Later Jaina mathematicians, Dharamanandana and Sundarasuri, continued explorations on magic squares and similar arrangements.
But from the fourteenth century, Kerala became the locus of several new siddhaantas, bhaashyas, karanas etc., that it is now called the Kerala school of mathematics.
The Kerala school
Shortly after Aryabhata, a mathematician called Haridatta had composed a work title Parahita Ganita based on Aryabhatiyam. After that we seem to have a vacuum in Kerala until Madhava (c 1340 AD) of Sangamagrama (Kudalur in Malayalam). What followed is an astonishing continuity of the guru-shishya parampara from Govinda Bhattatiri to Rajaraja Varma.
Madhava was famous as a skilled instrument maker. Indian historians of mathematics consider him the pitamaha of the Kerala school. Perhaps, his most famous contribution is sums of infinite series.
To calculate the circumference (paridhi) of a circle, said Madhava, we must multiply the diameter (vyaasa) by four times one minus tri-sharaa-aadi-vishama-samkhyaa-bhaktam-rNam. A phrase of compactness, which Aryabhata would have enjoyed. In other words, a sequence of odd (vishama samkhyaa) denominators (bhaktam) starting (aadi) with three (tri) and five (sharaa). The word sharaa here is a bhutasamkhya (see third essay) word for the five arrows of Manmatha, whose archery takes precedence over Rama and Arjuna and even Tripurantaka, in this case.
The sum of the series in the brackets adds up to /4, and is famous as the Gregory Leibniz series.
Madhavas series was quoted and a proof (upapatti) also given a century later by Jyeshtadeva.
Parameshvara Like Brahmagupta providing a sphuTam to Paitamaha Siddhaanta, Parameshvara observed that over time, predictions of earlier astronomers did not agree with observed positions based on calculations. In such a situation, he observed, one must adjust ones methods and calculations, because planets and stars will conform to them.
He titled his book Drg Ganitam. This title, which means Observed Calculations, is a popular phrase for jyotisham in south India, though the author himself has faded from public memory. His shishya NilakanTha referred to him as Paschimaam Bodhi, the western scholar.
Nilakantha Somayaaji More unknown than even Brahmagupta, was a polymath like Varahamihira. He was a scholar of Shad darshana (the six philosophies of Hinduism, and also in vyaakarana, chandas, the Bhagavata and various such literature. He also studied Vedanga Jyotisha, Pancha Siddhaantika, Brhat Samhita etc.
This historical curiosity and scholarship may have shined in other scholars, too, but in Nilakantha, we have contemporary evidence. He was also a prolific composer, of several texts.
He was a friend a Sundararaja, a jyotisha of the neighboring Tamil Nadu, and took the effort to compile a written list of answers for questions posed by the former, compiled into a book called Sundararaja Prashnottara.
Aficionados of European science may be reminded of the extensive correspondence of Franklin, Newton, Darwin, Humboldt etc.
A ninth century mathematician called Virasena in his commentary Dhavala gave this equation, that the sum of all powers of 1/4 is 1/3. One sees the reflection of this in Madhavas several infinite series. Nilakantha questioned this apparent absurdity. How does the sum of this infinite series increase to that finite value (1/3), and that it reaches finite value?
He reasoned and explained it by deriving the following sequence of results
As we add more terms, argued Nilakantha, the difference between 1/3 and the powers of 1/4 become extremely small, but never zero, unless we add terms up to infinity. In the 20th century, Ramanujan revelled in such series.
Quasi Heliocentric theory NilakanThas questioning of an assumption of planets latitudes (vikshepa), and his subsequent discovery was truly astronomical (pardon the pun). From the siddhantas through Bhaskaracharya, all astronomers used a slightly different method to calculate the latitudes of Mercury (Budha) and Venus (Shukra), than they did for the other planets. This niggled most of them, as inappropriate, especially Bhaskara, who then consoled himself with Prthudaka Svamis explanation.
But NilakanTha questioned this acceptance, and modified his computation, and effectively the orbital model for these planets. He came to the conclusion that these two planets revolve around the Sun (but in his model, the Sun still revolves around the Earth).
The geometrical argument is too complicated not only for this article, but even, perhaps, for those who are not astronomers, so I will only present a visual illustration of his model here.
The rest is here:
Indian Astronomy And Mathematics: When Kerala Became The Locus Of Genius - Swarajya