Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/164

 to the ides AC, AB, but any way inclined to them. In their place draw Pq, Pr parallel to AC; and Ps, Pr parallel to AB; and becaue the angles of the triangles PQq, PRr, PSs, PTt are given, the ratio's of PQ to Pq, PR to to Pr, PS to Ps, PT to Pt will be alo given; and therefore the compounded ratio's PQ X PR to Pq x Pr, and PS x PT to Ps x Pt are given. But from what we have demontrated before, the ratio of Pq x Pr to Ps x Pt is given; and therefore alo the ratio of PQ x PR to PS x PT. Q. E. D.

The ame things uppoed, it the rectangle PQ x PR of the lines drawn to the two oppoite des of the trapezium is to the rectangle PS x PT of thoe drawn to the other two ides, in a given ratio; the point P, from whence thoe lines are drawn, will be placed in a conic ection decribed about the trapezium. (PL 8. Fig. 7.)



Conceive a conic ction to he decribed paing through the points A, B, C, D, and any one of the infinite number of points P, as for example p; I ay the point P will be always placed in this ection. If you deny the thing, join AP cutting this conic ection omewhere ele if poible than in P, as in b. Therefore if from thoe points p and b, in the given angles to the ides of the trapezium, we draw the right lines pq, pr, ps, pt, and bk, bn, bf, bd, we hall have