Page:Newton's Principia (1846).djvu/225

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 lineolao 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 $\scriptstyle \times$ pf to PF $\scriptstyle \times$ 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 $\scriptstyle \times$ ps to PS $\scriptstyle \times$ pi as IQ to iq. And compounding the ratios PI² $\scriptstyle \times$ pf $\scriptstyle \times$ ps is to pi² $\scriptstyle \times$ PF $\scriptstyle \times$ PS, as IH $\scriptstyle \times$ IQ to ih $\scriptstyle \times$ 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 $\scriptstyle \times$ ps to PF $\scriptstyle \times$ PS. 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, ps, as PI to PQ, and pi to pq; that is (because of the like triangles PIQ and PSF, piq and psf), as PS to PF and ps to pf. Thence ex æquo, the attraction of the corpuscle P towards S is to the attraction of the corpuscle p towards s as $\scriptstyle {\frac {PF\times pf\times ps}{PS}}$ is to $\scriptstyle {\frac {pf\times PF\times ps}{ps}}$, that is, as ps² to PS² . And, by a like reasoning, the forces with which the superficies described by the revolution of the arcs KL, kl attract those corpuscles, will be as ps² to PS² . And in the same ratio will be the forces of all the circular superficies into which each of the sphærical 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 sphærical superficies exerted upon those corpuscles will be in the same ratio.   Q.E.D