VOL. LIU.] PHILOSOPHICAL TRANSACTIONS. 63 



rectangles under ktl and under kvl, being thus found, mo will be to ot, and 

 MR to Rv, in this given ratio (by prop. 30 1. 7 P^pp.) o and t being the vertexes 

 of the diameter mi. But the rectangles under ktl, kvl cannot be assigned, as 

 here required, unless the ratio given for that of the square of om to the rect- 

 angle under kol, or that of the square of rm to the rectangle under krl, be not 

 less than that of the rectangle under kml to the square of half kl ; that is, 

 when the ratio of the square of on to the rectangle under anb, and that of the 

 square of Ra to the rectangle under aqb, or that of the given ratio of the rect- 

 angle under peg tu that under agb, is not less than that of the rectangle under 

 ak, bl to the square of half ab, or of the rectangle under ac, bd to the square 

 of AB. ' 



But if one of the opposite sections pass through a and b, and the other through 

 c and D, the ratio of the rectangle under peg to that under agb, will be less 

 than that of the rectangle under ac, bd to the square of ab, fig. 7. For cl 

 being drawn parallel to ab, and ad joined and continued to m, the line dm falls 

 wholly within the section passing through c and d : therefore km is less than kl, 

 and the ratio of kd to kl less than that of kd to km, that is, of bd to ab; 

 whence bk being equal to ac, and ck to ab, the ratio of the rectangle under 

 BKD to that under ckl, being the ratio of the rectangle under egh, or peg, to 

 that under agb, will be less than the ratio of the rectangle under ac, bd to the 

 square of ab. And here the point l is given; for the given rectangle under bkd 

 is to that under ckl, in the given ratio of the rectangle under hge, or that under 

 PEG, to the rectangle under agb ; hence ck, equal to ab, being given, kl is 

 given, and consequently the point l. 



Again, bl being joined, and neop drawn parallel to ab, also gep continued 

 to a, as AG, equal to ca, is to pa so will ck be to dk, and op to eg, equal to 

 OB, as KL to bk ; consequently the rectangle under op, ag will be to that under 

 EG, PQ, as that under kl, ck to that under kb, dk, that is, as the rectangle 

 under agb to that under peg; and, by combining the antecedents and conse- 

 quents, the rectangle under pen will be to that under oeg in the same given 

 ratio. 



Moreover dk being to ac as km to cm, the ratio of dk to ac, that is, the ratio 

 of the rectangle under bkd to the square of ac, will be less than the ratio of kl 

 to CL, or the ratio of the rectangle under ckl to that under ab, cl; therefore, 

 by permutation and inversion, the ratio of the rectangle under ckl to the rect- 

 angle under bkd, that is, the given ratio of the rectangle under nep to that 

 under anc, equal to that under geo, is greater than the ratio of that under ab, 

 CL to the square of ac. And hence, the opposite sections passing through the 

 angles of the quadrilateral abcl, whose sides ab, cl are parallel, will be given 

 as before. 



