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mined "variable" as occurring in the use of "some" or "any" is the really important one in mathematics; that of the "unknown" in an equation, which is to be solved as quickly as possible, is only of subordinate use, though of course it is very important. One of the causes of the apparent triviality of much of elementary algebra is the preoccupation of the text-books with the solution of equations. The same remark applies to the solution of the inequality ${\displaystyle (3')}$ as compared to the original statement ${\displaystyle (3)}$.

But the majority of interesting formulae, especially when the idea of some is present, involve more than one variable. For example, the consideration of the pairs of numbers ${\displaystyle x}$ and ${\displaystyle y}$ (fractional or integral) which satisfy ${\displaystyle x+y=1}$ involves the idea of two correlated variables, ${\displaystyle x}$ and ${\displaystyle y}$. When two variables are present the same two main types of statement occur. For example, ${\displaystyle (1)}$ for any pair of numbers, ${\displaystyle x}$ and ${\displaystyle y}$, ${\displaystyle x+y=y+x}$, and ${\displaystyle (2)}$ for some pairs of numbers, ${\displaystyle x}$ and ${\displaystyle y}$, ${\displaystyle x+y=1}$.

The second type of statement invites consideration of the aggregate of pairs of numbers which are bound together by some fixed relation—in the case given, by the relation ${\displaystyle x+y=1}$. One use of formulae of the first type, true for any pair of numbers, is that by them formulae of the second type can be