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Giovanni Frattini. Eugenio Beltrami (16 November 1835 – 18 February 1900) was an Italian mathematician notable for his work concerning differential geometry and mathematical physics. His work was noted especially for clarity of exposition. He was the first to prove consistency of non-Euclidean geometry by modeling it on a surface of constant ...
The complete system of hyperbolic geometry was published by Lobachevsky in 1829/1830, while Bolyai discovered it independently and published in 1832. In 1868, Eugenio Beltrami provided models of hyperbolic geometry, and used this to prove that hyperbolic geometry was consistent if and only if Euclidean geometry was.
In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or n -dimensional unit ball) and lines are represented by the chords, straight line segments with ideal ...
The disk model was first described by Bernhard Riemann in an 1854 lecture (published 1868), which inspired an 1868 paper by Eugenio Beltrami. [2] Henri Poincaré employed it in his 1882 treatment of hyperbolic, parabolic and elliptic functions, [3] but it became widely known following Poincaré's presentation in his 1905 philosophical treatise, Science and Hypothesis. [4]
In non-Euclidean geometry, the Poincaré half-plane model is a way of representing the hyperbolic plane using points in the familiar Euclidean plane. Specifically, each point in the hyperbolic plane is represented using a Euclidean point with coordinates whose coordinate is greater than zero, the upper half-plane, and a metric ...
The model for hyperbolic geometry was answered by Eugenio Beltrami, in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model, which models the entirety of hyperbolic space, and used this to show that ...
In 1868 Eugenio Beltrami showed that the geometry of the pseudosphere was directly related to that of the more abstract hyperbolic plane, discovered independently by Lobachevsky (1830) and Bolyai (1832).
Advocates of the position that Euclidean geometry is the one and only "true" geometry received a setback when, in a memoir published in 1868, "Fundamental theory of spaces of constant curvature", [76] Eugenio Beltrami gave an abstract proof of equiconsistency of hyperbolic and Euclidean geometry for any dimension.