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.
Author(s): Ellery B. Golos
Edition: 1
Publisher: HOLT, RINEHART and WINSTON
Year: 1968
Language: English
Pages: C, XIV,225
Part 1 An Analysis of Axiomatic Systems
Introduction
History
Objectives
1 Ingredients and Tools
1.1 Definitions and Undefinitions
EXERCISES 1.1
1.2 Axioms
EXERCISES 1.2
1.3 Logic
EXERCISES 1.3
1.4 Sets
EXERCISES 1.4
2 Finite Geometries
2.1 Axiom Sets I and 2
2.2 An Axiomatic System, 3
EXERCISES 2.2
2.3 Direct and Indirect Proofs
2.4 Further Proofs in Axiomatic System 3
EXERCISES 2.4
2.5 Axiomatic System I
EXERCISES 2.5
2.6 The Systems of Young and Fano
3 Properties of Axiomatic Systems
3.1 Consistency
3.2 Models for Consistency
3.3 Independence
EXERCISES 3.3·
3.4 Completeness
3.5 Examples of Isomorphisms
EXERCISES 3.5
REVIEW EXERCISES
4 A Critique of Euclid
4.1 Tacit, or Unstated, Assumptions
4.2 More Unstated Assumptions, Flaws, and Omissions
4.3 The Danger in Diagrams
4.4 New Systems
References
Part 2 An Axiomatic Development of Elementary Geometry
Introduction
5 The Foundations of Geometry
5.1 Properties of Incidence and Existence
EXERCISES 5.1
5.2 An Order Relation
EXERCISES 5.2
5.3 Segments
EXERCISES 5.3
5.4 The Axiom of Pasch
EXERCISES 5.4
5.5 Convex Sets
EXERCISES 5.5
5.6 Interior and Exterior
EXERCISES 5.6
5.7 About Angles and Rays
EXERCISES 5.7
5.8 Convex Quadrilaterals
EXERCISES 5.8
ADDITIONAL EXERCISES
6 Congruence and Comparison
6.1 Axionl.s of Congruence for Segments
EXERCISES 6.1
6.2 Comparison of Segments
6.3 Congruences for Angles and Triangles
EXERCISES 6.3
6.4 Angle Addition and Subtraction
EXERCISES 6.4
6.5 Comparison of Angles
EXERCISES 6.5
7 Elementary Geometry
7.1 Euclid's Theorems Reproved
EXERCISES 7.1
7.2 The Exterior Angle Theorem
EXERCISES 7.2
7.3 Midpoints and Halves
EXERCISES 7.3
7.4 Right Angles and Non-right Angles
EXERCISES 7.4
7 .5 Constructions
EXERCISES 7.5
8 Synthetic Approach to a Metric Problem
8.1 Difficulties in Some Simple Theorems
8.2 Measure
8.3 Addition of Line Segments
8.4 Addition of Angles
EXERCISES 8.4
8.5 Epilogue
References
Part 3 Non-Euclidean Geometry
Introduction
9 The Concept of Parallelism
9.1 History
EXERCISES 9.1
9.2 The Geometry of Bolyai-Lohachevsky
EXERCISES 9.2
9.3 A New Kind of Triangle
EXERCISES 9.3
10 New Shapes
10.1 Absolute Geometry
EXERCISES 10.1
10.2 New Quadrilaterals
EXERCISES 10.2
10.3 Congruent Quadrilaterals
EXERCISES 10.3
10.4 A Comparison of Hyperbolic and Euclidean Properties
EXERCISES 10.4
11 A Model World
11.1 A Model
11.2 A Model World
11.3 Is Euclidean Geometry True?
References
Appendix Euclid´s Axioms and Common Notions and the Statements of Book I of Elements
The Definitions of Book I
The Postulates
Co1nmon Notions
The Forty-Eight Propositions of Book I
Index
