Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/292

 the vulgar cycloid. For if the diameter of the globe be infinitely increaed, its sphærical uperficies will be changed into a plane, and the centripetal force will act uniformly in the direction of lines perpendicular to that plane, and this cycloid of ours will become the ame with the common cycloid. But in that cae the length of the arc of the cycloid between that plane and the decribing point, will become equal to four times the vered ine of half the arc of the wheel between the ame plane and the decribing point as was dicovered by Sir Chritopher Wren. And a pendulum between two uch cycloids will ocillate in a imilar and equal cycloid in equal times as M. Huygens demontrated. The decent of heavy bodies alo in the time of one ocillation will be the ame as M. Huygens exhibited.

The propoitions here demontrated are adapted to the true contitution of the Earth, in o far as wheels moving in an of its great circles will decribe by the motions of nails fixed in their perimeters, cycloids without the globe; and pendulums in mines and deep caverns of the Earth mut ocillate in cycloids without the globe, that thoe ocillations may be performed in equal times. For gravity (as will be hewn in the third book) decreafes in its progres from the uperficies of the Earth; upwards in a duplicate ratio of the ditances from the centre of the earth, downwards in a imple ratio of the ame.