INVESTIGATION of PORISMS. 175 



20. We might next proceed to confider the particular Po- 

 rifms which Dr Sim son has reftored, and to (hew, that every- 

 one of them is the indeterminate cafe of fome problem. But 

 of this it is fo eafy for any one, who has attended to the prece- 

 ding remarks, to fatisfy himfelf, by barely examining the enun- 

 ciations of thofe propofitions, that the detail into which it would 

 lead feems to be unneceffary. I fhall therefore go on to make 

 fome obfervations on that kind of analyjis which is particularly 

 adapted to the inveftigation of Porifms. 



If the idea which we have given of thefe propofitions be 

 juft, it follows, that they are always to be difcovered, by con- 

 fidering the cafes in which the conftruction of a problem fails, 

 in confequence of the lines which, by their interfe&ion, or the 

 points which, by their pofition, were to determine the magni- 

 tude required, happening to coincide with one another. A Po- 

 rifm may therefore be deduced from the problem it belongs to, 

 in the fame manner that the propofitions concerning the maxima 

 and minima of quantities are deduced from the problems of 

 which they form the limitations ; and fuch no doubt is the 

 moft natural and moft obvious analyfis of which this clafs 

 of propofitions will admit. 



It is not, however, the only one that they will admit of; and 

 there are good reafons for wifhing to be provided with ano- 

 ther, by means of which, a Porifm that is any how fufpected 

 to exift, may be found out, independently of the general folu- 

 tion of the problem to which it belongs. Of thefe reafons, 

 one is, that the Porifm may perhaps admit of being invefti- 

 gated more eafily than the general problem admits of being re- 

 folved ; and another is, that the former, in almoft every cafe, 

 helps to difcover the fimpleft and moft elegant folution that can 

 be given of the latter. 



The truth of this laft obfervation has been already exempli- 

 fied in two of the preceding problems, where the Porifmatic 

 cafe, by determining the point K in the firft, and L in the 



fecond 



