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

THE GROUP CONCEPT 22
point nor line-to-line transformations but are at once
point-to-line and line-to-point transformations. These
are represented by the pair of formulas
ax +by +c
gx +hy +i
dx +ey +f
gx+hy+i
where the coefficients are again subject to a relation like
(i'). Any such transformation converts a point (x, y)
into a line (u, v); now operate on the points of this line
by the same transformation or another one of form (3);
the points are converted into lines constituting a pencil
having a vertex, say (x', y'); thus the combination converts point (x, y) into point (x', y')-it is a point-to-point
transformation and hence belongs to the group of collineations; you thus see that the set of transformations (3)
is not a group; but this set and the collineations together
constitute a group including the collineations as a subgroup. This large group is called the Group of Projective
Transformations of the Plane. Why? Because every
transformation in it and no other transformation leaves
all anharmonic ratios unchanged.
What is the projective geometry of the plane? The
group now in hand enables us to answer the question
perfectly. The answer is: Projective plane geometry is
that geometry which studies such and only such properties
of plane figures as remain invariant under the group of
projective transformations.
In reading the essays of the late Henri Poincaré you
have met the statement: " Euclidean space is simply a
group." The foregoing examples should enable you to