Table Of ContentA MODERN VIEW OF GEOMETRY
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A MODERN VIEW OF
GEOMETRY
Leonard M. Blumenthal
DOVER PUBLICATIONS, INC.
Mineola, New York
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Copyright
Copyright © 1961, 1989 by Leonard M. Blumenthal
All rights reserved.
Bibliographical Note
This Dover edition, first published in 1980 and reissued in 2017, is an
unabridged and corrected republication of the work originally published in 1961
by W. H. Freeman and Company, San Francisco.
Drawings by Evan Gillespie.
International Standard Book Number
ISBN-13: 978-0-486-63962-8
ISBN-10: 0-486-63962-2
Library of Congress Catalog Card Number: 79–56332
Manufactured in the United States by LSC Communications
63962205 2017
www.doverpublications.com
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Preface
T
HE TITLE of this book may recall to the reader a celebrated
work of three volumes written by the German mathematician,
Felix Klein, and published almost exactly fifty years ago. The
second volume of Klein’s Elementare Mathematik vom Höhere
Standpunkt aus is devoted to geometry, but the “modern”
viewpoint from which we shall examine certain parts of
geometry has almost nothing in common with the “higher
standpoint” from which that distinguished writer surveyed the
subject.
The abstract, postulational, method which has now permeated
nearly all parts of mathematics makes it difficult, if not
meaningless, to mark the boundary of that mathematical
province which is called geometry. There is wisdom as well as wit
in saying that, geometry is the mathematics that a geometer
does, for today perhaps geometry more properly describes a
point of view—a way of looking at a subject—than it denotes
any one part of mathematics.
Our principal concern is with the postulational geometry of
planes, with the greatest emphasis on the coordinatization of
affine and projective planes. It is shown in detail how the
algebraic structure of an abstract “coordinate” set is determined
by the geometric structure that postulates impose on an abstract
“point” set. It is seen that the process can be reversed: a
“geometric” entity (plane) arises from an initially given
“algebraic” entity (ternary ring), and the geometric properties of
the one are logical consequences of the algebraic properties of
the other. The essential unity of algebra and geometry is thus
made quite clear. The developments of Chapters IV, V, and VI
amount to a study of ordinary analytic geometry from a “higher
standpoint,” and should be very illuminating to the thoughtful
student.
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Chapter VII contains a detailed development of a simple set of
postulates for the euclidean plane, and Chapter VIII gives
postulates for the non-euclidean planes.
It is believed that much of the book’s content is accessible to
college students and to high school teachers of mathematics, and
that the first three chapters (at least) could be read with profit
by the celebrated man in the street. A postulational approach to
a subject usually makes very slight demands on the reader’s
technical knowledge, but it is likely to compensate for this by
exacting from him close attention and a genuine desire to learn.
It is hoped that this book might be useful in National Science
Foundation Summer Institutes. This is, indeed, an expanded
version of a course given by the author at such an Institute, held
at the University of Wyoming during the summer of 1959. If the
book is used as a text (or for supplementary reading) in a
graduate course devoted to modern postulational geometry, the
first three chapters might well be omitted. On the other hand,
Chapters I–V and Chapter VII constitute more than enough
material for an enlightening course in a summer institute for
high school teachers of mathematics.
The material of Chapters IV–VIII has been developed quite
recently. Though no attempt is made to credit the source of each
theorem, the author’s indebtedness to R. H. Brack’s excellent
expository article, “Recent Advances in the Foundations of
Euclidean Plane Geometry” (American Mathematical Monthly,
vol. 62, No. 7, 1955, pp. 2–24), is apparent in Chapters IV and V.
Chapter VI makes considerable use of L. A. Skornyakov’s
monograph, “Projective Planes” (American Mathematical
Society, Translation Number 99, 1953). Much of the material in
both of these articles has its source in the pioneering work of
Marshall Hall’s “Projective Planes” (Transactions American
Mathematical Society, vol. 54, 1943, pp. 229–277).
The content of Chapters VII and VIII stems from some
contributions made by the author to distance geometry. A
detailed account of this subject is given in his Theory and
Applications of Distance Geometry, Clarendon, 1953.
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The author wishes to record his deep appreciation of Mrs. Kay
Hunt’s excellent typing of the manuscript.
January 1961
Leonard M. Blumenthal
University of Missouri-Columbia
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Contents
CHAPTER I
Historical Development of the Modern View
1. The Rise of Postulational Geometry. Euclid’s Elements
2. Some Comments on Euclid’s System
3. The Fifth Postulate
4. Saccheri’s Contribution
5. Substitutes for the Fifth Postulate
6. Non-euclidean Geometry—a Nineteenth Century
Revolution
7. Later Developments
8. The Role of Non-euclidean Geometry in the Development
of Mathematics
CHAPTER II
Sets and Propositions
1. Abstract Sets
2. The Russell Paradox
3. Operations on Sets
4. One-to-one Correspondence. Cardinal Number
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5. Finite and Infinite Sets. The Trichotomy Theorem and the
Axiom of Choice
6. Propositions
7. Truth Tables
8. Forms of Argumentation
9. Deductive Theory
CHAPTER III
Postulational Systems
1. Undefined Terms and Unproved Propositions
2. Consistency, Independence, and Completeness of a
Postulational System
3. The Postulational System 7
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4. A Finite Affine Geometry
5. Hilbert’s Postulates for Three-dimensional Euclidean
Geometry
CHAPTER IV
Coordinates in an Affine Plane
Foreword
1. The Affine Plane
2. Parallel Classes
3. Coordinatizing the Plane π
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4. Slope and Equation of a Line
5. The Ternary Operation T
6. The Planar Ternary Ring [Γ, T]
7. The Affine Plane Defined by a Ternary Ring
8. Introduction of Addition
9. Introduction of Multiplication
10. Vectors
11. A Remarkable Affine Plane
CHAPTER V
Coordinates in an Affine Plane with Desargues and Pappus
Properties
Foreword (The First Desargues Property)
1. Completion of the Equivalence Definition for Vectors
2. Addition of Vectors
3. Linearity of the Ternary Operator
4. Right Distributivity of Multiplication Over Addition
5. Introduction of the Second Desargues Property
6. Introduction of the Third Desargues Property
7. Introduction of the Pappus Property
8. The Desargues Properties as Consequences of the Pappus
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