the vulgar cycloid. For if the diameter of the globe be infinitely increaſed, its sphærical ſuperficies 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 ſame with the common cycloid. But in that caſe the length of the arc of the cycloid between that plane and the deſcribing point, will become equal to four times the verſed ſine of half the arc of the wheel between the ſame plane and the deſcribing point as was diſcovered by Sir Chriſtopher Wren. And a pendulum between two ſuch cycloids will oſcillate in a ſimilar and equal cycloid in equal times as M. Huygens demonſtrated. The deſcent of heavy bodies alſo in the time of one oſcillation will be the ſame as M. Huygens exhibited.
The propoſitions here demonſtrated are adapted to the true conſtitution of the Earth, in ſo far as wheels moving in an of its great circles will deſcribe by the motions of nails fixed in their perimeters, cycloids without the globe; and pendulums in mines and deep caverns of the Earth muſt oſcillate in cycloids without the globe, that thoſe oſcillations may be performed in equal times. For gravity (as will be ſhewn in the third book) decreafes in its progreſs from the ſuperficies of the Earth; upwards in a duplicate ratio of the diſtances from the centre of the earth, downwards in a ſimple ratio of the ſame.
