On a new type of rational and highly convergent series, by which the ratio of the circumference to the diameter is able to be expressed

论Śankara Varman（正确）与Mādhava（错误）的圆周长数值

On Śankara Varman's (correct) and Mādhava's (incorrect) values for the circumferences of circles

本文考察喀拉拉学派两位数学家计算特殊圆周长的方法，证实Śankara Varman的结果可通过Mādhava-Gregory级数逐次取整得到，而Mādhava的结果无法用其已知的π级数推导，其计算方法仍是未解之谜。

Circular reasoning: who first proved that $C/d$ is a constant?

We answer the question: who first proved that $C/d$ is a constant? We argue that Archimedes proved that the ratio of the circumference of a circle to its diameter is a constant independent of the circle and that the circumference constant equals the area constant ($C/d=A/r^{2}$). He stated neither result explicitly, but both are implied by his work. His proof required the addition of two axioms beyond those in Euclid's \emph{Elements}; this was the first step toward a rigorous theory of arc length. We also discuss how Archimedes's work coexisted with the 2000-year belief -- championed by schol…

On the acceleration of the convergence of the "Mādhava-Leibniz series"

This paper expounds very innovative results achieved between the mid-14th century and the beginning of the 16th century by Indian astronomers belonging to the so-called "Mādhava school". These results were in keeping with researches in trigonometry: they concern the calculation of the eight of the circumference of a circle. They not only expose an analog of the series expansion of arctan(1) usually known as the "Leibniz series", but also other analogs of series expansions, the convergence of which is much faster. These series expansions are derived from evaluations of the rests of the partial …

How Archimedes showed that $π$ is approximately equal to 22/7

The ratio of the circumference, C, of a circle to its diameter, D, is a constant number denoted by $π$ and is independent of the size of the circle. It is known that $π$ is an irrational number and therefore cannot be expressed as a common fraction. Its value is approximately equal to 3.141592. Since Archimedes was one of the first persons to suggest a rational approximation of 22/7 for $π$, it is sometimes referred to as Archimedes' constant. In this article, we discuss how Archimedes came up with his formula. Archimedes in fact proved that 223/71 < $π$ < 22/7. Here we provide an improved low…