Page:Newton's Principia (1846).djvu/124

of PS, PI; that is (because of the parallels IH, PR, and the equal angles IPR, HPZ), of PS, PH, the difference of which is equal to the whole axis 2AC. Draw QT perpendicular to SP; and putting L for the principal latus rectum of the hyperbola (that is, for $\scriptstyle {\frac {2BC^{2}}{AC}}$, we shall have L $\scriptstyle \times$ QR to L $\scriptstyle \times$ Pv as QR to Pv, or Px to Pv, that is (because of the similar triangles Pxv, PEC), as PE to PC, or AC to PC. And L $\scriptstyle \times$ Pv will be to Gv $\scriptstyle \times$ Pv as L to Gv; and (by the properties of the conic sections) the rectangle GvP is to Qv² as PC² to CD²; and by (Cor. 2, Lem. VII.), Qv² to Qx² the points Q and P coinciding, becomes a ratio of equality; and Qx² or Qv² is to QT² as EP² to PF², that is, as CA² to PF², or (by Lem. XII.) as CD² to CB²: and, compounding all those ratios together, we shall have L $\scriptstyle \times$ QR to QT² as AC $\scriptstyle \times$ L $\scriptstyle \times$ PC² $\scriptstyle \times$ CD², or 2CB² $\scriptstyle \times$ PC² $\scriptstyle \times$ CD² to PC $\scriptstyle \times$ Gv $\scriptstyle \times$ CD² $\scriptstyle \times$ CB², or as 2PC to Gv. But the points P and Q coinciding, 2PC and Gv are equal. And therefore the quantities L $\scriptstyle \times$ QR and QT², proportional to them, will be also equal. Let those equals be drawn into $\scriptstyle {\frac {SP^{2}}{QR}}$, and we shall have L $\scriptstyle \times$ SP² equal to $\scriptstyle {\frac {SP^{2}\times QT^{2}}{QR}}$. And therefore (by Cor. I and 5, Prop. VI.) the centripetal force is reciprocally as L $\scriptstyle \times$ SP², that is, reciprocally in the duplicate ratio of the distance SP.   Q.E.I.

The same otherwise.

Find out the force tending from the centre C of the hyperbola. This will be proportional to the distance CP. But from thence (by Cor. 3, Prop. VII.) the force tending to the focus S will be as $\scriptstyle {\frac {PE^{3}}{SP^{2}}}$, that is, because PE is given reciprocally as SP².   Q.E.I.