INVESTIGATION of PORISMS. 203 
nius, the fubject of a treatife confifting 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- 
veftigation. 
Tue fame {crupulous 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 affuming the data 
in a particular ftate, yet the refult may be perfe@lly general, and 
will accommodate itfelf to every cafe with fuch wonderful verfati- 
lity, as is {carcely credible to the moft experienced mathematician, 
and fuch as often forces him to ftop, in the midft of his calculus, 
and to look back,with a mixture of diffidence and admiration, on 
the unforefeen harmony of his conclufions. 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 deferibed; 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: 
“Ge2 as: 
