One of the most famous schools of mathematics and astronomy in India was the Kerala school of maths and astronomy, founded by Madhava of Sangamagrama during the 14th century, who made pioneering studies in infinite series, calculus, trignometry, geometry and algebra. He was born around 1350, near to Aloor,a panchayat in Kerala’s Thrissur district. He was the first to use infinite series approximations for a range of trigonometric functions, which has often been regarded as a significant step to move from the finite step of ancient mathematics towards an inifinite limit.
Most of the mathematical discoveries of the Kerala school, came from an effort to solve astronomy problems. Their most important results, related to series expansion for trignometry was recorded in a book called Tantrasangraha. They also provided what is now considered the first example of a power series,especially for several infinite series expansions, including sinθ and arctanθ, two centuries before Europe invented calculus.
Madhava actually had a legacy to fall back upon. Much before him, there was Kǖţalur Kizhār, Vararuci who is called the father of astronomy in Kerala, and Śaṅkaranārāyaṇa who established an observatory at Kodangallur. The reason why he is often considered the founder of Kerala school was his extensive documentation of the work he did. One of his disciples Vatasseri Parameshvara Nambudiri was the one who came up with observational astronomy as a discipline.
Parameshvara’s son Damodara was the tutor for Nilkanth Somayaji, while Nilakantha’s disciple was Jyeshtadeva, who wrote the commentary on Tantrasangraha. His disciple Achyutha Pisharodi, discovered the techniques for reduction of the ecliptic. While Melpathur Narayana Bhattathiri was the disciple for Achyutha, very well known for his Narayaneeyam, a devotional composition in praise of Guruvayoor Krishna, that is sung even now at the temple there.
One of Madhava’s most important contribution to mathematics has been the discovery of infinite series for sine, cosine, tangent and arctangent, as well as the proof of power series for inverse tangent described in Yuktibhasha. The series was rediscovered by James Gregory 3 centuries later. Initially known as the Gregory series, it was later named as the Madhava-Gregory-Leibintz series.
He also composed an accurate table of sines, using a quarter circle at 24 equal intervals, his contribution to trignometry.
sin q = q – q3/3! + q5/5! – q7/7! +…
cos q = 1 – q2/2! + q4/4! – q6/6! +…
He also worked on the value of Pi, cited in the Mahajyānayana prakāra. The infinite series expansion of Pi is now called Madhava-Leibintz series.
Some of the other famous works of Madhava were Venvaroha- Describing the computation of the positions of the Moon Chandravakyas- Collection of numbers, related to the Moon and it’s orbit around the Earth.
One of the foremost mathematicians and astronomers of this school was Nilakantha Somajayi, known for his astronomical treatise Tantrasamgraha and a commentary on Aryabhatta. He was born in a small village near Tirur in Mallapuram district on June 14, 1444. Though he personally claimed e was born on a Kali Day, 1,660,181, a reference to the Kali Yuga, as per his account his father was Jatavedasa and a brother named Sankara. Again as per his own account Nilakantha Somayaji claimed he was a Bhatta belonging to the Gargya Gotra, and a follower of Asvalayana Sutra.
As per his own Laghuramayana, he belonged to the Kellalur family, his wife’s name was Arya, and he had two sons Rama and Dakshinamurthy. While he had learnt the Vedanta under Ravi, it was Kottesora Damodara who initiated Neelakantha into astronomy and also basic principles of Maths computations.Thunchaththu Ezhuthachan, the great Malayalam poet was believed to be his disciple. Nilalkantha got the title Somayaji, as he was assumed to have the performed the Somayajna ritual as per Namboodiri tradition.
He was a true polymath, with mastery over philosophy, grammar, astronomy, math.It’s believed that he could refer to a Mimamsa authority during a debate and also apply a grammatical dictum to same purpose. Sundaraja a contemporary Tamil astronomer called him sad-darshani-parangata, one who mastered 6 Systems of Indian philosophy.
His greatest work to date is Tantrasangraha, completed in 1501, which has around 8 chapters, written in Sanskrit. In this work he revised Aryabhatta’s model for the center of Mercury and Venus, and it remained the most accurate till Kepler came up with his own laws. His seminal work contains the following chapters
Madhyama-prakaranam- Astronomical computation
Sphuta-prakaranam-Primarily on planets, RSine, sum of series of natural numbers
Chhaya-prakaranam – Sun’s position, and it’s use in computations.
Chandragrahana-prakaranam- On the Lunar Eclipse,
Ravigrahana-prakaranam- On the Solar Eclipse
Vyatipata-prakaranam- Longitudes of Sun and Moon
Drikkarma-prakaranam- Rising and setting of moon and planets
Sringonnati-prakaranam- Lunar cusp.
His other work is Aryabhatiyabhasya, a commentary on Aryabhatta’s work where he developed a computational system for a partially heliocentric model, where the planets orbit the Sun, which in turn moves around the Earth, similiar to Tyco Brahe’s system. And also Jyotirmimamsa , where he outlines the methodology for astronomical and mathematical work, as also the Vedic influences in the astronomical work.
Some other important works by Nilakantha were
Golasara- Basic astronomical elements and process
Candrachayaganita- Calculation of time based on sun and moon’s shadow
Grahapariksakrama- Verification of astronomical observations.
The Kerala school is noted for it’s emphasis on practical astronomy, where study is devoted to astronomical computations, than just analysis of results. As well as the study of linguistics too.
What was apparent was that the Kerala School of Astronomy and Maths was a wonderful ecosystem lasting for 2 centuries, and produced some brilliant work in those fields, with learning passing down from generation to generation.
History Under Your Feet 