056 Analysis by Its History.pdf

Undergraduate Texts in Mathematics

Readings in Mathematics

Editors

S. Axler

K.A. Ribet

Graduate Texts in Mathematics

Readings in Mathematics

Ebbinghaus/Hermes/Hirzebruch/Koecher/Mainzer/Neukirch/Prestel/Remmert: Numbers

Fulton/Harris: Representation Theory: A First Course

Murty: Problems in Analytic Number Theory

Remmert: Theory of Complex Functions

Walter: Ordinary Differential Equations

Undergraduate Texts in Mathematics

Readings in Mathematics

Anglin: Mathematics: A Concise History and Philosophy

Anglin/Lambek: The Heritage of Thales

Bressoud: Second Year Calculus

Hairer/Wanner: Analysis by Its History

Hämmerlin/Hoffmann: Numerical Mathematics

Isaac: The Pleasures of Probability

Knoebel/Laubenbacher/Lodder/Pengelley: Mathematical Masterpieces: Further Chronicles

by the Explorers

Laubenbacher/Pengelley: Mathematical Expeditions: Chronicles by the Explorers

Samuel: Projective Geometry

Stillwell: Numbers and Geometry

Toth: Glimpses of Algebra and Geometry, Second Edition

E. Hairer G. Wanner

Analysis by

Its History

Editors

E. Hairer

G. Wanner

Department of Mathematics

University of Geneva

Geneva, Switzerland

Editorial Board

S. Axler

K.A. Ribet

Mathematics Department

San Francisco State

University of California

University

atBerkeley

San Francisco, CA 94132

Berkeley, CA 94720-3840

USA

axler@sfsu.edu

ribet@math.berkeley.edu

ISBN: 978-0-387-77031-4

e-ISBN: 978-0-387-77036-9

Library of Congress Control Number: 2008925883

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Preface

. . . that departed from the traditional dry-as-dust mathematics textbook.

(M. Kline, from the Preface to the paperback edition of Kline 1972)

Also for this reason, I have taken the trouble to make a great number of

drawings. (Brieskorn & Knorrer, Plane algebraic curves , p. ii)

. . . I should like to bring up again for emphasis . . . points, in which my

exposition differs especially from the customary presentation in the text-

books:

1. Illustration of abstract considerations by means of ﬁgures.

2. Emphasis upon its relation to neighboring ﬁelds, such as calculus of dif-

ferences and interpolation . . .

3. Emphasis upon historical growth.

It seems to me extremely important that precisely the prospective teacher

should take account of all of these. (F. Klein 1908, Engl. ed. p. 236)

Traditionally, a rigorous ﬁrst course in Analysis progresses (more or less) in the

following order:

sets,

mappings

⇒ limits,

⇒ derivatives ⇒ integration.

On the other hand, the historical development of these subjects occurred in reverse

order:

Cantor 1875

Dedekind

⇐ Cauchy 1821

Weierstrass

⇐ Newton 1665

Leibniz 1675

⇐ Archimedes

In this book, with the four chapters

Chapter I. Introduction to Analysis of the Inﬁnite

Chapter II. Differential and Integral Calculus

Chapter III. Foundations of Classical Analysis

Chapter IV. Calculus in Several Variables,

we attempt to restore the historical order, and begin in Chapter I with Cardano,

Descartes, Newton, and Euler’s famous Introductio . Chapter II then presents 17th

and 18th century integral and differential calculus “on period instruments” (as a

musician would say). The creation of mathematical rigor in the 19th century by

Cauchy, Weierstrass, and Peano for one and several variables is the subject of

Chapters III and IV.

This book is the outgrowth of a long period of teaching by the two authors.

In 1968, the second author lectured on analysis for the ﬁrst time, at the University

of Innsbruck, where the ﬁrst author was a ﬁrst-year student. Since then, we have

given these lectures at several universities, in German or in French, inﬂuenced by

many books and many fashions. The present text was ﬁnally written up in French

for our students in Geneva, revised and corrected each year, then translated into

English, revised again, and corrected with the invaluable help of our colleague

John Steinig. He has corrected so many errors that we can hardly imagine what

we would have done without him.

continuous

functions

Kepler 1615

Fermat 1638

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