Fluxion

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Glossary of calculus

The fluxion of a "fluent" (a time-varying quantity, or function) is its instantaneous rate of change, or gradient, 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: 6accdæ13eff7i319n4o4qrr4s8t12vz

The gibberish string was in fact an enciphered Latin phrase, 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}}=4+o

Here $o$ is an infinitely small amount of time^[6] and according to Newton, we can now ignore it because of its infinite smallness.^[7] 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, slammed 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 remain 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 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: Wikisource. p. 25.
↑ 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.

Calculus
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Isaac Newton
Publications	De analysi per aequationes numero terminorum infinitas (1669, published 1711) Method of Fluxions (1671) De motu corporum in gyrum (1684) Philosophiæ Naturalis Principia Mathematica (1687) General Scholium (1713) Opticks (1704) The Queries (1704) Arithmetica Universalis (1707)
Other writings	Notes on the Jewish Temple Quaestiones quaedam philosophicae The Chronology of Ancient Kingdoms Amended (1728) An Historical Account of Two Notable Corruptions of Scripture (1754)
Discoveries and inventions	Calculus Fluxions Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Newton's cradle Sextant
Theory expansions	Kepler's laws of planetary motion Problem of Apollonius
Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation Post-Newtonian expansion Parameterized post-Newtonian formalism Newton–Cartan theory Schrödinger–Newton equation Gravitational constant Newton's laws of motion Newtonian dynamics Newton's method in optimization Gauss–Newton algorithm Truncated Newton method Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Newtonian fluid Corpuscular theory of light Leibniz–Newton calculus controversy Newton's notation Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method Newton fractal Generalized Gauss–Newton method Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number Kissing number problem Power number Newton's quotient Parallelogram of force Newton–Puiseux theorem Solar mass Dynamics Absolute space and time Absolute theory Finite difference Table of Newtonian series Impact depth Structural coloration Inertia Spectrum
Phrases	"Hypotheses non fingo" "Standing on the shoulders of giants"
Life	Cranbury Park Woolsthorpe Manor Early life Later life Religious views Occult studies The Mysteryes of Nature and Art Scientific revolution Copernican Revolution
Friends and family	Catherine Barton John Conduitt William Clarke Benjamin Pulleyn William Stukeley William Jones Isaac Barrow Abraham de Moivre John Keill
Cultural depictions	Newton (Blake) Newton (Paolozzi) In popular culture
Related	Writing of Principia Mathematica List of things named after Newton Isaac Newton Institute Isaac Newton Medal Isaac Newton Group of Telescopes Isaac Newton Telescope Newton (unit) Elements of the Philosophy of Newton Isaac Newton S/O Philipose

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Fluxion

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