Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/307

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the times, by radii drawn both to that centre and to other mutually.


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 bodies P revolving round the other S remaining unmoved and deſcribe a figure ſimilar and equal to thoſe which the bodies deſcribe about each other mutually, in a ſubduplicate ratio of the other body S to the ſum of the bodies S + 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 it ſubduplicate ratio of the body S to the ſum of the bodies S + P. And by compoſition of ratio's the fums 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.