Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/154

 which produce to V, v, o that TV, tv may be equal to TS, tS. Bifect Vv in O, and erect the indefinite perpendicular OH, and cut the right Line VS infinitely produced in K and k, o that VK be to KS, and Vk to kS as the principal axe of the trajectory to be decribed is to the ditance of it's foci. On the diameter Kk decribe a circle cutting OH in H; and with the foci S, H, and principal axe equal to VH, decribe a trajectory. I ay the thing is done. For, biecting Kk in X, and joining HX, HS, HV, Hv, becaue VK is to KS, as Vk to kS; and by compoition, as Vk + Vk to KS + kS; and by diviion as Vk - VK to kS - KS that is, as 2VX to 2KX and 2KX to 2SX, and therefore as VX to HX and HX to SX the triangles VXH, HXS will be imilar; Therefore VH will be to SH, as VX to XH; and therefore as VK to KS. Wherefore VH the principal axe of the decribed trajectory has the ame ratio to SH the ditance of the foci, as the principal axe of the trajectory which was to be decribed has to the ditance of its foci; and is therefore of the ame pecies. And eeing VH, vH, are equal to the principal axe, and VS, vS are perpendicularly biected by the right lines TR, tr; 'tis evident (by lem. 15.) that thoe right lines touch the decribed trajectory. Q. E. F.

. About the focus S (Pl. 7. Fig. 6.) it is required to decribe a trajectory, which hall touch a right line TR in a given point R. On the right line TR let fall the perpendicular ST; which produce to K o that TV maybe equal to ST, join VR, and cut the right line VS indefinitely produced in K and k, o that VK may be to SK, and Vk to Sk as the principal axe of the ellipis to be decribed, to the ditance of itfoci; and on the diameter Kk decribing