The axioms of descriptive geometry, by A.N. Whitehead.

31, 32]

DEFINITION OF A RELATION

35

The class of all the referents in respect to a relation is called the
doman of the relation, and the class of all relata is the converse
domain., In mathematics a one-one relation is often spoken of as a transformation (or correspondence) of the members of its domain into (or
with) the corresponding members of the converse domain. The correspondence is definite and reversible, and constitutes a rule by which
we can pass from any member of one class to a corresponding definite
member of the other class.
For example, the equation,
2x + 3y = 4,
constitutes a one-one relation of all real numbers, positive or negative,
to the same class of real numbers. This brings out the fact that the
domain and the converse domain can be identical.
Again, a projective relation between all the points on one line of
projective space and all the points on another (or the same) line
constitutes a one-one relation, or transformation, or correspondence,
between the points of the two lines. Any one-one relation of which
both the domain and the converse domain are each of them all the
points of a projective space will be called a one-one point correspondence.
32. By reasoning* based upon the axioms of Projective Geometry,
without reference to any idea of distance or of congruence, coordinates
can be introduced, so that the ratios of four coordinates characterize
each point, and the equation of a plane is a homogeneous equation of
the first degree. Let X, Y, Z, U be the four coordinates of any
point; then it will be more convenient for us to work with nonhomogeneous coordinates found by putting x for XI/U, y for Y/U,
z for Z/ U. Accordingly the actual values of x, y, z are, as usual, the
coordinates characterizing a point. All points can thus be represented
by finite values of x, y, z, except points on the plane, U= 0. For these
points some or all of x, y, and z are infinite. In order to deal with
this plane either recourse must be had to the original homogeneous
coordinates, or the limiting values of x to y to z must be considered as
they become infinite.
The plane, x =0, is called the yz plane, the line, y =0, z= 0, is
called the axis of x, and the plane, U=0, is called the infinite
plane.
* Cf. Proj. Geom, ohs. vi, and vn.
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