Adam Adamandy Kochański's approximations of π: reconstruction of the algorithm

Details Adam Adamandy Kochański's approximations of π: reconstruction of the algorithm Adam Adamandy Kochański's approximations of π: reconstruction of the algorithm

2011-11-07

arxiv.org/abs/1111.1739v1

Mathematics

History and Overview

11 pages, 1 figure

Scientific paper

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.

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