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

 cutting off from the great circles AHB, ahb, the equal arcs HK, bk, IL, il; and to thoe 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 alo to the diameters the perpendiculars IQ, iq. Let now the angles DPE, dpe vanih; and becaue 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; becaue their lat ratio, when the angles DPE, dpe vanih together, is the ratio of equality. Thee things then uppoed, it will be, as PI to PF o is RI to DF, and, as pf to pi o is df or DF to ri; and ex æquo, as PI x pf to PF x pi o is RI to ri, that is (by cor. 3. lem. 7.) o 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 $$\scriptstyle PI^2 \times pf \times ps$$ is to $$\scriptstyle pi^2 \times PF \times PS$$, as IH x IQ to ib x iq; that is, as the circular uperficies which is decribed by the arc IH as the emicircle AKB revolves about the diameter AB, is to the circular uperficies decribed by the arch ih as the emicircle akb revolves about the diameter ab. And the forces with which thee uperficies attracts the corpucles P and p in the direction of lines tending to thoe uperficies are by the hypotheis as the uperficies themelves directly, and the quares of the ditances of the uperficies from thoe corpucles inverely; that is, as pf x ps to PF x PS. And thee 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