1907/01/01 by Oswald Veblen, J. H. Maclagan-Wedderburn · 100 citations
Mathematics · Engineering · #Mathematics and Applications #Structural Analysis and Optimization #Dynamics and Control of Mechanical Systems #Mathematics #Pure mathematics
paper · pdf · doi:10.1090/s0002-9947-1907-1500792-1
published in Transactions of the American Mathematical Society 8(3), 379-388 (American Mathematical Society)
openalex publication_date 1907/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/06/11
The object of this paper is to discuss certain non-pascalian and nondesarguesian geometries.Beside their intrinsic interest, these geometries are useful in forming independence proofs for systems of axioms for geometry.In a previous paper J it has been shown how to construct all finite projective geometries of three or more dimensions and all two dimensional projective geometries in which the Desargues theorem (cf.third footnote) is valid.The present paper supplements the other by showing the existence of non-desarguesian finite geometries.It is to be noted that, since every finite linear associative algebra in which every number possesses an inverse is commutative, therefore every non-pascalian finite geometry is necessarily non-desarguesian.Moreover, as has been shown by Hessenberg,|| every non-desarguesian geometry is also non-pascalian, whether finite or infinite.A projective plane geometry is a set of elements called points, finite or infinite in number, subject to the following conditions :1.If A and B are any two points, there is (a) one and () only one set of points called a line and containing both A and B.2. If a and 6 are any two lines, there is (a) one and () only onef point contained in both a and 6.3. Each line contains at least three points.