Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, Abhandlungen der Kniglichen Gesellschaft der Wissenschaften zu Göttingen, 13, 1867, On the Hypotheses which lie at the Bases of Geometry, Nature, no. Über eine Eigenschaft der konformen Abbildung kreisförmiger Bereiche. From Schwarz to Pick to Ahlfors and beyond. Application de l’Analyse à la Géométrie, 5. Hyperbolic Geometry: the first 150 years. What Frege Meant When He Said: Kant is Right about Geometry. Springer Undergraduate Mathematics Series. Worlds out of nothing: a course in the history of geometry in the 19th Century. Non-Euclidean geometry: a re-interpretation. Note sur l’impossibilité de démotrer par une construction plane le principe des parallèles, dit Postulatum d’Euclide, Nouvelles annales de mathématiques 2 esérie. Floyd, William J., Kenyon, Richard, Walter R. André, Häfliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften, Vol. Dover Publications, Inc.: New York.īridson, Martin R. The History of the Calculus and Its Conceptual Development. Un precursore italiano di Legendre e di Lobatschewsky. Sulla superficie di rotazione che serve di tipo alle superficie pseudosferiche. Beltrami “Sugli spazi di curvatura costante”. II, (Unknown Month 1868), 232–255 English transl., Théorie fondamentale des espace de courbure constante, Annales scientifiques de l’É.N.S. Teoria fondamentale degli spazii di curvatura costante, Ann. VI 284–312 English transl., Essai d’interpretation de la géométrie noneuclidéenne, Annales scientifiques de l’É.N.S. Saggio di interpretazione della geometria non-euclidea, Giornale di Matematiche. Delle variabili complesse sopra una superificie qualunque, Ann. Risoluzione del problema: “Riportare i punti di una superficie sopra un piano in modo che le linee geodetiche vengano rappresentate da linee rette”. II, 468 p., Hoepli: Milano.īeltrami, Eugenio. Opere matematiche di Eugenio Beltrami, pubblicate per cura della Facoltà di Scienze dalla Regia Università di Roma. Of course, in the projective-space model, often called Riemann’s non-Euclidean geometry, one not only has to give up the infinite extension of straight lines, but also the fact that a line divides the plane into two parts or, which is about the same, orientability of the plane.Īhlfors, Lars Valerian. Kelin realized the role of this model in the discussion of non-Euclidean geometries. Clifford used the two-to-one covering of the real projective plane by the sphere to exhibit a geometry with positive constant curvature in which (1) there was just one line through two points (2) space was homogeneous and isotropic and (3) there are no distinct, parallel straight lines. Some of the principles used by Saccheri had to be abandoned: uniqueness of the line through two points, it seemed but also the infinite extension of lines (Riemann seems to be the first to point out that the right geometric requirement is not that the straight lines have infinite length – what he calls “infinite extent of the line”, translated in “unboundedness” in modern netric space theory –, but that one finds no obstructions while following a straight line – a property he calls “unboundedness”, translated nowadays in “metrically complete and without boundary”). In the Habilitationschrift, Riemann offered the sphere as a model for a geometry in which no parallel existed. This was considered to be a major problem by Beltrami, who was looking for a geometry in which all principles of Euclidean geometry hold true, but the uniqueness of parallels, but not for Riemann. All rights reserved.The obtuse hypothesis holds on a sphere, using geodesics (great circles) as straight lines but on a sphere we do not have uniqueness of the geodesic through two points. Copyright © 2023, Columbia University Press. The Columbia Electronic Encyclopedia, 6th ed.
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