286 REASONING. 



ticular number of all things without distinction, but every 

 algebraical symbol does more, it represents all numbers with^ 

 out distinction. As soon as we conceive a thing divided into 

 equal parts, without knowing into what number of parts, we 

 may call it a or x, and apply to it, without danger of error, 

 every algebraical formula in the books. The proposition, 

 2 (a + b) = 2 a + % &, is a truth co- extensive with all nature. 

 Since then algebraical truths are true of all things whatever, 

 and not, like those of geometry, true of lines only or angles 

 only, it is no wonder that the symbols should not excite in 

 our minds ideas of any things in particular. When we de- 

 monstrate the forty-seventh proposition of Euclid, it is not 

 necessary that the words should raise in us an image of all 

 right-angled triangles, but only of some one right-angled 

 triangle : so in algebra we need not, under the symbol a, 

 picture to ourselves all things whatever, but only some one 

 thing; why not, then, the letter itself? The mere written 

 characters, a, b, x, y, z, serve as well for representatives of 

 Things in general, as any more complex and apparently 

 more concrete conception. That we are conscious of them 

 however in their character of things, and not of mere signs, 

 is evident from the fact that our whole process of reason- 

 ing is carried on by predicating of them the properties of 

 things. In resolving an algebraic equation, by what rules do 

 we proceed ? By applying at each step to a, b, and x, the 

 proposition that equals added to equals make equals ; that 

 equals taken from equals leave equals ; and other propositions 

 founded on these two. These are not properties of language, 

 or of signs as such, but of magnitudes, which is as much as 

 to say, of all things. The inferences, therefore, which are suc- 

 cessively drawn, are inferences concerning things, not sym- 

 bols ; though as any Things whatever will serve the turn, 

 there is no necessity for keeping the idea of the Thing at all 

 distinct, and consequently the process of thought may, in this 

 case, be allowed without danger to do what all processes of 

 thought, when they have been performed often, will do if per- 

 mitted, namely, to become entirely mechanical. Hence the 

 general language of algebra comes to be used familiarly with- 



