430 OPERATIONS SUBSIDIARY TO INDUCTION. 



made to approximate to the compactness, the entix-e unmeaningness, 

 and the capability of being used as counters without a thought of what 

 they represent, which are characteristic of the a and h, the x and y, of 

 algebra. This notion has led to sanguine views of the acceleration of 

 the progress of science by means which, as I conceive, cannot possi- 

 bly conduce to that end, and forms part of that exaggerated estimate 

 of the influence of signs, which has contributed in no small degree to 

 prevent the real laws of our intellectual ojaerations fi-om being kept in 

 view, or even rightly understood. 



In th-3 first place, a set of signs by which we reason without con- 

 sciousness of their meaning, can be seniceable, at most, only in our 

 deductive operations. In our direct inductions we cannot for a mo- 

 ment dispense 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 difler. But, further, this reasoning by 

 counters is only suitable to a very limited portion even of our deduc- 

 tive processes. In our reasonings respecting numbers, the only gen- 

 eral 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 tlfeir 

 various corollaries. Not only can no hesitation ever arise respecting 

 the applicability of these principles, since they are true of all mag- 

 nitudes whatever ; but every possible application, of which they are 

 susceptible, may be reduced to a technical rule ; suc"h 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 

 hare then to apj^ly theorems of geometry not tnie of all lines withovit 

 exception, and to select those which are true of the lines we are rea- 

 soning about. And how can we do this unless we keep completely in 

 mind what particular lines these are % Since additional geometrical 

 truths may be introduced into the ratiocination in any stage of its pro- 

 gress, we cannot suffer ourselves, during even the smallest part of it, 

 to use the names mechanically (as we use algebraical symbols) without 

 an image annexed to them. It is only after ascertaining that the so- 

 lution of a question concerning lines can be made to depend upon a 

 previous question concerning numbers, or in other words after the 

 question has been (to speak technically) reduced to an equation, that 

 the unmeaning signs become available, and that the nature of the facts 

 themselves to which the investigation relates can be dismissed from 

 the mind. Up to the establishment of the equation, the language in 

 which mathematicians cairy on their reasoning does not differ in char- 

 acter fi-om that employed by close reasoners on any other kind of 

 subject. 



I do not deny that every correct ratiocination, when thrown into the 

 syllo.gistic shape, is conclusive from the mere form of the expression, 

 provided none of the terms used be ambiguous; and this is one of the 

 circvmistances which have led some philosophers to think that if all 

 names were so judiciously constiiicted and so carefully defined as not 

 to admit of any ambiguity, the improvement thus made in language 

 would not only give to the conclusions of every deductive science the 

 same certainty with those of mathematics, but would reduce all reason- 

 ings to the application of a technical form, and enable their conclu- 

 siveness to be rationally assented to after a merely mechanical pro- 



