User:Tkuvho/Infinitesimal

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 $$\frac{1}{\infty}$$ in area calculations.

Pierre de Fermat, inspired by Diophantus, introduced the concept adequality, i.e. "adequate" or approximate equality (up to an infinitesimal error), which ultimately played a key role in a modern mathematical implementation of infinitesimal definitions of derivative and integral. 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 such greats 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 Emile 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.

Infinitésimal