CONTRIBUTIONS OF KELALLUR NILAKANTHA SOMAYAJI

    CONTRIBUTIONS OF KELALLUR     NILAKANTHA SOMAYAJI                   

      Sona Joseph , B.Ed Student 

                                                           John Paul Memorial B.Ed College, 

                                                                      Labbakkada,Idukki 

                                                                sonajoseph814@gmail.com      

                                     

Abstract

      The worlds says that the Indian mathematics darkness has disappeared after the arrival of famous Indian mathematicians ,but there are many mathematicians from kerala who represent the golden age of Indian mathematics.Madhava,Parameswaran,Neelakanta Somayaji are the famous kerala mathematicians which contributes a lot to the Indian mathematics and astronomy.But kerala mathematics has not received the support or attention it deserves.

       The kerala school of astronomy and mathematics or the kerala school was a school of mathematics and astronomy founded by Madhava of sangamagrama in Tirur, Malappuram, Kerala, India .Kelallur Nilakantha Somayaji is one of the famous mathematician and astronomer in Kerala.His most important contribution is the series expansion for trigonometric functions were described in Sanskrit verse in a book by Nilakanta called Tantrasangraha.In this work ,we describes the major contributions of Kelallur Nilakantha Somayaji .

Keywords:Kelallur Nilakantha Somayaji,Tantrasamgraha

Introduction

   Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501. He had also composed an elaborate commentary on Aryabhatiya called the Aryabhatiya Bhasya. In this Bhasya, Nilakantha had discussed infinite series expansions of trigonometric functions and problems of algebra and spherical geometry. Grahapariksakrama is a manual on making observations in astronomy based on instruments of the time. Known popularly as Kelallur Chomaathiri, he is considered an equal to Vatasseri Parameshwaran Nambudiri.

Objectives

• To discuss the biography of Kelallur Nilakantha Somayaji

• The main works of Kelallur Nilakantha Somayaji

• The detailed description of the text Tantrasamagra

Biography

Nilakantha was born into a Namputiri Brahmin family which came from South Malabar in Kerala. He was born on Wednesday, June 14, 1444, and was a resident of Trkkantiyur

(Sanskritised into Sri -Kundapura ), near Tirur, Ponnai taluk, South Malabar. The Nambudiri is main caste of Kerala. It is an orthodox caste whose members consider themselves descendants of the ancient Vedic religion. .

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.His teachers were Ravi with whom he studied Vedanta, and Damodara, son of Paramesvara, who initiated him into Astronomy and the underlying mathematical principles. That

Nilakantha lived upto a ripe old age, even to become a centenarian, is attested by a contemporary reference made to him in a Malayalam work on astrology Prasnasara composed in 1542-43.

Works

• Golasara : Description of basic astronomical elements and procedures

• Sidhhantadarpana : A short work in 32 slokas enunciating the astronomical constants with reference to the Kalpa and specifying his views on astronomical concepts and topics.

• Candrachayaganita : A work in 32 verses on the methods for the calculation of time from the measurement of the shadow of the gnomon cast by the moon and vice versa.

• Aryabhatiya-bhashya : Elaborate commentary on Aryabhatiya.

• Sidhhantadarpana-vyakhya : Commentary on his own Siddhantadarapana.

• Chandrachhayaganita-vyakhya : Commentary on his own Chandrachhayaganita.

• Sundaraja-prasnottara : Nilakantha's answers to questions posed by Sundaraja, a Tamil Nadu-based astronomer.

• Grahanadi-grantha : Rationale of the necessity of correcting old astronomical constants by observations.

• Grahapariksakrama : Description of the principles and methods for verifying astronomical computations by regular observations.

• Jyotirmimamsa: Analysis of astronomy

In his most notable work Tantra Samgraha (which 'spawned' a later anonymous commentary Tantrasangraha-vyakhya and a further commentary by the name Yuktidipaika, written in 1501) he elaborates and extends the contributions of Madhava. Sadly none of his mathematical works are extant, however it can be determined that he was a mathematician of some note. Nilakantha was also the author of Aryabhatiya-bhasa a commentary of the Aryabhatiya. of great significance is the presence of mathematical proof (inductive) in Nilakantha's work.

Furthermore, his demonstration of particular cases of the series

tan¯¹ t = t -t³/3 + t/5 -... ,

When t = 1 and t = 1/3, and remarkably good rational approximations of pi(using another Madhava series) are of great interest. Various results regarding infinite geometrically progressing convergent series are also attributed to Nilakantha.

Tantrasamgraha

Tantrasangraha, or Tantrasangraha, (literally, A Compilation of the System) is an important astronomical treatise written by Nilakantha Somayaji an astronomer/mathematician belonging to the Kerala school of astronomy and mathematics. The treatise was completed in 1501 CE. It consists of 432 verses in Sanskrit divided into eight chapters. Tantrasamgraha had spawned a few commentaries: Tantrasamgraha-vyakhya of anonymous authorship and Yuktibhāṣā authored by Jyeshtadeva in about 1550 CE. Tantrasangraha, together with its commentaries, bring forth the depths of the mathematical accomplishments the Kerala school of astronomy and mathematics, in particular the achievements of the remarkable mathematician of the school Sangamagrama Madhava. In his Tantrasangraha, Nilakantha revised Aryabhata's model for the planets Mercury and Venus. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century.

It was C.M. Whish, a civil servant of East India Company, who brought to the attention of the western scholarship the existence of Tantrasamgraha through a paper published in 1835. The other books mentioned by C.M. Whish in his paper were Yukthibhasha of Jyeshtadeva, Karanapaddhati of Puthumana Somayaji and Sadratnamala of Sankara Varman.

Nilakantha Somayaji, the author of Tantrasamgraha, was a Nambudiri belonging to the Gargya gotra and a resident of Trikkantiyur, near Tirur in central Kerala. The name of his Illamwas Kelallur. He studied under Damodara, son of Paramesvara. The first and the last verses in Tantrasamgraha contain chronograms specifying the dates, in the form Kali days, of the commencement and of the completion of book.

Synopsis of the book

A brief account of the contents of Tantrasamgraha is presented below. A descriptive account of the contents is available in Bharatheeya Vijnana/Sastra Dhara. Full details of the contents are available in an edition of Tantrasamgraha published in the Indian Journal of History of Science.

Chapter 1 (Madhyama-prakaranam): The purpose of the astronomical computation, civil and sidereal day measurements, lunar month, solar month, intercalary month, revolutions of the planets, theory of intercalation, planetary revolution in circular orbits, computation of kali days, mathematical operations like addition, subtraction, multiplication, division, squaring and determining square root, fractions, positive and negative numbers, computation of mean planets, correction for longitude, longitudinal time, positions of the planets at the beginning of Kali era, planetary apogees in degrees. (40 slokas)

Chapter 2 (Sphuta-prakaranam (On true planets)): Computation of risings, and arcs, construction of a circle of diameter equal to the side of a given square, computation of the circumference without the use of square and roots, sum of series, sum of the series of natural numbers, of squares of numbers, of cubes of numbers, processes relating to Rsines and arcs, computation of the arc of a given Rsine, computation of the circumference of a circle, derivation of Rsines for given Rversed sine and arc, computation of Rsine and arcs, accurate computation of the 24 ordained Rsines, sectional Rsines and Rsine differences, sum of Rsine differences, summation of Rsine differences computation of the arc of an Rsine according to Madhava, computation of Rsine and Reversed sine at desired point without the aid of the ordained Rsines, rules relating to triangles, rules relating to cyclic quadrilaterals, rules relating to the hypotenuse of a quadrilateral, computation of the diameter from the area of the cyclic quadrilateral, surface area of a sphere, computation of the desired Rsine, the accessional difference, sun's daily motion in minutes of arc, application of accessional difference to true planets, measure of day and night on applying accessional difference, conversion of the arc of Rsine of the accessional difference, etc. (59 slokas)

Chapter 3 (Chhaya-prakaranam (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. (116 slokas)

Chapter 4 (Chandragrahana-prakaranam (Treatise on the lunar eclipse)): Diameter of the Earth's shadow in minutes, Moon's latitude and Moon's rate of motion, probability of an eclipse, total eclipse and rationale of the explanation given for total eclipse, half duration and first and last contacts, points of contacts and points of release in eclipse, and their method of calculation, visibility of the contact in the eclipse at sunrise and sunset, contingency of the invisibility of an eclipse, possibility of the deflection, deflection due to latitude and that due to declination. (53 slokas)

Chapter 5 (Ravigrahana-prakaranam (Treatise on the solar eclipse)): Possibility of a solar eclipse, minutes of parallax in latitude of the sun, minutes of parallax in latitude of the moon,. maximum measure of the eclipse, middle of the eclipse, time of first contact and last contact, half duration and times of submergence and emergence, reduction to observation of computed eclipse, mid eclipse, known prediction of an eclipse. (63 slokas)

Chapter 6 (Vyatipata-prakaranam (On vyatipata)): Deals with the complete deviation of the longitudes of the sun and the moon. (24 slokas)

Chapter 7 (Drikkarma-prakaranam(On visibility computation)): Discusses the rising and setting of the moon and planets. (15 slokas)

Chapter 8 (Sringonnati-prakaranam (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. (40 slokas)

Conclusion

It is clear that Nilakantaha somayaji from kerala have contributed much to the growth of mathematics. Unfortunately, contributions from Kerala haven‘t got much care and concern. Even the Indian Mathematics and Astronomy also concentrated on Bhaskara , Aryabatta etc. But the kerala mathematicians such as Madhava,parameshawara,Jyesthadevaa etc are the mathematicians

from keral a that contributes a lot to the mathematics and astronomy.From this work we understand the biography,major works ,Contributions and his major book tantrasamagra of nilakantha Somayaji.

References

1 .Robortson E F(2015) ,Nilakantha Somayaji :Astronomer /Mathematician of Kerala School of

Astronomy Retrived from https://www-sanskritimagazine com.cdn.ampproject.org/v/s/www.sanskritmagazine.com/vedic_science/nilakantha-somayaji/? on 3 /02/2021

2 .Ramasubramanian K and Sriram S (2011),Tantrasangraha of Nilakantha Somayaji,Hindustan Book Agency 2011 New Delhi.

3 .Wikipedia