Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1111.1739

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

Henryk Fukś
View original source ↗Read source document ↗Export metadata ↓

Abstract

In his 1685 paper "Observationes cyclometricae" published in Acta Eruditorum, Adam Adamandy Kochański 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 π. Kochański 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 Kochański was very close to discovery of continued fractions and convergents of π.

Explore related subjects

math.HO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Henryk Fukś. 2014-05-29. Adam Adamandy Kochański's approximations of π: reconstruction of the algorithm. https://doi.org/10.1007/s00283-012-9312-1

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Come for the vibe, stay for the math

This article describes our experiences in mathematical outreach over the past decade. We talk about specific activities, but also general principles that we've learned along the way.

math.HO↗
arXiv2026

Graduate Mathematics in the Age of AI: Forming Mathematicians for Original, Independent, and Responsible Inquiry

Artificial intelligence can increasingly produce plausible, sophisticated mathematical material faster than a developing graduate student can understand or verify it. A sophisticated result or paper draft therefore becomes weaker evidence of the student's own mathematical development. This creates a formation gap between output and personal capacity, and a trust gap between a convincing argument and warranted acceptance. The formation gap can persist even when the student understands the output: understanding a supplied argument does not by itself establish the capacity to initiate and direct inquiry. These gaps are not the whole story. AI can also help students explore examples, compare approaches, enter unfamiliar areas, and undertake ambitious research. The task is to design an apprenticeship that realizes these possibilities while developing substantive mathematical command. The central purpose of a mathematics PhD is to form mathematicians capable of original, independent, and responsible inquiry, including inquiry conducted with AI. This document develops that objective through four connected capacities: competence, judgment, independence, and responsibility. It distinguishes a work's contribution to mathematics from the evidence it provides of a student's formation; explains how a known answer can initiate rather than end creative inquiry; and proposes changes in learning activities, assessment, doctoral originality, advising, and institutional support. Purposeful independent work and ambitious AI-assisted research are complementary parts of the model. Its recommendations include proportionate contribution statements, recognition of advising costs, and staged pilots evaluating both mathematical ability and effective human--AI collaboration. The aim is not to preserve an inherited sequence of training, but to improve mathematical formation as mathematical practice changes.

math.HO↗