Madhava of Sangamagrama

Information on Madhava, the famous Indian mathematician.

1. Madhava of Sangamagrama ഇരിഞ്ഞാറ്റപ്പിള്ളി മാധവൻ നമ്പൂതിരി A PRESENTATION ON THE CONTRIBUTIONS MADE BY THE GENIUS MATHEMATICIAN FROM KERALA, INDIA

2. We pay obeisance to the computational geniuses of the great Indian mathematicians and astronomers of yesteryears. This presentation is a tribute to their unparalleled achievements.

3. INDEX INTRODUCTION CONTRIBUTIONS Infinite series Trigonometry The value of π (pi) Calculus MADHAVA’S WORKS KERALA SCHOOL OF ASTRONOMY AND MATHEMATICS INFLUENCES

4. INTRODUCTION No personal details about Madhava has come to light. Sangamagrama is surmised to be a reference to his place of residence. Some historians have identified Sangamagrama as modern day Irinjalakuda in Thrissur District, Kerala. Madhava provided the creative impulse for the development of a rich mathematical tradition in medieval Kerala. However, except for a couple, most of Madhava's original works have been lost. 

5. Iringatappilli temple near Irinjalakkuda The granite slabs in the picture were said to have been used by Madhava for temple rituals and astronomical studies.

6. A flow-chart showing the disciples of Madhava

7. CONTRIBUTIONS INFINITE SERIES Among his many contributions, he discovered infinite series for the trigonometric functions of sine, cosine, arctangent, and many methods for calculating the circumference of a circle.  In the text Yuktibhasa ,  Jyeṣṭhadeva  describes the series in the following manner: “The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.”

8. The series is known as the Gregory series. Its discovery has been attributed to Gottfried Wilhelm Leibniz (1646 - 1716) and James Gregory (1638 1675). The series was known in Kerala more than two centuries before its European discovers were born.

9. TRIGONOMETRY Madhava composed an accurate table of sines . Marking a quarter circle at twenty-four equal intervals, he gave the lengths of the half-chord ( sines ) corresponding to each of them. It is believed that he may have computed these values based on the series expansions: sin q = q − q3/3! + q5/5! − q7/7! + ... cos  q = 1 − q2/2! + q4/4! − q6/6! + ...

10. The value of π (pi) Madhava's work on the value of the mathematical constant Pi is cited in the  Mahajyānayana prakāra  ("Methods for the great sines "). While some scholars such as Sarma feel that this book may have been composed by Madhava himself, it is more likely the work of a 16th-century successor.  This text attributes most of the expansions to Madhava , and gives the following infinite series expansion of π, now known as the  Madhava -Leibniz series:

11. He obtained the series 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, namely: R n  = (−1) n  / (4 n ), or R n  = (−1) n ⋅ n  / (4 n 2  + 1), or R n  = (−1) n ⋅( n 2  + 1) / (4 n 3  + 5 n ), where the third correction leads to highly accurate computations of π. It has long been speculated how Madhava found these correction terms.

12. They are the first three convergents of a finite continued fraction, which, when combined with the original Madhava's series evaluated to n terms, yields about 3n/2 correct digits: The absolute value of the correction term in next higher order is | R n | = (4 n 3  + 13 n ) / (16 n 4  + 56 n 2  + 9). Madhava also carried out investigations into other series for arc lengths 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. He also discovered the solutions of transcendental equations by iteration and found the approximation of transcendental numbers by continued fractions.

13. CALCULUS Madhava laid the foundations for the development of calculus, which were further developed by his successors at the Kerala school of astronomy and mathematics. (Certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those of  Bhāskara II. It is uncertain, however, whether any of these ideas were transmitted to the West, where calculus appears to have been developed independently by Isaac Newton and Leibniz at this time. This is an area of current research, and historians and mathematicians are currently researching whether results obtained by the Kerala School of Mathematicians influenced the development of Calculus through the transmission of their results to Europe.

14. MADHAVA’S WORKS K. V. Sarma  has identified Madhava as the author of the following works: Golavada Madhyamanayanaprakara Mahajyanayanaprakara  (Method of Computing Great Sines ) Lagnaprakarana  ( लग्नप्रकरण) Venvaroha  ( वेण्वारोह)[24] Sphutacandrapti  ( स्फुटचन्द्राप्ति) Aganita-grahacara  ( अगणित-ग्रहचार) Chandravakyani  ( चन्द्रवाक्यानि) ( Table of Moon-mnemonics)

15. KERALA SCHOOL OF ASTRONOMY AND MATHEMATICS The Kerala school of astronomy and mathematics flourished for at least two centuries beyond Madhava . In Jyeṣṭhadeva we find the notion of integration, termed  sankalitam , (lit. collection), as in the statement: ekadyekothara pada sankalitam samam padavargathinte pakuti , which translates as the integral of 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, Jyeshthadeva's   Yuktibhāṣā  may be considered the world's first calculus text. The group also did much other work in astronomy; indeed many more pages are developed to astronomical computations than are for discussing analysis related results.

16. The Kerala school also contributed much to linguistics (the relation between language and mathematics is an ancient Indian tradition). The  ayurvedic  and poetic traditions of Kerala can also be traced back to this school. The famous poem,  Narayaneeyam , was composed by  Narayana Bhattathiri , a prominent scholar of this school

17. INFLUENCES Madhava has been called "the greatest mathematician-astronomer of medieval India", or as "the founder of mathematical analysis; some of his discoveries in this field show him to have possessed extraordinary intuition“. The Kerala school was well known in the 15th and 16th centuries, in the period of the first contact with European navigators in the Malabar Coast. At the time, the port of  Muziris , near  Sangamagrama , was a major center for maritime trade, and a number of Jesuit missionaries and traders were active in this region. Given the fame of the Kerala school, and the interest shown by some of the Jesuit groups during this period in local scholarship, some scholars, including G. Joseph of the U. Manchester have suggested 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

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