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Thursday, April 12, 2018

Nila School of Mathematics

One has heard of the Kerala School of Mathematics and its lone stellar performer, Madhava of Sangramagrama. He and his disciples and grand disciples( if one may say so, to refer to the lineage of disciples) , Parameswara, Neelakantha Somayaji, Jyeshthadeva, Sankara Varier, Achyutha Pisharoti and others worked on trigonometry, calculus and geometry centuries before these branches were known to the western world.

But, Nila School? This was the topic of a stimulating lecture by Dr. N K Sundareswaran, Professor, Department of Sanskrit, Calicut University. Hailing from Palakkad, he was initiated into Sanskrit and Yajurveda by his father in the traditional manner. Later he graduated in Mathematics and obtained his Master’s and Doctorate in Sanskrit from Calicut University. He has been teaching and guiding research for more than a quarter of a century. His doctoral thesis is on Nilakantha Somayaji’s contributions to astronomy.
He took us through the main dramatis personae of the Nila School, which in effect, comprised all the above worthies, excluding Madhava of Sangramagramam. How did this happen? A lone genius from Irinjalakkuda and all his disciples and successors from a remote group of villages more than 50 kms. north of the place?
 We know about Madhava (c.1340-c1425) mostly from the work of later mathematicians. Traditionally, he was believed to have been an Embranthiri ( a Tuluva Brahmin) from Aloor, near Irinjalakkuda in Trissur district. Most earlier writers had given a convoluted interpretation of the name ‘Sangramagramam’ to mean Irinjalakkuda/Koodalmanikyam.
 But, Prof. Sundareswaran, following Prof. P P Divakaran (formerly of Tata Institute of Fundamental Research) would prefer to interpret Sangamagramam to mean Kootallur, a village close to Ponnani. The literal translation of Sangamagramam is Kootallur. Also, the village and more particularly, the Namboodiri Illam ( Kutalloor Mana) has a long tradition of learning even to the present century. It is situated at the confluence of the rivers Nila and Tootha, which suggest the name Kootallur, a village at the confluence.
If it is accepted that Madhava of Sangamagrama belonged to Kootallur, then everything falls into place, as all the other mathematicians were from neighbouring villages such as Tripparangodu, Trikkandiyur, Alathiyur etc. (Incidentally, Prof. PPD suggests that Tirunavaya also used to be and is sometimes still referred to a Trimurtisangamam on account of the presence, on either bank of the river, of temples dedicated to the Hindu trinity.)
Madhava’s chief contribution was the discovery of the infinite series for the trigonometric functions of sine, cosine, tangent and arctangent. The same series were developed in Europe for the first time by James Gregory in 1667, more than two hundred years after Madhava. The world of mathematics has acknowledged this and the series is now known as Madhava-Gregory-Lleibniz series.( Leibniz only re-obtained the formula for pi which had already been obtained by Madhava.)
Vatasseri Parameswara ( c.1380-1460) hailed from Alathiyur, near Tirur and was the direct disciple of Madhava. He was an astrologer as well and suggested improvements to the findings of Aryabhata, Bhaskara, Govindasvami and others through his copious commentaries. He was also known as the father of the Drig ganita system which is a system of astronomical computations considered more accurate that the then existing Parahita system. His contribution was in proving theories through observation. Damodara was the son and pupil of Parameswara and was also a notable mathematician and astronomer.
Nilakantha Somayaji ( 1444-1520?) was Damodara’s disciple. Born in Kelalloor mana in Trikkandiyur, Nilakantha was closely associated with the Sree Rama Temple in Alathiyur, (which is more famous for its upa devata, Hanuman.
Nilakantha’s major work was the Tantrasangraha written in 1500 which was principally an astronomical treatise. His theories are, however, without proofs or explanations of the logic. For this, we have to turn to his disciple, Jyesthadeva.
Jyesthadeva (c1500-c1575), the disciple of both Damodara and Nilakantha was the next great mathematician. He was also from Tripparangodu, another nearby temple-village. His biggest contribution was the Yuktibhasha, written in Malayalam. It offers detailed analytical commentary on Nilakantha’s Tantrasangraha. His contribution was in providing the process and the derivations for arriving at many of his guru’s theorems.
Sankara Varier, who was a contemporary of Jyeshtadeva, was the sole non-Brahmin mathematician whose works included Yuktideepika and Kriyakarmakari.

The question came up – did this knowledge travel from Kerala to the West, or were the scholarships parallel and unconnected?. Calicut Heritage Forum had the privilege of hosting,  a few years ago, Dr. George Gheverghese Joseph of Manchester University who was posed the same question. His reply was equivocal, to put it rather bluntly. But his own writing about there being plenty of opportunities for the Jesuits, who swarmed the area in the sixteenth century, to carry the new knowledge to the West, betrays his views. According to him, there was a strong motivation for this transmission: Pope Gregory XIII had set up a committee to look into modernising the Julian calendar. The German Jesuit, Clavius was on this committee and he had been repeatedly requesting his brethren spread over the world for information on how people constructed calendars in other parts of the world. The Jesuits who were in numbers in Vettathu kingdom ( they had even managed to convert the Vettath King to Christianity, and Antonio Gomez who replaced Francis Xavier in India was a frequent visitor to Tanur) could hardly have missed the opportunity to pick up the astronomically accurate calculations made popular by the Nila school of mathematicians.
Prof. Sundareswaran, however, preferred not to comment on this. Instead, he focused on how knowledge was being transmitted from generation to generation in an unbroken chain of succession. His concluding statement - that only around 7 per cent of this fund of knowledge had been deciphered, the rest waiting in numerous cadjan leaf manuscripts to be unravelled - reminded one of the poet A K Ramanujam's perceptive observation : Even one's own tradition is not one's birthright; it has to be earned, repossessed.


  1. thanks CK
    wonderful stuff..
    If you recall , i asked you a question once about why these mathematicians studied eclipses, and I then went on to study it - This is quite relevant to the Nila school as well, take a look at the first of my outputs.


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