View source for Fluxion

{{Short description|Historical mathematical concept; form of derivative}}
{{about|the mathematical concept}}
{{Calculus}}
[[File:The method of fluxions and infinite series p.20.gif|thumb|400px|Newton's introduction of the notions ''"fluent"'' and ''"fluxion"'' in his 1736 book]]
A '''fluxion''' is the [[instantaneous rate of change]], or [[gradient]], of a [[fluent (mathematics)|fluent]] (a time-varying quantity, or [[function (mathematics)|function]]) at a given point.<ref>{{cite book|last1=Newton|first1=Sir Isaac|title=The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines|date=1736|publisher=Henry Woodfall; and sold by John Nourse|url=https://archive.org/details/methodfluxionsa00newtgoog|access-date=6 March 2017|language=en}}</ref> Fluxions were introduced by [[Isaac Newton]] to describe his form of a [[time derivative]] (a [[derivative]] with respect to time). Newton introduced the concept in 1665 and detailed them in his [[mathematics|mathematical]] treatise, ''[[Method of Fluxions]]''.<ref>{{MathWorld|Fluxion}}</ref> Fluxions and fluents made up Newton's early [[calculus]].<ref>{{Britannica|fluxion}}</ref>

==History==
Fluxions were central to the [[Leibniz–Newton calculus controversy]], when Newton sent a letter to [[Gottfried Wilhelm Leibniz]] explaining them, but concealing his words in code due to his suspicion. He wrote:<ref>{{cite book|last1=Turnbull|first1=Isaac Newton. Ed. by H.W.|title=The correspondence of Isaac Newton|date=2008|publisher=Univ. Press|location=Cambridge [u.a.]|isbn=9780521737821|edition=Digitally printed version, pbk. re-issue.}}</ref>
{{quote|I cannot proceed with the explanations of the fluxions now, I have preferred to conceal it thus: 6accdæ13eff7i3l9n4o4qrr4s8t12vx.}}
The gibberish string was in fact a [[hash code]] (by denoting the frequency of each letter) of the [[Latin]] phrase ''Data æqvatione qvotcvnqve flventes qvantitates involvente, flvxiones invenire: et vice versa'', meaning: "Given an equation that consists of any number of flowing quantities, to find the fluxions: and vice versa".<ref name="clegg">{{cite book|last1=Clegg|first1=Brian|title=A brief history of infinity: the quest to think the unthinkable|date=2003|publisher=Constable|location=London|isbn=9781841196503|url-access=registration|url=https://archive.org/details/briefhistoryofin0000cleg}}</ref>

==Example==
If the fluent {{tmath|y}} is defined as <math>y=t^2</math> (where {{tmath|t}} is time) the fluxion (derivative) at <math>t=2</math> is:
:<math>\dot y = \frac{\Delta y}{\Delta t} = \frac{(2+o)^2-2^2}{(2+o)-2} = \frac{4+4o+o^2-4}{2+o-2} = \frac{4o+o^2}{o}</math>
Here {{tmath|o}} is an [[infinitesimal|infinitely small]] amount of time.<ref>{{cite web|last1=Buckmire|first1=Ron|title=History of Mathematics|url=https://sites.oxy.edu/ron/math/395/10/ws/23.pdf|access-date=28 January 2017}}</ref> So, the term {{tmath|o^2}} is second order infinite small term and according to Newton, we can now ignore {{tmath|o^2}} because of its second order infinite smallness comparing to first order infinite smallness of {{tmath|o}}.<ref>{{cite web|title=Isaac Newton (1642-1727)|url=http://www.mhhe.com/math/calc/smithminton2e/cd/tools/timeline/newton.html|website=www.mhhe.com|access-date=6 March 2017}}</ref> So, the final equation gets the form:
:<math>\dot y = \frac{\Delta y}{\Delta t}=\frac{4o}{o}=4</math>
He justified the use of {{tmath|o}} as a non-zero quantity by stating that fluxions were a consequence of movement by an object.

==Criticism==
Bishop [[George Berkeley]], a prominent [[philosopher]] of the time, denounced Newton's fluxions in his essay ''[[The Analyst]]'', published in 1734.<ref>{{cite wikisource |title=The Analyst: a Discourse addressed to an Infidel Mathematician |wslink=The Analyst: a Discourse addressed to an Infidel Mathematician#XVI |last=Berkeley |first=George |authorlink=George Berkeley |year=1734 |location=London |page=25}}</ref> Berkeley refused to believe that they were accurate because of the use of the [[infinitesimal]] {{tmath|o}}. He did not believe it could be ignored and pointed out that if it was zero, the consequence would be [[division by zero]]. Berkeley referred to them as "ghosts of departed quantities", a statement which unnerved mathematicians of the time and led to the eventual disuse of infinitesimals in calculus.

Towards the end of his life Newton revised his interpretation of {{tmath|o}} as [[infinitesimal|infinitely small]], preferring to define it as approaching [[zero]], using a similar definition to the concept of [[limit (mathematics)|limit]].<ref>{{cite journal|last1=Kitcher|first1=Philip|title=Fluxions, Limits, and Infinite Littlenesse. A Study of Newton's Presentation of the Calculus|journal=Isis|date=March 1973|volume=64|issue=1|pages=33–49|doi=10.1086/351042|s2cid=121774892}}</ref> He believed this put fluxions back on safe ground. By this time, Leibniz's derivative (and his [[Leibniz's notation|notation]]) had largely replaced Newton's fluxions and fluents, and remains in use today.

==See also==
*[[History of calculus]]
*[[Newton's notation]]
*[[Hyperreal number]]: A modern formalization of the reals that includes infinity and infinitesimals
*[[Nonstandard analysis]]

==References==
{{reflist}}

{{Calculus topics}}
{{Isaac Newton|state=collapsed}}
[[Category:Mathematical analysis]]
[[Category:Differential calculus]]
[[Category:History of calculus]]