178 On the ORIGIN and 
the line tobe found. Forif it be not parallel toK L, the point 
of their interfection muft be at a finite diftance from the point M, 
and therefore making as 6 to «, fo this diftance to a fourth pro- 
portional, the diftance from H, at which A E’ interfects F K, will 
be equal to that fourth proportional. But A E’ does not inter- 
fect FK, for they are parallel by conftruction ; therefore BE’ 
cannot interfec&t K L; K L is therefore parallel to BE’, a line 
given in pofition. . 
AealIn, let A E’} BE’ be infle@ted to E’, fo that AE” may |— 
pafs through the given point H; then it is plain, that BE’ 
mutt pafs through the point to be found M ; for if not, it may 
be demonftrated, juft as has been done above, that AE” 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- 
tion. 
Now, if from E there be drawn EP parallel to AE, and 
ES parallel to BE’, BSistoSE as BL to LN, and AP to 
PE as AF to FG; wherefore the ratio of FG to LN is com- 
pounded of the ratios of AF to BL, PE to SE, and BS to ~ 
AP. But the ratio of PE to SE is the fame with that of AE’ ~ 
to B FE, and the ratio of BS to AP is the fame with that of 
DB to DA, becaufe DB isto BS as DE to E’E, or as D A to 
AP. Therefore the ratio of FG to LN is compounded of the 
ratios of AFtoBL, AE’ to BE, and DB to DA. 
In like manner, becaufe E” is a point in the line DE, and 
AE’, BE” are inflected to it, the ratio of FH to LM, is com- 
pounded of the fame ratios of AF to B TEAsAtE to BE’. and 
DB to DA; and therefore the ratio of FH to I.M is the 
fame with that of FG to NL, and the fame confequently with 
that of HGto MN. But the ratio of HG to MN is given, 
being by fuppofition that of « to @; the ratio of FH to LM~ 
