Proposition LX. Theorem XXIII.
If two bodies S and P, attracting each other with forces reciprocally proportional to the ſquares of their diſtance, revolve about their common centre of gravity; I ſay that the principal axis of the ellipſis which either of the bodies as P deſcribesr by this motion about the other S, will be to the principal axis of the ellipſis, which the ſame body P may deſcribe in the ſame periodical time about the other body S quieſcent, as the ſum of the two bodies S + P to the firſt of two mean proportionals between that ſum and the other body S.
For if the ellipſes deſcribed were equal to each other, their periodic times by the laſt theorem would be in a ſubduplicate ratio of the body S to the ſum of the bodies S + P. Let the periodic time in the latter ellipſis be diminiſhed in that ratio, and the periodic times will become equal; but by prop. 15. the principal axis of the ellipſis will be diminiſhed in a ratio ſeſquiplicate to the former ratio; that is in a ratio, to which the ratio of S to S + P is triplicate; and therefore that axis will be to the principal axis of the other ellipſis. as the firſt of two mean proportionals between
