infinitesimal adj : infinitely or immeasurably small; "two minute whiplike threads of protoplasm"; "reduced to a microscopic scale" [syn: minute, microscopic]

Infinitesimal \In`fin*i*tes"i*mal\, a. [Cf. F. infinit['e]simal, fr. infinit['e]sime infinitely small, fr. L. infinitus. See Infinite, a.] Infinitely or indefinitely small; less than any assignable quantity or value; very small. Infinitesimal calculus, the different and the integral calculus, when developed according to the method used by Leibnitz, who regarded the increments given to variables as infinitesimal.

Infinitesimal \In`fin*i*tes"i*mal\, n. (Math.) An infinitely small quantity; that which is less than any assignable quantity.

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Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.

In common speech, an infinitesimal object is an object which is smaller than any feasible measurement, hence not zero size, but so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" in the vernacular means "extremely small".

Before the nineteenth century none of the mathematical concepts relating to infinitesimals as we know them today were formally defined, but many of these concepts were already used. The founders of infinitesimal calculus — Euler, Leibniz, Newton, Cauchy, and many others — calculated with infinitesimals and achieved correct results even though the definitions available at the time came to be thought of as insufficiently precise by the followers of Weierstrass in the 1870s. For example, Cauchy defined infinitesimals in terms of variable quantities (such as sequences) tending to zero.

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