|Born: 14 June 1444 in Trkkantiyur
(near Tirur), Kerala, India
Died: 1544 in India
Nilakantha was born into a Namputiri Brahmin family which came from
South Malabar in Kerala. The Nambudiri is the main caste of Kerala.
It is an orthodox caste whose members consider themselves descendants
of the ancient Vedic religion.
He was born in a house called
Kelallur which it is claimed coincides with the present Etamana in
the village of Trkkantiyur 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
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
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.
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
The series p/4 = 1 - 1/3 + 1/5 - 1/7 + ... is a
special case of the series representation for arctan, namely
tan-1x = x - x3/3 + x5/5 - x7/7 + ...
It is well known that
one simply puts x = 1 to obtain the series for p/4. The author of 
reports on the appearance of these series in the work of Leibniz and
James Gregory from the 1670s. The contributions of the two European
mathematicians to this series are well known but in  the results
on this series in the work of Madhava nearly three hundred years
earlier as presented by Nilakantha in the Tantrasamgraha is also
Nilakantha derived the series expansion
tan-1x = x - x3/3 + x5/5 - x7/7 + ...
by obtaining an
approximate expression for an arc of the circumference of a circle
and then considering the limit. An interesting feature of his work
was his introduction of several additional series for p/4 that
converged more rapidly than
p/4 = 1 - 1/3 + 1/5 - 1/7+ ... .
The author of  provides a reconstruction of how he may have
arrived at these results based on the assumption that he possessed a
certain continued fraction representation for the tail series
1/(n+2) - 1/(n+4) + 1/(n+6) - 1/(n+8) + .... .
Tantrasamgraha is not the only work of Nilakantha of which we have
the text. He also wrote Golasara which is written in fifty-six
Sanskrit verses and shows how mathematical computations are used to
calculate astronomical data. The Siddhanta Darpana is written in
thirty-two Sanskrit verses and describes a planetary model. The
Candracchayaganita is written in thirty-one Sanskrit verses and
explains the computational methods used to calculate the moon's
The head of the Nambudiri caste in
Nilakantha's time was Netranarayana and he became Nilakantha's patron
for another of his major works, namely the Aryabhatiyabhasya which is
a commentary on the Aryabhatiya of Aryabhata I. In this work
Nilakantha refers to two eclipses which he observed, the first on 6
March 1467 and the second on 28 July 1501 at Anantaksetra. Nilakantha
also refers in the Aryabhatiyabhasya to other works which he wrote
such as the Grahanirnaya on eclipses which have not survived.
Article by: J J O'Connor and E F Robertson