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The Origins of Cauchy's Rigorous Calculus The Origins of Cauchy's Rigorous Calculus Judith V. Grabiner Flora Sanborn Pitzer Professor of Mathematics Pitzer College Dover Publications, Inc. Mineola, New York Copyright Copyright © 1981 The Massachusetts Institute of Technology All rights reserved. Bibliographical Note This Dover edition, first published in 2005, is an unabridged republication of the edition published by The MIT Press, Cambridge, Massachusetts, 1981. Library of Congress Cataloging-in-Publication Data Grabiner, Judith V. The origins of Cauchy's rigorous calculus I Judith V. Grabiner. p.cm. Originally published: Cambridge, Mass. : MIT Press, c1981. Includes bibliographical references and index. ISBN 0-486-43815-5 (pbk.) 1. Calculus-History. 2. Cauchy, Augustin Louis, Baron, 1789-1857. I. Title. QA303.G742005 515'.09-dc22 2004059331 Manufactured in the United States of America Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501 To my parents, Alfred and Ruth Tofield Victor Contents Preface Vlll Abbreviations of Titles x Introduction Cauchy and the Nineteenth-Century Revolution in 1 Calculus 5 The Status of Foundations in Eighteenth-Century 2 Calculus 16 The Algebraic Background of Cauchy's New 3 Analysis 47 The Origins of the Basic Concepts of Cauchy's 4 Analysis: Limit, Continuity, Convergence 77 The Origins of Cauchy's Theory of the 5 Derivative 114 The Origins of Cauchy's Theory of the Definite 6 Integral 140 Conclusion 164 Appendix: Translations from Cauchy's Oeuvres 167 Notes 176 References 225 Index 241 Preface Augustin-Louis Cauchy gave the first reasonably success­ ful rigorous foundation for the calculus. Beginning with a precise definition of limit, he initiated the nineteenth­ century theories of convergence, continuity, derivative, and integral. The clear superiority of Cauchy's work over what had come before, and the apparent break with the past that such superiority implies, combine to raise an urgent historical question: Since no great work arises in a vacuum, what, in the thought of his predecessors, made Cauchy'S achievement possible? In this book I attempt to answer this question by giving the intellectual back­ ground of Cauchy's accomplishment. Understanding the way mathematical ideas develop has, besides its intrinsic interest, immediate application to our understanding of mathematics and to the teaching of mathematics. The history of the foundations of the calculus provides the real motivation for the basic ideas, and also helps us to see which ideas were-and thus are-really hard. For the convenience of teachers and students of mathematics, I have, whenever possible, cited the mathe­ matics of the past in readily accessible editions. In the appendix I provide English translations of some of Cauchy's major contributions to the foundations of the calculus. It is a pleasure to acknowledge the many sources of encouragement and support I received while preparing this study. The nucleus of the research was funded by a Fellowship from the American Council of Learned Societies. My PhD research on Joseph-Louis Lagrange, on which part of this study is based, was supported by a doctoral fellowship from the National Science Foundation, and some of the research for the final revision of this manuscript was supported by the National Science Foundation under Grant No. SOC 7907844. I would like to thank the Harvard College Library for the excellence orits collection and the assistance ofits staff. Dirk J. Struik has often given me the benefit of his vast Preface knowledge of eighteenth-century mathematics and has reminded me that the mathematician is a social being as well as a creator of mathematical ideas. Uta Merzbach systematically introduced me to many aspects of the history of mathematics, and first suggested to me that eighteenth-century approximation techniques were worth investigating. Joseph W. Dauben, my colleague at Historia Mathematica, gave me the benefit of his criticism of an early version of this book and was generally informative and encouraging. Also, several anonymous referees have provided useful comments at various stages of the prepa­ ration of the manuscript. I am grateful to the students in my classes on the history of mathematics at Harvard University, California State University at Los Angeles, Pomona College, and California State University, Dominguez Hills; they have provided both incisive ques­ tions and reflective comments. lowe significant intellec­ tual debts to the late Carl Boyer for his History of the Calculus, and to Thomas S. Kuhn for his The Structure of Scientific Revolutions. Above all I would like to record my gratitude to I. Bernard Cohen of Harvard University. He has been an unfailing source of inspiration as well as of material assis­ tance, and, most important, he taught me what it means to think like a historian. Finally, I would like to thank my husband Sandy Grabiner, who interrupted his own work to spend many hours reading various drafts of this book; his encouragement and mathematical insight have provided the necessary and sufficient conditions for its completion; and my son David, who helped proofread. Abbreviations of Titles B. Bolzano, Rein analytischer Beweis = Rein analytischer Beweis des Lehrsatzes dass zwischen je zwry Werthen, die ein entgegen­ gesetztes Resultat gewaehren, wenigstens eine reele Wurzel del' Gleichung liege, Prague, lS17. A.-L. Cauchy, Calcul infinitesimal = Resume des lefons donnees a l' ecole royale poly technique sur le caLcul irifinitesimal, vol. 1 [all published], Paris, IS23. In the edition of Cauchy's Oeuvres, series 2, vol. 4, pp. 5 - 261; all references will be to this edition. A.-L. Cauchy, Cours d'analyse = Cours d'analyse de l'ecole royale poly technique. Ire partie: analyse algebrique [all published], Paris, IS21. In the edi tion of Cauchy's Oeuvres, series 2, vol. 3; all page references will be to this edition. J .-L. Lagrange, Calcul des Jonctions = Lefons sur le calcul des fonctions, new ed., Paris, lS06. In the edition of Lagrange's Oeuvres, vol. 10. J.-L. Lagrange, Equations numeriques = Traite de la resolution des equations numeriques de tous les degres, 2nd ed., Paris, lS08. In the edition of Lagrange's Oeuvres, vol. 8. J.-L. Lagrange, Fonctions analytiques = Theorie des fonctions analytiques, contenant les principes du calcul differentiel, degages de loute consideration d'infiniment petits, d'evanouissans, de limites et dejiuxions, et reduits al'analyse algebrique des quantitesfinies, new ed., Paris, lS13. In the edition of Lagrange's Oeuvres, vol. 9. The Origins of Cauchy's Rigorous Calculus Introduction What is the calculus? When we ask modern mathe­ maticians, we get two different answers. One answer is, of course, that the calculus is the branch of mathematics which studies the relationships between functions, their derivatives, and their integrals. Its most important subject matter is its applications: to tangents, areas, volumes, arc lengths, speeds, and distances. The calculus can be fruit­ fully viewed and effectively taught as a set of intuitively understood problem-solving techniques, widely applicable to geometry and to physical systems. Through the gener­ ality of its basic concepts and through the heuristic qua­ lities ofits notation, the calculus demonstrates the power of mathematics to state and solve problems pertaining to every aspect of science. The calculus is something else as well, however: a set of theorems, based on precise definitions, about limits, continuity, series, derivatives, and integrals. The calculus may seem to be about speeds and distances, but its logical basis lies in an entirely different subject-the algebra of inequalities. The relationship between the uses of the calculus and the justification of the calculus is anything but obvious. A student who asked what speed meant and was answered in delta-epsilon terms might be forgiven for re­ sponding with shock, "How did anybody ever think of such an answer?" These two different aspects-use and justification­ of the calculus, simultaneously coexisting in the modern subject, are in fact the legacies of two different historical periods: the eighteenth and the nineteenth centuries. In the eighteenth century, analysts were engaged in exciting and fruitful discoveries about curves, infinite processes, and physical systems. The names we attach to important results in the calculus-Bernoulli's numbers, L'Hopital's rule, Taylor's series, Euler's gamma function, the Lagrange remainder, the Laplace transform-attest to the mathematical discoveries of eighteenth-century analysts. Though not indifferent to rigor, these researchers spent 2 Introduction most of their effort developing and applying powerful methods, some of which they could not justify, to solve problems; they did not emphasize the mathematical im­ portance of the foundations of the calculus and did not really see foundations as an important area of mathe­ matical endeavor. By contrast, a major task for nineteenth-century ana­ lysts like Cauchy, Abel, Bolzano, and Weierstrass was to give rigorous definitions of the basic concepts and, even more important, rigorous proofs of the results of the calcu­ lus. Their proofs made precise the conditions under which the relations between the concepts of the calculus held. Indeed, nineteenth-century precision made possible the discovery and application of concepts like those of uni­ form convergence, uniform continuity, summability, and asymptotic expansions, which could neither be studied nor even expressed in the conceptual fi'amework of eighteenth­ century mathematics. The very names we use for some basic ideas in analysis reflect the achievements of nine­ teenth-century mathematicians in the foundations of ana­ lysis: Abel's convergence theorem, the Cauchy criterion, the Riemann integral, the Bolzano-Weierstrass theorem, the Dedekind cut. And the symbols of nineteenth-century rigor-the ubiquitous delta and epsilon-first appear in their accustomed logical roles in Cauchy's lectures on the calculus in 1823. Of course nineteenth-century analysis owed much to eighteenth-century analysis. But the nineteenth-century foundations of the calculus cannot be said to have grown naturally or automatically out of earlier views.
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