Madhava of Sangamagrama

Madhava (മാധവന്) of Sangamagrama (1350–1425) was a prominent mathematician-astronomer from Kerala, India. He was the founder of the Kerala School of Mathematics and is considered the founder of mathematical analysis for having taken the decisive step from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis. He is considered as one of the greatest mathematician-astronomers of the Middle Ages due to his important contributions to the fields of mathematical analysis, infinite series, calculus, trigonometry, geometry and algebra.

Unfortunately, most of Madhava's original works have been lost in course of time, as they were written primarily on perishable material. However his works have been detailed by later scholars of the Kerala School, primarily Nilakantha Somayaji and Jyesthadeva.

Contributions

Perhaps Madhava's 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, and thus he is considered the founder of mathematical analysis. In particular, Madhava invented the fundamental ideas of:

Infinite series expansions of functions.
Power series.
Taylor series.
Maclaurin series.
Trigonometric series.
Rational approximations of infinite series.

Among 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 describes -

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.

This yields

r\theta={\frac {r\ sin  \theta  }{cos  \theta

}}-(1/3)\,r\,{\frac { \left(sin \theta \right) ^ {3}}{ \left(cos \theta \right) ^{3}}}+(1/5)\,r\,{\frac { \left(sin \theta \right) ^{5}}{ \left(cos \theta \right) ^{5}}}-(1/7)\,r\,{\frac { \left(sin \theta \right) ^{7}}{ \left(cos \theta \right) ^{ 7}}} + ...

which further yields the theorem

\theta = \tan \theta - (1/3) \tan^3 \theta + (1/5) \tan^5 \theta - \ldots

popularly attributed to James Gregory, three centuries after Madhava. This series was traditionally known as the Gregory series but scholars have recently begun referring to it as the Madhava-Gregory series, in recognition of Madhava's work.

Madhava also found the infinite series expansion of π:

\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots \pm \frac{1}{2n -1}

which he obtained from the power series expansion of the arctangent function.

Using a rational approximation of this series, he gave values of the number π as 3.14159265359 - correct to 11 decimals; and as 3.1415926535898 - correct to 13 decimals. These were the most accurate approximations of π given since the 5th century (see History of Pi).

He gave two methods for computing the value of π.

One of these methods is to obtain a rapidly converging series by transforming the original infinite series of π. By doing so, he obtained the infinite series

\pi = \sqrt{12}\left(1-{1\over 3\cdot3}+{1\over5\cdot 3^2}-{1\over7\cdot 3^3}+\cdots\right)

and used the first 21 terms to compute an approximation of π correct to 11 decimal places as 3.14159265359.

The other method he used was to add a remainder term to the original series of π. He used the remainder term

\frac{n^2 + 1}{4n^3 + 5n}

in the infinite series expansion of $\frac{\pi}{4}$ to improve the approximation of π to 13 decimal places of accuracy when n = 75.

Madhava was also responsible for many other original discoveries, including:

Mathematical analysis

Trigonometric series for sine, cosine, tangent, and arctangent functions
Additional Taylor series approximations of sine and cosine functions
Investigations into other series for arclengths and the associated approximations to rational fractions of π
Methods of polynomial expansion.
Tests of convergence of infinite series.
Analysis of infinite continued fractions.

Trigonometry

The analysis of trigonometric functions (as described above).
Sine table to 12 decimal places of accuracy.
Cosine table to 9 decimal places of accuracy.

Geometry

The analysis of the circle (as described above).
Many methods for calculating the circumference of a circle.
Computation of π correct to 13 decimal places.

Algebra

The solution of transcendental equations by iteration.
Approximation of transcendental numbers by continued fractions.

Calculus

Differentiation.
Integration.
Term by term integration.

Madhava laid the foundations for the development of calculus, including differential calculus and integral calculus, which were further developed by his successors at the Kerala School. (It should be noted that Archimedes also contributed to integral calculus, though not to differential calculus.) Madhava also extended some results found in earlier works, including those of Bhaskara.

Some scholars have suggested that Madhava's work was transmitted to Europe via traders and Jesuit missionaries, and as a result, had an influence on later European developments in analysis and calculus. (See Possible transmission of Kerala mathematics to Europe for further information.)

Kerala School of Astronomy and Mathematics

The Kerala School was a school of mathematics and astronomy founded by Madhava in Kerala (in South India) which included as its prominent members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries and has its intellectual roots with Aryabhatta who lived in the 5th century. The lineage continues down to modern times but the original research seems to have ended with Narayana Bhattathiri (1559-1632). These astronomers, in attempting to solve problems, invented revolutionary ideas of calculus. These discoveries included the theory of infinite series, tests of convergence (often attributed to Cauchy), differentiation, term by term integration, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral. They achieved most of these results up to several centuries before European mathematicians.

Jyeshtadeva consolidated the Kerala School's discoveries in the Yuktibhasa, the world's first calculus text.

The Kerala School also contributed much to linguistics. The ayurvedic and poetic traditions of Kerala were founded by this school. The famous poem, Narayaneeyam, was composed by Narayana Bhattathiri.

Bibliography

George Gheverghese Joseph. The Crest of the Peacock: Non-European Roots of Mathematics, 2nd Edition. Penguin Books, 2000.
T. Hayashi, T. Kusuba and M. Yano. 'The correction of the Madhava series for the circumference of a circle', Centaurus 33 (pages 149-174). 1990.
C. T. Rajagopal and M. S. Rangachari. 'On an untapped source of medieval Keralese mathematics', Archive for History of Exact Sciences 18 (pages 89-102). 1978.
St Andrews University, 2000.
Ian Pearce. Madhava of Sangamagrama at the MacTutor archive. St Andrews University, 2002.
Dr. Sarada Rajeev. Neither Newton nor Leibnitz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala. University of Rochester, 2005

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