>EC. XL] OF NATURAL PHILOSOPHY. 197 PROPOSITION LIX. THEOREM XXII. The periodic time of two bodies S and P revolving round their common centre of gravity C,is to the periodic time of one of the bwlies 1? re volving round the other S remaining unmoved, and describing a fig ure similar and equal to those which the bodies describe about each other mutually r, in a subduplicate ratio of the other body S to the sii/rn of the bodies S -f P. For, by the demonstration of the last Proposition, the times in which any similar arcs PQ and pq are described are in a subduplicate ratio of the distances CP and SP, or sp, that is, in a subduplicate ratio of the ody S to the sum of the bodies S + P. And by composition of ratios, the sums of the times in which all the similar arcs PQ and pq are described, that is, the whole times in which the whole similar figures are described are in the same subduplicate ratio. Q.E.D. PROPOSITION LX. THEOREM XXIII. If two bodies S and P, attracting each other with forces reciprocally pro portional 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, describes 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, propor tionals 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 4- P. Let the periodic time in the latter ellipsis be diminished in that ratio, and the periodic times will become equal ; but, by Prop. XV, 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 4- 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 S -f- P and S to S 4- P. And inversely the principal axis of the ellipsis de scribed about the movable body will be to the principal axis of that described round the immovable as S + P to the first of two mean proportionals be tween S 4- P and S. Q.E.D. PROPOSITION LXI. THEOREM XXIV. If two bodies attracting each other with any kind of forces, and not otherwise agitated or obstructed, are moved in any manner whatsoever, those 'motions will be the same as if they did not at all attract each other mutually, but were both attracted with the same forces by a third body placed in their common centre of gravity ; and the law of the