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INVESTIGATION of PORISMS. 159 
6. Tue fubje& of Porifms is not, however, exhaufted, nor is 
it yet placed in fo clear a light as to need no farther illuftration. 
It yet remains to enquire into the probable origin of thefe pro- 
pofitions, that is to fay, into the fteps by which the ancient geo- 
mieters appear to have been led to the difcovery of them. It 
remains alfo to point out the relations in which they ftand to 
the other claffes of geometrical truths ; to confider the fpecies 
of analyfis, whether geometrical or algebraical, that belongs to 
them ; and, if poflible, to affign the reafon why they have fo 
long efcaped the notice of modern mathematicians. It is to 
thefe points that the following obfervations are chiefly di- 
rected. 
I BEGIN with defcribing the fteps that appear to have led the 
ancient geometers to the difcovery of Porifms; and muft here 
fupply the want of exprefs teftimony by probable reafonings, 
_~fuch as are neceflary, whenever we would trace remote difco- 
veries to their fources, and which have more weight in mathe- 
matics than in any other of the fciences. 
7. Ir cannot be doubted, that it has been the folution of 
problems which, in all {tates of the mathematical fciences, has 
led to the difcovery of moft geometrical truths. The firft ma- 
‘thematical enquiries, in particular, muft have occurred in the 
form of queftions, where fomething was given, and fomething 
required to be done ; and by the reafonings neceffary to anfwer 
thefe queftions, or to difcover the relation between the things 
that were given, and thofe that were to be found, many truths 
were fuggefted, which came afterwards to be the fubjects of fepa- 
rate demonftration. ‘The number of thefe was the greater, that 
- the ancient geometers always undertook the folution of problems 
with a fcrupulous and minute attention, which would fcarcely 
fuffer any of the collateral truths to efcape their obfervation. 
We know from the examples which they have left us, that they 
never confidered a problem as refolved, till they had diftinguifh- 
ed all its varieties, and evolved feparately every different cafe 
that 
