Now available from Waveland Press, the Third Edition of Roads to Geometry is appropriate for several kinds of students. Pre-service teachers of geometry are provided with a thorough yet accessible treatment of plane geometry in a historical context. Mathematics majors will find its axiomatic development sufficiently rigorous to provide a foundation for further study in the areas of Euclidean and non-Euclidean geometry. By using the SMSG postulate set as a basis for the development of plane geometry, the authors avoid the pitfalls of many "foundations of geometry" texts that encumber the reader with such a detailed development of preliminary results that many other substantive and elegant results are inaccessible in a one-semester course. At the end of each section is an ample collection of exercises of varying difficulty that provides problems that both extend and clarify results of that section, as well as problems that apply those results. At the end of chapters 3-7, a summary list of the new definitions and theorems of each chapter is included.
Not-for-sale instructor's resource materials available online to college and university faculty only; contact publisher directly.
Author(s): Edward C. Wallace, Stephen F. West
Edition: 3
Publisher: Waveland Press
Year: 2015
Language: English
Commentary: Publisher PDF | Published: October 22, 2015
Pages: 510
City: Long Grove, Illinois
Tags: Geometry; Axiomatic Systems; Geometry Axiom Sets; Plane Euclidean Geometry; Analytical Geometry; Transformational Geometry; Non-Euclidean Geometries; Projective Geometry
Title Page
Contents
Preface
Chapter 1: Rules of the Road: Axiomatic Systems
1.1 Historical Background
Thales of Miletus
Pythagoras
Hippocrates of Chios
Plato
Eudoxus
Euclid
Exercises 1.1
1.2 Axiomatic Systems and Their Properties
The Axiomatic Method
Models
Properties of Axiomatic Systems
Exercises 1.2
1.3 Finite Geometries
Four-Point Geometry
The Geometries of Fano and Young
Exercises 1.3
1.4 Axioms for Incidence Geometry
Exercises 1.4
Chapter 2: Many Ways to Go: Axiom Sets for Geometry
2.1 Introduction
2.2 Euclid's Geometry and Euclid's Elements
Exercises 2.2
2.3 Modern Euclidean Geometries
Exercises 2.3
2.4 Hilbert's Axioms for Euclidean Geometry
Exercises 2.4
2.5 Birkhoff's Axioms for Euclidean Geometry
Exercises 2.5
2.6 The SMSG Postulates for Euclidean Geometry
Exercises 2.6
2.7 Non-Euclidean Geometries
Exercises 2.7
Chapter 3: Traveling Together: Neutral Geometry
3.1 Introduction
3.2 Preliminary Notions
Exercises 3.2
3.3 Congruence Conditions
Exercises 3.3
3.4 The Place of Parallels
Exercises 3.4
3.5 The Saccheri-Legendre Theorem
Exercises 3.5
3.6 The Search for a Rectangle
Exercises 3.6
3.7 Chapter 3 Summary
Chapter 4: One Way to Go: Euclidean Geometry of the Plane
4.1 Introduction
4.2 The Parallel Postulate and Some Implications
Exercises 4.2
4.3 Congruence and Area
Exercises 4.3
4.4 Similarity
Exercises 4.4
4.5 Some Euclidean Results Concerning Circles
Exercises 4.5
4.6 Some Euclidean Results Concerning Triangles
Exercises 4.6
4.7 More Euclidean Results Concerning Triangles
Exercises 4.7
4.8 The Nine-Point Circle
Exercises 4.8
4.9 Euclidean Constructions
Exercises 4.9
4.10 Laboratory Activities Using Dynamic Geometry Software
Chapter 4, Laboratory 1
Chapter 4, Laboratory 2
Chapter 4, Laboratory 3
4.11 Chapter 4 Summary
Chapter 5: Side Trips: Analytical and Transformational Geometry
5.1 Introduction
5.2 Analytical Geometry
Historical Perspectives
Coordinatization of the Plane
Distance in the Plane
Analytical Equations of Straight Lines and Circles
Applications of Analytical Geometry
Exercises 5.2
5.3 Transformational Geometry
Introduction
Mappings and Transformations
Isometries
Applications of Isometries to Theorem Proving
Similarities and Their Applications to Theorem Proving
Exercises 5.3
5.4 Analytical Transformations
Introduction
Analytical Equations for Isometries
Analytical Equations for Similarities
Applications of Isometries and Similarities Using Analytical Transformations
Exercises 5.4
5.5 Inversion
Introduction
Inversion in a Circle
Exercises 5.5
5.6 Chapter 5 Summary
Chapter 6: Other Ways to Go: Non-Euclidean Geometries
6.1 Introduction
6.2 A Return to Neutral Geometry: The Angle of Parallelism
Exercises 6.2
6.3 The Hyperbolic Parallel Postulate
Exercises 6.3
6.4 Hyperbolic Results Concerning Polygons
Exercises 6.4
6.5 Area in Hyperbolic Geometry
Exercises 6.5
6.6 Showing Consistency: A Model for Hyperbolic Geometry
Exercises 6.6
6.7 Classifying Theorems
Exercises 6.7
6.8 Elliptic Geometry: A Geometry with No Parallels?
Two Models
Some Results in Elliptic Geometry
Exercises 6.8
6.9 Geometry in the Real World
6.10 Laboratory Activities Using Dynamic Geometry Software
Chapter 6, Laboratory 1
Chapter 6, Laboratory 2
Chapter 6, Laboratory 3
Chapter 6, Laboratory 4
Summary
6.11 Chapter 6 Summary
Chapter 7: All Roads Lead to . . . : Projective Geometry
7.1 Introduction
7.2 The Real Projective Plane
Exercises 7.2
7.3 Duality
Exercises 7.3
7.4 Perspectivity
Exercises 7.4
7.5 The Theorem of Desargues
Exercises 7.5
7.6 Projective Transformations
Exercises 7.6
7.7 Chapter 7 Summary
Appendix A: Projective Geometry
Appendix B
Appendix C
Appendix D
Appendix E
Bibliography
Index
The SMSG Postulates for Euclidean Geometry
