Dirichlet: a mathematical biography

317 Pages·2018·1.476 MB·English

by  Merzbach, Uta C

Uta C. Merzbach
Dirichlet
A Mathematical Biography

Uta C. Merzbach
Dirichlet
A Mathematical Biography
UtaC. Merzbach (Deceased)
Georgetown, TX,USA
ISBN978-3-030-01071-3 ISBN978-3-030-01073-7 (eBook)
https://doi.org/10.1007/978-3-030-01073-7
LibraryofCongressControlNumber:2018958343
MathematicsSubjectClassification(2010): 01-XX,31-XX,30E25,11-XX
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Preface
Amongthequestionsapotentialreaderfrequentlyasksaboutabook,letmesingle
out the following three: Why this subject? What brings the author to the subject?
How is the subject presented?
Dirichlet was a mathematician who shared responsibility for significant trans-
formations in mathematics during the nineteenth century in Western Europe, a
notable period in the history of mathematics. Yet there is no book-length study of
his lifeandworkandonly asmall percentage ofhis publications tends tobe cited.
I first saw his name in conjunction with Dedekind’s Eleventh Supplement to
Dirichlet’s Lectures on Number Theory. As a graduate student in mathematics,
interestedinalgebra,IwaspuzzledbywhatthisSupplementmighthavetodowith
Dirichlet.Shortlythereafter,whileassistingGarrettBirkhoffwiththeSourcebookof
Classical Analysis, I studied Dirichlet’s own writings on convergence, functions,
andpotentialtheory.Thatdidnotanswermyoriginalquestionbutledmetogivea
series of talks on Dirichlet in the 1970s, including an invited lecture at a meeting
of the American Mathematical Society. This in turn led to an encouraging con-
versationwiththelateSaundersMacLaneandnumerouschatswiththelateWalter
Kaufmann-Bühler of Springer-Verlag New York. It was he who urged me to
expand my earlier research into a book. After some primary source studies in
libraries and archives, I became enthusiastic about such a project, realizing that
Dirichlet was a man who produced his conceptual contributions while living in an
unusually fascinating intellectual environment.
AlthoughIhad tolayasidethis intended, expanded study because ofnumerous
othercommitments,IrecentlywasabletoreturntoDirichlet,torelearnmanythings
I had forgotten, but to take the same pleasure in learning more about the man, his
mathematics, and his environment that I had been granted before.
Thepresentationofa“mathematicalbiography”producesthesamechallengesas
most biographies, especially with regard to the questions raised repeatedly among
historians concerning the relationship of history to biography, of internal versus
external studies, of psychohistory vis-a-vis a “factual” listing of events, or of the
thematic content of a historical study. I have tried to avoid such methodological
arguments. Rather, I have attempted to record as accurately as possible, using the
vii
viii Preface
tools in my possession, the life of a man who loved mathematics, who through
accidents of time and place was able to pursue it as a career, who did not promote
himself but, as noted in our concluding chapter, had a strong support system
throughout most of his life. Many of those who attended his lectures inBerlin and
Göttingen were proud to call themselves his students, no matter how widely their
subsequent fields of specialization diverged.
As I realized the rare opportunity Dirichlet presents to convey to the contem-
porary reader a sense for certain nineteenth-century lives, my originally intended
moremathematicallyorientedstudyturnedintoafullerbiography.Ihopetheresult
may interestnotonly mathematical specialists but readers with other backgrounds.
I have chosen to alternate the chapters dealing with Dirichlet’s publications with
those discussing the corresponding biographical aspects of his life. This decision
was based on an early conversation and the example of Walter Kaufmann-Bühler
whichpersuadedmethatthisapproachcanfacilitatethereadingforbothgroupsof
readers.
In writing about the individual publications, I chose to condense the main
threadsofDirichlet’smathematicaldiscussionssufficientlytoguidethosewhowish
toworkthroughhisargumentsandtosuggestchangesinhismethodologyvis-a-vis
his main predecessors and his successors. Readers may find access to the full
versions of his papers through the references in these chapters pointing to the
bibliography at the end of the book. At the same time, I have described in greater
detail his introductions to the mathematical memoirs, frequently non-technical and
oftenincludingreferences,thatreflecthisowninterestinhistory.Thesenotonlyare
largelyunderstandabletothegeneralreaderbutalsoshedconsiderablelightonthe
development of the subject and on Dirichlet’s own thought processes. Though at
timesrepetitive,theyarestylisticallymoreaccessiblethanthetechnicalpartsofhis
publications.
Incontrasttothelecturespublishedposthumouslybysomeofhisstudents,many
of Dirichlet’s memoirs have been ignored in the secondary literature. While cov-
ering his memoirs in the odd-numbered Chaps. 5 through 13, I have attempted to
conveythepurposeandmajorstatementsofeachwithonlyoccasionaldetailsofhis
proofs sufficient to convey his methodology and to interest the curious general
reader. More involved mathematical readers may attempt to fill in proof details or
read up on them in the extensive multilingual bibliography at the end of this
volume.
It should be noted that Dirichlet was not an emulable stylist. He was careful in
detailing his arguments, but this very care, combined with his adhering to termi-
nology quite different from the smoother, generalized one developed by later
generations, frequently resulted in a certain awkwardness. This may have been
reinforced by a lack of interest and consequent experience in writing. As will be
notedinthefollowing,thiswasmoreevidentinhisformalpresentationsthanitwas
in his lectures and conversations.
Preface ix
Acknowledgements
Beforeacknowledgingspecificsourcesofsupport,Ishouldliketocallthereader’s
attentiontosomecommentsatthebeginningofthebibliography,whereIsingleout
the meticulous publications of Kurt-R. Biermann and the more recent, concise,
biographical sketch by J. Elstredt.
The original motivation for a detailed study of Dirichlet’s mathematics arose
from a set of lectures I gave at the invitation in the 1970s from Judith Victor
Grabiner and the organizers of a Southern California lecture series.
A sabbatical leave from the Smithsonian Institution provided me with the
opportunity to do research in the relevant archives and manuscript collections of
Berlin,Göttingen,andKasselnamedattheendofthebibliography.Specialthanks
aredue toDr. Haenel inGöttingen for many helpful services and toRudolf Elvers
inBerlin,who,amongothercourtesies,providedmewithaccesstohistypescriptof
Fanny Hensel’s diaries before their publication in 2002.
NumerousconversationswithHaroldM.EdwardsinGöttingenandWashington,
DC, during the early phase of related studies furnished considerable food for
thought.
This volume benefitted considerably from the advice of Jeanne LaDuke, who
read over large portions of the draft and provided needed grammatical, ortho-
graphic, and stylistic advice. There would have been considerable delay in pro-
ducing a readable copy without the assistance of Judy Green, who devoted many
hours to translating my coded text to the appropriate LaTeX format.
I owe special thanks to the Birkhäuser editor, Sarah A. Goob, for her unfailing
patience and courtesy. It pleased me greatly that, in a conversation at one of the
Joint Mathematics meetings, Dr. Anna Mätzener had expressed immediate interest
in adopting this project that had been orphaned after the death of Walter
Kaufmann-Bühler, the former New York editor of Springer-Verlag, New York.
Georgetown, TX, USA Uta C. Merzbach
June 2017
’
Publisher s Acknowledgements
ThisbookwouldnothavebeenpossiblewithoutthededicationofProf.Dr.Jeanne
LaDuke and Prof. Dr. Judy Green. After Dr. Merzbach passed away in 2017,
Dr. LaDuke and Dr. Green continued to proofread the manuscript and transfer the
manuscriptfromMicrosoftWordtoLaTeX.Theyspentlonghoursreadingthrough
Dr.Merzbach’snotesandfillinginthegapstofinalizethebook,allinhonoroftheir
colleague and dear friend. We are grateful to these two mathematicians for pro-
vidingthemathematicalcommunitywithadeeperunderstandingofDirichletallthe
while adding to Dr. Merzbach’s legacy.
xi
Contents
1 Rhineland . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Düren. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Bonn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Cologne . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Paris . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.1 Early Reports Home . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Madame Lorge and the Deutgens. . . . . . . . . . . . . . . . . . . . . . 10
2.3 Professors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Smallpox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.5 Water Flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 First Employment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.7 Obligations at Home; Draft Call. . . . . . . . . . . . . . . . . . . . . . . 14
2.8 The Mysterious Research Project . . . . . . . . . . . . . . . . . . . . . . 15
3 First Success . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.1 Fermat’s Claim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Lacroix and Legendre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.3 The Draft Board and the Institut of the Académie. . . . . . . . . . 18
3.4 The Review Committee’s Report . . . . . . . . . . . . . . . . . . . . . . 19
3.5 Legendre’s Proof; Dirichlet’s “Addition” . . . . . . . . . . . . . . . . 20
4 Return to Prussia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1 Political Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 The Death of Foy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.3 Fourier and Humboldt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.4 Approaches to Prussia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.5 Gauss. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.6 The Cultural Ministry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.7 The Breslau Appointment . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.8 Bonn and the Doctorate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
xiii