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Article ID: CBB000930656

The Contributions of Hilbert and Dehn to Non-Archimedean Geometries and Their Impact on the Italian School (2007)

Cerroni, Cinzia (Author)

Revue d'Histoire des Mathématiques
Volume: 13
Pages: 259--299

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Non-euclidean geometry Geometry Mathematics 20th century, early Italy Enriques, Federigo Archimedes Hilbert, David Dehn, Max

In this paper we investigate the contribution of Dehn to the development of non-Archimedean geometries. We will see that it is possible to construct some models of non-Archimedean geometries in order to prove the independence of the continuity axiom and we will study the interrelations between Archimedes' axiom and Legendre's theorems. Some of these interrelations were also studied by Bonola, who was one of the very few Italian scholars to appreciate Dehn's work. We will see that, if Archimedes' axiom does not hold, the hypothesis on the existence and the number of parallel lines through a point is not related to the hypothesis on the sum of the inner angles of a triangle. Hilbert himself returned to this problem giving a very interesting model of a non-Archimedean geometry in which there are infinitely many lines parallel to a fixed line through a point while the sum of the inner angles of a triangle is equal to two right angles. Keywords : David Hilbert, Max Dehn, Federico Enriques, Roberto Bonola, Non-Archimedean geometry

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