INVESTIGATION of PORISMS. 195 
TueEN becaufe the triangles BFG, LFN are fimilar, GF is to 
EN as BF to FL, that is, in a given ratio; and therefore, fince 
FG alfo makes given angles with the two ftraight lines EG and 
EF, given in pofition, the point N is in a ftraight line, given in 
pofition, and pafling through E, viz. EN. 
Now, fince BF is to FL as BG to LN, and alfo as AE or BG 
to EO, LN and EO are equal, and being alfo parallel, OL is 
parallel to EN, that is, to a line given in pofition; and the 
point O, in OL, is given, therefore OL is given in pofition ; 
which was to be found. : 
Conftruction. FROM any two given points, A and B’, in the 
line AB, draw AE and B’F’ interfeGting CD and EF in C, E, D’ 
and F’, fo that AC may be to CE, and B’D’ to DF’ in the fame 
given ratio of «8 to By, (§ 32.) Produce alfo AE to O, and 
BY’ to L’, fo that AE may be to EO, and BF’ to FL’ in the 
fame given ratio of wy to 7d If OL’ be joined, it will be the 
line required. 
For let B be any point whatfoever in AB, and as AB’ to AB, 
fo let OL’ be to OL, and let BL be drawn, cutting CD’, and 
EF in D and F, the line BL is divided in thefe points, fimilarly 
to the given line wd. For fince the two lines AO and B’L’ are 
divided fimilarly by the three lines AB’, CD’ and OL’, and fince 
two of thefe laft, AB’ and OL’, are alfo divided fimilarly to one 
another by the three lines AO, B’L’ and BL, BL will be divided 
in D, in the fame ratio wherein B’L’ is divided in D’, or AO in C, 
(Lem. 1. Cony.). In the fame way, BL is divided in F, in the 
fame ratio wherein AO is divided in E; BL is therefore fimilarly 
divided to AO, or to «3, which was to be demonftrated. 
34. Hence it is plain, “ If two fimilarly divided lines, as AO 
and BL, be drawn any how, and if ftraight lines AB, CD, EF, 
OL, be drawn through the points of divifion of thefe lines, innu- 
merable lines may be placed between the lines AB, CD, EF and 
OL, which will be divided by them, fimilarly to the lines AO, and 
BL.” For, by what is here demonftrated, every line which cuts 
j Bb2 any: 
