2 
INVESTIGATION of PORISMS. ~ ‘x79 
_ magnitude. Now, LM is parallel to BF’, a line given in po- 
fition ; therefore M is ina line QM parallel to AB, and given 
in pofition. But the point M is alfo in another line B E” given 
in pofition; therefore the point M, and alfo the line KLM 
drawn through it parallel to BE’, are given in pofition, which 
were to be found. 
Bi Tue conftruction is thus: From A draw AE’ parallel to 
_ FK, meeting DE in E’; join B EF, and take in it B Q, fo that 
_ asato@foHFto BQ and through Q draw QM parallel to 
AB. Let HA be drawn, and produced till it meet D E in E’, 
and let BE” be drawn meeting QM in M. Through M draw 
K ML parallel to BE’; then is K ML the line, and M the 
point, which were to be found. 
Ir is plain, that there are two lines which will anfwer the 
conditions of the Porifm ; for if in QB, produced on the other 
fide of B, there be taken Bg equal to BQ; and if gm be drawn 
parallel to AB, interfecting M B in m; and if ma be drawn pa- 
rallel to BQ, the part mz, cut off by EB produced, will be 
equal to MN, and have to H G the ratio required. 
{r is plain alfo, that whatever be the ratio of « to @, and 
whatever be the magnitude of FH, if the other things given 
remain the fame, the lines found will be all parallel to BE’. 
But if the ratio of « to @ remain the fame likewife, and if only 
‘the point H vary, the pofition of KL will remain the fame, 
and the-point M will vary. 
4 _ 23. Tuts conftruction, from which, and the foregoing analyfis, 
the fynthetical demonftration follows readily, will be found to 
__ be more fimple than Dr Stmson’s, owing entirely to the ufe that 
has been made of the Jaw of continuity in the two extreme cafes, 
_ where, according to the language of the modern analyfis, HG 
becomes infinite, in the one, and equal to nothing, in the 
_ other. Had it been affirmed, agreeably to that fame lan- 
a, guage, that in the firft of thofe cafes, becaufe of the conftant 
1 : Le f ratio 
