294 OPERATIONS SUBSIDIARY TO INDUCTION. 



ence of signs, which has contributed in no small 

 degree to prevent the real laws of our intellectual 

 operations from being kept in view, or even rightly 

 understood. 



In the first place, a set of signs by which we 

 reason without consciousness of their meaning, can be 

 serviceable, at most, only in our deductive operations. 

 In our direct inductions we cannot for a moment dis- 

 pense with a distinct mental image of the phenomena, 

 since the whole operation turns upon a perception of 

 the particulars in which those phenomena agree and 

 differ. But, further, this reasoning by counters is 

 only suitable to a very limited portion even of our 

 deductive processes. In our reasonings respecting num- 

 bers, the only general principles which we ever have 

 occasion to introduce, are these. Things which are 

 equal to the same thing are equal to one another, and 

 The sums or differences of equal things are equal; 

 with their various corollaries. Not only can no hesi- 

 tation ever arise respecting the applicability of these 

 principles, since they are true of all magnitudes what- 

 ever; but every possible application, of which they 

 are susceptible, may be reduced to a technical rule; 

 such as, in fact, the rules of the calculus are. But 

 if the symbols represent any other things than mere 

 numbers, let us say even straight or curve lines, we 

 have then to apply theorems of geometry not true of 

 all lines without exception, and to select those which 

 are true of the lines we are reasoning about. And 

 how can we do this unless we keep completely in 

 mind what particular lines these are? Since addi- 

 tional geometrical truths may be introduced into the 

 ratiocination in any stage of its progress, we cannot 

 suffer ourselves, during even the smallest part of it, 

 to use the names mechanically (as we use algebraical 



