INFESTIGATION of PORIBMS, 197 



Let KT be that line, and afluming the points A and B', and 

 drawing the lines AO, B"L, fo that they may be fimilarly di- 

 vided to the line ct$, as in the conftruction of the Porifm, then 

 if OL be joined, it will be given in pofition, and the extremity 

 K, of the line KT, will be in the line OL, by the Porifm ; but 

 it is alfo in the line RS ; it is therefore given. Now, by the 

 lemma, AT is to TB' as OK to KL', and the lines- OK and KL' 

 being given, the ratio of AT to TB' is given, fo that T is given, 

 and therefore TK is given in pofition. Q. E. I. 



Now, it is evident, that if RS make a fmall angle with OL, 

 any error in the determination of that angle will make a great 

 variation in the pofition of the point K. A fmall change in it 

 may, for inftance, make RS parallel to OL, and confequently 

 will throw off K, to an infinite diftance, fo that the line, which is 

 fought, will be impoflible to be found; and in general, the varia- 

 tion of the pofition of K, correfponding to a given variation in 

 the angle RKO, will be, cceteris paribus, inverfely as the fquare 

 of the fine of that angle. The nearer, therefore, that the 

 problem is to the Porifm, the lefs is the folution of it to be de- 

 pended on, and the more does it partake of the indefinite cha- 

 racter of the latter. 



35. Sir Isaac Newton has extended the hypothefis of the 

 problem from which the preceding Porifm is derived, and has 

 formed from it one more general, which he has alfo refolved, 

 with a view to its application in aftronomy. It is this : " To 

 " defcribe a quadrilateral, given in fpecies, that (hall have its 

 81 angles upon four ftraight lines given in pofition *.'* 



As it is evident, that the former problem is but a particular 

 cafe of this lad, it is natural to expect, that a Porifm is alfo to 

 be derived from it, or that the lines given in pofition may be 

 fuch, that the problem will become indeterminate. On attempt- 

 ing the analyfis, I have accordingly found this conjecture veri- 

 fied ' ? 



* Prin. Math. lib. I. lem. 2?. 



