64 PHILOSOPHICAL TRANSACTIONS. [aNNO 1763. 



When the given ratio of the square of om to the rectangle under lok shall be 

 that of the rectangle under kml to the square of half kl, fig. 6, by which the 

 given ratio of the rectangle under feg to that under age shall be that of the 

 rectangle under AC, bd to the square of ab, the points t and v shall unite in 

 one, bisecting kl, and the points o and r shall also unite in one, dividing the 

 line KLM harmonically; and then the locus of the point e will be each of the 

 diagonals of the quadrilateral. 



In the last place, if the diagonals ad, bc of the quadrilateral were drawn, 

 cutting GE in i and k, fig. 8, and the ratio of the rectangle under kei to that 

 under aid were given, and not that of the rectangle under gef to that under 

 agb; then the intersection of these diagonals, as l, will be in the line drawn 

 from M bisecting ac, and bd, and the point l will fall within the quadrilateral, by 

 which tiie locus, when an ellipsis or single hyperbola, will be assigned by the 

 36th prop, of the aforesaid book of Pappus; and when opposite sections, by the 

 30th prop- or be reduced to the preceding cases thus: since kg will be to gb as 

 CA to ab, and ig to ga as bd to ab, the rectangle under kgi will be to that un- 

 der AGB, in the given ratio of the rectangle under ac, bd to the square of ab. 

 Therefore when the ratio of the rectangle under kei to that under aid is given, 

 the rectangle under aid also bearing a given ratio to that under agb, the ratio 

 of the rectangle under kei to that under agb will be given ; and in the last place 

 the ratio of the rectangle under gef to that under agb will be given, this rect- 

 angle under gef being the excess of that under kgi above that under kei, by 

 prop. 193, lib. 7, Papp. And thence the sections will be determined, as before. 



And thus may the locus of the point sought be assigned in all the cases of 

 this ancient problem, which Sir Isaac Newton has distinctly explained. The 

 other cases, he has alluded to, may be treated as follows. When 3 of the given 

 lines shall be parallel, as ac, bd, and hi, the 4th line being ab, fig. 9, and 

 KELM being parallel to ab, the ratio of the rectangle under kel to the rectangle 

 tinder eg and em shall be given ; that is, 3 points a, b, and h being given in the 

 Una AB, with the line ge insisting on ab in a given angle, that the rectangle 

 under agb shall be to that under gh and ge in a given ratio : then take an 

 equal to bh, and draw no parallel to ac, bd, and hi. — Then if np be drawn, 

 so that po be to on in the giveratio, np will be given in position, and po will 

 be to ON, that is, eg, as the rectangle under kel to that under meg; so that 

 the rectangle under kel will be equal to that under po, em. But the rectangle 

 under okm is equal to the excess of that imder oem above that under kel, by 

 prop. J 94, lib. 7j Papp.; therefore the rectangle under okm, or that under nam, 

 or under nbh, is equal to that under em and the excess of oe above op, that is, 

 to the rectangle under pem ; the point e therefore is in an hyperbola described to 

 the given asymptotes pn, mh, and passing through a and b. 



