Beltrami, Klein, and the Acceptance of Non-Euclidean Geometry

  • Jeremy Gray
Part of the Springer Undergraduate Mathematics Series book series (SUMS)


Eugenio Beltrami’s presentation of non-Euclidean geometry began the successful reception of non-Euclidean geometry. The influence of Gauss is another factor, but Kantian philosophy, contrary to what the historian Roberto Bonola suggested in 1906, was probably not a strong influence. In two major papers of 1871 and 1873, and also in his Erlangen program (1872), Felix Klein unified most of the existing geometries, including non-Euclidean geometry, by showing that they were special cases of projective geometry. Klein’s approach to non-Euclidean geometry is described, using his extension of the idea of Cayley metric. The influence of the Erlangen program seems to have been less than some mathematical historians have thought. The chapter ends with the Weierstrass-Killing hyperboloid model of non-Euclidean geometry, and a comparison of it with the Beltrami model.


Euclidean Geometry Projective Geometry Projective Transformation Reading Room Kantian Philosophy 
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Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.Department of MathematicsThe Open UniversityWalton Hall, Milton KeynesUnited Kingdom
  2. 2.The Mathematics InstituteThe University of WarwickWarwickUnited Kingdom

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