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equation in the form ${\displaystyle x^{2}+(0\times x)-4=0}$. For, by the laws stated above, ${\displaystyle x^{2}+(0x2)-4=x^{2}+0-4=x^{2}-4}$. Hence the equation ${\displaystyle x^{2}-4=0}$, is merely representative of a particular class of quadratic equations and belongs to the same general form as does ${\displaystyle x^{2}-3x+2=0}$.

For these three reasons the symbol ${\displaystyle 0}$, 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, and ${\displaystyle y}$, represents a particular correlation between the pair of variables. Thus ${\displaystyle x+y-1=0}$ represents one definite correlation, and ${\displaystyle 3x+2y-5=0}$ represents another definite correlation between the variables ${\displaystyle x}$ and ${\displaystyle y}$; 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 ${\displaystyle x}$ and ${\displaystyle y}$? Here we want to symbolize any linear correlation; just as a symbolizes any