Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/167

 to an infinite ditance, by which means the ides of the figure which converge to thoe points, will become parallel: And in this cae the conic ection will pas through the other points. and will go the ame way as the parallels in infinitum.

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 poition, each to each, at given angles, the rectangle PQ x PR, under any two of the lines drawn, hall be to the rectangle PS x PT, under the other two, in a given ratio.

Suppoe 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 poition, in the points A, B, C, D. From one of thoe as A, draw any right line AH, in which you would find the point P. Let this cut the oppoite lines BD, CD, in A and I; and, becaue 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 alo 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.

Hence alo a tangent may be drawn to any point D of the locus of all the points P. For