Proving Unproved Euclidean Propositions on a New Foundational Basis

A. Leon¹

Section:Research Paper, Product Type: Journal-Paper
Vol.7 , Issue.3 , pp.61-81, Jun-2020

Online published on Jun 30, 2020

Copyright © A. Leon . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
This article introduces a new foundation for Euclidean geometry more productive than other classical and modern alternatives. Some well-known classical propositions that were proved to be unprovable on the basis of other foundations of Euclidean geometry can now be proved within the new foundational framework. Ten axioms, 28 definitions and 40 corollaries are the key elements of the new formal basis. The axioms are totally new, except Axiom 5 (a light form of Euclid?s Postulate 1), and Axiom 8 (an extended version of Euclid?s Postulate 3). The definitions include productive definitions of concepts so far primitive, or formally unproductive, as straight line, angle or plane The new foundation allow to prove, among other results, the following axiomatic statements: Euclid`s First Postulate, Euclid`s Second Postulate, Hilbert`s Axioms I.5, II.1, II.2, II.3, II.4 and IV.6, Euclid`s Postulate 4, Posidonius-Geminus` Axiom, Proclus` Axiom, Cataldi`s Axiom, Tacquet`s Axiom 11, Khayyam`s Axiom, Playfair`s Axiom, and an extended version of Euclid`s Fifth Postulate.

Key-Words / Index Term :
foundation of Euclidean geometry, sidedness, straightness, orthogonality, parallelism, convergence

References :
[1] T. Heath, ?The Thirteen Books of Euclid`s Elements,? Second ed., vol. I, Dover Publications Inc, New York, pp. 153-374, 1956.
[2] J. Playfair, ?Elements of Geometry,? W.E. Dean Printer and Publisher, New York, pp. 5-11, 1846.
[3] D. Hilbert, ?The Foundations of Geometry,? The Open Court Publishing Company, La Salle, pp. 2-21, 1950.
[4] G. Birkhoff y R. Beatley, ?Basic Geometry,? American Mathematical Society, Providence, pp. 38-164, 2000.
[5] G. Cantor, ?Contributions to the founding of the theory of transfinite numbers,? Dover Publications Inc., New York, pp. 85-208, 1955.
[6] G. Cantor, ??ber verschiedene Theoreme asu der Theorie der Punktmengen in einem n-fach ausgedehnten stetigen Raume Gn,? Acta Mathematica, Vol. 7, pp. 105-124, 1885.