Madhava of Sangamagrama

Indian mathematician and astronomer (c.1340 c.1425)

infobox person
| native name =
| native name lang = ml
| name = Madhava of Sangamagrama
| image =
| alt =
| caption =
| birth name =
| birth date = c. 1340 (or c. no – c. 1350 (c. 1340 – c. 1425) was an Indian mathematician and astronomer who is considered as the founder of the Kerala school of astronomy and mathematics in the Late 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".

Little is known about Mādhava's life with certainty. However, from scattered references to Mādhava 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 south 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. 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.

There are several known astronomers who preceded Madhava, including Kǖṭalur Kizhār (2nd century), Vararuci (4th century), and Śaṅkaranārāyaṇa (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. Achyutha Pisharadi of Trikkantiyur is mentioned as a disciple of Jyeṣṭhadeva, and the grammarian Melpathur Narayana Bhattathiri as his disciple. 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.

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! +

Madhava developed the power series expansion for some trigonometry functions 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.

The Kerala school of astronomy and mathematics was founded by Madhava of Sangamagrama in Kerala, South India and included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th centuries. They gave three important results, series expansion of three trigonometry functions of sine, cosine and arctant and the proof of their results where later given in Yuktibhasa text. 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.