THE MATHEMATICAL PRINCIPLES [BOOK ). and mathematical proportions of them ; as I observed before ir (lie Defi nitions. In mathematics we are to investigate the quantities of forces with their proportions consequent upon any conditions supposed ; then, when we enter upon physics, we compare those proportions with the phe nomena of Nature, that we may know what conditions of those forces an swer to the several kinds of attractive bodies. And this preparation being made, we argue more safely concerning the physical species, causes, and proportions of the forces. Let us see, then, with what forces spherical bodies consisting of particles endued with attractive powers in the manner above spoken of must act mutually upon one another : and what kind of motions will follow from thence. SECTION XII. Of the attractive forces of sphcerical bodies. PROPOSITION LXX. THEOREM XXX. If to every point of a spherical surface there tend equal centripetal forces decreasing in, the duplicate ratio of the distances from those points ; I say, that a corpuscle placed within that superficies will not be attract ed by those forces any way. Let HIKL, be that sphaerical superficies, and P a corpuscle placed within. Through P let there be drawn to this superficies to two lines HK, IL, intercepting very small arcs HI, KL ; and because (by Cor. 3, Lem. VII) the triangles HPI,LPK are alike, those arcs will be proportional to the distances HP LP ; and any particles at HI and KL of the spheri cal superficies, terminated by right lines passing through P, will be in the duplicate ratio of those distances. Therefore the forces of these particles exerted upon the body P are equal between themselves. For the forces are as the particles directly, and the squares of the distances inversely. And these two ratios compose the ratio of equality. The attractions therefore, being made equally towards contrary parts, destroy each other. And by a like reasoning all the attractions through the whole spherical superficies are destroyed by contrary attractions. Therefore the body P will not be any way impelled by those attractions. Q.E.D. PROPOSITION LXXI. THEOREM XXXI. The same things supposed as above, I say, that a cor pu vie placed with out the sph(eric»l superficies is attracted towards the centre of tht sphere wiih a force reciprocally proportional to the square of its dis tance from that centre. Let AHKB, ahkb, be two equal sphaerical superficies described about