It is natural to supersede the statements (2) and (3) by the questions:

(2') For what number $x$ is $x + 2 = 3$ ;

(3') For what numbers $x$ is $x + 2 > 3$ .

Considering (2'), $x + 2 = 3$ is an equation, and

it is easy to see that its solution is $x = 3 - 2 = 1$ . When we have asked the question implied in the statement of the equation $x + 2 = 3$ , $x$ is called the unknown. The object of the solution of the equation is the determination of the unknown. Equations are of great importance in mathematics, and it seems as

though (2') exemplified a much more thoroughgoing and fundamental idea than the original statement (2). This, however, is a complete mistake. The idea of the undetermined

"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 (3') as compared to the original statement (3).

But the majority of interesting formulæ,

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

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 $x + y = 1$ . One use of formulæ of the first type, true for any pair of numbers, is that by them formulæ of the second type can be

thrown into an indefinite number of equivalent forms. For example, the relation $x + y = 1$ is equivalent to the relations $y + x = 1, (x - y) + 2 y = 1, 6 x + 6 y = 6,$ and so on. Thus a skilful mathematician uses that equivalent form of the relation under consideration which is most convenient for his immediate purpose.

It is not in general true that, when a pair of terms satisfy some fixed relation, if one of the terms is given the other is also definitely determined. For example, when $x$ and $y$ satisfy $y^{2} = x$ , if $x = 4$ , $y$ can be $\pm 2$ , thus, for any positive value of $x$ there are alternative values for $y$ . Also in the relation $x + y > 1$ , when either $x$ or $y$ is given, an indefinite number of values remain open for the other.

Again there is another important point to be noticed. If we restrict ourselves to positive numbers, integral or fractional, in considering the relation $x + y = 1$ , then, if either $x$ or $y$ be greater than $1$ , there is no positive number which the other can assume so as to satisfy the relation. Thus the "field" of the relation for $x$ is restricted to numbers less than $1$ , and similarly for the "field" open to $y$ . Again, consider integral numbers only, positive or negative, and take the relation