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.
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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 unifiedyet
straightforward and accessibleexposition of the foundations
of Euclidean, hyperbolic, and spherical geometry. Unique in approach,
it combines an extended themethe study of a generalized absolute
plane from axioms through classification into the three fundamental
classical planeswith 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
Focuses on one main topicThe axiomatic development
of the absolute planewhich is pursued through a classification
into Euclidean, hyperbolic, and spherical planes.
The theme of simultaneous study of different types of
Plane geometryFollowed 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
Presents the axioms for absolute plane geometry gradually
Forces students to consider familiar words in a new light.
- Students are gently introduced to axiomatic development
without interrupting the mathematical progress of the main
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 examplese.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 axiomsFacilitates
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
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 topicse.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
All proofs are written in detail in paragraph style.
Informal explanationsand references to specific modelsare
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
10 to 20 exercises varied in length and difficulty, at
the end of each chapter.
- Ch. 3 uses logical puzzles as a non-intimidating way
of gaining practice in creating and writing proofs.
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 exerciseeffective in developing comprehension
and articulationasks students to rewrite the statement of a particular
axiom or theorem entirely in words without the use of symbols or letters-as-symbols.
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.
11. Pencils and Angles.
12. The Crossbar Theorem.
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.
Appendix I. Definitions and Assumptions from Book I of Euclid's
Appendix II. The Side-Angle-Side Axiom in the Hyperbolic