[Book Cover]

Roads to Geometry, 2/e

Edward C. Wallace, SUNY, Geneseo
Stephen F. West, SUNY, Geneseo

Published August, 1997 by Prentice Hall Engineering/Science/Mathematics

Copyright 1998, 447 pp.
ISBN 0-13-181652-7

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This book provides a geometrical experience that unifies a mostly Euclidean approach with various non-Euclidean views of the world. It offers the reader a “map” for a voyage through plane geometry and its various branches, as well as side-trips that discuss analytic and transformational geometry.


Takes an informal tone while presenting the material in a reasonably rigorous manner.
Organizes the material into a logical progression.

  • Chapters are separated into independent units so students can learn information in bites, while offering instructors the flexibility to cover information at their own discretion.
  • Contains a summary at the conclusion of each chapter that includes a list of new definitions and theorems to aid in the organization of the material.
  • A set of exercises at the end of each section includes both elementary and advanced problems linked to material covered in the presentation allowing for review and reinforcement.
Presents Euclidean and non-Euclidean geometries with a significant amount of background information, allowing the instructor to place much of the development in an historical context.
Offers a “tear-out” page that lists the SMSG axioms on the back cover so students do not have to turn to an appendix each time an axiom is invoked.
Places ample diagrams where appropriate to provide visual cues for the steps in the proofs.
NEW— Focuses a spotlight on key geometers throughout history so students obtain a feel for the progress in geometric thinking from Euclid through modern geometers such as Hilbert and Birkhoff.
NEW— Exercise sets have been expanded.
NEW— The book's overall readability has been improved.
NEW— Includes suggestions for explorations using graphics calculators and computers.

Table of Contents
    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.

      Introduction. Euclid's Geometry and Euclid's Elements. An Introduction to Modern Euclidean Geometries. Hilbert's Model for Euclidean Geometry. Birkhoff's Model for Euclidean Geometry. SMSG Postulates for Euclidean Geometry. Non-Euclidean Geometries.

    3. Traveling Together (Neutral Geometry).

      Introduction. Preliminary Notions. Congruence Conditions. The Place of Parallels. The Saccheri-Legendre 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. Euclidean Results Concerning Circles. Some Euclidean Results Concerning Triangles. More Euclidean Results Concerning Triangles. The Nine-Point Circle. Euclidean Constructions. Summary.

    5. Side Trips (Analytic and Transformational Geometry).

      Introduction. Analytic Geometry. Transformational Geometry. Analytic 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. Summary.

    7. All Roads Lead To . . . Projective Geometry.

      Introduction. The Real Projective Plane. Duality. Perspectivity. The Theorem of Desargues. Projective Transformations. Summary.

    Appendix A.

      Euclid's Definitions and Postulates Book I.

    Appendix B.

      Hilbert's Axioms for Euclidean Plane Geometry.

    Appendix C.

      Birkhoff's Postulates for Euclidean Plane Geometry.

    Appendix D.

      The SMSG Postulates for Euclidean Geometry.

    Appendix E.

      Geometer's SketchPad…• Scripts for Poincaré Model of Hyperbolic Geometry.



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