Mathematical Expeditions: Chronicles by the Explorers

288 Pages·1999·11.119 MB·English

by  Reinhard Laubenbacher, David Pengelley (auth.)

Undergraduate Texts in Mathematics
Readings in Mathematics
Editors
S. Axler
F.w. Gehring
K.A. Ribet
Springer Science+Business Media, LLC
Graduate Texts in Mathematics
Readings in Mathematics
Ebbinghaus/Hermes/HirzebruchiKoecher/MainzerlNeukirchIPrestel!Remmert: Numbers
Fulton/Harris: Representation Theory: A First Course
Remmert: Theory ofC omplex Functions
Walter: Ordinary Differential Equations
Undergraduate Texts in Mathematics
Readings in Mathematics
Anglin: Mathematics: A Concise History and Philosophy
Anglin/Lambek: The Heritage ofThales
Bressoud: Second Year Calculus
Hairer/Wanner: Analysis by Its History
HammerlinIHoffmann: Numerical Mathematics
Isaac: The Pleasures ofP robability
Laubenbacher/Pengelley: Mathematical Expeditions: Chronicles by the Explorers
Samuel: Projective Geometry
Stillwell: Numbers and Geometry
Toth: Glimpses ofA lgebra and Geometry
Reinhard Laubenbacher
David Pengelley
Mathelllatical Expeditions
Chronicles by the Explorers
With 94 I1lustrations
, Springer Science+Business Media, LLC
Reinhard Laubenbacher
David pengelley
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003-0001
USA
Editorial Board
S. Axler F.w. Gehring K.A. Ribet
Mathematics Department Mathematics Department Mathematics Department
San Francisco State East Hall University of California
University University of Michigan at Berkeley
San Francisco, CA 94132 Ann Arbor, MI 48109 Berkeley, CA 94720-3840
USA USA USA
FRONT CO"ER ILLVSTRATIONS: 8ackground: Sophie Germain's 1819 letter to Cari F. Gauss about
her work on Fermat's Last Theorem. Foreground (cloekwise from top left): Georg Cantor, Pierre
de Fermat, Evariste Galois, Gottfried Leibniz; at eenter, Nikolai Lobaehevsky.
Mathematics Subjeet Classifieation (1991): 01-01, 04-01,11-01,12-01,26-01,51-01
Library of Congress Cataloging-in-Publieation Data
Laubenbaeher, Reinhard.
Mathematieal expeditions : ehronicles by the explorers /
Reinhard Laubenbaeher, David Pengelley.
p. em. - (Undergraduate texts in mathematics. Readings in mathematies)
Includes bibliographical referenees (p. - ) and index.
ISBN 978-0-387-98433-9 ISBN 978-1-4612-0523-4 (eBook)
DOI 10.1007/978-1-4612-0523-4
1. Mathematies-History-Sourees. 1. pengelley, David. II. Title.
III. Series.
QA21.L34 1998
51O'.9-de21 98-3889
Printed on acid-free paper.
© 1999 Springer Science+Business Media New York
Originally published by Springer-Verlag New York in 1999
Softcover reprint ofthe hardcover IsI edilion 1999
AII 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
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Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
Production managed by Steven Pisano; manufacturing supervised by Thomas M. King.
'lYpeset from the authors' LaThX files by The Bartlett Press, Ine., Marietta, GA.
9 8 7 6 5 4 3 2 (Corrected second printing, 2000)
ISBN 978-0-387-98433-9
Preface
Nothing captures the excitement of discovery as authentically as a description by
the discoverers themselves. What better place to read about the search for the ori
gin of the Nile than the report of Sir Richard Burton on his amazing journey into
the depths of the African continent? The overwhelming awe and beauty of the
Grand Canyon comes alive when we hear John Wesley Powell tell of his thrilling
ride down its unexplored rapids. The same holds true for the explorers of the un
known territories of the mathematical world. While a second-hand account of a
mathematical expedition might seem more orderly than the description of the ex
plorer, it will likely lack the excitement, immediacy, and insights of a story told by
someone who was there. It is this excitement and immediacy of mathematical dis
covery that we want to convey. This book contains the stories offive mathematical
journeys into new realms, pieced together through the writings of the explorers
themselves. Some were guided by mere curiosity and the thrill of adventure, others
by more practical motives. In all cases the outcome was a vast expansion of the
known mathematical world and the realization that still greater vistas remain to be
explored.
In the development of calculus, one great goal was to find methods for com
puting areas and volumes beyond the achievements of classical Greek geometry.
The potential for applications was enormous, and once success was achieved, the
sciences were revolutionized. Likewise, the prospect for methods to solve general
algebraic equations fostered high hopes for important applications within math
ematics, as well as in the sciences and engineering. In contrast, the quest for a
solution to the so-called Fermat's Last Theorem in number theory seems whim
sical, apparently motivated by nothing more than the desire to meet a challenge.
Nonetheless, when the goal was finally reached, mathematics had profited immea
surably from the effort. The same outcome crowned the attempt to conquer the
treacherous world of infinite sets. What first seemed liked a hopeless fight against
VI Preface
paradoxes and the tenets of theology turned into one of the greatest revolutions
mathematics has ever seen. Finally, the apparently placid waters of geometry were
hiding a rich world beyond the imagination of all but a handful of brave souls who
dared explore the possibilities ofa non-Euclidean view of geometry and the physi
cal world. Begun out of a desire for mathematical elegance and completeness, this
two-thousand-year quest led to a vast expansion of mathematics and fundamental
applications to the theory of relativity. We will tell these stories as much as possible
by guiding the reader through the very words of the mathematicians at the heart
of these events.
This book is more about mathematics than it is about its history. Our goal is
to throw light on the mathematical world we live in today, and we believe that
its history is essential to understanding and appreciation. Our project began as a
freshman honors course we have taught at New Mexico State University since 1989.
The aim of the course is to introduce students from a wide variety of majors to the
exciting world of mathematical discovery. Typically, some subsequently decide to
major in mathematics. In the course we try to get across the thrill of exploring the
unknown that motivates most mathematicians. Students see the mighty mountains
that the community of mathematicians scales, sometimes through the joint effort
of many generations. In the end a better understanding of the mathematical present
is paired with the realization that mathematics is a living, breathing subject, facing
new challenges every day. We hope that this book serves the same goals.
The book can be used in a variety of ways. The five chapters are completely
independent of one another, as are largely the individual sections within each chap
ter. Necessarily, the level of difficulty within a chapter varies considerably. The
chapter introduction and first sections can be appreciated and understood by some
one with a good high school education in mathematics. The later sections require
considerably more mathematical maturity. It is our vision that the book will be
enticing both to the intellectually curious reader and to instructors and students as
a course text. The introduction to each chapter summarizes the story historically
and mathematically, and subsequent sections feature the original writings of ma
jor explorers in that particular story of discovery. The five introductions, together
with selections from the other sections, can be used as a text for a mathematically
oriented history of mathematics course. Individual chapters can be used in a seri
ous mathematics appreciation course or as a supplement to another mathematics
course. Most importantly, it is our hope that the text will encourage the creation
of courses like the one from which it originated. In our one-semester course, we
usually focus on just two or three chapters. There is enough material in the book
for at least two semesters.
Our initial inspiration to create courses in which students learn mathematics in
its historical context was William Dunham's "Great Theorems" course [44, 45].
Unlike Dunham we insist on reading primary sources. (Our one compromise is
the use of English translations.) We have discussed the feasibility and the many
benefits of this approach in [103, 104, 105]. The resources [24, 156, 167, 168]
as well as the newsletter [90] also contain much information on using history in
teaching mathematics.
Preface vii
Our primary sources trace five central themes in the evolution ofmathematics.
Our selection criteria were the importance of the source as a milestone of progress
and its accessibility without extensive prior preparation. In these choices and in
our own commentary we make no claim to be comprehensive in breadth, detail, or
the contributions of various individuals, groups, or cultures. Ours is not a history
of mathematics, but rather an exploration of some exciting mathematics through
its historical artifacts.
How do we use these malerials in our own teaching? Usually, we work through
the introduction together with the students and jump to the later sections as the
sources arc mentioned. The annotalion after each source is there to help with
sticky points, but is used sparingly. We have included many exercises based on
the original sources, and welcome more from our readers. A most useful exercise
is to rewrite a source in one's own word.. using modem notation, filling in all the
missing details.
Finally, we integrate prose readings about mathematics into the course, many
from the wonderful collection [130]. We provide students with questions about
these readings, and written answers then fonn the basis for class discussion.
We strongly encourage the reader to go beyond this book to explore the rich and
rewarding world of primary sources. There arc substamial collections of original
sources available in English, sueh as [13, 14, 58, 87, 122, 160, 166]. Collected
works are, ofc ourse, also a great resource {142]. We have provided many references
in the text for further reading.
This book has been in the making for almost ten years. It might never have been
completed without the help of many people and institutions. The directors Tom
Hoeksema and Bill Eamon of the NMSU honors program provided extensive sup
port and encouragement for the course from which this book grew. Our department
heads, Carol Walker and Doug Kurtz, believed cnough in our approach to tcaehing
to help us make it imo a pennanent addition to our curriculum. A grant from thc
Division of Undergraduate Educationatlhe National Science Foundation provided
extensive resources. The NSF advisory committee, consisting of Judy Grabiner,
Tom Hoeksema, and Fred Rickey, gave lots of great advice, diligent reading, and
editorial suggestions on several drafts. In addition, Folrence Fasanelli provided
sage words of wisdom at crucial timcs. Thc grant also allowed us to involve a
graduate assistant, Xenia Kramer, in the project. We owe her special thanks for
her extensive contributions to research and writing, and for testing earlier drafts
in the classroom as an apprentice teacher. The lion's share of the credit must go
to our students, however, without whom this book would never have been written.
We used early versions of the manuscript in classes at NMSU, as well as Cornell
University and the two-year-Iong NSF-sponsored Young Scholars Mathematics
Workshop in the Rockies, at Colorado College. Our students' enthusiasm con
vinced us that teaching with original sources can work, and their feedback greatly
improved the book.
Many people have helped us in locating original sources. We could always
rely on the expertise of Keith Dennis, as well as his wonderful private collection.
Our research on Sophie Gennain would have been impossible without the help of
VIII Preface
Larry Bucciarelli, Catherine Goldstein, Helmut Rohlfing of the Niedersachsische
Staatsbibliothek in Gbttingen, and the Bibliotheque Nationale in Paris. Thanks
are also due to Dave Bayer, Don Davis, and Anne-Michel Pajus. We received
assistance with translations from Helene Barcelo and Mai Gehrke. Bill Donahue,
Danny Otero, Kim Plofker, and Frank Williams contributed their expertise in Latin.
A number of teachers here and elsewhere have used earlier versions of the
manuscript with their students and provided many useful suggestions for improve
ment. We are especially grateful to Helene Barcelo, David Arnold (and his student
Charles McCoy), Danny Otero, Jamie Pommersheim, Mike Siddoway, and Irena
Swanson. We thank all those who volunteered to read drafts and gave suggestions,
especially Otto Bekken and Klaus Barner. We have also been most fortunate to
benefit from the ideas and expertise of fellow instructors at programs where we
have presented our approach and materials, including a minicourse for the Math
ematical Association of America, and the NSF/MAA Institute in the History of
Mathematics and its Use in Teaching, especially the detailed comments of David
Dennis and Ed Sandifer.
We are very grateful to Ina Lindemann from Springer-Verlag, who showed great
interest in our project and supported us with just the right mixture of patience and
prodding. Her enthusiasm and great forbearance provided much encouragement.
We also thank the other staff at Springer, and our copyeditor, David Kramer, for
their expert assistance and excellent suggestions. Finally, we thank Rose Marquez
for her expert secretarial assistance.
The second author thanks his wife, Pat Penfield, for her enduring and invaluable
love, encouragement, and support for this endeavor, excellent ideas, and incisive
critiques; and his parents Daphne and Ted, for their constant love, encouragement
and support, inspiration, and interest in history.
October, 1998 Reinhard Laubenbacher
David Pengelley
Contents
Preface v
Geometry: The Parallel Postulate 1
1.1 Introduction ........ .
1.2 Euclid's Parallel Postulate .. 18
1.3 Legendre's Attempts to Prove the Parallel Postulate 24
1.4 Lobachevskian Geometry . . . . . . . . . . . . . . 31
1.5 Poincare's Euclidean Model for Non-Euclidean Geometry 43
2 Set Theory: Taming the Infinite 54
2. I Introduction.. . ..... 54
2.2 Bolzano's Paradoxes of the Infinite 69
2.3 Cantor's Infinite Numbers 74
2.4 Zermelo's Axiomatization . . . . . 89
3 Analysis: Calculating Areas and Volumes 95
3.1 Introduction . . ....... .. . . 95
3.2 Archimedes' Quadrature of the Parabola 108
3.3 Archimedes' Method ......... . 118
3.4 Cavalieri Calculates Areas of Higher Parabolas. 123
3.5 Leibniz's Fundamental Theorem of Calculus 129
3.6 Cauchy's Rigorization of Calculus 138
3.7 Robinson Resurrects Infinitesimals 150
3.8 Appendix on Infinite Series ... . 154