"The newest version is typo-free. It is a great book for students. I have already assigned it." – Ioannis Argyros, Cameron University
"This is the only undergraduate text I know which has reasonable coverage of both hyperbolic and elliptic geometry, along with Euclidean geometry. I am also delighted that it is now available at a lower cost." — Tevian Dray, Oregon State University
510 pages, $69.95 list
1-4786-3163-5
978-1-4786-3163-7
© 2004
paperback
Instructor's resource materials available here
To obtain a username and password to access these materials, contact comps@waveland.com.
To obtain a username and password to access these materials, contact comps@waveland.com.
eBook availability
Roads to Geometry
Third Edition
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.
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.
Reactions
1. Rules of the Road: Axiomatic Systems
Historical Background / Axiomatic Systems and Their Properties / Finite Geometries / Axioms for Incidence Geometry
2. Many Ways to Go: Axiom Sets for Geometry
Introduction / Euclid's Geometry and Euclid's Elements / Modern Euclidean Geometry / Hilbert's Axioms for Euclidean Geometry / Birkhoff's Axioms for Euclidean Geometry / The SMSG Postulates for Euclidean Geometry / Non-Euclidean Geometry
3. Traveling Together: Neutral Geometry
Introduction / Preliminary Notions / Congruence Conditions / The Place of Parallels / The Saccheri-Lengendre Theorem / The Search for a Rectangle / Summary
4. One Way to Go: Euclidean Geometry of the Plane
Introduction / The Parallel Postulate and Some Implications / Congruence and Area / Similarity / Some Euclidean Results Concerning Circles / Some Euclidean Results Concerning Triangles / More Euclidean Results Concerning Triangles / The Nine-Point Circle / Euclidean Constructions / Laboratory Activities Using Dynamic Geometry Software / Summary
5. Side Trips: Analytical and Transformational Geometry
Introduction / Analytical Geometry / Transformational Geometry / Analytical Transformations / Inversion / Summary
6. Other Ways to Go: Non-Euclidean Geometries
Introduction / A Return to Neutral Geometry: The Angle of Parallelism / The Hyperbolic Parallel Postulate / Hyperbolic Results Concerning Polygons / Area in Hyperbolic Geometry / Showing Consistency: A Model for Hyperbolic Geometry / Classifying Theorems / Elliptic Geometry: A Geometry with No Parallels? / Geometry in the Real World / Laboratory Activities Using Dynamic Geometry Software / Summary
7. All Roads Leads to . . . : Projective Geometry
Introduction / The Real Projective Plane / Duality / Perspectivity / The Theorem of Desargues / Projective Transformations / Summary
Appendix A: Euclid's Definitions, Postulates, and First Ten Propositions, Book I
Appendix B: Hilbert's Axioms for Euclidean Plane Geometry
Appendix C: Birkhoff's Postulates for Euclidean Plane Geometry / Undefined Terms and Relations
Appendix D: The SMSG Postulates for Euclidean Geometry
Appendix E: Using Dynamic Geometry Software to Explore the Poincare Model of Hyperbolic Geometry / Shortcuts for The Geometer's SketchPad / Cabri Macros for the Poincare Model of Hyperbolic Geometry
Historical Background / Axiomatic Systems and Their Properties / Finite Geometries / Axioms for Incidence Geometry
2. Many Ways to Go: Axiom Sets for Geometry
Introduction / Euclid's Geometry and Euclid's Elements / Modern Euclidean Geometry / Hilbert's Axioms for Euclidean Geometry / Birkhoff's Axioms for Euclidean Geometry / The SMSG Postulates for Euclidean Geometry / Non-Euclidean Geometry
3. Traveling Together: Neutral Geometry
Introduction / Preliminary Notions / Congruence Conditions / The Place of Parallels / The Saccheri-Lengendre Theorem / The Search for a Rectangle / Summary
4. One Way to Go: Euclidean Geometry of the Plane
Introduction / The Parallel Postulate and Some Implications / Congruence and Area / Similarity / Some Euclidean Results Concerning Circles / Some Euclidean Results Concerning Triangles / More Euclidean Results Concerning Triangles / The Nine-Point Circle / Euclidean Constructions / Laboratory Activities Using Dynamic Geometry Software / Summary
5. Side Trips: Analytical and Transformational Geometry
Introduction / Analytical Geometry / Transformational Geometry / Analytical Transformations / Inversion / Summary
6. Other Ways to Go: Non-Euclidean Geometries
Introduction / A Return to Neutral Geometry: The Angle of Parallelism / The Hyperbolic Parallel Postulate / Hyperbolic Results Concerning Polygons / Area in Hyperbolic Geometry / Showing Consistency: A Model for Hyperbolic Geometry / Classifying Theorems / Elliptic Geometry: A Geometry with No Parallels? / Geometry in the Real World / Laboratory Activities Using Dynamic Geometry Software / Summary
7. All Roads Leads to . . . : Projective Geometry
Introduction / The Real Projective Plane / Duality / Perspectivity / The Theorem of Desargues / Projective Transformations / Summary
Appendix A: Euclid's Definitions, Postulates, and First Ten Propositions, Book I
Appendix B: Hilbert's Axioms for Euclidean Plane Geometry
Appendix C: Birkhoff's Postulates for Euclidean Plane Geometry / Undefined Terms and Relations
Appendix D: The SMSG Postulates for Euclidean Geometry
Appendix E: Using Dynamic Geometry Software to Explore the Poincare Model of Hyperbolic Geometry / Shortcuts for The Geometer's SketchPad / Cabri Macros for the Poincare Model of Hyperbolic Geometry