- Madhava of Sangamagrama
Infobox Person

name=Mādhava of Sangamagrama

caption=

dead=dead

birth_date=1350

birth_place=Kerala ,India

death_date=1425

death_place=**Mādhava of Sangamagrama**(born as**Irinjaatappilly Madhavan Namboodiri**) (c.1350–c.1425) was a prominent Hindu mathematician-astronomer from the town ofIrinjalakkuda , nearCochin ,Kerala ,India , which was at the time known as "Sangamagrama " (lit. "sangama" = union, "grāma"=village). He is considered the founder of theKerala school of astronomy and mathematics . He is the first to have developed 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-passage toinfinity ".cite journal

title = On an untapped source of medieval Keralese mathematics

author = C T Rajagopal and M S Rangachari

journal = Archive for History of Exact Sciences

url = http://www.springerlink.com/content/mnr38341u762u544/?p=a9e26ffde91946b288bcb6deebac245c&pi=0

volume = 18 (2)

month = June | year = 1978

pages = 89–102] His discoveries opened the doors to what has today come to be known asmathematical analysis .cite web

publisher=School of Mathematics and Statistics University of St Andrews, Scotland | title=Biography of Madhava

author = J J O'Connor and E F Robertson

url=http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Madhava.html

title=Madhava of Sangamagrama

accessdate=2007-09-08] One of the greatest mathematician-astronomers of theMiddle Ages , Madhava contributed toinfinite series ,calculus ,trigonometry ,geometry andalgebra .Some scholars have also suggested that Madhava's work, through the writings of the Kerala school, may have been transmitted to Europe via

Jesuit missionaries and traders who were active around the ancient port of Kochi at the time. As a result, it may have had an influence on later European developments in analysis and calculus.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

number=1

year =2001

pages=77–104]**Historiography**Although there is some evidence of Mathematical work in Kerala prior to Madhava (e.g. "Sadratnamala" c.1300, a set of fragmentary results), it is clear from citations that Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala. However, most of Madhava's original work (possibly excepting an astronomy text) is 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 "sinθ" and "arctanθ". The 16th c. text "Mahajyānayana prakāra" cites Madhava as the source for several series derivations for π. InJyesthadeva 's "Yuktibhasa " (c.1530cite web

publisher=K V Sharma & S Hariharan

work=Yuktibhasa of Jyesthadeva

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

accessdate=2006-07-09

format=PDF] ), written in Malayalam, these series are presented with proofs in terms of theTaylor series expansions for polynomials like 1/(1+x^{2}), with x = "tanθ", etc.Thus, what is explicitly Madhava's work is a source of some debate. The "Yukti-dipika" (also called the "Tantrasangraha-vyakhya"), possibly composed Sankara Variyar, a student of Jyesthadeva, presents several versions of the series expansions for "sinθ", "cosθ", and "arctanθ", as well as some products with radius and arclength, most versions of which appear in Yuktibhasa. For those that do not, Rajagopal and Rangachari have argued, quoting extensively from the original Sanskrit, that since some of these have been attributed by Nilakantha to Madhava, possibly some of the other forms might also be the work of Madhava.

Others have speculated that the early text "Karana Paddhati" (c.1375-1475), or the "Mahajyānayana prakāra" might have been written by Madhava, but this is unlikely.

"Karana Paddhati", along with the even earlier Keralese mathematics text "Sadratnamala", as well as the "Tantrasangraha" and "Yuktibhasa", were considered in an 1835 article by Charles Whish, which was the first to draw attention to their priority over Newton in discovering the Fluxion (Newton's name for differentials)cite journal

author =Charles Whish

year = 1835

title = On the Hindu Quadrature of the Circle, and the infinite Series ofthe proportion of the circumference to the diameter exhibited in the four shastras: the Tantra Sangraham, Yucti Bhasha, Carana Padhati, and Sadratnamala

journal = Transactions of the Royal Asiatic Society of Great Britain and Ireland

volume = 3

pages = 509–523] . In the mid-20th century, the Russian scholar Jushkevich revisited the legacy of Madhava [*cite book*] , and a comprehensive look at the Kerala school was provided by Sarma in 1972cite book

title = Geschichte der Mathematik im Mittelalter (German translation, Leipzig, 1964, of the Russian original, Moscow, 1961).

author = A.P. Jushkevich,

year = 1961

address = Moscow

title = A History of the Kerala School of Hindu Astronomy

author = K V Sarma

year = 1972

address = Hoshiarpur] .**Lineage**[

200px|thumb|Explanation_of_the_sine rule in "Yuktibhasa "]Before Madhava, there is a large gap in the Indian mathematical tradition, and in particular, there is little known about any tradition of Mathematics in Kerala. It is possible that other unknown figures may have preceded him. However, we have a clearer record of the tradition after Madhava.

Parameshvara Namboodri was possibly a direct disciple. According to a palmleaf manuscript of a Malayalam commentary on theSurya Siddhanta , Parameswara's son Damodara (c. 1400-1500) had both Nilakantha and Jyesthadeva as his disciples.Achyuta Pisharati of Trikkantiyur is mentioned as a disciple of Jyeshtadeva, and the grammarianMelpathur Narayana Bhattathiri as his disciple.**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 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)cite journal

title = On medieval Keralese mathematics,

author = C T Rajagopal and M S Rangachari

journal = Archive for History of Exact Sciences

url = http://www.springerlink.com/content/t1343xktl7g52003/

volume = 35

year = 1986

pages = 91–99

doi = 10.1007/BF00357622 ] . This implies that the limit nature of the infinite series was quite well understood by him. Thus, Madhava may have invented the ideas underlyinginfinite series expansions of functions,power series ,Trigonometric series , and rational approximations of infinite series.However, as stated above, which results are precisely Madhava's and which are those of his successors, are somewhat difficult to determine. The following presents a summary of results that have been attributed to Madhava by various scholars.

**Infinite series**Among his many contributions, he discovered the infinite series for the

trigonometric function s ofsine ,cosine , tangent andarctangent , and many methods for calculating thecircumference of acircle . One of Madhava's series is known from the text "Yuktibhasa ", which contains the derivation and proof of thepower series for inverse tangent, discovered by Madhava.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

accessdate=2006-07-09] In the text,Jyesthadeva 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.cite journal

author = R C Gupta

title = The Madhava-Gregory series

journal = Math. Education

volume = 7

year = 1973

pages = B67–B70] This yields$r\; heta=\{frac\; \{rsin\; heta\; \}\{cos\; heta\; -(1/3),r,\{frac\; \{\; left(sin\; heta\; ight)\; ^\{3\{\; left(cos\; heta\; ight)\; ^\{3\}+(1/5),r,\{frac\; \{\; left(sin\; heta\; ight)\; ^\{5\{\; left(cos\; heta\; ight)\; ^\{5\}-(1/7),r,\{frac\; \{\; left(sin\; heta\; ight)\; ^\{7\{\; left(cos\; heta\; ight)\; ^\{7\}\; +\; ...$which further yields the result::$heta\; =\; an\; heta\; -\; (1/3)\; an^3\; heta\; +\; (1/5)\; an^5\; heta\; -\; ldots$This series was traditionally known as the Gregory series (after James Gregory, who discovered it three centuries after Madhava). Even if we consider this particular series as the work of

Jyeshtadeva , 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 Madhava-Gregory seriescite 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

accessdate=2006-07-09

format=PDF] .**Trigonometry**Madhava also gave a most accurate table of sines, defined in terms of the values of the half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is believed that he may have found these highly accurate tables based on these series expansions:

: sin q = q - q

^{3}/3! + q^{5}/5! - ...: cos q = 1 - q^{2}/2! + q^{4}/4! - ...**The value of π (pi)**We find Madhava's work on the value of π cited in the "Mahajyānayana prakāra" ("Methods for the great sines"). While some scholars such as Sarma feel that this book may have been composed by Madhava himself, it is more likely the work of a 16th century successor . This text attributes most of the expansions to Madhava, and gives the following infinite series expansion of π, now known as the Madhava-Leibniz series: [

*citation|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=*] [Cambridge University Press |year=1999|isbn=0521789885|page=58*citation|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*]:$frac\{pi\}\{4\}\; =\; 1\; -\; frac\{1\}\{3\}\; +\; frac\{1\}\{5\}\; -\; frac\{1\}\{7\}\; +\; cdots\; +\; frac\{(-1)^n\}\{2n\; +\; 1\}\; +\; cdots$

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

_{n}", for the error after computing the sum up to n terms. Madhava gave three forms of Rn which improved the approximation, namely: R

_{n}= 1/(4n), or: R_{n}= n/ (4n^{2}+ 1), or: R_{n}= (n^{2}+ 1) / (4n^{3}+ 5n).where the third correction leads to highly accurate computations of π.

It is not clear how Madhava might have found these correction terms [

*T. Hayashi, T. Kusuba and M. Yano. 'The correction of the Madhava series for the circumference of a circle', "Centaurus"*] . The most convincing is that they come as the first three convergents of a continued fraction which can itself be derived from the standard Indian approximation to π namely 62832/20000 (for the original 5th c. computation, see**33**(pages 149-174). 1990.Aryabhata ).He also gave a more rapidly converging series by transforming the original infinite series of π, obtaining the infinite series

:$pi\; =\; sqrt\{12\}left(1-\{1over\; 3cdot3\}+\{1over5cdot\; 3^2\}-\{1over7cdot\; 3^3\}+cdots\; ight)$By using the first 21 terms to compute an approximation of π, he obtains a value correct to 11 decimal places (3.14159265359)cite journal

author = R C Gupta

title = Madhava's and other medieval Indian values of pi

journal = Math. Education

volume = 9 (3)

year = 1975

pages = B45–B48] .The value of3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava [*The 13-digit accurate value of π, 3.1415926535898, can be reached using the infinite series expansion of π/4 (the first sequence) by going up to n = 76*] ,but may be due to one of his followers. These were the most accurate approximations of π given since the 5th century (seeHistory of numerical approximations of π ).The text "Sadratnamala", usually considered as prior to Madhava, appears to give the astonishingly accurate value of π =3.14159265358979324 (correct to 17 decimal places). Based on this, R. Gupta has argued that this text may also have been composed by Madhava.

**Algebra**Madhava also carried out investigations into other series for arclengths and the associated approximations to rational fractions of π, found methods of

polynomial expansion , discovered tests of convergence of infinite series, and the analysis of infinitecontinued fraction s.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 .] He also discovered the solutions of transcendental equations byiteration , and found the approximation oftranscendental number s by continued fractions.**Calculus**Madhava laid the foundations for the development of

calculus , which were further developed by his successors at theKerala school of astronomy and mathematics .cite web

publisher=Canisius College

work=MAT 314

url=http://www.canisius.edu/topos/rajeev.asp

title=Neither Newton nor Leibniz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala

accessdate=2006-07-09] 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

accessdate=2006-07-07] (It should be noted that certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those ofBhaskara .In

calculus , he used early forms of differentiation, integration, and either he, or his disciples developed integration for simple functions.**Kerala School of Astronomy and Mathematics**The Kerala school of astronomy and mathematics flourished for at least two centuries beyond Madhava. In Jyesthadeva we find the notion of integration, termed "sankalitam", (lit. collection), as in the statement:

:"ekadyekothara pada sankalitam samam padavargathinte pakuti",

which translates as the integration a variable ("pada") equals half thatvariable squared ("varga"); i.e. The integral of x dx is equal tox

^{2}/ 2. This is clearly a start to the process ofintegral calculus . A related result states that the area under a curve is itsintegral . Most of these results pre-date similar results in Europe by several centuries. In many senses, Jyeshtadeva's "Yuktibhasa " may be considered the world's firstcalculus text.The group also did much other work in astronomy; indeed many more pages are developed to astronomical computations than are for discussing analysis related results.

The Kerala school also contributed much to linguistics (the relation between language and mathematics is an ancient Indian tradition, see

Katyayana ). The ayurvedic and poetic traditions ofKerala can also be traced back to this school. The famous poem,Narayaneeyam , was composed by Narayana Bhattathiri.**Influence**Madhava has been called "the greatest mathematician-astronomer of medieval India", or as "the founder of mathematical analysis; some of his discoveries in this field show him to have possessed extraordinary intuition."cite book

title = The crest of the peacock

author = G G Joseph

year = 1991

address = London] . O'Connor and Robertson state that a fair assessment of Madhava is thathe took the decisive step towards modern classical analysis.**Propagation to Europe?**The Kerala school was well known in the 15th-16th c., in the period of the first contact with European navigators in the

Malabar coast. At the time, the port of Kochi, nearSangamagrama , was a major center for maritime trade, and a number ofJesuit 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 suggestedcite 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 =2007-08-13

accessdate = 2007-09-05] 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. While no European translations have been discovered of these texts, it is possible that these ideas may still have had an influence on later European developments in analysis and calculus. (See Kerala school for more details).**References****ee also***

Indian mathematics

*List of Indian mathematicians

*Kerala school of astronomy and mathematics

*History of calculus

*Wikimedia Foundation.
2010.*

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**Mādhava of Sañgamāgrama**— Madhava Born c.1350 Died c.1425 Residence Sangamagrama (Irinjalakuda (?) in Kerala) Nationality Indian Ethnicity … Wikipedia**Madhava De Sangamagrama**— (1350 1425) était un mathématicien indien, père de l analyse mathématique. Portail du monde indien … Wikipédia en Français**Madhava de Sangamagrama**— (1350 1425) était un mathématicien indien, père de l analyse mathématique. Calcul de pi Vers 1400, Madhava de Sangamagrama trouve une série permettant de calculer π, la première : Cette série, qui est en fait un cas particulier de : ,… … Wikipédia en Français**Madhava de Sangamagrama**— Madhava (माधव) de Sangamagrama (1350 1425), fue un importante matemático de Kerala, India. Madhava fue fundador de la Escuela de Kerala, y es considerado el padre del análisis matemático, por haber dado el paso decisivo desde los procedimientos… … Wikipedia Español**Madhava**— Mādhava (the Sanskrit vrddhi derivation of madhu sweet ) may be a Sanskrit patronymic, descendant of Madhu (a man of the Yadu tribe) . especially of Krishna or Parashurama as incarnations of Vishnu, see Madhava (Vishnu) an icon of Krishna Madhava … Wikipedia**Sangamagrama**— Sangamagrama, a town in medieval Kerala believed to be the town of Irinjalakkuda [cite journal title = On an untapped source of medieval Keralese mathematics author = C T Rajagopal and M S Rangachari journal = Archive for History of Exact… … Wikipedia**Kerala school of astronomy and mathematics**— The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta… … Wikipedia**Candravakyas**— Candravākyās are a collection of numbers, arranged in the form of a list, related to the motion of the Moon in its orbit around the Earth. These numbers are couched in the katapayadi system of representation of numbers and so apparently appear… … Wikipedia**List of Indian inventions**— [ thumb|200px|right|A hand propelled wheel cart, Indus Valley Civilization (3000–1500 BCE). Housed at the National Museum, New Delhi.] [ 200px|thumb|Explanation of the sine rule in Yuktibhasa .] List of Indian inventions details significant… … Wikipedia**Trigonometric functions**— Cosine redirects here. For the similarity measure, see Cosine similarity. Trigonometry History Usage Functions Generalized Inverse functions … Wikipedia