198 THE MATHEMATICAL PRINCIPLES [BOOK I attracting Jones will be the sam# in respect of the distance of the. bodies from, the common centre, as in respect of the distance between the two bodies. For those forces with which the bodies attract each other mutually, by tending to the bodies, tend also to the common centre of gravity lying di rectly between them ; and therefore are the same as if they proceeded from 'an intermediate body. QJG.D. And because there is given the ratio of the distance of either body from that common centre to the distance between the two bodies, there is given, of course, the ratio of any power of one distance to the same power of the . ther distance ; and also the ratio of any quantity derived in any manner from one of the distances compounded any how with given quantities, to another quantity derived in like manner from the other distance, and as many given quantities having that given ratio of the distances to the first Therefore if the force with which one body is attracted by another be di rectly or inversely as the distance of the bodies from each other, or as any power of that distance ; or, lastly, as any quantity derived after any man ner from that distance compounded with given q-uantities ; then will the same force with which the same body is attracted to the common centre of gravity be in like manner directly or inversely as the distance of the at tracted body from the common centre, or as any power of that distance ; or, lastly, as a quantity derived in like sort from that distance compounded with analogous given quantities. That is, the law of attracting force will be the same with respect to both distances. Q,.E.D. PROPOSITION LXII. PROBLEM XXXVIII. To determine the motions of two bodies which attract each other with forces reciprocally proportional to the squares of the distance between them, aflid are, let fall from given places. The bodies, by the last Theorem, will be moved in the same manner as if they were attracted by a third placed in the common centre of their gravity ; and by the hypothesis that centre will be quiescent at the begin ning of their motion, and therefore (by Cor. 4, of the Laws of Motion) will be always quiescent. The motions of the bodies are therefore to be deter mined (by Prob. XXV) in the same manner as if they were impelled by forces tending to that centre: and then we shall have the motions of the bodies attracting each other mutually. Q.E.I. PROPOSITION LXIII. PROBLEM XXXIX. To determine the motions of two bodies attracting each other with forces reciprocally proportional to the squares of their distance, and going off from given places in, given directions with given velocities. The motions of the bodies at the beginning being given, there is given