# KELALLUR NILAKANTHA SOMAYAJIPADU

### Nilakantha was born in a house called Kelallur which it is claimed coincides with the present Edamana in the village of **Trikkantiyur** (Thrikkandiyur) near Tirur in south India. His father was Jatavedas and the family belonged to the Gargya gotra, which was a Indian caste that prohibits marriage to anyone outside the caste. The family followed the Ashvalayana sutra which was a manual of sacrificial ceremonies in the Rigveda, a collection of Vedic hymns. He worshipped the personified deity Soma who was the “master of plants” and the healer of disease. This explains the name Somayaji which means he was from a family qualified to conduct the Soma ritual.

### Nilakantha studied astronomy and Vedanta, one of the six orthodox systems of Indian Hindu philosophy, under the teacher Ravi. He was also taught by Damodra who was the son of Paramesvara. Paramesvara was a famous Indian astronomer and Damodra followed his father’s teachings. This led Nilakantha also to become a follower of Paramesvara. A number of texts on mathematical astronomy written by Nilakantha have survived. In all he wrote about ten treatises on astronomy.

### The Tantrasamgraha is his major astronomy treatise written in 1501. It consists of 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion. The first two chapters deal with the motions and longitudes of the planets. The third chapter Treatise on shadow deals with various problems related with the sun’s position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates.

### The fourth and fifth chapters are Treatise on the lunar eclipse and On the solar eclipse and these two chapters treat various aspects of the eclipses of the sun and the moon. The sixth chapter is On vyatipata and deals with the complete deviation of the longitudes of the sun and the moon. The seventh chapter On visibility computation discusses the rising and setting of the moon and planets. The final chapter On elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.

### The Tantrasamgraha is very important in terms of the mathematics Nilakantha uses. In particular he uses results discovered by Madhava and it is an important source of the remarkable mathematical results which he discovered. However, Nilakantha does not just use Madhava’s results, he extends them and improves them. An anonymous commentary entitled Tantrasangraha-vakhya appeared and, somewhat later in about 1550, Jyesthadeva published a commentary entitled Yuktibhasa that contained proofs of the earlier results by Madhava and Nilakantha. This is quite unusual for an Indian text in giving mathematical proofs.

The series Ï€/4 = 1 – 1/3 + 1/5 – 1/7 + … is a special case of the series representation for arctan, namely

The series Ï€/4 = 1 – 1/3 + 1/5 – 1/7 + … is a special case of the series representation for arctan, namely