Editing History of calculus (section)

==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>