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

This page has been proofread, but needs to be validated.

to the ſides AC, AB, but any way inclined to them. In their place draw Pq, Pr parallel to AC; and Ps, Pr parallel to AB; and becauſe 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 alſo 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 demonſtrated before, the ratio of Pq x Pr to Ps x Pt is given; and therefore alſo the ratio of PQ x PR to PS x PT. Q. E. D.


Lemma XVIII.

The ſame things ſuppoſed, it the rectangle PQ x PR of the lines drawn to the two oppoſite ſdes of the trapezium is to the rectangle PS x PT of thoſe drawn to the other two ſides, in a given ratio; the point P, from whence thoſe lines are drawn, will be placed in a conic ſection deſcribed about the trapezium. (PL 8. Fig. 7.)

Plate 8, Figure 7
Plate 8, Figure 7

Conceive a conic ſction to he deſcribed paſſing through the points A, B, C, D, and any one of the infinite number of points P, as for example p; I ſay the point P will be always placed in this ſection. If you deny the thing, join AP cutting this conic ſection ſomewhere elſe if poſſible than in P, as in b. Therefore if from thoſe points p and b, in the given angles to the ſides of the trapezium, we draw the right lines pq, pr, ps, pt, and bk, bn, bf, bd, we ſhall have