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

seen(from left).

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 , Katyayana , Manava , Panini , Pingala and Yajnavalkya) , classical ( Vararuchi, Aryabhata, Varahamihira, Brahmagupta ),medieval( Narayana Pandita, Bhaskaracharya, Samgamgrammadhava, Nilakanda Somayaji, Jyestadeva, Achuta Pisharoti, Melpathur Narayan Bhattathiri, Sankaravarman) and modern periods ( Srinivas Ramanujan, Harish Chandra, Narendra karmakar S Chandrasekhar, S N Bose). 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. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, 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), Aryabhata, Varahamihira, Brahmagupta, Bhaskaracharya, Samgamagram madhava, Nilakandaa Somayaji, Jyeshtadeva, Sankaravarman extending to those of Srinivasa Ramanujan, S N Bose, Harish Chandra Prasanta Chandra Mahalanobis,and reaching to the current period of Narendra Karmakar, Jayant Narlikar, S. R. Srinivasa Varadhan , E C G Sudarsan and Thanu Padmanabhan.

Why to rediscover Samgamagrama Madhavan?

Political chaos caused halting of further generation of new knowledge in North India while Kerala, the south western tip of India, escaped the majority of such political upheaval, 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, 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.. Kerala mathematics was strongly influenced by astronomy leading to the derivation of mathematical results of very high importance. 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. It is quite probable that there are still further discoveries of 'Kerala mathematics' to be made, 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, Leibniz, Gregory, Taylor and Euler were first developed in India. This further demands the necessity of mining out the unexplored landscapes

of Kerala Mathematics so that we may by lucky enough to get gems of high values and qualities. In this context we should remember a self taught mathematician from Trivandrum, Mr P Padmakumar, who discovered astonishing properties of magic square called Srirama Chakram which is now known as strongly Magic Square. We should promote the works of such people among us who are capable of carrying out such wonderful jobs of deciphering our ancient knowledge.

Of all the mathematicians of medieval period, 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. 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. In his series of papers Whish showed that works of Newton, Leibniz, Gregory and others (who lived duirng 17th to 18th century) were just rediscoveries of the mathematics contributed by Kerala School. However his works did not get much attention from the academicians and researchers of the west. 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, Mr C Rajagopal and his colleagues. One of the members of the Kerala School namely Jyeshtadeva needs a special mention. While the rest of the scholars wrote their works in Sanskrit, Jyeshta Deva wrote his book Yukti Bhasha, a treatise in mathematics and Astronomy, in Malayalam for wider accessibility of the knowledge.

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 visishyateGrihanamanisoyam syannigenamanimadhava.

He , known as, Madhavan belongs to the house described as the bekuladhishtita Vihar or in malayalam Iranji ( Bakulam ) ninna Palli . Even to this date there is a house named Iringatappally in Kallettunkara near Iringalakkuda. Ulloor describes Sangama Grama Madhavan as belonging to Iringatappally house in Sangama Grama ( village of Snagameswara, diety of Koodal Manikya Temple-Iringalakkuda) . From the writings of his disciples, the period of his life time can be fixed as 1350 -1425 , three hundred years before the life time of Newton , Gregory and Leibnitz.

It is a fact that even the village of Kallettunkara does not know Samgamagrama Madhavan, one of the stalwarts of Mathematics and Astronomy. It is high time that appropriate steps to be taken to rediscover him . There is a temple of Krishna in Kallettunkara where the Great Acharya used to sit for hours watching the stars . 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.

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 , particularly, in the mind of People of Kerala. Following are some of the programme envisaged

1. Naming the Panchayat Library as Samgamagrama Madhavan Grantha Sala 2. Naming the road leading to the ancestral house of the Acharya as Samgamagrama Madhavan Road. 3. 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. This will stamp the name of Samgamagrama Madhavan in the mind of people

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Some of the future activities are 1. On October 17 Vijayadasami day organizing vidyarambham, public function and a function of face to face with scientists for school children.2. Establishing Samgamagrama Madhavan Research Centre in Kallettinkara / Iringalakkuda to promote studies on history of Mathematics and Astronomy with special reference to Kerala School.3. Organizing monthly programme on Science, Technology and Mathematics for the benefit of public and students.4. Taking up projects with the help of various funding agencies.5. 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. Media form an important component in promoting this noble act and we request the help from media of all format- print and electronic– in reviving the legacy and glory of Iringalakkuda.

Prof. V.P.N. NampooriPresident

Swadeshi Science Movement

Madhava of Sangamagrama - Kerala Mathematician

Posted Date: 04-Nov-2012 Category: General

Author: T.M.Sankaran Member Level: Gold Points: 50 (Rs 50)

Read about Madhava of Sangamagrama,Kerala mathematician. Kerala has a very rich history in subject Mathematics. Though less details have been discovered the available materials show that several stalwarts in it lived in this geographically small area, made contributions comparable to modern mathematicians. Sangama grama Madhavan is one among them, about whom certain available facts are presented below.

Our country, India, was rich in the field of mathematics from very early periods dating back to B.C. 3000 or more. Numbers with decimal system, geometry, 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. Vedas also provide light into the application of various mathematical methods. Connected with yagas (a ritual to please Gods) and other rituals, determination of suitable time, construction of yaga place, etc. are decided using mathematical methods. The counting numbers including zero (0,1,2, ......,9) which are now used throughout the world are said to have originated here. It is referred sometimes as Arabic numerals, since the Europeans got them from Arabs who had commercial relationships with India from very early periods. Through this connection Arabs got these numerals from India.

When talking about Indian mathematicians the name of Arya bhata I (475 – 550 AD) comes first. According to certain historians the birth place of Arya bhata was in Kerala, though he later shifted to Patna. However after this there is a big gap in the mathematical history of Kerala. However, Kerala has a golden history of Mathematics between 14 and 17 centuries. There are various names already identified as Mathematicians belonging to Kerala, such as Sangamagrama Madhavan, Vadassery Parameswaran, Kelallur Neelakanta Somayaji, Chithrabhanu, Narayanan, Jyeshtadevan, Puthumana chomathiri, Achutha Pisharoti, Sankara Variyar, Sakara varman and so on. Only through some of the known works certain details about these great people are known.

Four works related to Kerala mathematics and astronomy of those periods are generally considered as

'Thantra samgraha' of Neelakantan, 'Yukthi bhasha' of Jyeshtadevan, 'Karana paddhathy' of Puthumana chomathiri and 'Sadrathna mala' of Sankara varman.

One of the names among the Kerala Mathematicians whose works are well recognised is that of Sangama grama Madhavan. His name is connected with the village to where he belonged. It is Sangama Gramam, it is related to the name of a temple 'Sangameswara kshethram' near Irinjalakkuda in Thrissur district. His life period is considered to be during 1350 to 1425 AD. Much biographical details of Madhavan are not available. He belonged to Embranthiri community, a sub- group of Brahmin community. It is believed that these people migrated from coastal Karnataka regions. In his book 'Venvaroham' certain mentions are there about his village and house. This information helped to identify the village as Sangama gramam. 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).

Sangamagrama Madhavan is one of the greatest names among the ancient Indian astronomers. He is recognised also as a pure mathematician. His known works include 'Venvaroham', 'Lagnaprakaram', 'sphutachandrapthi', 'Mahagyanayana prakaram', 'Madhyayanayanaprakaram', 'Aganitham' and 'Aganitha panchangam'. Most of this works are based on Vararuchi's 'Chandravakya Padhathi'. Vararuchi's method only could calculate the position of moon to the nearest minute, where as Madhavan did it to seconds correctness, which is described in 'Venvaroham'. The mathematical principles involved in this method were far advanced than what were available during those periods. He has used the principles of integration and infinite series.

From the works of Madhavan's disciples and successors in the field such as Neelakantan, Jyeshtadevan, Narayanan and Sankara Varier more references to Madhavan's contributions are available. Some of them are very important results in modern mathematics. His results include the derivation of infinite series for circular and trigonometric functions (it is popular as Gregory series for arc-tangent), 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).

Similarly the series related to the value of 'Pi' invented by Madhavan in the 14th century was re-invented by Gregory in 1671 and Leibniz in 1673. 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, but now having seen that the same was invented about two and a half century back by Madhavan, the series has been re-named as Madhava – Gregory series. Similarly the Newton's power series expansion for sine and cosine is now known as Madhava – Newton series.

Madhavan is referred to in certain references as 'Golavid' meaning 'a learned man about the globe', mainly because of his works in the field. His works have highly influenced the works of Parameswaran, Neelakantan, Jyeshtadevan and others. Madhavan can be even considered as the introducer of topic Mathematical Analysis, since his attempts in this field are notable. His intuitive approach in solving

problems can be compared with that of great Sreenivasa Ramanujan (1887 – 1920).

Further studies are needed to find out the Madhavan's total contributions in the field of Mathematics. 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. He had discovered several results in trigonometry and infinite series much earlier than these were rediscovered by European Mathematicians.

INDIAN MATHEMATICS - MADHAVA

Madhava sometimes called the greatest mathematician-astronomer of medieval India. He came from the town of Sangamagrama in Kerala, near the southern tip of India, and founded the Kerala School of Astronomy and Mathematics in the late 14th Century.

Although almost all of Madhava's original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine, cosine, tangent and arctangent functions and the value of π), representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.

Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancient Egyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions.

Madhava of Sangamagrama (c.1350-1425)

But Madhava went further and linked the idea of an infinite series with geometry and trigonometry. He realized that, by successively adding and subtracting different odd number fractions to infinity, 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). Through his application of this series, Madhava obtained a value for π correct to an astonishing 13 decimal places.

He went on to use the same mathematics to obtain infinite series expressions for the sine formula, which could then be used to calculate the sine of any angle to any degree of accuracy, as well as for other trigonometric functions like cosine, tangent and arctangent. Perhaps even more remarkable, though, is that he also gave estimates of the error term or correction term, implying that he quite understood the limit nature of the infinite series.

Madhava’s use of infinite series to approximate a range of trigonometric functions, which were further developed by his successors at the Kerala School, effectively laid the foundations for the later development of calculus and analysis, and either he or his disciples developed an early form of integration for simple functions. Some historians have 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 Cochin (Kochi) at the time, and may have had an influence on later European developments in calculus.

Among his other contributions, Madhava discovered the solutions of some transcendental equations by a process of iteration, and found approximations for some transcendental numbers by continued fractions. In astronomy, he discovered a procedure to determine the positions of the Moon every 36 minutes, and methods to estimate the motions of the planets.

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Information about Madhava of Sangamagrama

Mādhava of Sangamagrama (Malayalam: മാ�ധവന്� , Mādhavan) (c.1350–c.1425) was a prominent Hindu mathematician-astronomer from the town of Irinjalakkuda, near Cochin, Kerala, India, which was at the time known as Sangamagrama (lit. sangama = union, grāma=village). He is considered the

Madhava’s method for approximating π by an infinite series of fractions

founder of the Kerala 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 to infinity"[1]. His discoveries opened the doors to what has today come to be known as mathematical analysis.[2]. One of the greatest mathematician-astronomers of the Middle Ages, Madhava contributed to infinite series, calculus, trigonometry, geometry and algebra.

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[3].

Historiography

Although there is some evidence of Mathematical work in Kerala prior to Madhava (e.g. Sadratnamala c.1300, a set of fragmentary results[3]), 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[3]) 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. In Jyesthadeva's Yuktibhasa (c.1530[4]), written in Malayalam, these series are presented with proofs in terms of the Taylor series expansions for polynomials like 1/(1+x2), 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[1], 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[1].

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)[3]. In the mid-20th century, the Russian scholar Jushkevich revisited the legacy of Madhava[6], and a comprehensive look at the Kerala school was provided by Sarma in 1972[4].

Lineage

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 the Surya 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 grammarian Melpathur Narayana Bhattathiri as his disciple[4].

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)[8]. This implies that the limit nature of the infinite series was quite well understood by him. Thus, Madhava may have invented the ideas underlying infinite series expansions of functions, power series, Trigonometric series, and rational approximations of infinite series.[8]

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 seriesAmong his many contributions, he discovered the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text Yuktibhasa, which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.[9] In the text, Jyesthadeva describes the series in the following manner:

Insert the text of the quote here, without quotation marks.

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). 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 occasionally referred to as the Madhava-Gregory series[9][10].

TrigonometryMadhava 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[2]:

sin q = q - q3/3! + q5/5! - ...cos q = 1 - q2/2! + q4/4! - ...

The value of (pi)πWe 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[4] feel that this book may have been composed by Madhava himself, it is more likely the work of a 16th century successor [2]. This text attributes most of the expansions to Madhava, and gives the following infinite series expansion of π:

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, Rn, for the error after computing the sum up to n terms. Madhava gave three forms of Rn which improved the approximation[2], namely

Rn = 1/(4n), orRn = n/ (4n2 + 1), or

Rn = (n2 + 1) / (4n3 + 5n).

where the third correction leads to highly accurate computations of π.

It is not clear how Madhava might have found these correction terms[11]. 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 Aryabhata).

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, he obtains a value correct to 11 decimal places (3.14159265359)[12]. The value of 3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava[13], but may be due to one of his followers. These were the most accurate approximations of π given since the 5th century (see History 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[12][4].

AlgebraMadhava 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 infinite continued fractions.[4] He also discovered the solutions of transcendental equations by iteration, and found the approximation of transcendental numbers by continued fractions.[4]

CalculusMadhava laid the foundations for the development of calculus, which were further developed by his successors at the Kerala school of astronomy and mathematics.[8][16] (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 of Bhaskara.

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, [10]

which translates as the integration a variable (pada) equals half that variable squared (varga); i.e. The integral of x dx is equal to x2 / 2. This is clearly a start to the process of integral calculus. A related result states that the area under a curve is its integral. 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 first calculus text.[8][16][3]

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[4].

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 of Kerala 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"[4], or as "the founder of mathematical analysis; some of his discoveries in this field show him to have possessed extraordinary intuition."[3]. O'Connor and Robertson state that a fair assessment of Madhava is that he took the decisive step towards modern classical analysis[2].

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, near Sangamagrama, was a major center for maritime trade, and a number of Jesuit missionaries and traders were active in this region. Given the fame of the Kerala school, and the interest shown by some of the Jesuit groups during this period in local scholarship, Some scholars, including G. Joseph of the U. Manchester have suggested[18] 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[3]. 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).

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 3.14159265359. By adding a remainder term to the original power series of π /4, he was able to compute π to an accuracy of 13 decimal places.

he number π (pi) is a mathematical constant that is the ratio of a circle's circumference to its diameter. The constant, sometimes written pi, is approximately equal to 3.14159. It has been represented by the Greek letter "π" since the mid-18th century. π is an irrational number, 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); consequently, its decimal representation never ends and never repeats. Moreover, π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. The transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge. The digits in the decimal representation of π appear to be random, although no proof of this supposed randomness has yet been discovered.

For thousands of years, mathematicians have attempted to extend their understanding of π, sometimes by computing its value to a high degree of accuracy. Before the 15th century, mathematicians such as Archimedes and Liu Hui used geometrical techniques, based on polygons, to estimate the value of π. Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of π, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.

ow known as the Gregory-Leibniz series since it was rediscovered by James Gregory and Gottfried Leibniz in the 17th century. Unfortunately, the rate of convergence is too slow to calculate many digits in practice; about 4,000 terms must be summed to improve upon Archimedes' estimate. However, by transforming the series into

\pi = \sqrt{12} \, \left(1-\frac{1}{3 \cdot 3} + \frac{1}{5 \cdot 3^2} - \frac{1}{7 \cdot 3^3} + \cdots\right)

Madhava was able to calculate π as 3.14159265359, correct to 11 decimal places. The record was beaten in 1424 by the Persian astronomer Ghiyath al-Kashi, who determined 16 decimals of π.

development of integral calculus.[6] In the 12th century, the Persian mathematician Sharaf al-Din al-Tusi discovered the derivative of cubic polynomials, an important result in differential calculus.[7] In the 14th century, Madhava of Sangamagrama, along with other mathematician-astronomers of the Kerala school of astronomy and mathematics, described special cases of Taylor series,[8] which are treated in the text Yuktibhasa.[9][10][11]

The Kerala School, European Mathematics and NavigationBy D.P. Agrawal

The National Geographic has declared Kerala, the south-west coast near the tip of the Indian peninsula, as God's Own Country. It has been a centre of maritime trade, with its rich variety of spices greatly in demand, even as early as the time of the Babylonians. Famous travellers and explorers such as Ibn Battuta and Vasco da Gama came from across the Arabian Sea. In recent years, Kerala has gained recognition for its role in the reconstruction of medieval Indian mathematics.

Joseph (1994) has very emphatically brought out the significance of the Kerala school of Maths in his The Crest of the Peacock, though the Eurocentric scholars have severely criticized it. C.K. Raju, the well known mathematician and historian of science, has also written a good deal not only on the famous work, Yuktibhasa by Jyesthadeva, but also on the export of Maths from India to Europe. Until recently there was a misconception that mathematics, in India made no progress after Bhaskaracharya, that later scholars seemed 'content to chew the cud, writing endless commentaries on the works of the venerated mathematicians who preceded them, until they were introduced to modern mathematics by the British. Though the picture about the rest of India is not clear, in Kerala, the period between the fourteenth and seventeenth centuries marked a high point in the indigenous development of astronomy and mathematics. The quality of the mathematics available from the texts that have been studied is of such a high level, compared with the earlier period that it is impossible to bridge the gap between the two periods. Nor can one invoke a 'convenient' external agency, like Greece or Babylonia to explain the Kerala phenomenon. There were later discoveries in European mathematics, which were anticipated by Kerala astronomer-mathematicians two hundred to three hundred years earlier. And this leads us to ask whether the developments in Kerala had any influence on European mathematics. The only scholar who has dealt with this issue to my mind is C.K. Raju, whose views would also be discussed in this essay.

Joseph informs that in 1835, Charles Whish published an article in which he referred to four works – Nilakantha's Tantra Samgraha, Jyesthadeva's Yuktibhasa, Putumana Somayaji's Karana Paddhati and Sankara Varman' s Sadratnamala – as being among the main astronomical and mathematical texts of the Kerala school. While there were some doubts about Whish's views on the dating and authorship of these works, his main conclusions are still broadly valid. Writing about Tantra Samgraha, 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, or function, which he called a 'fluent']. The Sadratnamala, a summary of a number of earlier works, he says 'abounds with fluxional forms and series to be found in no work of foreign countries'. The Kerala discoveries include the Gregory and Leibniz series for the inverse tangent, the Leibniz power series for p, and the Newton power series for the sine and cosine, as well as certain remarkable rational approximations of trigonometric functions, including the well-known Taylor series approximations for the sine and cosine functions. And these results had apparently been obtained without the use of infinitesimal calculus.

In the 1940s it was Rajagopal and his collaborators who highlighted the contributions of Kerala mathematics, though none of their results has as yet percolated into the standard Western histories of mathematics. For example, Boyer (1968, p. 244) writes that 'Bhaskara was the last significant medieval mathematician from India, and his work represents the culmination of earlier Hindu contributions.' And according to Eves (1983, p. 164), 'Hindu mathematics after Bhaskara made only spotty progress until modem times.'

Madhava's work on power series for p and for sine and cosine functions is referred to by a number of the later writers, although the original sources remain undiscovered or unstudied. Nilakantha (1445-1555) was mainly an astronomer, but his Aryabhatiya Bhasya and Tantra Samgraha contain work on infinite-series expansions, problems of algebra and spherical geometry. Jyesthadeva (c. 1550) wrote, in a regional language rather than in Sanskrit, Yuktibhasa, one of those rare texts in Indian mathematics or astronomy that gives detailed derivations of many theorems and formulae in use at the time.This work is mainly based on the Tantra Samgraha of Nilakantha. A joint commentary on Bhaskaracharya's Lilavati by Narayana (c. 1500-75) and Sankara Variar (c. 1500-1560), entitled Kriyakramakari, also contains a discussion of Madhava's work. The Karana Paddhati by Putumana Somayaji (c. 1660-1740) provides a detailed discussion of the various trigonometric series. Finally there is Sankara Varman, the author of Sadratnamala, 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. His work in five chapters contains, appropriately, a summary of most of the results of the Kerala school, without any proofs though.

Astronomy provided the main motive for the study of infinite-series expansions of p and rational approximations for different trigonometric functions. For astronomical work, it was necessary to have both an accurate value for p and highly detailed trigonometric tables. In this area Kerala mathematicians made the following discoveries:

1. The power series for the inverse tangent, usually attributed to Gregory andLeibniz;

2. The power series for p, usually attributed to Leibniz, and a number of rationalapproximations to p; and

3. The power series for sine and cosine, usually attributed to Newton, and approximations for sine and cosine functions (to the second order of small quantities), usually attributed to Taylor; this work was extended to a third-order series approximation of the sine function, usually attributed to Gregory.

Apart from the work on infinite series, there were extensions of earlier work notably of Bhaskaracharya:

1. The discovery of the formula for the circum-radius of a cyclic quadrilateral,which goes under the name of l'Huilier's formula;

2. The use of the Newton-Gauss interpolation formula (to the second order) by Govindaswami; and3. The statement of the mean value theorem of differential calculus, first recorded by Paramesvara

(1360-1455) in his commentary on Bhaskaracharya's Lilavati.

Here it may be relevant to note some points of the debate that CK Raju has been carrying out with the West in general, and with Whiteside (the famous historian of Maths) in particular, about the export of Maths to Europe.

Raju's Encounter with Eurocentric scholars

Raju (personal communication) explains that Whiteside, while conceding Madhava's priority for the development of infinite series, distorts the dates of both Madhava and the Yuktibhasa, by about a century in each case. (Madhava was 14th-15th c. CE,not 13th, while the Tantrasangraha [1501 CE] and Yuktibhasa [ca. 1530 CE] are both 16th c. CE texts, not 17th.) In fact, in the 16th c. CE Jesuits were busy translating and transmitting very many Indian texts to Europe; during the 16th c. CE, their activities were especially concentrated in the vicinity of their Cochin College, where they were teaching Malayalam to the local children (especially Syrian Christians) whose mother tongue it was, and where copies of the Yuktibhasa and several other related texts were and still are in common use, for calendar-making for example.

After the trigonometric values in the 16th and early 17th c. CE, exactly the infinite series in these Indian texts started appearing in the works, from 1630 onwards, of Cavalieri, Fermat, Pascal, Gregory etc. who had access in various ways to the Jesuit archives at the Collegio Romano. 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 theYuktibhasa contemporaneous!

Raju has no doubt that in the course of "the fabrication of ancient Greece" (in Martin Bernal's words), some Western historians acquired ample familiarity with this technique of juggling the dates of key texts. Having anticipated this, 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.

While the case for the origin of the calculus in India, and its transmission to Europe is otherwise clear, there remains the important question of epistemology ("Was it really the calculus that Indians discovered?"). For, while European mathematicians accepted the practical value of the Indian infinite series as a technique of calculation, many of them did not, even then, accept the accompanying methods of proof. Hence, like the algorismus which took some five centuriesto be assimilated in Europe, the calculus took some three centuries to be assimilated within the European frame of mathematics.Raju has discussed this question in depth, in relation to formalist mathematical epistemology from Plato to Hilbert, in an article "Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa".

In this paper, Raju proposes a new understanding of mathematics. He argues that formal deductive proof does not incorporate certainty, since the underlying logic is arbitrary, and the theorems that can be derived from a particular set of axioms would change if one were to use Buddhist logic, or, say, Jain logic.

Raju further states, "Indeed, I should point out that my interest in all this is not to establish priority, as Western historians have unceasingly sought to do, but to understand the historical development of mathematics and its epistemology. The development of the infinite series and more precise computations of the circumference of the circle, by Aryabhata's school, over several hundredyears, is readily understood as a natural consequence of Aryabhata's work, which first introduced the trigonometric functions and methods of calculating their approximate numerical values. The transmission of the calculus to Europe is also readily understood as a natural consequence of the European need to learn about navigation, the calendar, and the circumference of the earth. The centuries of difficulty in accepting the calculus in Europe is more naturally understood inanalogy with the centuries of difficulty in accepting the algorismus, due, in both cases, to the difficulty in assimilating an imported epistemology. Though such an understanding of the past varies strikingly from the usual "heroic" picture that has been propagated by Western historians, it is far more real, hence more futuristically oriented, for it also helps us to understand e.g. how to tacklethe epistemological challenge posed today in interpreting the validity of the results of large-scale numerical computation, and hence to decide, e.g., how mathematics education must today be conducted.

'I would not like to go further here into the difficult question of epistemology, and the interaction between history and philosophy of mathematics, except to link it to Whiteside's use of the phrase "Hindu matmatics" [sic]. Am I to understand that Whiteside now implicitly accepts also the possible influence of Newton's theology on his mathematics, and is alluding, albeit indirectly, to some subtle new changes brought about by Newton in the prevailing atmosphere of, shall we say, "Christianmathematics"? Probably not. I presume instead that, despite his protestations to the contrary, Whiteside is really referring to the Eurocentric belief that there is only one "mainstream" mathematics, and everything else needs to be qualified as "Hindu mathematics","Islamic mathematics" etc.

'Now it is true that I have commented on formalist mathematical epistemology from the perspective of Buddhist, Jain, Nyaya,and Lokayata notions of proof (pramana),in my earlier cited paper and book. I have also commented elsewhere, from the perspective of Nagarjuna's sunyavada, on the re-interpretation of sunya as zero in formal arithmetic, and the difficulties that this created in the European understanding of both algorismus and calculus, difficulties that persist tothis day in e.g. the current way of handling division by zero in the Java computing language. Nevertheless, having also scanned the OED for the meaning of "Hindu", I still don't quite know what this term "Hindu" means, especially in Whiteside's "ruggedly individualistic" non-Eurocentric sense, and especially when it is linked with mathematics! Given the fundamental differences between the four schools listed above, it is very hard for me to dump them all, like Whiteside, into a single category of "Hindu"; on the other hand, if we exclude some, which counts as "Hindu" and which not, and why? And exactly how does that relate to mathematics?

'A key element of the Project of History of Indian Science, Philosophy, and Culture, as I stated earlier, is to get rid of this sort of conceptual clutter ,authoritatively sought to be imposed by

colonialists (and their victims/collaborators), and to rewrite history from a fresh, pluralisticperspective. In my case, 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. Since my objective is truth and understanding, I am ever willing to correct myself, and I remain open to all legitimate criticism, but I do not recognize dramatic poses, assertions of authority, abuse, cavil, misleading circumlocutions, etc. as any part of such legitimate criticism.

'There are numerous other points in Whiteside's prolix response, to which it would be inappropriate to provide detailed corrections here. [E.g., I do not share the historical view needed to speak of the "re-birth" of European mathematics in the 16th and 17th c., which view Whiteside freely attributes to me, though I would accept that direct trade with India in spices also created a direct route for Indian mathematics, bypassing the earlier Arab route.] For the record, I deny as similarly inaccurate all the interpolations and distortions he has introduced into what I have said.

'There is, however, one issue, which remains puzzling, even from a purely Eurocentric perspective. In what sense did Newton invent the calculus? Clearly, the calculus as a method of calculation preceded Newton, even in Europe. Clearly, also, the calculus/analysis as something epistemologically secure, within the formalist frame of _mathematics as proof_, postdates Dedekind and the formalist approach to real numbers. While Newton did apply the calculus to physics, that would no more make him the inventor of the calculus than the application of the computer to a difficult problem of genetics, and possible adaptations to its design, would today make someone the inventor of the computer. Doubtless Newton's authority conferred a certain social respectability on the calculus. The credit that Newton gets for the calculus depends also upon his quarrel with Leibniz, and the rather dubious methods of "debate" he used in the process. But none of this convincingly establishes the credit for calculus given to Newton, even within the Eurocentric (as distinct from Anglocentric) frame. So what basis is there to give credit to Newton for originating the calculus, while denying it, for example, to Cavalieri, Fermat, Pascal, 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, starting from the "Newtonian Revolution". According to the "standard" story, the calculus was invented independently by Leibniz and Newton. This story of indigenous development, ab initio, is now beginning to totter, like the story of the "Copernican Revolution". The English- speaking world has known for over one and a half centuries that "Taylor" series expansions for sine, cosine and arctangent functions were found in Indian mathematics/astronomy/timekeeping (jyotisa) texts, and specifically the works of Madhava, Neelkantha, Jyeshtadeva etc. No one else, however, has so far studied the connection of these Indian developments to European mathematics.

The relation is provided by the requirements of the European navigational problem, the foremost problem of the time in Europe. Columbus and Vasco da Gama used dead reckoning and were ignorant of celestial navigation. Navigation, however, was both strategically and economically

the key to the prosperity of Europe of that time. Accordingly, various European governments acknowledged their ignorance of navigation, while announcing huge rewards to anyone who developed an appropriate technique of navigation.

The Jesuits, of course, needed to understand how the local calendar was made, especially since their own calendar was then so miserably off the mark, partly because the clumsy Roman numerals had made it difficult to handle fractions. Moreover, European navigational theorists like Nunes, Mercator, Stevin, and Clavius were then well aware of the acute need not only for a good calendar, but also for precise trigonometric values, at a level of precision then found only in these Indian texts. This knowledge was needed to improve European navigational techniques, as European governments desperately sought to develop reliable trade routes to India, for direct trade with India was then the big European dream of getting rich. At the start of this period, Vasco da Gama, lacking knowledge of celestial navigation, could not navigate the Indian ocean, and needed an Indian pilot to guide him across the sea from Melinde in Africa, to Calicut in India.

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, the Dutch prize of 1636, Mazarin's prize to Morin of 1645, the French offer (through Colbert) of 1666, and the British prize legislated in 1711. Many key scientists of the time (Huygens, Galileo, etc.) were involved in these efforts: the navigational problem was the specific objective of the French Royal Academy, and a key concern for starting the British Royal Society.

Prior to the clock technology of the 18th century, attacks on the navigational problem in the 16th and 17th c. focused on mathematics and astronomy, which were (correctly) believed to hold the key to celestial navigation, and it was widely (and correctly) believed by navigational theorists and mathematicians (e.g. by Stevin and Mersenne) that this knowledge was to be found in ancient mathematical and astronomical or time-keeping (jyotisa) texts of the east. Though the longitude problem has recently been highlighted, this was preceded by a latitude problem, and the problem of loxodromes.

The solution of the latitude problem required a reformed calendar: the European calendar was off by 10 days, and this led to large inaccuracies (more than 3 degrees) in calculating latitude from measurement of solar altitude at noon, using e.g. the method described in the Laghu Bhaskariya of Bhaskara I. However, reforming the calendar required a change in the dates of the equinoxes, hence a change in the date of Easter, and this was authorised by the Council of Trent in 1545. This period saw the rise of the Jesuits. Clavius studied in Coimbra under the mathematician, astronomer and navigational theorist Pedro Nunes, and Clavius subsequently reformed the Jesuit mathematical syllabus at the Collegio Romano. Clavius also headed the committee which authored the Gregorian Calendar Reform of 1582, and remained in correspondence with his teacher Nunesduring this period.

Jesuits, like Matteo Ricci, who trained in mathematics and astronomy, under Clavius' new syllabus [Matteo Ricci also visited Coimbra and learnt navigation], were sent to India. In a 1581 letter, Ricci explicitly acknowledged that he was trying to understand local methods of

timekeeping (jyotisa), from "an intelligent Brahmin or an honest Moor", in the vicinity of Cochin, which was, then, the key centre for mathematics and astronomy, since the Vijaynagar empire had sheltered it from the continuous onslaughts of raiders from the north. Language was not a problem, since the Jesuits had established a substantial presence in India, had a college in Cochin, and had even started printing presses in local languages, like Malayalam and Tamil by the 1570's.

In addition to the latitude problem, settled by the Gregorian Calendar Reform, there remained the question of loxodromes, which were the focus of efforts of navigational theorists like Nunes, Mercator etc. The problem of calculating loxodromes is exactly the problem of the fundamental theorem of calculus. Loxodromes were calculated using sine tables, and Nunes, Stevin, Clavius etc. were greatly concerned with accurate sine values for this purpose, and each of them published lengthy sine tables. Madhava's sine tables, using the series expansion of the sine function were then the most accurate way to calculate sine values.

Europeans encountered difficulties in using these precise sine value for determining longitude, as in Indo-Arabic navigational techniques or in the Laghu Bhaskariya, because this technique of longitude determination also required an accurate estimate of the size of the earth, and Columbus had underestimated the size of the earth to facilitate funding for his project of sailing West. Columbus' incorrect estimate was corrected, in Europe, only towards the end of the 17th c. CE. Even so, the Indo-Arabic navigational technique required calculation, while Europeans lacked the ability to calculate, since algorismus texts had only recently triumphed over abacus texts, and the European tradition of mathematics was "spiritual" and "formal" rather than practical, as Clavius had acknowledged in the 16th c. and as Swift (Gulliver's Travels) had satirized in the 18th c. This led to the development of the chronometer, an appliance that could be mechanically used without application of the mind.

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. We should all be thankful to both G.G. Joseph and C.K. Raju for their valuable contributions in this regard.

Kerala school of astronomy and mathematicsFrom Wikipedia, the free encyclopedia

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For other uses of this name, see Kerala school (disambiguation).

The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school flourished between the 14th and 16th centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri (1559–1632). In attempting to solve astronomical problems, the Kerala school independently created a number of important mathematics concepts. Their most important results

—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called Tantrasangraha, and again in a commentary on this work, called Tantrasangraha-vakhya, of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work Yuktibhasa (c.1500-c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.[1]

Their work, completed two centuries before the invention of calculus in Europe, provided what is now considered the first example of a power series (apart from geometric series).[2] However, they did not formulate a systematic theory of differentiation and integration, nor is there any direct evidence of their results being transmitted outside Kerala.[3][4][5][6]

Contents

1 Contributions o 1.1 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. These include the following (infinite) geometric series:

for [7]

This formula, 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).[8]

The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs.[1] They used this to discover a semi-rigorous proof of the result:

for large n. This result was also known to Alhazen.[1]

They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for , , and .[9] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:[1]

where

where, for , the series reduce to the standard power series for these trigonometric functions, for example:

and

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

The Kerala school made use of the rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature (i.e. computation of area under the arc of the circle), was not yet developed.)[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. For example, the error, , (for n odd, and i = 1, 2, 3) for the series:

where

They manipulated the terms, using the partial fraction expansion of : to obtain a more rapidly converging series for :[1]

They used the improved series to derive a rational expression,[1] for correct up to nine decimal places, i.e. . They made use of an intuitive notion of a limit to compute these results.[1] The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,[10] though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.

The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835, though there exists another work, namely Kala Sankalita by J. Warren from 1825[11] which briefly mentions the discovery of infinite series by Kerala astronomers. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries."[12] However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in Yuktibhasa given in two papers,[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, sin, and cosine (with English translation and commentary).[16][17]

The genius of Puliyoor Purushotaman Namboothiri condensed the thousands of slokas of the 18 Siddhantas into 1000 odd slokas, called the Ganitha Nirnayam. It is one of the best books on Mathematical Astrology ever written. He was ably assisted by the Maths Professor, Prof Krishna Warrier.

In the Kerala system, 14 major perturbations of the Moon are highlighted and 14 trignometric corrections are done for the 14 lunar anomalies.

Chathur dasebhyebhyo balanyabhibhyo Neethva Thulasadi Vasa Dhanarnam

Krithva tad Indor Apaneeya Thungam Thado Mridujya phala Samskruthendu

( Ganitha Nirnayam )

After computing the Chandra Madhyamam, the mean longitude of the Moon, after 14 trignometric corrections, the longitude is called Samskrutha Chandra Madhyamam, Samskruthendu.

Then the Parinathi Kriya or Parinathi Samskara, Reduction to the Ecliptic is done.

Vikshepa Vritheeya Gatho Vipatha Thasmannayel Jyam Parinathyabhikhyam Yugmau pada swarnam idam Vidheyam Syal Kranti Vritteeya Ehaisha Chandra.

( Ganitha Nirnaya )

This is the Longitude corrected thrice Method of Kerala Astronomy

While the Western method is brilliant, finding out the True Anomaly of the Planet ( Theta = v + w ) and then converting it into Cartesian coordinates, x,y and z and then converting them into Spherical Coordinates, r, Theta and Phi ( vide Paul Schlyter, Computing Planetary Positions.htm ), the Kerala Method is no less effective.

In the Kerala Method, after computing the Mean Longitude of the planet, Graha Madhyamam, the Manda Anomaly, the angle between Position ( of the planet ) and Aphelion is computed. The formula used is Manda Anomaly = Mean Longitude of the Planet - Aphelion. Then Manda jya phalam, x, the angle between the planet on the Mean Circle and the planet on the Heliocentric circle, 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.

Then Vipata Kendra, the angle between Position and the Node is computed ( Vipata Kendra = Ecliptic longitude - Ascending Node ) and h, the Parinathi Phalam, 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. And finally the Sheeghra Anomaly, the angle between Position and the Earth Sun Vector ( Sheegra Anomaly = Ecliptic Longitude - Longitude of Sun ) is computed to get the true longitudes of planets and x, the Sheegra phalam, 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.

Vikshepa Vritta = Heliocentric Circle.

Vikshepa Vritteeya Manda Sphutam = Heliocentric Longitude, l

Vikshepa Vritteya Manda Karna = the Radius Vector, r

Kranti Vritta = The Ecliptic

Kranti Vritteeya Manda Sphuta = Ecliptic Longitude Kranti Vritteya Manda Karna = Ecliptic Vector

The Sheeghra Pratimandala = Geocentric Circle.

The Sheeghra Karna = Geocentric Vector, Delta The Sheeghra Sphuta = Geocentric Longitude, Lamda Vikshepa = Celestial Latitude.

Possibility of transmission of Kerala School results to Europe

A. K. Bag suggested in 1979 that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries.[18] Kerala was in continuous contact with China and Arabia, and Europe. The suggestion of some communication routes and a chronology by some scholars[19][20] could make such a transmission a possibility, however, there is no direct evidence by way of relevant manuscripts that such a transmission took place.[20] In fact, according to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century."[9][21]

Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.[10] However, they were not able, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today."[10] The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;[10] however, it is not known with certainty whether the immediate predecessors of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources of which we are not now aware."[10] This is an active area of current research, especially in the manuscript collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre national de la recherche scientifique in Paris.[10]

See also

Indian astronomy Indian mathematics Indian mathematicians History of mathematics

Notes

1. ^ a b c d e f g h i Roy, Ranjan. 1990. "Discovery of the Series Formula for by Leibniz, Gregory, and Nilakantha." Mathematics Magazine (Mathematical Association of America) 63(5):291-306.

2. ̂ (Stillwell 2004, p. 173)3. ̂ (Bressoud 2002, p. 12) Quote: "There is no evidence that the Indian work on series was known

beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down

through several generations of disciples, but they remained sterile observations for which no one could find much use."

4. ̂ Plofker 2001, p. 293 Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here"

5. ̂ Pingree 1992, p. 562 Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."

6. ̂ Katz 1995, pp. 173–174 Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. Thy were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the

connection between them, and turn the calculus into the great problem-solving tool we have today."

7. ̂ Singh, A. N. (1936). "On the Use of Series in Hindu Mathematics". Osiris 1: 606–628. doi:10.1086/368443.

8. ̂ Edwards, C. H., Jr. 1979. The Historical Development of the Calculus. New York: Springer-Verlag.

9. ^ a b Bressoud, David. 2002. "Was Calculus Invented in India?" The College Mathematics Journal (Mathematical Association of America). 33(1):2-13.

10. ^ a b c d e f Katz, V. J. 1995. "Ideas of Calculus in Islam and India." Mathematics Magazine (Mathematical Association of America), 68(3):163-174.

11. ̂ Current Science,12. ̂ Charles Whish (1835), Transactions of the Royal Asiatic Society of Great Britain and Ireland13. ̂ Rajagopal, C.; Rangachari, M. S. (1949). "A Neglected Chapter of Hindu Mathematics". Scripta

Mathematica 15: 201–209.14. ̂ Rajagopal, C.; Rangachari, M. S. (1951). "On the Hindu proof of Gregory's series". Scripta

Mathematica 17: 65–74.15. ̂ Rajagopal, C.; Venkataraman, A. (1949). "The sine and cosine power series in Hindu

mathematics". Journal of the Royal Asiatic Society of Bengal (Science) 15: 1–13.16. ̂ Rajagopal, C.; Rangachari, M. S. (1977). "On an untapped source of medieval Keralese

mathematics". Archive for the History of Exact Sciences 18: 89–102.17. ̂ Rajagopal, C.; Rangachari, M. S. (1986). "On Medieval Kerala Mathematics". Archive for the

History of Exact Sciences 35: 91–99.18. ̂ A. K. Bag (1979) Mathematics in ancient and medieval India. Varanasi/Delhi: Chaukhambha

Orientalia. page 285.19. ̂ Raju, C. K. (2001). "Computers, Mathematics Education, and the Alternative Epistemology of

the Calculus in the Yuktibhasa". Philosophy East and West 51 (3): 325–362.20. ^ a b Almeida, D. F.; John, J. K.; Zadorozhnyy, A. (2001). "Keralese Mathematics: Its Possible

Transmission to Europe and the Consequential Educational Implications". Journal of Natural Geometry 20: 77–104.

21. ̂ Gold, D.; Pingree, D. (1991). "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine". Historia Scientiarum 42: 49–65.

References

Bressoud, David (2002), "Was Calculus Invented in India?", The College Mathematics Journal (Math. Assoc. Amer.) 33 (1): 2–13, JSTOR 1558972.

Gupta, R. C. (1969) "Second Order of Interpolation of Indian Mathematics", Ind, J.of Hist. of Sc. 4 92-94

Hayashi, Takao (2003), "Indian Mathematics", in Grattan-Guinness, Ivor, Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 1, pp. 118-130, Baltimore, MD: The Johns Hopkins University Press, 976 pages, ISBN 0-8018-7396-7.

Joseph, G. G. (2000), The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ: Princeton University Press, 416 pages, ISBN 0-691-00659-8.

Katz, Victor J. (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine (Math. Assoc. Amer.) 68 (3): 163–174, JSTOR 2691411.

Parameswaran, S., ‘Whish’s showroom revisited’, Mathematical gazette 76, no. 475 (1992) 28-36

Pingree, David (1992), "Hellenophilia versus the History of Science", Isis 83 (4): 554–563, doi:10.1086/356288, JSTOR 234257

Plofker, Kim (1996), "An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text", Historia Mathematica 23 (3): 246–256, doi:10.1006/hmat.1996.0026.

Plofker, Kim (2001), "The "Error" in the Indian "Taylor Series Approximation" to the Sine", Historia Mathematica 28 (4): 283–295, doi:10.1006/hmat.2001.2331.

Plofker, K. (20 July 2007), "Mathematics of India", in Katz, Victor J., The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages (published 2007), pp. 385–514, ISBN 0-691-11485-4.

C. K. Raju. 'Computers, mathematics education, and the alternative epistemology of the calculus in the Yuktibhâsâ', Philosophy East and West 51, University of Hawaii Press, 2001.

Roy, Ranjan (1990), "Discovery of the Series Formula for by Leibniz, Gregory, and Nilakantha", Mathematics Magazine (Math. Assoc. Amer.) 63 (5): 291–306, JSTOR 2690896.

Sarma, K. V. and S. Hariharan: Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy - an analytical appraisal, Indian J. Hist. Sci. 26 (2) (1991), 185-207

Singh, A. N. (1936), "On the Use of Series in Hindu Mathematics", Osiris 1: 606–628, doi:10.1086/368443, JSTOR 301627

Stillwell, John (2004), Mathematics and its History (2 ed.), Berlin and New York: Springer, 568 pages, ISBN 0-387-95336-1.

Tacchi Venturi. 'Letter by Matteo Ricci to Petri Maffei on 1 Dec 1581', Matteo Ricci S.I., Le Lettre Dalla Cina 1580–1610, vol. 2, Macerata, 1613.