets from the sun's centre. The former part of the Proposition is manifest from Phasn. V, and Prop. II, Book I ; the latter from Phaen. IV, and Cor. 6, Prop. IV, of the same Book. But this part of the Proposition is, with great accuracy, de monstrable from the quiescence of the aphelion points ; for a very small aberration from the reciprocal duplicate proportion would (by Cor. 1, Prop. XLV, Book I) produce a motion of the apsides sensible enough in every single revolution, and in many of them enormously great. PROPOSITION III. THEOREM III. That the force by which the moon is retained in its orbit tends to the earth ; and is reciprocally as the square of the distance of its plac>>, from the earths centre. The former part of the Proposition is evident from Pha3n. VI, and Prop. II or III, Book I ; the latter from the very slow motion of the moon's apo gee; which in every single revolution amounting but to 3° 3' in consequentia, may be neglected. For (by Cor. 1. Prop. XLV, Book I) it ap pears, that, if the distance of the moon from the earth's centre is to the semi-diameter of the earth as D to 1, the force, from which such a motion will result, is reciprocally as D2^f 3, i. e., reciprocally as the power of D, whose exponent is 2^^ ; that is to say, in the proportion of the distance something greater than reciprocally duplicate, but which comes 59f time? nearer to the duplicate than to the triplicate proportion. But in regard that this motion is owinsr to the action of the sun (as we shall afterwards