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.