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A fluxion is the instantaneous rate of change, or gradient, of a fluent (a time-varying quantity, or function) at a given point.^[1] 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 mathematical treatise, Method of Fluxions.^[2] Fluxions and fluents made up Newton's early calculus.^[3]

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:^[4]

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".^[5]

Example

If the fluent ⁠ $y$ ⁠ is defined as $y=t^{2}$ (where ⁠ $t$ ⁠ is time) the fluxion (derivative) at $t=2$ is:

{\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}}

Here ⁠ $o$ ⁠ is an infinitely small amount of time.^[6] So, the term ⁠ $o^{2}$ ⁠ is second order infinite small term and according to Newton, we can now ignore ⁠ $o^{2}$ ⁠ because of its second order infinite smallness comparing to first order infinite smallness of ⁠ $o$ ⁠.^[7] So, the final equation gets the form:

{\dot {y}}={\frac {\Delta y}{\Delta t}}={\frac {4o}{o}}=4

He justified the use of ⁠ $o$ ⁠ 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.^[8] Berkeley refused to believe that they were accurate because of the use of the infinitesimal ⁠ $o$ ⁠. 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 ⁠ $o$ ⁠ as infinitely small, preferring to define it as approaching zero, using a similar definition to the concept of limit.^[9] He believed this put fluxions back on safe ground. By this time, Leibniz's derivative (and his notation) had largely replaced Newton's fluxions and fluents, and remains in use today.

References

^ Newton, Sir Isaac (1736). The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines. Henry Woodfall; and sold by John Nourse. Retrieved 6 March 2017.
^ Weisstein, Eric W. "Fluxion". MathWorld.
^ Fluxion at the Encyclopædia Britannica
^ Turnbull, Isaac Newton. Ed. by H.W. (2008). The correspondence of Isaac Newton (Digitally printed version, pbk. re-issue. ed.). Cambridge [u.a.]: Univ. Press. ISBN 9780521737821.
^ Clegg, Brian (2003). A brief history of infinity: the quest to think the unthinkable. London: Constable. ISBN 9781841196503.
^ Buckmire, Ron. "History of Mathematics" (PDF). Retrieved 28 January 2017.
^ "Isaac Newton (1642-1727)". www.mhhe.com. Retrieved 6 March 2017.
^ Berkeley, George (1734). The Analyst: a Discourse addressed to an Infidel Mathematician . London. p. 25 – via Wikisource.
^ Kitcher, Philip (March 1973). "Fluxions, Limits, and Infinite Littlenesse. A Study of Newton's Presentation of the Calculus". Isis. 64 (1): 33–49. doi:10.1086/351042. S2CID 121774892.

[1] Newton, Sir Isaac (1736). The Method of Fluxions and Infinite Series: With Its Application to the Geometry of Curve-lines. Henry Woodfall; and sold by John Nourse. Retrieved 6 March 2017.

[2] Weisstein, Eric W. "Fluxion". MathWorld.

[3] Fluxion at the Encyclopædia Britannica

[4] Turnbull, Isaac Newton. Ed. by H.W. (2008). The correspondence of Isaac Newton (Digitally printed version, pbk. re-issue. ed.). Cambridge [u.a.]: Univ. Press. ISBN 9780521737821.

[clegg-5] Clegg, Brian (2003). A brief history of infinity: the quest to think the unthinkable. London: Constable. ISBN 9781841196503.

[6] Buckmire, Ron. "History of Mathematics" (PDF). Retrieved 28 January 2017.

[7] "Isaac Newton (1642-1727)". www.mhhe.com. Retrieved 6 March 2017.

[8] Berkeley, George (1734). The Analyst: a Discourse addressed to an Infidel Mathematician . London. p. 25 – via Wikisource.

[9] Kitcher, Philip (March 1973). "Fluxions, Limits, and Infinite Littlenesse. A Study of Newton's Presentation of the Calculus". Isis. 64 (1): 33–49. doi:10.1086/351042. S2CID 121774892.

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@@ Line 1: / Line 1: @@
+{{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]]
-The '''fluxion''' of a "[[fluent (mathematics)|fluent]]" (a time-varying quantity, or [[function (mathematics)|function]]) is its instantaneous [[Instantaneous rate of change|rate of change]], or [[gradient]], 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://books.google.com/books?id=WyQOAAAAQAAJ|accessdate=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>
+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: 6accdæ13eff7i319n4o4qrr4s8t12vz}}
+{{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 an [[cipher|enciphered]] [[Latin]] phrase, 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}}</ref>
+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} = 4+o</math>
+:<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|accessdate=28 January 2017}}</ref> and according to Newton, we can now ignore it because of its infinite smallness.<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|accessdate=6 March 2017}}</ref> He justified the use of {{tmath|o}} as a non-zero quantity by stating that fluxions were a consequence of movement by an object.
+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, slammed 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.
+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}}</ref> He believed this put fluxions back on safe ground. By this time, Leibniz's derivative (and his notation) had largely replaced Newton's fluxions and fluents, and remain in use today.
+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 notation) had largely replaced Newton's fluxions and fluents, and remains in use today.
 ==See also==
-{{div col|colwidth=22em}}
-*''[[Method of Fluxions]]
 *[[History of calculus]]
-*[[Leibniz–Newton calculus controversy]]
-*[[Derivative]]
 *[[Newton's notation]]
+*[[Hyperreal number]]: A modern formalization of the reals that includes infinity and infinitesimals
-*[[Fluent (mathematics)]]
+*[[Nonstandard analysis]]
-{{div col end}}
-{{portal bar|Mathematics}}
 ==References==

v t e Sir Isaac Newton
Publications	Fluxions (1671) De Motu (1684) Principia (1687) Opticks (1704) Queries (1704) Arithmetica (1707) De Analysi (1711)
Other writings	Quaestiones (1661–1665) "standing on the shoulders of giants" (1675) Notes on the Jewish Temple (c. 1680) "General Scholium" (1713; "hypotheses non fingo" ) Ancient Kingdoms Amended (1728) Corruptions of Scripture (1754)
Contributions	Calculus fluxion Impact depth Inertia Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Spectrum Structural coloration
Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation post-Newtonian expansion parameterized gravitational constant Newton–Cartan theory Schrödinger–Newton equation Newton's laws of motion Kepler's laws Newtonian dynamics Newton's method in optimization Apollonius's problem truncated Newton method Gauss–Newton algorithm Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Corpuscular theory of light Leibniz–Newton calculus controversy Newton's notation Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method generalized Gauss–Newton method Newton fractal Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number kissing number problem Newton's quotient Parallelogram of force Newton–Puiseux theorem Absolute space and time Luminiferous aether Newtonian series table
Personal life	Woolsthorpe Manor (birthplace) Cranbury Park (home) Early life Later life Apple tree Religious views Occult studies Scientific Revolution Copernican Revolution
Relations	Catherine Barton (niece) John Conduitt (nephew-in-law) Isaac Barrow (professor) William Clarke (mentor) Benjamin Pulleyn (tutor) Roger Cotes (student) William Whiston (student) John Keill (disciple) William Stukeley (friend) William Jones (friend) Abraham de Moivre (friend)
Depictions	Newton by Blake (monotype) Newton by Paolozzi (sculpture) Isaac Newton Gargoyle Astronomers Monument
Namesake	Newton (unit) Newton's cradle Isaac Newton Institute Isaac Newton Medal Isaac Newton Telescope Isaac Newton Group of Telescopes XMM-Newton Sir Isaac Newton Sixth Form Statal Institute of Higher Education Isaac Newton Newton International Fellowship
Categories	Isaac Newton