Cor. 9. From the ſame demonſtration it likewiſe follows, that the arc which a body, uniformly revolving in a circle by means of a given force, deſcribe ſin any time, is a mean proportioanl between the diameter of the circle, and the ſpace which the ſame body falling by the ſame given ſpace would deſcend thro' in the ſame given time.
Scholium.
The caſe of the 6th corollary obtains in the celeſtial bodies, (as Sir Chriſtopher Wren, Dr. Hooke, Dr. Halley have ſeverally obſerved) and therefore in what follows, I intend to treat more at large of thoſe things which relate to centripetal force decreaſing in a duplicate ratio of the diſtances from the centres. Moreover, by means of the preceding propoſition and its corollaries, we may diſcover the proportion of a centripetal force to any other known force, ſuch as that of gravity. For if a body by means of its gravity revolves in a circle concentric to the Earth, this gravity is the centripetal force of that body. But from the deſcent of heavy bodies, the time of one entire revolution, as well as the arc deſcribed in any given time, is given, (by cor. 9. of this prop.) And by ſuch propoſitions, Mr. Huygens, in his excellent book De Horlogie Oſcillatorio, has compared the force of gravity with the centripetal forces of revolving bodies.
The preceding propoſition may be likewiſe demonſtrated after this manner. In any circle ſuppoſe a polygon to be inſcribed of any number of ſides. And if a body, moved with a given velocity along the ſides of the polygon, is reflectedfrom the circle at the ſeveral angular points; the force, with which at
