Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/349

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cutting off from the great circles AHB, ahb, the equal arcs HK, bk, 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 x pf to PF x pi ſo is RI to ri, that is (by cor. 3. lem. 7.) ſo is the arc IH to the arc ih. Again PI is to PS as IQ to SE, and ps ro pi as se or SE to iq; and ex æquo PI x ps to PS x pi as IQ to iq. And compounding the ratio's is to , as IH x IQ to ib x iq; that is, as the circular ſuperficies which is deſcribed by the arc IH as the ſemicircle AKB revolves about the diameter AB, is to the circular ſuperficies deſcribed by the arch ih as the ſemicircle akb revolves about the diameter ab. And the forces with which theſe ſuperficies attracts 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 x PS. And theſe forces again are to the oblique parts of them which (by the revolution 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