Vol. liii.] raiLOSOPHicAL transactions. 6s 



Again if two of the given lines only are parallel, but the rectangles otherwise 

 related to them, than as above. Suppose the ratio of the rectangle under 

 AG, EF to that under bg, ge is given. Let cd meet ab in l, and let hei, mfn 

 be drawn parallel to ab, and lk parallel to ac and bd, fig. 10. Then the 

 parallelogram em will be to the parallelogram eb in the given ratio. Take ao 

 to OB in that ratio, and draw op parallel to ac and bd. Here the point o will 

 be given, and the parallelogram pa will be in the given ratio to the parallelogram 

 pb ; whence ab will be to Bo as the parallelogram bh to the parallelogram bp, 

 and as the difference between the parallelogram em and eb to the parallelogram 

 EB, consequently as the parallelogram gm to the parallelogram pg ; therefore 

 the ratio of the rectangle under ag, fg to the rectangle under eg, ep or og 

 will be given ; and in the last place the ratio of pg to gl being given, the ratio 

 of the rectangle under ag and gl to that under eg, og will be given. And 

 thus 3 points a, l, o, will be given, with ge insisting on ab in a given angle, 

 as in the preceding case. 



Moreover, ac and bd being parallel, ab and cd may be also parallel, fig. 11. 

 And then, when the ratio of the rectangle under agb to that under gef is 

 given, the determination of the locus is so obvious as not to have required a 

 distinct explanation. But when the rectangle under ag, ef bears a given ratio 

 to that under bg, ge ; let the diagonals ad, bc be drawn, and helk drawn pa- 

 rallel to AD. Then the rectangle under hel will be to that under kei in the 

 same given ratio ; and if cm be taken to mb in the same ratio, the lines mnp, 

 MOQ drawn, the first parallel to ac, bd, and the other parallel to ab, cd, will be 

 given in position, and the diagonal bm will bigect both ik, no, and hl ; there- 

 fore the rectangle under hel being to that under kei as mc to mb, that is, as 

 NH to nk, here by division the rectangle under hbl will be to that under ihk, 

 by the prop, of Papp. before cited, as nh to hk. ; therefore equal to that under 

 NH and iH or kl. But the rectangle under nbo is equal to the sum of the 

 rectangles under hnl and under hel, by the same; therefore the rectangle un- 

 der NEO is equal to that under nh, nk, equal to that under apd, that is, equal 

 to that under paq, or that under pdq, the diagonal bm bisecting both pq and 

 ad. But thus the point e is in an hyperbola described to the asymptotes mn, 

 mo, and passing through a and d. 



The determination of this locus for 3 lines is solved almost explicitly by 

 Apollonius in the last 3 propositions of his 3d book of Conies. For if the 3 

 lines proposed were ab, ac, bc, fig. 12, 13, 14, and the point sought d, so that 

 the ratio of the rectangle under edf (the line ep being drawn parallel to bc) 

 should be in a given ratio to the square of a line drawn from d to bc in a given 

 angle, the square of which line will be in a given ratio to the rectangle under 

 be, cf ; then if bh, ci are drawn parallel to ac and ab respectively, also bdl, 



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