TERMINOLOGY AND NOMENCLATURE. 



463 



susceptible of being numbered) have 

 no property in common. There is 

 nothing, therefore, to distract the 

 mind from the set of mechanical 

 operations which are to be performed 

 upon the symbols, such as squaring 

 both sides of the equation, multiplying 

 or dividing them by the same or by 

 equivalent symbols, and so forth. 

 Each of these operations, it is true, 

 corresponds to a syllogism ; represents 

 one step of ratiocination, relating not 

 to the symbols, but to the things sig- 

 nified by them. But as it has been 

 found practicable to frame a technical 

 form, by conforming to which we can 

 make sure of finding the conclusion 

 of the ratiocination, our end can be 

 completely attained without our ever 

 thinking of anything but the symbols. 

 Being thus intended to work merely 

 as mechanism, they have the qualities 

 which mechanism ought to have. 

 They are of the least possible bulk, 

 so that they take up scarcely any 

 room, and waste no time in their 

 manipulation ; they are compact, and 

 fit so closely together that the eye 

 can take in the whole at once of 

 almost every operation which they 

 are employed to perform. 



These admirable properties of the 

 symbolical language of mathematics 

 have made so strong an impression on 

 the minds of many thinkers, as to 

 have led them to consider the symbo- 

 lical language in question as the ideal 

 type of philosophical language gener- 

 ailly ; to think that names in general, 

 or (as they are fond of calling them) 

 signs, are fitted for the purposes of 

 thought in proportion as they can be 

 made to approximate to the compact- 

 ness, the entire unmeaningness, and 

 the capability of being used as counters 

 without a thought of what they repre- 

 .sent, 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, I conceive, 

 cannot possibly conduce to that end, 

 and forms part of that exaggerated 

 ^timate of the influence of signs 



which has contributed in no small 

 degree to prevent the real laws of 

 our intellectual operations from being 

 rightly understood. 



In the first place, a set of signs by 

 which we reason without conscious- 

 ness of their meaning, can be service- 

 able, at most, only in our deductive 

 operations. In our direct inductions 

 we cannot for a moment dispense with 

 a distinct mental image of the pheno- 

 mena, since the whole operation turns 

 on 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 respect- 

 ing numbers, the only general prin- 

 ciples 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 dif- 

 ferences of equal things are equal ; 

 with their various corollaries. Not 

 only can no hesitation ever arise re- 

 specting the applicability of these 

 principles, since they are true of all 

 magnitudes whatever, but every pos- 

 sible application of which they are 

 susceptible may be reduced to a tech- 

 nical rule ; and such, in fact, the rules 

 of the calculus are. But if the sym- 

 bols 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 additional geometrical 

 truths may be introduced into the 

 ratiocination in any stage of its pro- 

 gress, we cannot suffer ourselves, dur- 

 ing 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 solution of a 

 question concerning lines can be made 

 to depend on a previous question con 

 cei-ning numbers, or, in other wt)rd8, 



