Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/171



If the two oppoite angular points A and P (Pl. 9. Fig. 2.) of any parallelogram ASPQ touch any conic ection in the points A and P; and the ides AQ, AS of one of thoe angles, indefinitely produced, meet the ame conic ection in B and C; and from the points of concoure B and C to any fifth point D of the conic ection, two right lines BD, CD are drawn meeting the two other ides PS, PQ of the parallelogram, indefinitely produced, in T and R; the parts PR and PT, ''cut off from the ides, will always be one to the other in a given ratio. And vice vera, if thoe parts cut of are one to the other in a given ratio, the locus of the point D will be a conic ection, paing through the the four points'' A, B, C, P.

Join BP, CP, and from the point D draw the two right lines DG, DE, of which the firt DG hall be parallel to AB, and meet PB, PQ, CA in H, I, G; and the other DE hall be parallel to AC, and meet PC, PS, AB, in F, K, E; and (by Lem. 17.) the rectangle DE x DF will be to the rectangle DG x DH, in a given ratio. But PQ in to DE (or IQ) as PB to HB, and conequently as PT to DH; and by permutation, PQ