cted is proportional to the ſemi-diameter of the sphere.
For conceive two corpuſcles to be ſeverally attracted by two ſpheres, one by one the other by the other, and their diſtances from the centres of the ſpheres to be proportional to the diameters of the ſpheres reſpectively; and the ſpheres to be reſolved into like particles diſpoſed in a like ſituation to the corpuſcles. Then the attractions of one corpuſcle towards the ſeveral particles of one ſphere, will be to the attractions of the other towards as many analogous particles of the other ſphere in a ratio compounded of the ratio of the particles directly and the duplicate ratio, of the diſŧances inverſely. But the particles are as the ſpheres, that is in a triplicate ratio of the diameters, and the diſtances are as the diameters; and the firſt ratio directly with the laſt ratio taken twice inverſely, becomes the ratio of diameter to diameter. Q. E. D.
Cor. 1. Hence if corpuſcles revolve in circles about ſpheres compoſed of matter equally attracting; and the diſtances from the centres of the ſpheres be proportional to their diameters; the periodic times will be equal.
Cor. 2. And vice versa, if the periodic times are equal, the diſtances will be proportional to the diameters. Theſe two corollaries appear from cor. 5. prop. 4.
Cor. 3. If to the ſeveral points of any two ſolids whatever, of like figure and equal denſity, there tend equal centripetal forces decreaſing in a duplicate ratio of the diſtances from thoſe points; the forces with which corpuſcles placed in a like
