SEC. V.] OF NATURAL PHILOSOPHY. 131 AS. will be either an ellipsis, a parabola, or an hyperbola ; the point a in the first case falling on the same side of the line GP as the point A ; in the second, going oft* to an infinite distance ; in the third, falling on the other side of the line GP. For if on GF the perpendiculars CI, DK are let fall, TC will be to HB as EC to EB ; that is, as SO to SB ; and by permutation, 1C 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. Where fore 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 perpendiculars 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 Conies, Prop. XXV., Lib. VIII. SECTION V. How the orbits are to be found when neither focus is given. LEMMA XVII. If from any point P of a given conic section, to the four produced sides AB, CD, AC, DB, of any trapezium ABDC inscribed in that section, as many right lines PQ, PR, PS, PT are drawn in given ang7ei, each line to each side ; the rectangle PQ, X PR of those on the opposite sides AB, CD, will be to the rectangle PS X PT of those on tie other two opposite sides AC, BD, in a given ratio. CASE 1. Let us suppose, first, that the lines drawn to one pair of opposite sides are parallel to either of I ^^ p ;T the other sides ; as PQ and PR to the side AC, and s