Madhava of Sangamagrama

Mādhava of Sangamagrama
Born	1350 Kerala, India
Died	1425

Mādhava of Sangamagrama (Malayalam: മാധവന്‍, Mādhavan) (born as Irinjaatappilly Madhavan Namboodiri) (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 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". His discoveries opened the doors to what has today come to be known as mathematical analysis.. 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.

Historiography

Although there is some evidence of Mathematical work in Kerala prior to Madhava (e.g. Sadratnamala c.1300, a set of fragmentary results), 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) 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), 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, 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.

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

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.

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). 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.

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 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 contains the derivation and proof of the power series for inverse tangent, discovered by Madhava. In the text, Jyesthadeva 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.

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 result:

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

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 referred to as the Madhava-Gregory series.

Trigonometry

Madhava 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:

sin q = q - q³/3! + q⁵/5! - ...

cos q = 1 - q²/2! + q⁴/4! - ...

The value of π (pi)

We find Madhava's work on the value of π cited in the Mahajyānayana prakāra ("Methods for the great sines"). 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:

${\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots +{\frac {(-1)^{n}}{2n+1}}+\cdots$

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

R_n = 1/(4n), or

R_n = n/ (4n² + 1), or

R_n = (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. 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

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

By using the first 21 terms to compute an approximation of π, he obtains a value correct to 11 decimal places (3.14159265359). The value of 3.1415926535898, correct to 13 decimals, is sometimes attributed to Madhava, 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.

Algebra

Madhava 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. He also discovered the solutions of transcendental equations by iteration, and found the approximation of transcendental numbers by continued fractions.

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. (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

Main article: 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,

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.

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.

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

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 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. 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).

See also

• Indian mathematics
• List of Indian mathematicians
• Kerala school of astronomy and mathematics
• History of calculus