Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/172

 is to PT, as DE to DH. Likewie PR is to DF as RG to DC, and therefore as (IG or) PS to DG; and, by permutation, PR is to PS as DF to DG; and, by compounding thoe ratio's, the rectangle PQ x PR will be to tie rectangle PS x PT as the rectangle DE x DF is to the rectangle DG x DH, and conequently in a given ratio. But PQ and PS are given, and therefore the ratio of PR to PT is given. Q. E. D.

But if PR and PT are uppoed to be in a given ratio one to the other, then by going back again by a like reaoning, it will follow that the rectangle DE x DF is to the rectangle DG x DH in a given ratio; and o the point D (by lem. 18.) will lie in a conic ection paing thro' the points A, B, C, P, as its locus. Q. E. D.

Hence if we draw BC cutting PQ in r, and in PT take Pt to Pr in the ame ratio which PT has to PR: Then Pr will touch the conic ection in the point B. For uppoe the point D to coalece with the point B, o that the chord BD vanihing, BT hall become a tangent, and CD and BT will coincide with CB and Br. Co tt. z. And vice verla, if B r is atangent, and the lines B D, C D meet in any point D of a conic ection; P R will be to P Tas Pr to Pr. And on the contrary, if.PR is to PTaPr to Pr, (hm BD, and CD will meet in ome point D of a conic ection.

One conic ection cannot cut another conic ection in more than four points. For, if it is poible, let two conic ections pas thro' the five points A, B, C, P, O; and let the right line BD cut them in the points D, d, and the right line Cd cut the right line PQ in q. Therefore PR is to PT as Pq to PT: