The Foundations of Geometry and the Non-Euclidean Plane / Edition 1

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Overview

The Foundations of Geometry and the Non-Euclidean Plane is a self-contained text for junior, senior, and first-year graduate courses. Historical material is interwoven with a rigorous ruler- and protractor axiomatic development of the Euclidean and hyperbolic planes. Additional topics include the classical axiomatic systems of Euclid and Hilbert, axiom systems for three and four dimensional absolute geometry, and Pieri's system based on rigid motions. Models, such as Taxicab Geometry, are used extensively to illustrate theory.

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Product Details

  • ISBN-13: 9780387906942
  • Publisher: Springer New York
  • Publication date: 3/22/1982
  • Series: Undergraduate Texts in Mathematics Series
  • Edition description: 1st ed. 1982. Corr. 4th printing 1997
  • Edition number: 1
  • Pages: 509
  • Product dimensions: 9.21 (w) x 6.14 (h) x 1.19 (d)

Table of Contents

1. Equivalence Relations.- 1.1 Logic.- 1.2 Sets.- 1.3 Relations.- 1.4 Exercises.- Graffiti.- 2 Mappings.- 2.1 One-to-One and Onto.- 2.2 Composition of Mappings.- 2.3 Exercises.- Graffiti.- 3 The Real Numbers.- 3.1 Binary Operations.- 3.2 Properties of the Reals.- 3.3 Exercises.- Graffiti.- 4 Axiom Systems.- 4.1 Axiom Systems.- 4.2 Incidence Planes.- 4.3 Exercises.- Graffiti.- One Absolute Geometry.- 5 Models.- 5.1 Models of the Euclidean Plane.- 5.2 Models of Incidence Planes.- 5.3 Exercises.- Graffiti.- 6 Incidence Axiom and Ruler Postulate.- 6.1 Our Objectives.- 6.2 Axiom 1: The Incidence Axiom.- 6.3 Axiom 2: The Ruler Postulate.- 6.4 Exercises.- Graffiti.- 7 Betweenness.- 7.1 Ordering the Points on a Line.- 7.2 Taxicab Geometry.- 7.3 Exercises.- Graffiti.- 8 Segments, Rays, and Convex Sets.- 8.1 Segments and Rays.- 8.2 Convex Sets.- 8.3 Exercises.- Graffiti.- 9 Angles and Triangles.- 9.1 Angles and Triangles.- 9.2 More Models.- 9.3 Exercises.- Graffiti.- 10 The Golden Age of Greek Mathematics (Optional).- 10.1 Alexandria.- 10.2 Exercises.- 11 Euclid’S Elements (Optional).- 11.1 The Elements.- 11.2 Exercises.- Graffiti.- 12 Pasch’s Postulate and Plane Separation Postulate.- 12.1 Axiom 3: PSP.- 12.2 Pasch, Peano, Pieri, and Hilbert.- 12.3 Exercises.- Graffiti.- 13 Crossbar and Quadrilaterals.- 13.1 More Incidence Theorems.- 13.2 Quadrilaterals.- 13.3 Exercises.- Graffiti.- 14 Measuring Angles and the Protractor Postulate.- 14.1 Axiom 4: The Protractor Postulate.- 14.2 Peculiar Protractors.- 14.3 Exercises.- 15 Alternative Axiom Systems (Optional).- 15.1 Hilbert’s Axioms.- 15.2 Pieri’s Postulates.- 15.3 Exercises.- 16 Mirrors.- 16.1 Rulers and Protractors.- 16.2 MIRROR and SAS.- 16.3 Exercises.- Graffiti.- 17 Congruence and the Penultimate Postulate.- 17.1 Congruence for Triangles.- 17.2 Axiom 5: SAS.- 17.3 Congruence Theorems.- 17.4 Exercises.- Graffiti.- 18 Perpendiculars and Inequalities.- 18.1 A Theorem on Parallels.- 18.2 Inequalities.- 18.3 Right Triangles.- 18.4 Exercises.- Graffiti.- 19 Reflections.- 19.1 Introducing Isometries.- 19.2 Reflection in a Line.- 19.3 Exercises.- Graffiti.- 20 Circles.- 20.1 Introducing Circles.- 20.2 The Two-Circle Theorem.- 20.3 Exercises.- Graffiti.- 21 Absolute Geometry and Saccheri Quadrilaterals.- 21.1 Euclid’s Absolute Geometry.- 21.2 Giordano’s Theorem.- 21.3 Exercises.- Graffiti.- 22 Saccherfs Three Hypotheses.- 22.1 Omar Khayyam’s Theorem.- 22.2 Saccheri’s Theorem.- 22.3 Exercises.- Graffiti.- 23 Euclid’s Parallel Postulate.- 23.1 Equivalent Statements.- 23.2 Independence.- 23.3 Exercises.- Graffiti.- 24 Biangles.- 24.1 Closed Biangles.- 24.2 Critical Angles and Absolute Lengths.- 24.3 The Invention of Non-Euclidean Geometry.- 24.4 Exercises.- Graffiti.- 25 Excursions.- 25.1 Prospectus.- 25.2 Euclidean Geometry.- 25.3 Higher Dimensions.- 25.4 Exercises.- Graffiti.- Two Non-Euclidean Geometry.- 26 Parallels and the Ultimate Axiom.- 26.1 Axiom 6: HPP.- 26.2 Parallel Lines.- 26.3 Exercises.- Graffiti.- 27 Brushes and Cycles.- 27.1 Brushes.- 27.2 Cycles.- 27.3 Exercises.- Graffiti.- 28 Rotations, Translations, and Horolations.- 28.1 Products of Two Reflections.- 28.2 Reflections in Lines of a Brush.- 28.3 Exercises.- Graffiti.- 29 The Classification of Isometries.- 29.1 Involutions.- 29.2 The Classification Theorem.- 29.3 Exercises.- Graffiti.- 30 Symmetry.- 30.1 Leonardo’s Theorem.- 30.2 Frieze Patterns.- 30.3 Exercises.- Graffiti.- 31 HOrocircles.- 31.1 Length of Arc.- 31.2 Hyperbolic Functions.- 31.3 Exercises.- Graffiti.- 32 The Fundamental Formula.- 32.1 Trigonometry.- 32.2 Complementary Segments.- 32.3 Exercises.- Graffiti.- 33 Categoricalness and Area.- 33.1 Analytic Geometry.- 33.2 Area.- 33.3 Exercises.- Graffiti.- 34 Quadrature of the Circle.- 34.1 Classical Theorems.- 34.2 Calculus.- 34.3 Constructions.- 34.4 Exercises.- Hints and Answers.- Notation Index.

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