';~ o{&~ 1tM-&eedtdeeue ad ~t'Zfl 7~ ol&~ 1tM-Eeedtetea1e 4'ett Ellery B. Golos Ohio University HOLT, RINEHART and WINSTON INC. New York • Chicago • San Francisco • Atlanta • Dallas Montreal • Toronto • London Copyright© 1968 by Holt, Rinehart and Winston, Inc. All Rights Reserved Library of Congress Catalog Card Number: 68-18413 2684959 Printed in the United States of America 1 2 3 4 5 6 7 8 9 To LILA 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 comprehen sive 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 high school 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 l\Iathematics 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 mathe matics 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. The first part is designed to give the student the tools, insight, and motivation to approach elementary geometry from a new perspective and with an open mind. While portions of this part of the book may seem a bit too abstract for the beginner, the author has found that there are enough concrete interpretations to keep most students on a firm footing, and enough new challenges to stimulate them. vii viii Prefa ce The second part is meant to introduce the student to a modern, rigorous approach to the foundations of geometry. It attempts to introduce and pursue many important topics that are often slighted; at the same time, it does not, as is so ofteq the case in studies in foundations, introduce an axiom system which is so delicately pure that it seems never to arrive at the standard theorems of elementary geometry. Thus it combines both rigor and intuition. The third part is devoted to non-Euclidean geometry, and as such, forces the student to reason without placing too much reliance on his intuition. It is to be hoped that the first two parts have prepared him for this. It is best, perhaps, to attempt to devote at least two academic quarters to the book. We have found, in several years of teaching this material, that it can be covered in one semester if, instead of lecturing, the in structor uses class time in clarifying and answering questions about content and "problems." This technique is not merely a time-saving device. It forces the student to read; it stimulates class discussion; it sometimes even leads to "discovering" new truths. Because we have used this approach, we have attempted to make the text self-contained and readable. The partitioning of the text offers a greater degree of flexibility in its use. Part II can be used by itself (perhaps with some of Part I as outside reading). Other possibilities are to use Parts I and II; Parts II and III (Chapter 4 as outside reading). The time one spends on each part will depend greatly upon the time one wishes to devote to proofs and exer cises; they should not be slighted, because they enrich the course considerably. Because the emphasis throughout the text is on its approach to the material rather than on its specific content, and because the text is available to students at an elementary level, it has been our experience that it has a great deal of "transfer" value. We have found this to hold both in preparing the student for the next geometry course, and in just giving him an edge in gaining that nebulous goal called "mathematical maturity." But even if the text achieves none of the foregoing, the author will be pleased if it stimulates any reader to pursue one of the topics on his own. The author wishes to express his appreciation to the Cambridge Uni versity Press for giving him permission to quote from its definitive pub lication on Euclid: The Thirteen Books of Euclid's Elements, by T. L. Heath. The proofs of several theorems are reproduced in their entirety Preface ix within the text; and statements of the axioms, definitions, and theorems of Book I are quoted in the text and listed in the Appendix. If there are any errors in the book, they are present in spite of the many helpful suggestions and constructive criticisms made by colleagues and friends who read and used a preliminary version of the work. The author wishes especially to thank Professor W. T. Fishback, who first encouraged him to write the book and used a preliminary manuscript at Earlham College; Robert Lifsey, who used the same version of the manuscript at Ohio University; Professor V. Klee, who, as an editorial consultant for Holt, Rinehart and Winston, offered many constructive suggestions along with comments which inspired the author to try harder to achieve his aims; and finally, the author wishes to express his gratitude to E. D. Goodrich, who, with painstaking care, has read all of the versions through which the manuscript evolved. Thanks are also due to Paula Smith, who did such a fine job of typing the final manuscript. E. B. G. Athens, Ohio February 1968