An introduction to mathematics, by A. N. Whitehead.

VARIABLES 19
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)+2y=l, 6x+6y=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 y2=x, if x=4, y can be =-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-=l, 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