On the equivalence of Lagrange's axiom to the Lotschnittaxiom

Victor Pambuccian

doi:10.1007/s00022-009-0018-2

On the equivalence of Lagrange's axiom to the Lotschnittaxiom

Victor Pambuccian

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

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 language	English (US)
Pages (from-to)	165-171
Number of pages	7
Journal	Journal of Geometry
Volume	95
Issue number	1-2
DOIs	https://doi.org/10.1007/s00022-009-0018-2
State	Published - 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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10.1007/s00022-009-0018-2

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AB - 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."

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KW - Bachmann's Lotschnittaxiom

KW - Lagrange's axiom

KW - Metric planes

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On the equivalence of Lagrange's axiom to the Lotschnittaxiom

Abstract

Keywords

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