TRAINS OF REASONING. 283 



to follow, if all reasoning be induction, that the diffi- 

 culties of philosophical investigation must lie in the 

 inductions exclusively, and that when these were easy, 

 and susceptible of no doubt or hesitation, there could 

 be no science, or, at least,, no difficulties in science. 

 The existence, for example, of an extensive Science 

 of Mathematics, requiring the highest scientific genius 

 in those who contributed to its creation, and calling 

 for a most continued and vigorous exertion of intellect 

 in order to appropriate it when created, may seem 

 hard to be accounted for on the foregoing theory. 

 But the considerations more recently adduced remove 

 the mystery, by showing, that even when the induc- 

 tions themselves are obvious, there may be much 

 difficulty in finding whether the particular case which 

 is the subject of inquiry comes within them ; and 

 ample room for scientific ingenuity in so combining 

 various inductions, as, by means of one within which 

 the case evidently falls, to bring it within others in 

 which it cannot be directly seen to be included. 



When the more obvious of the inductions which 

 can be made in any science from direct observations, 

 have been made, and general formulas have been 

 framed, determining the limits within which these 

 inductions are applicable ; as often as a new case can 

 be at once seen to come within one of the formulas, 

 the induction is applied to the new case, and the 

 business is ended. But new cases are continually 

 arising, which do not obviously come within any 

 formula whereby the questions we want solved in 

 respect of them could be answered. Let us take an 

 instance from geometry ; and as it is taken only for 

 illustration, let the reader concede to us for the 

 present, what we shall endeavour to prove in the next 

 chapter, that the first principles of geometry are 



