Geometry: Euclid and Beyond

535 Pages·2000·15.844 MB·English

by  Robin Hartshorne (auth.)

Undergraduate Texts in Mathematics
Editors
S.Axler
F.WGehring
K.A. Ribet
Undergraduate Texts in Mathematics
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(continued after index)
Robin Hartshorne
Geometry:
Euclid and Beyond
With 550 Illustrations
~ Springer
Robin Hartshorne
Department of Mathematics
University of California
Berkeley, CA 94720
USA
[e-mail]
Editorial Board
S.Axler F.W. Gehring KA. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East HaU University of California
University University of Michigan at Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 9472().3840
USA USA USA
[e-mail] [e-mail]. [e-mail]
umich.edu
Mathematics Subject Classification (2000): 51-01
Library of Congress Cataloging-in-Publication Data
Hartshorne, Robin.
Geometry: Euclid and beyond/Robin Hartshorne.
p. cm. - (Undergraduate texts in mathematics)
Includes bibliographical references and index.
ISBN 978-1-4419-3145-0 ISBN 978-0-387-22676-7 (eBook)
DOI 10.1007/978-0-387-22676-7
1. Geometry 1. Title. II. Series.
QA451 .H37 2000
51621-dc21 99-044789
© 1997 Robin Hartshorne. An earlier version of this book appeared as Companion to
EucZid: A Course of Geometry, based on Euclid' s Elements and its modern descendants.
Berkeley Mathematics Lecture Notes, volume 9, American Mathematical Society &
Berkeley Center for Pure and Applied Mathematics.
© 2000 Robin Hartshorne
Originally published by Springer-Verlag New York, Inc. in 2000
Softcover reprint ofthe hardcover Ist edition 2000
AH rights reserved. This work may not be translated or copied in whole or in part without
the written permission ofthe publisher (Springer Science+Business Media, LLC), except for
brief excerpts in connection with reviews or scholarly analysis. Use in connection with any
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similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar
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9 8 765 4
springeronline.com
For
Edie, Ben, and Joemy
and
In Loving Memory of
Jonathan Churchill Hartshorne
1972-1992
I have not found anything in Lobatchevski's
work that is new to me, but the development is
made in a different way from the way I had
started and to be sure masterfully done by Lo
batchevski in the pure spirit of geometry.
- letter from Gauss to Schumacher (1846)
Preface
In recent years, I have been teaching a junior-senior-level course on the classi
cal geometries. This book has grown out of that teaching experience. I assume
only high-school geometry and some abstract algebra. The course begins in
Chapter 1 with a critical examination of Euclid's Elements. Students are expected
to read concurrently Books I-IV of Euclid's text, which must be obtained sepa
rately. The remainder of the book is an exploration of questions that arise natu
rally from this reading, together with their modern answers. To shore up the
foundations we use Hilbert's axioms. The Cartesian plane over a field provides
an analytic model of the theory, and conversely, we see that one can introduce
coordinates into an abstract geometry. The theory of area is analyzed by cutting
figures into triangles. The algebra of field extensions provides a method for
deciding which geometrical constructions are possible. The investigation of the
parallel postulate leads to the various non-Euclidean geometries. And in the last
chapter we provide what is missing from Euclid's treatment of the five Platonic
solids in Book XIII of the Elements.
For a one-semester course such as I teach, Chapters 1 and 2 form the core
material, which takes six to eight weeks. Then, depending on the taste of the in
structor, one can follow a more geometric path by going directly to non-Euclidean
geometry in Chapter 7, or a more algebraic one, exploring the relation between
geometric constructions and field extensions, by doing Chapters 3, 4, and 6. For
me, one of the most interesting topics is the introduction of coordinates into an
abstractly given geometry, which is done for a Euclidean plane in Section 21,
and for a hyperbolic plane in Section 41.
Throughout this book, I have attempted to choose topics that are accessible
vii
viii
Preface
to undergraduates and that are interesting in their own right. The exercises are
meant to be challenging, to stimulate a sense of curiosity and discovery in the
student. I purposely do not indicate their difficulty, which varies widely.
I hope this material will become familiar to every student of mathematics,
and in particular to those who will be future teachers.
lowe thanks to Marvin Greenberg for reading and commenting on large
portions of the text, to Hendrik Lenstra for always having an answer to my
questions, and to Victor Pambuccian for valuable references to the literature.
Thanks to Faye Yeager for her patient typing and retyping of the manuscript.
And special thanks to my wife, Edie, for her continual loving support.
Of all the works of antiquity which have
been transmitted to the present times, none are
more universally and deservedly esteemed than
the Elements of Geometry which go under the
name of Euclid. In many other branches of
science the moderns have far surpassed their
masters; but, after a lapse of more than two
thousand years, this performance still maintains
its original preeminence, and has even acquired
additonal celebrity from the fruitless attempts
which have been made to establish a different
system.
- from the preface to
Bonnycastle's Euclid
London (1798)
Contents
Introduction 1
Chapter 1. Euclid's Geometry 7
1. A First Look at Euclid's Elements 8
2. Ruler and Compass Constructions 18
3. Euclid's Axiomatic Method 27
4. Construction of the Regular Pentagon 45
5. Some Newer Results 51
Chapter 2. Hilbert's Axioms 65
6. Axioms of Incidence 66
7. Axioms of Betweenness 73
8. Axioms of Congruence for Line Segments 81
9. Axioms of Congruence for Angles 90
10. Hilbert Planes 96
11. Intersection of Lines and Circles 104
12. Euclidean Planes 112
Chapter 3. Geometry over Fields 117
l3. The Real Cartesian plane 118
14. Abstract Fields and Incidence 128
15. Ordered Fields and Betweenness 135
ix
X Contents
16. Congruence of Segments and Angles 140
17. Rigid Motions and SAS 148
18. Non-Archimedean Geometry 158
Chapter 4. Segment Arithmetic 165
19. Addition and Multiplication of Line Segments 165
20. Similar Triangles 175
21. Introduction of Coordinates 186
Chapter 5. Area 195
22. Area in Euclid's Geometry 196
23. Measure of Area Functions 205
24. Dissection 212
25. Quadratura Circuli 221
26. Euclid's Theory of Volume 226
27. Hilbert's Third Problem 231
Chapter 6. Construction Problems and Field Extensions 241
28. Three Famous Problems 242
29. The Regular 17-Sided Polygon 250
30. Constructions with Compass and Marked Ruler 259
31. Cubic and Quartic Equations 270
32. Appendix: Finite Field Extensions 280
Chapter 7. Non-Euclidean Geometry 295
33. History of the Parallel Postulate 296
34. Neutral Geometry 304
35. Archimedean Neutral Geometry 319
36. Non-Euclidean Area 326
37. Circular Inversion 334
38. Digression: Circles Determined by Three Conditions 346
39. The Poincare Model 355
40. Hyperbolic Geometry 373
41. Hilbert's Arithmetic of Ends 388
42. Hyperbolic Trigonometry 403
43. Characterization of Hilbert planes 415
Chapter 8. Polyhedra 435
44. The Five Regular Solids 436
45. Euler's and Cauchy's Theorems 448