Madhava of Sangamagrama

Mādhava of Sangamagrāma (Mādhavan) was an Indian mathematician and astronomer (jyotirvid) who founded the Kerala school of astronomy and mathematics. Following the traditional Indian approach of Gaṇita-Jyotiṣa (mathematical astronomy), Madhava made pioneering contributions to the study of infinite series, trigonometry, Spherical geometry, algebra, and analysis. 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".

Biography Little is known about Madhava's life with certainty. However, from scattered references to Madhava found in diverse manuscripts, historians of Kerala school have pieced together information about the mathematician. In a manuscript preserved in the Oriental Institute, Baroda, Madhava has been referred to as Mādhavan vēṇvārōhādīnām karttā ... Mādhavan Ilaññippaḷḷi Emprān. This is clearly Sanskrit for Ilaññippaḷḷi. Ilaññi is the Malayalam name of the evergreen tree Mimusops elengi and the Sanskrit name for the same is Bakuḷa. Palli is a term for village. The Sanskrit house name bakuḷādhiṣṭhita . . . vihāra has also been interpreted as a reference to the Malayalam house name Iraññi ninna ppaḷḷi and some historians have tried to identify it with one of two currently existing houses with names Iriññanavaḷḷi and Iriññārapaḷḷi both of which are located near Irinjalakuda town in central Kerala.

Most of the writers of astronomical and mathematical works who lived after Madhava's period have referred to Madhava as "Sangamagrama Madhava" and as such it is important that the real import of the word "Sangamagrama" be made clear. The general view among many scholars is that Sangamagrama is the town of Irinjalakuda some 70 kilometers south of the Nila river and about 70 kilometers north of Cochin. Thus the most plausible possibility is that the forefathers of Madhava migrated from the Tulu land or thereabouts to settle in Kudallur village, which is situated on the southern banks of the Nila river not far from Tirunnavaya, a generation or two before his birth and lived in a house known as Ilaññippaḷḷi whose present identity is unknown.

Date There are also no definite evidence to pinpoint the period during which Madhava flourished. In his Venvaroha, Madhava gives a date in 1400 CE as the epoch. Madhava's pupil Parameshvara Nambudiri, the only known direct pupil of Madhava, is known to have completed his seminal work Drigganita in 1430 and the Paramesvara's date has been determined as -1455. From such circumstantial evidences historians have assigned the date to Madhava.

Historiography Mathematical astronomy has been practised over a long time in Kerala through Aryabhatiya tradition and the Parahita system introduced earlier by Haridatta in the 7th century CE. However, the references from later scholars suggest that Madhava provided the creative impulse for advanced analysis based rich mathematical tradition in Kerala school. 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 . 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+x2), 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 Variar, 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, 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.

Scholarly lineage (Guru-śiṣya paramparā)

The educational system in Kerala ran through Guru-śiṣya paramparā. Earlier astronomers included Kǖṭalur Kizhār (2nd century CE), Vararuci (who initiated the Kaṭapayādi systems, 4th century CE) and Śaṅkaranārāyaṇa (author of the Laghubhāskarīyavivaraṇa, 869 CE). After Madhava, the succession was carried out by several direct and indirect disciples for multiple generations. 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. Achyutha Pisharadi of Trikkantiyur is mentioned as a disciple of Jyeṣṭhadeva, and the grammarian Melpathur Narayana Bhattathiri as his disciple.. In Europe, the first such series was developed by James Gregory in 1667. This yields:

or equivalently:

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. Madhava's values are accurate to the seventh decimal place. 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! + ...

The value of (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

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 Rn for the error after computing the sum up to n 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 3n/2 correct digits:

The absolute value of the correction term in next higher order is

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

By using the first 21 terms to compute an approximation of , 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 given since the 5th century (see History of numerical approximations of ).

The text Sadratnamala appears to give the astonishingly accurate value of = 3.14159265358979324 (correct to 17 decimal places). Based on this, R. Gupta has suggested that this text was also composed by Madhava. (Certain ideas of calculus were known to earlier mathematicians.) Madhava also extended some results found in earlier works, including those of Bhāskara II.

Lagnaprakarana Lagnaprakarana is an astronomical treatise that deals with the calculation of the ascendant (lagna) and the planetary positions. Palm-leaf manuscripts of the work have been catalogued and preserved at research archives in Kerala. This includes the Manuscript Research & Preservation Section (MRPC) of Saint Joseph College, Irijalakuda, where the palm-leaf scripture of Lagnaprakarana was recently digitalised confirming to British Standards, under guidance of Professor Litty Chacko. Preservation of the original palm-leaf script has also been ensured by dusting and application of eucalyptus oil to prevent further damage.

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 Sphuṭacandrāpti Aganita-grahacara Chandravakyani (Table of Moon-mnemonics)

Kerala School of Astronomy and Mathematics

The Kerala school of astronomy and mathematics, founded by Madhava, flourished between the 14th and 16th centuries, and included among its members Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The group is known for series expansion of three trigonometric functions of sine, cosine and arctan and proofs of their results where later given in the Yuktibhasa. The group also did much other work in astronomy: more pages are devoted to astronomical computations than purely mathematical results. O'Connor and Robertson state that a fair assessment of Madhava is that he took the decisive step towards modern classical analysis. 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. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place.