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

24 INTRODUCTION TO MATHEMATICS
miles, we may then conclude that the train
was running at the rate of 60 miles per hour.
But five miles is a long distance, and we
cannot be sure that just at noon the train
was moving at this pace. At noon it may
have been running 70 miles per hour, and
afterwards the break may have been put on.
It will be safer to work with a smaller interval,
say one minute, which includes noon, and to
measure the space traversed during that
period. But for some purposes greater
accuracy may be required, and one minute
may be too long. In practice, the necessary
inaccuracy of our measurements makes it
useless to take too small a period for measurement. But in theory the smaller the period
the better, and we are tempted to say that
for ideal accuracy an infinitely small period
is required. The older mathematicians, in
particular Leibniz, were not only tempted,
but yielded to the temptation, and did say
it. Even now it is a useful fashion of speech,
provided that we know how to interpret it
into the language of common sense. It is
curious that, in his exposition of the foundations of the calculus, Newton, the natural
scientist, is much more philosophical than
Leibniz, the philosopher, and on the other
hand, Leibniz provided the admirable notation which has been so essential for the progress of the subject.