182 On the ORIGIN and 



and AH^AB 1 :: AL : N, ex equo BK 1 : AB ! ::LB:N. From 

 L draw L O, L M perpendicular to A C, C B ; L O and L M are 

 given in magnitude. 



Now, becaufe A B* : B K 2 :: A D 2 : D F 2 , N : L B :: A D 2 : D F 2 , 



LB 

 fo that D F 2 = ' A D 2 , and for the fame reafon, D E 2 ss 



N 



AL , LB 



• B D 2 . But [Loci Plant, append. Lem. i.) . AD J + 



N V N 



AL LB AL AB 



--BD 2 = _. ALl + __. BLl + __. DL ,. that ih 



AB 

 DE l +DF l = L0 2 -f-LM 2 -f--^--DL\ 



Join L G ; then by hypothecs, L O 1 + L M 2 has to L G 2 

 the fame ratio which DF'-fDE 1 has to D G J ; and if this ra- 



tio be that of R to N, L O 2 -{- L M 2 z= — LG J ; and therefore 



N 



R AB 



DE' + DP = -*LG J + -— DL\ But DE 2 -f DF 2 = 



R R AB R 



-•DG l ; therefore — • LG 2 + — — ■• D L 2 = — • D G 2 , and 



N N N N ' 



• DL l = -(D G 2 — L G 2 ). The excefs of the fquare of 



N N 4 



D G above the fquare of L G, has therefore a conftant ratio to 

 the fquare of D L, viz. that of A B to R. The angle DLG 

 is therefore a right angle, and the ratio of A B to R, the ratio 

 of equality, otherwife L D would be given in magnitude, 

 which is contrary to the fuppofition. The line L G is there- 

 fore given in pofition ; and fince R is to N, that is, A B to N, 

 as the fquares of L O and L M to the fquare of L G, therefore 

 the fquare of LG, and confequently the line LG, is given in 

 magnitude. The point G is therefore given, and alfo the ratio 



of 



