Abstract
An axiom system for Euclidean geometry with Euclid’s version of the parallel postulate, in which the order axioms are introduced in terms of the separation a line introduces in the plane, as pioneered by Sperner (Math Ann 121:107–130, 1949), in which the compass can be used only to transport segments, which lacks the Pasch axiom, is shown to imply the Pasch axiom due to the very form in which Euclid chose to express his fifth postulate. This shows, as first noted without proof by Salvatore di Noi in that same year 1949, that Euclid did not need the Pasch axiom.
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This paper was largely written while enjoying the wonderful hospitality of the University of Urbino “Carlo Bo” during the Spring semester of 2022, a stay made possible by Pierluigi Graziani, for which I am filled with gratitude.
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Pambuccian, V. Why did Euclid not need the Pasch axiom?. J. Geom. 115, 13 (2024). https://doi.org/10.1007/s00022-024-00712-x
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DOI: https://doi.org/10.1007/s00022-024-00712-x