Seventeenth Century Arguments for the Impossibility of the Indefinite and the Definite Circle Quadrature (2014)
Lützen, Jesper (Author)
Revue d'Histoire des Mathématiques
Volume: 20, no. 2
Issue: 2
Pages: 211-251
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 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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