Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/163

 the right line which bi€cts thoe parallel ides will be one of the diameters of the conic ection. and will likewie biect RQ. Let O be the point in which RQ is biected, and PO will be an ordinate to that diameter. Produce PO to K o that OK may be equal to PO, and OK will be an ordinate on the other ide of that diameter. Since therefore the points A, B, P, and K are placed in the conic ection, and PK cuts AB in a given angle, the rectangle PQK (by prop. 17. 19. 21. & 23. book 3. of Apolloniu's conics) will be to the rectangle AQB in a given ratio. But QK and PR are equal, as being the differences of the equal lines OK, OP, and OQ, OR; whence the rectangles PQK and PQ x PR are equal; and therefore the rectangle PQ x PR is to the rectangle AQB, that is, to the rectangle PS x PT in a given ratio. Q. E. D.

Let us next uppoe that the oppoite ides AC and BD (Pl. 8. Fig.. 5.) of the trapezium, are not parallel. Draw Bd parallel to AC and meeting as well the right line ST in r, as the conic ection in d. loin Cd cutting PQ in r, and draw DM parallel to PQ, cutting Cd in M and AB in N. Then (becaue of the imilar triangles BTt, DBN) Bt or PQ is to Tt as DN to NB. And o Rr is to AQ or PS as DM to AN. Wherefore. by multiplying the antecedents by the antecedents and the conequents by the conequents, as the rectangle PQ x Rr is to the rectangle PS x Tt. o will the rectangle NDM be to the rectangle ANB, and (by cae 1.) o is the rectangle PQ x Pr to the rectangle Ps x Pt, and by diviion, o is the rectangle PQ x PT to the rectangle PS x PT. Q. E. D.

Let us uppoe latly the four lines PQ, PR, PS, PT (Pl. 8. Fig. 6.) not to be parallel