*21. Correspondence between points and numbers.* In the theory of
analytic geometry a one-to-one correspondence is assumed to exist between
points on a line and numbers. In order to justify this assumption a very
extended definition of number must be made use of. A one-to-one
correspondence is then set up between points in the plane and pairs of
numbers, and also between points in space and sets of three numbers. A
single constant will serve to define the position of a point on a line;
two, a point in the plane; three, a point in space; etc. In the same
theory a one-to-one correspondence is set up between loci in the plane and
equations in two variables; between surfaces in space and equations in
three variables; etc. The equation of a line in a plane involves two
constants, either of which may take an infinite number of values. From
this it follows that there is an infinity of lines in the plane which is
of the second order if the infinity of points on a line is assumed to be
of the first. In the same way a circle is determined by three conditions;
a sphere by four; etc. We might then expect to be able to set up a
one-to-one correspondence between circles in a plane and points, or planes
in space, or between spheres and lines in space. Such, indeed, is the
case, and it is often possible to infer theorems concerning spheres from
theorems concerning lines, and vice versa. It is possibilities such as
these that, give to the theory of one-to-one correspondence its great
importance for the mathematician. It must not be forgotten, however, that
we are considering only _continuous_ correspondences. It is perfectly
possible to set, up a one-to-one correspondence between the points of a
line and the points of a plane, or, indeed, between the points of a line
and the points of a space of any finite number of dimensions, if the
correspondence is not restricted to be continuous.