The Foundations of Geometry and the Non-Euclidean Plane

1. Absolute Geometry

   1. Perpendiculars and Inequalities

      • George E. Martin

      Pages 204-215

   2. Reflections

      • George E. Martin

      Pages 216-225

   3. Circles

      • George E. Martin

      Pages 226-238

   4. Absolute Geometry and Saccheri Quadrilaterals

      • George E. Martin

      Pages 239-254

   5. Saccheri’s Three Hypotheses

      • George E. Martin

      Pages 255-268

   6. Euclid’s Parallel Postulate

      • George E. Martin

      Pages 269-291

   7. Biangles

      • George E. Martin

      Pages 292-316

   8. Excursions

      • George E. Martin

      Pages 317-331

2. Non-Euclidean Geometry

   1. Front Matter

      Pages 333-333

      PDF

   2. Parallels and the Ultimate Axiom

      • George E. Martin

      Pages 334-346

   3. Brushes and Cycles

      • George E. Martin

      Pages 347-359

   4. Rotations, Translations, and Horolations

      • George E. Martin

      Pages 360-370

   5. The Classification of Isometries

      • George E. Martin

      Pages 371-385

   6. Symmetry

      • George E. Martin

      Pages 386-401

   7. Horocircles

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      Pages 402-420

   8. The Fundamental Formula

      • George E. Martin

      Pages 421-443

   9. Categoricalness and Area

      • George E. Martin

      Pages 444-463

   10. Quadrature of the Circle

       • George E. Martin

       Pages 464-493

3. Back Matter

   Pages 494-512

   PDF

This book is a text for junior, senior, or first-year graduate courses traditionally titled Foundations of Geometry and/or Non­ Euclidean Geometry. The first 29 chapters are for a semester or year course on the foundations of geometry. The remaining chap­ ters may then be used for either a regular course or independent study courses. Another possibility, which is also especially suited for in-service teachers of high school geometry, is to survey the the fundamentals of absolute geometry (Chapters 1 -20) very quickly and begin earnest study with the theory of parallels and isometries (Chapters 21 -30). The text is self-contained, except that the elementary calculus is assumed for some parts of the material on advanced hyperbolic geometry (Chapters 31 -34). There are over 650 exercises, 30 of which are 10-part true-or-false questions. A rigorous ruler-and-protractor axiomatic development of the Euclidean and hyperbolic planes, including the classification of the isometries of these planes, is balanced by the discussion about this development. Models, such as Taxicab Geometry, are used exten­ sively to illustrate theory. Historical aspects and alternatives to the selected axioms are prominent. The classical axiom systems of Euclid and Hilbert are discussed, as are axiom systems for three­ and four-dimensional absolute geometry and Pieri's system based on rigid motions. The text is divided into three parts. The Introduction (Chapters 1 -4) is to be read as quickly as possible and then used for ref­ erence if necessary.

• Absolute Geometrie
• Area
• Congruence
• Geometry
• Martin
• Nichteuklidische Geometrie
• Non-Euclidean Geometry
• Plane

• Department of Mathematics and Statistics, State University of New York at Albany, Albany, USA

  George E. Martin

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