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Non-Euclidean Geometry in the Theory of Automorphic Functions

Jacques Hadamard
American Mathematical Society
Publication Date: 
Number of Pages: 
History of Mathematics 17
[Reviewed by
Fernando Q. Gouvêa
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It is cause for joy and celebration that the "Sources" subseries of the AMS/LMS "History of Mathematics" series continues to grow. This book, which is volume 17 in the AMS/LMS history series, contains a small book by Jacques Hadamard on the connections between non-Euclidean geometry and the theory of automorphic functions. Hadamard's book was written in the 1920s for publication in Russia, appearing in a series entitled "The Geometry of Lobachevskii and the Development of Its Ideas". It was translated into Russian, and the original French text seems to have been lost; what we have here, accordingly, is a translation from the Russian edition. It is preceded in this edition by a "Brief History of Automorphic Function Theory, 1880-1930" by Jeremy Gray. This is short but helpful, emphasizing the very distinct approaches to the subject used by Poincaré and by Klein, leading to differing French and German approaches to the subject. Hadamard's essay belongs firmly in the French camp, with a strong emphasis on solving differential equations and also with characteristic foundational weaknesses. This little book should be of interest both to historians seeking to understand the evolution of the theory of automorphic functions and to mathematicians working in the area, and thus it is a valuable addition to the (rather short) list of original source material available in English translation. Keep them coming, AMS!


  • Historical introduction
  • A brief history of automorphic function theory, 1880-1930
  • The group of motions of the hyperbolic plane and its properly discontinuous subgroups
  • Discontinuous groups in three geometries. Fuchsian functions
  • Fuchsian functions
  • Kleinian groups and functions
  • Algebraic functions and linear algebraic differential equations
  • Fuchsian groups and geodesics
  • References
  • Additional references