Editing History of calculus (section)

===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>