TERMINOLOGY AND NOMENCLATURE. 495 



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 a ratiocination i*elating not to the sym- 

 bols, but to the things signified by them. But as it has been found practi- 

 cable 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 at- 

 tained without our ever thinking of any thing 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 tip 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 symbolical 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 compact- 

 ness, the entire unmeaningness, and the capability of being used as counters 

 without a thought of what they represent, which are characteristic of the a 

 and 5, 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, can 

 not possibly conduce to that end, and forms part of that exaggerated esti- 

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

 prevent the real laws of our intellectual operations from being rightly un- 

 derstood. 



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 opera- 

 tions. In our direct inductions we can not for a moment dispense with a 

 distinct mental image of the phenomena, since the whole operation turns 

 on a perception of the particulars in which those phenomena agree and dif- 

 fer. But, further, this reasoning by counters is only suitable to a very lim- 

 ited portion even of our deductive processes. In our reasonings respecting 

 numbers, the only general principles which we ever have occasion to intro- 

 duce 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 ap- 

 plicability 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; and such, 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 geom- 

 etry 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 ad- 

 ditional geometrical truths may be introduced into the ratiocination in any 

 stage of its progress, we can not 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 

 solution of a question concerning lines can be made to depend on 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 be- 

 come available, and that the nature of the facts themselves to which the in- 



