### 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 | |

State | Published - 2009 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

**On the equivalence of Lagrange's axiom to the Lotschnittaxiom.** / Pambuccian, Victor.

Research output: Contribution to journal › Article

*Journal of Geometry*, vol. 95, no. 1-2, pp. 165-171. https://doi.org/10.1007/s00022-009-0018-2

}

TY - JOUR

T1 - On the equivalence of Lagrange's axiom to the Lotschnittaxiom

AU - Pambuccian, Victor

PY - 2009

Y1 - 2009

N2 - 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."

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."

KW - Absolute geometry

KW - Bachmann's Lotschnittaxiom

KW - Lagrange's axiom

KW - Metric planes

UR - http://www.scopus.com/inward/record.url?scp=77952955648&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77952955648&partnerID=8YFLogxK

U2 - 10.1007/s00022-009-0018-2

DO - 10.1007/s00022-009-0018-2

M3 - Article

AN - SCOPUS:77952955648

VL - 95

SP - 165

EP - 171

JO - Journal of Geometry

JF - Journal of Geometry

SN - 0047-2468

IS - 1-2

ER -