178 On the ORIGIN and 



the line to be found. For if it be not parallel to K L, the point 

 of their interferon muft be at a finite diftance from the point M, 

 and therefore making as (2 to a. fo this diftance to a fourth pro- 

 portional, the diftance from H, at which A E' interfecls F K, will 

 be equal to that fourth proportional. But A E' does not inter- 

 fect F K, for they are parallel by conftruclion ; therefore B E' 

 cannot interfecl KL; K L is therefore parallel to B E', a line 

 given in pofition. 



Again, let A E"; B E" be inflected to E", fo that A E" may 

 pafs through the given point H ; then it is plain, that B E" 

 muft pafs through the point to be found M ; for if not, it may 

 be demonftrated, juft as has been done above, that A E" does 

 not pafs through H, contrary to the fuppofition. The point to 

 be found is therefore in the line E"B, which is given in pofi- 

 tiom 



Now, if from E there be drawn E P parallel to A E', and 

 E S parallel to B E\ B S is to S E as B L to L N, and A P to 

 P E as A F to F G ; wherefore the ratio of F G to L N is com- 

 pounded of the ratios of A F to B L, PE to S E, and B S to 

 A P. But the ratio of P E to S E is the fame with that of A E' 

 to B E', and the ratio of B S to A P is the fame with that of 

 DBtoD A, becaufe D B is to B S as D E' to E' E, or as I A to 

 A P. Therefore the ratio of F G to L N is compounded of the 

 ratios of A F to B L, A E' to B E', and D B to D A. 



In like manner, becaufe E" is a point in the line D E, and 

 A E", B E" are inflected to it, the ratio of F H to L M, is com- 

 pounded of the fame ratios of AF to B L, A £' to BE', and 

 DB to DA; and therefore the ratio of F H to LM is the 

 fame with that of F G to N L, and the fame confequently with 

 that of H G to M N. But the ratio of H G to M N is given, 

 being by fuppofition that of a to (B ; the ratio of F H to LM 

 is therefore alfo given, and F H being given, LM is given in 



magnitude. 



