166 REASONING. 



distinction, but every algebraical symbol does more, it represents ail 

 numbers without distinction. As soon as we conceive a thing divided 

 into equal parts, without knowing into what numbers of parts, we may 

 call it a or x, and apply to it, without danger of eiTor, every alge- 

 braical formula in the books. The proposition, 2(« -\-b) = 2a-^ 2b, 

 is a ti-uth coextensive with the creation. Since then algebraical 

 truths are true of all things whatever, and not, like those of geometry, 

 true of Hues 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, pictiire to ourselves all 

 things whatever, but only some one thing ; why not, then, the letter 

 itself] The mere written characters, a, b, x, ?/, 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 tlie fact that our whole process of reasoning is carried on 

 by predicating of them the properties of things. In resolving an 

 alo-ebraic 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 upon 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 successively 

 drawn, are inferences conceniing Things, not symbols ; although 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 permitted, namely, to become entirely mechanical. Hence 

 the general language of algebra comes to be used familiarly without 

 exciting ideas, as all other general language is prone to do from mere 

 habit, though in no other case than this can it be done with complete 

 safety. But when we look back to see from whence the probative 

 force of die process is derived, we find that at every single step, 

 unless we suppose ourselves to be thinking and talking of the things, 

 and not the mere symbols, the evidence fails. 



There is another circumstance, which, still more than that which we 

 have now mentioned, gives plausibility to the notion that the proposi- 

 tions of arithmetic and algebra are merely verbal. This is, that when 

 considered as propositions respecting Things, tliey all have the appear- 

 ance of being identical propositions. The assertion. Two and one 

 are equal to three, considered as an assertion respecting objects, as 

 for instance " Two pebbles and one pebble are equal to three peb- 

 bles," does not affirm equality between two collections of pebbles, but 

 absolute identity. It affirms that if we put one pebble to two pebbles, 

 those very pebbles are three. The objects, therefore, being the very 

 same, and the mere assertion that " objects are themselves" being in- 

 significant, it seems but natural to consider the proposition, Two and 

 one are equal to three, as asserting mere identity of signification be- 

 tvi'een the two names. 



