Page:Newton's Principia (1846).djvu/120

 (by Lemma XI) will be changed in the duplicate ratio of PR or QT. Therefore the ratio $$\scriptstyle \frac{QT^{2}}{QR}$$ remains the same as before, that is, as SP. And $$\scriptstyle \frac{QT^{2}\times SP^{2}}{QR}$$ is as SP³, and therefore (by Corol. 1 and 5, Prop. VI) the centripetal force is reciprocally as the cube of the distance SP. Q.E.I.

The perpendicular SY let fall upon the tangent, and the chord PV of the circle concentrically cutting the spiral, are in given ratios to the height SP; and therefore SP³ is as SY² $$\scriptstyle \times$$ PV, that is (by Corol. 3 and 5, Prop. VI) reciprocally as the centripetal force.

This is demonstrated by the writers on the conic sections.

Suppose CA, CB to be semi-axes of the ellipsis; GP, DK, conjugate diameters; PF, QT perpendiculars to those diameters; Qv an ordinate to the diameter GP; and if the parallelogram QvPR be completed, then (by the properties of the conic sections) the rectangle PvG will be to Qv² as PC² to CD²; and (because of the similar triangles QvT, PCF), Qv² to QT² as PC² to PF²; and, by composition, the ratio of PvG to QT² is compounded of the ratio of PC² to CD², and of the ratio of PC² to PF², that is, vG to $$\scriptstyle \frac{QT^{2}}{Pv}$$ as PC² to $$\scriptstyle \frac{CD^{2}\times PF^{2}}{PC^{2}}$$. Put QR for Pv, and (by Lem. XII) BC $$\scriptstyle \times$$ CA for CD $$\scriptstyle \times$$ PF; also (the points P and Q coinciding) 2PC for vG; and multiplying