# Indian Mathematics - Redressing the balance

### Ian G Pearce

### Keralese mathematics: II. 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

Although the

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 I will discuss later and 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

Furthermore, his demonstration of particular cases of the series

when

Citabhanu (1475-1550) has yet to find a place in books on Indian mathematics. His work on the solution of equations is quoted in a work called

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

*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 (see later).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*Nx*^{2}+ 1 =*y*^{2}(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 I will discuss later and 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^{-1}t=t-t^{3}/3 +t^{5}/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 NilakanthaCitabhanu (1475-1550) has yet to find a place in books on Indian 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, a regional language of Kerala). Similarly to the work of Nilakantha it is almost unique in the history of Indian mathematics, in that it contains both proofs of theorems and derivations of rules. He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using*kuttaka*.