On Pappian and Desarguesian affine planes

V. O. Pshyk

doi:10.30970/ms.64.1.92-98

V. O. Pshyk Ivan Franko National University of Lviv, Lviv, Ukraine https://orcid.org/0009-0000-8019-5191

DOI: https://doi.org/10.30970/ms.64.1.92-98

Анотація

We present a complete proof of the classical Heisenberg’s Theorem on the Desarguesian property of Pappian affine planes. In the proof we consider two cases, one of which was omitted in the original Hessenberg’s proof, making it incomplete.

Біографія автора

V. O. Pshyk, Ivan Franko National University of Lviv, Lviv, Ukraine

Ivan Franko National University of Lviv, Lviv, Ukraine

Посилання

A. Cronheim, A proof of Hessenberg’s theorem, Proc. Amer. Math. Soc., 4 (1953), 219–221. doi:10.2307/2031794.

G. Hessenberg, Beweis des Desarguesschen Satzes aus dem Pascalschen, Math. Ann., 61 (1905), №2, 161–172. doi:10.1007/BF01457558.

E.A. Marchisotto, The theorem of Pappus: a bridge between algebra and geometry, Amer. Math. Monthly, 109 (2002), №6, 497–516. doi:10.2307/2695440.

V. Pambuccian, C Schacht, The axiomatic destiny of the theorems of Pappus and Desargues In Geometry in history, Springer, Cham, (2019), 355–399.

J. Playfair, Elements of Geometry. New York: W. E. Dean, Printer & Publisher, 1846. URL: https://archive.org/details/elementsgeometr05playgoog/page/n6/mode/2up

Proclus, A commentary on the first book of Euclid’s Elements. Princeton University Press, Princeton, NJ, 1970. Translated with introduction and notes by G.R. Morrow. doi:10.2307/j.ctv10vm1rv.

S. Weinlos, Sur lindependance des axiomes de coıncidence et de parallelite dans un systeme axiomes de la geometrie euclidienne a trois dimensions, Fundamenta Mathematicae, 11 (1928), №1, 206–221.