1、Cauchys Cours danalyseAn Annotated TranslationFor other titles published in this series, go tohttp:/ and Studiesin the History of Mathematicsand Physical SciencesEditorial BoardL. BerggrenJ.Z. BuchwaldJ. LutzenRobert E. Bradley, C. Edward SandiferCauchys Cours danalyseAn Annotated Translation123Robe
2、rt E. BradleyDepartment of Mathematics andComputer ScienceAdelphi UniversityGarden CityNY 11530USAbradleyadelphi.eduC. Edward SandiferDepartment of MathematicsWestern Connecticut State UniversityDanbury, CT 06810USAsandiferewcsu.eduSeries Editor:J.Z. BuchwaldDivision of the Humanities and Social Sci
3、encesCalifornia Institute of TechnologyPasadena, CA 91125USAbuchwaldits.caltech.eduISBN 978-1-4419-0548-2 e-ISBN 978-1-4419-0549-9DOI 10.1007/978-1-4419-0549-9Springer Dordrecht Heidelberg London New YorkLibrary of Congress Control Number: 2009932254Mathematics Subject Classification (2000): 01A55,
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7、ld Calinger,Victor Katz and Frederick Rickey, who taughtus the importance and satisfaction of readingoriginal sources, and to our friends inARITHMOS, with whom we enjoy puttingthose lessons into practice.Translators PrefaceModern mathematics strives to be rigorous. Ancient Greek geometers had simila
8、rgoals, to prove absolute truths by using perfect deductive logic starting from incon-trovertible premises.Often in the history of mathematics, we see a pattern where the ideas and appli-cations come first and the rigor comes later. This happened in ancient times, whenthe practical geometry of the M
9、esopotamians and Egyptians evolved into the rigor-ous efforts of the Greeks. It happened again with calculus. Calculus was discovered,some say invented, almost independently by Isaac Newton (16421727) about 1666and by Gottfried Wilhelm von Leibniz (16461716) about 10 years later, but its rig-orous f
10、oundations were not established, despite several attempts, for more than 150years.In 1821, Augustin-Louis Cauchy (17891857) published a textbook, the Coursdanalyse, to accompany his course in analysis at the Ecole Polytechnique. It is oneofthemostinfluentialmathematicsbookseverwritten.NotonlydidCauc
11、hyprovidea workable definition of limits and a means to make them the basis of a rigoroustheory of calculus, but also he revitalized the idea that all mathematics could be seton such rigorous foundations. Today, the quality of a work of mathematics is judgedin part on the quality of its rigor; this
12、standard is largely due to the transformationbrought about by Cauchy and the Cours danalyse.The 17th century brought the new calculus. Scientists of the age were convincedof the truth of this calculus by its impressive applications in describing and predict-ingtheworkingsofthenaturalworld,especially
13、inmechanicsandthemotionsoftheplanets. The foundations of calculus, what Colin Maclaurin (16981746) and Jeanle Rond dAlembert (17171783) later called its metaphysics, were based on theintuitive geometric ideas of Leibniz and Newton. Some of their contemporaries, es-pecially Bishop George Berkeley (16
14、851753) in England and Michel Rolle (16521719) in France, recognized the problems in the foundations of calculus. Rolle, forexample, said that calculus was “a collection of ingenious fallacies,” and Berkeleyridiculed infinitely small quantities, one of the basic notions of early calculus, as“the gho
15、sts of departed quantities.” Both Berkeley and Rolle freely admitted thepracticality of calculus, but they challenged its lack of rigorous foundations. Weviiviii Translators Prefaceshould note that Rolles colleagues at the Paris Academy eventually convinced himto change his mind, but Berkeley remain
16、ed skeptical for his entire life.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,three main schools of thought developed: infinitesimals, limits, and formal algebraof series. We could consi
17、der the British ideas of fluxions and evanescent quanti-ties either to be a fourth school or to be an ancestor of these others. LeonhardEuler (17071783) Euler 1755 was the most prominent exponent of infinitesi-mals, though he devoted only a tiny part of his immense scientific corpus to issuesof foun
18、dations. Colin Maclaurin Maclaurin 1742 and Jean le Rond dAlembertDAlembert 1754 favored limits. Maclaurins ideas on limits were buried deepin his Treatise of Fluxions, and they were overshadowed by the rest of the opus.DAlemberts works were very widely read, but even though they were publishedat al
19、most the same time as Eulers contrary views, they did not stimulate much of adialog.We suspect that the largest school of thought on the foundations of calculus wasin fact a pragmatic school calculus worked so well that there was no real incentiveto worry much about its foundations.In An V of the Fr
20、ench Revolutionary calendar, 1797 to the rest of Europe, Joseph-Louis Lagrange (17361813) Lagrange 1797 returned to foundations with hisbook, the full title of which was Theorie des fonctions analytiques, contenant lesprincipes du calcul differentiel, degages de toute consideration dinfiniment petit
21、sou devanouissans, de limites ou de fluxions, et reduits a lanalyse algebrique desquantitesfinies(Theoryofanalyticfunctionscontainingtheprinciplesofdifferentialcalculus, without any consideration of infinitesimal or vanishing quantities, of limitsor of fluxions, and reduced to the algebraic analysis
22、 of finite quantities). The bookwas based on his analysis lectures at the Ecole Polytechnique. Lagrange used powerseries expansions to define derivatives, rather than the other way around. Lagrangekept revising the book and publishing new editions. Its fourth edition appeared in1813, the year Lagran
23、ge died. It is interesting to note that, like the Cours danalyse,Lagranges Theorie des fonctions analytiques contains no illustrations whatsoever.Just two years after Lagrange died, Cauchy joined the faculty of the Ecole Poly-techniqueasprofessorofanalysisandstartedtoteachthesamecoursethatLagrangeha
24、d taught. He inherited Lagranges commitment to establish foundations of calcu-lus, but he followed Maclaurin and dAlembert rather than Lagrange and soughtthose foundations in the formality of limits. A few years later, he published hislecture notes as the Cours danalyse de lEcole Royale Polytechniqu
25、e; I.re Partie.Analyse algebrique. The book is usually called the Cours danalyse, but some cat-alogs and secondary sources call it the Analyse algebrique. Evidently, Cauchy hadintended to write a second part, but he did not have the opportunity. The year afterits publication, the Ecole Polytechnique
26、 changed the curriculum to reduce its em-phasis on foundations Lutzen 2003, p. 160. Cauchy wrote new texts, Resume deslecons donnees a lEcole Polytechnique sur le calcul infinitesimal, tome premier in1823 and Lecons sur le calcul differentiel in 1829, in which he reduced the materialin the Cours dan
27、alyse about foundations to just a few dozen pages.Translators Preface ixBecause it became obsolete as a textbook just a year after it was published, theCours danalyse saw only one French edition in the 19th century. That first edition,published in 1821, was 568 pages long. The second edition, publis
28、hed as Volume 15(also identified as Series 2, Volume III) of Cauchys Oeuvres completes, appearedin 1897. Its content is almost identical to the 1821 edition, but its pagination isquite different, there are some different typesetting conventions, and it is only 468pages long. The Errata noted in the
29、first edition are corrected in the second, and anumber of new typographical errors are introduced. At least two facsimiles of thefirst edition were published during the second half of the 20th century, and digitalversionsofbotheditionsareavailableonline,forexample,throughtheBibliothequeNationale de
30、France. There were German editions published in 1828 and 1885, anda Russian edition published in Leipzig in 1864. A Spanish translation appeared in1994, published in Mexico by UNAM. The present edition is apparently the firstedition in any other language.The Cours danalysebeginswithashortIntroductio
31、n,inwhichCauchyacknowl-edges the inspirationof his teachers, particularlyPierre SimonLaplace (17491827)and Simeon Denis Poisson (17811840), but most especially his colleague and for-mer tutor Andre Marie Ampere (17751836). It is here that he gives his oft-citedintent in writing the volume, “As for t
32、he methods, I have sought to give them all therigor which one demands from geometry, so that one need never rely on argumentsdrawn from the generality of algebra.”The Introduction is followed by 16 pages of “Preliminaries,” what today mightbe called “Chapter Zero.” Here, Cauchy takes pains to define
33、 his terms, carefullydistinguishing, for example, between number and quantity. To Cauchy, numbershad to be positive and real, but a quantity could be positive, negative or zero, real orimaginary, finite, infinite or infinitesimal.Beyond the Preliminaries, the book naturally divides into three major
34、parts anda couple of short topics. The first six chapters deal with real functions of one andseveral variables, continuity, and the convergence and divergence of series.In the second part, Chapters 7 to 10, Cauchy turns to complex variables, whathe calls imaginary quantities. Much of this parallels
35、what he did with real numbers,but it also includes a very detailed study of roots of imaginary equations. We findhere the first use of the words modulus and conjugate in their modern mathematicalsenses. Chapter 10 gives Cauchys proof of the fundamental theorem of algebra, thata polynomial of degree
36、n has n real or complex roots.Chapters 11 and 12 are each short topics, partial fraction decomposition of ra-tional expressions and recurrent series, respectively. In this, Cauchys structure re-minds us of Leonhard Eulers 1748 text, the Introductio in analysin infinitorum Eu-ler 1748, another classi
37、c in the history of analysis. In Euler, we find 11 chapters onreal functions, followed by Chapters 12 and 13, “On the expansion of real functionsinto fractions,” i.e., partial fractions, and “On recurrent series,” respectively.The third major part of the Cours danalyse consists of nine “Notes,” 140
38、pagesin the 1897 edition. Cauchy describes them in his Introduction as “.several notesplaced at the end of the volume where I have presented the derivations which mayx Translators Prefacebe useful both to professors and students of the Royal Colleges, as well as to thosewho wish to make a special st
39、udy of analysis.”Though Cauchy was only 32 years old when he published the Cours danalyse,and had been only 27 when he began teaching the analysis course on which it wasbased, he was already an accomplished mathematician. This should not be surpris-ing, as it was not easy to earn an appointment as a
40、 professor at the Ecole Polytech-nique. Indeed, by 1821, Cauchy had published 28 memoirs, but the Cours danalysewas his first full-length book.Cauchys first original mathematics concerned the geometry of polyhedra andwas done in 1811 and 1812. Louis Poinsot (17771859) had just established theexisten
41、ce of three new nonconvex regular polyhedra. Cauchy, encouraged to studythe problem by Lagrange, Adrien-Marie Legendre (17521833) and Etienne LouisMalus (17751812), Belhoste 1991, pp. 2526 extended Poinsots results, discov-ered a generalization of Eulers polyhedral formula,V E +F = 2, and proved tha
42、taconvexpolyhedronwithrigidfacesmustberigid.Theseresultsbecamehisearliestpapers, the two-part memoir “Recherches sur les polyedres” and “Sur les polygoneset les polyedres.” Cauchy 1813 Despite his early success, Cauchy seldom returnedto geometry, and these are his only significant results in the fie
43、ld.After Cauchys success with the problems of polyhedra, his father encouragedhim to work on one of Fermats (16011665) problems, to show that every integeris the sum of at most three triangular numbers, at most four squares, at most fivepentagonal numbers, and, in general, at most n n-gonal numbers.
44、 He presented hissolution to the Institut de France on November 13, 1815 and published it under thetitle “Demonstration generale du theoreme de Fermat sur les nombres polygones”Cauchy 1815. Belhoste Belhoste 1991, p. 46 tells us that this was the article “thatmade him famous,” and suggests that “the
45、 announcement of his proof may havesupported his appointment to the Ecole Polytechnique a few days later.”Just a month later, on December 26, 1815, the Academys judgment was con-firmed when Cauchy won the Grand Prix de Mathematiques of the Institut deFrance, and its prize of 3000 francs, for an essa
46、y on the theory of waves.With his career established, Cauchy married Alose de Bure (1795?1863) in1818. They had two daughters. It is a measure of Cauchys later fame and successthat one of his daughters married a count, the other a viscount. Indeed, FreudenthalDSB Cauchy, p. 135 says that Cauchy “was
47、 one of the best known people of histime.”The de Bure family were printers and booksellers. The title page of the Coursdanalyse, published by de Bure freres, describes them as “Libraires du Roi et de labibliotheque du Roi.”It seems that Cauchy was an innovative but unpopular teacher at the Ecole Pol
48、y-technique. He, along with Ampere and Jacques Binet (17861856), proposed sub-stantial revisions in the analysis, calculus and mechanics curricula. Cauchy wrotethe Cours danalyse to support the new curriculum.In 1820, though, before the Cours danalyse was published, but apparently af-ter it had been
49、 written and the publisher had committed to printing it, the ConseilTranslators Preface xiFig. 1 Cauchy, by Susan Petry, 1828 cm, bas relief in tulip wood, 2008. An interpretation ofportraits by Boilly (1821) and Roller ( 1840). Photograph by Eliz Alahverdian, 2008. Reprintedwith permission of Susan Petry and Aliz Alahverdian. All rights reserved.xii Translators PrefacedInstruction, more or
