INVESTIGATION of PORISMS. 203 



nius, the fubject: of a treatife confiding of two books. The 

 firft book has feven general divifions, and twenty-four cafes ; 

 the fecond, fourteen general divifions, and feventy-three cafes, 

 each of which cafes is feparately confidered. Nothing, it is 

 evident, that was any way connected with the problem, could 

 efcape a geometer, who proceeded with fuch minutenefs of in- 

 vestigation. 



The fame fcrupulous exactnefs may be remarked in all the 

 other mathematical refearches of the ancients ; and the reafon 

 doubtlefs is, that the geometers of thofe ages, however expert 

 they were in the ufe of their analyfis, had not fufficient expe- 

 rience in its powers, to truft to the more general applications of 

 it. That principle which we call the law of continuity, and 

 which connects the whole fyftem of mathematical truths by 

 a chain of infenfible gradations, was fcarcely known to them, 

 and has been unfolded to us, only by a more extenfive know- 

 ledge of the mathematical fciences, and by that moft perfect: 

 mode of exprefling the relations of quantity, which forms the 

 language of algebra ; and it is this principle alone which has 

 taught us, that though in the folution of a problem, it may be 

 impoflible to conduct: the inveftigation without afluming the data 

 in a particular ftate, yet the refult may be perfectly general, and 

 will accommodate itfelf to every cafe with fuch wonderful verfati- 

 lity,as is fcarcely credible to the moft experienced mathematician, 

 and fuch as often forces him to flop, in the midft of his calculus, 

 and to look back,with a mixture of diffidence and admiration, on 

 the unforefeen harmony of his conclusions. All this was unknown 

 to the ancients ; and therefore they had no refource, but to ap- 

 ply their analyfis feparately to each particular cafe, with that 

 extreme caution which has juft been defcribed ; and in doing 

 fo, they were likely to remark many peculiarities, which more 

 extenfive views, and more expeditious methods of inveftigation, 

 might perhaps have induced them to overlook. 



39. To reft fatisfied, indeed, with too general refults, and not 

 to defcend fufficiently into particular details, may be confidered' 



C c 2 as 



