INFINITESIMAL

nfinitesimals 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, but not zero in size; or, 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".

Archimedes exploited infinitesimals in The Method to find areas of regions and volumes of solids. The classical authors tended to seek to replace infinitesimal arguments by arguments by exhaustion which they felt were more reliable. The 15th century saw the pioneering work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. Simon Stevin developed a continuum of decimals in the 16th century. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus. He exploited an infinitesimal denoted in area calculations.

The use of infinitesimals in Leibniz relied upon a heuristic principle called the Law of Continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa. The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph Lagrange. Augustin-Louis Cauchy exploited infinitesimals in defining continuity and an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstact versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions. Du Bois-Reymond's work inspired both Émile Borel and Thoralf Skolem. Skolem developed the first non-standard models of arithmetic in 1934. A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed non-standard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The standard part function implements Fermat's adequality.