# 1911 Encyclopædia Britannica/Infinitesimal Calculus

**INFINITESIMAL CALCULUS.** 1. The infinitesimal calculus
is the body of rules and processes by means of which continuously
varying magnitudes are dealt with in mathematical analysis.
The name “infinitesimal” has been applied to the calculus
because most of the leading results were first obtained by means
of arguments about “infinitely small” quantities; the “infinitely
small” or “infinitesimal” quantities were vaguely conceived
as being neither zero nor finite but in some intermediate,
nascent or evanescent, state. There was no necessity for this
confused conception, and it came to be understood that it can
be dispensed with; but the calculus was not developed by its
first founders in accordance with logical principles from precisely
defined notions, and it gained adherents rather through the
impressiveness and variety of the results that could be obtained
by using it than through the cogency of the arguments by which
it was established. A similar statement might be made in
regard to other theories included in mathematical analysis, such,
for instance, as the theory of infinite series. Many, perhaps all,
of the mathematical and physical theories which have survived
have had a similar history—a history which may be divided
roughly into two periods: a period of construction, in which
results are obtained from partially formed notions, and a period
of criticism, in which the fundamental notions become progressively
more and more precise, and are shown to be adequate
bases for the constructions previously built upon them. These
periods usually overlap. Critics of new theories are never lacking.
On the other hand, as E. W. Hobson has well said, “pertinent
criticism of fundamentals almost invariably gives rise to new
construction.” In the history of the infinitesimal calculus the
17th and 18th centuries were mainly a period of construction,
the 19th century mainly a period of criticism.