On the equivalence of Lagrange's axiom to the Lotschnittaxiom

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Abstract

We prove that, in Hilbert's plane absolute geometry, an axiom used by Lagrange in a proof of the Euclidean parallel postulate in a paper read on 3 February 1806 at the Institut de France, which states that "If a and b are two parallels from P to g, then the reflection of a in b is parallel to g as well", is equivalent to F. Bachmann's Lotschnittaxiom, which states that "The perpendiculars on the sides a right angle always intersect."

Original languageEnglish (US)
Pages (from-to)165-171
Number of pages7
JournalJournal of Geometry
Volume95
Issue number1-2
DOIs
StatePublished - Dec 15 2009

Keywords

  • Absolute geometry
  • Bachmann's Lotschnittaxiom
  • Lagrange's axiom
  • Metric planes

ASJC Scopus subject areas

  • Geometry and Topology

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