220 THE MATHEMATICAL PRINCIPLES [BOOK 1 PROPOSITION LXXIL THEOREM XXXII. If to the several points of a sphere there tend equal centripetal forces de creasing in a duplicate ratio of the distances from those points ; and there be given both the density of the sphere and the ratio of the di ameter of the sphere to the distance of the corpuscle from its centre ; I say, that the force with which the corpuscle is attracted is propor tional 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 sev eral 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 distances inversely. But the particles are as the spheres, that is, in a triplicate ra tio 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. 3, Prop. IV. COR. 3. If to the several points of an^ two solids whatever, of like figare 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 situation to those two solids will be attracted by them, will be to each other as the diameters of the solids. PROPOSITION LXXIII. THEOREM XXXIII. If to the several points of a given sphere there tend equal centripetal forces decreasing in a duplicate ratio of the distances from the points ; 1 say, that a corpuscle placed within the sphere is attracted by a force proportional to its distance from the centre. In the sphere ABCD, described about the centre S, let there be placed the corpuscle P ; and about the same centre S, with the interval SP? conceive de