This is the second article in the series Mathematicians of Kerala
Narayana Pandit (c. 1340-1400), the earliest of the notable Keralese mathematicians, is known to have definitely written two works, an arithmetical treatise called Ganita Kaumudi and an algebraic treatise called Bijganita Vatamsa. He was strongly influenced by the work of Bhaskara II, which proves work from the classic period was known to Keralese mathematicians and was thus influential in the continued progress of the subject. Due to this influence Narayana is also thought to be the author of an elaborate commentary of Bhaskara II’s Lilavati, titled Karmapradipika (or Karma-Paddhati). It has been suggested that this work was written in conjunction with another scholar, Sankara Variyar, while others attribute the work to Madhava.
Although the Karmapradipika contains very little original work, seven different methods for squaring numbers are found within it, a contribution that is wholly original to the author. Narayana’s other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, using the second order indeterminate equation Nx2 + 1 = y2 (Pell’s equation). Mathematical operations with zero, several geometrical rules and discussion of magic squares and similar figures are other contributions of note. Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II’s work.
R Gupta has also brought to light Narayana’s contributions to the topic of cyclic quadrilaterals. Subsequent developments of this topic, found in the works of Sankara Variyar and Ganesa interestingly show the influence of work of Brahmagupta.
Paramesvara (c. 1370-1460) is known to have been a pupil of Narayana Pandit, and also Madhava of Sangamagramma, who is thought to have been a significant influence. He wrote commentaries on the work of Bhaskara I, Aryabhata I and Bhaskara II, and his contributions to mathematics include an outstanding version of the mean value theorem. Furthermore Paramesvara gave a mean value type formula for inverse interpolation of sine, and is thought to have been the first mathematician to give the radius of circle with inscribed cyclic quadrilateral, an expression that is normally attributed to Lhuilier (1782).
In turn, Nilakantha Somayaji (1444-1544) was a disciple of Paramesvara and was educated by his son Damodra. 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 -1t = t – t3/3 + t5/5 – … ,
when t = 1 and t = 1/
3, and remarkably good rational approximations of p (using another Madhava series) are of great interest. Various results regarding infinite geometrically progressing convergent series are also attributed to Nilakantha.
Citabhanu (1475-1550) has yet to find a place in books on Bharatiya mathematics. His work on the solution of equations is quoted in a work called Kriya-krama-kari, by the scholar Sankara Variar, who is also relatively little known (although R Gupta mentions a further text, written by him).
Jyesthadeva (c. 1500-1575) was a member of the Kerala School, which was founded on the work of Madhava, Nilakantha, Paramesvara and others. His key work was the Yukti-bhasa (written in Malayalam). Similarly to the work of Nilakantha, it is almost unique in the history of Bharatiya mathematics, in that it contains both proofs of theorems and derivations of rules. He also studied various topics found in many previous Bharatiya works, including integer solutions of systems of first degree equations solved using kuttaka.
(To be continued…The articles in this series appeared on the portal of University of St. Andrews.)
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