DEMONSTRATION, AND NECESSARY TRUTHS. 189 



Nevertheless, it will appear on consideration, that this apparently so de- 

 cisive instance is no instance at all ; that there is in every step of an arith- 

 metical or algebraical calculation a real induction, a real inference of facts 

 from facts; and that what disguises the induction is simply its compre- 

 hensive nature, and the consequent extreme generality of the language. 

 All numbers must be numbers of something: there are no such things as 

 numbers in the abstract. Ten must mean ten bodies, or ten sounds, or ten 

 beatings of the pulse. But though numbers must be numbers of some- 

 thing, they may be numbers of any thing. Propositions, therefore, con- 

 cerning numbers, have the remarkable peculiarity that they are proposi- 

 tions concerning all things whatever; all objects, all existences of every 

 kind, known to our experience. All things possess quantity ; consist of 

 parts which can be numbered ; and in that character possess all the prop- 

 erties which are called properties of numbers. That half of four is two, 

 must be true whatever the word four represents, whether four hours, four 

 miles, or four pounds weight. We need only conceive a thing divided into 

 four equal parts (and all things may be conceived as so divided), to be able 

 to predicate of it every property of the number four, that is, every arith- 

 metical proposition in which the number four stands on one side of the 

 equation. Algebra extends the generalization still farther : every number 

 represents that particular number of all things without distinction, but ev- 

 ery algebraical symbol does more, it represents all numbers without dis- 

 tinction. As soon as we conceive a thing divided into equal parts, without 

 knowing into what number of parts, we may call it a or a;, and apply to it, 

 without danger of error, every algebraical formula in the books. The 

 proposition, 2 (a + 6) = 2 « + 2 5, is a truth co-extensive Avith all nature. 

 Since then algebraical truths are true of all things whatever, and not, like 

 those of geometry, true of lines only or of angles only, it is no wonder that 

 the symbols should not excite in our minds ideas of any things in particu- 

 lar. When we demonstrate the forty-seventh proposition of Euclid, it is 

 lot necessary that the words should raise in us an image of all right-angled 

 jriangles, but only of some one right-angled triangle : so in algebra we 

 leed not, under the symbol a, picture to ourselves all things whatever, but 

 )nly some one thing; why not, then, the letter itself? The mere written 

 characters, a, 6, x, y, z, serve as well for representatives of Things in general, 

 IS any more complex and apparently more concrete conception. That we 

 ire conscious of them, however, in their character of things, and not of mere 

 dgns, is evident from the fact that our whole process of reasoning is cai- 

 •ied on by predicating of them the properties of things. In resolving an 

 ilgebraic equation, by what rules do we proceed ? By applying at each 

 tep to a, h, and x, the proposition that equals added to equals make equals ; 

 hat equals taken from equals leave equals ; and other propositions founded 



• »n these two. These are not properties of language, or of signs as such, 

 )ut of magnitudes, which is as much as to say, of all things. The infer- 

 nces, therefore, which are successively drawn, are inferences concerning 

 hings, not symbols ; though as any Things whatever will serve the turn, 

 here is no necessity for keeping the idea of the Thing at all distinct, and 



• onsequently the process of thought may, in this case, be allowed without 

 ' langer to do what all processes of thought, when they have been performed 

 ' ften, will do if permitted, namely, to become entirely mechanical. Hence 

 ■ he general language of algebra comes to be used familiarly without excit- 



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

 liough in no other case than this can it be done with complete safety. 



