An introduction to mathematics, by A. N. Whitehead.

152 INTRODUCTION TO MATHEMATICS
value suddenly for a big, heavy train. The
train certainly cannot be running at forty
miles per hour from 11.45 A.M. up to noon,
and then suddenly, without any lapse of time,
commence running at 50 miles per hour. We
at once admit that the change of velocity
will be a gradual process. But how about
sudden blows of adequate magnitude? Suppose two trains collide; or, to take smaller
objects, suppose a man kicks a football. It
certainly appears to our sense as though the
football began suddenly to move. Thus, in
the case of velocity our senses do not revolt
at the idea of its being a discontinuous function of the time, as they did at the idea of the
train being instantaneously transported from
Bletchley to Rugby. As a matter of fact,
if the laws of motion, with their conception
of mass, are true, there is no such thing as
discontinuous velocity in nature. Anything
that appears to our senses as discontinuous
change of velocity must, according to them,
be considered to be a case of gradual change
which is too quick to be perceptible to us.
It would be rash, however, to rush into the
generalization that no discontinuous functions are presented to us in nature. A man
who, trusting that the mean height of the
land above sea-level between London and
Paris was a continuous function of the distance from London, walked at night on