Foundations of Euclidean and Non-Euclidean Geometry

Description

This book is an attempt to present, at an elementary level, an approach to geometry in keeping with the spirit of Euclid, and in keeping with the modern developments in axiomatic mathematics. It is not a comprehensive study of Euclidean geometry-far from it; it is not a survey of various types of geometry. And while it is designed to meet prerequisites in geometry for secondary school teachers, it is neither a review of highschool topics, nor an extension of such topics, as has been customarily covered over the years in books designed for college geometry. The emphasis of the book is on the method of presentation rather than on presenting a mass of Information. It is in the spirit of Euclid in two senses: it is a synthetic approach to geometry; it is an axiomatic approach. The author recognizes the fact that, in high schools, there are many advantages to developing geometry by the method now coming into use. We refer to the method credited to Birkhoff and Beatley, and adopted and modified by the School Mathematics Study Group. It is, at this time, the simplest and most practical way to present the subject matter at that level. However, there is much to be said for presenting the college mathematics major and prospective teacher with a different approach. It is hoped that this book will serve this purpose. The book attempts to blend several aspects of mathematical thought: namely, intuition, cre.ative thinking, abstraction, rigorous deduction, and the excitement of discovery. It is designed to do so at an elementary level. In covering these aspects, it falls rather naturally into three parts.