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Goldstein·Schappacher·Schwermer TheShapingofArithmeticafterC.F.Gauss’sDisquisitionesArithmeticae Catherine Goldstein Norbert Schappacher Joachim Schwermer Editors The Shaping of Arithmetic C. F. Gauss’s after Disquisitiones Arithmeticae With36Figures 123 CatherineGoldstein JoachimSchwermer Histoiredessciencesmathématiques FakultätfürMathematik InstitutdemathématiquesdeJussieu UniversitätWien 175rueduChevaleret Nordbergstraße15 75013Paris,France 1090Wien,Austria E-mail:[email protected] E-mail:[email protected] NorbertSchappacher UFRdemathématique etd’informatique/IRMA 7rueRenéDescartes 67084StrasbourgCedex,France E-mail:[email protected] LibraryofCongressControlNumber:2006932291 ISBN 978-3-540-20441-1 SpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplication ofthispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfrom Springer.ViolationsareliableforprosecutionundertheGermanCopyrightLaw. SpringerisapartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply, even in the absence of a specific statement, thatsuch names are exempt from the relevant protectivelawsandregulationsandthereforefreeforgeneraluse. Typesetbytheeditors Production:LE-TEXJelonek,Schmidt&VöcklerGbR,Leipzig Coverdesign:WMXDesignGmbH,Heidelberg Printedonacid-freepaper 44/3100/YL 543210 Inmemoriam MARTINKNESER 1928–2004 Foreword In1998,theeditorsconvincedthemselvesthatitwastherighttimetotakestockof recentresearchconcerningthemodernhistoryofnumbertheory,andtoevaluatein itslightourcomprehensionofthedevelopmentofthisdisciplineasawhole. One issueatstakewastobringtogetherhistoriographicalresultscomingfromdifferent disciplinesandlinguisticdomainswhich,wefelt,hadremainedtoooftenunaware ofeachother. WeorganizedtwomeetingsattheMathematischesForschungsinstitutOberwol- fach: firstasmallRIP-workshopheldJune14–19,1999,amonghistoriansofnumber theoryandhistoriansofrelatedtopics,andthenalargerconferencewhichtookplace June17–23,2001,twohundredyearsafterthepublicationofCarlFriedrichGauss’s DisquisitionesArithmeticae. Thelatterbroughttogetherhistoriansandphilosophers ofmathematicswithnumbertheoristsinterestedintherecenthistoryoftheirfield. Two further meetings, organized by one of us in Vienna and Zürich the following years,continuedourventure. Twoconcreteprojectsarosefromtheseactivities. Oneconcernedthecreation of resources, for scholars and students: we initiated a bibliography of secondary literatureontheHistoryofNumberTheorysince1800.1 Thepresentvolumeisthesecondresultofourwork. Itaimsatansweringthe question,alreadyraisedduringthefirstworkshop,ontheroleofGauss’sDisquisi- tiones Arithmeticae in the definition and evolution of number theory. This role is hereappraisedinacomparativeperspective,withattentionbothtothemathematical receptionofthetreatise,andtoitsroleasamodelfordoingmathematics. Thevolume istheresultofacollectivework. Althoughallauthorshavekepttheirpropervoices, theyhavealsoacceptedquiteabitofeditorialinterferencewithaviewtomakingthe volumeascoherentaspossible. Wehavenonethelessleftroomfororiginalanalyses andresults,includingnewlydiscovereddocuments. 1. ApreliminaryversionofthisbibliographyhasbeenkindlyputonlinebyFranzLem- mermeyeronawebsitehostedbytheUniversityofHeidelberg(http://www.rzuser.uni- heidelberg.de/ hb3/HINTbib.html). ∼ vii viii During its rather long elaboration, the present book has greatly profited from thehelpofmanyindividualsandinstitutionswhichweheregratefullyacknowledge: theMathematischesForschungsinstitutOberwolfachanditsdirectoratthetimeof the meetings, Matthias Kreck; the Erwin-Schroedinger International Institute for Mathematical Physics at Vienna; the ETH at Zürich, and special encouragement providedbyUrsStammbach;theCIRMatLuminy,whichgaveusaweek’srefuge foroureditorialworkinthesummerof2003;theLaboratoiredemathématiquesde l’UniversitéParis-Sud,theInstitutdemathématiquesdeJussieu,aswellastheInsti- tutdelarecherchemathématiqueavancéeatStrasbourgwhichactivelysupportedour jointworkforthepreparationofthisbook. SpecialthanksgototheAbteilungHand- schriftenundAlteDruckederNiedersächsischenStaats-undUniversitätsbibliothek Göttingen, andinparticulartoJürgenRohlfing, fortheexpertcollaborationwhich madesomanydocumentsavailabletous,someofthemashighqualityscans. We also express our sincere gratitude to Springer-Verlag and their associated staff at HeidelbergandLeipzig,especiallytoJoachimHeinze,whobelievedinthisproject atanearlystage. OurwarmestthanksgotoFrazerJarvisforlinguisticworkonthe texts,andtoJimRitterforconstanttechnicalandmoralsupport. Andlastbutnotleast,wethankalltheparticipantsoftheOberwolfachmeetings whohavesharedtheirinsightsandknowledgewithusforthebenefitoftheproject: besides the authors of this book, Leo Corry, Hélène Gispert, Jeremy Gray, Ralf Haubrich, HelmutKoch, MartinaSchneider, TakaseMasahito, ErhardScholz, Urs Stammbach,andHansWussing. Oneoftheparticipantsatthe2001conferencewasMartinKneser. Despitehis serious illness, his intense, visible passion for number theory and its history was a challenging inspiration to all of us, historians and mathematicians alike. Martin KneserdiedonFebruary16,2004. Wededicatethisbooktohismemory. July2006 CatherineGoldstein NorbertSchappacher JoachimSchwermer Table of Contents Foreword vii TableofContents ix EditionsofC.F.Gauss’sDisquisitionesArithmeticae xi PartI.ABook’sHistory 1 ChapterI.1 ABookinSearchofaDiscipline(1801–1860) 3 CatherineGoldstein&NorbertSchappacher ChapterI.2 SeveralDisciplinesandaBook(1860–1901) 67 CatherineGoldstein&NorbertSchappacher PartII.AlgebraicEquations,QuadraticForms,HigherCongruences: KeyMathematicalTechniquesoftheDisquisitiones 105 ChapterII.1 TheDisquisitionesArithmeticaeandtheTheoryofEquations 107 OlafNeumann ChapterII.2 CompositionofBinaryQuadraticFormsandthe FoundationsofMathematics 129 HaroldM.Edwards ChapterII.3 CompositionofQuadraticForms: AnAlgebraicPerspective 145 DellaFenster&JoachimSchwermer ChapterII.4 TheUnpublishedSectionEight: OntheWaytoFunction FieldsoveraFiniteField 159 GüntherFrei PartIII.TheGermanReceptionoftheDisquisitionesArithmeticae: InstitutionsandIdeas 199 ChapterIII.1 ANetworkofScientificPhilanthropy: Humboldt’sRelations withNumberTheorists 201 HerbertPieper ChapterIII.2 ^O Je‰c >Arijmht–zei: TheRiseofPureMathematics asArithmeticwithGauss 235 Jose´ Ferreiro´s PartIV.ComplexNumbersandComplexFunctionsinArithmetic 269 ChapterIV.1 FromReciprocityLawstoIdealNumbers: An(Un)Known ManuscriptbyE.E.Kummer 271 ReinhardBölling ix x ChapterIV.2 EllipticFunctionsandArithmetic 291 ChristianHouzel PartV.NumbersasModelObjectsofMathematics 313 ChapterV.1 TheConceptofNumberfromGausstoKronecker 315 JacquelineBoniface ChapterV.2 OnArithmetization 343 BirgitPetri&NorbertSchappacher PartVI.NumberTheoryandtheDisquisitionesinFranceafter1850 375 ChapterVI.1 TheHermitianFormofReadingtheDisquisitiones 377 CatherineGoldstein ChapterVI.2 NumberTheoryattheAssociationfrançaisepour l’avancementdessciences 411 Anne-MarieDécaillot PartVII.SpotlightingSomeLaterReactions 429 ChapterVII.1 AnOverviewonItalianArithmeticaftertheDisquisitiones Arithmeticae 431 AldoBrigaglia ChapterVII.2 Zolotarev’sTheoryofAlgebraicNumbers 453 PaolaPiazza ChapterVII.3 GaussGoesWest: TheReceptionoftheDisquisitiones ArithmeticaeintheUSA 463 DellaFenster PartVIII.Gauss’sTheoremsintheLongRun: ThreeCaseStudies 481 ChapterVIII.1 ReductionTheoryofQuadraticForms: TowardRäumliche AnschauunginMinkowski’sEarlyWork 483 JoachimSchwermer ChapterVIII.2 GaussSums 505 SamuelJ.Patterson ChapterVIII.3 TheDevelopmentofthePrincipalGenusTheorem 529 FranzLemmermeyer ListofIllustrations 563 Index 565 Authors’Addresses 577 Editions of Carl Friedrich Gauss’s Disquisitiones Arithmeticae TheDisquisitionesArithmeticaehasbeenomittedfromthelistofreferencesoftheindividual chapters:welistunderneathitsvariouseditions.Throughoutthisbook,passagesfromGauss’s DisquisitionesArithmeticaearereferredtoonlybythearticlenumber. ThetitleofGauss’s workisroutinelyabbreviatedas“D.A.”Forallworks,amentionof[Author1801a]refersto theitem“AUTHOR.1801a”inthebibliography,amentionof[Author1801/1863]referstothe 1863editioninthisitem. 1801. Disquisitiones Arithmeticae. Leipzig: Fleischer. Repr. Bruxelles: Culture et civilisation, 1968. Repr. Hildesheim: Olms, 2006. Rev. ed. in Werke, vol. 1, ed. Königliche Gesellschaft zu Göttingen [E. Schering]. Göttingen: Universitäts- Druckerei,1863;2ndrev.ed.,1870;repr.Hildesheim: Olms,1973. http://gallica.bnf.fr http://dz-srv1.sub.uni-goettingen.de/cache/toc/D137206.html 1807. Recherches arithmétiques. French transl. A.-C.-M. Poullet-Delisle. Paris: Courcier. Repr.Sceaux: Gabay,1989. http://gallica.bnf.fr 1889. ArithmetischeUntersuchungen. Germantransl.H.Maser. InUntersuchungen über höhere Arithmetik, pp. 1–453. Berlin: Springer. Repr. New York: Chelsea, 1965;2nded.,1981. http://dz-srv1.sub.uni-goettingen.de/cache/toc/D232699.html 1959. Arifmeticˇeskie issledovaniya. Russian transl. V. B. Dem’yanov. In Trudi po teorii cˇisel [Works on number theory], ed. I. M. Vinogradov, B. N. Delone, pp.7–583. Moscow: AcademyofSciencesUSSR. 1966. DisquisitionesArithmeticae. Englishtransl.A.A.Clarke. NewHaven: Yale UniversityPress. Rev. ed. W.C.Waterhouse. NewYork: Springer,1986. 1995. DisquisitionesArithmeticae. Spanishtransl.H.BarrantesCampos,M.Josephy, A. Ruiz Zu`ñiga. Coleccio´n Enrique Pérez Arbelaez 10. Santa Fe de Bogota´: AcademiaColombianadeCienciasExactas,FisicasyNaturales. 1995. Seisuuron. Japanesetransl.TakaseMasahito. Tokyo: Asakura-Shoten. 1996. Disquisicions aritmètiques. Catalan transl. G. Pascual Xufré. Barcelona: Institutd’EstudisCatalans,SocietatCatalanadeMatemàtiques. xi Fig.I. TitlepageofDisquisitionesArithmeticae,1801edition (Privatecopy)

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imply, even in the absence of a specific statement, that such names are its light our comprehension of the development of this discipline as a whole. l'Université Paris-Sud, the Institut de mathématiques de Jussieu, as well as the Insti- Chapter VI.1 The Hermitian Form of Reading the Disquisit
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