Editing Calculus (section)

=== Medieval ===
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| image1            = Ibn al-Haytham crop.jpg
| caption1          = [[Ibn al-Haytham]], 11th-century Arab mathematician and physicist
| image2            = भास्कराचार्य.jpg
| 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/>