INVESTIGATION of PORISMS. 179 



magnitude. Now, L M is parallel to B E", a line given in po- 

 rtion ; therefore M is in a line QJV1 parallel to A B, 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 K L M 

 drawn through it parallel to B E", are given in pofition, which 

 were to be found. 



The conftruclion is thus: From A draw AE' parallel to 

 F K, meeting D E in E' ; join B E', and take in it B Q^ fo that 

 as a, to /3 fo H F to B Q_ and through Qjiraw QJV1 parallel to 

 A B. Let H A be drawn, and produced till it meet D E in E", 

 and let B E" be drawn meeting QJV1 in M. Through M draw 

 K M L parallel to BE'; then is K M L the line, and M the 

 point, which were to be found. 



It 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 B q equal to B Qj and if q m be drawn 

 parallel to AB, interfering M B in m ; and if m X be drawn pa- 

 rallel to B Q^ the part m n, cut off by E B produced, will be 

 equal to M N, and have to H G the ratio required. 



It is plain alfo, that whatever be the ratio of a to (2, and 

 whatever be the magnitude of F H, if the other things given 

 remain the fame, the lines found will be all parallel to B E'. 

 But if the ratio of a. to (B remain the fame likewife, and if only 

 the point H vary, the pofition of K L will remain the fame, 

 and the point M will vary. 



23. This conftru&ion, from which, and the foregoing analyfis, 

 the fynthetical demonflration follows readily, will be found to 

 be more fimple than Dr Sim son's, owing entirely to the ufe that 

 has been made of the law of continuity in the two extreme cafes 

 where, according to the language of the modern analyfis, H G 

 becomes infinite, in the one, and equal to nothing, in the 

 other. Had it been affirmed, agreeably to that fame lan- 

 guage, that in the firft of thofe cafes, becaufe of the conflant 



Z 2 rat io 



