9 III.Madhava of Sangamagramma Although born in Cochin on the Keralese coast before the previous four scholars I have chosen to save my discussion of Madhava of Sangamagramma (c. 1340 - 1425) till last, as I consider him to be the greatest mathematician-astronomer of medieval India. Sadly all of his mathematical works are currently lost, although it is possible extant work may yet be 'unearthed'. It is vaguely possible that he may have written Karana Paddhati a work written sometime between 1375 and 1475, but this is only speculative. All we know of Madhava comes from works of later scholars, primarily Nilakantha and Jyesthadeva. G Joseph also mentions surviving astronomical texts, but there is no mention of them in any other text I have consulted. His most significant contribution was in moving on from the finite procedures of ancient mathematics to 'treat their limit passage to infinity', which is considered to be the essence of modern classical analysis. Although there is not complete certainty it is thought Madhava was responsible for the discovery of all of the following results: 1) = tan - (tan3 )/3 + (tan5 )/5 - ... , equivalent to Gregory series. 2) r = {r(rsin )/1(rcos )}-{r(rsin )3/3(rcos )3}+{r(rsin )5/5(rcos )5}- ... 3) sin = 3 /3! + 5 /5! - ..., Madhava-Newton power series. 4) cos = 1 - 2/2! + 4/4! - ..., Madhava-Newton power series. Remembering that Indian sin = rsin , and Indian cos = rcos . Both the above results are occasionally attributed to Maclaurin. 5) /4 1 - 1/3 + 1/5 - ... 1/n (-fi(n+1)), i = 1,2,3, and where f1 = n/2, f2 = (n/2)/(n2 + 1) and f3 = ((n/2)2 + 1)/((n/2)(n2 + 4 + 1))2 (a power series for , attributed to Leibniz) 6) /4 = 1 - 1/3 + 1/5 - 1/7 + ... 1/n {-f(n+1)}, Euler's series. A particular case of the above series when t =1/ 3 gives the expression: 7) p = 12 (1 - {1/(3 3)} + {1/(5 32)} - {1/(7 33)} + ...} In generalisation of the expressions for f2 and f3 as continued fractions, the scholar D Whiteside has shown that the correcting function f(n) which makes 'Euler's' series (of course it is not in fact Euler's series) exact can be represented as an infinite continued fraction. There was no European parallel of this until W Brouncker's celebrated reworking in 1645 of J Wallis's related continued product. A further expression involving : 8) d 2d + 4d/(22 - 1) - 4d/(42 - 1) + ... 4d/(n2 + 1) etc, this resulted in improved approximations of , a further term was added to the above expression, allowing Madhava to calculate to 13 decimal places. The value p = 3.14159265359 is unique to Kerala and is not found in any other mathematical literature. A value correct to 17 decimal places (3.14155265358979324) is found in the work Sadratnamala. R Gupta attributes calculation of this value to Madhava, (so perhaps he wrote this work, although this is pure conjecture). Of great interest is the following result: 9) tan -1x = x - x3/3 + x5/5 - ..., Madhava-Gregory series, power series for inverse tangent, still frequently attributed to Gregory and Leibniz. It is also expressed in the following way: 10) rarctan(y/x) = ry/x - ry3/3x3 + ry5/5x5 - ..., where y/x 1 The following results are also attributed to Madhava of Sangamagramma: 11) sin(x + h) sin x + (h/r)cos x - (h2/2r2)sin x 12) cos(x + h) cos x - (h/r)sin x - (h2/2r2)cos x Both the approximations for sine and cosine functions to the second order of small quantities, (see over page) are special cases of Taylor series, (which are attributed to B Taylor). Finally, of significant interest is a further 'Taylor' series approximation of sine: 13) sin(x + h) sin x + (h/r)cos x - (h2/2r2)sin x + (h3/6r3)cos x. Third order series approximation of the sine function usually attributed to Gregory. With regards to this development R Gupta comments: ...It is interesting that a four-term approximation formula for the sine function so close to the Taylor series approximation was known in India more than two centuries before the Taylor series expansion was discovered by Gregory about 1668. [RG5, P 289] Although these results all appear in later works, including the Tantrasangraha of Nilakantha and the Yukti-bhasa of Jyesthadeva it is generally accepted that all the above results originated from the work of Madhava. Several of the results are expressly attributed to him, for example Nilakantha quotes an alternate version of the sine series expansion as the work of Madhava. Further to these incredible contributions to mathematics, Madhava also extended some results found in earlier works, including those of Bhaskaracarya. The work of Madhava is truly remarkable and hopefully in time full credit will be rewarded to his work, as C Rajagopal and M Rangachari note: ...Even if he be credited with only the discoveries of the series (sine and cosine expansions, see above, 3) and 4)) at so unexpectedly early a date, assuredly merits a permanent place among the great mathematicians of the world. [CR /MR1, P 101] Similarly G Joseph states: ...We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition. [GJ, P 293] With regards to Keralese contributions as a whole, M Baron writes (in D Almeida, J John and A Zadorozhnyy): ...Some of the results achieved in connection with numerical integration by means of infinite series anticipate developments in Western Europe by several centuries. [DA/JJ/AZ1, P 79] There remains a final Kerala work worthy of a brief mention, Sadrhana-Mala an astronomical treatise written by Sankara Varman 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 as the last notable name in Keralese mathematics. In recent histories of mathematics there is acknowledgement that some of Madhava's remarkable results were indeed first discovered in India. This is clearly a positive step in redressing the imbalance but it seems unlikely that full 'credit' will be given for some time, as that will possibly require the re-naming of various series, which seems unlikely to happen! Still in many quarters Keralese contributions go unnoticed, D Almeida, J John and A Zadorozhnyy note that a well known historian of mathematics makes: ...No acknowledgement of the work of the Keralese school. [DA/JJ/AZ1, P 78] (Despite several Western publications of Keralese work.) Discovering Sangamagrama Madhavan Written by Prof. V.P.N. Nampoori Shri. A. Jayakumar, Secretary General, Vijnana Bharati addressing media at Ernakulam Press Club. Shri. K. Vijayaraghavan, Dr. V.P.N.Nampoori, Dr. K. Ravindran, Rajkumar are also R. Taylor and Euler were first developed in India. Jyeshtadeva. Katyayana . Harish Chandra Prasanta Chandra Mahalanobis. Aryabhata. Samgamagram madhava. Leibniz. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. Melpathur Narayan Bhattathiri. and a full analysis has yet to be carried out even though several findings have already been showed that several major concepts of renaissance European mathematics attributed to stalwarts like Newton. Sankaravarman) and modern periods ( Srinivas Ramanujan. It is quite probable that there are still further discoveries of 'Kerala mathematics' to be made. Archimedes and Apollonius It was Einstein who said we should be grateful to Indians who taught us how to count While rest of the world was in dark ages India made strides in Mathematics the last 3000 years of legacy through the works of Sulbakaras ( 800-600 BC). Jyestadeva. Why to rediscover Samgamagrama Madhavan? Political chaos caused halting of further generation of new knowledge in North India while Kerala. Jayant Narlikar. Panini .medieval( Narayana Pandita. E C G Sudarsan and Thanu Padmanabhan. Kerala mathematics was strongly influenced by astronomy leading to the derivation of mathematical results of very high importance. Varahamihira. The beautiful number system ( zero and decimel system) invented by the Indians on which mathematical development has rested is complimented by Laplace as The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. Nilakandaa Somayaji. Achuta Pisharoti. Nilakanda Somayaji. Harish Chandra. S. As a result of the untiring works of people like Prof K V Sharma who found that only about 1% of the total available manuscripts in mathematics and Astronomy in Kerala is deciphered and made known to the world while the rest is still under the vast unexplored ocean of knowledge.and reaching to the current period of Narendra Karmakar. This further demands the necessity of mining out the unexplored landscapes . S N Bose. Sankaravarman extending to those of Srinivasa Ramanujan. Brahmagupta ). The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Samgamgrammadhava. classical ( Vararuchi. allowing a generally peaceful existence to continue causing the pursuit of scientific development to continue 'uninterrupted' and is hailed as the second Golden age of Indian Mathematics.seen(from left).. Aryabhata. S N Bose). Varahamihira. Brahmagupta. escaped the majority of such political upheaval. Gregory. the south western tip of India. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity. first being the period of 5th century AD to 10th Century AD It has come to light only during the last few decades of 20th century that mathematics (and astronomy) continued to flourish in Kerala for several hundred years during medieval era especially from 14th to 18th century. Narendra karmakar S Chandrasekhar. Bhaskaracharya. Pingala and Yajnavalkya) . Introduction It is without doubt that mathematics today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years divided into ancient (Apastamba . Baudhayana . Bhaskaracharya. Srinivasa Varadhan . Manava . of Kerala Mathematics so that we may by lucky enough to get gems of high values and qualities. Madhavan belongs to the house described as the bekuladhishtita Vihar or in malayalam Iranji ( Bakulam ) ninna Palli . One of the members of the Kerala School namely Jyeshtadeva needs a special mention. . the period of his life time can be fixed as 1350 -1425 . one of the stalwarts of Mathematics and Astronomy. Place of birth and Period of Sangamagram Madhava Place of birth of Sangamagrama Madhavan can be known from the 13th sloka of his only surviving book called Venuaroham which runs as follows: Bekuladhishtitatwena viharoyo visishyate Grihanamanisoyam syannigenamanimadhava. Gregory and others (who lived duirng 17th to 18th century) were just rediscoveries of the mathematics contributed by Kerala School. in Malayalam for wider accessibility of the knowledge. Leibniz. Of all the mathematicians of medieval period. Mr C Rajagopal and his colleagues. diety of Koodal Manikya Temple-Iringalakkuda) . In his series of papers Whish showed that works of Newton. three hundred years before the life time of Newton . There is a temple of Krishna in Kallettunkara where the Great Acharya used to sit for hours watching the stars . While the rest of the scholars wrote their works in Sanskrit. He . From the writings of his disciples. Jyeshta Deva wrote his book Yukti Bhasha. Mr P Padmakumar. We should promote the works of such people among us who are capable of carrying out such wonderful jobs of deciphering our ancient knowledge. known as. Sangamagrama Madhavan and his school were known to the western world through the series of papers published by Mr Charles Whish duirng 1834 in the journal called Transactions of Asiatic Society of Great Britain and Ireland. It is high time that appropriate steps to be taken to rediscover him . There are two stone slabs in the temple used by the Acharya for the sky watch. People of Kerala should come together to work for regaining the glory of Kerala School of Mathematics. In this context we should remember a self taught mathematician from Trivandrum. name of Sangamagrama Madhavan is the most important who founded a continuous chain of Guru Shihya parampara from 14th century to 18th century and is generally known as Kerala School of Mathematics. It is a fact that even the village of Kallettunkara does not know Samgamagrama Madhavan. Even to this date there is a house named Iringatappally in Kallettunkara near Iringalakkuda. Only after one century of Whish's works that world started knowing and admiring the valuable contributions of Kerala Mathematics through Prof S K Sharma. who discovered astonishing properties of magic square called Srirama Chakram which is now known as strongly Magic Square. However his works did not get much attention from the academicians and researchers of the west. Gregory and Leibnitz. Ulloor describes Sangama Grama Madhavan as belonging to Iringatappally house in Sangama Grama ( village of Snagameswara. a treatise in mathematics and Astronomy. P.Programme Swadeshi Science Movement Kerala jointly with the Panchayat and the people of Kallettinkara and Iringalakkuda wish to formulate variety of programme in reviving the memory of Samgamagrama Madhavan in the mind of people of India . On October 17 Vijayadasami day organizing vidyarambham. V. Requesting Calicut university to establish a Study Centre for Mathematics and Statistics in Kallettinkara / Iringalakkuda Let Iringalakkuda may once again become samgamam (union) of scholars and students revitalizing the broken chain of the Guru Sishya Parampara. Nampoori President Swadeshi Science Movement .N. 2. Media form an important component in promoting this noble act and we request the help from media of all format. This will stamp the name of Samgamagrama Madhavan in the mind of people  Some of the future activities are 1. 3. 3. Following are some of the programme envisaged 1. Naming the Panchayat Library as Samgamagrama Madhavan Grantha Sala 2. particularly. Prof. 4. Organizing monthly programme on Science. Technology and Mathematics for the benefit of public and students. Taking up projects with the help of various funding agencies.print and electronicâ€― in reviving the legacy and glory of Iringalakkuda. Naming the road leading to the ancestral house of the Acharya as Samgamagrama Madhavan Road. 5. in the mind of People of Kerala. public function and a function of face to face with scientists for school children. Establishing Samgamagrama Madhavan Research Centre in Kallettinkara / Iringalakkuda to promote studies on history of Mathematics and Astronomy with special reference to Kerala School. Step will be Taken to install a board in the Railway station indicating "to visit the birth palce of Samgamagrama Madhavan step down here" A brief write-up about Samgamgram Madhan may be displayed in the railway station and the Panchayat office. . Achutha Pisharoti. astronomy and related fields had developed during those periods as per the excavations made at places such as Mohenjadaro and Harappa on the banks of river Sindh.. The counting numbers including zero (0. made contributions comparable to modern mathematicians. Our country. Vedas also provide light into the application of various mathematical methods. There are various names already identified as Mathematicians belonging to Kerala.1. Numbers with decimal system. Only through some of the known works certain details about these great people are known.Madhava of Sangamagrama .. Vadassery Parameswaran. However. Narayanan.C. since the Europeans got them from Arabs who had commercial relationships with India from very early periods. though he later shifted to Patna. Sankara Variyar.. determination of suitable time. Kerala has a very rich history in subject Mathematics. . Sangama grama Madhavan is one among them. 3000 or more. such as Sangamagrama Madhavan. Kerala has a golden history of Mathematics between 14 and 17 centuries. Connected with yagas (a ritual to please Gods) and other rituals. Puthumana chomathiri. Jyeshtadevan. Though less details have been discovered the available materials show that several stalwarts in it lived in this geographically small area.2.Sankaran Category: General Member Level: Gold Points: 50 (Rs 50) Read about Madhava of Sangamagrama.. India. etc. It is referred sometimes as Arabic numerals. When talking about Indian mathematicians the name of Arya bhata I (475 – 550 AD) comes first. Chithrabhanu.Kerala mathematician. Four works related to Kerala mathematics and astronomy of those periods are generally considered as . However after this there is a big gap in the mathematical history of Kerala. Through this connection Arabs got these numerals from India. According to certain historians the birth place of Arya bhata was in Kerala. was rich in the field of mathematics from very early periods dating back to B.. Kelallur Neelakanta Somayaji.9) which are now used throughout the world are said to have originated here. about whom certain available facts are presented below.Kerala Mathematician Posted Date: 04-Nov-2012 Author: T.M. geometry. are decided using mathematical methods. construction of yaga place. Sakara varman and so on. 'Aganitham' and 'Aganitha panchangam'. 'Karana paddhathy' of Puthumana chomathiri and 'Sadrathna mala' of Sankara varman. His known works include 'Venvaroham'. but now having seen that the same was invented about two and a half century back by Madhavan. It is believed that these people migrated from coastal Karnataka regions. Vararuchi's method only could calculate the position of moon to the nearest minute. The infinite series for arc-tangent was developed by James Gregory of Scotland in the year 1667 and hence it is known as Gregory series. a sub.group of Brahmin community. it is related to the name of a temple 'Sangameswara kshethram' near Irinjalakkuda in Thrissur district. Similarly the Newton's power series expansion for sine and cosine is now known as Madhava – Newton series. 'sphutachandrapthi'. 'Lagnaprakaram'. This information helped to identify the village as Sangama gramam. Sangamagrama Madhavan is one of the greatest names among the ancient Indian astronomers. In his book 'Venvaroham' certain mentions are there about his village and house. Madhavan can be even considered as the introducer of topic Mathematical Analysis. His life period is considered to be during 1350 to 1425 AD. His name is connected with the village to where he belonged. the series has been re-named as Madhava – Gregory series. His results include the derivation of infinite series for circular and trigonometric functions (it is popular as Gregory series for arc-tangent). He has used the principles of integration and infinite series. which is described in 'Venvaroham'. Neelakantan.'Thantra samgraha' of Neelakantan. His intuitive approach in solving . since his attempts in this field are notable. One of the names among the Kerala Mathematicians whose works are well recognised is that of Sangama grama Madhavan. Most of this works are based on Vararuchi's 'Chandravakya Padhathi'. Madhavan is referred to in certain references as 'Golavid' meaning 'a learned man about the globe'. 'Yukthi bhasha' of Jyeshtadevan. Jyeshtadevan. He is recognised also as a pure mathematician. 'Madhyayanayanaprakaram'. He belonged to Embranthiri community. where as Madhavan did it to seconds correctness. 'Mahagyanayana prakaram'. Similarly the series related to the value of 'Pi' invented by Madhavan in the 14th century was reinvented by Gregory in 1671 and Leibniz in 1673. From the works of Madhavan's disciples and successors in the field such as Neelakantan. infinite series for the mathematical constant 'Pi' (the ratio of the circumference of a circle to its diameter) and the Newton Power series expansion for sine and cosine (This is known in Newton's name because in Europe this appeared in 1676 in a letter written by Newton to the Secretary of Royal Society). Narayanan and Sankara Varier more references to Madhavan's contributions are available. Jyeshtadevan and others. His works have highly influenced the works of Parameswaran. The house name as indicated in the above work is 'Ilaininna palli' (Two houses with slightly modified names are still there near Irinjalakkuda Railway station at Kallettumkara in Irijalakkuda). It is Sangama Gramam. mainly because of his works in the field. The mathematical principles involved in this method were far advanced than what were available during those periods. Some of them are very important results in modern mathematics. Much biographical details of Madhavan are not available. tangent and arctangent functions and the value of π). he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine. He had discovered several results in trigonometry and infinite series much earlier than these were rediscovered by European Mathematicians.problems can be compared with that of great Sreenivasa Ramanujan (1887 – 1920). near the southern tip of India. etc. (as even the ancient Egyptians and Greeks had known). which had been rather nervous about the concept of infinity. Madhava was more than happy to play around Madhava of Sangamagrama (c.13501425) with infinity. Unlike most previous cultures.MADHAVA Madhava sometimes called the greatest mathematician-astronomer of medieval India. cosine. Although almost all of Madhava's original work is lost. INDIAN MATHEMATICS . He showed how. He came from the town of Sangamagrama in Kerala. the exact total of one can only be achieved by adding up infinitely many fractions. particularly infinite series. . One thing is certain that he stood above all mathematicians of his period and had shown his intuitive powers in the formation of theories and solving them. with its implications for the future development of calculus and mathematical analysis. although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth. Further studies are needed to find out the Madhavan's total contributions in the field of Mathematics. and founded the Kerala School of Astronomy and Mathematics in the late 14th Century. representing the first steps from the traditional finite processes of algebra to considerations of the infinite. 1350–c.1425) was a prominent Hindu mathematician-astronomer from the town of Irinjalakkuda. Kerala. and found approximations for some transcendental numbers by continued fractions. Mādhavan) (c. tangent and arctangent. He realized that. Madhava discovered the solutions of some transcendental equations by a process of iteration. he discovered a procedure to determine the positions of the Moon every 36 minutes. Perhaps even more remarkable. which was . he could home in on an exact formula for π (this was two centuries before Leibniz was to come to the same conclusion in Europe). Some historians have suggested that Madhava's work. In astronomy. He went on to use the same mathematics to obtain infinite series expressions for the sine formula. by successively adding and subtracting different odd number fractions to infinity. Madhava obtained a value for π correct to an astonishing 13 decimal places. Among his other contributions. implying that he quite understood the limit nature of the infinite series. and methods to estimate the motions of the planets. is that he also gave estimates of the error term or correction term. Madhava of Sangamagrama Tweet Information about Madhava of Sangamagrama Mādhava of Sangamagrama (Malayalam: . Madhava’s method for approximating π by an infinite series of fractions Madhava’s use of infinite series to approximate a range of trigonometric functions. which could then be used to calculate the sine of any angle to any degree of accuracy. and may have had an influence on later European developments in calculus. may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Cochin (Kochi) at the time. Through his application of this series. India.But Madhava went further and linked the idea of an infinite series with geometry and trigonometry. and either he or his disciples developed an early form of integration for simple functions. effectively laid the foundations for the later development of calculus and analysis. though. as well as for other trigonometric functions like cosine. which were further developed by his successors at the Kerala School. through the writings of the Kerala School. near Cochin. as well as some products with radius and arclength. For those that do not. cosθ. Others have speculated that the early text Karana Paddhati (c. He is considered the founder of the Kerala school of astronomy and mathematics. presents several versions of the series expansions for sinθ. which was the first to draw attention to their priority over Newton in discovering the Fluxion (Newton's name for differentials)[3]. may have been transmitted to Europe via Jesuit missionaries and traders who were active around the ancient port of Kochi at the time. that since some of these have been attributed by Nilakantha to Madhava. Thus. Sadratnamala c. as well as the Tantrasangraha and Yuktibhasa. quoting extensively from the original Sanskrit[1]. written in Malayalam.1530[4]).[2]. along with the even earlier Keralese mathematics text Sadratnamala. However. He is the first to have developed infinite series approximations for a range of trigonometric functions. As a result. it may have had an influence on later European developments in analysis and calculus[3]. etc. text Mahajyānayana prakāra cites Madhava as the source for several series derivations for π.g. these series are presented with proofs in terms of the Taylor series expansions for polynomials like 1/(1+x2). a set of fragmentary results[3]). The Yukti-dipika (also called the Tantrasangraha-vyakhya). . calculus. Rajagopal and Rangachari have argued. most versions of which appear in Yuktibhasa. through the writings of the Kerala school. which has been called the "decisive step onward from the finite procedures of ancient mathematics to treat their limit-passage to infinity"[1]. with x = tanθ. and a comprehensive look at the Kerala school was provided by Sarma in 1972[4]. In Jyesthadeva's Yuktibhasa (c. it is clear from citations that Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala. were considered in an 1835 article by Charles Whish. as the source for several infinite series expansions.1300. including sinθ and arctanθ. possibly some of the other forms might also be the work of Madhava. possibly composed Sankara Variyar. the Russian scholar Jushkevich revisited the legacy of Madhava[6]. trigonometry. One of the greatest mathematician-astronomers of the Middle Ages. geometry and algebra. particularly in Nilakantha Somayaji's Tantrasangraha (c. The 16th c. but this is unlikely[1].at the time known as Sangamagrama (lit. Karana Paddhati. or the Mahajyānayana prakāra might have been written by Madhava. grāma=village). Madhava contributed to infinite series. In the mid-20th century. and arctanθ. Historiography Although there is some evidence of Mathematical work in Kerala prior to Madhava (e. what is explicitly Madhava's work is a source of some debate. sangama = union.1375-1475). His discoveries opened the doors to what has today come to be known as mathematical analysis. a student of Jyesthadeva.1500). Some scholars have also suggested that Madhava's work. most of Madhava's original work (possibly excepting an astronomy text[3]) is lost. He is referred to in the work of subsequent Kerala mathematicians. Parameshvara Namboodri was possibly a direct disciple. which results are precisely Madhava's and which are those of his successors. In Europe. there is a large gap in the Indian mathematical tradition.Lineage Explanation of the sine rule in Yuktibhasa Before Madhava. and the grammarian Melpathur Narayana Bhattathiri as his disciple[4].[8] However. and in particular. power series. The following presents a summary of results that have been attributed to Madhava by various scholars. as stated above. Madhava may have invented the ideas underlying infinite series expansions of functions. are somewhat difficult to determine. we have a clearer record of the tradition after Madhava. Parameswara's son Damodara (c. It is possible that other unknown figures may have preceded him. Contributions If we consider mathematics as a progression from finite processes of algebra to considerations of the infinite. This implies that the limit nature of the infinite series was quite well understood by him. However. 1400-1500) had both Nilakantha and Jyesthadeva as his disciples. there is little known about any tradition of Mathematics in Kerala. but what is truly remarkable is his estimate of an error term (or correction term)[8]. the first such series were developed by James Gregory in 1667. Madhava's work is notable for the series. Trigonometric series. then the first steps towards this transition typically come with infinite series expansions. According to a palmleaf manuscript of a Malayalam commentary on the Surya Siddhanta. It is this transition to the infinite series that is attributed to Madhava. Achyuta Pisharati of Trikkantiyur is mentioned as a disciple of Jyeshtadeva. and rational approximations of infinite series. . Thus. q2/2! + q4/4! . and certainly other infinite series of a similar nature had been worked out by Madhava.. which contains the derivation and proof of the power series for inverse tangent. it is occasionally referred to as the Madhava-Gregory series[9][10]. It is believed that he may have found these highly accurate tables based on these series expansions[2]: sin q = q . Today. what is most impressive is that he also gave a correction term. discovered by Madhava... without quotation marks. Jyesthadeva describes the series in the following manner: Insert the text of the quote here. 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"). Even if we consider this particular series as the work of Jyeshtadeva. Rn. cosine..[9] In the text. tangent and arctangent.. While some scholars such as Sarma[4] feel that this book may have been composed by Madhava himself. it is more likely the work of a 16th century successor [2]. cos q = 1 .Infinite series Among his many contributions.q3/3! + q5/5! . However. Madhava gave three forms of Rn which improved the approximation[2]. This text attributes most of the expansions to Madhava. This yields which further yields the result: This series was traditionally known as the Gregory series (after James Gregory. who discovered it three centuries after Madhava). namely .. it would pre-date Gregory by a century. and many methods for calculating the circumference of a circle. and gives the following infinite series expansion of π: which he obtained from the power series expansion of the arc-tangent function. he discovered the infinite series for the trigonometric functions of sine. 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. for the error after computing the sum up to n terms. Trigonometry Madhava also gave a most accurate table of sines. One of Madhava's series is known from the text Yuktibhasa. 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. and either he. and the analysis of infinite continued fractions. The value of 3. he used early forms of differentiation. and found the approximation of transcendental numbers by continued fractions. see Aryabhata).[4] He also discovered the solutions of transcendental equations by iteration. including those of Bhaskara. where the third correction leads to highly accurate computations of π. The text Sadratnamala. R. or Rn = (n2 + 1) / (4n3 + 5n). These were the most accurate approximations of π given since the 5th century (see History of numerical approximations of π). He also gave a more rapidly converging series by transforming the original infinite series of π. obtaining the infinite series By using the first 21 terms to compute an approximation of π.14159265358979324 (correct to 17 decimal places). or Rn = n/ (4n2 + 1).Rn = 1/(4n). found methods of polynomial expansion. It is not clear how Madhava might have found these correction terms[11]. appears to give the astonishingly accurate value of π =3. discovered tests of convergence of infinite series. is sometimes attributed to Madhava[13]. correct to 13 decimals. but may be due to one of his followers.14159265359)[12].1415926535898. In calculus. computation. he obtains a value correct to 11 decimal places (3. . integration. Gupta has argued that this text may also have been composed by Madhava[12][4]. which were further developed by his successors at the Kerala school of astronomy and mathematics.[4] Calculus Madhava laid the foundations for the development of calculus. or his disciples developed integration for simple functions.) Madhava also extended some results found in earlier works. usually considered as prior to Madhava. Algebra Madhava also carried out investigations into other series for arclengths and the associated approximations to rational fractions of π.[8][16] (It should be noted that certain ideas of calculus were known to earlier mathematicians. Based on this. The Kerala school also contributed much to linguistics (the relation between language and mathematics is an ancient Indian tradition. in the period of the first contact with European navigators in the Malabar coast. [10] which translates as the integration a variable (pada) equals half that variable squared (varga). Most of these results pre-date similar results in Europe by several centuries. it is possible that these ideas may still have had an influence on later European developments in analysis and calculus. was a major center for maritime trade. At the time. This is clearly a start to the process of integral calculus. Influence Madhava has been called "the greatest mathematician-astronomer of medieval India"[4]. see Katyayana).Kerala School of Astronomy and Mathematics The Kerala school of astronomy and mathematics flourished for at least two centuries beyond Madhava. Narayaneeyam. While no European translations have been discovered of these texts. as in the statement: ekadyekothara pada sankalitam samam padavargathinte pakuti."[3].e. A related result states that the area under a curve is its integral.. collection). In many senses. Manchester have suggested[18] that the writings of the Kerala school may have also been transmitted to Europe around this time. i. Jyeshtadeva's Yuktibhasa may be considered the world's first calculus text. near Sangamagrama. some of his discoveries in this field show him to have possessed extraordinary intuition. O'Connor and Robertson state that a fair assessment of Madhava is that he took the decisive step towards modern classical analysis[2]. Joseph of the U. including G. and the interest shown by some of the Jesuit groups during this period in local scholarship. (See Kerala school for more details). . termed sankalitam. which was still about a century before Newton[3]. Some scholars. or as "the founder of mathematical analysis. indeed many more pages are developed to astronomical computations than are for discussing analysis related results[4]. (lit. was composed by Narayana Bhattathiri. In Jyesthadeva we find the notion of integration.[8][16][3] The group also did much other work in astronomy. The integral of x dx is equal to x2 / 2. The famous poem. Given the fame of the Kerala school. and a number of Jesuit missionaries and traders were active in this region. the port of Kochi. Propagation to Europe? The Kerala school was well known in the 15th-16th c. The ayurvedic and poetic traditions of Kerala can also be traced back to this school. about 4.14159. π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. Starting around the 15th century. by transforming the series into \pi = \sqrt{12} \. based on polygons.14159265359. and were used by mathematicians including Madhava of Sangamagrama.[6] In the 12th century. sometimes written pi. the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials. Carl Friedrich Gauss. However. the rate of convergence is too slow to calculate many digits in practice. mathematicians have attempted to extend their understanding of π. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. For thousands of years. The record was beaten in 1424 by the Persian astronomer Ghiyath al-Kashi.[8] which are treated in the text Yuktibhasa.000 terms must be summed to improve upon Archimedes' estimate. and Srinivasa Ramanujan.[7] In the 14th century. to estimate the value of π. he number π (pi) is a mathematical constant that is the ratio of a circle's circumference to its diameter. Isaac Newton. although no proof of this supposed randomness has yet been discovered. π is an irrational number. Before the 15th century. consequently. Moreover.[9][10][11] . is approximately equal to 3.\frac{1}{7 \cdot 3^3} + \cdots\right) Madhava was able to calculate π as 3. \left(1-\frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3^2} . By adding a remainder term to the original power series of π /4. new algorithms based on infinite series revolutionized the computation of π. he was able to compute π to an accuracy of 13 decimal places.14159265359. an important result in differential calculus. It has been represented by the Greek letter "π" since the mid-18th century. Leonhard Euler. correct to 11 decimal places. Unfortunately. sometimes by computing its value to a high degree of accuracy. development of integral calculus. ow known as the Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. along with other mathematician-astronomers of the Kerala school of astronomy and mathematics. The constant. Madhava of Sangamagrama. who determined 16 decimals of π. mathematicians such as Archimedes and Liu Hui used geometrical techniques.The Indian mathematician and astronomer Madhava of Sangamagrama in the 14th century computed the value of π after transforming the power series expansion of π /4 into the form and using the first 21 terms of this series to compute a rational approximation of π correct to 11 decimal places as 3. its decimal representation never ends and never repeats. described special cases of Taylor series. which means that it cannot be expressed exactly as a ratio of two integers (such as 22/7 or other fractions that are commonly used to approximate π). The digits in the decimal representation of π appear to be random. like Greece or Babylonia to explain the Kerala phenomenon. though the Eurocentric scholars have severely criticized it. and the Newton power series for the sine and cosine. but also on the export of Maths from India to Europe. The Kerala discoveries include the Gregory and Leibniz series for the inverse tangent. Joseph (1994) has very emphatically brought out the significance of the Kerala school of Maths in his The Crest of the Peacock. compared with the earlier period that it is impossible to bridge the gap between the two periods. in Kerala. Nor can one invoke a 'convenient' external agency. Kerala has gained recognition for its role in the reconstruction of medieval Indian mathematics. Though the picture about the rest of India is not clear. in India made no progress after Bhaskaracharya. In recent years. While there were some doubts about Whish's views on the dating and authorship of these works. including the well-known Taylor series approximations for the sine and cosine functions.P. whose views would also be discussed in this essay. the period between the fourteenth and seventeenth centuries marked a high point in the indigenous development of astronomy and mathematics. . European Mathematics and Navigation By D. until they were introduced to modern mathematics by the British.The Kerala School. writing endless commentaries on the works of the venerated mathematicians who preceded them. he claimed that this work laid the foundation for a complete system of fluxions ['Fluxion' was the term used by Isaac Newton for the rate of change (derivative) of a continuously varying quantity. Charles Whish published an article in which he referred to four works – Nilakantha's Tantra Samgraha. which he called a 'fluent']. And these results had apparently been obtained without the use of infinitesimal calculus. the Leibniz power series for p. The quality of the mathematics available from the texts that have been studied is of such a high level. or function. It has been a centre of maritime trade. The only scholar who has dealt with this issue to my mind is C. C. which were anticipated by Kerala astronomer-mathematicians two hundred to three hundred years earlier. a summary of a number of earlier works. Jyesthadeva's Yuktibhasa. Until recently there was a misconception that mathematics. The Sadratnamala. Putumana Somayaji's Karana Paddhati and Sankara Varman' s Sadratnamala – as being among the main astronomical and mathematical texts of the Kerala school. has also written a good deal not only on the famous work. as God's Own Country. as well as certain remarkable rational approximations of trigonometric functions. his main conclusions are still broadly valid.K. Joseph informs that in 1835. Raju. he says 'abounds with fluxional forms and series to be found in no work of foreign countries'. And this leads us to ask whether the developments in Kerala had any influence on European mathematics. with its rich variety of spices greatly in demand. Raju. the well known mathematician and historian of science. Famous travellers and explorers such as Ibn Battuta and Vasco da Gama came from across the Arabian Sea. the south-west coast near the tip of the Indian peninsula. that later scholars seemed 'content to chew the cud. Agrawal The National Geographic has declared Kerala. Writing about Tantra Samgraha. There were later discoveries in European mathematics.K. even as early as the time of the Babylonians. Yuktibhasa by Jyesthadeva. and 3. 1500-1560). this work was extended to a third-order series approximation of the sine function. and approximations for sine and cosine functions (to the second order of small quantities). first recorded by Paramesvara (1360-1455) in his commentary on Bhaskaracharya's Lilavati. and 3. which goes under the name of l'Huilier's formula. it was necessary to have both an accurate value for p and highly detailed trigonometric tables. Astronomy provided the main motive for the study of infinite-series expansions of p and rational approximations for different trigonometric functions.' And according to Eves (1983. appropriately. In this area Kerala mathematicians made the following discoveries: 1. Yuktibhasa. The use of the Newton-Gauss interpolation formula (to the second order) by Govindaswami. though none of their results has as yet percolated into the standard Western histories of mathematics. usually attributed to Newton. p. 1500-75) and Sankara Variar (c. A joint commentary on Bhaskaracharya's Lilavati by Narayana (c. Boyer (1968. who lived at the beginning of the nineteenth century and may be said to have been the last of the notable names in Kerala mathematics. 1660-1740) provides a detailed discussion of the various trigonometric series.In the 1940s it was Rajagopal and his collaborators who highlighted the contributions of Kerala mathematics. His work in five chapters contains. Nilakantha (1445-1555) was mainly an astronomer.This work is mainly based on the Tantra Samgraha of Nilakantha. The Karana Paddhati by Putumana Somayaji (c. 1550) wrote. The discovery of the formula for the circum-radius of a cyclic quadrilateral. p. entitled Kriyakramakari. 164). The power series for sine and cosine. usually attributed to Gregory. in a regional language rather than in Sanskrit. . problems of algebra and spherical geometry. 244) writes that 'Bhaskara was the last significant medieval mathematician from India. usually attributed to Taylor. The power series for . also contains a discussion of Madhava's work. The power series for the inverse tangent. 'Hindu mathematics after Bhaskara made only spotty progress until modem times. the author of Sadratnamala. usually attributed to Gregory and Leibniz. The statement of the mean value theorem of differential calculus. there were extensions of earlier work notably of Bhaskaracharya: 1. Jyesthadeva (c. although the original sources remain undiscovered or unstudied. For example. without any proofs though. and a number of rational approximations to . a summary of most of the results of the Kerala school. but his Aryabhatiya Bhasya and Tantra Samgraha contain work on infinite-series expansions. 2. and his work represents the culmination of earlier Hindu contributions. 2. Finally there is Sankara Varman. usually attributed to Leibniz.' Madhava's work on power series for p and for sine and cosine functions is referred to by a number of the later writers. one of those rare texts in Indian mathematics or astronomy that gives detailed derivations of many theorems and formulae in use at the time. For astronomical work. Apart from the work on infinite series. 1530 CE] are both 16th c.) In fact. in an article "Computers. there remains the important question of epistemology ("Was it really the calculus that Indians discovered?"). . of Cavalieri. who had access in various ways to the Jesuit archives at the Collegio Romano. CE. Jain logic. accept the accompanying methods of proof. and its transmission to Europe is otherwise clear. by about a century in each case. where they were teaching Malayalam to the local children (especially Syrian Christians) whose mother tongue it was. CE. in the 16th c. CE. He argues that formal deductive proof does not incorporate certainty. Fermat. exactly the infinite series in these Indian texts started appearing in the works. Hence. while European mathematicians accepted the practical value of the Indian infinite series as a technique of calculation.not 13th. many of them did not. and the Alternative Epistemology of the Calculus in the Yuktibhasa". since the underlying logic is arbitrary. for calendar-making for example. say. While the case for the origin of the calculus in India. and the theorems that can be derived from a particular set of axioms would change if one were to use Buddhist logic. not 17th. their activities were especially concentrated in the vicinity of their Cochin College. CE Jesuits were busy translating and transmitting very many Indian texts to Europe. Since Whiteside has a copy of the printed commentary on the Yuktibhasa. he could hardly have failed to notice this similarity with the European works with which he seeks to make the Yuktibhasa contemporaneous! Raju has no doubt that in the course of "the fabrication of ancient Greece" (in Martin Bernal's words). the evidence for the transmission of the calculus from India to Europe is far more robust than the sort of evidence on which "Greek" history is built – it cannot be upset by quibbling about the exact date of a single well-known manuscript like the Yuktibhasa. the calculus took some three centuries to be assimilated within the European frame of mathematics. Having anticipated this. Mathematics Education.Here it may be relevant to note some points of the debate that CK Raju has been carrying out with the West in general. distorts the dates of both Madhava and the Yuktibhasa. Gregory etc. in relation to formalist mathematical epistemology from Plato to Hilbert. and with Whiteside (the famous historian of Maths) in particular. Raju's Encounter with Eurocentric scholars Raju (personal communication) explains that Whiteside. For. even then. After the trigonometric values in the 16th and early 17th c. or. from 1630 onwards. while conceding Madhava's priority for the development of infinite series.Raju has discussed this question in depth. while the Tantrasangraha [1501 CE] and Yuktibhasa [ca. Pascal. some Western historians acquired ample familiarity with this technique of juggling the dates of key texts. Raju proposes a new understanding of mathematics. and where copies of the Yuktibhasa and several other related texts were and still are in common use. In this paper. during the 16th c. like the algorismus which took some five centuries to be assimilated in Europe. CE texts. about the export of Maths to Europe. (Madhava was 14th-15th c. on the other hand.g. it is very hard for me to dump them all. e. I still don't quite know what this term "Hindu" means. from the perspective of Nagarjuna's sunyavada. which first introduced the trigonometric functions and methods of calculating their approximate numerical values. "Christian mathematics"? Probably not. I should point out that my interest in all this is not to establish priority. and especially when it is linked with mathematics! Given the fundamental differences between the four schools listed above. in both cases.in my earlier cited paper and book. Nyaya. except to link it to Whiteside's use of the phrase "Hindu matmatics" [sic].and Lokayata notions of proof (pramana). Though such an understanding of the past varies strikingly from the usual "heroic" picture that has been propagated by Western historians. into a single category of "Hindu". is to get rid of this sort of conceptual clutter . having also scanned the OED for the meaning of "Hindu". how mathematics education must today be conducted.g.g. Philosophy. I presume instead that. Jain. I have also commented elsewhere. hence more futuristically oriented. and the difficulties that this created in the European understanding of both algorismus and calculus. The development of the infinite series and more precise computations of the circumference of the circle.authoritatively sought to be imposed by . for it also helps us to understand e. by Aryabhata's school. and the circumference of the earth. 'Now it is true that I have commented on formalist mathematical epistemology from the perspective of Buddhist. as Western historians have unceasingly sought to do. the current way of handling division by zero in the Java computing language. it is far more real. and is alluding. The transmission of the calculus to Europe is also readily understood as a natural consequence of the European need to learn about navigation. due. 'I would not like to go further here into the difficult question of epistemology. and the interaction between history and philosophy of mathematics. which counts as "Hindu" and which not. difficulties that persist to this day in e. The centuries of difficulty in accepting the calculus in Europe is more naturally understood in analogy with the centuries of difficulty in accepting the algorismus. if we exclude some. especially in Whiteside's "ruggedly individualistic" nonEurocentric sense."Islamic mathematics" etc. the calendar. on the re-interpretation of sunya as zero in formal arithmetic. Whiteside is really referring to the Eurocentric belief that there is only one "mainstream" mathematics. how to tackle the epistemological challenge posed today in interpreting the validity of the results of large-scale numerical computation. and Culture. Nevertheless. and everything else needs to be qualified as "Hindu mathematics". Am I to understand that Whiteside now implicitly accepts also the possible influence of Newton's theology on his mathematics.Raju further states.. like Whiteside. "Indeed. despite his protestations to the contrary. is readily understood as a natural consequence of Aryabhata's work. and why? And exactly how does that relate to mathematics? 'A key element of the Project of History of Indian Science. but to understand the historical development of mathematics and its epistemology. as I stated earlier. shall we say. over several hundred years. albeit indirectly. to some subtle new changes brought about by Newton in the prevailing atmosphere of. to the difficulty in assimilating an imported epistemology. and hence to decide. though I would accept that direct trade with India in spices also created a direct route for Indian mathematics. the calculus as a method of calculation preceded Newton. The relation is provided by the requirements of the European navigational problem. assertions of authority. Fermat..g. and possible adaptations to its design. one issue. Pascal. however. etc. I am ever willing to correct myself. has so far studied the connection of these Indian developments to European mathematics. and I remain open to all legitimate criticism. I do not share the historical view needed to speak of the "re-birth" of European mathematics in the 16th and 17th c. abuse. however. cavil. bypassing the earlier Arab route.colonialists (and their victims/collaborators).. the foremost problem of the time in Europe. So what basis is there to give credit to Newton for originating the calculus. while denying it. Since my objective is truth and understanding. I deny as similarly inaccurate all the interpolations and distortions he has introduced into what I have said. however. [E. would today make someone the inventor of the computer. and to rewrite history from a fresh. But none of this convincingly establishes the credit for calculus given to Newton. the calculus/analysis as something epistemologically secure.] For the record. within the formalist frame of _mathematics as proof_. Jyeshtadeva etc. The credit that Newton gets for the calculus depends also upon his quarrel with Leibniz. pluralistic perspective. Columbus and Vasco da Gama used dead reckoning and were ignorant of celestial navigation. even within the Eurocentric (as distinct from Anglocentric) frame. In my case. 'There is. The English. was both strategically and economically . is now beginning to totter. which view Whiteside freely attributes to me. Navigation. that would no more make him the inventor of the calculus than the application of the computer to a difficult problem of genetics. Clearly. as any part of such legitimate criticism. While Newton did apply the calculus to physics. Neelkantha. Doubtless Newton's authority conferred a certain social respectability on the calculus. and Leibniz? Navigation and Calculus In his recent talk (2000) Raju emphasised that the calculus has played a key role in the development of the sciences.speaking world has known for over one and a half centuries that "Taylor" series expansions for sine. for example. even from a purely Eurocentric perspective. postdates Dedekind and the formalist approach to real numbers. like the story of the "Copernican Revolution". misleading circumlocutions. it is part of this fresh perspective to redefine the nature of present-day university mathematics by shifting away from formal and spiritual mathematics-as-proof to practical and empirical mathematics-as-calculation. This story of indigenous development. but I do not recognize dramatic poses. to which it would be inappropriate to provide detailed corrections here. 'There are numerous other points in Whiteside's prolix response. In what sense did Newton invent the calculus? Clearly. which remains puzzling. No one else. even in Europe. According to the "standard" story. and the rather dubious methods of "debate" he used in the process. ab initio. cosine and arctangent functions were found in Indian mathematics/astronomy/timekeeping (jyotisa) texts. also. starting from the "Newtonian Revolution". the calculus was invented independently by Leibniz and Newton. and specifically the works of Madhava. to Cavalieri. the key to the prosperity of Europe of that time. the Dutch prize of 1636. and Clavius were then well aware of the acute need not only for a good calendar. These rewards spread over time from the appointment of Nunes as Professor of Mathematics in 1529. to the Spanish government's prize of 1567 through its revised prize of 1598. Galileo. and this led to large inaccuracies (more than 3 degrees) in calculating latitude from measurement of solar altitude at noon. of course. but also for precise trigonometric values. who trained in mathematics and astronomy. Prior to the clock technology of the 18th century. Mazarin's prize to Morin of 1645. focused on mathematics and astronomy. and remained in correspondence with his teacher Nunes during this period. The solution of the latitude problem required a reformed calendar: the European calendar was off by 10 days.g. at a level of precision then found only in these Indian texts. and a key concern for starting the British Royal Society. could not navigate the Indian ocean. using e. Vasco da Gama.) were involved in these efforts: the navigational problem was the specific objective of the French Royal Academy. and the British prize legislated in 1711. attacks on the navigational problem in the 16th and 17th c. Moreover. At the start of this period. as European governments desperately sought to develop reliable trade routes to India. by Stevin and Mersenne) that this knowledge was to be found in ancient mathematical and astronomical or time-keeping (jyotisa) texts of the east. The Jesuits. and needed an Indian pilot to guide him across the sea from Melinde in Africa. In a 1581 letter. the method described in the Laghu Bhaskariya of Bhaskara I. to Calicut in India. This period saw the rise of the Jesuits. partly because the clumsy Roman numerals had made it difficult to handle fractions. hence a change in the date of Easter. under Clavius' new syllabus [Matteo Ricci also visited Coimbra and learnt navigation]. and it was widely (and correctly) believed by navigational theorists and mathematicians (e. Many key scientists of the time (Huygens. Accordingly. Stevin. this was preceded by a latitude problem. Mercator. Jesuits. various European governments acknowledged their ignorance of navigation. European navigational theorists like Nunes. However. etc. Though the longitude problem has recently been highlighted. while announcing huge rewards to anyone who developed an appropriate technique of navigation. for direct trade with India was then the big European dream of getting rich. and Clavius subsequently reformed the Jesuit mathematical syllabus at the Collegio Romano. This knowledge was needed to improve European navigational techniques. and this was authorised by the Council of Trent in 1545. Clavius studied in Coimbra under the mathematician. which were (correctly) believed to hold the key to celestial navigation. lacking knowledge of celestial navigation. needed to understand how the local calendar was made. Ricci explicitly acknowledged that he was trying to understand local methods of . like Matteo Ricci. were sent to India. reforming the calendar required a change in the dates of the equinoxes. especially since their own calendar was then so miserably off the mark. astronomer and navigational theorist Pedro Nunes.g. the French offer (through Colbert) of 1666. and the problem of loxodromes. Clavius also headed the committee which authored the Gregorian Calendar Reform of 1582. Melpathur Narayana Bhattathiri and Achyuta Panikkar. the Kerala school independently created a number of important mathematics concepts. Europeans encountered difficulties in using these precise sine value for determining longitude. and had even started printing presses in local languages. Columbus' incorrect estimate was corrected. Their most important . were greatly concerned with accurate sine values for this purpose. which included among its members: Parameshvara. and as Swift (Gulliver's Travels) had satirized in the 18th c. as in Indo-Arabic navigational techniques or in the Laghu Bhaskariya. search For other uses of this name. an appliance that could be mechanically used without application of the mind. Language was not a problem. in the vicinity of Cochin. because this technique of longitude determination also required an accurate estimate of the size of the earth. then. Stevin. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). the free encyclopedia Jump to: navigation. The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala. Achyuta Pisharati. In attempting to solve astronomical problems. and each of them published lengthy sine tables. only towards the end of the 17th c. see Kerala school (disambiguation). Joseph and C. while Europeans lacked the ability to calculate. Kerala school of astronomy and mathematics From Wikipedia. This led to the development of the chronometer. using the series expansion of the sine function were then the most accurate way to calculate sine values. Raju for their valuable contributions in this regard. India. Thus we see that the great Kerala School of Maths needs a fuller treatment in the history of Indian science than has been given so far. in Europe. In addition to the latitude problem. The problem of calculating loxodromes is exactly the problem of the fundamental theorem of calculus.K. had a college in Cochin. Clavius etc. Even so.timekeeping (jyotisa).G. and the European tradition of mathematics was "spiritual" and "formal" rather than practical. since the Jesuits had established a substantial presence in India. the Indo-Arabic navigational technique required calculation. and Columbus had underestimated the size of the earth to facilitate funding for his project of sailing West. there remained the question of loxodromes. settled by the Gregorian Calendar Reform. as Clavius had acknowledged in the 16th c. Jyeshtadeva. which were the focus of efforts of navigational theorists like Nunes. the key centre for mathematics and astronomy. since the Vijaynagar empire had sheltered it from the continuous onslaughts of raiders from the north. Mercator etc. Madhava's sine tables. since algorismus texts had only recently triumphed over abacus texts. Loxodromes were calculated using sine tables. which was. Neelakanta Somayaji. We should all be thankful to both G. from "an intelligent Brahmin or an honest Moor". like Malayalam and Tamil by the 1570's. and Nunes. CE. results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha. provided what is now considered the first example of a power series (apart from geometric series).[2] However. however. was already known in the work of the 10th century Iraqi mathematician Alhazen (the Latinized form of the name Ibn al-Haytham) (965-1039). This result was also known to Alhazen. written in Malayalam. they did not formulate a systematic theory of differentiation and integration. These include the following (infinite) geometric series: for [7] This formula. but proofs for the series for sine. and also in a commentary on Tantrasangraha.[1] . cosine.[1] They used this to discover a semi-rigorous proof of the result: for large n. nor is there any direct evidence of their results being transmitted outside Kerala. and again in a commentary on this work.1500-c. by Jyesthadeva.1 Infinite Series and Calculus 2 Possibility of transmission of Kerala School results to Europe 3 See also 4 Notes 5 References 6 External links Contributions Infinite Series and Calculus The Kerala school has made a number of contributions to the fields of infinite series and calculus.[1] Their work. of unknown authorship. completed two centuries before the invention of calculus in Europe. and inverse tangent were provided a century later in the work Yuktibhasa (c. The theorems were stated without proof. though the inductive hypothesis was not yet formulated or employed in proofs.1610).[8] The Kerala school made intuitive use of mathematical induction.[3][4][5][6] Contents 1 Contributions o 1. called Tantrasangraha-vakhya. was not yet developed. . . computation of area under the arc of the circle). (The later method of Leibniz. the series reduce to the standard power series for these trigonometric functions. using quadrature (i. which when translated to mathematical notation. and i = 1.They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for . For example. (for n odd.e. and .) The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results.)[1] They also made use of the series expansion of to obtain an infinite series expression (later known as Gregory series) for :[1] Their rational approximation of the error for the finite sum of their series are of particular interest. the error. for .[9] The Tantrasangrahavakhya gives the series in verse. can be written as:[1] where where. 2. 3) for the series: where . for example: and (The Kerala school themselves did not use the "factorial" symbolism. 14 major perturbations of the Moon are highlighted and 14 trignometric corrections are done for the 14 lunar anomalies. sin. the mean longitude of the Moon. Rajagopal and his associates. It is one of the best books on Mathematical Astrology ever written.[16][17] The genius of Puliyoor Purushotaman Namboothiri condensed the thousands of slokas of the 18 Siddhantas into 1000 odd slokas. i. was not yet formulated. They made use of an intuitive notion of a limit to compute these results. Whish in 1835. Chathur dasebhyebhyo balanyabhibhyo Neethva Thulasadi Vasa Dhanarnam Krithva tad Indor Apaneeya Thungam Thado Mridujya phala Samskruthendu ( Ganitha Nirnayam ) After computing the Chandra Madhyamam. In the Kerala system.[1] for correct up to nine decimal places. M. 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. the longitude is called Samskrutha Chandra Madhyamam. when the discoveries of the Kerala school were investigated again by C. Samskruthendu. using the partial fraction expansion of : rapidly converging series for :[1] to obtain a more They used the improved series to derive a rational expression.They manipulated the terms. and cosine (with English translation and commentary). .[1] The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions. According to Whish. after 14 trignometric corrections. Prof Krishna Warrier. Whish's results were almost completely neglected.e. namely Kala Sankalita by J. He was ably assisted by the Maths Professor. though there exists another work. until over a century later. The works of the Kerala school were first written up for the Western world by Englishman C. Warren from 1825[11] which briefly mentions the discovery of infinite series by Kerala astronomers. called the Ganitha Nirnayam.[10] though the notion of a function. or of exponential or logarithmic functions."[12] However. . Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers.[13][14] a commentary on the Yuktibhasa's proof of the sine and cosine series[15] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan. Then the Parinathi Kriya or Parinathi Samskara.Ascending Node ) and h. the angle between the planet on the Mean Circle and the planet on the Heliocentric circle. Vikshepa Vritteeya Manda Sphutam = Heliocentric Longitude. is computed by the formula x = R e Sin M and x is then added ( if long > 180 ) or subtracted ( if long <180 ) to the Mean Longitude of the planet. ( Ganitha Nirnaya ) This is the Longitude corrected thrice Method of Kerala Astronomy While the Western method is brilliant. the Kerala Method is no less effective. the angle between Position ( of the planet ) and Aphelion is computed. r. l Vikshepa Vritteya Manda Karna = the Radius Vector. Vikshepa Vritta = Heliocentric Circle. Computing Planetary Positions. Graha Madhyamam. In the Kerala Method. finding out the True Anomaly of the Planet ( Theta = v + w ) and then converting it into Cartesian coordinates. x. And finally the Sheeghra Anomaly. the Manda Anomaly. the angle between Position and the Node is computed ( Vipata Kendra = Ecliptic longitude . Reduction to the Ecliptic is done. x. Vikshepa Vritheeya Gatho Vipatha Thasmannayel Jyam Parinathyabhikhyam Yugmau pada swarnam idam Vidheyam Syal Kranti Vritteeya Ehaisha Chandra.htm ). the angle between Position and the Earth Sun Vector ( Sheegra Anomaly = Ecliptic Longitude . Theta and Phi ( vide Paul Schlyter.Aphelion. the angle between the planet on the Heliocentric Circle and the planet of the Ecliptic is computed and then h is added or subtracted ( added if mean longitude is in even signs and subtracted if mean longitude is in odd signs ) to the Mean Longitude of the planet. the Sheegra phalam.y and z and then converting them into Spherical Coordinates.Longitude of Sun ) is computed to get the true longitudes of planets and x. after computing the Mean Longitude of the planet. the Parinathi Phalam. r Kranti Vritta = The Ecliptic Kranti Vritteeya Manda Sphuta = Ecliptic Longitude Kranti Vritteya Manda Karna = Ecliptic Vector . Then Manda jya phalam. The formula used is Manda Anomaly = Mean Longitude of the Planet . the angle between the planet on the Ecliptic and the Geocentric Circle is added ( if long >180 ) or subtracted ( if long <180 ) to the mean longitude ! These are the terms used in Kerala algorithms for the calculation of planetary longitudes. Then Vipata Kendra. especially in the manuscript collections of Spain and Maghreb. cosine. 1990. and turn calculus into the great problem-solving tool we have today. or even outside of Kerala. there is no direct evidence by way of relevant manuscripts that such a transmission took place. 2. learned of some of the ideas of the Islamic and Indian mathematicians through sources of which we are not now aware. Ranjan. 12) Quote: "There is no evidence that the Indian work on series was known beyond India.[10] However. K. "including. Lamda Vikshepa = Celestial Latitude."[10] This is an active area of current research. they had.[18] Kerala was in continuous contact with China and Arabia. and Nilakantha. ^ a b c d e f g h i Roy. Delta The Sheeghra Sphuta = Geocentric Longitude. until the nineteenth century. according to David Bressoud. research that is now being pursued. ^ (Stillwell 2004. been lost to India. show the connection between the two. among other places. Gregory." Mathematics Magazine (Mathematical Association of America) 63(5):291-306. Fermat and Roberval. The expansions of the sine. The Sheeghra Karna = Geocentric Vector."[10] The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own.The Sheeghra Pratimandala = Geocentric Circle. or even outside Kerala.[10] See also Indian astronomy Indian mathematics Indian mathematicians History of mathematics Notes 1. "Discovery of the Series Formula for by Leibniz. until the nineteenth century. at the Centre national de la recherche scientifique in Paris. p.[10] however. in particular. ^ (Bressoud 2002. for all practical purposes. it is not known with certainty whether the immediate predecessors of Newton and Leibniz."[9][21] Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. Possibility of transmission of Kerala School results to Europe A. 173) 3. 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. p. "there is no evidence that the Indian work of series was known beyond India. to "combine many differing ideas under the two unifying themes of the derivative and the integral. they were not able. and Europe. however.[20] In fact. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe. as Newton and Leibniz were. The suggestion of some communication routes and a chronology by some scholars[19][20] could make such a transmission a possibility. and arc tangent had been passed down . pp. They were apparently only interested in specific cases in which these ideas were needed. . in (Bag 1979. The points of resemblance. and now we have the Sanskrit texts properly edited. at least in any of the material that has come down to us.D. they were not interested in any polynomial of degree higher than four. show the .g. p. ^ Katz 1995. in the 1830s. in the 10th century)" [Joseph 1991. therefore. they also knew how to calculate the differentials of these functions. 293). But. in which Whish's article was published. The matter resurfaced in the 1950s. 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.. is irrelevant here" 5. ^ Pingree 1992. as in the examples we have seen. however. The differential "principle" was not generalized to arbitrary functions—in fact. 294).." 4. 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. remained within that specific trigonometric context. 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...through several generations of disciples.. There is no danger. . 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. and we understand the clever way that Mādhava derived the series without the calculus.. p. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. it appears. Indian scholars... however. not to mention that of its derivative or an algorithm for taking the derivative. When this was first described in English by Charles Whish.. it was heralded as the Indians' discovery of the calculus. 300]. particularly between early European calculus and the Keralese work on power series. 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." 6. that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus.. that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. on the other hand. but they remained sterile observations for which no one could find much use. 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. 562 Quote:"One example I can give you relates to the Indian Mādhava's demonstration. . 285)). the explicit notion of an arbitrary function. Thy were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral. in about 1400 A. It does not appear. presumably at first because they could not admit that an Indian discovered the calculus. 1000—and evidently could find such a formula for any polynomial in which they were interested. It should be borne in mind. or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991. 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. but later because no one read anymore the Transactions of the Royal Asiatic Society. 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. 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