to an infinite diſtance, by which means the ſides of the figure which converge to thoſe points, will become parallel: And in this caſe the conic ſection will paſs through the other points. and will go the ſame way as the parallels in infinitum.
Lemma XIX.
To find a point P (Pl. 8. Fig. 8.) from which if four right line: PQ, PR, PS, PT are drawn to as many other right lines AB, CD, AC, BD given by poſition, each to each, at given angles, the rectangle PQ x PR, under any two of the lines drawn, ſhall be to the rectangle PS x PT, under the other two, in a given ratio.
Suppoſe the lines AB, CD, to which the two right lines PQ, PR, containing one of the rectangles, are drawn to meet two other lines, given by poſition, in the points A, B, C, D. From one of thoſe as A, draw any right line AH, in which you would find the point P. Let this cut the oppoſite lines BD, CD, in A and I; and, becauſe all the angles of the figure are given, the ratio of PQ to PA, and PA to PS, and therefore of PQ to PS will be alſo given. Subducting this ratio from the given ratio of PQ x PR to PS x PT the ratio of PR to PT will be given; and adding the given ratio's of PI to PR, and PT to PH the ratio of PI to PH, and therefore the point P will be given. Q. E. I.
Cor. 1. Hence alſo a tangent may be drawn to any point D of the locus of all the points P. For
