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The Foundations of Geometry and the Non-Euclidean Plane : Undergraduate Texts in Mathematics - G. E. Martin

The Foundations of Geometry and the Non-Euclidean Plane

Undergraduate Texts in Mathematics


Published: 19th December 1997
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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.

Foreword to the Student
Equivalence Relationsp. 2
Mappingsp. 10
The Real Numbersp. 20
Axiom Systemsp. 34
Absolute Geometry
Modelsp. 50
Incidence Axiom and Ruler Postulatep. 65
Betweennessp. 73
Segments, Rays, and Convex Setsp. 84
Angles and Trianglesp. 95
The Golden Age of Greek Mathematics (Optional)p. 111
Euclid's Elements (Optional)p. 121
Pasch's Postulate and Plane Separation Postulatep. 131
Crossbar and Quadrilateralsp. 144
Measuring Angles and the Protractor Postulatep. 155
Alternative Axiom Systems (Optional)p. 172
Mirrorsp. 182
Congruence and the Penultimate Postulatep. 192
Perpendiculars and Inequalitiesp. 204
Reflectionsp. 216
Circlesp. 226
Absolute Geometry and Saccheri Quadrilateralsp. 239
Saccheri's Three Hypothesesp. 255
Euclid's Parallel Postulatep. 269
Bianglesp. 292
Excursionsp. 317
Non-Euclidean Geometry
Parallels and the Ultimate Axiomp. 334
Brushes and Cyclesp. 347
Rotations, Translations, and Horolationsp. 360
The Classification of Isometriesp. 371
Symmetryp. 386
Horocirclesp. 402
The Fundamental Formulap. 421
Categoricalness and Areap. 444
Quadrature of the Circlep. 464
Hints and Answersp. 494
Notation Indexp. 503
Indexp. 504
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780387906942
ISBN-10: 0387906940
Series: Undergraduate Texts in Mathematics
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 512
Published: 19th December 1997
Publisher: Springer-Verlag New York Inc.
Country of Publication: US
Dimensions (cm): 23.4 x 15.6  x 3.18
Weight (kg): 0.91
Edition Number: 3