idea worthy to be placed beside the notions of the Variable and of Form so far as concerns

its importance in governing mathematical procedure. Any limitation whatsoever upon the generality of theorems, or of proofs, or of interpretation is abhorrent to the mathematical instinct. These three notions, of the variable, of form, and of generality, compose a sort of mathematical trinity which preside over the whole subject. They all really spring from the same root, namely from the abstract nature of the science.

Let us see how generality is gained by the introduction of this idea of operations. Take the equation $x + 1 = 3$ ; the solution is $x = 2$ . Here we can interpret our symbols as mere numbers, and the recourse to "operations" is entirely unnecessary. But, if $x$ is a mere number, the equation $x + 3 = 1$ is nonsense. For $x$ should be the number of things which remain when you have taken $3$ things away from $1$ thing; and no such procedure is possible. At this point our idea of algebraic form steps in, itself only generalization under another aspect. We consider, therefore, the

general equation of the same form as $x + 1 = 3$ . This equation is $x + a = b$ , and its solution is $x = b - a$ . Here our difficulties become acute; for this form can only be used for the numerical interpretation so long as $b$ is greater than $a$ , and we cannot say without qualification that $a$ and $b$ may be any constants. In other words we have introduced a limitation on the variability of the "constants" $a$ and $b$ , which we must drag like a chain throughout all our reasoning. Really prolonged mathematical investigations would be impossible under such conditions. Every equation would at last be buried under a pile of limitations. But if we now interpret our symbols as "operations," all limitation vanishes like magic. The equation $x + 1 = 3$ gives $x = + 2$ , the equation $x + 3 = 1$ gives $x = - 2$ , the equation $x + a = b$ gives $x = b - a$ which is an operation of addition or subtraction as the case may be. We need never decide whether $b - a$ represents the operation of addition or of subtraction, for the rules of procedure with the symbols are the same in either case.

It does not fall within the plan of this work to write a detailed chapter of elementary algebra. Our object is merely to make plain the fundamental ideas which guide the formation of the science. Accordingly we do not further explain the detailed rules by which the "positive and negative numbers" are

multiplied and otherwise combined. We have explained above that positive and negative numbers are operations. They have also been called "steps." Thus $+ 3$ is the step by which we go from $2$ to $5$ , and $- 3$ is the step backwards by which we go from $5$ to $2$ . Consider the line $O X$ divided in the way explained in the earlier part of the chapter, so that its points represent numbers. Then $+ 2$

pg86 is the step from $O$ to $B$ , or from $A$ to $C$ , or (if the divisions are taken backwards along $O X^{'}$ ) from $C^{'}$ to $A^{'}$ , or from $D^{'}$ to $B^{'}$ , and so on. Similarly $- 2$ is the step from $O$ to $B^{'}$ , or from $B^{'}$ to $D^{'}$ , or from $B$ to $O$ , or from $C$ to $A$ .

We may consider the point which is reached by a step from $O$ , as representative of that step. Thus $A$ represents $+ 1$ , $B$ represents $+ 2$ , $A^{'}$ represents $- 1$ , $B^{'}$ represents $- 2$ , and so on. It will be noted that, whereas previously with the mere "unsigned" real numbers the points on one side of $O$ only, namely along $O X$ , were representative of numbers, now with steps every point on the whole line stretching on both sides of $O$ is representative of a step. This is a pictorial representation of the superior generality introduced by the positive and negative numbers, namely the