The arithmetic of infinitesimals series#
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. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. 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 the transcendental law of homogeneity that specifies procedures for replacing expressions involving inassignable quantities, by expressions involving only assignable ones. He exploited an infinitesimal denoted 1/∞ in area calculations. 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. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. Simon Stevin's work on decimal representation of all numbers in the 16th century prepared the ground for the real continuum. 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. In his formal published treatises, Archimedes solved the same problem using the method of exhaustion. 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. The concept of infinitesimals was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Infinitely many infinitesimals are summed to produce an integral. To give it a meaning, it usually must be compared to another infinitesimal object in the same context (as in a derivative). Hence, when used as an adjective, "infinitesimal" means "extremely small". In common speech, an infinitesimal object is an object that 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.
Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the " infinite -th" item in a sequence. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. In mathematics, infinitesimals are things so small that there is no way to measure them.See: Infinity, Integer Number, Integral.An Infinitesimal Number is a Real Number that is a Fraction with Infinity as the Denominator.