A Course in Old and New Geometry Volume I: Axiomatic and Neutral Geometry
The front cover shows the figure from page 132 showing the construction by Hilbert tools to erect an isosceles triangle onto a given segment. Since Hilbert tools are weaker than Euclidean tools, one cannot construct an equilateral triangle in neutral geometry. A decade long experience of teaching the course Fundamental of Geometry at the University of North Carolina at Charlotte, many notes for exercises, reading and research are the bases for this bulky work. The present first volume begins with Hilbert's axioms from the Foundations of Geometry [12]. After some discussion of logic and axioms in general, we go on with a short section about incidence geometries in two and three dimensions. As in Hilbert's system, there follow detailed sections about the axioms of order, next about congruence in neutral geometry,-which means without assumption of the axiom of parallelism. I deal with the axioms of measurement and of completeness, and deviating from Hilbert, with axioms about circles. The last two sections deal with the theory of area in neutral geometry, and neutral triangle geometry. My heartfelt thanks go to everyone involved, this means famous authors as well as my former students.
Franz Rothe has received his doctoral degree in mathematics from the university of T ̈ubingen, Germany. He has been professor at the University of North Carolina at Charlotte, and has published about 40 articles and a lecture notes in mathematics, and two further books on number theory, modern algebra, and graph theory. Dr. Rothe is retired since several years, and is now emeritus professor.
