Cauchy’s Cours d’analyse: An Annotated Translation PDF
Preview Cauchy’s Cours d’analyse: An Annotated Translation
Cauchy’s Cours d’analyse An Annotated Translation Forothertitlespublishedinthisseries,goto http://www.springer.com/series/4142 Sources and Studies in the History of Mathematics and Physical Sciences EditorialBoard L.Berggren J.Z.Buchwald J.Lu¨tzen Robert E. Bradley, C. Edward Sandifer Cauchy’s Cours d’analyse An Annotated Translation 123 RobertE.Bradley C.EdwardSandifer DepartmentofMathematicsand DepartmentofMathematics ComputerScience WesternConnecticutStateUniversity AdelphiUniversity Danbury,CT06810 GardenCity USA NY11530 [email protected] USA [email protected] SeriesEditor: J.Z.Buchwald DivisionoftheHumanitiesandSocialSciences CaliforniaInstituteofTechnology Pasadena,CA91125 USA [email protected] ISBN978-1-4419-0548-2 e-ISBN978-1-4419-0549-9 DOI10.1007/978-1-4419-0549-9 SpringerDordrechtHeidelbergLondonNewYork LibraryofCongressControlNumber:2009932254 MathematicsSubjectClassification(2000):01A55,01A75,00B50,26-03,30-03 (cid:2)c SpringerScience+BusinessMedia,LLC2009 Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsorscholarlyanalysis.Usein connection with any form of information storage and retrieval, electronic adaptation, computer software,orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not theyaresubjecttoproprietaryrights. Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) WededicatethisvolumetoRonaldCalinger, VictorKatzandFrederickRickey,whotaught ustheimportanceandsatisfactionofreading originalsources,andtoourfriendsin ARITHMOS,withwhomweenjoyputting thoselessonsintopractice. Translators’ Preface Modern mathematics strives to be rigorous. Ancient Greek geometers had similar goals,toproveabsolutetruthsbyusingperfectdeductivelogicstartingfromincon- trovertiblepremises. Ofteninthehistoryofmathematics,weseeapatternwheretheideasandappli- cationscomefirstandtherigorcomeslater.Thishappenedinancienttimes,when thepracticalgeometryoftheMesopotamiansandEgyptiansevolvedintotherigor- ouseffortsoftheGreeks.Ithappenedagainwithcalculus.Calculuswasdiscovered, somesayinvented,almostindependentlybyIsaacNewton(1642–1727)about1666 andbyGottfriedWilhelmvonLeibniz(1646–1716)about10yearslater,butitsrig- orousfoundationswerenotestablished,despiteseveralattempts,formorethan150 years. In 1821, Augustin-Louis Cauchy (1789–1857) published a textbook, the Cours d’analyse,toaccompanyhiscourseinanalysisattheE´colePolytechnique.Itisone ofthemostinfluentialmathematicsbookseverwritten.NotonlydidCauchyprovide a workable definition of limits and a means to make them the basis of a rigorous theoryofcalculus,butalsoherevitalizedtheideathatallmathematicscouldbeset onsuchrigorousfoundations.Today,thequalityofaworkofmathematicsisjudged inpartonthequalityofitsrigor;thisstandardislargelyduetothetransformation broughtaboutbyCauchyandtheCoursd’analyse. The17thcenturybroughtthenewcalculus.Scientistsoftheagewereconvinced ofthetruthofthiscalculusbyitsimpressiveapplicationsindescribingandpredict- ingtheworkingsofthenaturalworld,especiallyinmechanicsandthemotionsofthe planets.Thefoundationsofcalculus,whatColinMaclaurin(1698–1746)andJean le Rond d’Alembert (1717–1783) later called its metaphysics, were based on the intuitivegeometricideasofLeibnizandNewton.Someoftheircontemporaries,es- peciallyBishopGeorgeBerkeley(1685–1753)inEnglandandMichelRolle(1652– 1719)inFrance,recognizedtheproblemsinthefoundationsofcalculus.Rolle,for example,saidthatcalculuswas“acollectionofingeniousfallacies,”andBerkeley ridiculed infinitely small quantities, one of the basic notions of early calculus, as “the ghosts of departed quantities.” Both Berkeley and Rolle freely admitted the practicality of calculus, but they challenged its lack of rigorous foundations. We vii viii Translators’Preface shouldnotethatRolle’scolleaguesattheParisAcademyeventuallyconvincedhim tochangehismind,butBerkeleyremainedskepticalforhisentirelife. Later in the 18th century, only a few mathematicians tried to address the ques- tions of foundations that had been raised by Berkeley and Rolle. Over the years, threemainschoolsofthoughtdeveloped:infinitesimals,limits,andformalalgebra of series. We could consider the British ideas of fluxions and evanescent quanti- ties either to be a fourth school or to be an ancestor of these others. Leonhard Euler (1707–1783) [Euler 1755] was the most prominent exponent of infinitesi- mals,thoughhedevotedonlyatinypartofhisimmensescientificcorpustoissues of foundations. Colin Maclaurin [Maclaurin 1742] and Jean le Rond d’Alembert [D’Alembert 1754] favored limits. Maclaurin’s ideas on limits were buried deep in his Treatise of Fluxions, and they were overshadowed by the rest of the opus. D’Alembert’s works were very widely read, but even though they were published atalmostthesametimeasEuler’scontraryviews,theydidnotstimulatemuchofa dialog. Wesuspectthatthelargestschoolofthoughtonthefoundationsofcalculuswas infactapragmaticschool–calculusworkedsowellthattherewasnorealincentive toworrymuchaboutitsfoundations. InAnVoftheFrenchRevolutionarycalendar,1797totherestofEurope,Joseph- Louis Lagrange (1736–1813) [Lagrange 1797] returned to foundations with his book, the full title of which was The´orie des fonctions analytiques, contenant les principes du calcul diffe´rentiel, de´gage´s de toute conside´ration d’infiniment petits ou d’e´vanouissans, de limites ou de fluxions, et re´duits a` l’analyse alge´brique des quantite´sfinies(Theoryofanalyticfunctionscontainingtheprinciplesofdifferential calculus,withoutanyconsiderationofinfinitesimalorvanishingquantities,oflimits oroffluxions,andreducedtothealgebraicanalysisoffinitequantities).Thebook wasbasedonhisanalysislecturesattheE´colePolytechnique.Lagrangeusedpower seriesexpansionstodefinederivatives,ratherthantheotherwayaround.Lagrange kept revising the book and publishing new editions. Its fourth edition appeared in 1813,theyearLagrangedied.Itisinterestingtonotethat,liketheCoursd’analyse, Lagrange’sThe´oriedesfonctionsanalytiques containsnoillustrationswhatsoever. JusttwoyearsafterLagrangedied,CauchyjoinedthefacultyoftheE´colePoly- techniqueasprofessorofanalysisandstartedtoteachthesamecoursethatLagrange hadtaught.HeinheritedLagrange’scommitmenttoestablishfoundationsofcalcu- lus, but he followed Maclaurin and d’Alembert rather than Lagrange and sought those foundations in the formality of limits. A few years later, he published his lecture notes as the Cours d’analyse de l’E´cole Royale Polytechnique; I.re Partie. Analysealge´brique.ThebookisusuallycalledtheCoursd’analyse,butsomecat- alogsandsecondarysourcescallittheAnalysealge´brique.Evidently,Cauchyhad intendedtowriteasecondpart,buthedidnothavetheopportunity.Theyearafter its publication, the E´cole Polytechnique changed the curriculum to reduce its em- phasisonfoundations[Lu¨tzen2003,p.160].Cauchywrotenewtexts,Re´sume´ des lec¸onsdonne´esal’E´colePolytechniquesurlecalculinfinitesimal,tomepremierin 1823andLec¸onssurlecalculdiffe´rentielin1829,inwhichhereducedthematerial intheCoursd’analyseaboutfoundationstojustafewdozenpages. Translators’Preface ix Becauseitbecameobsoleteasatextbookjustayearafteritwaspublished,the Coursd’analysesawonlyoneFrencheditioninthe19thcentury.Thatfirstedition, publishedin1821,was568pageslong.Thesecondedition,publishedasVolume15 (also identified as Series 2, Volume III) of Cauchy’s Oeuvres comple`tes, appeared in 1897. Its content is almost identical to the 1821 edition, but its pagination is quitedifferent,therearesomedifferenttypesettingconventions,anditisonly468 pageslong.TheErratanotedinthefirsteditionarecorrectedinthesecond,anda number of new typographical errors are introduced. At least two facsimiles of the first edition were published during the second half of the 20th century, and digital versionsofbotheditionsareavailableonline,forexample,throughtheBibliothe`que NationaledeFrance.TherewereGermaneditionspublishedin1828and1885,and a Russian edition published in Leipzig in 1864. A Spanish translation appeared in 1994, published in Mexico by UNAM. The present edition is apparently the first editioninanyotherlanguage. TheCoursd’analysebeginswithashortIntroduction,inwhichCauchyacknowl- edgestheinspirationofhisteachers,particularlyPierreSimonLaplace(1749–1827) andSime´onDenisPoisson(1781–1840),butmostespeciallyhiscolleagueandfor- mer tutor Andre´ Marie Ampe`re (1775–1836). It is here that he gives his oft-cited intentinwritingthevolume,“Asforthemethods,Ihavesoughttogivethemallthe rigorwhichonedemandsfromgeometry,sothatoneneedneverrelyonarguments drawnfromthegeneralityofalgebra.” The Introduction is followed by 16 pages of “Preliminaries,” what today might be called “Chapter Zero.” Here, Cauchy takes pains to define his terms, carefully distinguishing, for example, between number and quantity. To Cauchy, numbers hadtobepositiveandreal,butaquantitycouldbepositive,negativeorzero,realor imaginary,finite,infiniteorinfinitesimal. BeyondthePreliminaries,thebooknaturallydividesintothreemajorpartsand a couple of short topics. The first six chapters deal with real functions of one and severalvariables,continuity,andtheconvergenceanddivergenceofseries. In the second part, Chapters 7 to 10, Cauchy turns to complex variables, what hecallsimaginaryquantities.Muchofthisparallelswhathedidwithrealnumbers, but it also includes a very detailed study of roots of imaginary equations. We find herethefirstuseofthewordsmodulusandconjugateintheirmodernmathematical senses.Chapter10givesCauchy’sproofofthefundamentaltheoremofalgebra,that apolynomialofdegreenhasnrealorcomplexroots. Chapters 11 and 12 are each short topics, partial fraction decomposition of ra- tionalexpressionsandrecurrentseries,respectively.Inthis,Cauchy’sstructurere- mindsusofLeonhardEuler’s1748text,theIntroductioinanalysininfinitorum[Eu- ler1748],anotherclassicinthehistoryofanalysis.InEuler,wefind11chapterson realfunctions,followedbyChapters12and13,“Ontheexpansionofrealfunctions intofractions,”i.e.,partialfractions,and“Onrecurrentseries,”respectively. ThethirdmajorpartoftheCoursd’analyseconsistsofnine“Notes,”140pages inthe1897edition.CauchydescribestheminhisIntroductionas“...severalnotes placedattheendofthevolume[where]Ihavepresentedthederivationswhichmay
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