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

160 MATHEMATICAL PHILOSOPHY
Theorem: that any segment, however long, can be converted,
by a point-to-point transformation, into any one of its parts,
however short.
Before further illustrating the concept of mathematical
transformation, let us ask what we mean by the law of
such a transformation. The answer is pretty evident.
It is that the law of a transformation is any rule, formula,
scheme or device which, given any one of the elements or
Q.
FIG. 2z.
objects we are dealing with, determines its transform (or
transforms, if it have more than one). It is, you should
note, a psychological affair, the law being a device for
guiding the transfer of attention from a given object to a
definite other.object (or objects). Such a law may be
variously expressed: if the class to be transformed be
small, it is practicable to express the law by tabulation,
as by the foregoing tables (I), (2),..., (o0); this is theoretically possible for any finite class, but is impracticable
if the class, though finite, be very large; to express a law