DEMONSTRATION, AND NECESSARY TRUTHS. 333 



out 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 propo- 

 sition, 2 (a + &) = 2 a+2 b, is a truth co-extensive with 

 the creation. Since then algebraical truths are true 

 of all things whatever, and not, like those of geo- 

 metry, 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 

 demonstrate 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, &, 

 x, y, #, 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 reasoning is carried on by predicating of them the 

 properties of things. In resolving an algebraic equa- 

 tion, by what rules do we proceed ? By applying at 

 each step to a, b, and #, the proposition that equals 

 added to equals make equals; that equals taken from 

 equals leave equals ; and other propositions founded 

 upon these two. These are not properties of lan- 

 guage, or of signs as such, but of magnitudes, which 

 is as much as to say, of all things. The inferences, 

 therefore, which are successively drawn, are inferences 

 concerning Things, not symbols ; although as any 

 Things whatever will serve the turn, there is no neces- 

 sity for keeping the idea of the Thing at all distinct, 

 and consequently the process of thought may, in this 



