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

220 MATHEMATICAL PHILOSOPHY
the second converts x' into point x":
a2x'+b2
(3) x"=
in (3) replace x' by its value given by (2), simplify and then
notice that you have a transformation of form (I) converting x directly into x". This shows that the set of
transformations have the group property. To show that
they obey the associative law, it is sufficient to perform
the operations
(4) (ai, bl, c1, dl)o[(a2, b2, c2, d2)o(a3, b3, C3, d3)],
(5) [(ai, bl, ci, dl)o(a2, b2, c2, d2)]o(aa, b3, C3 d),
and then to observe that the results are the same. The
identical element i is (a, o, o, a)-that is, the transformation, x' = x. The inverse of any transformation (a, b, c, d)
is (-d, b, c, -a) for you can readily show that combination of these gives (a, O, O, a).
The fact to be specially noted is that this group of
so-called homographic transformations defines a certain
kind of geometry in the line L-namely, its projective
geometry. In a line there are various geometries; among
these the projective geometry is characterized by its subject-matter, and its subject-matter consists of such properties of point sets, or figures, as remain invariant under its
homographic group.
And now I come to the example alluded to a moment
ago-the one to be taken from geometries in (or of) a
plane. The foregoing homographic group-in a line, a
one-dimensional space-has an analogue in a projective
plane, another in ordinary 3-dimensional projective space,
another in a projective space of four dimensions, and so on