17^ On the ORIGIN and 



fecond of them, became necefTary to the general folution. In 

 more difficult problems, the fame will be found to hold ftill 

 more remarkably, and this is evidently what Pappus had in 

 view, when, in a pafTage already quoted, he called Porifms, 

 <f Collectio artificioiimma multarum rerum quae fpeclant ad 

 " analyfin difficiliorum et generalium problematum." 



On this account, it is defirable to have a method of investi- 

 gating Porifms, which does not require, that we fliould have 

 previoufly refolved the problems they are connected with, and 

 which may always ferve to determine, whether to any given 

 problem there be attached a Porifm, or not. Dr Simson's ana- 

 lysis may be considered as anfwering to this defcription ; for as 

 that geometer did not regard thefe proportions at all in the 

 light that is done here, nor in relation toVheir origin, an inde- 

 pendent analyfis of this kind, was the only one that could oc- 

 cur to him ; and he has accordingly given one which is extreme- 

 ly ingenious, and by no means eafy to be invented, but which 

 he ufes with great fkilfulnefs and dexterity throughout the 

 whole of his Restoration. 



It is not eafy to afcertain whether this be the precife me- 

 thod ufed by the ancients. Dr Simson had here nothing to 

 direct him but his genius, and has the full merit of the firft 

 inventor. It feems probable, however, that there is at lead a 

 great affinity between the methods, fince the lemmata given by 

 Pappus as necefTary to Euclid's demonstrations, are fubfervient 

 alfo to thofe of our modern geometer. 



21. I shall employ the fame fort of analysis in the Po- 

 rifms that follow, at least till we come to treat of them alge- 

 braically, where a method of invefligating thefe propositions 

 will prefent itfelf, which is perhaps more simple and direct than 

 any ether. The following Porifm is the firft of Euclid's, and 

 the firft alfo that was reftored. It is given here to exem- 

 plify the advantage which, in investigations of this kind, may 

 be derived from employing the law of continuity in its utmost 



extent, 



