Science Networks Historical Studies 50 Robert E. Bradley Salvatore J. Petrilli C. Edward Sandifer L'Hôpital's Analyse des infiniments petits An Annotated Translation with Source Material by Johann Bernoulli ScienceNetworks.HistoricalStudies FoundedbyErwinHiebertandHansWußing Volume50 EditedbyEberhardKnobloch,HelgeKraghandVolkerRemmert Editorial Board: K.Andersen,Amsterdam S.Hildebrandt, Bonn H.J.M.Bos,Amsterdam D.KormosBuchwald,Pasadena U.Bottazzini, Roma Ch.Meinel,Regensburg J.Z.Buchwald,Pasadena J.Peiffer,Paris K.Chemla,Paris W.Purkert,Bonn S.S.Demidov,Moskva D.Rowe,Mainz M.Folkerts,München Ch.Sasaki,Tokyo P.Galison,Cambridge, Mass. R.H.Stuewer,Minneapolis J.Gray,Milton Keynes V.P.Vizgin,Moskva R.Halleux,Liège Moreinformationaboutthisseriesathttp://www.springer.com/series/4883 Robert E. Bradley (cid:129) Salvatore J. Petrilli C. Edward Sandifer L’Hôpital’s Analyse des infiniments petits An Annotated Translation with Source Material by Johann Bernoulli RobertE.Bradley SalvatoreJ.Petrilli MathematicsandComputerScience MathematicsandComputerScience AdelphiUniversity AdelphiUniversity GardenCity,NY,USA GardenCity,NY,USA C.EdwardSandifer Newtown,CT,USA ISSN1421-6329 ISSN2296-6080 (electronic) ScienceNetworks.HistoricalStudies ISBN978-3-319-17114-2 ISBN978-3-319-17115-9 (eBook) DOI10.1007/978-3-319-17115-9 LibraryofCongressControlNumber:2015935196 MathematicsSubjectClassification(2010):01A45,01A75,26-03 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2015 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Coverillustration:FromWallerMsde-00215,AugustBeer:ÜberdieCorrectiondesCosinusgesetzes beiderAnwendungdesNicol’schenPrismasinderPhotometrie,after1850.Withfriendlypermissionby TheWallerManuscriptCollection(partoftheUppsalaUniversityLibraryCollections) Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) Translators’ Preface Analyse des infiniment petits, pour l’intelligence des lignes courbes was the first textbookofthedifferentialcalculus.ThetitletranslatesasAnalysisoftheinfinitely small, for the understanding of curved lines. It was published anonymously in Parisin1696,althoughmembersoftheFrenchmathematicalcommunitywerewell aware that the author was Guillaume François Antoine de l’Hôpital,1 the Marquis of Saint-Mesme (1661–1704). The textbook was successful, as evidenced by the appearance of a posthumous second edition (L’Hôpital 1716), which identified the author.2 Pierre Varignon (1646–1722), who was professor of mathematics at Collège des Quatre-Nations in Paris and a friend of l’Hôpital, created a collection ofclarificationsandadditionstotheAnalyse.Thesewerepublishedposthumously (Varignon1725),afewyearsafterthe1716editionoftheAnalyse.Latereditionsof the Analyse included similar commentary and continued to appear throughout the 18thcentury(L’Hôpital1768,1781). Differential and integral calculus are generally considered to have their origins intheworksofSirIsaacNewton(1642–1727)andWilhelmGottfriedvonLeibniz (1646–1716)3 in the late 17th century, although the roots of the subject reach far backintothatcenturyand,arguably,evenintoantiquity.Leibnizfirstdescribedhis new calculus in a cryptic article more than a decade before the publication of the Analyse (Leibniz 1684). For allpractical purposes, Leibniz’ early papers were not 1L’Hôpitalspelledhisname“Hospital”andthiswasthespellingusedintheposthumoussecond edition of the text in 1715/1716. Fontenelle used the spelling “Hôpital,” which is the standard modern spelling of the name, in his Eulogy (p. 295). There is no consensus among English- languageauthorsoftheearly21stcenturyastowhichspellingoughttobeused. 2Technically, this was the third edition, despite the words “Seconde Edition” on the title page. Therewasa“SecondeEdition”oneyearearlier(L’Hôpital1715),withmanytypographicalerrors, ofwhichthe1716editionisacorrectedversion(Bernoulli1955,pp.499–500). 3Formoreontheprioritydisputeoverthediscoveryofthecalculus,whichisnotamatterofinterest forthisvolume,seeHall(1980). v vi Translators’Preface understood, until Jakob Bernoulli (1654–1705) and his younger brother Johann4 (1667–1748) began studying them in about 1687 and making discoveries of their ownusinghistechniques. Bernard de Fontenelle (1657–1757) became the secretary of the Académie des Sciences in Paris in 1697 and wrote the eulogy of l’Hôpital for the academy’s journal.Hesaidthatin1696, ...theGeometryoftheInfinitelysmallwasstillnothingbutakindofMystery,and,soto speak,aCabalisticSciencesharedamongfiveorsixpeople.TheyoftengavetheirSolutions intheJournalswithoutrevealingtheMethodthatproducedthem,andevenwhenonecould discoverit,itwasonlyafewfeebleraysofthisSciencethathadescaped,andtheclouds immediatelyclosedagain.5(Fontenelle1708,pp.133–134) Later on, Jean Etienne Montucla (1725–1799) went one step further and listed theonlypeoplethathebelievedunderstoodLeibniz’calculusbefore1696:Leibniz himself, Jakob and Johann Bernoulli, Pierre Varignon and l’Hôpital (Montucla 1799, p. 397). L’Hôpital’s Analyse changed all of this and for much of the 18th century,hisbookservedaspiringFrenchmathematiciansastheirfirstintroduction tothenewcalculus. For all that the Analyse was a popular and successful introduction to the differentialcalculus,it’sremarkablethatthereisnoaccountoftheintegralcalculus inthebook.InhisPreface,l’Hôpitalexplainedwhy InallofthisthereisonlythefirstpartofMr.Leibniz’calculus,...Theotherpart,whichwe callintegralCalculus,consistsingoingbackfromtheseinfinitelysmallquantitiestothe magnitudesorthewholesofwhichtheyarethedifferences,thatistosayinfindingtheir sums.Ihadalsointendedtopresentthis.However,Mr.Leibniz,havingwrittenmethathe isworkingonaTreatisetitledDeScientiâinfiniti,Itookcarenottodeprivethepublicof suchabeautifulWork...[p.liii]. Unfortunately,LeibniznevercompletedthisbookOntheScienceoftheInfinite. The Analyse consists of ten chapters, which L’Hôpital called “sections.” We consider it to have three parts. The first part, an introduction to the differential calculus,consistsofthefirstfourchapters: 1. InwhichwegivetheRulesofthiscalculus. 2. Use of the differential calculus for finding the Tangents of all kinds of curved lines. 3. Useofthedifferentialcalculusforfindingthegreatestandtheleastordinates,to whicharereducedquestionsDemaximis&minimis. 4. Useofthedifferentialcalculusforfindinginflectionpointsandcusps. Taken together, these chapters provide a thorough introduction to the differential calculus in about 70 pages. The next five chapters are devoted to what can only be described as an advanced text on differential geometry, motivated in part by whatwerethencutting-edgeresearchproblemsinopticsandotherfields.Thefinal 4OftenreferredtoasJohann(I)BernoullitodistinguishhimfromhissonJohannandgrandson Johann,whowerealsosuccessfulmathematicians. 5Thisquotationfromp.299ofthistext. Translators’Preface vii chapterismildlypolemical,demonstratingthesuperiorityofLeibniz’newcalculus, when compared to the methods of René Descartes (1596–1660) and Johann van WaverenHudde(1628–1704). The RoleofJohannBernoulli MostbiographicalinformationknownabouttheMarquisdel’Hôpitalcomesfrom Fontenelle’s eulogy (Fontenelle 1708, pp. 116–146), a translation of which is includedinChapter13ofthisvolume.However,Fontenelleknewlittleornothingat thetimeofl’Hôpital’sdeathabouttheroleofJohannBernoulliinthecomposition of the Analyse. L’Hôpital himself acknowledged a debt to Johann Bernoulli in his preface: IacknowledgehavingreceivedmuchfromtheilluminationsofMessrs.Bernoulli,particu- larlythoseoftheyounger,presentlyProfessoratGroningen.Ihavemadeplainuseoftheir discoveriesandthoseofMr.Leibniz.ThisiswhyIgrantthattheymayclaimasmuchof thisastheymaywish,beingcontentwiththatwhichtheyarewillingtoleaveforme[p.liv]. Inlightofthis,itseemssomewhatstrangethatMontuclawouldwrite“Wemay onlyfindfaultinthatMr.del’Hôpitaldidnotmakewellenoughknownthedebthe owedtoMr.Bernoulli”(Montucla1799,p.397),buttherecordshowsthatJohann Bernoulli’s influence on the structure and content of the Analyse was much more significantthanthesewordsofrecognitionwouldsuggest. Amongthefewdetailsknownaboutl’Hôpital’searlylife,Fontenellerecounted thathesolvedoneof Pascal’sproblemsinvolvingthecycloidattheageof15.The Marquis became a cavalry officer, but had only attained the rank of captain when he resigned his commission due to poor eyesight. He devoted all of his energy to mathematics from that point onward. Some time around 1690, he joined Nicolas Malebranche’s(1638–1715)circle,whichwasengaged,amongotherthings,inthe studyofhighermathematics.ItwasthereinNovember1691thathemetthe24-year- oldJohannBernoulli,whowasvisitingParisandhadbeeninvitedbyMalebranche to present his construction of the catenary at the salon. Although Fontenelle made no mention of this meeting, it is documented by Spiess in his introduction to the Bernoulli-l’Hôpital correspondence, which contains what may be considered adefinitivebiographyoftheMarquisdel’Hôpital(Bernoulli1955,pp.123–130). There is no contemporary account of this meeting. Bernoulli wrote of the encounter in his autobiography, which he composed in 1741, but Spiess considers an earlier account that he gave in a letter to Pierre Rémond de Montmort (1678– 1719) to be more reliable; see Bernoulli (1955, p. 135–137). In May 21, 1718, Bernoulli told Montmort that upon meeting the Marquis, he soon found him to beagoodenoughmathematician withregardtoordinarymathematics,butthathe knew nothing of the differential calculus, other than its name, and had not even heardoftheintegralcalculus.L’HôpitalhadapparentlymasteredFermat’smethod of finding maxima and minima and told Bernoulli that he had used it to invent a rule for determining the radius of curvature for arbitrary curves. The method was viii Translators’Preface unwieldy and actually could only be used at local extrema of algebraic curves. Bernoullishowedhimtheformulafortheradiusofcurvaturethathehaddeveloped withhisbrotherJakob,whichemployedsecond-orderdifferentials.Apparently,this so impressed the Marquis that he visited Bernoulli the very next day and engaged himashistutorinthedifferentialandintegralcalculus. TheMarquisdeL’Hôpital(1661–1704)–portraitinwalnutbySusanPetry,2014,25(cid:2)19(cid:2)3 cm.;photographbytheartist Translators’Preface ix BernoullitutoredtheMarquisinhisParisapartmentfourtimesaweekfromlate 1691 through the end of July 1692. We are fortunate that l’Hôpital insisted that Bernoulli commit his lessons to paper. Bernoulli typically composed each lesson, whichhewroteinLatin,thenightbeforehegavethemtotheMarquis.Fortunately, his friend and later colleague at the University of Basel, Johann Heinrich von JohannBernoulli(1667–1748)–portraitincherrybySusanPetry,2014,26(cid:2)19(cid:2)4cm.; photographbytheartist