62 PHILOSOPHICAL TRANSACTIONS. [aNNO 1763, 



the given ratio of the rectangle under peg to that under agb, must be greater' 

 than the ratio of the square of half the difference between ac and bd to the 

 square of ab. But in the hyperbola the square of mo is less than the rectangle 

 under- kol, by which the ratio of the rectangle under peg to that under agb, 

 shall be less than that of the square of half the difference between ac and bd to 

 the square of ab,* fig. 4. 



In both cases, if the point t be such, that the rectangle under mot be equal 

 to that under lok, fig. 3, 4, by which mo shall be to ot in the given ratio of 

 the square of mo to the rectangle under lok, the given rectangle under kml will 

 be to the rectangle under ltk (by prop. 35, 1. 7, Papp.) in this given ratio, and 

 therefore given; consequently the points t and o will be given. In like manner, 

 if the rectangle under mrv be equal to that under lrk, so that mr be to rv in 

 the given ratio of the square of rm to the rectangle under lrk, the given rect- 

 angle under kml (by prop. 22, 1. 7, Papp-) will be to the rectangle under lvk, in 

 the same given proportion, whence the points v and r will be given. Thus in 

 both cases the points t and v will be found by applying to the given line kl a 

 rectangle exceeding by a square, to which the given rectangle under kml shall 

 be in the given ratio of the square of mo to the rectangle under kol, or of the 

 square of mr to the rectangle under krl; mo being to ot, and mr to rv, in 

 that given ratio. 



But in the last place, if this given ratio be that of equality, so that the square 

 of em be equal to the rectangle under krl, fig. 5, by adding to both the rect- 

 angle under mrl, that under rml will be equal to that under km, lr, and mr to 

 RL as KM to ML, and the vertex r of the diameter ri will be given, the conic 

 section being here a parabola, this diameter having thus but one vertex. 



Hitherto the point e, when the line efg falls between ac and bd, is without 

 the quadrilateral, and within the lines ab, cd, when efg is without the quadri- 

 lateral. But when e is within the lines ac, bd in the first case, and without in 

 the second, the locus of the point e will be opposite sections, each passing 

 through two angles of the quadrilateral. 



When one section passes through a and c, and the other through b and d, 

 then if the diameter mi be drawn, as before, and to kl be applied a rectangle 

 deficient by a square, to which the given rectangle under kml, fig. 6, shall be 

 in the given ratio of the square of mo to the rectangle under kol, or of the 

 square of mr to the rectangle under krl, the points t and v, constituting the 



• As the square of om shall be greater or less than the rectangle under kol, the square of nm 

 will be respectively greater or less than the rectangle under anb; therefore the ratio of the square 

 of NO to the rectangle under anb, that is, of the rectangle under feg to that under age, will be 

 accordingly greater or less than the ratio of the square of no to the square of n m, which is the same 

 with that of the square of the difference between ak, bl to the square of ab. — Orig. 



