TRAINS OF REASONING. 141 



caiTy on tlve same metaplior) of several chains united at the extremity, 

 as thus : a a mark of d, h of e, c o^f, d ef of n; therefure ahc a mark 

 of«. Suppose, for example, the following combination of circum- 

 stances : 1st, rays of light impinging on a reflecting surface; 2(1, that 

 sijrface parabolic ; 3d, those rays parallel to each other and to the 

 axis of the surface. It is to be proved that the concourse of these 

 three circumstances is a mark that the reflected rays will pass through 

 the focus of the parabolic surface. Now each of the three circum- 

 stances is singly a mark of somQthing material, to the case. Rays of 

 light impinging on a reflecting surface, are a mark that those rays will 

 be reflected at an angle equal to the angle of incidence. The para- 

 bolic form of the surface is a mark that, fi'om any point of it, a line 

 drawn to the focus and a line parallel to the axis will make equal an- 

 gles with the surface. And finally, the parallelism of the rays to the 

 axis is a mark that theu- angle of incidence coincides with one of these 

 equal angles. The three marks taken together are therefore a mark 

 of .all these three things united. But the three United are evidently a 

 mark that the angle of reflection must coincide with the other of the 

 two equal angles, that fonned by a line drawn to t!ie focus ; and this 

 again, by the fundamental axiom concerning sti'aight 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 ai'e abundant, as in all propositions where the hypothesis includes 

 numerous conditions : " If a circle be taken, and //"within that circle 

 a point be taken, not the centre, and //"straight lings be drawn from 

 that point to the circumference, then," &c. 



§ 4. The considerations now stated remove a serious difficulty from 

 the view we have taken of reasoning ; which view might othervidse 

 have seemed not easily reconcilable with the fact that there are De- 

 ductive or Ratiocinative Sciences. It might seem to follow, if all rea- 

 soning be induction, that the difficulties 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 difficvilties in science. The existence, for example, of an 

 extensive Science of Mathematics, requiring the highest scientific ge- 

 nius 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 inductions themselves are 

 obvious, there may be much difficulty in finding whether the partic- 

 .ular case which is the subject of inquiry comes within them ; and am-, 

 pie room for scientific ingenuity in so combining various inductions, 

 as, by means of one within which the case e^^dently 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, detennining 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 ap- 

 plied to the new case, and the business is ended. But new cases 

 ai'e continually arising, which do not obviously come within any 



