Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/229

and ſuppoſe the latus rectum of the figure RPB to be L. From cor. 9. prop. 16. it is manifeſt that the velocity of a body, moving in the line RPB about the centre S, in any place P, is to the velocity of a body deſcribing a circle about the ſame centre, at the diſtance SP, in the ſubduplicate ratio of the rectangle $\scriptstyle {\frac {1}{2}}\times SP$ to $\scriptstyle SY^{2}$.. For by the properties of the conic ſections ACB is to $\scriptstyle CP^{2}$ as 2AO to L and therefore $\textstyle {\frac {2CP^{2}\times AO}{AP}}$ is equal to L. Therefore thoſe velocities are to each other in the ſubduplicate $\textstyle {\frac {CP^{2}\times AO\times SP}{ACB}}$ to $\scriptstyle SY^{2}$. Moreover by the properties of the conic sections, CO is to BO as BO to TO, and (by compoſition or diviſion) as CB to BT. Whence (by diviſion or compoſition) as CB to BT. Whence (by diviſion or compoſition) BO - or + CO will be to BO as CT to BT, that is AC will be to AO as CP to BQ; and therefore $\textstyle {\frac {CP^{2}\times AO\times SP}{ACB}}$ is equal to $\textstyle {\frac {BQ^{2}\times AC\times SP}{AO\times BC}}$. Now ſuppose CP, the breadth of the figure RPB, to be diminiſhed in infinitum, ſo as the point P may come to coincide with the point C, and the point S with the point B, and the line SP with the line BC, and the line ST with the line BQ; and the velocity of the body now deſcending perpendicularly in the line CB will be to the velocity of a body deſcribing a circle about the centre B at the diſtance BC, in the ſubduplicate ratio of $\textstyle {\frac {BQ^{2}\times AC\times SP}{AO\times BC}}$ to $\scriptstyle SY^{2}$, that is (neglecting the ratio's of equality of