Kerala school of astronomy and mathematics
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The '''Kerala school of astronomy and mathematics''' was a school of [[Indian mathematics|mathematics]] and [[Indian astronomy|astronomy]] founded by [[Madhava of Sangamagrama]] in [[Kerala]], [[South 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 the original discoveries of the school seems to have ended with [[Melpathur Narayana Bhattathiri|Narayana Bhattathiri]] (1559-1632). In attempting to solve astronomical problems, the Kerala school ''independently'' created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in [[Sanskrit]] verse in a book by Neelakanta called ''Tantrasangraha'', 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]]'' (c.1500-c.1610), 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 discovery of these three important series expansions of calculus—several centuries before calculus was developed in Europe by [[Leibniz]] and [[Newton]]—was a landmark achievement in mathematics. However, the Kerala School cannot be said to have invented ''calculus'',<ref name=bressoud/> because, while they were able to develop [[Taylor series]] expansions for the important trigonometric functions, they developed neither a comprehensive theory of differentiation or integration, nor the fundamental theorem of calculus.<ref name=katz/>
==Contributions==
===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:
:<math> \frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots </math> for <math>|x|<1 </math><ref name =singh>Singh, A. N. Singh. 1936. "On the Use of Series in Hindu Mathematics." ''Osiris'' 1:606-628.</ref>
This formula, however, was already known in the work of the 10th century [[Iraq]]i [[Islamic mathematics|mathematician]] [[Ibn al-Haytham|Alhazen]] (the [[Latin]]ized form of the name Ibn al-Haytham) (965-1039).<ref>Edwards, C. H., Jr. 1979. ''The Historical Development of the Calculus''. New York: Springer-Verlag.</ref>
The Kerala school made intuitive use of [[mathematical induction]], though the [[Inductive hypothesis#Formal 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:
:<math>1^p+ 2^p + \cdots + n^p \approx \frac{n^{p+1}}{p+1}</math> for large ''n''. This result was also known to Alhazen.<ref name=roy/>
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/>
:<math>r\arctan(\frac{y}{x}) = \frac{1}{1}\cdot\frac{ry}{x} -\frac{1}{3}\cdot\frac{ry^3}{x^3} + \frac{1}{5}\cdot\frac{ry^5}{x^t} - \cdots , </math> where <math>y/x \leq 1. </math>
:<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} - \cdot </math>
:<math> r - \cos x = r\cdot \frac{x^2}{(2^2-2)r^2} - r\cdot \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:
::<math>\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots </math> and
::<math>\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots </math> (The Kerala school themselves 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/>
:<math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \ldots </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:
:<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>
::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 error term to derive a faster converging series for <math>\pi</math>:<ref name=roy/>
:<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." ''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.
The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835, though there exists some other works, namely '''Kala Sankalita''' by J.Warren in 1825<ref>[http://www.physics.iitm.ac.in/~labs/amp/kerala-astronomy.pdf Current Science]</ref> which briefly mentions the discovery of infinite series by Kerala astronomers . 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="charles">{{cite book
| author =Charles Whish
| year = 1835
| title = Transactions of the Royal Asiatic Society of Great Britain and Ireland
| publisher =
}}
</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. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in ''Yuktibhasa'' given in two papers,<ref>Rajagopal, C. and M. S. Rangachari. 1949. "A Neglected Chapter of Hindu Mathematics." ''Scripta Mathematica''. 15:201-209.</ref><ref>Rajagopal, C. and M. S. Rangachari. 1951. "On the Hindu proof of Gregory's series." ''Ibid.'' 17:65-74.</ref> a commentary on the ''Yuktibhasa'''s proof of the sine and cosine series<ref>Rajagopal, C. and A. Venkataraman. 1949. "The sine and cosine power series in Hindu mathematics." ''Journal of the Royal Asiatic Society of Bengal (Science)''. 15:1-13.</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>Rajagopal, C. and M. S. Rangachari. 1977. "On an untapped source of medieval Keralese mathematics." ''Archive for the History of Exact Sciences''. 18:89-102.</ref><ref>Rajagopal, C. and M. S. Rangachari. 1986. "On Medieval Kerala Mathematics." ''Archive for the History of Exact Sciences''. 35:91-99.</ref>
===Geometry, Arithmetic, and Algebra===
In the fields of [[geometry]], [[arithmetic]], and [[algebra]], the Kerala school discovered a formula for the [[ecliptic]],{{Fact|date=February 2007}} [[Lhuilier]]'s formula for the circumradius of a [[cyclic quadrilateral]] by [[Parameshvara]],<ref name=Paramesvara>J. J. O'Connor and E. F. Robertson (2000). [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Paramesvara.html Paramesvara], ''[[MacTutor History of Mathematics archive]]''.</ref><ref name=Mathematicians-of-Kerala>Ian G. Pearce (2002). [http://www-gap.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_2.html Mathematicians of Kerala]. ''[[MacTutor History of Mathematics archive]]''. [[University of St Andrews]].</ref> decimal [[floating point]] numbers,<ref>D. F. Almeida, G. G. Joseph (2004). "Eurocentrism in the History of Mathematics: The Case of the Kerala School", ''Race and Class''.</ref> the [[secant method]] and [[iterative method]]s for solution of [[non-linear]] equations by [[Parameshvara]],<ref name=Paramesvara/><ref>K. Plofker (1996). "An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text", ''Historia Mathematica'' '''23''' (3), p. 246-256.</ref> and the [[Isaac Newton|Newton]]-[[Carl Friedrich Gauss|Gauss]] interpolation formula by Govindaswami.{{Fact|date=February 2007}}
===Astronomy===
In [[Indian astronomy|astronomy]], Madhava discovered a procedure to determine the positions of the [[Moon]] every 36 minutes, and methods to estimate the motions of the planets.<ref name=Kak>[[Subhash Kak|S. Kak]] (2002). [http://www.ece.lsu.edu/kak/grolier.pdf History of Indian Science], p. 6. [[Louisiana State University]].</ref> Late Kerala school astronomers gave a formulation for the equation of the center of the planets,<ref name=Kak/><ref>Joseph (2000), p. 298-300.</ref> and a [[heliocentrism|heliocentric]] model of the solar system.<ref name=Kak/>
In 1500, [[Nilakantha Somayaji|Nilakanthan Somayaji]] (1444-1544) of the [[Kerala school of astronomy and mathematics]], in his ''[[Tantrasangraha]]'', revised [[Aryabhata]]'s model for the planets [[Mercury (planet)|Mercury]] and [[Venus]]. His equation of the [[Center of mass|centre]] for these planets remained the most accurate until the time of [[Johannes Kepler]] in the 17th century.<ref name=Joseph408>George G. Joseph (2000), p. 408.</ref>
Nilakanthan Somayaji, in his ''Aryabhatiyabhasya'', a commentary on Aryabhata's ''Aryabhatiya'', developed his own computational system for a partially [[heliocentrism|heliocentric]] planetary model, in which Mercury, Venus, [[Mars]], [[Jupiter]] and [[Saturn]] orbit the [[Sun]], which in turn orbits the [[Earth]], similar to the [[Tychonic system]] later proposed by [[Tycho Brahe]] in the late 16th century. Nilakantha's system, however, was mathematically more efficient than the Tychonic system, due to correctly taking into account the equation of the centre and [[latitude|latitudinal]] motion of Mercury and Venus. Most astronomers of the [[Kerala school of astronomy and mathematics]] who followed him accepted his planetary model.<ref name=Joseph408/><ref>K. Ramasubramanian, M. D. Srinivas, M. S. Sriram (1994). "Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion", ''[[Current Science]]'' '''66''', p. 784-790.</ref>
===Linguistics===
In [[linguistics]], the [[Ayurveda|ayurvedic]] and poetic traditions of [[Kerala]] were founded by this school, and the famous poem, ''[[Narayaneeyam]]'', was composed by [[Melpathur Narayana Bhattathiri|Narayana Bhattathiri]].{{Fact|date=February 2007}}
==Prominent mathematicians==
===Madhavan of Sangamagrama===
[[Madhava of Sangamagrama]] (c. 1340-1425) was the founder of the [[Kerala School]]. Although it is possible that he wrote ''Karana Paddhati'' a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars.
Little is known about Madhava, who lived at Irinjalakuda,Thrissur district between the years 1340 and 1425.the famous Bharatha temple is Koodalmaanikyam is at Irinjalakuda.'Koodal'in Sanskrit means Sangamam. Nilkantha attributes the series for ''sine'' to Madhava. It is not known if Madhava discovered the other series as well, or whether they were discovered later by others in the Kerala school.
Madhava's discoveries include the Taylor series for the sine,<ref name=katz/> cosine, tangent and arctangen functions,<ref name=MacTutor>{{MacTutor Biography|id=Madhava}} [[St Andrews University]], 2000.</ref> the second-order Taylor series approximations of the sine and cosine functions and the third-order Taylor series approximation of the sine function, the power series of [[π]] (usually attributed to [[Leibniz]]),<ref name=Madhava>Ian G. Pearce (2002). [http://www-gap.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_3.html Madhava of Sangamagramma]. ''[[MacTutor History of Mathematics archive]]''. [[University of St Andrews]].</ref> the solution of [[Transcendental function|transcendental equations]] by [[iteration]],{{Fact|date=February 2007}} and the approximation of [[transcendental number]]s by [[continued fraction]]s.<ref name=Madhava/> Madhava correctly computed the value of <math>\pi</math> to 9 decimal places<ref name=roy/> and 13 decimal places,<ref name=Madhava/> and produced sine and cosine tables to 9 decimal places of accuracy.<ref>Joseph (2000), p. 293.</ref> He also extended some results found in earlier works, including those of [[Bhaskara]].<ref name=Madhava/>
===Narayanan Pandit===
[[Narayana Pandit]] (1340-1400), had written two works, an arithmetical treatise called ''Ganita Kaumudi'' and an [[algebra]]ic treatise called ''Bijganita Vatamsa''. Narayanan is also thought to be the author of an elaborate commentary of [[Bhaskara II]]'s [[Lilavathi]], titled ''Karmapradipika'' (or ''Karma-Paddhati'').<ref name=Narayana>J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Narayana.html Narayana], ''[[MacTutor History of Mathematics archive]]''.</ref>
Although the ''Karmapradipika'' contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and [[magic square]]s.<ref name=Narayana/>
Narayanan's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order [[indeterminate equation]] ''nq''<sup>2</sup> + 1 = ''p''<sup>2</sup> ([[Pell's equation]]), solutions of indeterminate [[Degree of a polynomial|higher-order equations]], mathematical operations with [[0 (number)|zero]], several [[geometry|geometrical]] rules, and a discussion of [[magic square]]s and similar figures.<ref name=Narayana/> Evidence also exists that Narayana made minor contributions to the ideas of [[Derivative|differential calculus]] found in Bhaskara II's work. Narayana has also made contributions to the topic of [[cyclic quadrilateral]]s.<ref name=Mathematicians-of-Kerala/>
===Parameshvaran===
[[Parameshvara]] (1370-1460), the founder of the Drigganita system of Astronomy, was a prolific author of several important works. He belonged to the Alathur village situated on the bank of Bharathappuzha.He is stated to have made direct astronomical observations for fifty-five years before writing his famous work, Drigganita. He also wrote commentaries on the works of [[Bhaskara I]], [[Aryabhata]] and [[Bhaskara II]]. His ''Lilavathi Bhasya'', a commentary on Bhaskara II's ''Lilavathi'', contains one of his most important discoveries: an early version of the [[mean value theorem]].<ref name=Paramesvara/> This is considered one of the most important results in differential calculus and one of the most important theorems in mathematical analysis, and was later essential in proving the [[fundamental theorem of calculus]].
The ''Siddhanta-Deepika'' by Paramesvara is a commentary on the commentary of [[Govindsvamin]] on [[Bhaskara I]]'s ''Maha-bhaskareeya''. This work contains some of his eclipse observations, including one made at Navakshethra in 1422 and two made at Gokarna in 1425 and 1430. It also presents a mean value type formula for inverse interpolation of the sine function, a one-point iterative technique for calculating the sine of a given angle, and a more efficient approximation that works using a two-point iterative algorithm, which is essentially the same as the modern [[secant method]].<ref name=Paramesvara/>
Parameshvaran was also the first mathematician to give the radius of a circle with an inscribed [[cyclic quadrilateral]], an expression that is normally attributed to [[Simon Antoine Jean L'Huillier|L'Huilier]] (1782).<ref name=Paramesvara/>
===Nilakanthan Somayaji===
[[Nilakantha Somayaji|Nilakantha]] (1444-1544) was a disciple of Govinda, son of Parameshvara. He was a brahmin from Trkkantiyur in Ponnani taluk. His younger brother Sankara was also a scholar in astronomy. Nilakantha's most notable work ''Tantra Samgraha'' (which 'spawned' a later anonymous commentary ''Tantrasangraha-vyakhya'' and a further commentary by the name ''Yukthideepika'', written in 1501) he elaborates and extends the contributions of Madhava.<ref name=Nilakantha>J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Nilakantha.html Nilakantha], ''[[MacTutor History of Mathematics archive]]''.</ref>
Nilakantha was also the author of ''Aryabhatiya-bhashya'', a commentary of the ''[[Aryabhatiya]]''. Of great significance in Nilakantha's work includes the presence of [[Mathematical induction|inductive]] mathematical proofs, a derivation and proof of the Madhava-Gregory series of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, an improved series expansion of π that converges more rapidly, and the relationship between the power series of π and arctangent.<ref name=Nilakantha/> He also gave sophisticated explanations of the irrationality of π, the correct formulation for the equation of the center of the planets, and a [[heliocentrism|heliocentric]] model of the solar system.<ref name=Kak/>
===Chitrabhanu===
[[Citrabhanu]] (c. 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two [[simultaneous equation|simultaneous]] [[Diophantine]] equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:<ref name=Indian-mathematics>J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html An overview of Indian mathematics], ''[[MacTutor History of Mathematics archive]]''.</ref>
:<math>\ x + y = a, x - y = b, xy = c, x^2 + y^2 = d, x^2 - y^2 = e, x^3 + y^3 = f, x^3 - y^3 = g.</math>
For each case, Chitrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.<ref name=Indian-mathematics/>
===Jyesthadevan===
[[Jyesthadeva]] (c. 1500-1600) was another member of the Kerala School. His key work was the ''[[Yuktibhasa]]'' (written in [[Malayalam]], a regional language of the Indian state of [[Kerala]]), the world's first [[Calculus]] text. It contained most of the developments of earlier Kerala School mathematicians, particularly from Madhava. Similar to the work of Nilakantha, it is unique in the history of Indian mathematics, in that it contains proofs of theorems, derivations of rules and series, a derivation and proof of the Madhava-Gregory series of the arctangent function, proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other mathematicians of the Kerala School. It also contains a proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions.<ref name=Nilakantha>J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Nilakantha.html Nilakantha], ''[[MacTutor History of Mathematics archive]]''.</ref>
He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using ''kuttaka'' method, and rules of finding the sines and the cosines of the sum and difference of two angles.<ref name=Nilakantha/> Jyesthadevan also gave the earliest statement of [[John Wallis|Wallis]]' theorem, and geometrical derivations of infinite series.
===Sankaran Varma===
There remains a final Kerala work worthy of a brief mention, ''Sadratnamala'' an astronomical treatise written by [[Shankara Variyar|Sankara Varma]] (1800-1838) that serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands out as the last notable name in Keralan mathematics. A remarkable contribution was his compution of π correct to 17 decimal places.<ref name=Madhava/>
==Possibility of transmission of Kerala School results to Europe==
Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from [[Kerala]] by traders and [[Jesuit]] missionaries.<ref name=almeida/> Kerala was in continuous contact with [[China]] and [[Arabia]], and [[Europe]]. The existence of [[circumstantial evidence]]<ref>C. K. Raju (2001). "Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa", ''Philosophy East and West'' '''51''' (3), p. 325-362.</ref> such as communication routes and a suitable chronology certainly 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>Almeida, D. F., J. K. John, and A. Zadorozhnyy. 2001. "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications." ''Journal of Natural Geometry'', 20:77-104.</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> Gold, D. and D. Pingree. 1991. "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine." ''Historia Scientiarum''. 42:49-65.</ref>
Both [[Islamic mathematics|Arab]] and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.<ref name=katz/> However, they were not able to, as [[Isaac Newton|Newton]] and [[Gottfried Leibniz|Leibniz]] were, 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/> 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, 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 [[Centre national de la recherche scientifique]] in [[Paris]].<ref name=katz/>
==Notes==
<div style="font-size: 90%">
<references/>
</div>
==References==
<!--<div style="font-size: 90%">-->
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| title=Discovery of the Series Formula for <math> \pi </math> by Leibniz, Gregory, and Nilakantha
| journal=Mathematics Magazine (Math. Assoc. Amer.)
| volume=63
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*Sarma, K. V. and S. Hariharan: ''Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy - an analytical appraisal'', Indian J. Hist. Sci. 26 (2) (1991), 185-207
*{{Harvard reference
| last=Singh
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*{{Harvard reference
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*Tacchi Venturi. 'Letter by Matteo Ricci to Petri Maffei on 1 Dec 1581', ''Matteo Ricci S.I., Le Lettre Dalla Cina 1580–1610'', vol. 2, Macerata, 1613.
</div>
==External links==
* ''[http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_kerala.htm The Kerala School, European Mathematics and Navigation]'', 2001.
*[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.canisius.edu/topos/rajeev.asp Neither Newton nor Leibnitz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala], 2005.
*[http://www.physorg.com/news106238636.html "Indians predated Newton 'discovery' by 250 years"] ''phys.org,'' 2007
==See also==
*[[Indian astronomy]]
*[[Indian mathematics]]
*[[Indian mathematicians]]
*[[History of mathematics]]
[[Category:Hindu astronomy]]
[[Category:History of mathematics]]
[[Category:History of astronomy]]
[[Category:Kerala school|*]]
[[es:Escuela de Kerala]]