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Euclidean Geometry and its Subgeometries

  • Book
  • © 2015

Overview

Authors:
  1. Edward John Specht

    1. Indiana University South Bend, SOUTH BEND, USA

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  2. Harold Trainer Jones

    1. Andrews University, Berrien Springs, USA

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  3. Keith G. Calkins

    1. Andrews University, Berrien Springs, USA

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  4. Donald H. Rhoads

    1. Andrews University, Bloomington, USA

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  • Provides a complete and rigorous axiomatic treatment of Euclidean geometry.

  • Proofs for many theorems are worked out in detail.

  • Takes a modern approach by replacing congruence axioms with a transformational definition of congruence

  • Includes supplementary material: sn.pub/extras

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xix
    Download chapter PDF

  2. Preliminaries and Incidence Geometry (I)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 1-35

  3. Affine Geometry: Incidence with Parallelism (IP)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 37-44

  4. Collineations of an Affine Plane (CAP)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 45-61

  5. Incidence and Betweenness (IB)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 63-77

  6. Pasch Geometry (PSH)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 79-138

  7. Ordering a Line in a Pasch Plane (ORD)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 139-148

  8. Collineations Preserving Betweenness (COBE)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 149-154

  9. Neutral Geometry (NEUT)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 155-224

  10. Free Segments of a Neutral Plane (FSEG)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 225-233

  11. Rotations About a Point of a Neutral Plane (ROT)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 235-249

  12. Euclidean Geometry Basics (EUC)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 251-264

  13. Isometries of a Euclidean Plane (ISM)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 265-279

  14. Dilations of a Euclidean Plane (DLN)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 281-304

  15. Every Line in a Euclidean Plane Is an Ordered Field (OF)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 305-317

  16. Similarity on a Euclidean Plane (SIM)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 319-333

  17. Axial Affinities of a Euclidean Plane (AX)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 335-345

  18. Rational Points on a Line (QX)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 347-359

  19. A Line as Real Numbers (REAL); Coordinatization of a Plane (RR)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 361-390

  20. Belineations on a Euclidean/LUB Plane (AA)

    • Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads
    Pages 391-400
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Keywords

About this book

In this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert. This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving metric notions and that explores congruence and isometries by means of reflection mappings. The authors present thirteen axioms in sequence, proving as many theorems as possible at each stage and, in the process, building up subgeometries, most notably the Pasch and neutral geometries. Standard topics such as the congruence theorems for triangles, embedding the real numbers in a line, and coordinatization of the plane are included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The final chapter covers consistency and independence of axioms, as well as independence of definition properties.
There are over 300 exercises; solutions to many of these, including all that are needed for this development, are available online at the homepage for the book at www.springer.com. Supplementary material is available online covering construction of complex numbers, arc length, the circular functions, angle measure, and the polygonal form of the Jordan Curve theorem.
Euclidean Geometry and Its Subgeometries is intended for advanced students and mature mathematicians, but the proofs are thoroughly worked out to make it accessible to undergraduate students as well. It can be regarded as a completion, updating, and expansion of Hilbert's work, filling a gap in the existing literature.

Reviews

“This is the most detailed undergraduate textbook on the axiomatic foundation of Euclidean geometry ever written.” (Victor V. Pambuccian, Mathematical Reviews, July, 2016)

“The authors do a commendable job of writing out proofs in detail and attempting to make the text accessible to undergraduates. … It makes a very useful reference source, and … there aren’t very many current textbooks that discuss geometry from this particular point of view. I commend this book to the attention of instructors with an interest in the foundations of geometry, and to university librarians.” (Mark Hunacek, MAA Reviews, maa.org, March, 2016)

Authors and Affiliations

  • Indiana University South Bend, SOUTH BEND, USA

    Edward John Specht

  • Andrews University, Berrien Springs, USA

    Harold Trainer Jones, Keith G. Calkins

  • Andrews University, Bloomington, USA

    Donald H. Rhoads

Bibliographic Information

  • Book Title: Euclidean Geometry and its Subgeometries

  • Authors: Edward John Specht, Harold Trainer Jones, Keith G. Calkins, Donald H. Rhoads

  • DOI: https://doi.org/10.1007/978-3-319-23775-6

  • Publisher: Birkhäuser Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer International Publishing Switzerland 2015

  • Hardcover ISBN: 978-3-319-23774-9Published: 12 January 2016

  • Softcover ISBN: 978-3-319-79533-1Published: 30 March 2018

  • eBook ISBN: 978-3-319-23775-6Published: 31 December 2015

  • Edition Number: 1

  • Number of Pages: XIX, 527

  • Number of Illustrations: 59 b/w illustrations

  • Topics: Geometry, History of Mathematical Sciences

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