Editing Calculus (section)

=== 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]].