Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

222 MATHEMATICAL PHILOSOPHY
we say, however, that F and F' are projective because,
as can be proved, the anharmonic ratio of any 4 points
(or lines) of either is equal to that of the corresponding
(transform) points (or lines) of the other. By the method
indicated for the homographic transformations of a line,
you can prove that the oo8 transformations of form (i)
constitute a group.
Just as a point of the plane has two coordinates (x, y),
so a line depends on two coordinates; there are various
ways to see that such is the case; an easy way is this:
the line, ax+by+c-O, depends solely upon the ratios
(a: b: c) of the coefficients; these three ratios are not
independent-two of them determine the third one; you
thus see that the line depends upon only two independent
variables-it has, like the point, two coordinates; let us
denote them by (u, v). Now consider the transformations, Ju+Kv+L
Pu+Qv+R
(2)
( _Mu+Nv+O
Pu+Qv+R
where the coefficients are subject to a relation like (i').
We saw that a transformation (i) converts points into
points directly and lines into lines indirectly; just so, a
transformation (2) converts lines into lines directly and
points into points indirectly; hence the group of line-toline transformations (2) is essentially the same as the
foregoing group of point-to-point transformations (I).
This latter group is called the group of collineations of
(or in) the plane.
I am going now to ask you to notice an ensemble of
transformations (of the plane) that are neither point-to