or an hyperbola; the point a in the firſt caſe falling on the ſame ſide of the line (GF as the point A; in the ſecond, going off to an infinite diſtance; in the third, falling on the other ſide 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 ſame ratio. Wherefore the points B, C, D lie in a conic ſection deſcribed about the focus S, in ſuch manner that all the right lines drawn from the focus S to the ſeveral points of the ſection, and the perpendicular: let fall from the ſame points on the right line GF are in that given ratio.
That excellent geometer M. De la Hire has ſolved this problem much after the ſame way in his conics, prop. 25. lib. 8.
