Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/258

VCp to the angle VCP, that is as the tranverſ motion of the body p, to the tranſverſe motion of the body P: it is manifeſt that the body p, at the expiration of that time, will be found in the place m. Theſe things will be ſo, if the bodies p and P are equally moved in the directions of the lines pC and PC, and are therefore urged with equal forces in thoſe directions. But if we take an angle pCn that is to the angle pCk as the angle VCp to the angle VCP, and nC be equal to kC in that caſe the body p at the expiration of the time will really be in n; and is therefore urged with a greater force than the body P, if the angle nCp is greater than the angle kCp, that is, if the orbit upk move either in conſequantia, or in antecedentia with a celerity greater than the double of that with which the line CP moves in conſequentia; and with a leſs force if the orbit moves flower in antecedentia. And the difference of the forces will be as the interval mn of the places through which the body would be carried by the action of that difference in that given ſpace of time. About the centre C with the interval Cn or Ck ſuppoſe a circle deſcribed cutting the lines mr, mn produced in s and r, and the rectangle mn x mt will be equal to the rectangle mk x ms, and therefore mn will be equal to $\textstyle {\frac {mk\times ms}{mt}}$. But ſince the triangles pCk, pCn, in a given time, are of a given magnitude, kr and mr, and their difference mk, and their ſum mr, are reciprocally as the altitude pC, and therefore the rectangle 'mk x rm is reciprocally as the ſquare of the altitude pC. But moreover mr is directly as $\scriptstyle {\frac {1}{2}}$ms, that is, as the altitude