Introduction

Abstract

Over the centuries many attempts were made to prove the Parallel Postulate of Euclid. The discovery that an alternative is possible was a major breakthrough that lead to the development of hyperbolic geometry as a subject in its own right. The fact that the hyperbolic axioms are as consistent as the Euclidean axioms can be established by models within Euclidean geometry, but the fact that the hyperbolic axioms are categorical must be proved by careful development from the axioms. It is hoped that this book will encourage the development of geometric intuition and also the recognition that mathematics itself is not compartmentalized in spite of the fact that names are given to various branches. The close relationship between analysis and geometry allows the use of analysis to simplify many derivations and proofs.