240 REASONING. 



the fundamental axiom concerning straight lines, is a mark 

 that the reflected rays pass through the focus. Most chains of 

 physical deduction are of this more complicated type ; and even 

 in mathematics such are abundant, as in all propositions where 

 the hypothesis includes numerous conditions : &quot; // a circle be 

 taken, and if within that circle a point be taken, not the 

 centre, and if straight lines be drawn from that point to the 

 circumference, then,&quot; &c. 



4. The considerations now stated remove a serious diffi- 



9 



culty from the view we have taken of reasoning ; which view 

 might otherwise have seemed not easily reconcilable with the 

 fact that there are Deductive or Ratiocinative Sciences. It 

 might seem to follow, if all reasoning be induction, that the 

 difficulties of philosophical investigation must lie in the induc 

 tions exclusively, and that when these were easy, and suscep 

 tible 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 considera 

 tions more recently adduced remove the mystery, by showing, 

 that even when the inductions 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, 



