VOL. LIII.] PHILOSOPHICAL TRANSACTIONS. Qv 



contained under qp, pr to that under sp, pt, shall be given; and this by pursuing 

 the steps, by which he proves that the point p will in every quadrilateral be in a 

 conic section, may be readily reduced to the case of a quadrilateral with 2 sides 

 parallel, after this manner. Draw Bt and dn parallel to ac ; then find the point 

 M in nd, so that the rectangle under mdn be to that under anb in the ratio given ; 

 and draw cr md. Here Rr will be to a«, or sp, as md to an, and Bt, or ap, to 

 Tt as nd to NB ; whence the rectangle under Rr, ap, will be to that under sp, t^, 

 as that under mdn to that under anb, that is, in the ratio given of the rectangle 

 under rpq to that under spt. Therefore, by taking the sum of the antecedents 

 and of the consequents, the rectangle under rra will be to that under sp/, that 

 is, to the rectangle under aob, in the quadrilateral ABcd, whose two sides ac, 

 hd, are parallel, in the given ratio. 



In like manner, if 3 of the given lines passed through one point, as the lines 

 CA, CB, CD, fig. 2, and the rectangle under qpr be to that under spt in a given 

 ratio, this case is with the same facility reduced to the like quadrilateral thus. 



Draw BE parallel to ac, that shall cut st produced in /; and let the point p be 

 taken, so that the rectangle under ca, ep, be to the square of ab, in the ratio 

 given; then crp being drawn, b/, or qp, will be to Tt, as ac to ar, and Rr to aq, 

 or SP, as EF to ab; whence the rectangle under qp, Rr will be to that under Tt, 

 SP, as that under ac, ef, to the square of ab, that is, in the given ratio of the 

 rectangle under qpr to that under spt ; and the rectangle under QPr will be to 

 that under sp< or aqb, in the quadrilateral abcf, whose two sides ac, bf are 

 parallel, in the same given ratio. 



Now let ABCD be a quadrilateral having the two sides ac, bd parallel, with any 

 conic section passing through the 4 points a, b, c, d, fig. 3, 4, 5 ; also, the 

 point E being taken in the section, and efg being drawn parallel to ac or bd, 

 let the ratio of the rectangle under agb to the rectangle under peg be given: 

 then the conic section will be given. — Let the sides ab, cd meet in m, and draw 

 MI bisecting ac and bd in k and l. Then the diameter of the section, to which 

 AC and bd are lines ordinately applied, will be in the line mi ; and if np, qs be 

 tangents to the section, and parallel to ac and bd, fig. 3, 4, the points o, r, in 

 which they intersect mi, will be the points of their contact, and the vertexes of 

 that diameter. But the square of no is to the rectangle under anb, and the 

 square of qr to the rectangle under aqb, as the rectangle under egh or peg, to 

 that under agb, therefore in a given ratio; but the ratio of nm to no, the same 

 as that of qm to qr. is also given ; whence the ratio of the square of nm to the 

 rectangle under anb, or of the square of om to the rectangle under kol, is 

 given, as also the ratio of the square of rm to the rectangle under krl. 



Now in the ellipsis the square of mo, the distance of the remoter vertex of 

 the diameter or from m, fig. 3, is greater than the rectangle under kol; that is, 



