An Introductory Study of Hyperbolic Geometry

Abstract

The object of this paper is to develop Hyperbolic geometry using, as tools, only the postulates and propositions of Euclid (excluding the ones dealing with parallel lines) and the tools of logic. To the best of my knowledge and throughout my research I have found no one work which contains a total picture of Hyperbolic geometry. There are bits and pieces scattered through a multitude of manuscripts. In the University of Michigan's Math Library there is a set of the complete works of Lobachevsky, which unfortunately has not been translated from the original Russian. In recent years there has been an increased interest in Hyperbolic geometry because it is felt that 1t is a better description of physical space than is Euclidean geometry. If it can be shown, logically, rather than by measurement, wh1ch even at best contain an error; that the theorems of Hyperbolic geometry follow from our set of axioms and postulates, then the error measurement and the error due to something appearing to be "right" is avoided.