Euler: The Master of Us All

205 Pages·1999·9.817 MB·English

by  William Dunham

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The Dolciani Mathematical Expositions
NUMBER TWENTY-TWO
Euler
ThMea stoefUr sA ll
WillDiuanmh am
Truman Koehler Professor of Mathematics
Muhlenberg College
Published and Distributed by
THE MATHEMATICAL ASSOCIATION OF AMERICA
THE
DOLCIANI MATHEMATICAL EXPOSITIONS
Published by
THE MATHEMATICAL ASSOCIATION OF AMERICA
Committee on Publications
JAMES W. DANIEL, Chair
Dolciani Mathematical Expositions Editorial B()ard
BRUCE P. PALKA, Editor
EDWARD J. BARBEAU
IRL C. BIVENS
SUSAN C. GELLER
The DOLCIANI MATHEMATICAL EXPOSITIONS series of the Mathematical
Association of Amenca was established through a generous gift to the Association from
Mary P. Dolciani, Professor of Mathematics at Hunter College of the City University
of New York. In making the gift, Professor Dolciani, herself an exceptionally talented
and successful expositor of mathematics, had the purpose of furthenng the ideal of
excellence in mathematical exposition.
The Association, for its part, was delighted to accept the gracious gesture initi­
ating the revolving fund for this series from one who has served the Association with
distinction, both as a member of the Committee on Publications and as a member of
the Board of Governors. It was with genuine pleasure that the Board chose to name the
senes in her honor.
The books in the series are selected for their lucid expository style and stimulat­
ing mathematical content. Typically, they contain an ample supply of exercises, many
with accompanying solutions. They are intended to be sufficiently elementary for the
undergraduate and even the mathematically inclined high-school student to understand
and enjoy, but also to be interesting and sometimes challenging to the more advanced
mathematician.
DOLCIANI MATHEMATICAL EXPOSITIONS
Mathematical Gems.
I. Mathematical Gems IRI, oss Honsberger
2. Mathematical Morsels, Ross Honsberger
3. Mathematical Plums, Ross Honsberger
4. Great Moments in MaRthoesms aHtiocns s(bBeerfgoerre (1e6d5.)0 ),
5. Maxima and Minima without Calculus, Howard Eves
6. Great Moments in Mathematics (After 16Iv5a0n) ,N iven
7. Map Coloring, Polyhedra, and the Four-ColorH Porwobarledm E, ves
8. Mathematical Gems Ill, David Barnette
9. More Mathematical MorRseolss,s Honsberger
10. Old and New Unsolved ProblRemosss i nH oPnlasbneer gGeero metry and Number Theory,
11. Victor
PKrloeeb laenmds Sfotarn M Waathgeomn aticians. Young and Old,
12. Excursions in Calculus. An Interplay of the CoPnatuinl uRo.u Hs aalnmdo tsh e Discrete,
13. Robert
TMh.e Y Wouonhga scum County Problem Book,
14. George T. Gilbert, Mark Krusemeyer, and
LLioorne nH Cu. nLtianrgs oann d Other Mathematical Pursuits: A Collection of Mathematics.
15. Verse, and Stories by Ralph P. Boas, Jr..
edited by Gerald L. Alexanderson and
Dale H. Mugler
Linear Algebra Problem Book,
16. From Erdos to Kiev: Problems oPfa Oull yRm. pHiaaldm Coasl iber.
1187 . Which Way Did the Bicycle Go?. . and Other IntriguRinogss M Haotnhsebmeargtiecra l Mysteries.
JIno sPeop/hy aD '.s EF.o Kotosntehpasu:s Meri, sDcealnla Vneeloleums Panro, balnedm Ss taann dW Easgsoany s,
19. Diophantus and Diophantine Equations, Ross Honsberger
20. Logic as Algebra, I. G. Bashmakova
21. Euler: The Master Poafu Ul Hs Aallml, os and Steven Givant
22. William Dunham
Service Center
MAA
P. 0. Box 9 I 112
Washington, DC 20090-1112
1-800-331-1622 fax: [phone]
For Penny,
who has made all the difference.
Contents
Acknowledgments . . xi
Preface ........ xv
Biographical Sketch . xix
1.
Euler and Number Theory I
2. Euler and Logarithms . . 17
3. Euler and Infinite Series . 39
4.
Euler and Analytic Number Theory 61
5. Euler and Complex Variables 81
6.
Euler and Algebra . . . . 103
7.
Euler and Geometry . . . 125
8. Euler and Combinatorics . 149
Conclusion . . . . . . . . . 171
Appendix: Euler's Opera Omnia 175
Index ............. . 181
ix
Acknowledgments
During the 1997-98 academic year, I was privileged to hold the Donald B.
Hoffman Research Fellowship at Muhlenberg College. For their part in granting
me this award, I thank Muhlenberg's Faculty Development and Scholarship
Committee, Dean Curtis Dretsch, and President Arthur Taylor.W ithout such
support, I could never have begun-let alone completed-this project.
Likewise, I acknowledge the two libraries whose resources proved es­
sential over the past year: the Trexler Library of Muhlenberg College, where
this manuscript was written; and the Fairchild-Martindale Library of Lehigh
University, where Euler's Opera Omnia can be found in all its glory.
Among those who encouraged this work, foremost is Don Albers, Direc­
tor of Publications and Programs of the Mathematical Association of America.
Don has been a constant source of good advice and good cheer. For his profes­
sionalism and his friendship, I shall always be grateful.
Others made significant contributions, most notably Bruce Palka, who ed­
its the Dolciani Mathematical Expositions; Elaine Pedreira and Beverly Ruedi,
who shepherded the manuscript through production; and Jerry Alexanderson
and Leon Varjian, whose careful reading of the document generated a number
of helpful comments.
I also wish to express my appreciation to friends and colleagues in Muh­
lenberg's Department of Mathematical Sciences: George Benjamin, Roland
Dedekind, Margaret Dodson, Linda Luckenbill, John Meyer, David Nelson,
Bob Stump, and Bob Wagner.
Beyond the Muhlenberg campus are individuals deserving special men­
tion.O ne is George Poe, Professor of French at the University of the South, a
dear friend without whose translations I would have committed one faux pas
after another.L ikewise, I thank Claramae and Carol Dunham and Ruth and
Bob Evans, all of whom have been steadfast in their support.A nd there are our
sons Brendan and Shannon, who remain the greatest.
xi
xii Euler: The Master of Us All
Finally, and most sincerely, I wish to recognize my wife and colleague
Penny Dunham, who made numerous improvements to the final manuscript and
who applied her computer wizardry to generate the figures contained herein.
For such assistance-and for so much more-I dedicate this book to her with
love and thanks.
W.
AllentoPwenn,n sylvania
DUNHAM
"Read Euler, read Euler. He is the master of us all."
-Laplace