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
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.
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. 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.
Euclid's Definitions and Postulates Book I.
Hilbert's Axioms for Euclidean Plane Geometry.
Birkhoff's Postulates for Euclidean Plane Geometry.
The SMSG Postulates for Euclidean Geometry.
Geometer's SketchPad Scripts for Poincaré Model of Hyperbolic Geometry.