17th century arguments for the impossibility of the indefinite and the definite circle quadrature

Jesper Lützen

17th century arguments for the impossibility of the indefinite and the definite circle quadrature

Bidragets oversatte titel: 1600-tals argumenter for umuligheden af det indefinite og det definite cirkelkvadratur

Jesper Lützen

Institut for Matematiske Fag

7 Citationer (Scopus)

Abstract

En analyse af argumenter fra 1600-tallet for umuligheden af at kvadrere cirklen. Argumenterne angår først og fremmest umuligheden af at finde et algebraisk udtryk for cirklens kvadratur. Spørgsmålet om det indefinite cirkelkvadratur handler om muligheden af at finde et udtryk for arealet af et vilkårligt sirkeludsnit, medens det definite cirkelkvadratur omhandler arealet af hele cirklen. I artiklen analyseres bidrag af Wallis, Gregory, Leibniz og Newton.

Bidragets oversatte titel	1600-tals argumenter for umuligheden af det indefinite og det definite cirkelkvadratur
Originalsprog	Engelsk
Tidsskrift	Revue d'histoire des Mathematiques
Vol/bind	20
Udgave nummer	2
Sider (fra-til)	211-251
ISSN	1262-022X
Status	Udgivet - 2014

Emneord

Det Natur- og Biovidenskabelige Fakultet
Matematikkens historie
Cirklens kvadratur
Wallis
Gregory
Leibniz
Newton

Citationsformater

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abstract = "The classical problem of the quadrature (or equivalently the rectification) of the circle enjoyed a renaissance in the second half of the 17th century. The new analytic methods provided the means for the discovery of infinite expressions of π and for the first attempts to prove impossibility statements related to the quadrature of the circle. In this paper the impossibility arguments put forward by Wallis, Gregory, Leibniz and Newton are analyzed and the controversies they gave rise to are discussed. They all deal with the impossibility of finding an algebraic expression of the area of a sector of a circle in terms of its radius and cord, or of the area of the entire circle. It is argued that the controversies were partly due to a lack of precision in the formulation of the results. The impossibility results were all part of a constructive problem solving mathematical enterprise. They were intended to show that certain solutions of the quadrature problem were the best possible because simpler (analytic) solutions were impossible.",

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AB - The classical problem of the quadrature (or equivalently the rectification) of the circle enjoyed a renaissance in the second half of the 17th century. The new analytic methods provided the means for the discovery of infinite expressions of π and for the first attempts to prove impossibility statements related to the quadrature of the circle. In this paper the impossibility arguments put forward by Wallis, Gregory, Leibniz and Newton are analyzed and the controversies they gave rise to are discussed. They all deal with the impossibility of finding an algebraic expression of the area of a sector of a circle in terms of its radius and cord, or of the area of the entire circle. It is argued that the controversies were partly due to a lack of precision in the formulation of the results. The impossibility results were all part of a constructive problem solving mathematical enterprise. They were intended to show that certain solutions of the quadrature problem were the best possible because simpler (analytic) solutions were impossible.

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KW - Matematikkens historie

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KW - Wallis

KW - Gregory

KW - Leibniz

KW - Newton

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VL - 20

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JO - Revue d'Histoire des Mathematiques

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17th century arguments for the impossibility of the indefinite and the definite circle quadrature

Abstract

Emneord

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