132 THE MATHEMATICAL PRINCIPLES [BOOK I CASE 2. Let us next suppose that the oppo site sides AC and BD of the trapezium are not parallel. Draw Be/ parallel to AC, and meeting as well the right line ST in /, as the conic section in d. Join Cd cutting PQ in r, and draw DM parallel to PQ, cutting Cd in M, and AB in N. Then (because of the similar triangles BTt, DBN), Et or PQ is to Tt as DN to NB. And ^^ Q N so Rr is to AQ or PS as DM to AN. Wherefore, by multiplying the antecedents by the antecedents, and the consequents by the consequents, as the rectangle PQ X Rr is to the rectangle PS X Tt, so will the rectangle N i)M be to the rectangle ANB ; and (by Case 1) so is the rectangle PQ X Pr to the rectangle PS X Pt : and by division, so is the rectangle PQ X PR to the rectangle PS X PT. Q.E.D. CASE 3. Let us suppose, lastly, the four lines ?Q, PR, PS, PT, not to be parallel to the sides AC, AB, but any way inclined to them. In their place draw Pq, Pr, parallel to AC ; and Ps, Pt parallel to AB ; and because the angles of the triangles PQ, PRr, PSs, PTt are given, the ratios of IQ to Pq, PR to Pr, PS to P*, PT to Pt will b? also given; and therefore the compound ed ratios Pk X PR to P? X Pr, and PS X PT to Ps X Pt are given. But from what we have demonstrated before, the ratio of Pq X Pi 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 s 'niL things supposed, if the rectangle PQ X PR of the lines drawn to the two opposite sides 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. Conceive a conic section to be described pas sing 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 c1 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 ^ B to the sides of the trapezium, we draw the right lines pq, pr, ps, pt, and bk, bn, bf, bd, we shall have, as bk X bn to bf X bd,