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

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.