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[[Calculus]], originally called [[infinitesimal]] calculus, is a mathematical discipline focused on [[limit (mathematics)|limits]], [[continuity (mathematics)|continuity]], [[derivative]]s, [[integral]]s, and [[infinite series]]. Many elements of calculus appeared in ancient Greece, then in China and the Middle East, and still later again in medieval Europe and in India. Infinitesimal calculus was developed in the late 17th century by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] independently of each other. An argument over priority led to the [[Leibniz–Newton calculus controversy]] which continued until the death of Leibniz in 1716. The development of calculus and its uses within the sciences have continued to the present.

==Etymology==

In [[mathematics education]], ''calculus'' denotes courses of elementary [[mathematical analysis]], which are mainly devoted to the study of [[Function (mathematics)|functions]] and limits. The word ''calculus'' is [[Latin]] for "small pebble" (the [[diminutive]] of ''[[wikt:calx|calx]],'' meaning "stone"), a meaning which still [[Calculus (medicine)|persists in medicine]]. Because such pebbles were used for counting out distances,<ref>See, for example:
*{{Cite web|title=history - Were metered taxis busy roaming Imperial Rome?|url=https://skeptics.stackexchange.com/questions/8841/were-metered-taxis-busy-roaming-imperial-rome|access-date=2022-02-13|date=2020-06-17|website=Skeptics Stack Exchange}}
*{{Cite book|last=Cousineau|first=Phil|url=https://books.google.com/books?id=m8lJVgizhbQC&q=Ancient+Roman+taximeter+calculus&pg=PT80|title=Wordcatcher: An Odyssey into the World of Weird and Wonderful Words|date=2010-03-15|publisher=Simon and Schuster|isbn=978-1-57344-550-4|oclc=811492876|pages=58|language=en}}</ref> tallying votes, and doing [[abacus]] arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.<ref>{{cite OED|calculus}}</ref>

In addition to the differential calculus and integral calculus, the term is also used widely for naming specific methods of calculation. Examples of this include [[propositional calculus]] in logic, the [[calculus of variations]] in mathematics, [[process calculus]] in computing, and the [[felicific calculus]] in philosophy.

==Early precursors of calculus==
{{see also|History of mathematics}}

===Ancient===
[[Image:Archimedes pi.svg|thumb|Archimedes used the [[method of exhaustion]] to compute the area inside a circle.]]

====Egypt and Babylonia====
{{see also|Ancient Egyptian mathematics|Babylonian mathematics}}
The ancient period introduced some of the ideas that led to [[integral]] calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian [[Moscow Mathematical Papyrus|Moscow papyrus]] ({{Circa|1820&nbsp;BC}}), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning.<ref>{{cite book | first1=Morris | last1=Kline | title=Mathematical thought from ancient to modern times | volume=1 | publisher=Oxford University Press | isbn=978-0-19-506135-2 | pages=18–21 | date=1990-08-16 }}</ref> [[Babylon]]ians may have discovered the [[trapezoidal rule]] while doing astronomical observations of [[Jupiter]].<ref>{{cite journal|last1=Ossendrijver|first1=Mathieu|title=Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph|journal=[[Science (journal)|Science]]|date=29 January 2016|volume=351|issue=6272|pages=482–484|doi=10.1126/science.aad8085|pmid=26823423|bibcode=2016Sci...351..482O|s2cid=206644971}}</ref><ref>{{cite journal |title=Signs of Modern Astronomy Seen in Ancient Babylon |journal=New York Times |year=2016 |first=Kenneth |last=Chang|url=https://www.nytimes.com/2016/01/29/science/babylonians-clay-tablets-geometry-astronomy-jupiter.html?action=click&contentCollection=science&region=rank&module=package&version=highlights&contentPlacement=1&pgtype=sectionfront}}</ref>

====Greece====
{{See also|Ancient Greek mathematics}}
[[File:Parabolic segment and inscribed triangle.svg|thumb|upright=.7|Archimedes used the [[method of exhaustion]] to calculate the area under a parabola in his work ''[[Quadrature of the Parabola]]''.]]
From the age of Greek mathematics, [[Eudoxus of Cnidus|Eudoxus]] (c. 408–355&nbsp;BC) used the [[method of exhaustion]], which foreshadows the concept of the limit, to calculate areas and volumes, while [[Archimedes]] (c. 287–212&nbsp;BC) [[The Method of Mechanical Theorems|developed this idea further]], inventing [[heuristics]] which resemble the methods of integral calculus.<ref>Archimedes, ''Method'', in ''The Works of Archimedes'' {{isbn|978-0-521-66160-7}}</ref> Greek mathematicians are also credited with a significant use of [[infinitesimal]]s. [[Democritus]] is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone's smooth slope prevented him from accepting the idea. At approximately the same time, [[Zeno of Elea]] discredited infinitesimals further by his articulation of the [[Zeno's paradoxes|paradoxes]] which they seemingly create.

Archimedes developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat in his ''[[The Quadrature of the Parabola]]'', ''[[Archimedes use of infinitesimals|The Method]]'', and ''[[On the Sphere and Cylinder]]''.<ref>MathPages — [http://mathpages.com/home/kmath343.htm Archimedes on Spheres & Cylinders] {{Webarchive|url=https://web.archive.org/web/20100103045422/http://mathpages.com/home/kmath343.htm |date=2010-01-03 }}</ref> It should not be thought that infinitesimals were put on a rigorous footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept a proposition as true. It was not until the 17th century that the method was formalized by [[Bonaventura Cavalieri|Cavalieri]] as the [[method of Indivisibles]] and eventually incorporated by [[Isaac Newton|Newton]] into a general framework of [[integral calculus]]. Archimedes was the first to find the tangent to a curve other than a circle, in a method akin to differential calculus. While studying the spiral, he separated a point's motion into two components, one radial motion component and one circular motion component, and then continued to add the two component motions together, thereby finding the tangent to the curve.<ref>{{cite book |first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of Mathematics |edition=2nd |publisher=Wiley |year=1991 |isbn=978-0-471-54397-8 |chapter=Archimedes of Syracuse |pages=[https://archive.org/details/historyofmathema00boye/page/127 127] |quote=Greek mathematics sometimes has been described as essentially static, with little regard for the notion of variability; but Archimedes, in his study of the spiral, seems to have found the tangent to a curve through kinematic considerations akin to differential calculus. Thinking of a point on the spiral 1=''r'' = ''aθ'' as subjected to a double motion — a uniform radial motion away from the origin of coordinates and a circular motion about the origin — he seems to have found (through the parallelogram of velocities) the direction of motion (hence of the tangent to the curve) by noting the resultant of the two component motions. This appears to be the first instance in which a tangent was found to a curve other than a circle.<br/>Archimedes' study of the spiral, a curve that he ascribed to his friend [[Conon of Samos|Conon of Alexandria]], was part of the Greek search for the solution of the three famous problems. |chapter-url=https://archive.org/details/historyofmathema00boye/page/127 }}</ref>

====China====
{{see also|Chinese mathematics}}
The [[method of exhaustion]] was independently invented in [[Chinese mathematics|China]] by [[Liu Hui]] in the 4th century AD in order to find the area of a circle.<ref>{{cite book|series=Chinese studies in the history and philosophy of science and technology|volume=130|title=A comparison of Archimdes' and Liu Hui's studies of circles|first1=Liu|last1=Dun|first2=Dainian|last2=Fan|first3=Robert Sonné|last3=Cohen|publisher=Springer|year=1966|isbn=978-0-7923-3463-7|page=279|url=https://books.google.com/books?id=jaQH6_8Ju-MC}}, [https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 Chapter, p. 279]</ref> In the 5th century, [[Zu Chongzhi]] established a method that would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]].<ref>{{cite book|title=Calculus: Early Transcendentals|edition=3|first1=Dennis G.|last1=Zill|first2=Scott|last2=Wright|first3=Warren S.|last3=Wright|publisher=Jones & Bartlett Learning|year=2009|isbn=978-0-7637-5995-7|page=xxvii|url=https://books.google.com/books?id=R3Hk4Uhb1Z0C}} [https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of page 27]</ref>

===Medieval===
====Middle East====
{{See also|Mathematics in the medieval Islamic world}}
[[file:Ibn al-Haytham crop.jpg|thumb|upright=.7|Ibn al-Haytham, 11th-century Arab mathematician and physicist]]
In the Middle East, [[Ibn al-Haytham|Hasan Ibn al-Haytham]], Latinized as Alhazen ({{c.|lk=no|965|1040}}&nbsp;AD) derived a formula for the sum of [[fourth power]]s. He determined the equations to calculate the area enclosed by the curve represented by <math>y=x^k</math> (which translates to the integral <math>\int x^k \, dx</math> in contemporary notation), for any given non-negative integer value of <math>k</math>.<ref>{{Cite journal |last=Dennis |first=David |last2=Kreinovich |first2=Vladik |last3=Rump |first3=Siegfried M. |date=1998-05-01 |title=Intervals and the Origins of Calculus |url=https://doi.org/10.1023/A:1009989211143 |journal=Reliable Computing |language=en |volume=4 |issue=2 |pages=191–197 |doi=10.1023/A:1009989211143 |issn=1573-1340}}</ref> He used the results to carry out what would now be called an [[Integral|integration]], where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a [[paraboloid]].<ref name=katz>{{Cite journal|last=Katz |first=Victor J. |author-link=Victor J. Katz |date=June 1995 |title=Ideas of Calculus in Islam and India |url=https://www.tandfonline.com/doi/full/10.1080/0025570X.1995.11996307 |journal=[[Mathematics Magazine]] |language=en |volume=68 |issue=3 |pages=163–174 |doi=10.1080/0025570X.1995.11996307 |issn=0025-570X |jstor=2691411}}</ref> [[Roshdi Rashed]] has argued that the 12th century mathematician [[Sharaf al-Dīn al-Tūsī]] must have used the derivative of cubic polynomials in his ''Treatise on Equations''. Rashed's conclusion has been contested by other scholars, who argue that he could have obtained his results by other methods which do not require the derivative of the function to be known.<ref>{{cite journal |last1=Berggren |first1=J. L. |last2=Al-Tūsī |first2=Sharaf Al-Dīn |last3=Rashed |first3=Roshdi |last4=Al-Tusi |first4=Sharaf Al-Din |title=Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt |journal=[[Journal of the American Oriental Society]] |date=April 1990 |volume=110 |issue=2 |pages=304–309 |doi=10.2307/604533|jstor=604533 }}</ref>

====India====
{{see also|Indian mathematics}}
Evidence suggests [[Bhāskara II]] was acquainted with some ideas of differential calculus.<ref>50 Timeless Scientists von K.Krishna Murty</ref> Bhāskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of '[[infinitesimal]]s'.<ref>{{cite journal |last=Shukla |first=Kripa Shankar |year=1984 |title=Use of Calculus in Hindu Mathematics |journal=Indian Journal of History of Science |volume=19 |pages=95–104}}</ref> There is evidence of an early form of [[Rolle's theorem]] in his work, though it was stated without a modern formal proof.<ref>{{Cite web |title=Rolle’s theorem {{!}} Definition, Equation, & Facts {{!}} Britannica |url=https://www.britannica.com/science/Rolles-theorem |access-date=2025-03-02 |website=www.britannica.com |language=en}}</ref><ref>{{cite book |first=Roger |last=Cooke |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |year=1997 |chapter=The Mathematics of the Hindus |pages=[https://archive.org/details/historyofmathema0000cook/page/213 213–215] |isbn=0-471-18082-3 |chapter-url=https://archive.org/details/historyofmathema0000cook/page/213}}</ref> In his astronomical work, Bhāskara gives a result that looks like a precursor to infinitesimal methods: if <math>x \approx y</math> then <math>\sin(y) - \sin(x) \approx (y - x)\cos(y)</math>. This leads to the derivative of the sine function, although he did not develop the notion of a derivative.<ref>{{cite book |first=Roger |last=Cooke |title=The History of Mathematics: A Brief Course |publisher=Wiley-Interscience |year=1997 |chapter=The Mathematics of the Hindus |pages=[https://archive.org/details/historyofmathema0000cook/page/213 213–215] |isbn=0-471-18082-3 |chapter-url=https://archive.org/details/historyofmathema0000cook/page/213}}</ref>

Some ideas on calculus later appeared in Indian mathematics, at the [[Kerala school of astronomy and mathematics]].<ref name=katz/> [[Madhava of Sangamagrama]] in the 14th century, and later mathematicians of the Kerala school, stated components of calculus such as the [[Taylor series]] and [[infinite series]] approximations.<ref>[http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html Indian mathematics<!-- Bot generated title -->]</ref> They considered series equivalent to the Maclaurin expansions of {{tmath|\sin(x)}}, {{tmath|\cos(x)}}, and {{tmath|\arctan(x)}} more than two hundred years before they were studied in Europe. But they did not combine many differing ideas under the two unifying themes of the [[derivative]] and the [[integral]], show the connection between the two, and turn calculus into the powerful problem-solving tool we have today.<ref name=katz/>

====Europe====
The mathematical study of continuity was revived in the 14th century by the [[Oxford Calculators]] and French collaborators such as [[Nicole Oresme]]. They proved the "Merton [[mean speed theorem]]": that a uniformly accelerated body travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body.<ref>{{cite book|first=Carl B. |last=Boyer |author-link=Carl Benjamin Boyer |title=A History of the Calculus and Its Conceptual Development |publisher=Dover |year=1959 |isbn=978-0-486-60509-8 |chapter-url=https://books.google.com/books?id=KLQSHUW8FnUC&pg=PA79 |chapter=III. Medieval Contributions |pages=79–89 |url=https://books.google.com/books?id=KLQSHUW8FnUC}}</ref>

==Modern precursors==
=== Integrals ===
[[Johannes Kepler]]'s work ''Stereometrica Doliorum'' published in 1615 formed the basis of integral calculus.<ref>{{cite web |title=Johannes Kepler: His Life, His Laws and Times |date=24 September 2016 |url=https://www.nasa.gov/kepler/education/johannes |accessdate=2021-06-10 |publisher=NASA |archive-date=2021-06-24 |archive-url=https://web.archive.org/web/20210624003856/https://www.nasa.gov/kepler/education/johannes/ |url-status=dead }}</ref> Kepler developed a method to calculate the area of an ellipse by adding up the lengths of many radii drawn from a focus of the ellipse.<ref name=EB1911>{{cite EB1911 |wstitle=Infinitesimal Calculus/History |display=Infinitesimal Calculus § History |volume=14 |page=537}}</ref>

A significant work was a treatise inspired by Kepler's methods<ref name=EB1911/> published in 1635 by [[Bonaventura Cavalieri]] on his [[method of indivisibles]]. He argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. He discovered [[Cavalieri's quadrature formula]] which gave the area under the curves ''x''<sup>''n''</sup> of higher degree. This had previously been computed in a similar way for the parabola by Archimedes in ''[[The Method of Mechanical Theorems|The Method]]'', but this treatise is believed to have been lost in the 13th century, and was only rediscovered in the early 20th century, and so would have been unknown to Cavalieri. Cavalieri's work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

[[Evangelista Torricelli|Torricelli]] extended Cavalieri's work to other curves such as the [[cycloid]], and then the formula was generalized to fractional and negative powers by Wallis in 1656. In a 1659 treatise, Fermat is credited with an ingenious trick for evaluating the integral of any power function directly.<ref name=quadrature>{{Cite web | last1 = Paradís | first1 = Jaume | last2 = Pla | first2 = Josep | last3 = Viader | first3 = Pelagrí | title = Fermat's Treatise On Quadrature: A New Reading | url = http://www.econ.upf.edu/docs/papers/downloads/775.pdf | access-date = 2008-02-24 | archive-date = 2007-01-07 | archive-url = https://web.archive.org/web/20070107221624/http://www.econ.upf.edu/docs/papers/downloads/775.pdf | url-status = dead }}</ref> Fermat also obtained a technique for finding the centers of gravity of various plane and solid figures, which influenced further work in quadrature.

=== Derivatives ===
In the 17th century, European mathematicians [[Isaac Barrow]], [[René Descartes]], [[Pierre de Fermat]], [[Blaise Pascal]], [[John Wallis]] and others discussed the idea of a [[derivative]]. In particular, in ''Methodus ad disquirendam maximam et minima'' and in ''De tangentibus linearum curvarum'' distributed in 1636, Fermat introduced the concept of [[adequality]], which represented equality up to an infinitesimal error term.<ref>{{cite book|author-link=André Weil |last=Weil |first=André |title=Number theory: An approach through History from Hammurapi to Legendre |location=Boston |publisher=Birkhauser Boston |year=1984 |isbn=0-8176-4565-9 |page=28}}</ref> This method could be used to determine the maxima, minima, and tangents to various curves and was closely related to differentiation.<ref name=Pellegrino>{{cite web | last = Pellegrino | first = Dana | title=Pierre de Fermat | url=http://www.math.rutgers.edu/~cherlin/History/Papers2000/pellegrino.html| access-date=2008-02-24}}</ref>

[[Isaac Newton]] would later write that his own early ideas about calculus came directly from "Fermat's way of drawing tangents."<ref name=Simmons>{{cite book | last = Simmons | first = George F. | title = Calculus Gems: Brief Lives and Memorable Mathematics | url = https://archive.org/details/calculusgemsbrie0000simm | url-access = registration | publisher = Mathematical Association of America | year = 2007 | page = [https://archive.org/details/calculusgemsbrie0000simm/page/98 98] | isbn = 978-0-88385-561-4}}</ref>

=== Fundamental theorem of calculus ===
The formal study of calculus brought together Cavalieri's infinitesimals with the [[calculus of finite differences]] developed in Europe at around the same time, and Fermat's adequality. The combination was achieved by [[John Wallis]], [[Isaac Barrow]], and [[James Gregory (astronomer and mathematician)|James Gregory]], the latter two proving predecessors to the second [[fundamental theorem of calculus]] around 1670.<ref>{{Cite journal|last=Hollingdale|first=Stuart|date=1991|title=Review of Before Newton: The Life and Times of Isaac Barrow|url=https://www.jstor.org/stable/531707|journal=[[Notes and Records of the Royal Society of London]]|volume=45|issue=2|pages=277–279|doi=10.1098/rsnr.1991.0027|issn=0035-9149|jstor=531707|s2cid=165043307|quote=The most interesting to us are Lectures X-XII, in which Barrow comes close to providing a geometrical demonstration of the fundamental theorem of the calculus... He did not realize, however, the full significance of his results, and his rejection of algebra means that his work must remain a piece of mid-17th century geometrical analysis of mainly historic interest.}}</ref><ref>{{Cite journal|last=Bressoud|first=David M.|author-link=David Bressoud|date=2011|title=Historical Reflections on Teaching the Fundamental Theorem of Integral Calculus|url=https://www.tandfonline.com/doi/full/10.4169/amer.math.monthly.118.02.099|journal=[[The American Mathematical Monthly]]|volume=118|issue=2|pages=99|doi=10.4169/amer.math.monthly.118.02.099|s2cid=21473035}}</ref>

[[James Gregory (astronomer and mathematician)|James Gregory]], influenced by Fermat's contributions both to tangency and to quadrature, was then able to prove a restricted version of the second fundamental theorem of calculus, that integrals can be computed using any of a function's antiderivatives.<ref name= sherlock >
See, e.g., Marlow Anderson, Victor J. Katz, Robin J. Wilson, ''Sherlock Holmes in Babylon and Other Tales of Mathematical History'', Mathematical Association of America, 2004, [https://books.google.com/books?id=BKRE5AjRM3AC&pg=PA114 p.&nbsp;114].
</ref><ref name=geometriae>{{cite book| last=Gregory | first=James | title=Geometriae Pars Universalis | url=https://archive.org/details/gregory_universalis | publisher= Patavii: typis heredum Pauli Frambotti | year=1668 | location=[[Museo Galileo]] }}</ref>

The first full proof of the fundamental theorem of calculus was given by [[Isaac Barrow]].<ref name= barrowGeomLect >{{cite book |title=The geometrical lectures of Isaac Barrow, translated, with notes and proofs, and a discussion on the advance made therein on the work of his predecessors in the infinitesimal calculus |publisher=Open Court |location=Chicago |year=1916 |url=https://archive.org/details/geometricallectu00barruoft }} Translator: J. M. Child (1916)</ref>{{rp|p.61 when arc ME ~ arc NH at point of tangency F fig.26}}<ref name= revChildsTranslat >[https://www.ams.org/journals/bull/1918-24-09/S0002-9904-1918-03122-4/S0002-9904-1918-03122-4.pdf Review of J.M. Child's translation (1916) The geometrical lectures of Isaac Barrow] reviewer: Arnold Dresden (Jun 1918) p.454 Barrow has the fundamental theorem of calculus</ref>

=== Other developments ===
[[File:hyperbola E.svg|thumb|Shaded area of one unit square measure when ''x'' = 2.71828... The discovery of [[Euler's number]] e, and its exploitation with functions e<sup>x</sup> and natural logarithm, completed integration theory for calculus of rational functions.]]
One prerequisite to the establishment of a calculus of functions of a [[real number|real]] variable involved finding an [[antiderivative]] for the [[rational function]] <math>f(x) \ = \ \frac{1}{x} .</math> This problem can be phrased as [[quadrature (mathematics)|quadrature]] of the rectangular hyperbola ''xy'' = 1. In 1647 [[Gregoire de Saint-Vincent]] noted that the required function ''F'' satisfied <math>F(st) = F(s) + F(t) ,</math> so that a [[geometric sequence]] became, under ''F'', an [[arithmetic sequence]]. [[A. A. de Sarasa]] associated this feature with contemporary algorithms called ''logarithms'' that economized arithmetic by rendering multiplications into additions. So ''F'' was first known as the [[hyperbolic logarithm]]. After [[Euler]] exploited e = 2.71828..., and ''F'' was identified as the [[inverse function]] of the [[exponential function]], it became the [[natural logarithm]], satisfying <math>\frac{dF}{dx} \ = \ \frac{1}{x} .</math>

The first proof of [[Rolle's theorem]] was given by [[Michel Rolle]] in 1691 using methods developed by the Dutch mathematician [[Johann van Waveren Hudde]].<ref>{{cite book
|title=A Transition to Advanced Mathematics: A Survey Course
|first1=William
|last1=Johnston
|first2=Alex
|last2=McAllister
|publisher=Oxford University Press US
|year=2009
|isbn=978-0-19-531076-4
|page=333
|url=https://books.google.com/books?id=LV21vHwnkpIC}}, [https://books.google.com/books?id=LV21vHwnkpIC&pg=PA333 Chapter 4, p. 333]
</ref> The mean value theorem in its modern form was stated by [[Bernard Bolzano]] and [[Augustin-Louis Cauchy]] (1789–1857) also after the founding of modern calculus. Important contributions were made by Barrow, [[Christiaan Huygens|Huygens]], and many others.

Barrow has been credited by some authors as having invented calculus, however, Swiss mathematician [[Florian Cajori]] notes that while Barrow did work out a set of "geometric theorems suggesting to us constructions by which we can find lines, areas and volumes whose magnitudes are ordinarily found by the analytical processes of the calculus", he did not create "what by common agreement of mathematicians has been designated by the term differential and integral calculus", and further notes that "Two processes yielding equivalent results are not necessarily the same". Cajori finishes with stating that "The invention belongs rightly belongs to Newton and Leibniz".<ref>{{Cite journal |last=Cajori |first=Florian |author-link=Florian Cajori |date=1919 |title=Who Was the First Inventor of the Calculus? |url=http://www.jstor.org/stable/2974042?origin=crossref |journal=The American Mathematical Monthly |volume=26 |issue=1 |pages=15 |doi=10.2307/2974042}}</ref>

==Newton and Leibniz==
{{See also|Leibniz–Newton calculus controversy}}
[[File:GodfreyKneller-IsaacNewton-1689.jpg|thumb|upright=.7|[[Isaac Newton]]]]
[[File:Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg|thumb|upright=.7|[[Gottfried Leibniz]]]]

Before [[Isaac Newton|Newton]] and [[Gottfried Wilhelm Leibniz|Leibniz]], the word "calculus" referred to any body of mathematics, but in the following years, "calculus" became a popular term for a field of mathematics based upon their insights.<ref>{{harvnb|Reyes|2004|p=160}}</ref> Newton and Leibniz, building on this work, independently developed the surrounding theory of infinitesimal calculus in the late 17th century. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton provided some of the most important applications to physics, especially of [[integral calculus]].

By the middle of the 17th century, European mathematics had changed its primary repository of knowledge. In comparison to the last century which maintained [[Hellenistic]] mathematics as the starting point for research, Newton, Leibniz and their contemporaries increasingly looked towards the works of more modern thinkers.<ref>Such as Kepler, Descartes, Fermat, Pascal and Wallis. {{harvnb|Calinger|1999|p=556}}</ref>

Newton came to calculus as part of his investigations in [[physics]] and [[geometry]]. He viewed calculus as the scientific description of the generation of motion and [[magnitude (mathematics)|magnitude]]s. In comparison, Leibniz focused on the [[tangent problem]] and came to believe that calculus was a [[metaphysics|metaphysical]] explanation of change. Importantly, the core of their insight was the formalization of the inverse properties between the [[integral]] and the [[differential of a function]]. This insight had been anticipated by their predecessors, but they were the first to conceive calculus as a system in which new rhetoric and descriptive terms were created.<ref>Foremost among these was [[Isaac Barrow|Barrow]] who had created formulas for specific cases and Fermat who created a similar definition for the derivative. For more information; Boyer 184</ref>

===Newton===
Newton completed no definitive publication formalizing his [[fluxion]]al calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as the ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' and ''[[Opticks]]''. Newton would begin his mathematical training as the chosen heir of [[Isaac Barrow]] in [[Cambridge]]. His aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing the [[binomial theorem]], which he had extended to include fractional and negative [[Exponentiation|exponent]]s. Newton succeeded in expanding the applicability of the binomial theorem by applying the algebra of finite quantities in an analysis of [[infinite series]]. He showed a willingness to view infinite series not only as approximate devices, but also as alternative forms of expressing a term.<ref>{{harvnb|Calinger|1999|p=610}}</ref>

Newton's formulated his calculus between 1664-66,<ref>{{Cite book |last=Grattan-Guinness |first=Ivor |author-link=Ivor Grattan-Guinness |url=https://books.google.com/books?id=oej5DwAAQBAJ&pg=PA49 |title=From the Calculus to Set Theory 1630-1910: An Introductory History |date= |publisher=[[Princeton University Press]] |year=1980 |isbn=978-0-691-07082-7 |location= |pages=49 |language=en}}</ref><ref>{{Cite book |last=Guicciardini |first=Niccolò |author-link=Niccolò Guicciardini |url=https://books.google.com/books?id=uqnuDwAAQBAJ&pg=PA331 |title=Isaac Newton on Mathematical Certainty and Method |date=2009 |publisher=MIT Press |isbn=978-0-262-01317-8 |series= |location= |pages=331}}</ref><ref>{{cite web|last=Newton|first=Isaac|title=Waste Book|url=http://cudl.lib.cam.ac.uk/view/MS-ADD-04004/|access-date=10 January 2012}}</ref> later describing them as, "the prime of my age for invention and minded mathematics and [natural] philosophy more than at any time since." A manuscript dated May 20, 1665 showed that Newton "had already developed the calculus to the point where he could compute the tangent and the curvature at any point of a continuous curve."<ref>{{Cite book |last=Press |first=S. James |url=https://books.google.com/books?id=aAJYCwAAQBAJ&pg=PA88 |title=The Subjectivity of Scientists and the Bayesian Approach |last2=Tanur |first2=Judith M. |date=2016 |publisher=Dover Publications, Inc |isbn=978-0-486-80284-8 |edition= |location= |pages=88}}</ref> It was during his plague-induced isolation that the first written conception of [[Method of Fluxions|fluxionary calculus]] was recorded in the unpublished ''[[De analysi per aequationes numero terminorum infinitas|De Analysi per Aequationes Numero Terminorum Infinitas]]''. In this paper, he determined the area under a [[curve]] by first calculating a momentary rate of change and then extrapolating the total area. He began by reasoning about an indefinitely small triangle whose area is a function of ''x'' and ''y''. He then reasoned that the [[infinitesimal]] increase in the abscissa will create a new formula where {{nowrap|1=''x'' = ''x'' + ''o''}} (importantly, ''o'' is the letter, not the [[Numerical digit|digit]] 0). He then recalculated the area with the aid of the binomial theorem, removed all quantities containing the letter ''o'' and re-formed an algebraic expression for the area. Significantly, Newton would then "blot out" the quantities containing ''o'' because terms "multiplied by it will be nothing in respect to the rest".

At this point Newton had begun to realize the central property of inversion. He had created an expression for the area under a curve by considering a momentary increase at a point. In effect, the [[fundamental theorem of calculus]] was built into his calculations. While his new formulation offered incredible potential, Newton was well aware of its logical limitations at the time. He admits that "errors are not to be disregarded in mathematics, no matter how small" and that what he had achieved was "shortly explained rather than accurately demonstrated".

In an effort to give calculus a more rigorous explication and framework, Newton compiled in 1671 the ''[[Methodus Fluxionum et Serierum Infinitarum]]''. In this book, Newton's strict [[empiricism]] shaped and defined his fluxional calculus. He exploited [[instantaneous velocity|instantaneous]] [[Motion (physics)|motion]] and infinitesimals informally. He used math as a [[methodological]] tool to explain the physical world. The base of Newton's revised calculus became continuity; as such he redefined his calculations in terms of continual flowing motion. For Newton, variable magnitudes are not aggregates of infinitesimal elements, but are generated by the indisputable fact of motion. As with many of his works, Newton delayed publication. ''Methodus Fluxionum'' was not published until 1736.<ref>{{cite book|last=Eves|first=Howard|title=An introduction to the history of mathematics, 6th edition|page=400}}</ref>

Newton attempted to avoid the use of the infinitesimal by forming calculations based on [[ratio]]s of changes. In the ''Methodus Fluxionum'' he defined the rate of generated change as a [[fluxion]], which he represented by a dotted letter, and the quantity generated he defined as a [[fluent (mathematics)|fluent]]. For example, if <math>{x}</math> and <math>{y}</math> are fluents, then <math>\dot{x}</math> and <math>\dot{y}</math> are their respective fluxions. This revised calculus of ratios continued to be developed and was maturely stated in the 1676 text ''De Quadratura Curvarum'' where Newton came to define the present day derivative as the ultimate ratio of change, which he defined as the ratio between evanescent increments (the ratio of fluxions) purely at the moment in question. Essentially, the ultimate ratio is the ratio as the increments vanish into nothingness. Importantly, Newton explained the existence of the ultimate ratio by appealing to motion:<ref>''Principia'', [[Florian Cajori]] 8</ref><blockquote>For by the ultimate velocity is meant that, with which the body is moved, neither before it arrives at its last place, when the motion ceases nor after but at the very instant when it arrives... the ultimate ratio of evanescent quantities is to be understood, the ratio of quantities not before they vanish, not after, but with which they vanish</blockquote>Newton developed his fluxional calculus in an attempt to evade the informal use of infinitesimals in his calculations.

Historian [[A. Rupert Hall]] noted Newton's rapid development of calculus in comparison to contemporaries, stating that Newton "well before 1690 . . . had reached roughly the point in the development of the calculus that Leibniz, the two Bernoullis, L’Hospital, Hermann and others had by joint efforts reached in print by the early 1700s".<ref>{{Cite book |last=Hall |first=A. Rupert |url=https://books.google.com/books?id=DJnquszl8CUC&pg=PA136 |title=Philosophers at War: The Quarrel Between Newton and Leibniz |date=1980 |publisher=Cambridge University Press |isbn=978-0-521-22732-2 |location= |pages=136}}</ref>

===Leibniz===
[[File:Leibniz-Acta-1684-NovaMethodus.png|thumb|Leibniz: ''Nova methodus pro maximis et minimis'', Acta Eruditorum, Leipzig, October 1684. First page of Leibniz' publication of the differential calculus.]]
[[File:Leibniz, Gottfried Wilhelm von – Nova methodus pro maximis et minimis - Acta Eruditorum - Tabula XII - Graphs, 1684.jpg|thumb|Graphs referenced in Leibniz' article of 1684]]
While Newton began development of his fluxional calculus in 1665–1666 his findings did not become widely circulated until later. In the intervening years Leibniz also strove to create his calculus. In comparison to Newton who came to math at an early age, Leibniz began his rigorous math studies with a mature intellect. He was a [[polymath]], and his intellectual interests and achievements involved [[metaphysics]], [[law]], [[economics]], [[politics]], [[logic]], and [[mathematics]]. In order to understand Leibniz's reasoning in calculus his background should be kept in mind. Particularly, his [[metaphysics]] which described the universe as a [[Monadology]], and his plans of creating a precise formal logic whereby, "a general method in which all truths of the reason would be reduced to a kind of calculation".<ref>{{Cite book|chapter-url=https://plato.stanford.edu/entries/leibniz/|title = The Stanford Encyclopedia of Philosophy|chapter = Gottfried Wilhelm Leibniz|year = 2020|publisher = Metaphysics Research Lab, Stanford University}}</ref>

In 1672, Leibniz met the mathematician [[Christiaan Huygens|Huygens]] who convinced Leibniz to dedicate significant time to the study of mathematics. By 1673 he had progressed to reading [[Blaise Pascal|Pascal]]'s ''[[Traité des sinus du quart de cercle]]'' and it was during his largely [[autodidactic]] research that Leibniz said "a light turned on". Like Newton, Leibniz saw the tangent as a ratio but declared it as simply the ratio between [[ordinate]]s and [[abscissa]]s. He continued this reasoning to argue that the [[integral]] was in fact the sum of the ordinates for infinitesimal intervals in the abscissa; in effect, the sum of an infinite number of rectangles. From these definitions the inverse relationship or differential became clear and Leibniz quickly realized the potential to form a whole new system of mathematics. Where Newton over the course of his career used several approaches in addition to an approach using [[infinitesimal]]s, Leibniz made this the cornerstone of his notation and calculus.<ref>{{Cite web|url=https://mathshistory.st-andrews.ac.uk/Biographies/Leibniz/|title = Gottfried Leibniz - Biography}}</ref><ref>{{Cite web|url=https://www.britannica.com/biography/Gottfried-Wilhelm-Leibniz|title = Gottfried Wilhelm Leibniz {{pipe}} Biography & Facts}}</ref>

In the manuscripts of 25 October to 11 November 1675, Leibniz recorded his discoveries and experiments with various forms of notation. He was acutely aware of the notational terms used and his earlier plans to form a precise logical [[symbol]]ism became evident. Eventually, Leibniz denoted the infinitesimal increments of abscissas and ordinates ''dx'' and ''dy'', and the summation of infinitely many infinitesimally thin rectangles as a [[long s#Modern usage|long s]] (∫&nbsp;), which became the present integral symbol <math>\scriptstyle\int</math>.

While Leibniz's notation is used by modern mathematics, his logical base was different from our current one. Leibniz embraced infinitesimals and wrote extensively so as, "not to make of the infinitely small a mystery, as had Pascal."<ref>{{Cite book |last=Boyer |first=Carl |url=https://archive.org/details/the-history-of-the-calculus-carl-b.-boyer/page/209 |title=The History of the Calculus and Its Conceptual Development |publisher=Courier Corporation |year=1939 |isbn=9780486605098 |pages=209}}</ref> According to [[Gilles Deleuze]], Leibniz's zeroes "are nothings, but they are not absolute nothings, they are nothings respectively" (quoting Leibniz' text "Justification of the calculus of infinitesimals by the calculus of ordinary algebra").<ref name=Deleuze>{{cite web|last=Deleuze|first=Gilles|title=DELEUZE / LEIBNIZ Cours Vincennes - 22/04/1980|url=http://www.webdeleuze.com/php/texte.php?cle=53&groupe=Leibniz&langue=2|access-date=30 April 2013|archive-date=11 September 2012|archive-url=https://web.archive.org/web/20120911094850/http://www.webdeleuze.com/php/texte.php?cle=53&groupe=Leibniz&langue=2|url-status=dead}}</ref> Alternatively, he defines them as, "less than any given quantity". For Leibniz, the world was an aggregate of infinitesimal points and the lack of scientific proof for their existence did not trouble him. Infinitesimals to Leibniz were ideal quantities of a different type from appreciable numbers. The truth of continuity was proven by existence itself. For Leibniz the principle of continuity and thus the validity of his calculus was assured. Three hundred years after Leibniz's work, [[Abraham Robinson]] showed that using infinitesimal quantities in calculus could be given a solid foundation.<ref>{{Cite web|url=https://www.sjsu.edu/faculty/watkins/infincalc.htm|title = The Calculus of Infinitesimals}}</ref>

===Legacy===
The rise of calculus stands out as a unique moment in mathematics. Calculus is the mathematics of motion and change, and as such, its invention required the creation of a new mathematical system. Importantly, Newton and Leibniz did not create the same calculus and they did not conceive of modern calculus. While they were both involved in the process of creating a mathematical system to deal with variable quantities their elementary base was different. For Newton, change was a variable quantity over time and for Leibniz it was the difference ranging over a sequence of infinitely close values. Notably, the descriptive terms each system created to describe change was different. 

Historically, there was much debate over whether it was Newton or Leibniz who first "invented" calculus. This argument, the [[Leibniz and Newton calculus controversy]], involving Leibniz, who was German, and the Englishman Newton, led to a rift in the European mathematical community lasting over a century. Leibniz was the first to publish his investigations; however, it is well established that Newton had started his work several years prior to Leibniz and had already developed a theory of [[tangent]]s by the time Leibniz became interested in the question.
It is not known how much this may have influenced Leibniz. The initial accusations were made by students and supporters of the two great scientists at the turn of the century, but after 1711 both of them became personally involved, accusing each other of [[plagiarism]].

The priority dispute had an effect of separating English-speaking mathematicians from those in continental Europe for many years. Only in the 1820s, due to the efforts of the [[Analytical Society]], did [[Leibnizian analytical calculus]] become accepted in England. Today, both Newton and Leibniz are given credit for independently developing the basics of calculus. It is Leibniz, however, who is credited with giving the new discipline the name it is known by today: "calculus". Newton's name for it was "the science of [[fluent (mathematics)|fluent]]s and [[Method of Fluxions|fluxion]]s".

While neither of the two offered convincing logical foundations for their calculus according to mathematician [[Carl Benjamin Boyer|Carl B. Boyer]], Newton came the closest, with his best attempt coming in ''Principia'', where he described his idea of "prime and ultimate ratios" and came extraordinarily close to the [[Limit (mathematics)|limit]], and his ratio of velocities corresponded to a single [[real number]], which would not be fully defined until the late nineteenth century.<ref name=":0">{{Cite journal |last=Boyer |first=Carl B. |date=1970 |title=The History of the Calculus |url=https://www.jstor.org/stable/3027118 |journal=The Two-Year College Mathematics Journal |volume=1 |issue=1 |pages=60–86 |doi=10.2307/3027118 |jstor=3027118 |issn=0049-4925}}</ref> On the other hand, the calculus of Leibniz was from a "logical point of view, distinctly inferior to that of Newton, for it never transcended the view of <math>\frac{dy}{dx}</math> as a quotient of infinitely small changes or differences in ''y'' and ''x''."<ref name=":0" /> However, heuristically, it was a success, despite being a "failure" from a logical point of view.<ref name=":0" />

[[File:Maria Gaetana Agnesi.jpg|thumb|upright=.7|[[Maria Gaetana Agnesi]]]]
The work of both Newton and Leibniz is reflected in the notation used today. Newton introduced the notation <math>\dot{f}</math> for the [[derivative (mathematics)|derivative]] of a function ''f''.<ref>The use of prime to denote the [[derivative]], <math> f'\left(x\right),</math> is due to Lagrange.</ref> Leibniz introduced the symbol <math>\int</math> for the [[integral]] and wrote the [[derivative]] of a function ''y'' of the variable ''x'' as <math>\frac{dy}{dx}</math>, both of which are still in use.

Since the time of Leibniz and Newton, many mathematicians have contributed to the continuing development of calculus. One of the first and most complete works on both infinitesimal and [[integral calculus]] was written in 1748 by [[Maria Gaetana Agnesi]].<ref>{{cite book |title=A Biography of Maria Gaetana Agnesi, an Eighteenth-century Woman Mathematician |edition=illustrated |first1=Antonella |last1=Cupillari |publisher=Edwin Mellen Press |year=2007 |isbn=978-0-7734-5226-8 |page=iii |title-link=A Biography of Maria Gaetana Agnesi|contributor-last=Allaire|contributor-first=Patricia R.|contribution=Foreword}}</ref><ref>{{cite web| url=http://www.agnesscott.edu/lriddle/women/agnesi.htm| title=Maria Gaetana Agnesi| first=Elif| last=Unlu|date=April 1995| publisher =[[Agnes Scott College]]}}</ref>

==Developments==

===Calculus of variations===
The [[calculus of variations]] began with the work of [[Isaac Newton]], such as with [[Newton's minimal resistance problem]],<ref>{{Cite journal |last=Torres |first=Delfim F. M. |date=2021-07-29 |title=On a Non-Newtonian Calculus of Variations |journal=Axioms |language=en |volume=10 |issue=3 |pages=171 |arxiv=2107.14152 |doi=10.3390/axioms10030171 |issn=2075-1680 |doi-access=free}}</ref> which Newton formulated and solved in 1685, and later published in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia]]'' in 1687,<ref name=":2">{{Cite book |last=Goldstine |first=Herman H. |url=https://books.google.com/books?id=_iTnBwAAQBAJ&pg=PA7 |title=A History of the Calculus of Variations from the 17th Through the 19th Century |date=1980 |publisher=Springer New York |isbn=978-1-4613-8106-8 |series= |location= |pages=7–21}}</ref> and which was the first problem in the field to be formulated and correctly solved, and was one of the most difficult problems tackled by variational methods prior to the twentieth century.<ref name=":2" /><ref name=":02">{{Citation |last=Ferguson |first=James |title=A Brief Survey of the History of the Calculus of Variations and its Applications |date=2004 |arxiv=math/0402357 |bibcode=2004math......2357F}}</ref><ref name=":1">{{Cite book |last=Rowlands |first=Peter |url=https://books.google.com/books?id=ipA4DwAAQBAJ&pg=PA36 |title=Newton and the Great World System |date=2017 |publisher=[[World Scientific Publishing]] |isbn=978-1-78634-372-7 |pages=36–39 |language=en |doi=10.1142/q0108}}</ref> It was followed by the [[brachistochrone curve]] problem of [[Johann Bernoulli]] (1696), which Bernoulli solved, using the principle of least time, but not the calculus of variations, whereas Newton did to solve it in 1697, thus he pioneered the field with his work on the two problems.<ref name=":1" /> The problem was similar to one raised by [[Galileo Galilei]] in 1638, but he did not solve the problem explicity nor did he use the methods based on calculus.<ref name=":02" /> It immediately occupied the attention of [[Jakob Bernoulli]] but [[Leonhard Euler]] first elaborated the subject. His contributions began in 1733, and his ''Elementa Calculi Variationum'' gave to the science its name. [[Joseph Louis Lagrange]] contributed extensively to the theory, and [[Adrien-Marie Legendre]] (1786) laid down a method, not entirely satisfactory, for the discrimination of maxima and minima. To this discrimination Brunacci (1810), [[Carl Friedrich Gauss]] (1829), [[Siméon Denis Poisson]] (1831), [[Mikhail Vasilievich Ostrogradsky]] (1834), and [[Carl Gustav Jacob Jacobi]] (1837) have been among the contributors. An important general work is that of Sarrus (1842) which was condensed and improved by [[Augustin Louis Cauchy]] (1844). Other valuable treatises and memoirs have been written by Strauch (1849), Jellett (1850), [[Otto Hesse]] (1857), [[Alfred Clebsch]] (1858), and Carll (1885), but perhaps the most important work of the century is that of [[Karl Weierstrass]]. His course on the theory may be asserted to be the first to place calculus on a firm and rigorous foundation.

===Operational methods===
{{Main|Operational calculus}}
[[Antoine Arbogast]] (1800) was the first to separate the symbol of operation from that of quantity in a differential equation. [[Francois-Joseph Servois]] (1814) seems to have been the first to give correct rules on the subject. [[Charles James Hargreave]] (1848) applied these methods in his memoir on differential equations, and [[George Boole]] freely employed them. [[Hermann Grassmann]] and [[Hermann Hankel]] made great use of the theory, the former in studying [[equation]]s, the latter in his theory of [[complex number]]s.

===Integrals===
[[Niels Henrik Abel]] seems to have been the first to consider in a general way the question as to what [[differential equation]]s can be integrated in a finite form by the aid of ordinary functions, an investigation extended by [[Joseph Liouville|Liouville]]. [[Augustin Louis Cauchy|Cauchy]] early undertook the general theory of determining [[definite integral]]s, and the subject has been prominent during the 19th century. [[Frullani integral]]s, [[David Bierens de Haan]]'s work on the theory and his elaborate tables, [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]]'s lectures embodied in [[Friedrich Wilhelm Franz Meyer|Meyer]]'s treatise, and numerous memoirs of [[Adrien-Marie Legendre|Legendre]], [[Siméon Denis Poisson|Poisson]], [[Giovanni Antonio Amedeo Plana|Plana]], [[Joseph Ludwig Raabe|Raabe]], [[Leonhard Sohncke|Sohncke]], [[Oscar Xavier Schlömilch|Schlömilch]], [[Edwin Bailey Elliott|Elliott]], [[Charles Leudesdorf|Leudesdorf]] and [[Leopold Kronecker|Kronecker]] are among the noteworthy contributions.

[[Euler integral (disambiguation)|Eulerian integrals]] were first studied by [[Leonhard Euler|Euler]] and afterwards investigated by Legendre, by whom they were classed as Eulerian integrals of the first and second species, as follows:

:<math>\int_0^1 x^{n-1}(1 - x)^{n-1} \, dx</math>

:<math>\int_0^\infty e^{-x} x^{n-1} \, dx</math>

although these were not the exact forms of Euler's study.

If ''n'' is a positive [[integer]]:
:<math>\int_0^\infty e^{-x}x^{n-1}dx = (n-1)!,</math>
but the integral converges for all positive real <math>n</math> and defines an [[analytic continuation]] of the [[factorial]] function to all of the [[complex plane]] except for poles at zero and the negative integers. To it Legendre assigned the symbol <math>\Gamma</math>, and it is now called the [[gamma function]]. Besides being analytic over positive reals <math>\mathbb{R}^{+}</math>, <math>\Gamma</math> also enjoys the uniquely defining property that <math>\log \Gamma</math> is [[Convex function|convex]], which aesthetically justifies this analytic continuation of the factorial function over any other analytic continuation. To the subject [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]] has contributed an important theorem (Liouville, 1839), which has been elaborated by [[Joseph Liouville|Liouville]], [[Eugène Charles Catalan|Catalan]], [[Leslie Ellis]], and others. [[Joseph Ludwig Raabe|Raabe]] (1843–44), Bauer (1859), and [[Christoph Gudermann|Gudermann]] (1845) have written about the evaluation of <math>\Gamma (x)</math> and <math>\log \Gamma (x)</math>. Legendre's great table appeared in 1816.

==Applications==
The application of the [[infinitesimal calculus]] to problems in [[physics]] and [[astronomy]] was contemporary with the origin of the science. All through the 18th century these applications were multiplied, until at its close [[Pierre-Simon Laplace|Laplace]] and [[Joseph Louis Lagrange|Lagrange]] had brought the whole range of the study of forces into the realm of analysis. To [[Joseph Louis Lagrange|Lagrange]] (1773) we owe the introduction of the theory of the potential into dynamics, although the name "[[Potential function (disambiguation)|potential function]]" and the fundamental memoir of the subject are due to [[George Green (mathematician)|Green]] (1827, printed in 1828). The name "[[potential]]" is due to [[Carl Friedrich Gauss|Gauss]] (1840), and the distinction between potential and potential function to [[Rudolf Julius Emanuel Clausius|Clausius]]. With its development are connected the names of [[Peter Gustav Lejeune Dirichlet|Lejeune Dirichlet]], [[Bernhard Riemann|Riemann]], [[John von Neumann|von Neumann]], [[Eduard Heine|Heine]], [[Leopold Kronecker|Kronecker]], [[Rudolf Lipschitz|Lipschitz]], [[Elwin Bruno Christoffel|Christoffel]], [[Gustav Kirchhoff|Kirchhoff]], [[Eugenio Beltrami|Beltrami]], and many of the leading physicists of the century.

It is impossible in this article to enter into the great variety of other applications of analysis to physical problems. Among them are the investigations of Euler on vibrating chords; [[Sophie Germain]] on elastic membranes; Poisson, [[Gabriel Lamé|Lamé]], [[Jean Claude Saint-Venant|Saint-Venant]], and [[Alfred Clebsch|Clebsch]] on the [[theory of elasticity|elasticity]] of three-dimensional bodies; [[Jean Baptiste Joseph Fourier|Fourier]] on [[heat]] diffusion; [[Augustin-Jean Fresnel|Fresnel]] on [[light]]; [[James Clerk Maxwell|Maxwell]], [[Hermann von Helmholtz|Helmholtz]], and [[Heinrich Rudolf Hertz|Hertz]] on [[electricity]]; Hansen, Hill, and [[Hugo Gyldén|Gyldén]] on [[astronomy]]; Maxwell on [[spherical harmonic]]s; [[John Strutt, 3rd Baron Rayleigh|Lord Rayleigh]] on [[acoustics]]; and the contributions of Lejeune Dirichlet, [[Wilhelm Eduard Weber|Weber]], [[Gustav Kirchhoff|Kirchhoff]], [[Franz Ernst Neumann|F. Neumann]], [[William Thomson, 1st Baron Kelvin|Lord Kelvin]], [[Rudolf Julius Emanuel Clausius|Clausius]], [[Vilhelm Bjerknes|Bjerknes]], [[James MacCullagh|MacCullagh]], and Fuhrmann to physics in general. The labors of Helmholtz should be especially mentioned, since he contributed to the theories of dynamics, electricity, etc., and brought his great analytical powers to bear on the fundamental axioms of mechanics as well as on those of pure mathematics.

Furthermore, infinitesimal calculus was introduced into the social sciences, starting with [[Neoclassical economics]]. Today, it is a valuable tool in mainstream economics.

==See also==
* [[Analytic geometry]]
* [[History of logarithms]]
* [[Nonstandard calculus]]

==Notes==
{{Reflist}}

===Sources===
*{{cite book |last=Calinger |first=Ronald |title=A Contextual History of Mathematics |publisher=Prentice-Hall |year=1999 |isbn=978-0-02-318285-3 |oclc=40479696}}
*{{cite journal |last=Reyes |first=Mitchell |title=The Rhetoric in Mathematics: Newton, Leibniz, the Calculus, and the Rhetorical Force of the Infinitesimal |journal=[[Quarterly Journal of Speech]] |volume=90 |issue=2 |pages=159–184 |year=2004 |doi=10.1080/0033563042000227427 |s2cid=145802382 }}

==Further reading==
*{{cite book |last1=Grattan-Guinness |first1=Ivor |author-link=Ivor Grattan-Guinness |title=The Rainbow of Mathematics : A History of the Mathematical Sciences |date=2000 |publisher=W.W. Norton |isbn=978-0-393-32030-5 |oclc=44155819 |zbl=0876.01003}}. Ch. 5 The age of trigonometry : Europe, 1540–1660 and Ch. 6 The calculus and its consequences, 1660–1750.
*{{cite thesis |type=M.A. |last=Hoffman |first= Ruth Irene |title=On the development and use of the concepts of the infinitesimal calculus before Newton and Leibniz |date=1937 |publisher=University of Colorado |oclc=48160073}}
*{{cite book |last=Roero |first=C.S. |chapter-url=https://books.google.com/books?id=UdGBy8iLpocC&pg=PA46 |chapter=Gottfried Wilhelm Leibniz, first three papers on the calculus (1684, 1686, 1693) |editor=Grattan-Guinness, I. |title=Landmark writings in Western mathematics 1640–1940 |url=https://books.google.com/books?id=UdGBy8iLpocC |year=2005 |publisher=Elsevier |isbn=978-0-444-50871-3 |pages=46–58 |editor-link=Ivor Grattan-Guinness |zbl=1090.01002}}
*{{cite journal |last=Roero |first=C.S. |title=Jakob Bernoulli, attentive student of the work of Archimedes: marginal notes to the edition of Barrow |journal=Boll. Storia Sci. Mat. |volume=3 |issue=1 |pages=77–125 |year=1983 |issn=0392-4432 }}

==External links==
{{wikiquotes}}
* [http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html A history of the calculus in The MacTutor History of Mathematics archive], 1996.
* [http://www.economics.soton.ac.uk/staff/aldrich/Calculus%20and%20Analysis%20Earliest%20Uses.htm Earliest Known Uses of Some of the Words of Mathematics: Calculus & Analysis]
* [http://cudl.lib.cam.ac.uk/collections/newton Newton Papers, Cambridge University Digital Library]
* {{in lang|en|ar}} [http://www.wdl.org/en/item/4327/ The Excursion of Calculus], 1772
{{History of science}}
{{History of mathematics}}

[[Category:History of calculus| ]]