INVESTIGATION of PORISMS. 197 
Let KT be that line, and affuming the points A and B’, and 
drawing the lines AO, B’L, fo that they may be fimilarly di- 
vided to the line ad, 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; itis 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 {mall 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 impoffible 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, c@teris paribus, inverfely as the {quare 
of the fine of that angle. The nearer, therefore, that the 
problem is to the Porifin, 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 fhall have its 
“ angles upon four ftraight lines given in pofition *.”’ 
’ As it is evident, that the former problem is but a particular 
cafe of this laft, 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. x. lem. 29. 
