Adam Adamandy Kocha\'nski's approximations of \pi: reconstruction of the algorithm

TL;DR

This paper reconstructs Adam Adamandy Kochański's 17th-century algorithm for approximating pi, revealing its connection to continued fractions and providing insights into its properties and historical significance.

Contribution

We reconstruct the complete algorithm Kochański used for approximating pi, which was previously only partially understood, and analyze its relation to continued fractions.

Findings

Reconstructed the full approximation algorithm from Kochański's partial description.

Identified the algorithm's close relation to continued fractions of pi.

Provided insights into the properties and historical context of Kochański's approximations.

Abstract

In his 1685 paper "Observationes cyclometricae" published in Acta Eruditorum, Adam Adamandy Kocha\'nski presented an approximate ruler-and-compass construction for rectification of the circle. It is not generally known that the first part of this paper included an interesting sequence of rational approximations of \pi. Kocha\'nski gave only a partial explanation of the algorithm used to produce these approximations, while promising to publish details at a later time, which has never happened. We reconstruct the complete algorithm and discuss some of its properties. We also argue that Kocha\'nski was very close to discovery of continued fractions and convergents of \pi.