This is an idea to which we shall have continually to recur; it is not going too far to say that no part of modern mathematics can be properly understood without constant recurrence to it. The conception of form is so general that it is difficult to characterize it in abstract terms. At this stage we shall do better merely to consider examples. Thus the equations $2 x - 3 = 0$ , $x - 1 = 0$ , $5 x - 6 = 0$ , are all equations of the same form, namely, equations involving one unknown $x$ , which is not multiplied by itself, so that $x^{2}$ , $x^{3}$ , etc., do not appear. Again $3 x^{2} - 2 x + 1 = 0$ , $x^{2} - 3 x + 2 = 0$ , $x^{2} - 4 = 0$ , are all equations of the same form, namely, equations involving one unknown $x$ in which $x \times x$ , that is $x^{2}$ , appears. These equations are called quadratic equations. Similarly cubic equations, in which $x^{3}$ appears, yield another form, and so on. Among the three quadratic equations given above there is a minor difference between the last equation,

$x^{2} - 4 = 0$ , and the preceding two equations, due to the fact that $x$ (as distinct from $x^{2}$ ) does not appear in the last and does in the other two. This distinction is very unimportant in comparison with the great fact that they are all three quadratic equations.

Then further there are the forms of equation stating correlations between two variables; for example, $x + y - 1 = 0$ , $2 x + 3 y - 8 = 0$ , and so on. These are examples of what is called the linear form of equation. The reason for this name of "linear" is that the graphic method of representation, which is explained at the end of II., always represents such equations by a straight line. Then there are other forms for two variables–-for example, the quadratic form, the cubic form, and so on. But the point which we here insist upon is that this study of form is facilitated, and, indeed, made possible, by the standard method of writing equations with the symbol $0$ on the right-hand side.

There is yet another function performed by $0$ in relation to the study of form. Whatever number $x$ may be, $0 \times x = 0$ , and $x + 0 = x$ . By means of these properties minor differences of form can be assimilated. Thus the difference mentioned above between the quadratic equations $x^{2} - 3 x + 2 = 0$ , and $x^{2} - 4 = 0$ , can be obliterated by writing the latter

equation in the form $x^{2} + (0 \times x) - 4 = 0$ . For, by the laws stated above, $x^{2} + (0 \times x) - 4 = x^{2} + 0 - 4 = x^{2} - 4$ . Hence the equation $x^{2} - 4 = 0$ is merely representative of a particular class of quadratic equations and belongs to the same general form as does $x^{2} - 3 x + 2 = 0$ .

For these three reasons the symbol $0$ , representing the number zero, is essential to modern mathematics. It has rendered possible types of investigation which would have been impossible without it.

The symbolism of mathematics is in truth the outcome of the general ideas which dominate the science. We have now two such general ideas before us, that of the variable and that of algebraic form. The junction of these concepts has imposed on mathematics another type of symbolism almost quaint in its character, but none the less effective. We have seen that an equation involving two variables, $x$ and $y$ , represents a particular correlation between the pair of variables. Thus $x + y - 1 = 0$ represents one definite correlation, and $3 x + 2 y - 5 = 0$ represents another definite correlation between the variables $x$ and $y$ ; and both correlations have the form of what we have called linear correlations. But now, how can we represent any linear correlation between the variable numbers $x$ and $y$ ? Here we want to symbolize any linear correlation; just as $x$ symbolizes any

number. This is done by turning the numbers which occur in the definite correlation 3 x + 2 y − 5 = 0 into letters. We obtain a x + b y − c = 0 . Here a , b , c , stand for variable numbers just as do x and y : but