View source for Kerala school of astronomy and mathematics

{{Short description|Hindu astronomy, mathematics, science school in India}}
{{Use dmy dates|date=July 2022}}
{{Infobox school
| name                    = Kerala school of astronomy and mathematics
| image                   = Kerala school chain of teachers.jpg
| alt                     = 
| caption                 = Chain of teachers of the Kerala school
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| location                = Central and Northern [[Kerala]], India
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| other_name              = <!-- or | other_names = -->
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| founder                 = [[Madhava of Sangamagrama]] 
| type                    = [[Astronomy]], [[Mathematics]], [[Science]]
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}}

The '''Kerala school of astronomy and mathematics''' or the '''Kerala school''' was a school of [[Indian mathematics|mathematics]] and [[Indian astronomy|astronomy]] founded by [[Madhava of Sangamagrama]] in [[Kingdom of Tanur|Tirur]], [[Malappuram district|Malappuram]], [[Kerala]], India, which included among its members: [[Parameshvara]], [[Neelakanta Somayaji]], [[Jyeshtadeva]], [[Achyuta Pisharati]], [[Melpathur Narayana Bhattathiri]] and [[Achyuta Panikkar]]. The school flourished between the 14th and 16th centuries and its original discoveries seem to have ended with [[Melpathur Narayana Bhattathiri|Narayana Bhattathiri]] (1559–1632).  In attempting to solve astronomical problems, the Kerala school independently discovered a number of important mathematical concepts. Their most important results—series expansion for trigonometric functions—were described in [[Sanskrit]] verse in a book by Neelakanta called ''[[Tantrasangraha]]'' (around 1500), and again in a commentary on this work, called ''Tantrasangraha-vakhya'', of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work ''[[Yuktibhasa]]'' ({{circa|1530}}), written in [[Malayalam]], by Jyesthadeva, and also in a commentary on ''Tantrasangraha''.<ref name=roy>Roy, Ranjan.  1990. "Discovery of the Series Formula for <math> \pi </math> by Leibniz, Gregory, and Nilakantha."  ''Mathematics Magazine'' (Mathematical Association of America) 63(5):291–306.</ref>

Their work, completed two centuries before the invention of [[calculus]] in Europe, provided what is now considered the first example of a [[power series]] (apart from [[geometric series]]).<ref>{{Harv|Stillwell|2004|p=173}}</ref><ref>{{Harv|Bressoud|2002|p=12}} Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."</ref><ref>{{Harvnb|Plofker|2001|p=293}} 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—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"</ref>

==Background==
[[Mathematics in the medieval Islamic world|Islamic scholars]] nearly developed a general [[formula]] for finding [[Integral|integrals]] of [[Polynomial|polynomials]] by 1000 AD —and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has been found to date. [[List of Indian mathematicians|Indian scholars]], on the other hand, were by the year 1600 able to use formula similar to [[Ibn al-Haytham]]'s sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and [[Indian mathematics|India]] many centuries before [[Isaac Newton]]. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed.<ref>{{Harvnb|Pingree|1992|p=562}} 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 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."</ref><ref>{{Harvnb|Katz|1995|pp=173–174}} Quote: "How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested.  But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us.  Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested.  By the same time, they also knew how to calculate the differentials of these functions.  So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton.  It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus.  They were apparently only interested in specific cases in which these ideas were needed.<br>{{quad}}There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented the calculus.  They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today."</ref>

==Contributions==
[[File:Pages_from_Yuktibhasa.jpg|thumb|420x420px|Pages from the [[Yuktibhāṣā|Yuktibhasa]] c.1530]]

=== Infinite series and calculus ===
The Kerala school has made a number of contributions to the fields of [[Series (mathematics)|infinite series]] and [[calculus]]. These include the following infinite geometric series:

{{block indent|<math> \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots \text{ for } |x|<1 </math><ref name =singh>{{cite journal | last1 = Singh | first1 = A. N. | date = 1936 | title = On the Use of Series in Hindu Mathematics | journal = Osiris | volume = 1 | pages = 606–628 |doi = 10.1086/368443 | s2cid = 144760421 }}</ref>}}

The Kerala school made intuitive use of [[mathematical induction]], though the [[Inductive hypothesis#Description|inductive hypothesis]] was not yet formulated or employed in proofs.<ref name=roy/> They used this to discover a semi-rigorous proof of the result:

{{block indent|<math>1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1}</math>}}

for large ''n''.

They applied ideas from (what was to become) [[Derivative|differential]] and [[integral]] [[calculus]] to obtain ([[Taylor series|Taylor–Maclaurin]]) infinite series for <math>\sin x</math>, <math>\cos x</math>, and <math> \arctan x</math>.<ref name=bressoud>Bressoud, David.  2002.  "Was Calculus Invented in India?"  ''The College Mathematics Journal'' (Mathematical Association of America).  33(1):2–13.</ref> The ''Tantrasangraha-vakhya'' gives the series in verse, which when translated to mathematical notation, can be written as:<ref name=roy/>

{{block indent|<math>r\arctan\left(\frac{y}{x}\right) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^5} - \cdots, \text{ where } \frac y x \leq 1. </math>}}

{{block indent|<math>r\sin \frac{x}{r} = x - x\cdot\frac{x^2}{(2^2+2)r^2} + x\cdot \frac{x^2}{(2^2+2)r^2}\cdot\frac{x^2}{(4^2+4)r^2} - \cdots </math>}}

{{block indent|<math> r \left( 1 - \cos \frac{x}{r} \right) = r \frac{x^2}{(2^2-2)r^2} - r \frac{x^2}{(2^2-2)r^2}\cdot \frac{x^2}{(4^2-4)r^2} + \cdots 
</math>}}

where, for <math> r = 1, </math> the series reduce to the standard power series for these trigonometric functions, for example:

{{block indent|<math>\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots </math> and}}

{{block indent|<math>\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots </math>}}

(The Kerala school did not use the "factorial" symbolism.)

The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results.  (The later method of Leibniz, using quadrature (''i.e.'' computation of area under the arc of the circle), was not yet developed.)<ref name=roy/> They also made use of the series expansion of <math>\arctan x</math> to obtain an infinite series expression (later known as Gregory series) for <math>\pi</math>:<ref name=roy/>
{{block indent|<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots  </math>}}

Their rational approximation of the ''error'' for the finite sum of their series are of particular interest.  For example, the error, <math>f_i(n+1)</math>, (for ''n'' odd, and ''i = 1, 2, 3'') for the series:

{{block indent| <math>\frac{\pi}{4} \approx 1 - \frac{1}{3}+ \frac{1}{5} - \cdots (-1)^{(n-1)/2}\frac{1}{n} + (-1)^{(n+1)/2}f_i(n+1)</math>
{{block indent|where <math>f_1(n) = \frac{1}{2n}, \ f_2(n) = \frac{n/2}{n^2+1}, \ f_3(n) = \frac{(n/2)^2+1}{(n^2+5)n/2}.</math>}}}}

They manipulated the terms, using the partial fraction expansion of :<math>\frac{1}{n^3-n}</math> to obtain a more rapidly converging series for <math>\pi</math>:<ref name=roy/>

{{block indent|<math>\frac{\pi}{4} = \frac{3}{4} + \frac{1}{3^3-3} - \frac{1}{5^3-5} + \frac{1}{7^3-7} - \cdots  </math>}}

They used the improved series to derive a rational expression,<ref name=roy/> <math>104348/33215</math> for <math>\pi</math> correct up to nine decimal places, i.e. <math>3.141592653</math>. They made use of an intuitive notion of a [[Limit (mathematics)|limit]] to compute these results.<ref name=roy/> The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,<ref name="katz">Katz, V. J.  1995.  "Ideas of Calculus in Islam and India." ([https://maa.org/sites/default/files/0025570x37767.di021190.02p0147a.pdf pdf]) ''Mathematics Magazine'' (Mathematical Association of America), 68(3):163-174.</ref> though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.

===Recognition===
In 1825 John Warren published a memoir on the division of time in southern India,<ref>John Warren (1825) [https://books.google.com/books?id=nttCAAAAcAAJ A Collection of Memoirs on Various Modes According to which Nations of the Southern Part of India Divide Time] from [[Google Books]]</ref> called the ''Kala Sankalita'', which briefly mentions the discovery of infinite series by Kerala astronomers.

The works of the Kerala school were first written up for the Western world by Englishman [[C. M. Whish]] in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries".<ref name=whish>{{cite journal |last=Whish |first=Charles M. |year=1835 |title=XXXIII. On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, the Yucti Bháshá, Carana Padhati, and Sadratnamáka |journal=Transactions of the Royal Asiatic Society |volume=3 |pages=509–523 |url=https://archive.org/details/transactionsofro03asia/page/509/ }}</ref> However, Whish's results were almost completely neglected until over a century later, when the discoveries of the Kerala school were investigated again by [[C. T. Rajagopal]] and his associates. Their work includes commentaries on the proofs of the arctan series in ''Yuktibhasa'' given in two papers,<ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S.  | date = 1949 | title = A Neglected Chapter of Hindu Mathematics | journal = Scripta Mathematica | volume = 15 | pages = 201–209 }}</ref><ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S.  | date = 1951 | title = On the Hindu proof of Gregory's series | journal = Scripta Mathematica | volume = 17 | pages = 65–74 }}</ref> a commentary on the ''Yuktibhasa''{{'}}s proof of the sine and cosine series<ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Venkataraman | first2 = A.  | date = 1949 | title = The sine and cosine power series in Hindu mathematics | journal = Journal of the Royal Asiatic Society of Bengal (Science) | volume = 15 | pages = 1–13 }}</ref> and two papers that provide the [[Sanskrit]] verses of the ''Tantrasangrahavakhya'' for the series for arctan, sin, and cosine (with English translation and commentary).<ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S.  | date = 1977 | title = On an untapped source of medieval Keralese mathematics | doi = 10.1007/BF00348142 | journal = Archive for History of Exact Sciences | volume = 18 | issue = 2 | pages = 89–102 | s2cid = 51861422 }}</ref><ref>{{cite journal | last1 = Rajagopal | first1 = C. | last2 = Rangachari | first2 = M. S.  | date = 1986 | title = On Medieval Kerala Mathematics | journal = Archive for History of Exact Sciences | volume = 35 | issue = 2| pages = 91–99 | doi=10.1007/BF00357622| s2cid = 121678430 }}</ref>

In 1972 [[K. V. Sarma]] published his ''[[A History of the Kerala School of Hindu Astronomy]]'' which described features of the School such as the continuity of knowledge transmission from the 13th to the 17th century:  [[Govinda Bhattathiri]] to [[Parameshvara]] to [[Damodara]] to [[Nilakantha Somayaji]] to [[Jyesthadeva]] to [[Acyuta Pisarati]]. Transmission from teacher to pupil conserved knowledge in "a practical, demonstrative discipline like astronomy at a time when there was not a proliferation of printed books and public schools."

In 1994 it was argued that the [[heliocentric model]] had been adopted about 1500 A.D. in Kerala.<ref>K. Ramasubramanian, M. D. Srinivas & M. S. Sriram (1994) [https://web.archive.org/web/20101223145939/http://www.physics.iitm.ac.in/~labs/amp/kerala-astronomy.pdf Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 A.D.) and the implied heliocentric picture of planetary motion], ''[[Current Science]]'' [https://www.currentscience.ac.in/show.issue.php?volume=66&issue=10 '''66''' (10)]: 784–90</ref>

==Possible transmission of Kerala school results to Europe==
A. K. Bag suggested in 1979 that knowledge of these results might have been transmitted to Europe through the trade route from [[Kerala]] by traders and [[Jesuit]] missionaries.<ref>A. K. Bag (1979) ''Mathematics in ancient and medieval India''. Varanasi/Delhi: Chaukhambha Orientalia. page 285.</ref> Kerala was in continuous contact with China and [[Arabia]], and [[Europe]].  The suggestion of some communication routes and a chronology by some scholars<ref>{{cite journal | last1 = Raju | first1 = C. K. |author-link=C. K. Raju | date = 2001 | title = Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa | journal = Philosophy East and West | volume = 51 | issue = 3| pages = 325–362 | doi=10.1353/pew.2001.0045| s2cid = 170341845 }}</ref><ref name=almeida/> could make such a transmission a possibility; however, there is no direct evidence by way of relevant manuscripts that such a transmission took place.<ref name=almeida>{{cite journal | last1 = Almeida | first1 = D. F. | last2 = John | first2 = J. K. | last3 = Zadorozhnyy | first3 = A. | date = 2001 | title = Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications | journal = Journal of Natural Geometry | volume = 20 | pages = 77–104 }}</ref> According to [[David Bressoud]], "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century".<ref name=bressoud/><ref name=gold>{{cite journal | last1 = Gold | first1 = D. | last2 = Pingree | first2 = D.  | date = 1991 | title = A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine | journal = Historia Scientiarum | volume = 42 | pages = 49–65 }}</ref> V. J. Katz notes some of the ideas of the Kerala school have similarities to the work of 11th-century Iraqi scholar [[Ibn al-Haytham]],<ref name=katz/> suggesting a possible transmission of ideas from [[Islamic mathematics]] to Kerala.<ref>{{Harvnb|Katz|1995|p=174}}.</ref>

Both Indian and [[Islamic mathematics|Arab]] scholars made discoveries before the 17th century that are now considered a part of calculus.<ref name=katz/> According to Katz, they were yet 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", like [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]].<ref name=katz/> The intellectual careers of both Newton and Leibniz are well documented and there is no indication of their work not being their own;<ref name=katz/> however, it is not known with certainty whether the immediate ''predecessors'' of Newton and Leibniz, "including, in particular, [[Pierre de Fermat|Fermat]] and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources of which we are not now aware".<ref name=katz/> This is an active area of current research, especially in the manuscript collections of Spain and [[Maghreb]], research that is now being pursued, among other places, at the {{lang|fr|[[Centre national de la recherche scientifique]]|italic=no}} in [[Paris]].<ref name=katz/>

==See also==
*[[Indian astronomy]]
*[[Indian mathematics]]
*[[Indian mathematicians]]
*[[History of mathematics]]
*[[A History of the Kerala School of Hindu Astronomy]]
*[[List of astronomers and mathematicians of the Kerala school]]

==Notes==
{{reflist}}

==References==
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==External links==
{{commons category|Kerala school of astronomy and mathematics}}
*[https://web.archive.org/web/20021015003732/http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html An overview of Indian mathematics], ''[[MacTutor History of Mathematics archive]]'', 2002.
*[http://www-history.mcs.st-and.ac.uk/history/Projects/Pearce/index.html Indian Mathematics: Redressing the balance], ''MacTutor History of Mathematics archive'', 2002.
*[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_1.html Keralese mathematics], ''MacTutor History of Mathematics archive'', 2002.
*[http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/Chapters/Ch9_4.html Possible transmission of Keralese mathematics to Europe], ''MacTutor History of Mathematics archive'', 2002.
*[http://www.physorg.com/news106238636.html "Indians predated Newton 'discovery' by 250 years"] ''phys.org,'' 2007

{{Indian mathematics}}
{{Indian astronomy}}
{{Kerala School}}
{{Ancient Dharmic centres of Higher Learning}}

[[Category:Kerala school of astronomy and mathematics| ]]
[[Category:Indian mathematics]]
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