On M\=adhava and his correction terms for the M\=adhava-Leibniz series for $\pi$

Abstract

This paper is intended to serve two purposes: one, to present an account of the life of Sangamagr\=ama M\=adhava, the founder of the Kerala school of astronomy and mathematics which flourished during the 15th - 18th centuries, based on modern historical scholarship and two, to present a critical study of the three enigmatic correction terms, attributed to M\=adhava, for obtaining more accurate values of $\pi$ while computing its value using the M\=adhava-Leibniz series. For the second purpose, we have collected together the original Sanskrit verses describing the correction terms, their English translations and their presentations in modern notations. The Kerala rationale for these correction terms are also critically examined. The general conclusion in this regard is that, even though the correction terms give high precision approximations to the value of $\pi$, the rationale presented by Kerala authors is not strong enough to convince modern mathematical scholarship. The author has extended M\=adhava's results by presenting higher order correction terms which yield better approximations to $\pi$ than the correction terms attributed to M\=adhava. The various infinite series representations of $\pi$ obtained by M\=adhava and his disciples from the basic M\=adhava-Leibniz series using M\=adhava's correction terms are also discussed. A few more such series representations using the better correction terms developed by the author are also presented. The various conjectures regarding how M\=adhava might have originally arrived at the correction terms are also discussed in the paper.