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On J. Bolyai's parallel construction

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Abstract

Given a model of Hilbert's incidence, order and congruence axioms (an H-plane) in which the Line-Circle Principle and the hypothesis of the acute angle hold, J. Bolyai's parallel construction is shown to yield the two parallels through the given point that have a “common perpendicular at infinity” with the given line through the ideal points at which they meet the given line; these need not be asymptotic parallels, for the plane need not be hyperbolic. Two new characterizations of hyperbolic planes are given, based on W. Pejas' classification of H-planes. The role of Archimedes' axiom is clarified.

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Authors and Affiliations

  1. Natural Sciences II, University of California, 95064, Santa Cruz, California, USA

    Marvin J. Greenberg

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  1. Marvin J. Greenberg
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Greenberg, M.J. On J. Bolyai's parallel construction. J Geom 12, 45–64 (1979). https://doi.org/10.1007/BF01920232

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  • DOI: https://doi.org/10.1007/BF01920232

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