Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/161

 or an hyperbola; the point a in the firt cae falling on the ame ide of the line (GF as the point A; in the econd, going off to an infinite ditance; in the third, falling on the other ide of the line GF. For if on GF, the perpendiculars Ct, DK are let fall, IC will be to HB as EC to EB; that is, as SC to SB; and by permutation IC to SC as HB to SB, or as GA to SA. And, by the like argument, we may prove that KD is to SD in the ame ratio. Wherefore the points B, C, D lie in a conic ection decribed about the focus S, in uch manner that all the right lines drawn from the focus S to the everal points of the ection, and the perpendicular: let fall from the ame points on the right line GF are in that given ratio.

That excellent geometer M. De la Hire has olved this problem much after the ame way in his conics, prop. 25. lib. 8.