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Parallel Lines

Published online by Cambridge University Press:  20 November 2018

H. S. M. Coxeter*
Affiliation:
Department of Mathematics, University of Toronto, Toronto, Ont. M5S 1A1
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About a hundred years ago, the author of Through the Looking-Glass wrote another book called Euclid and his Modern Rivals. Were these rivals Lobachevsky and Bolyai, Riemann and Schlafli? No, they were merely the authors of dull school textbooks that would soon be forgotten. The sad truth is that, in 1873, hardly anyone in England knew of the breakthrough that had occurred on the Continent some fifty years before: only Cayley in Cambridge, Clifford in London, and a few students. Even if Cayley or Clifford had visited Oxford and given a lecture there, it is doubtful that he would have succeeded in convincing the conservative Dodgson that Euclid's postulates could be modified to yield two new worlds, surpassing in strangeness the worlds of the two Alice books and yet just as logically consistent as Euclid.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Coxeter, H. S. M., Non-Euclidean Geometry (5th éd.), University of Toronto Press, 1965.Google Scholar
2. Coxeter, H. S. M., The inversive plane and hyperbolic space, Abh. Math. Sem. Univ. Hamburg, 29 (1966), pp. 217-241*.Google Scholar
3. Coxeter, H. S. M., The problem of Apollonius, Amer. Math. Monthly, 75 (1968), pp. 5-15.Google Scholar
4. Kulczycki, Stefan, Non-Euclidean Geometry (translated by Knapowski, S.), Pergamon, 1961.Google Scholar
5. Liebmann, Heinrich, Nichteuklidische Géométrie, Leipzig, 1905.Google Scholar