Oshin Albrecht

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updated by Oshin Albrecht -- Fri 27 Jul 2012 - 01:40

Infinitesimal

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 insight with exploiting infinitesimals was that objects could still retain certain defined properties, such as angle or slope, even though

these objects were quantitatively small.[1] The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which

originally referred to the "infinite-th" item in a series. It was originally introduced around 1670 by either Nicolaus Mercator or

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". An infinitesimal object by itself is often useless and not very well defined; in order to give it a

meaning it usually has to be compared to another infinitesimal object in the same context (as in a derivative) or added together with an extremely large (an infinite) amount of other infinitesimal objects (as in an integral).

Archimedes used what eventually came to be known as the Method of indivisibles in his work The Method of Mechanical Theorems to find

areas of regions and volumes of solids.[3] In his formal published treatises, Archimedes solved the same problem using the Method of

Exhaustion. The 15th century saw the 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's work on decimal representation of all

numbers in the 16th century prepared the ground for the real continuum.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 $\frac{1}{\infty}$ in area calculations.

The use of infinitesimals by Leibniz relied upon heuristic principles, such as the Law of Continuity: what succeeds for the finite numbers

succeeds also for the infinite numbers and vice versa; and theTranscendental Law of Homogeneity that specifies procedures for replacing

expressions involving inassignable quantities, by expressions involving only assignable ones. The 18th century saw routine use of

infinitesimals by mathematicians such as Leonhard Euler and Joseph LagrangeAugustin-Louis Cauchy exploited infinitesimals in

defining continuity and an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract 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. Borel explicitly linked du Bois-Reymond's work to

Cauchy's work on rates of growth of infinitesimals. 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