The proper Euclidean geometry is considered to be metric space and described in terms of only metric and finite metric subspaces (σ-immanent description). Constructing the geometry, one does not use topology and topological properties. For instance, the straight, passing through points A and B, is defined as a set of such points R that the area S(A,B,R) of triangle ABR vanishes. The triangle ar...

460 NOTICES OF THE AMS VOLUME 47, NUMBER 4 I n the fall semester of 1988, I taught an undergraduate course on Euclidean and nonEuclidean geometry. I had previously taught courses in projective geometry and algebraic geometry, but this was my first time teaching Euclidean geometry and my first exposure to non-Euclidean geometry. I used the delightful book by Greenberg [8], which I believe my stu...

By recasting metrical geometry in a purely algebraic setting, both Euclidean and non-Euclidean geometries can be studied over a general field with an arbitrary quadratic form. Both an affine and a projective version of this new theory are introduced here, and the main formulas extend those of rational trigonometry in the plane. This gives a unified, computational model of both spherical and hyp...

According to Felix Klein's Erlanger program (1872), a (classical) geometry is the study of properties of a space X invariant under a group G of transformations of X. In practice G will be a Lie group which acts transitively on X, so that X is represented as a homogeneous space G=H, where H G is a closed subgroup. For example Euclidean geometry is the geometry of n-dimensional Euclidean space R ...