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

DIFFERENTIAL CALCULUS 227
matical procedure was substantially right.
In fact, the subject was right, though the explanations were wrong. It is this possibility
of being right, albeit with entirely wrong explanations as to what is being done, that so
often makes external criticism-that is so far
as it is meant to stop the pursuit of a method
-singularly barren and futile in the progress
of science. The instinct of trained observers,
and their sense of curiosity, due to the fact
that they are obviously getting at something,
are far safer guides. Anyhow the general
effect of the success of the Differential Calculus was to generate a large amount of bad
philosophy, centring round the idea of the infinitely small. The relics of this verbiage
may still be found in the explanations of
many elementary mathematical text-books on
the Differential Calculus. It is a safe rule to
apply that, when a mathematical or philosophical author writes with a misty profundity,
he is talking nonsense.
Newton would have phrased the question
by saying that, as h approaches zero, in tlie
limit 2x+h becomes 2x. It is our task so to
explain this statement as to show that it does
not in reality covertly assume the existence
of Leibniz's infinitely small quantities. In
reading over the Newtonian method of statement, it is tempting to seek simplicity by