Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/114



From the ame demontration it likewie follows, that the arc which a body, uniformly revolving in a circle by means of a given force, decribe in any time, is a mean proportioanl between the diameter of the circle, and the pace which the ame body falling by the ame given pace would decend thro' in the ame given time.

The cae of the 6th corollary obtains in the celetial bodies, (as Sir Chritopher Wren, Dr. Hooke, Dr. Halley have everally oberved) and therefore in what follows, I intend to treat more at large of thoe things which relate to centripetal force decreaing in a duplicate ratio of the ditances from the centres. Moreover, by means of the preceding propoition and its corollaries, we may dicover the proportion of a centripetal force to any other known force, uch 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 decent of heavy bodies, the time of one entire revolution, as well as the arc decribed in any given time, is given, (by cor. 9. of this prop.) And by uch propoitions, Mr. Huygens, in his excellent book De Horlogie Ocillatorio, has compared the force of gravity with the centripetal forces of revolving bodies.

The preceding propoition may be likewie demontrated after this manner. In any circle uppoe a polygon to be incribed of any number of ides. And if a body, moved with a given velocity along the ides of the polygon, is reflectedfrom the circle at the everal angular points; the force, with which at