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

THE GROUP CONCEPT 217
F, and A'B' and C'D' being the distances between the
corresponding points of F', AB/CD =A'B'/C'D'. Two
figures having the same shape will be said to be similar,
and conversely. Having defined sameness of shape, or
similarity, of figures, I will define the term " shape of a
given figure " as follows: if F be a given, or specific, figure,
the shape of F is the class o of all figures similar to F; it
is evident that, if F and F' are not similar, the clais a
and the class -'-the shape of F'-have no figures in
common; it is evident, moreover, that there are as many
's as there are figure shapes. And now what do we
mean by the general term shape, or-what is tantamount
-shape of a figure? What the answer must be is pretty
obvious: shape is the class z of all the -'s. Note that
2 is a class of classes and that any a is a class of (similar)
figures. Having defined the general term shape, I have,
you see, virtually answered your question: what is the
geometry of shape?
Let us now see how the question may be answered by
means of the group concept. Two congruent figures are
clearly similar, and so similarity is invariant under the
group of displacements. But you readily see that there
are many other transformations under which similarity is
invariant. Let O be a point; consider the bundle of
straight lines,-all the lines through 0,-having O for its
vertex; every point of space is on some line of the bundle;
let k be any real number (except zero); let P be any
point and let P' be such a point on the line OP that the
segment OP' = k X segment OP; you see that each point P
is transformed into a point P'; the transformation is
called homothetic; its effect, if k be positive and exceed I,
is a uniform expansion of space from O outward in all
directions; if k be positive and less than I, the effect is