Let HIKL, be that sphærical 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 sphærical 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 sphærical 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 corpuscle placed without the sphærical superficies is attracted towards the centre of the sphere with a force reciprocally proportional to the square of its distance from that centre. [Pg 220] Let AHKB, ahkb, be two equal sphærical superficies described about the centre S, s; their diameters AB, ab; and let P and p be two corpuscles situate without the spheres in those diameters produced. Let there be drawn from the corpuscles the lines PHK, PIL, phk, pil, cutting off from the great circles AHB, ahb, the equal arcs HK, hk, IL, il; and to those lines let fall the perpendiculars SD, sd, SE, se, IR, ir; of which let SD, sd, cut PL, pl, in F and f. Let fall also to the diameters the perpendiculars IQ, iq. Let now the angles DPE, dpe, vanish; and because DS and ds, ES and es are equal, the lines PE, PF, and pe, pf, and the lineolæ DF, df may be taken for equal; because their last ratio, when the angles DPE, dpe vanish together, is the ratio of equality. These things then supposed, it will be, as PI to PF so is RI to DF, and as pf to pi so is df or DF to ri; and, ex æquo, as PI × pf to PF × pi so is RI to ri, that is (by Cor. 3, Lem VII), so is the arc IH to the arc ih. Again, PI is to PS as IQ to SE, and ps to pi as se or SE to iq; and, ex æquo, PI × ps to PS × pi as IQ to iq. And compounding the ratios PI2 × pf × ps is to pi2 × PF × PS, as IH × IQ to ih × iq; that is, as the circular superficies which is described by the arc IH, as the semi-circle AKB revolves about the diameter AB, is to the circular superficies described by the arc ih as the semi-circle akb revolves about the diameter ab. And the forces with which these superficies attract the corpuscles P and p in the direction of lines tending to those superficies are by the hypothesis as the superficies themselves directly, and the squares of the distances of the superficies from those corpuscles inversely; that is, as pf × ps to PF × PS.