Changes

==Origin==

==Origin==

New Latin ''infinitesimus'' [[infinite]] in rank, from [[Latin]] ''infinitus''

New Latin ''infinitesimus'' [[infinite]] in rank, from [[Latin]] ''infinitus''

*[http://en.wikipedia.org/wiki/18th_century 1706]

*[https://en.wikipedia.org/wiki/18th_century 1706]

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 [http://en.wikipedia.org/wiki/Nicolaus_Mercator Nicolaus Mercator] or [http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz Gottfried Wilhelm Leibniz].

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 [https://en.wikipedia.org/wiki/Nicolaus_Mercator Nicolaus Mercator] or [https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz Gottfried Wilhelm Leibniz].

==Definitions==

==Definitions==

*1: an infinitesimal [[quantity]] or variable  

*1: an infinitesimal [[quantity]] or variable  

==Description==

==Description==

'''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 [http://en.wikipedia.org/wiki/Slope slope], even though these objects were quantitatively small.  

'''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 [https://en.wikipedia.org/wiki/Slope slope], even though these objects were quantitatively small.  

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 [http://en.wikipedia.org/wiki/Integral integral]).

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 [https://en.wikipedia.org/wiki/Integral integral]).

[http://en.wikipedia.org/wiki/Archimedes Archimedes] used what [[eventually]] came to be known as the [http://en.wikipedia.org/wiki/Method_of_indivisibles Method of indivisibles] in his work [http://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems ''The Method of Mechanical Theorems''] to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the [http://en.wikipedia.org/wiki/Method_of_Exhaustion Method of Exhaustion]. The 15th century saw the work of [http://en.wikipedia.org/wiki/Nicholas_of_Cusa Nicholas of Cusa], further developed in the 17th century by [http://en.wikipedia.org/wiki/Johannes_Kepler Johannes Kepler], in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. [http://en.wikipedia.org/wiki/Simon_Stevin Simon Stevin]'s work on decimal representation of all [[numbers]] in the 16th century prepared the ground for the real [[continuum]]. [http://en.wikipedia.org/wiki/Bonaventura_Cavalieri 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.[http://en.wikipedia.org/wiki/John_Wallis 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 [[File:Infinite_-_Version_2.jpg]]in area calculations.

[https://en.wikipedia.org/wiki/Archimedes Archimedes] used what [[eventually]] came to be known as the [https://en.wikipedia.org/wiki/Method_of_indivisibles Method of indivisibles] in his work [https://en.wikipedia.org/wiki/The_Method_of_Mechanical_Theorems ''The Method of Mechanical Theorems''] to find areas of regions and volumes of solids. In his formal published treatises, Archimedes solved the same problem using the [https://en.wikipedia.org/wiki/Method_of_Exhaustion Method of Exhaustion]. The 15th century saw the work of [https://en.wikipedia.org/wiki/Nicholas_of_Cusa Nicholas of Cusa], further developed in the 17th century by [https://en.wikipedia.org/wiki/Johannes_Kepler Johannes Kepler], in particular calculation of area of a circle by representing the latter as an infinite-sided polygon. [https://en.wikipedia.org/wiki/Simon_Stevin Simon Stevin]'s work on decimal representation of all [[numbers]] in the 16th century prepared the ground for the real [[continuum]]. [https://en.wikipedia.org/wiki/Bonaventura_Cavalieri 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.[https://en.wikipedia.org/wiki/John_Wallis 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 [[File:Infinite_-_Version_2.jpg]]in area calculations.

The use of infinitesimals by [http://en.wikipedia.org/wiki/Leibniz Leibniz] relied upon heuristic principles, such as the [http://en.wikipedia.org/wiki/Law_of_Continuity Law of Continuity]: what succeeds for the [[finite]] numbers succeeds also for the [[infinite]] numbers and vice versa; and the [http://en.wikipedia.org/wiki/Transcendental_Law_of_Homogeneity Transcendental 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 [http://en.wikipedia.org/wiki/Leonhard_Euler Leonhard Euler] and [http://en.wikipedia.org/wiki/Joseph_Lagrange Joseph Lagrange]. [http://en.wikipedia.org/wiki/Augustin-Louis_Cauchy Augustin-Louis Cauchy] exploited infinitesimals in defining continuity and an early form of a [http://en.wikipedia.org/wiki/Dirac_delta_function Dirac delta function]. As Cantor and Dedekind were developing more [[abstract]] versions of Stevin's continuum, [http://en.wikipedia.org/wiki/Paul_du_Bois-Reymond 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 [http://en.wikipedia.org/wiki/%C3%89mile_Borel Émile Borel] and [http://en.wikipedia.org/wiki/Thoralf_Skolem 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 [http://en.wikipedia.org/wiki/Abraham_Robinson Abraham Robinson] in 1961, who developed [http://en.wikipedia.org/wiki/Non-standard_analysis non-standard analysis] based on earlier work by [http://en.wikipedia.org/wiki/Edwin_Hewitt Edwin Hewitt] in 1948 and Jerzy Łoś in 1955. The [http://en.wikipedia.org/wiki/Hyperreal_number hyperreals] implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The [http://en.wikipedia.org/wiki/Standard_part_function standard part function] implements Fermat's [http://en.wikipedia.org/wiki/Adequality adequality].[http://en.wikipedia.org/wiki/Infinitesimal]

The use of infinitesimals by [https://en.wikipedia.org/wiki/Leibniz Leibniz] relied upon heuristic principles, such as the [https://en.wikipedia.org/wiki/Law_of_Continuity Law of Continuity]: what succeeds for the [[finite]] numbers succeeds also for the [[infinite]] numbers and vice versa; and the [https://en.wikipedia.org/wiki/Transcendental_Law_of_Homogeneity Transcendental 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 [https://en.wikipedia.org/wiki/Leonhard_Euler Leonhard Euler] and [https://en.wikipedia.org/wiki/Joseph_Lagrange Joseph Lagrange]. [https://en.wikipedia.org/wiki/Augustin-Louis_Cauchy Augustin-Louis Cauchy] exploited infinitesimals in defining continuity and an early form of a [https://en.wikipedia.org/wiki/Dirac_delta_function Dirac delta function]. As Cantor and Dedekind were developing more [[abstract]] versions of Stevin's continuum, [https://en.wikipedia.org/wiki/Paul_du_Bois-Reymond 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 [https://en.wikipedia.org/wiki/%C3%89mile_Borel Émile Borel] and [https://en.wikipedia.org/wiki/Thoralf_Skolem 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 [https://en.wikipedia.org/wiki/Abraham_Robinson Abraham Robinson] in 1961, who developed [https://en.wikipedia.org/wiki/Non-standard_analysis non-standard analysis] based on earlier work by [https://en.wikipedia.org/wiki/Edwin_Hewitt Edwin Hewitt] in 1948 and Jerzy Łoś in 1955. The [https://en.wikipedia.org/wiki/Hyperreal_number hyperreals] implement an infinitesimal-enriched continuum and the transfer principle implements Leibniz's law of continuity. The [https://en.wikipedia.org/wiki/Standard_part_function standard part function] implements Fermat's [https://en.wikipedia.org/wiki/Adequality adequality].[https://en.wikipedia.org/wiki/Infinitesimal]

[[Category: Mathematics]]

[[Category: Mathematics]]