INVESTIGATION of PORISMS. 159 



6. The fubjecl 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- 

 portions, that is to fay, into the fteps by which the ancient geo- 

 meters appear to have been led to the difcovery of them. It 

 remains alfo to point out the relations in which they (land to 

 the other claffes of geometrical truths ', to confider the fpecies 

 of analyfis, whether geometrical or algebraical, that belongs to 

 them ; and/ if poffible, to afhgn 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 mufl here 

 fupply the want of exprefs teflimony 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. It cannot be doubted, that it has been the folution of 

 problems which, in all ftates of the mathematical fciences, has 

 led to the difcovery of mod geometrical truths. The firft ma- 

 thematical enquiries, in particular, mufl 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 fubjecls of fepa- 

 rate demonrtration. 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 



