On \'Sankara Varman's (correct) and M\=adhava's (incorrect) values for the circumferences of circles

TL;DR

This paper investigates the computational methods used by Sankara Varman and Madhava to estimate circle circumferences, revealing that Madhava's series do not produce his claimed values, leaving his actual method unknown.

Contribution

It demonstrates that Madhava's series do not yield his reported circumference values, and explores how Sankara Varman's values can be derived from Madhava's series with specific assumptions.

Findings

Sankara Varman's values can be derived from Madhava's series with certain approximations.

Madhava's series do not produce his claimed circumference values in general.

The actual method Madhava used remains unknown.

Abstract

This paper examines what computational procedures \'Sankara Varman (1774-1839) and Sangamagrama M\=adhava (c. 1340 - 1425), astronomer-mathematicians of the Kerala school, might have used to arrive at their respective values for the circumferences of certain special circles (a circle of diameter $10^{17}$ by the former and a circle of diameter $9\times 10^{11}$ by the latter). It is shown that if we choose the M\=adhava-Gregory series for $\tfrac{\pi}{6}=\arctan (\tfrac{1}{\sqrt{3}})$ to compute $\pi$ and then use it compute the circumference of a circle of diameter $10^{17}$ and perform the computations by ignoring the fractional parts in the results of every operation we get the value stated by \'Sankara Varman. It is also shown that, except in an unlikely case, none of the series representations of $\pi$ attributed to M\=adhava produce the value for the circumference attributed to him. The question how M\=adhava did arrive at his value still remains unanswered.