Editing Calculus (section)

=== 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
| direction         = horizontal
| total_width       = 330
| 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>