SEC. XII.J OF NATURAL PHILOSOPHY. the centre S, s ; their diameters AB, ab ; and let P and p be two corpus cles situate without the gpheres 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, SP, 1R, ir ; of which let SD, sd, cut PL, pi, in F and f. Let fall also to the diameters the perpen diculars IQ, iq. Let now the angles DPE, dpe, vanish; and because DS and ds, ES and es are equal, the lines PE, PP, and pe, pf, and the lineolso I )F, 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 cequo, as PI X pf to PF X pi so is R I 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 ceqno, PI X ps to PS X pi as IQ. to iq. And compounding the ratios PI2 X pf X ps is to pi2 X PF X PS, as IH X IQ to ih X iq ; that is, as the circular super ficies 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 X ps to PF XPS. And these forces again are to the oblique parts of them which (by the resolution of forces as in Cor. 2, of the Laws) tend to the centres in the directions of the lines PS, JDS-, as PI to PQ, and pi to pq ; that is (because of the like trian gles PIQ and PSF, piq and psf\ as PS to PF and ps to pf. Thence ex cequO) the attraction of the corpuscle P towards S is to the attraction of PF XpfXps. pf X PF X PS . the corpusclejo towards 5 as ~ =— is to — , that is, as ps2 to PS2. And, by a like reasoning, the forces with which the su perficies described by the revolution of the arcs KL, kl attract those cor puscles, will be as jDS2 to PS2. And in the same ratio will be the foroes of all the circular superficies into which each of the sphaerical superficies may be divided by taking sd always equal to SD, and se equal to SE. And therefore, by composition, the forces of the entire spherical superficies ex erted upon those corpuscles will be in the same ratio. Q.E.D