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The Foundations of Geometry and the Non-Euclidean Plane

  • Textbook
  • © 1975

Overview

Authors:
  1. George E. Martin

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

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Part of the book series: Undergraduate Texts in Mathematics (UTM)

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Table of contents (34 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Introduction

    1. Front Matter

      Pages 1-1
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    2. Equivalence Relations

      • George E. Martin
      Pages 2-9

    3. Mappings

      • George E. Martin
      Pages 10-19

    4. The Real Numbers

      • George E. Martin
      Pages 20-33

    5. Axiom Systems

      • George E. Martin
      Pages 34-47

  3. Absolute Geometry

    1. Front Matter

      Pages 49-49
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    2. Models

      • George E. Martin
      Pages 50-64

    3. Incidence Axiom and Ruler Postulate

      • George E. Martin
      Pages 65-72

    4. Betweenness

      • George E. Martin
      Pages 73-83

    5. Segments, Rays, and Convex Sets

      • George E. Martin
      Pages 84-94

    6. Angles and Triangles

      • George E. Martin
      Pages 95-110

    7. The Golden Age of Greek Mathematics

      • George E. Martin
      Pages 111-120

    8. Euclid’s Elements

      • George E. Martin
      Pages 121-130

    9. Pasch’s Postulate and Plane Separation Postulate

      • George E. Martin
      Pages 131-143

    10. Crossbar and Quadrilaterals

      • George E. Martin
      Pages 144-154

    11. Measuring Angles and the Protractor Postulate

      • George E. Martin
      Pages 155-171

    12. Alternative Axiom Systems

      • George E. Martin
      Pages 172-181

    13. Mirrors

      • George E. Martin
      Pages 182-191

    14. Congruence and the Penultimate Postulate

      • George E. Martin
      Pages 192-203
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About this book

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.

Authors and Affiliations

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

    George E. Martin

Bibliographic Information

  • Book Title: The Foundations of Geometry and the Non-Euclidean Plane

  • Authors: George E. Martin

  • Series Title: Undergraduate Texts in Mathematics

  • DOI: https://doi.org/10.1007/978-1-4612-5725-7

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York, Inc. 1975

  • Hardcover ISBN: 978-0-387-90694-2Published: 22 March 1982

  • Softcover ISBN: 978-1-4612-5727-1Published: 06 December 2011

  • eBook ISBN: 978-1-4612-5725-7Published: 06 December 2012

  • Series ISSN: 0172-6056

  • Series E-ISSN: 2197-5604

  • Edition Number: 1

  • Number of Pages: XVI, 512

  • Additional Information: Originally published by Intext Educational Publishers, New York, 1972

  • Topics: Geometry

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