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

TRANSFORMATION 159
O is x into the point whose distance is y, or ~x; the point
0, its distance being zero, is transformed into itself, for
the half of zero is zero; the point i is transformed into the
point 2; and the points between O and i have for their
transforms the points between O and ~; nothing, as you
see, could be clearer; yet our plan of joint reading had
to be here abandoned, for my fellow student, prophet of
philosophy, would not follow Bolzano's reasoning and
remained invincible to the bitter end. Can you beat that?
For a simple example illustrating both the concept of
transformation and that of infinite class geometrically,
consider Fig. 21. We are going to transform the class
composed of the points of segment AD into the subclass
composed of the points of segment A"D". Let all the
points of AD be joined to P; any such join, say, PB,
contains a point, B' of A'D'; associate B with B'; in
this way segment AD is transformed, point for point, into
segment A'D'; next join Q to all the points of A'B'; any
such join, say, QB', contains a point, B" of A"D";
associate B' with B"; in this way segment A'D' is transformed, point for point, into segment A"D"; now observe
that, starting with any point B of AD, the first transformation gives us B' of A'D', and the second leads to
B" of A"D"; finally, associate each such initial point B
with the final point B"; the result, as you see, is a pointto-point transformation of the entire segment AD into
one of its parts, the segment A"D"; and so we see, in the
light of the last transformation, that segment AD is an
infinite class of points; the same is, of course, true of any
other segment, however short, for, in the foregoing argument, AD is any segment you please. In passing, we may
notice also that we can choose Q so that segment A"D"
shall be any part we please of AD, and we have the