Mādhava Of Saṅgamagrāma

Iriññāttappiḷḷi Mādhavan known as Mādhava of Sangamagrāma () was an Indian mathematician and astronomer from the town believed to be present-day Kallettumkara, Aloor Panchayath, Irinjalakuda in Thrissur District, Kerala, India. He is considered the founder of the Kerala school of astronomy and mathematics. One of the greatest mathematician-astronomers of the Middle Ages, Madhava made pioneering contributions to the study of infinite series, calculus, trigonometry, geometry, and algebra. He was the first to use 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".

Historiography

Madhavan was born in an embranthiri brahmin family of tulu origin on 1340 in kingdom of Cochin. 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, except for a couple, most of Madhava's original works have been 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 sin ''θ'' and arctan ''θ''. The 16th-century text ''Mahajyānayana prakāra'' (Method of Computing Great Sines) cites Madhava as the source for several series derivations for π. In Jyeṣṭhadeva's ''Yuktibhāṣā'' (c. 1530),

written in Malayalam, these series are presented with proofs in terms of the Taylor series expansions for polynomials like 1/(1+''x''²), with ''x'' = tan ''θ'', etc.

Thus, what is explicitly Madhava's work is a source of some debate. The ''Yukti-dipika'' (also called the ''Tantrasangraha-vyakhya''), possibly composed by Sankara Variyar, a student of Jyeṣṭhadeva, presents several versions of the series expansions for sin ''θ'', cos ''θ'', and arctan ''θ'', as well as some products with radius and arclength, most versions of which appear in Yuktibhāṣā. 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, some of the other forms might also be the work of Madhava.

Others have speculated that the early text ''Karanapaddhati'' (c. 1375–1475), or the ''Mahajyānayana prakāra'' was written by Madhava, but this is unlikely.

''Karanapaddhati'', along with the even earlier Keralite mathematics text ''Sadratnamala'', as well as the ''Tantrasangraha'' and ''Yuktibhāṣā'', were considered in an 1834 article by Charles Matthew 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

There are several known astronomers who preceded Madhava, including Kǖţalur Kizhār (2nd century), Vararuci (4th century), and Sankaranarayana (866 AD). It is possible that other unknown figures preceded him. However, we have a clearer record of the tradition after Madhava. Parameshvara was a direct disciple. According to a palm leaf manuscript of a Malayalam commentary on the Surya Siddhanta, Parameswara's son Damodara (c. 1400–1500) had Nilakantha Somayaji as one of his disciples. Jyeshtadeva was a disciple of Nilakantha. Achyuta Pisharati of
Trikkantiyur is mentioned as a disciple of Jyeṣṭhadeva, 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).Madhava extended Archimedes' work on the geometric Method of Exhaustion to measure areas and numbers such as π, with arbitrary accuracy and error ''limits'', to an algebraic infinite series with a completely separate error ''term''.
This implies that he understood very well the limit nature of the infinite series. 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 is 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 infinite series for the trigonometric functions of sine, cosine, arctangent, and many methods for calculating the circumference of a circle. One of Madhava's series is known from the text ''Yuktibhāṣā'', which contains the derivation and proof of the power series for inverse tangent, discovered by Madhava.
In the text, Jyeṣṭhadeva describes the series in the following manner:

This yields:
:$r\theta=-(1/3)\,r\,+(1/5)\,r\,-(1/7)\,r\, + \cdots$
or equivalently:
:$\theta = \tan \theta - \frac + \frac - \frac + \cdots$

This series is Gregory's series (named after James Gregory, who rediscovered it three centuries after Madhava). Even if we consider this particular series as the work of Jyeṣṭhadeva, 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-Leibniz series.

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'' − ''q''³/3! + ''q''⁵/5! − ''q''⁷/7! + ...
: cos ''q'' = 1 − ''q''²/2! + ''q''⁴/4! − ''q''⁶/6! + ...

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:

: $\frac = 1 - \frac + \frac - \frac + \cdots = \sum_^\infty \frac,$

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, 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. 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 3''n''/2 correct digits:

: $\frac \approx 1 - \frac + \frac - \frac + \cdots + \frac + \cfrac.$

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).

He also gave a more rapidly converging series by transforming the original infinite series of π, obtaining the infinite series

: $\pi = \sqrt\left(1 - \frac + \frac - \frac + \cdots\right).$

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'' appears to give the astonishingly accurate value of π = 3.14159265358979324 (correct to 17 decimal places). Based on this, R. Gupta has suggested that this text was also composed by Madhava.

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.Ian G. Pearce (2002)
Madhava of Sangamagramma
''MacTutor History of Mathematics archive''. University of St Andrews.
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.

(Certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those of Bhāskara II.. However, they did not 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 powerful problem-solving tool we have today.

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'' ()
# ''Venvaroha'' ()
# '' Sphutacandrapti'' ()
# ''Aganita-grahacara'' ()
# '' Chandravakyani'' () (Table of Moon-mnemonics)

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.

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.

Possible propagation to Europe

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.

See also

*Madhava's sine table
*Madhava series
*Venvaroha
* Ganita-yukti-bhasa
*Kerala school of astronomy and mathematics
*List of Indian mathematicians
*Indian mathematics
*History of calculus

References

External links

Biography on MacTutor

{{DEFAULTSORT:Madhava of Sangamagrama
1340s births
1420s deaths
Scientists from Kerala
History of calculus
Indian Hindus
Kerala school of astronomy and mathematics
14th-century Indian mathematicians
15th-century Indian mathematicians
People from Irinjalakuda
15th-century Indian astronomers
14th-century Indian astronomers
Scholars from Kerala
Mathematical series