The New Foundation of Hyperbolic Geometry

The observation that the geometry of Bolyai and Lobachevsky and related geometries can be derived from mere axioms about connection [7–10]1 and the actual development of the hyperbolic plane geometry in terms of concepts of alignment by the Notre Dame school of geometry (1937–46) [1, 3, 4, 5, 13] seem to have attracted little attention.2 Euclidean geometry cannot be developed from axioms about connection (the first of Hilbert’s five groups of axioms) and, in fact, requires postulates about congruence, which are usually supplemented by assumptions about alignment, parallelism , order, and perpendicularity. The aforementioned observation thus reveals an inherent elementary character of hyperbolic geometry which does not seem to bear out Poincaré’s claim that, compared to non-Euclidean geometries, the Euclidean geometry is distinguished by simplicity (it is ‘athe simplest in itself, just as a polynomial of the first degree is simpler than a polynomial of the second degree’).