[Book Cover]

Foundations of Plane Geometry, 1/e

Harvey I. Blau, Northern Illinois University
John E. Wetzel, University of Illinois, Urbana

Coming February, 2000 by Prentice Hall Engineering/Science/Mathematics

Copyright 2000, 420 pp.
Cloth
ISBN 0-13-591405-1


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Summary

For junior/senior level courses in Geometry. Ideal for students who may have little previous experience with abstraction and proof, this text provides a rigorous and unified—yet straightforward and accessible—exposition of the foundations of Euclidean, hyperbolic, and spherical geometry. Unique in approach, it combines an extended theme—the study of a generalized absolute plane from axioms through classification into the three fundamental classical planes—with a leisurely development that allows ample time for students' mathematical growth. It is purposefully structured to facilitate the development of analytic and reasoning skills and to promote an awareness of the depth, power, and subtlety of the axiomatic method in general, and of Euclidean and non-Euclidean plane geometry in particular.

Features


Focuses on one main topic—The axiomatic development of the absolute plane—which is pursued through a classification into Euclidean, hyperbolic, and spherical planes.
The theme of simultaneous study of different types of Plane geometry—Followed throughout the book.
Presents specific models early (including the sphere, the Klein-Betrami hyperbolic model, and the “gap” plane in Chapter 1) and cites them often.
—This gives meaning and depth to concepts such as betweenness and separation which can be dismissed as all too obvious in a strictly Euclidean context.
Presents the axioms for absolute plane geometry gradually (Chapters 5-13).

  • Students are gently introduced to axiomatic development without interrupting the mathematical progress of the main topic.
Forces students to consider familiar words in a new light.
Defines concepts as soon as they make sense in the general development, but not necessarily before all axioms which relate to them have been introduced.
—This often leads to some thought-provoking examples—e.g., when “segment” is defined in Chapter 6, an example (the “Inside-Out” model) and a result (Proposition 6.4) show that at this stage, “length” is an invariant of a segment but “endpoints” are not. Unique approach to the standard sets of axioms—Facilitates incremental learning, e.g.:
—The “Betweenness” and “Quadrichotomy” axioms for points are simply stated assumptions which permit the development of much of the theory of betweenness for collinear points even though there are many non-standard (even finite) models which satisfy these postulates.
—The relatively natural and terse “Real Ray Axiom” allows the completion of the theory of betweenness of points.
—Most of the axioms for coterminal rays are formulated as exact analogs of previous axioms for collinear points. This analogy (“duality”) is invoked to instantly establish many properties of coterminal rays. The articulation of these properties reinforces understanding of the previous results about collinear points.
Original treatments of most topics—e.g.:
—The only polygons used in the development are triangles. The notions of rectangles and Saccheri quadrilaterals are confined to exercises. This leads to a novel proof of the theorem on angle sums of triangles in spherical planes (18.1).
—The separate introduction of halfplanes in Chapter 10 and fans of coterminal rays in Chapter 11 and the proof of their relationship in Theorem 12.2. The Crossbar Theorem (12.4) is then an almost immediate consequence. States and discusses the Ruler and Protractor Axioms (commonly used in secondary school geometry texts) as theoremsAfter all the axioms have been introduced (Chapter 13) — and gives their proofs as structured exercises.
Careful but not overly wordy explanations throughout the text.
—All proofs are written in detail in paragraph style.
Informal explanations—and references to specific models—are interspersed frequently.
—Figures
illustrate virtually all of the definitions and the key points of most proofs.
Informal chapter on logic (Ch. 2)—Addresses students' most common blunders and misconceptions and introduces basic ideas about proofs.
  • Ch. 3 uses logical puzzles as a non-intimidating way of gaining practice in creating and writing proofs.
10 to 20 exercises varied in length and difficulty, at the end of each chapter.
—Some call for proofs of results which are stated in the book without proof; some are fairly immediate applications of the theorems in the text; some involve longer chains of reasoning and often include hints; and others develop concepts which are omitted from the text such as Saccheri quadrilaterals.
—A simple type of exercise—effective in developing comprehension and articulation—asks students to rewrite the statement of a particular axiom or theorem entirely in words without the use of symbols or letters-as-symbols.


Table of Contents
    0. The Question of Parallels.
    1. Five Examples.
    2. Some Logic.
    3. Practice Proofs.
    4. Set Terminology and Sets of Real Numbers.
    5. An Axiom System for Plane Geometry: First Steps.
    6. Betweenness, Segments and Rays.
    7. Three Axioms for the Line.
    8. The Real Ray Axiom and Its Consequences.
    9. Antipodes and Opposite Rays.
    10. Separation.
    11. Pencils and Angles.
    12. The Crossbar Theorem.
    13. Side-Angle-Side.
    14. Perpendiculars.
    15. The Exterior Angle Inequality and Triangle Inequality.
    16. Further Inequalities Concerning Triangles.
    17. Parallels and the Diameter of the Plane.
    18. Angle Sums of Triangles.
    19. Parallels and Angle Sums.
    20. Concurrence Theorems.
    21. Circles.
    22. Similarity.
    Bibliography.
    Appendix I. Definitions and Assumptions from Book I of Euclid's Elements.
    Appendix II. The Side-Angle-Side Axiom in the Hyperbolic Plane.
    Index.


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