Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/124

 the ratio of $$\scriptstyle PC^2$$ to $$\scriptstyle CD^2$$ and of the ratio of $$\scriptstyle PC^2$$ to $$\scriptstyle PF^2$$, that is, vG to $$\textstyle \frac {QT^2}{Pv}$$ as $$\scriptstyle PC^2$$ to $$\textstyle \frac {CD^2 \times PF^1}{PC^2}$$. Put QR for Pv, and (by lem. 11.) BC x CA for CD x PF PM-C, alo (the points P and Q coinciding,) 2PC for vG; and multiplying the extremes and means together, we hall have $$\textstyle \frac {QT^2 \times PC^2}{QR}$$ equal to $$\textstyle \frac {2BC^2 \times CA^2}{PC}$$. Therefore (by cor. 5. prop. 6.) the centripetal force us reciprocally as $$\textstyle \frac {2BC^2 \times CA^2}{PC}$$; that is (becaue $$\scriptstyle 2BC^2 \times CA^2$$ is given) reciprocally as $$\textstyle \frac 1{PC}$$; that is, directly as the diŧance PC. Q. E. I.

The ame otherwie.

In the right line PG on the other ide of the point T; take the point u o that Tu may be eqal to Tv; then take uV, uch as hall be to vG as $$\scriptstyle DC^2$$ to $$\scriptstyle PC^2$$. And becaue $$\scriptstyle Qc^2$$ is to PvG as $$\scriptstyle DC^2$$ to $$\scriptstyle PC^2$$ (by the conic ections) we hall have $$\scriptstyle Qv^2 = Pv \times uV$$. Add the rectangle uPv to both ides, and the quare of the chord of the arc PQ will be equal to the rectangle VPv; and therefore a circle, which touches the conic ection in P, and paes thro' the point Q will pals alo thro' the point V. Now let the points P and Q meet, and the ratio of uV to vG, which is the ame with the ratio of $$\scriptstyle DC^2$$ to $$\scriptstyle PC^2$$, will become the ratio of PV to PG or PV to 2PC; and therefore PV will be equal to $$\textstyle \frac {2DC^2}{PC}$$. And therefore the force, by which the body P revolves in the ellipes, will be reciprocally as