196 - On the ORIGIN and 
fecond of them, became neceflary 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 paflage already quoted, he called Porifms, 
“ Colleétio artificiofifima multarum rerum que fpeétant ad 
“ analyfin difficiliorum et generalium problematum.”’ 
On this account, it is defirable to have a method of invefti- 
gating Porifms, which does not require, that we fhould 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 Srmson’s ana- 
lyfis may be confidered as anfwering to this defcription ; for as _ 
that geometer did not regard thefe propofitions at all in the 
light that is done here, nor in relation to“their 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 Reftoration. : 
Ir 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 leaft a 
great affinity between the methods, fince the /emmata given by 
Parpus as neceflary to Euciip’s demonttrations, are fubfervient 
alfo to thofe of our modern geometer. 
21. I sHALL employ the fame fort of analyfis in the Po- 
rifms that follow, at leaft till we come to treat of them alge- 
braically, where a method of inveftigating thefe propofitions 
will prefent itfelf, which is perhaps more fimple and direct than 
any other. The following Porifm is the firft of Euciip’s, and 
the firft alfo that was reftored. It is given here to exem- 
plify the advantage which, in inveftigations of this kind, may 
be derived from employing the Jaw of continuity in its utmoft 
extent, 
