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${\displaystyle y^{2}=x}$, satisfied by pairs of such numbers. Then whatever integral value is given to ${\displaystyle y}$, ${\displaystyle x}$ can assume one corresponding integral value. So the "field" for ${\displaystyle y}$ is unrestricted among these positive or negative integers. But the "field" for ${\displaystyle x}$ is restricted in two ways. In the first place ${\displaystyle x}$ must be positive, and in the second place, since ${\displaystyle y}$ is to be integral, a must be a perfect square. Accordingly, the "field" of ${\displaystyle x}$ is restricted to the set of integers ${\displaystyle 1^{2}}$, ${\displaystyle 2^{2}}$, ${\displaystyle 3^{2}}$, ${\displaystyle 4^{2}}$, and so on, i.e., to ${\displaystyle 1}$, ${\displaystyle 4}$, ${\displaystyle 9}$, ${\displaystyle 16}$, and so on.

The study of the general properties of a relation between pairs of numbers is much facilitated by the use of a diagram constructed as follows:

Draw two lines ${\displaystyle OX}$ and ${\displaystyle OY}$ at right angles; let any number ${\displaystyle x}$ be represented by ${\displaystyle x}$ units