TERMINOLOGY AND NOMENCLATURE. 261 



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 generally; 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 compactness, the entire unmeaningness, and the capa 

 bility of being used as counters without a thought of what 

 they represent, which are characteristic of the a and b, 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 estimate 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 

 consciousness of their meaning, can be serviceable, at most, 

 only in our deductive operations. In our direct inductions 

 we cannot for a moment dispense with a distinct mental image 

 of the phenomena, since the whole operation turns on a per 

 ception 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 pro 

 cesses. In our reasonings respecting numbers, 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 

 hesitation ever arise respecting the applicability of these prin 

 ciples, since they are true of all magnitudes whatever ; but 

 every possible application of which they are susceptible, may 



