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

216 MATHEMATICAL PHILOSOPHY
two questions are not equivalent. It studies all the
figures in space but it does not study all their properties.
Its subject-matter consists of those properties which it
does study. What are these? They are those and only
those properties (of figures) that remain invariant under
all displacements but under no other transformations of
space. The geometry of displacements might be called
congruence geometry. It includes the greater part of the
ordinary geometry of high school but not all of it, for the
latter deals, for example, with similarity of figures;
similarity is indeed invariant under displacements, but
it is also invariant under other transformations-the
so-called similitude transformations, to be mentioned
presently.
For a second example, consider the following. I may
wish to confine my study of spatial figures to their shape.
The doctrine thus arising may be called the geometry of
shape, or shape geometry. If I tell you that I am studying shape geometry and you ask me what I mean by the
geometry of shape, there are two ways in which I may
answer your question. One of the ways requires me to
define the term shape-shape of a geometric figure; the
other way,-the group way,-does not. Let us examine
them a little. I have never seen a mathematical definition
of shape, but it may, I believe, be precisely defined as
follows. We must distinguish the three things: sameness
of shape; shape of a given figure; and shape of a figure.
I will define the first; then the second in terms of the
first; and, finally, the third in terms of the second.
Two figures F and F' will be said to have the same shape
if and only if it is possible to set up a one-to-one correspondence between the points of F and those of F', such
that, AB and CD being any distances between points of