A First Course in Geometry

Edward T Walsh

• Publisher: Dover Publications
• Year: 2014
• Print ISBN: 9780486780207
• eBook ISBN: 9780486796680
• Topic: Mathematics
• Subtopic: Geometry
• Index: Mathematics

This introductory text is designed to help undergraduate students develop a solid foundation in geometry. Early chapters progress slowly, cultivating the necessary understanding and self-confidence for the more rapid development that follows. The extensive treatment can be easily adapted to accommodate shorter courses.
Starting with the language of mathematics as expressed in the algebra of logic and sets, the text covers geometric sets of points, separation and angles, triangles, parallel lines, similarity, polygons and area, circles, space geometry, and coordinate geometry. Each chapter includes a problem set arranged in order of increasing difficulty as well as review exercises and annotated references suggesting sources for further study. In addition to three helpful Appendixes, the book concludes with answers and hints for selected problems.

The essence of mathematics is its freedom.

G. CANTOR (1845–1918)

CHAPTER ONE

THE LANGUAGE OF MATHEMATICS

1.1 INTRODUCTION

If your intuition about physical space is fairly well developed, many of the ideas developed in this book will seem quite familiar to you. For although the geometry presented here is not the only way to think about space (more will be said about this in Chapter 5), it coincides with the notions most of us come to take for granted as we are growing up. On the other hand, if your geometric intuition is not very good, your formal study of geometry here should improve the situation.

In order that we may proceed with some confidence, our approach will be systematic. Specifically, we will employ what is often called “the axiomatic method.” This method was apparently invented by the Greeks of the fourth, fifth, and sixth centuries B.C. to securely systematize the geometry they were developing, in the face of some paradoxes they were unable to resolve.¹

The elements of the axiomatic method are few. We begin with some primitive or undefined terms, some definitions, and a set of non-contradictory assumptions called postulates or axioms.

We then endeavor to make plausible conjectures about relationships among objects of our study, employing the defined and undefined terms. Our attempts to establish the validity of these conjectures are referred to as proofs, and the valid conjectures are called theorems.

Since we want to discuss these conjectures with some measure of clarity and precision, it is obvious why we should first agree on the meanings of technical terms involved (definitions), as well as how these terms may be related to each other (axioms). But why must we also begin with some undefined terms? It would certainly be convenient to have a definition for every mathematical term that we encounter. But is this possible? Consider the following definitions from Webster’s Collegiate Dictionary

happening, n. Occurrence.

occurrence, n. Appearance or happening.

appearance, n. The action or an instance of appearing; an occurrence.

Note that these definitions haven’t really told us anything since each of the words is defined in terms of the others. Such sets of definitions are referred to as circular definitions, and although our trio is far less poetic, it is about as logically meaningful as Gertrude Stein’s “A rose is a rose is a rose . . .”

This problem of circularity arises whenever we attempt to define every term we use. To avoid this difficulty we will agree that certain terms will remain undefined, and, in particular, that definitions and other general statements (axioms, theorems, etc.) can be expressed using these primitive undefined terms. The meanings of the undefined terms will, however, become clearer as the subject unfolds.

In t...

Table of contents

• Cover
• Title Page
• Copyright Page
• Contents
• Preface to the Dover Edition
• Preface
• Errata
• Chapter 1 The Language of Mathematics
• Chapter 2 Geometric Sets Of Points: Lines, Planes, and Distance
• Chapter 3 Separation and Angles
• Chapter 4 Triangles
• Chapter 5 Parallel Lines
• Chapter 6 Similarity
• Chapter 7 Polygons and Area
• Chapter 8 Circles
• Chapter 9 Space Geometry
• Chapter 10  Coordinate Geometry
• Appendix A Properties of the Real Number System
• Appendix B Table of Squares and Square Roots
• Appendix C List of Postulates, Theorems, and Corollaries
• Answers and Hints for Selected Problems
• Index
• About the Author