Editing History of mathematics (section)

== Indian ==
{{Main|Indian mathematics}}
{{Further|History of science#Indian mathematics}}
{{See also|History of the Hindu–Arabic numeral system}}
[[File:Bakhshali numerals 2.jpg|thumb|right|upright=1.5|The numerals used in the [[Bakhshali manuscript]], dated between the 2nd century BC and the 2nd century AD.]]
{{multiple image
 | align = right
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 | image1 = 1911 sketch of numerals script history ancient India, mathematical symbols shapes.jpg
 | alt1 = Numerals evolution in India
 | caption1 = Indian numerals in stone and copper inscriptions<ref name=britnanaghat>[https://www.britannica.com/topic/numeral#ref797082 Development Of Modern Numerals And Numeral Systems: The Hindu-Arabic system], Encyclopaedia Britannica, Quote: "The 1, 4, and 6 are found in the Ashoka inscriptions (3rd century BC); the 2, 4, 6, 7, and 9 appear in the Nana Ghat inscriptions about a century later; and the 2, 3, 4, 5, 6, 7, and 9 in the Nasik caves of the 1st or 2nd century AD – all in forms that have considerable resemblance to today’s, 2 and 3 being well-recognized cursive derivations from the ancient = and ≡."</ref>
 | image2 = Indian numerals 100AD.svg
 | alt2 = Brahmi numerals
 | caption2 = Ancient Brahmi numerals in a part of India
}}
The earliest civilization on the Indian subcontinent is the [[Indus Valley civilization]] (mature second phase: 2600 to 1900 BC) that flourished in the [[Indus river]] basin. Their cities were laid out with geometric regularity, but no known mathematical documents survive from this civilization.<ref>{{Harv|Boyer|1991|loc="China and India" p. 206}}</ref>

The oldest extant mathematical records from India are the [[Sulba Sutras]] (dated variously between the 8th century BC and the 2nd century AD),<ref name="Boyer 1991 loc=China and India p. 207">{{Harv|Boyer|1991|loc="China and India" p. 207}}</ref> appendices to religious texts which give simple rules for constructing altars of various shapes, such as squares, rectangles, parallelograms, and others.<ref>{{Cite book |first=T.K. |last=Puttaswamy |chapter=The Accomplishments of Ancient Indian Mathematicians |pages=411–12 |title=Mathematics Across Cultures: The History of Non-western Mathematics |editor1-first=Helaine |editor1-last=Selin |editor1-link=Helaine Selin |editor2-first=Ubiratan |editor2-last=D'Ambrosio |editor2-link=Ubiratan D'Ambrosio |year=2000 |publisher=[[Springer Science+Business Media|Springer]] |isbn=978-1-4020-0260-1 }}</ref> As with Egypt, the preoccupation with temple functions points to an origin of mathematics in religious ritual.<ref name="Boyer 1991 loc=China and India p. 207"/> The Sulba Sutras give methods for constructing a [[squaring the circle|circle with approximately the same area as a given square]], which imply several different approximations of the value of π.<ref>{{cite journal |first=R.P. |last=Kulkarni |url=http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af9_32.pdf |title=The Value of π known to Śulbasūtras |archive-url=https://web.archive.org/web/20120206150545/http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005af9_32.pdf |archive-date=2012-02-06 |journal=Indian Journal of History of Science |volume=13 |issue=1 |date=1978 |pages=32–41}}</ref><ref name="Indian_sulbasutras">{{cite web |first1=J.J. |last1=Connor |first2=E.F. |last2=Robertson |title=The Indian Sulbasutras |publisher=Univ. of St. Andrew, Scotland |url=http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Indian_sulbasutras.html}}</ref>{{efn|1=The approximate values for π are 4 x (13/15)<sup>2</sup> (3.0044...), 25/8 (3.125), 900/289 (3.11418685...), 1156/361 (3.202216...), and 339/108 (3.1389)}} In addition, they compute the [[square root]] of 2 to several decimal places, list Pythagorean triples, and give a statement of the [[Pythagorean theorem]].<ref name="Indian_sulbasutras"/> All of these results are present in Babylonian mathematics, indicating Mesopotamian influence.<ref name="Boyer 1991 loc=China and India p. 207"/> It is not known to what extent the Sulba Sutras influenced later Indian mathematicians. As in China, there is a lack of continuity in Indian mathematics; significant advances are separated by long periods of inactivity.<ref name="Boyer 1991 loc=China and India p. 207"/>

[[Pāṇini]] (c. 5th century BC) formulated the rules for [[Sanskrit grammar]].<ref>{{Cite journal
 | last=Bronkhorst
 | first=Johannes
 | author-link= Johannes Bronkhorst
 | title=Panini and Euclid: Reflections on Indian Geometry
 | journal=Journal of Indian Philosophy
 |volume=29
 |issue=1–2
 | year=2001
 | pages=43–80
 | doi=10.1023/A:1017506118885
 | s2cid=115779583
 }}</ref> His notation was similar to modern mathematical notation, and used metarules, [[Transformation (geometry)|transformations]], and [[recursion]].<ref>{{Cite journal|last=Kadvany|first=John|date=2008-02-08|title=Positional Value and Linguistic Recursion|journal=Journal of Indian Philosophy|language=en|volume=35|issue=5–6|pages=487–520|doi=10.1007/s10781-007-9025-5|issn=0022-1791|citeseerx=10.1.1.565.2083|s2cid=52885600}}</ref> [[Pingala]] (roughly 3rd–1st centuries BC) in his treatise of [[Prosody (poetry)|prosody]] uses a device corresponding to a [[binary numeral system]].<ref>{{Cite book
|last1=Sanchez
|first1=Julio
|last2=Canton
|first2=Maria P.
|title=Microcontroller programming : the microchip PIC
|year=2007
|publisher=CRC Press
|location=Boca Raton, Florida
|isbn=978-0-8493-7189-9
|page=37
}}</ref><ref>Anglin, W. S. and J. Lambek (1995). ''The Heritage of Thales'', Springer, {{ISBN|0-387-94544-X}}</ref> His discussion of the [[combinatorics]] of [[Metre (music)|meters]] corresponds to an elementary version of the [[binomial theorem]]. Pingala's work also contains the basic ideas of [[Fibonacci number]]s (called ''mātrāmeru'').<ref>{{cite journal | last1 = Hall | first1 = Rachel W. | year = 2008 | title = Math for poets and drummers | url = http://people.sju.edu/~rhall/mathforpoets.pdf | journal = Math Horizons | volume = 15 | issue = 3| pages = 10–11 | doi = 10.1080/10724117.2008.11974752 | s2cid = 3637061 }}</ref>

The next significant mathematical documents from India after the ''Sulba Sutras'' are the ''Siddhantas'', astronomical treatises from the 4th and 5th centuries AD ([[Gupta period]]) showing strong Hellenistic influence.<ref>{{Harv|Boyer|1991|loc="China and India" p. 208}}</ref> They are significant in that they contain the first instance of trigonometric relations based on the half-chord, as is the case in modern trigonometry, rather than the full chord, as was the case in Ptolemaic trigonometry.<ref name=autogenerated2>{{Harv|Boyer|1991|loc="China and India" p. 209}}</ref> Through a series of translation errors, the words "sine" and "cosine" derive from the Sanskrit "jiya" and "kojiya".<ref name=autogenerated2 />

[[Image:Yuktibhasa.svg|upright|left|thumb|Explanation of the [[Law of sines|sine rule]] in ''[[Yuktibhāṣā]]'']]
Around 500 AD, [[Aryabhata]] wrote the ''[[Aryabhatiya]]'', a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.<ref>{{Harv|Boyer|1991|loc="China and India" p. 210}}</ref> It is in the ''Aryabhatiya'' that the decimal place-value system first appears. Several centuries later, the [[Islamic mathematics|Muslim mathematician]] [[Abu Rayhan Biruni]] described the ''Aryabhatiya'' as a "mix of common pebbles and costly crystals".<ref>{{Harv|Boyer|1991|loc="China and India" p. 211}}</ref>

In the 7th century, [[Brahmagupta]] identified the [[Brahmagupta theorem]], [[Brahmagupta's identity]] and [[Brahmagupta's formula]], and for the first time, in ''[[Brahmasphutasiddhanta|Brahma-sphuta-siddhanta]]'', he lucidly explained the use of [[0 (number)|zero]] as both a placeholder and [[decimal digit]], and explained the [[Hindu–Arabic numeral system]].<ref name="Boyer Siddhanta">{{cite book|last=Boyer|ref=none|author-link=Carl Benjamin Boyer|title=History of Mathematics|url=https://archive.org/details/historyofmathema00boye|url-access=registration|year=1991|chapter=The Arabic Hegemony|page=[https://archive.org/details/historyofmathema00boye/page/226 226]|publisher=Wiley |isbn=9780471543978|quote=By 766 we learn that an astronomical-mathematical work, known to the Arabs as the ''Sindhind'', was brought to Baghdad from India. It is generally thought that this was the ''Brahmasphuta Siddhanta'', although it may have been the ''Surya Siddhanata''. A few years later, perhaps about 775, this ''Siddhanata'' was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological ''[[Tetrabiblos]]'' was translated into Arabic from the Greek.}}</ref> It was from a translation of this Indian text on mathematics (c. 770) that Islamic mathematicians were introduced to this numeral system, which they adapted as [[Arabic numerals]]. Islamic scholars carried knowledge of this number system to Europe by the 12th century, and it has now displaced all older number systems throughout the world. Various symbol sets are used to represent numbers in the Hindu–Arabic numeral system, all of which evolved from the [[Brahmi numeral]]s. Each of the roughly dozen major scripts of India has its own numeral glyphs. In the 10th century, [[Halayudha]]'s commentary on [[Pingala]]'s work contains a study of the [[Fibonacci sequence]]<ref>{{Cite journal |last=Singh |first=Parmanand |date=1985-08-01 |title=The so-called fibonacci numbers in ancient and medieval India |url=https://dx.doi.org/10.1016/0315-0860%2885%2990021-7 |journal=Historia Mathematica |volume=12 |issue=3 |pages=229–244 |doi=10.1016/0315-0860(85)90021-7 |issn=0315-0860}}</ref> and [[Pascal's triangle]],<ref>{{Cite book |last=Ramasubramanian |first=K. |url=https://books.google.com/books?id=AEe9DwAAQBAJ&dq=Selected+Works+of+Radha+Charan+Gupta+on+History+of+Mathematics+pascal&pg=PA289 |title=Gaṇitānanda: Selected Works of Radha Charan Gupta on History of Mathematics |date=2019-11-08 |publisher=Springer Nature |isbn=978-981-13-1229-8 |language=en}}</ref> and describes the formation of a [[matrix (mathematics)|matrix]].{{Citation needed|date=April 2010}}

In the 12th century, [[Bhāskara II]],<ref>Plofker 2009 182–207</ref> who lived in southern India, wrote extensively on all then known branches of mathematics. His work contains mathematical objects equivalent or approximately equivalent to infinitesimals, [[Mean value theorem|the mean value theorem]] and 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><ref>Plofker 2009 pp. 197–98; George Gheverghese Joseph, ''The Crest of the Peacock: Non-European Roots of Mathematics'', Penguin Books, London, 1991 pp. 298–300; Takao Hayashi, "Indian Mathematics", pp. 118–30 in ''Companion History of the History and Philosophy of the Mathematical Sciences'', ed. I. Grattan. Guinness, Johns Hopkins University Press, Baltimore and London, 1994, p. 126.</ref> In the 14th century, [[Narayana Pandita (mathematician)|Narayana Pandita]] completed his ''[[Ganita Kaumudi]]''.<ref>{{Cite web |title=Narayana - Biography |url=https://mathshistory.st-andrews.ac.uk/Biographies/Narayana/ |access-date=2022-10-03 |website=Maths History |language=en}}</ref>

Also in the 14th century, [[Madhava of Sangamagrama]], the founder of the [[Kerala School of Astronomy and Mathematics|Kerala School of Mathematics]], found the [[Leibniz formula for pi|Madhava–Leibniz series]] and obtained from it a [[Approximations of π#Middle Ages|transformed series]], whose first 21 terms he used to compute the value of π as 3.14159265359. Madhava also found [[Gregory's series|the Madhava-Gregory series]] to determine the arctangent, the Madhava-Newton [[power series]] to determine sine and cosine and [[Taylor series|the Taylor approximation]] for sine and cosine functions.<ref>Plofker 2009 pp. 217–53.</ref> In the 16th century, [[Jyesthadeva]] consolidated many of the Kerala School's developments and theorems in the ''Yukti-bhāṣā''.<ref name="rajujournal">
{{cite journal| author1=Raju, C. K. | title=Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhāṣā 
|url=http://ckraju.net/papers/Hawaii.pdf
| journal=Philosophy East & West 
| volume=51
| issue=3
| date=2001
| pages=325–362 
| doi=10.1353/pew.2001.0045 
| s2cid=170341845 
| access-date=2020-02-11
}}
</ref><ref>Divakaran, P. P. (2007). "The first textbook of calculus: Yukti-bhāṣā", ''Journal of Indian Philosophy'' 35, pp. 417–33.</ref> It has been argued that certain ideas of calculus like infinite series and taylor series of some trigonometry functions, were transmitted to Europe in the 16th century<ref name=":2" /> via [[Jesuit]] missionaries and traders who were active around the ancient port of [[Muziris]] at the time and, as a result, directly influenced later European developments in analysis and calculus.<ref name=almeida>{{cite journal
|author = Almeida, D. F.; J. K. John and A. Zadorozhnyy
|title = Keralese mathematics: its possible transmission to Europe and the consequential educational implications
| journal = Journal of Natural Geometry
|volume= 20
|year =2001
|pages=77–104
|issue=1
}}</ref> However, other scholars argue that the Kerala School did not formulate a systematic theory of [[derivative|differentiation]] and [[integral|integration]], and that there is not any direct evidence of their results being transmitted outside Kerala.<ref>{{Cite journal | last = Pingree | first = David | author-link = David Pingree | title = Hellenophilia versus the History of Science | journal = Isis | volume = 83 | issue = 4 | pages = 554–563 |date=December 1992  | jstor = 234257 | doi = 10.1086/356288 | quote = One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by [[C. M. Whish|Charles Whish]], in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the ''Transactions of the Royal Asiatic Society'', in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series ''without'' the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution.| bibcode = 1992Isis...83..554P | s2cid = 68570164 }}</ref><ref>{{Cite journal | last = Bressoud | first = David | author-link = David Bressoud | title = Was Calculus Invented in India? | journal = College Mathematics Journal | volume = 33 | issue = 1 | pages = 2–13 | year = 2002 | doi=10.2307/1558972| jstor = 1558972 }}</ref><ref>{{Cite journal | last = Plofker | first = Kim | author-link = Kim Plofker | title = The 'Error' in the Indian "Taylor Series Approximation" to the Sine | journal = Historia Mathematica | volume = 28 | issue = 4 | page = 293 |date=November 2001 | doi = 10.1006/hmat.2001.2331 | quote =It is not unusual to encounter in discussions of Indian mathematics such assertions as that 'the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)' [Joseph 1991, 300], or that 'we may consider Madhava to have been the founder of mathematical analysis' (Joseph 1991, 293), or that Bhaskara II may claim to be 'the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus' (Bag 1979, 294).... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285))... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian 'discovery of the principle of the differential calculus' somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential 'principle' was not generalized to arbitrary functions&nbsp;– in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here| doi-access = free }}</ref><ref>{{Cite journal | last = Katz | first = Victor J. | title = Ideas of Calculus in Islam and India | journal = Mathematics Magazine | volume = 68 | issue = 3 | pages = 163–74 |date=June 1995  | url = http://www2.kenyon.edu/Depts/Math/Aydin/Teach/Fall12/128/CalcIslamIndia.pdf | jstor = 2691411 | doi=10.2307/2691411 }}</ref>