Table Of ContentCauchy’s Cours d’analyse
An Annotated Translation
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Sources and Studies
in the History of Mathematics
and Physical Sciences
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L.Berggren
J.Z.Buchwald
J.Lu¨tzen
Robert E. Bradley, C. Edward Sandifer
Cauchy’s Cours d’analyse
An Annotated Translation
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RobertE.Bradley C.EdwardSandifer
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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
