Editing Calculus (section)

== History ==
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{{Main|History of calculus}}

Modern calculus was developed in 17th-century Europe by [[Isaac Newton]] and [[Gottfried Wilhelm Leibniz]] (independently of each other, first publishing around the same time) but elements of it first appeared in ancient Egypt and later Greece, then in China<!-- Alphabetically, so please don't change the order, thank you --> and the Middle East, and still later again in medieval Europe and India.

=== Ancient precursors ===

==== Egypt ====
Calculations of [[volume]] and [[area]], one goal of integral calculus, can be found in the [[Egyptian mathematics|Egyptian]] [[Moscow Mathematical Papyrus|Moscow papyrus]] ({{circa|1820{{nbsp}}BC}}), but the formulae are simple instructions, with no indication as to how they were obtained.<ref>{{Cite book |last=Kline |first=Morris |url=https://books.google.com/books?id=wKsYrT691yIC |title=Mathematical Thought from Ancient to Modern Times: Volume 1 |year=1990 |publisher=Oxford University Press |isbn=978-0-19-506135-2 |pages=15–21 |language=en |author-link=Morris Kline |access-date=20 February 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150420/https://books.google.com/books?id=wKsYrT691yIC |url-status=live }}</ref><ref>{{Cite book |last=Imhausen |first=Annette |title=Mathematics in Ancient Egypt: A Contextual History |title-link=Mathematics in Ancient Egypt: A Contextual History |date=2016 |publisher=Princeton University Press |isbn=978-1-4008-7430-9 |page=112 |oclc=934433864 |author-link=Annette Imhausen}}</ref>

==== Greece ====
{{See also|Greek mathematics}}
[[File:Parabolic segment and inscribed triangle.svg|thumb|upright|right|Archimedes used the [[method of exhaustion]] to calculate the area under a parabola in his work ''[[Quadrature of the Parabola]]''.]]
Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician [[Eudoxus of Cnidus]] ({{circa|390–337{{nbsp}}BC|lk=no}}) developed  the [[method of exhaustion]] to prove the formulas for cone and pyramid volumes.

During the [[Hellenistic period]], this method was further developed by [[Archimedes]] ({{c.|lk=no|287|212}}{{nbsp}}BC), who combined it with a concept of the [[Cavalieri's principle|indivisibles]]—a precursor to [[Archimedes use of infinitesimals|infinitesimals]]—allowing him to solve several problems now treated by integral calculus. In ''[[The Method of Mechanical Theorems]]'' he describes, for example, calculating the [[center of gravity]] of a solid [[Sphere|hemisphere]], the center of gravity of a [[frustum]] of a circular [[paraboloid]], and the area of a region bounded by a [[parabola]] and one of its [[secant line]]s.<ref>See, for example:
* {{Cite web |last=Powers |first=J. |date=2020 |title="Did Archimedes do calculus?" |url=https://www.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2020-Jeffery%20Powers.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.maa.org/sites/default/files/images/upload_library/46/HOMSIGMAA/2020-Jeffery%20Powers.pdf |archive-date=9 October 2022 |url-status=live |website=[[Mathematical Association of America]] }}
* {{cite book |last=Jullien |first=Vincent |chapter=Archimedes and Indivisibles |date=2015 |doi=10.1007/978-3-319-00131-9_18 |title=Seventeenth-Century Indivisibles Revisited |pages=451–457 |place=Cham |publisher=Springer International Publishing |series=Science Networks. Historical Studies |volume=49 |isbn=978-3-319-00130-2  |issn = 1421-6329}}
* {{Cite web |last=Plummer |first=Brad |date=9 August 2006 |title=Modern X-ray technology reveals Archimedes' math theory under forged painting |url=http://news.stanford.edu/news/2006/august9/arch-080906.html |access-date=28 February 2022 |website=Stanford University |language=en |archive-date=20 January 2022 |archive-url=https://web.archive.org/web/20220120065134/https://news.stanford.edu/news/2006/august9/arch-080906.html |url-status=live }}
* {{cite book|author=Archimedes |title=The Works of Archimedes, Volume 1: The Two Books On the Sphere and the Cylinder |isbn=978-0-521-66160-7 |translator-first=Reviel |translator-last=Netz |publisher=Cambridge University Press |year=2004}}
* {{Cite journal |last1=Gray |first1=Shirley |last2=Waldman |first2=Cye H. |date=20 October 2018 |title=Archimedes Redux: Center of Mass Applications from The Method |journal=The College Mathematics Journal |volume=49 |issue=5 |pages=346–352 |doi=10.1080/07468342.2018.1524647 |issn=0746-8342 |s2cid=125411353}}</ref>

==== China ====
The method of exhaustion was later discovered independently in [[Chinese mathematics|China]] by [[Liu Hui]] in the 3rd century AD 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|year=1966|isbn=978-0-7923-3463-7|page=279|publisher=Springer |url=https://books.google.com/books?id=jaQH6_8Ju-MC|access-date=15 November 2015|archive-date=1 March 2023|archive-url=https://web.archive.org/web/20230301150353/https://books.google.com/books?id=jaQH6_8Ju-MC|url-status=live}},[https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279  pp. 279ff] {{Webarchive |url=https://web.archive.org/web/20230301150353/https://books.google.com/books?id=jaQH6_8Ju-MC&pg=PA279 |date=1 March 2023 }}</ref><ref name=":0" /> In the 5th century AD, [[Zu Gengzhi]], son of [[Zu Chongzhi]], established a method<ref>{{cite book|last1=Katz |first1=Victor J.|title=A history of mathematics|date=2008|location=Boston, MA|publisher=Addison-Wesley|isbn=978-0-321-38700-4 |edition=3rd|pages=203|author-link=Victor J. Katz}}</ref><ref>{{cite book|title=Calculus: Early Transcendentals|first1=Dennis G. |last1=Zill |first2=Scott|last2=Wright|first3=Warren S.|last3=Wright |publisher=Jones & Bartlett Learning|year=2009 |edition=3rd |isbn=978-0-7637-5995-7|page=xxvii |url=https://books.google.com/books?id=R3Hk4Uhb1Z0C|access-date=15 November 2015|archive-date=1 March 2023|archive-url=https://web.archive.org/web/20230301150357/https://books.google.com/books?id=R3Hk4Uhb1Z0C|url-status=live}} [https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 Extract of page 27] {{Webarchive |url=https://web.archive.org/web/20230301150353/https://books.google.com/books?id=R3Hk4Uhb1Z0C&pg=PR27 |date=1 March 2023 }}</ref> that would later be called [[Cavalieri's principle]] to find the volume of a [[sphere]].

=== Medieval ===
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| caption1          = [[Ibn al-Haytham]], 11th-century Arab mathematician and physicist
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| caption2          = Indian mathematician and astronomer [[Bhāskara II]]
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==== Middle East ====
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]] of this function, where the formulae 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 |journal=[[Mathematics Magazine]] |volume=68 |issue=3 |pages=163–174 |doi=10.1080/0025570X.1995.11996307 |issn=0025-570X |jstor=2691411}}</ref>


====India====
[[Bhāskara II]] ({{c.|lk=no|1114–1185}}) was acquainted with some ideas of differential calculus and suggested that the "differential coefficient" vanishes at an extremum value of the function.<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> In his astronomical work, he gave a procedure that looked like a precursor to infinitesimal methods. Namely, if <math>x \approx y</math> then <math>\sin(y) - \sin(x) \approx (y - x)\cos(y).</math> This can be interpreted as the discovery that [[cosine]] is the derivative of [[sine]].<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 the 14th century, Indian mathematicians gave a non-rigorous method, resembling differentiation, applicable to some trigonometric functions. [[Madhava of Sangamagrama]] and the [[Kerala School of Astronomy and Mathematics]] stated components of calculus. They studied series equivalent to the Maclaurin expansions of {{tmath|\sin(x)}}, {{tmath|\cos(x)}}, and {{tmath|\arctan(x)}} more than two hundred years before their introduction in Europe.<ref>{{Cite web |title=Madhava - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Madhava/ |access-date=2025-02-18 |website=Maths History |language=en}}</ref> According to [[Victor J. Katz]] they were not able to "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 great problem-solving tool we have today".<ref name=katz/>

=== Modern ===
[[Johannes Kepler]]'s work ''Stereometria Doliorum'' (1615) formed the basis of integral calculus.<ref>{{cite web |title=Johannes Kepler: His Life, His Laws and Times |date=24 September 2016 |publisher=NASA |url=https://www.nasa.gov/kepler/education/johannes |accessdate=10 June 2021 |archive-url=https://web.archive.org/web/20210624003856/https://www.nasa.gov/kepler/education/johannes/ |archive-date=24 June 2021 |url-status=live}}</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>

Significant work was a treatise, the origin being Kepler's methods,<ref name=EB1911/> written by [[Bonaventura Cavalieri]], who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to 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.

The formal study of calculus brought together Cavalieri's infinitesimals with the [[calculus of finite differences]] developed in Europe at around the same time. [[Pierre de Fermat]], claiming that he borrowed from [[Diophantus]], 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> The combination was achieved by [[John Wallis]], [[Isaac Barrow]], and [[James Gregory (astronomer and mathematician)|James Gregory]], the latter two proving predecessors to the [[Fundamental theorem of calculus|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|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 |journal=[[The American Mathematical Monthly]]|volume=118|issue=2|pages=99 |doi=10.4169/amer.math.monthly.118.02.099|s2cid=21473035}}</ref>

The [[product rule]] and [[chain rule]],<ref>{{cite book |title=Calculus: Single Variable, Volume 1 |edition=Illustrated |first1=Brian E. |last1=Blank |first2=Steven George |last2=Krantz |publisher=Springer Science & Business Media |year=2006 |isbn=978-1-931914-59-8 |page=248 |url=https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA248 |access-date=31 August 2017 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150354/https://books.google.com/books?id=hMY8lbX87Y8C&pg=PA248 |url-status=live }}</ref> the notions of [[higher derivative]]s and [[Taylor series]],<ref>{{cite book |title=The Rise and Development of the Theory of Series up to the Early 1820s |edition=Illustrated |first1=Giovanni |last1=Ferraro |publisher=Springer Science & Business Media |year=2007 |isbn=978-0-387-73468-2 |page=87 |url=https://books.google.com/books?id=vLBJSmA9zgAC&pg=PA87 |access-date=31 August 2017 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150355/https://books.google.com/books?id=vLBJSmA9zgAC&pg=PA87 |url-status=live }}</ref> and of [[analytic function]]s<ref>{{cite book|last=Guicciardini|first=Niccolò|chapter=Isaac Newton, Philosophiae naturalis principia mathematica, first edition (1687)|date=2005|title=Landmark Writings in Western Mathematics 1640–1940|pages=59–87|publisher=Elsevier |doi=10.1016/b978-044450871-3/50086-3|isbn=978-0-444-50871-3|quote=[Newton] immediately realised that quadrature problems (the inverse problems) could be tackled via infinite series: as we would say nowadays, by expanding the integrand in power series and integrating term-wise.}}</ref> were used by [[Isaac Newton]] in an idiosyncratic notation which he applied to solve problems of [[mathematical physics]]. In his works, Newton rephrased his ideas to suit the mathematical idiom of the time, replacing calculations with infinitesimals by equivalent geometrical arguments which were considered beyond reproach. He used the methods of calculus to solve the problem of planetary motion, the shape of the surface of a rotating fluid, the oblateness of the earth, the motion of a weight sliding on a [[cycloid]], and many other problems discussed in his ''[[Philosophiæ Naturalis Principia Mathematica|Principia Mathematica]]'' (1687). In other work, he developed series expansions for functions, including fractional and irrational powers, and it was clear that he understood the principles of the [[Taylor series]]. He did not publish all these discoveries, and at this time infinitesimal methods were still considered disreputable.<ref name=":1" />

{{multiple image
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| image1            = Gottfried Wilhelm Leibniz, Bernhard Christoph Francke.jpg
| caption1          = [[Gottfried Wilhelm Leibniz]] was the first to state clearly the rules of calculus.
| image2            = GodfreyKneller-IsaacNewton-1689.jpg
| caption2          = [[Isaac Newton]] developed the use of calculus in his [[Newton's laws of motion|laws of motion]] and [[Newton's law of universal gravitation|universal gravitation]].
}}

These ideas were arranged into a true calculus of infinitesimals by [[Gottfried Wilhelm Leibniz]], who was originally accused of [[plagiarism]] by Newton.<ref name=leib>{{cite book |last=Leibniz |first=Gottfried Wilhelm |title=The Early Mathematical Manuscripts of Leibniz |publisher=Cosimo, Inc. |year=2008 |page=228 |url=https://books.google.com/books?id=7d8_4WPc9SMC&pg=PA3 |isbn=978-1-605-20533-5 |access-date=5 June 2022 |archive-date=1 March 2023 |archive-url=https://web.archive.org/web/20230301150355/https://books.google.com/books?id=7d8_4WPc9SMC&pg=PA3 |url-status=live }}</ref> He is now regarded as an [[Multiple discovery|independent inventor]] of and contributor to calculus. His contribution was to provide a clear set of rules for working with infinitesimal quantities, allowing the computation of second and higher derivatives, and providing the [[product rule]] and [[chain rule]], in their differential and integral forms. Unlike Newton, Leibniz put painstaking effort into his choices of notation.<ref>{{cite book|first=Joseph |last=Mazur |author-link=Joseph Mazur |title=Enlightening Symbols / A Short History of Mathematical Notation and Its Hidden Powers|year=2014|publisher=Princeton University Press |isbn=978-0-691-17337-5 |page=166 |quote=Leibniz understood symbols, their conceptual powers as well as their limitations. He would spend years experimenting with some—adjusting, rejecting, and corresponding with everyone he knew, consulting with as many of the leading mathematicians of the time who were sympathetic to his fastidiousness.}}</ref>

Today, Leibniz and Newton are usually both given credit for independently inventing and developing calculus. Newton was the first to apply calculus to general [[physics]]. Leibniz developed much of the notation used in calculus today.<ref name="TMU" />{{Rp|pages=51–52}} The basic insights that both Newton and Leibniz provided were the laws of differentiation and integration, emphasizing that differentiation and integration are inverse processes, second and higher derivatives, and the notion of an approximating polynomial series.

When Newton and Leibniz first published their results, there was [[Newton v. Leibniz calculus controversy|great controversy]] over which mathematician (and therefore which country) deserved credit. Newton derived his results first (later to be published in his ''[[Method of Fluxions]]''), but Leibniz published his "[[Nova Methodus pro Maximis et Minimis]]" first. Newton claimed Leibniz stole ideas from his unpublished notes, which Newton had shared with a few members of the [[Royal Society]]. This controversy divided English-speaking mathematicians from continental European mathematicians for many years, to the detriment of English mathematics.<ref>{{Cite journal|last=Schrader|first=Dorothy V.|date=1962|title=The Newton-Leibniz controversy concerning the discovery of the calculus|journal=The Mathematics Teacher|volume=55|issue=5|pages=385–396 |doi=10.5951/MT.55.5.0385|jstor=27956626 |issn=0025-5769}}</ref> A careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation. It is Leibniz, however, who gave the new discipline its name. Newton called his calculus "[[Method of fluxions|the science of fluxions]]", a term that endured in English schools into the 19th century.<ref>{{cite book|first=Jacqueline |last=Stedall |author-link=Jackie Stedall |title=The History of Mathematics: A Very Short Introduction |title-link=The History of Mathematics: A Very Short Introduction |year=2012 |isbn=978-0-191-63396-6 |publisher=Oxford University Press}}</ref>{{Rp|100}} The first complete treatise on calculus to be written in English and use the Leibniz notation was not published until 1815.<ref>{{Cite journal |last=Stenhouse |first=Brigitte |date=May 2020 |title=Mary Somerville's early contributions to the circulation of differential calculus |journal=[[Historia Mathematica]] |volume=51 |pages=1–25 |doi=10.1016/j.hm.2019.12.001 |s2cid=214472568|url=http://oro.open.ac.uk/68466/1/accepted_manuscript.pdf }}</ref>

[[File:Maria Gaetana Agnesi.jpg|thumb|upright|right|[[Maria Gaetana Agnesi]]]]
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  |first1=Antonella |last1=Cupillari |author-link=Antonella Cupillari |location=[[Lewiston, New York]] |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]]| access-date=7 December 2010| archive-date=3 December 1998| archive-url=https://web.archive.org/web/19981203075738/http://www.agnesscott.edu/lriddle/women/agnesi.htm| url-status=live}}</ref>

=== Foundations ===
In calculus, ''foundations'' refers to the [[Rigorous#Mathematical rigor |rigorous]] development of the subject from [[axiom]]s and definitions.  In early calculus, the use of [[infinitesimal]] quantities was thought unrigorous and was fiercely criticized by several authors, most notably [[Michel Rolle]] and [[George Berkeley|Bishop Berkeley]]. Berkeley famously described infinitesimals as the [[ghosts of departed quantities]] in his book ''[[The Analyst]]'' in 1734.  Working out a rigorous foundation for calculus occupied mathematicians for much of the century following Newton and Leibniz, and is still to some extent an active area of research today.<ref name="Bell-SEP">{{cite web |url=https://plato.stanford.edu/entries/continuity/ |title=Continuity and Infinitesimals |date=6 September 2013 |website=[[Stanford Encyclopedia of Philosophy]] |first=John L. |last=Bell |access-date=20 February 2022 |author-link=John Lane Bell |archive-date=16 March 2022 |archive-url=https://web.archive.org/web/20220316170134/https://plato.stanford.edu/entries/continuity/ |url-status=live }}</ref>

Several mathematicians, including [[Colin Maclaurin|Maclaurin]], tried to prove the soundness of using infinitesimals, but it would not be until 150 years later when, due to the work of [[Augustin-Louis Cauchy|Cauchy]] and [[Karl Weierstrass|Weierstrass]], a way was finally found to avoid mere "notions" of infinitely small quantities.<ref>{{Cite book |last=Russell |first=Bertrand |author-link=Bertrand Russell |year=1946 |title=History of Western Philosophy |location=London |publisher=[[George Allen & Unwin Ltd]] |page=[https://archive.org/stream/westernphilosoph035502mbp#page/n857/mode/2up 857] |quote=The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibniz believed in actual infinitesimals, but although this belief suited his metaphysics it had no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish calculus without infinitesimals, and thus, at last, made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite number. "Continuity" had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated. |title-link= A History of Western Philosophy }}</ref> The foundations of differential and integral calculus had been laid. In Cauchy's ''[[Cours d'Analyse]]'', we find a broad range of foundational approaches, including a definition of [[continuous function|continuity]] in terms of infinitesimals, and a (somewhat imprecise) prototype of an [[(ε, δ)-definition of limit]] in the definition of differentiation.<ref>{{cite book |first=Judith V. |last=Grabiner |author-link=Judith Grabiner |title=The Origins of Cauchy's Rigorous Calculus |url=https://archive.org/details/originsofcauchys00judi |url-access=registration |location=Cambridge |publisher=MIT Press |year=1981 |isbn=978-0-387-90527-3 }}</ref> In his work, Weierstrass formalized the concept of [[Limit of a function|limit]] and eliminated infinitesimals (although his definition can validate [[nilsquare]] infinitesimals). Following the work of Weierstrass, it eventually became common to base calculus on limits instead of infinitesimal quantities, though the subject is still occasionally called "infinitesimal calculus". [[Bernhard Riemann]] used these ideas to give a precise definition of the integral.<ref>{{cite book|first=Tom |last=Archibald |chapter=The Development of Rigor in Mathematical Analysis |pages=117–129 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref> It was also during this period that the ideas of calculus were generalized to the [[complex plane]] with the development of [[complex analysis]].<ref>{{cite book|first=Adrian |last=Rice |chapter=A Chronology of Mathematical Events |pages=1010–1014 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref>

In modern mathematics, the foundations of calculus are included in the field of [[real analysis]], which contains full definitions and [[mathematical proof|proofs]] of the theorems of calculus. The reach of calculus has also been greatly extended. [[Henri Lebesgue]] invented [[measure theory]], based on earlier developments by [[Émile Borel]], and used it to define integrals of all but the most [[Pathological (mathematics)|pathological]] functions.<ref>{{cite book|first=Reinhard |last=Siegmund-Schultze |chapter=Henri Lebesgue |pages=796–797 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref> [[Laurent Schwartz]] introduced [[Distribution (mathematics)|distributions]], which can be used to take the derivative of any function whatsoever.<ref>{{Cite journal |last1=Barany |first1=Michael J. |last2=Paumier |first2=Anne-Sandrine |last3=Lützen |first3=Jesper |date=November 2017 |title=From Nancy to Copenhagen to the World: The internationalization of Laurent Schwartz and his theory of distributions |journal=[[Historia Mathematica]] |volume=44 |issue=4 |pages=367–394 |doi=10.1016/j.hm.2017.04.002|doi-access=free }}</ref>

Limits are not the only rigorous approach to the foundation of calculus. Another way is to use [[Abraham Robinson]]'s [[non-standard analysis]]. Robinson's approach, developed in the 1960s, uses technical machinery from [[mathematical logic]] to augment the real number system with [[infinitesimal]] and [[Infinity|infinite]] numbers, as in the original Newton-Leibniz conception. The resulting numbers are called [[hyperreal number]]s, and they can be used to give a Leibniz-like development of the usual rules of calculus.<ref>{{cite book|first=Joseph W. |last=Daubin |chapter=Abraham Robinson |pages=822–823 |title=The Princeton Companion to Mathematics |title-link=The Princeton Companion to Mathematics |editor-first1=Timothy |editor-last1=Gowers |editor-link1=Timothy Gowers |editor-first2=June |editor-last2=Barrow-Green |editor-link2=June Barrow-Green |editor-first3=Imre |editor-last3=Leader |editor-link3=Imre Leader |publisher=Princeton University Press |year=2008 |isbn=978-0-691-11880-2 |oclc=682200048}}</ref> There is also [[smooth infinitesimal analysis]], which differs from non-standard analysis in that it mandates neglecting higher-power infinitesimals during derivations.<ref name="Bell-SEP"/> Based on the ideas of [[F. W. Lawvere]] and employing the methods of [[category theory]], smooth infinitesimal analysis views all functions as being [[continuous function|continuous]] and incapable of being expressed in terms of [[Discrete mathematics|discrete]] entities. One aspect of this formulation is that the [[law of excluded middle]] does not hold.<ref name="Bell-SEP" /> The law of excluded middle is also rejected in [[constructive mathematics]], a branch of mathematics that insists that proofs of the existence of a number, function, or other mathematical object should give a construction of the object. Reformulations of calculus in a constructive framework are generally part of the subject of [[constructive analysis]].<ref name="Bell-SEP"/>

=== Significance ===
While many of the ideas of calculus had been developed earlier in [[Greek mathematics|Greece]], [[Chinese mathematics|China]], [[Indian mathematics|India]], [[Islamic mathematics|Iraq, Persia]], and [[Japanese mathematics|Japan]], the use of calculus began in Europe, during the 17th century, when Newton and Leibniz built on the work of earlier mathematicians to introduce its basic principles.<ref name=":0">{{Cite book|title=Chinese studies in the history and philosophy of science and technology|date=1996 |publisher=Kluwer Academic Publishers|author1=Dainian Fan|author2=R. S. Cohen|isbn=0-7923-3463-9|location=Dordrecht|oclc=32272485}}</ref><ref name=":1">{{Cite book|title=Landmark writings in Western mathematics 1640–1940|date=2005 |publisher=Elsevier|editor-first1=I.|editor-last1=Grattan-Guinness|editor-link1=Ivor Grattan-Guinness |isbn=0-444-50871-6 |location=Amsterdam |oclc=60416766}}</ref><ref>{{Cite book|last=Kline |first=Morris|author-link=Morris Kline|title=Mathematical thought from ancient to modern times |volume=3|date=1990 |publisher=Oxford University Press|isbn=978-0-19-977048-9 |location=New York|oclc=726764443}}</ref> The Hungarian polymath [[John von Neumann]] wrote of this work,
{{blockquote|The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.<ref>{{cite book|last=von Neumann |first=J. |author-link=John von Neumann |chapter=The Mathematician |editor-last=Heywood |editor-first=R. B. |title=The Works of the Mind |publisher=University of Chicago Press |year=1947 |pages=180–196}}  Reprinted in {{cite book|editor-last1=Bródy |editor-first1=F. |editor-last2=Vámos |editor-first2=T. |title=The Neumann Compendium |publisher=World Scientific Publishing Co. Pte. Ltd. |year=1995 |isbn=981-02-2201-7 |pages=618–626}}</ref>}}

Applications of differential calculus include computations involving [[velocity]] and [[acceleration]], the [[slope]] of a curve, and [[Mathematical optimization|optimization]].<ref name=":5" />{{Rp|pages=341–453}} Applications of integral calculus include computations involving area, [[volume]], [[arc length]], [[center of mass]], [[work (physics)|work]], and [[pressure]].<ref name=":5" />{{Rp|pages=685–700}} More advanced applications include [[power series]] and [[Fourier series]].

Calculus is also used to gain a more precise understanding of the nature of space, time, and motion. For centuries, mathematicians and philosophers wrestled with paradoxes involving [[division by zero]] or sums of infinitely many numbers. These questions arise in the study of [[Motion (physics)|motion]] and area. The [[ancient Greek]] philosopher [[Zeno of Elea]] gave several famous examples of such [[Zeno's paradoxes|paradoxes]]. Calculus provides tools, especially the [[Limit (mathematics)|limit]] and the [[infinite series]], that resolve the paradoxes.<ref>{{cite book|first=Eugenia |last=Cheng |author-link=Eugenia Cheng |title=Beyond Infinity: An Expedition to the Outer Limits of Mathematics |title-link=Beyond Infinity (mathematics book) |pages=206–210 |publisher=Basic Books |year=2017 |isbn=978-1-541-64413-7 |oclc=1003309980}}</ref>