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

THE GROUP CONCEPT 219
projective line-an ordinary line endowed with an
"ideal " point, or point at infinity, where the line meets
all lines parallel to it. Let L be a projective line. In
the preceding lecture, we gained some acquaintance with
the transformations of the form
ax+b
(I) x='
cx +d
where the coefficients are any real numbers such that
ad-bcso; we saw that there are oo3 such transformations
and that each of them converts the points of L into the
points of L in such a way that the anharmonic ratio of
any four points is equal to the anharmonic ratio of their
transforms. Distances are not preserved; neither are the
ordinary ratios of distances preserved; hence neither
congruence nor similarity is invariant; no relation among
points-that is, no property of figures (for here a figure is
simply a set of points on L)-is invariant except anharmonic ratio and properties expressible in terms of the
latter; no other transformations leave these properties
invariant. By a little finger work you can prove in a
formal way that these transformations constitute a group.
I will merely indicate the procedure, leaving it to you to
carry it out if you desire to do so. The transformations
differ only in their coefficients. Let (ai, bi, c, di),
(a2, b2, C2, d2), (a3, b3, c3, d3) be any three of the transformations; consider the first and second; the rule o
of combination is to be: operate with the first and then
on the result with the second. The first converts point
x into point x':
( aix +bi
) cx+di'