Madhava of Sangamagrama

Madhava (माधव) of Sangamagrama (1350-1425) was a major mathematician from Kerala, in South India. Madhava was the founder of the Kerala School, and is considered the father 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 also considered one of the greatest mathematician-astronomers of the Middle Ages, due to his important contributions to the fields of mathematical analysis, infinite series, calculus, and trignonometry.

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 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.
• 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 before him. 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.

Using a rational approximation of the Madhava-Gregory series, he gave a value of the number π as 3.14159265359 - correct to 11 decimals, the most accurate approximation of π after almost a thousand years.

Madhava was also responsible for a number of other original discoveries, including:

• Taylor series of the sine and cosine functions (Madhava-Newton power series).

• Taylor series of the tangent function.

• Taylor series of the arctangent function (Madhava-Gregory series).

• Second-order Taylor series approximations of the sine and cosine functions.

• Third-order Taylor series approximation of the sine function.

• Power series of π (usually attributed to Leibniz).

• Power series of π/4 (Euler's series).

• Power series for any angle θ (equivalent to the Gregory series).

• Power series for the radius of a circle.

• Power series for the diameter of a circle.

• Power series for the circumference of a circle.

• Many methods for calculating the circumference of a circle.

• Infinite continued fractions.

• Term by term integration.

• Laying the foundations for the development of calculus, which was then further developed by his successors at the Kerala School.

Kerala School of Astronomy and Mathematics

Main article: Kerala School

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.

See also

• Indian mathematics
• List of Indian mathematicians
• Kerala School
• Mathematical analysis
• Calculus
• History of Calculus
• Parameswara
• Neelakanta Somayaji
• Jyeshtadeva
• Achyuta Pisharati
• Melpathur Narayana Bhattathiri
• Achyuta Panikkar.

External links

• Madhava's mathematical contributions by J J O'Connor and E F Robertson
• Neither Newton nor Leibnitz - The Pre-History of Calculus and Celestial Mechanics in Medieval Kerala
• Madhava's Contributions by Ian Pearce