Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/165

 as bk x bn to bf x bd, o (by Lem. I7.) pq x pr to ps x pt; and o (hy uppoition) PQ x PR to PS x PT. And becaue of the imilar trapezia bkAf, PQAS, as bk to bf; o PQ to PS. Wherefore by dividing the terms of the preceding proportion by the correpondent terms of this, we hall have bm to bd as PR to PT. And therefore the equiangular trapezia Dubd, DRPT are imilar, and conequently their diagonal Dk, DP do coincide. Wherefore b falls in the interection of the right lines AP, DP, and conequently coincides with the point P. And therefore the point P where-ever it is taken falls to be in the aigned conic ection. Q. E. D.

Hence if three right lines PQ, PR, PS, are drawn from a common point P to as many other right lines given in poition AB, CD, AC, each to each, in as many angles repectively given, and the rectangle PQ x PR under any two of the lines drawn be to the quare $$\scriptstyle PS^2$$ of the third in a given ratio: The point P, from which the right lines are drawn, will be placed in a conic ection that touches the lines AB, CD in A and C; and the contrary. For the poition of the three right lines AB, CD, AC remain the ame, let the line BD approach to and coincide with the line AC; then let the line PT come likewie to coincide with the line PS; and the rectangle PS; PT will become $$\scriptstyle PS^2$$, and the right lines AB, CD, which before did cut the curve in the points A and B, C, and D, can no longer cut, but only touch, the curve in thoe co-inciding points.