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Accueil > Publications > Ouvrages parus > Ouvrages des chercheurs de SPHERE : 2015–... > The Impossibility of Squaring the Circle in the 17th Century.

The Impossibility of Squaring the Circle in the 17th Century.

A Debate Among Gregory, Huygens and Leibniz.

Davide Crippa

This book is about James Gregory’s attempt to prove that the quadrature of the circle, the ellipse and the hyperbola cannot be found algebraically. Additionnally, the subsequent debates that ensued between Gregory, Christiaan Huygens and G.W. Leibniz are presented and analyzed. These debates eventually culminated with the impossibility result that Leibniz appended to his unpublished treatise on the arithmetical quadrature of the circle.

The author shows how the controversy around the possibility of solving the quadrature of the circle by certain means (algebraic curves) pointed to metamathematical issues, particularly to the completeness of algebra with respect to geometry. In other words, the question underlying the debate on the solvability of the circle-squaring problem may be thus phrased : can finite polynomial equations describe any geometrical quantity ? As the study reveals, this question was central in the early days of calculus, when transcendental quantities and operations entered the stage.

Undergraduate and graduate students in the history of science, in philosophy and in mathematics will find this book appealing as well as mathematicians and historians with broad interests in the history of mathematics.

"– Delivers an unprecedented perspective on this topic
– Gives a fresh picture of the mathematics in the 17th Century based on previously unstudied documents
– Conveys mathematics with minimal technical requirements and thorough explanation so that the narrative can be followed by graduate and undergraduate students in sciences and humanities"

: : Birkhäuser, collection : "Frontiers in the History of Science" [part of Springer Nature]
: : mai 2019
: : VIII, 184 p., 32 illus., 29 illus. in color
: : ISBN : 978-3-030-01638-8

TABLE DES MATIERES [ télécharger ]

1 Introduction, p. 1
1.1 The Quadrature of the Circle and Its Impossibility, p. 1
1.2 The Famous Problems of Classical Geometry and Their Impossibilities, p. 2
1.3 Impossibility Results in Classical Mathematics, p. 5
1.4 Pappus on the Conditions of Solvability of Problems, p. 10
1.5 On the Impossibility of the Classical Problems in the Seventeenth Century, p. 15
1.6 The Problem of Squaring the Circle Until Descartes, p. 18
1.7 Are All Rectifications Algebraically Impossible ?, p. 27

2 James Gregory and the Impossibility of Squaring the Central Conic Sections, p. 35
2.1 A Seventeenth Century Controversy on the Impossibility of Squaring the Circle, p. 35
2.2 Gregory’s “Second Kind of Analysis”, p. 41
2.3 Introducing Convergent Sequences, p. 46
2.3.1 Polygonal Approximations, p. 46
2.3.2 Analytical and Non-analytical Quantities and Operations, p. 49
2.3.3 The Convergent Series of Polygonal Approximations, p. 53
2.3.4 Proving the Convergence of the Series, p. 59
2.3.5 Computing the Limit of a Series, p. 63
2.4 The Taming of the Impossible, p. 67
2.4.1 An Argument of Impossibility, p. 67
2.4.2 A New Operation, p. 70
2.5 Reception and Criticism of Gregory’s Impossibility Argument, p. 74
2.5.1 Huygens’ First Objections, p. 74
2.5.2 Wallis Against Gregory’s Impossibility Theorem, p. 79
2.5.3 Gregory’s Last Reply, p. 84
2.6 Concluding Remarks, p. 87

3 Leibniz’s Arithmetical Quadrature of the Circle, p. 93
3.1 Introduction, p. 93
3.2 The Manuscripts of De quadratura arithmetica, p. 95
3.3 Leibniz’s Knowledge of Gregory’sWorks, p. 97
3.4 The Arithmetical Quadrature of the Circle : ItsMain Results, p. 99
3.5 A Digression : Early Modern Techniques for the Quadrature of the Hyperbola, p. 103
3.5.1 Brouncker’s Quadrature of the Hyperbola, p. 106
3.5.2 Mercator’s and Wallis’s Quadrature of the Hyperbola, p. 109
3.6 Leibniz’s Arithmetical Quadrature : The Geometrical Reduction, p. 117
3.7 Towards the Arithmetical Quadrature of the Circle : The Analytical Solution, p. 125
3.8 First Communication of the New Results : Huygens and Oldenburg, p. 134
3.8.1 Criticism of Gregory, p. 138
3.9 The Classification of Quadratures and the Impossibility of Squaring the Circle, p. 143
3.9.1 An Impossibility Proof, p. 146
3.10 The Impossibility of Finding the Universal Quadrature of the Hyperbola, p. 152
3.11 The Significance of Leibniz’s Impossibility Result, p. 154

4 Conclusion, p. 157
4.1 The Role and Goals of Impossibility Results in Early Modern Geometry, p. 157
4.2 Two Early Modern Conceptualisations of Impossibility Results, p. 164
4.3 Concluding Remarks, p. 173
References, p. 175
Index, p. 183