VOL. Lirr.] PHILOSOPHICAL TRANSACTIONS. 6Q 



which with the given points shall form 4 segments such, that the rectangle under 

 two shall bear a given proportion to the rectangle under the other two. The 

 various cases of this problem appear to have been the subject of the 2d book, of 

 the mentioned treatise of Apollonius; and according to the character given by 

 Pappus of those propositions, these lemmas serve to reduce them to problems in 

 the first book, not those above mentioned, but those where 3 points being given, 

 the rectangle under the segments included by two, and a 4th point, shall bear a 

 given ratio to the rectangle under the segment formed by the 3d point, and a 

 given line. 



For instance, the 46th prop, is this: in the line ab, 4 points a, c, e, b, being 

 given; and the point f assumed between e and 

 B; also D taken, according to the 41st prop., so [ ^ ^ ~l ^ 



that the rectangle under adc be equal to that j 1 



under bde; if g be equal to the sum of ae, cb, 



the rectangle under afc, together with that under epb, will be equal to the 

 rectangle under o and dp. Here if it were proposed to find the point p, so that 

 the ratio of the rectangle under apc to that under epb should be given, the ratio 

 of the rectangle under apc to that under dp and the given line g would be given. 



But this analysis may be carried on to a compleat solution of the problem 

 thus. If CN be taken to g in the given ratio of the rectangle under apc to that 

 under dp and g, the point n will be given, and 

 the rectangle under ap, cn will be to that under I I I l~i I ~~l 



, . . r ..ACDEFB \ 



AP, g, m this ratio of CN to g; consequently the i i 



excess of the rectangle under af, cn sbove that ** 



under apc, that is, the rectangle under apn, will be to the excess of the rect- 

 angle under ap and g above that under dp and g. or the given rectangle under 

 AD, G, in the same given ratio ; and in the last place the rectangle under apn will 

 equal the given rectangle under ad and cn. 



Here I have chosen this prop, in particular, because the case of the problem 

 to which it is subservient, is subject to a determination, when pn shall be equal 

 to af. And then the rectangle under apn being equal to that under ad and cn, 

 as CN to FN so is af to AD, and by division as CF to pn so dp to ad; therefore 

 when AF is equal to pn, cp will be to ap as pd to ad: consequently cd to fd as 

 PD to AD, and the square of dp equal to the rectangle under adc, when the 

 problem admits of a single solution only, where the rectangle under afc will bear 

 to that under epb a less ratio than in any other situation of the point f between 

 E and b. 



Moreover cn is to g as the rectangle under apc to the sum of the rectangles 

 under afc and efb; therefore pn being equal to af, when the problem is limited 

 to this single solution, the rectangle under apc shall be to the rectangles under 



