View source for Madhava of Sangamagrama

{{Short description|Indian mathematician and astronomer (1340-1425)}}
{{pp|small=yes}} 
{{Use Indian English|date=March 2013}}{{Use dmy dates|date=July 2022}}
{{infobox person
| native_name        = 
| native_name_lang   = ml
| name               = Mādhava of Sangamagrāma
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| birth_date         = {{circa|1340}}<ref name=rajag78/><ref name="Roy1990">{{cite journal| first=Ranjan| last=Roy| year=1990| title=The Discovery of the Series Formula for {{pi}} by Leibniz, Gregory and Nilakantha| journal=Mathematics Magazine| volume=63| issue=5| pages=291–306| url=http://mathdl.maa.org/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf| doi=10.2307/2690896| jstor=2690896| archive-url=https://web.archive.org/web/20120224013439/http://mathdl.maa.org/images/upload_library/22/Allendoerfer/1991/0025570x.di021167.02p0073q.pdf| archive-date=24 February 2012| url-status=dead}}</ref><ref name=Pearce>Ian G. Pearce (2002). [https://web.archive.org/web/20030430190329/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>
| birth_place        = [[Sangamagrama]], [[Kingdom of Cochin]] <br/> (modern day [[Irinjalakuda]], [[Kerala]], [[India]])
| death_date         = {{circa|1425}} (aged 75–85)
| death_place        = [[Kingdom of Cochin|Cochin]], [[Vijayanagara Empire]] <br/> (modern day [[Kerala]], [[India]])
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| known_for          = Discovery of [[power series]] <br/> Expansions of trigonometric [[Sine]], [[Cosine]] and [[Arctangent]] functions <br/>[[Infinite series]] summation formulae for {{pi}}
| education          = 
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| employer           = 
| notable works      = ''Golavāda'', ''Madhyāmanayanaprakāra'', ''[[Venvaroha|Veṇvāroha]]'', ''[[Sphuṭacandrāpti]]''
| occupation         = [[Astronomer]]-[[mathematician]]
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| title              = ''Golavid'' (Master of Spherics)
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}}
'''Mādhava of Sangamagrāma''' ('''Mādhavan''')<ref name="KVS1">{{cite book |last1=K. V. Sarma |title=[[A History of the Kerala School of Hindu Astronomy]] (in perspective) |date=1972 |publisher=Vishveshvaranand Institute of Sanskrit & Indological Studies, [[Panjab University]] |location=Hoshiarpur |page=51 |bibcode=1972hksh.book.....S |ref=KVS}} Available [https://archive.org/download/KeralaSchoolOfAstronomy/Kerala%20School%20of%20Astronomy.pdf]</ref> ({{Circa|1340|1425}}) was an Indian [[mathematician]] and [[astronomer]] who is considered to be the founder of the [[Kerala school of astronomy and mathematics]] in the [[Medieval India|Late Middle Ages]]. Madhava made pioneering contributions to the study of [[Series (mathematics)|infinite series]], [[calculus]], [[trigonometry]], [[geometry]] and [[algebra]]. He was the first to use infinite series approximations for a range of trigonometric functions, which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their [[Limit (mathematics)|limit]]-passage to [[infinity]]".<ref name="rajag78">{{cite journal |last1=C. T. Rajagopal & M.S.Rangachari |title=On an Untapped Source of Medieval Keralese Mathematics |journal=Archive for History of Exact Sciences |date=1978 |volume=18 |issue=2 |page=101|doi=10.1007/BF00348142 |s2cid=51861422 }}</ref>

==Biography==
Little is known about Madhava's life with certainty. However, from scattered references to Madhava found in diverse manuscripts, historians of Kerala school have pieced together information about the mathematician. In a manuscript preserved in the Oriental Institute, Baroda, Madhava has been referred to as ''Mādhavan vēṇvārōhādīnām karttā ... Mādhavan Ilaññippaḷḷi Emprān''.<ref name ="KVS1"/> It has been noted that the epithet 'Emprān' refers to the  [[Embranthiri|Emprāntiri]] community, to which Madhava might have belonged.<ref name="PPD"/>

The term "Ilaññippaḷḷi" has been identified as a reference to the residence of Madhava. This is corroborated by Madhava himself. In his short work on the moon's positions titled ''[[Venvaroha|Veṇvāroha]]'', Madhava says that he was born in a house named ''bakuḷādhiṣṭhita . . . vihāra''.<ref name="Sphuta">{{cite book |last1=K. V. Sarma |title=Computation of the True Moon by Madhava of sangamagrama |date=1973 |publisher=Vishveshvaranand Institute of Sanskrit and Indological Studies, Panjab University |location=Hoshiarpur |page=12}} Available: [https://archive.org/download/SphutaChandrapti/Sphuta-Chandrapti.pdf] (Accessed on 1 January 2023)</ref> This is clearly Sanskrit for ''Ilaññippaḷḷi''. ''Ilaññi'' is the Malayalam name of the evergreen tree  ''[[Mimusops elengi]]'' and the Sanskrit name for the same is ''Bakuḷa''. Palli is a term for village. The Sanskrit house name ''bakuḷādhiṣṭhita . . . vihāra'' has also been interpreted as a reference to the Malayalam house name ''Iraññi ninna ppaḷḷi'' and some historians have tried to identify it with one of two currently existing houses with names ''Iriññanavaḷḷi'' and ''Iriññārapaḷḷi'' both of which are located near [[Irinjalakuda]] town in central Kerala.<ref name="Sphuta"/> This identification is far fetched because both names have neither phonetic similarity nor semantic equivalence to the word "Ilaññippaḷḷi".<ref name="PPD">{{cite book |last1=P. P. Divakaran |title=The Mathematics of India: Concepts, Methods, Connections |date=2018 |publisher=Springer - Hindustan Book Agency |location=Cochin |isbn=978-981-13-1773-6 |pages=282–290}}</ref>

Most of the writers of astronomical and mathematical works who lived after Madhava's period have referred to Madhava as "Sangamagrama Madhava" and as such it is important that the real import of the word "Sangamagrama" be made clear. The general view among many scholars is that Sangamagrama is the town of [[Irinjalakuda]] some 70 kilometers south of the Nila river and about 70 kilometers south of [[Cochin]].<ref name="PPD"/> It seems that there is not much concrete ground for this belief except perhaps the fact that the presiding deity of an early medieval temple in the town, the [[Koodalmanikyam Temple]], is worshiped as Sangameswara meaning the Lord of the Samgama and so Samgamagrama can be interpreted as the village of Samgameswara. But there are several places in [[Karnataka]] with ''samgama'' or its equivalent ''kūḍala'' in their names and with a temple dedicated to Samgamḗsvara, the lord of the confluence. ([[Kudalasangama]] in [[Bagalkot district]] is one such place with a celebrated temple dedicated to the Lord of the Samgama.)<ref name="PPD"/>

There is a small town on the southern banks of the Nila river, around 10 kilometers upstream from [[Tirunavaya]], called Kūḍallūr. The exact literal Sanskrit translation of this place name is Samgamagram: ''kūṭal'' in Malayalam means a confluence (which in Sanskrit is ''samgama'') and ''ūr'' means a village (which in Sanskrit is ''grama''). Also the place is at the confluence of the Nila river and its most important tributary, namely, the Kunti river. (There is no confluence of rivers near Irinjalakuada.) Incidentally there is still existing a [[Nambudiri]] (Malayali Brahmin) family by name ''Kūtallūr Mana'' a few kilometers away from the Kudallur village. The family has its origins in Kudallur village itself. For many generations this family hosted a great ''[[Gurukulam]]'' specialising in [[Vedanga]].<ref name="PPD"/> That the only available manuscript of ''[[Sphuṭacandrāpti]]'', a book authored by Madhava, was obtained from the manuscript collection of ''Kūtallūr Mana'' might strengthen the conjecture that Madhava might have had some association with ''Kūtallūr Mana''.<ref>{{cite book |last1=K. V. Sarma |title=Sputachandrapti: Computation of the True Moon by Madhava of Sangamagrama |date=1973 |publisher=Vishveshvaranand Institute of Sanskrit and Indological Studies, Panjab University |location=Hoshiarpur, Punjab |page=8}}</ref> Thus the most plausible possibility is that the forefathers of Madhava migrated from the Tulu land or thereabouts to settle in Kudallur village, which is situated on the southern banks of the Nila river not far from Tirunnavaya, a generation or two before his birth and lived in a house known as ''Ilaññippaḷḷi'' whose present identity is unknown.<ref name="PPD"/>

===Date===
There are also no definite evidences to pinpoint the period during which Madhava flourished. In his Venvaroha, Madhava gives a date in 1400 CE as the epoch. Madhava's pupil [[Parameshvara Nambudiri]], the only known direct pupil of Madhava, is known to have completed his seminal work [[Drigganita]] in 1430 and the Paramesvara's date has been determined as {{Circa|1360}}-1455. From such circumstantial evidences historians have assigned the date {{Circa|1340|1425}} to Madhava.

== Historiography ==
Although there is some evidence of mathematical work in Kerala prior to Madhava (''e.g.'', ''Sadratnamala'' {{which|date=December 2020}} c. 1300, a set of fragmentary results<ref name=whish/>), it is clear from citations that Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala.  However, except for a couple, most of Madhava's original works have been lost. He is referred to in the work of subsequent Kerala mathematicians, particularly in [[Nilakantha Somayaji]]'s ''Tantrasangraha'' (c. 1500), as the source for several infinite series expansions, including sin ''θ'' and arctan ''θ''.  The 16th-century text ''Mahajyānayana prakāra'' (Method of Computing Great Sines) cites Madhava as the source for several series derivations for {{pi}}.  In [[Jyeṣṭhadeva]]'s ''[[Yuktibhāṣā]]'' (c. 1530),<ref name="sarma">
{{cite web
| editor=K. V. Sarma
| editor-link=K. V. Sarma
| editor2=S. Hariharan
| work=Yuktibhāṣā of Jyeṣṭhadeva
| url=http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf
| title=A book on rationales in Indian Mathematics and Astronomy—An analytic appraisal
| access-date=2006-07-09
| archive-url = https://web.archive.org/web/20060928203221/http://www.new.dli.ernet.in/insa/INSA_1/20005ac0_185.pdf <!-- Bot retrieved archive --> |archive-date = 28 September 2006}}
</ref> written in [[Malayalam]], these series are presented with proofs in terms of the [[Taylor series]] expansions for polynomials like 1/(1+''x''<sup>2</sup>), with ''x'' = tan&nbsp;''θ'', etc.

Thus, what is explicitly Madhava's work is a source of some debate.  The ''Yukti-dipika'' (also called the ''Tantrasangraha-vyakhya''), possibly composed by [[Sankara Variar]], a student of Jyeṣṭhadeva, presents several versions of the series expansions for sin ''θ'', cos ''θ'', and arctan ''θ'', as well as some products with radius and arclength, most versions of which appear in Yuktibhāṣā.  For those that do not, Rajagopal and Rangachari have argued, quoting extensively from the original Sanskrit,<ref name=rajag78/> that since some of these have been attributed by Nilakantha to Madhava, some of the other forms might also be the work of Madhava.

Others have speculated that the early text ''[[Karanapaddhati]]'' (c. 1375–1475), or the ''Mahajyānayana prakāra'' was written by Madhava, but this is unlikely.<ref name=Pearce/>

''Karanapaddhati'', along with the even earlier Keralite mathematics text ''Sadratnamala'', as well as the ''Tantrasangraha'' and ''Yuktibhāṣā'', were considered in an 1834 article by [[C. M. Whish]], which was the first to draw attention to their priority over Newton in discovering the [[Method of Fluxions|Fluxion]] (Newton's name for differentials).<ref name=whish>{{Cite journal
 | author    = Charles Whish
 | year      = 1834
 | title     = On the Hindu Quadrature of the circle and the infinite series of the proportion of the circumference to the diameter exhibited in the four Sastras, the Tantra Sahgraham, Yucti Bhasha, Carana Padhati and Sadratnamala
 | journal   = Transactions of the Royal Asiatic Society of Great Britain and Ireland
 | publisher = [[Royal Asiatic Society of Great Britain and Ireland]]
 | doi       = 10.1017/S0950473700001221
 | volume    = 3
 | issue     = 3
 | pages     = 509–523
 | jstor     = 25581775
 | url       = https://zenodo.org/record/2223599
 }}
</ref> In the mid-20th century, the Russian scholar Jushkevich revisited the legacy of Madhava,<ref>{{cite book
 | title  = ''Geschichte der Mathematik im Mittelalter'' (German translation, Leipzig, 1964, of the Russian original, Moscow, 1961).
 | author = A.P. Jushkevich
 | year   = 1961
 | place  = Moscow
}}</ref> and a comprehensive look at the Kerala school was provided by Sarma in 1972.<ref name=sarma1972>{{cite book
 | title       = A History of the Kerala School of Hindu Astronomy
 | author      = K V Sarma
 | author-link = K V Sarma
 | year        = 1972
 | location    = Hoshiarpur
 | title-link  = A History of the Kerala School of Hindu Astronomy
 }}</ref>

=== Lineage ===
[[Image:Yuktibhasa.svg|200px|thumb|Proof of the [[Pythagorean theorem]] in ''[[Yuktibhāṣā]]'']]

There are several known astronomers who preceded Madhava, including Kǖṭalur Kizhār (2nd century),<ref>Purananuru 229</ref> [[Vararuci#Vararuci.2C the astronomer|Vararuci (4th century)]], and [[Śaṅkaranārāyaṇa]] (866 AD).  It is possible that other unknown figures preceded him.  However, we have a clearer record of the tradition after Madhava.  [[Parameshvara]] was a direct disciple.  According to a [[Palm-leaf manuscript|palm leaf manuscript]] of a Malayalam commentary on the [[Surya Siddhanta]], Parameswara's son Damodara (c. 1400–1500) had Nilakantha Somayaji as one of his disciples. Jyeshtadeva was a disciple of Nilakantha. [[Achyutha Pisharadi]] of Trikkantiyur is mentioned as a disciple of Jyeṣṭhadeva, and the grammarian [[Melpathur Narayana Bhattathiri]] as his disciple.<ref name=sarma/>

==Contributions==

If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite, then the first steps towards this transition typically come with infinite series expansions. It is this transition to the infinite series that is attributed to Madhava. In Europe, the first such series were developed by [[James Gregory (mathematician)|James Gregory]] in 1667. Madhava's work is notable for the series, but what is truly remarkable is his estimate of an error term (or correction term).<ref name=rajag86>Madhava extended Archimedes' work on the geometric Method of Exhaustion to measure areas and numbers such as {{pi}}, with arbitrary accuracy and error ''limits'', to an algebraic infinite series with a completely separate error ''term''.
{{cite journal
 | title   = On medieval Keralese mathematics
 | author  = C T Rajagopal and M S Rangachari
 | journal = Archive for History of Exact Sciences
 | volume  = 35
 | issue   = 2
 | year    = 1986
 | pages   = 91–99
 | doi     = 10.1007/BF00357622
 | s2cid   = 121678430
 }}</ref> This implies that he understood very well the limit nature of the infinite series.  Thus, Madhava may have invented the ideas underlying [[infinite series]] expansions of functions, [[power series]], [[trigonometric series]], and rational approximations of infinite series.<ref name="MAT 314">{{cite web |title=Neither Newton nor Leibniz – The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala |url=http://www.canisius.edu/topos/rajeev.asp |url-status=dead |archive-url=https://web.archive.org/web/20060806040307/http://www.canisius.edu/topos/rajeev.asp |archive-date=6 August 2006 |access-date=2006-07-09 |work=MAT 314 |publisher=Canisius College}}
</ref>

However, as stated above, which results are precisely Madhava's and which are those of his successors is difficult to determine.  The following presents a summary of results that have been attributed to Madhava by various scholars.

===Infinite series===
{{main article|Madhava series}}

Among his many contributions, he discovered infinite series for the [[trigonometric function]]s of [[sine]], [[cosine]], [[arctangent]], and many methods for calculating the [[circumference]] of a [[circle]]. One of Madhava's series is known from the text ''[[Yuktibhāṣā]]'', which contains the derivation and proof of the [[power series]] for [[Inverse trigonometric function|inverse tangent]], discovered by Madhava.<ref name="infinity">{{cite web
| publisher=D.P. Agrawal—Infinity Foundation
|work=Indian Mathemematics
| url=http://www.infinityfoundation.com/mandala/t_es/t_es_agraw_kerala.htm
| title=The Kerala School, European Mathematics and Navigation
| access-date=2006-07-09
}}
</ref> In the text, [[Jyeṣṭhadeva]] describes the series in the following manner:
{{cquote|The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.<ref name=Gupta>
{{cite journal
 | author  = R C Gupta
 | title   = The Madhava-Gregory series
 | journal = Math. Education
 | volume  = 7
 | year    = 1973
 | pages   = B67–B70
}}</ref>}}
This yields:
:<math> r\theta={\frac {r\sin  \theta  }{\cos  \theta
 }}-(1/3)\,r\,{\frac { \left(\sin \theta   \right) ^
{3}}{ \left(\cos  \theta   \right) ^{3}}}+(1/5)\,r\,{\frac {
 \left(\sin \theta  \right) ^{5}}{ \left(\cos
\theta  \right) ^{5}}}-(1/7)\,r\,{\frac { \left(\sin \theta
 \right) ^{7}}{ \left(\cos \theta  \right) ^{
7}}} + \cdots</math>
or equivalently:
:<math>\theta = \tan \theta - \frac{\tan^3 \theta}{3} + \frac{\tan^5 \theta}{5} - \frac{\tan^7 \theta}{7} + \cdots</math>

This series is [[Gregory's series]] (named after [[James Gregory (mathematician)|James Gregory]], who rediscovered it three centuries after Madhava).  Even if we consider this particular series as the work of [[Jyeṣṭhadeva]], it would pre-date Gregory by a century, and certainly other infinite series of a similar nature had been worked out by Madhava.  Today, it is referred to as the [[Gregory's series|Madhava-Gregory-Leibniz series]].<ref name=Gupta/><ref name=nair>{{cite web
 |publisher    = Prof. C.G.Ramachandran Nair
 |work         = Government of Kerala—Kerala Call, September 2004
 |url          = http://www.kerala.gov.in/keralcallsep04/p22-24.pdf
 |title        = Science and technology in free India
 |access-date  = 2006-07-09
 |url-status   = dead
 |archive-url  = https://web.archive.org/web/20060821195309/http://www.kerala.gov.in/keralcallsep04/p22-24.pdf
 |archive-date = 21 August 2006
}}
</ref>

===Trigonometry===

{{main article|Madhava's sine table}}

Madhava composed an accurate table of sines. Madhava's values are accurate to the seventh decimal place. Marking a quarter circle at twenty-four equal intervals, he gave the lengths of the half-chord (sines) corresponding to each of them. It is believed that he may have computed these values based on the series expansions:<ref name=mact-biog>{{cite web
 |publisher    = School of Mathematics and Statistics, [[University of St Andrews]], Scotland
 |title        = Madhava of Sangamagramma
 |author       = J. J. O'Connor and E. F. Robertson
 |year         = 2000
 |url          = http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html
 |archive-url  = https://web.archive.org/web/20060514012903/http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html
 |url-status   = dead
 |archive-date = 2006-05-14
 |work         = [[MacTutor History of Mathematics archive]]
 |access-date  = 2007-09-08
}}</ref>

:   sin ''q'' = ''q'' − ''q''<sup>3</sup>/3! + ''q''<sup>5</sup>/5! − ''q''<sup>7</sup>/7! + ...
:   cos ''q'' = 1 − ''q''<sup>2</sup>/2! + ''q''<sup>4</sup>/4! − ''q''<sup>6</sup>/6! + ...

===The value of {{pi}} (pi)===
{{main article|Madhava's correction term}}

Madhava's work on the value of the mathematical [[Pi|constant Pi]] is cited in the ''Mahajyānayana prakāra'' ("Methods for the great sines").{{citation needed|date=September 2012}}  While some scholars such as Sarma<ref name=sarma/> feel that this book may have been composed by Madhava himself, it is more likely the work of a 16th-century successor.<ref name=mact-biog/> This text attributes most of the expansions to Madhava, and gives the following [[Series (mathematics)|infinite series]] expansion of [[Pi|{{pi}}]], now known as the [[Leibniz formula for π|Madhava-Leibniz series]]:<ref>{{Cite book |title=Special Functions |url=https://archive.org/details/specialfunctions00andr_631 |url-access=limited |last=George E. Andrews, Richard Askey |first=Ranjan Roy |publisher=[[Cambridge University Press]] |year=1999 |isbn=0-521-78988-5 |page=[https://archive.org/details/specialfunctions00andr_631/page/n74 58]}}</ref><ref>{{Cite journal |first=R. C. |last=Gupta |title=On the remainder term in the Madhava-Leibniz's series |journal=Ganita Bharati |volume=14 |issue=1–4 |year=1992 |pages=68–71}}</ref>

: <math>\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots = \sum_{n=1}^\infty \frac{(-1)^{n-1}}{2n - 1},</math>

which he obtained from the power-series expansion of the arc-tangent function. However, what is most impressive is that he also gave a correction term ''R<sub>n</sub>'' for the error after computing the sum up to ''n'' terms,<ref name=mact-biog/> namely:

: ''R<sub>n</sub>'' = (−1)<sup>''n''</sup> / (4''n''), or
: ''R<sub>n</sub>'' = (−1)<sup>''n''</sup>⋅''n'' / (4''n''<sup>2</sup> + 1), or
: ''R<sub>n</sub>'' = (−1)<sup>''n''</sup>⋅(''n''<sup>2</sup> + 1) / (4''n''<sup>3</sup> + 5''n''),

where the third correction leads to highly accurate computations of {{pi}}.

It has long been speculated how Madhava found these correction terms.<ref>T. Hayashi, T. Kusuba and M. Yano. "The correction of the Madhava series for the circumference of a circle", ''[[Centaurus (journal)|Centaurus]]'' '''33''' (pages 149–174). 1990.</ref> They are the first three convergents of a finite continued fraction, which, when combined with the original Madhava's series evaluated to ''n'' terms, yields about 3''n''/2 correct digits:

: <math>\frac{\pi}{4} \approx 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots + \frac{(-1)^{n-1}}{2n - 1} + \cfrac{(-1)^n}{4n + \cfrac{1^2}{n + \cfrac{2^2}{4n + \cfrac{3^2}{n + \cfrac{4^2}{\dots + \cfrac{\dots}{\dots + \cfrac{n^2}{n[4 - 3(n \bmod 2)]}}}}}}}.
</math>

The absolute value of the correction term in next higher order is

: |''R<sub>n</sub>''| = (4''n''<sup>3</sup> + 13''n'') / (16''n''<sup>4</sup> + 56''n''<sup>2</sup> + 9).

He also gave a more rapidly converging series by transforming the original infinite series of {{pi}}, obtaining the infinite series

: <math>\pi = \sqrt{12}\left(1 - \frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3^2} - \frac{1}{7 \cdot 3^3} + \cdots\right).</math>

By using the first 21 terms to compute an approximation of {{pi}}, he obtains a value correct to 11 decimal places (3.14159265359).<ref name=gupta-pi>
{{cite journal
 | author  = R. C. Gupta
 | title   = Madhava's and other medieval Indian values of pi
 | journal = Math. Education
 | volume  = 9 | issue = 3
 | year    = 1975
 | pages   = B45–B48
}}</ref>
The value of 3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava,<ref>The 13-digit accurate value of {{pi}}, 3.1415926535898, can be reached using the infinite series expansion of {{pi}}/4 (the first sequence) by going up to n&nbsp;=&nbsp;76.</ref> but may be due to one of his followers. These were the most accurate approximations of {{pi}} given since the 5th century (see [[Approximations of π#Middle Ages|History of numerical approximations of {{pi}}]]).

The text ''Sadratnamala'' appears to give the astonishingly accurate value of {{pi}}&nbsp;=&nbsp;3.14159265358979324 (correct to 17 decimal places). Based on this, R. Gupta has suggested that this text was also composed by Madhava.<ref name=Pearce/><ref name=gupta-pi/>

Madhava also carried out investigations into other series for arc lengths and the associated approximations to rational fractions of {{pi}}.<ref name=Pearce>Ian G. Pearce (2002). [https://web.archive.org/web/20030430190329/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>

===Calculus===
Madhava  developed the [[power series]] expansion for some trigonometry functions which were further developed by his successors at the [[Kerala school of astronomy and mathematics]].<ref name="scotlnd">{{cite web
| publisher=School of Mathematics and Statistics University of St Andrews, Scotland
|work=Indian Maths
| url=http://www-history.mcs.st-andrews.ac.uk/HistTopics/Indian_mathematics.html
| title=An overview of Indian mathematics
| access-date=2006-07-07
}}
</ref> (Certain ideas of calculus were known to [[History of calculus|earlier mathematicians]].) Madhava also extended some results found in earlier works, including those of [[Bhāskara II]].<ref name="scotlnd" /> However, they did not combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, or turn calculus into the powerful problem-solving tool we have today.<ref name=":0">{{Cite journal |last=Katz |first=Victor J. |date=1995-06-01 |title=Ideas of Calculus in Islam and India |url=https://doi.org/10.1080/0025570X.1995.11996307 |journal=Mathematics Magazine |volume=68 |issue=3 |pages=163–174 |doi=10.1080/0025570X.1995.11996307 |issn=0025-570X}}</ref>

==Madhava's works==
[[K. V. Sarma]] has identified Madhava as the author of the following works:<ref>{{cite book |last=Sarma |first=K. V. |title=Contributions to the study of Kerala school of Hindu astronomy and mathematics |publisher=V V R I |location=Hoshiarpur |year=1977}}</ref><ref>{{cite book |last=David Edwin Pingree |title=Census of the exact sciences in Sanskrit |publisher=American Philosophical Society |location=Philadelphia |year=1981 |volume=4 |pages=414–415}}</ref>

# ''Golavada''
# ''Madhyamanayanaprakara''
# ''Mahajyanayanaprakara'' (Method of Computing Great Sines)
# ''Lagnaprakarana'' ({{lang|sa|लग्नप्रकरण}})
# ''[[Venvaroha]]'' ({{lang|sa|वेण्वारोह}})<ref>{{cite journal |last=K. Chandra Hari |year=2003 |title=Computation of the true moon by Madhva of Sangamagrama |journal=Indian Journal of History of Science |volume=38 |issue=3 |pages=231–253 |url=https://www.scribd.com/doc/14648892/Venvaroha-Computation-of-Moon-Madhava-of-a |access-date=27 January 2010}}</ref>
# ''[[Sphuṭacandrāpti]]'' ({{lang|sa|स्फुटचन्द्राप्ति}})
# ''Aganita-grahacara'' ({{lang|sa|अगणित-ग्रहचार}})
# ''[[Chandravakyas|Chandravakyani]]'' ({{lang|sa|चन्द्रवाक्यानि}}) (Table of Moon-mnemonics)

==Kerala School of Astronomy and Mathematics==
{{Main|Kerala school of astronomy and mathematics}}
The Kerala school of astronomy and mathematics, founded by Madhava, flourished between the 14th and 16th centuries, and included among its members [[Parameshvara]], [[Neelakanta Somayaji]], [[Jyeshtadeva]], [[Achyuta Pisharati]], [[Melpathur Narayana Bhattathiri]] and Achyuta Panikkar. The group is known for series expansion of three trigonometric functions of sine, cosine and arctant and proofs of their results where later given in the ''[[Yuktibhasa]]''.<ref name=whish/><ref name="scotlnd" /><ref name=":0"/> The group also did much other work in astronomy: more pages are devoted to astronomical computations than purely mathematical results.<ref name=sarma/>

The Kerala school also contributed to linguistics (the relation between language and mathematics is an ancient Indian tradition, see [[Kātyāyana]]). The [[Ayurveda|ayurvedic]] and poetic traditions of [[Kerala]] can be traced back to this school.  The famous poem, [[Narayaniyam]], was composed by [[Melpathur Narayana Bhattathiri|Narayana Bhattathiri]].

==Influence==

Madhava has been called "the greatest mathematician-astronomer of medieval India",<ref name=Pearce/>  some of his discoveries in this field show him to have possessed extraordinary intuition".<ref name=jos>{{Cite book |last=Joseph |first=George Gheverghese |orig-year=1991 |date=October 2010 |title=The Crest of the Peacock: Non-European Roots of Mathematics |edition=3rd |publisher=Princeton University Press |isbn=978-0-691-13526-7 |url=http://press.princeton.edu/titles/9308.html}}</ref> O'Connor and Robertson state that a fair assessment of Madhava is that
he took the decisive step towards modern classical analysis.<ref name=mact-biog/>

===Possible propagation to Europe===

The Kerala school was well known in the 15th and 16th centuries, in the period of the first contact with European navigators in the [[Malabar Coast]].  At the time, the port of [[Muziris]], near [[Sangamagrama]], was a major center for maritime trade, and a number of [[Jesuit]] missionaries and traders were active in this region.  Given the fame of the Kerala school, and the interest shown by some of the Jesuit groups during this period in local scholarship, some scholars, including G. Joseph of the U. Manchester have suggested<ref name=predated>
{{cite news
 |title        = Indians predated Newton 'discovery' by 250 years
 |publisher    = press release, University of Manchester
 |url          = http://www.humanities.manchester.ac.uk/aboutus/news/display/?id=121685
 |date         = 13 August 2007
 |access-date  = 2007-09-05
 |url-status   = dead
 |archive-url  = https://web.archive.org/web/20080321094302/http://www.humanities.manchester.ac.uk/aboutus/news/display/?id=121685
 |archive-date = 21 March 2008
}}</ref> that the writings of the Kerala school may have also been transmitted to Europe around this time, which was still about a century before Newton.<ref name=almeida>{{cite journal
|author = D F Almeida, 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, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place.<ref name="almeida" /> 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=gold>{{Citation | last1 = Gold | first1 = D. | last2 = Pingree | first2 = D. | year = 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 | postscript = . }}</ref>

==See also==
{{div col|colwidth=30em}}
*[[Madhava Observatory]]
*[[Madhava's sine table]] 
*[[Madhava series]]
*[[Madhava's correction term]]
*[[Venvaroha]]
*[[Yuktibhāṣā]]                   
*[[Kerala school of astronomy and mathematics]]
*[[List of astronomers and mathematicians of the Kerala school]]
*[[List of Indian mathematicians]]                                    
*[[Indian mathematics]]                         
*[[History of calculus]]
{{div col end}}

==References==
{{reflist}}

==External links==
* [http://www-history.mcs.st-and.ac.uk/Biographies/Madhava.html Biography on MacTutor]

{{Kerala School}}
{{Indian mathematics}}

{{Authority control}}

{{DEFAULTSORT:Madhava of Sangamagrama}}
[[Category:1340s births]]
[[Category:1420s deaths]]
[[Category:Scientists from Kerala]]
[[Category:History of calculus]]
[[Category:Indian Hindus]]
[[Category:Kerala school of astronomy and mathematics]]
[[Category:14th-century Indian mathematicians]]
[[Category:15th-century Indian mathematicians]]
[[Category:People from Irinjalakuda]]
[[Category:15th-century Indian astronomers]]
[[Category:14th-century Indian astronomers]]
[[Category:Scholars from Kerala]]
[[Category:Mathematical series]]