Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/121



If a body moves in the emi-circumference PQA; it is propoed to find the law of the centripetal force tending to a point S, o remote, that all the lines PS, RS drawn thereto, may be taken for parallels. Pl. 3. Fig. 5.

From C the centre of the emi-circle, let the emidiameter CA be drawn, cutting the parallels at right angles in M and M and join CP. Becaue of the imilar triangles CPM, PZT and RZQ we hall have $$\scriptstyle CP^2$$ to $$\scriptstyle PM^2$$ as $$\scriptstyle PR^2$$ to $$\scriptstyle QT^2$$; and from the nature of the circle, $$\scriptstyle PR^2$$ is equal to the rectangle $$\scriptstyle QR \times \overline {RN + QN}$$, or the points P, Q coinciding, to the rectangle $$\scriptstyle QR \times 2PMCP^2$$. Therefore $$\scriptstyle CP^2$$ is to $$\scriptstyle PM^2$$ as $$\scriptstyle QR \times 2PMPR^2$$ to $$\scriptstyle QT^2$$; and $$\textstyle \frac {QT^2}{QR} = \frac {2PM^2}{CP^2}$$ and $$\scriptstyle \frac {QT^2 \times SP^2}{QR} = \frac {2PM^2 \times SP^2}{CP^2}$$ that is, (neglecting the given ration $$\textstyle \frac {2SP^2}{CP^2}$$ reciprocally as $$\scriptstyle PM^2$$. Q. E. I.

And the ame thing is likewie ealy inferred from the preceding Propoition.