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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 ©SpringerNatureSwitzerlandAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfrom therelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authorsortheeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinor for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaimsinpublishedmapsandinstitutionalaffiliations. ThisbookispublishedundertheimprintBirkhäuser,www.birkhauser-science.combytheregistered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland ImagemadeavailablebytheUniversityofToronto—GersteinScienceInformationCentre 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