# Analysis by Its History

Springer New York, Oct 1, 2000 - Mathematics - 379 pages
. . . 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. (Brieskom & 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, Eng\. ed. p. 236) Traditionally, a rigorous first course in Analysis progresses (more or less) in the following order: limits, sets, '* continuous '* derivatives '* integration. mappings functions On the other hand, the historical development of these subjects occurred in reverse order: Archimedes Cantor 1875 Cauchy 1821 Newton 1665 . ;::: Kepler 1615 Dedekind . ;::: Weierstrass . ;::: Leibniz 1675 Fermat 1638 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.

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I am not a mathematician and never have I taken real analysis course. My knowledge is restricted to "Advanced Mathematics by Kreyszig".
I am interested however, in knowing more about this topic. The
book I tried to read was "Real Analysis by Rudin" (which is ofcourse more advanced than - functional analysis. It is a bible but I gave up. Could not get the real picture - the overall storyline or - why the concepts were being presented the way they were, in that order. But this book "Analysis by its History" is simply wonderful. I am thoroughly enjoying reading it. Next step would be to solve problems and finally move on to "Real Analysis". Every feature - even the old methods to solve algebraic equations are very fascinating and many concepts are well clarified and embellished by mathematical diagrams. Moreover, I did not notice many typographical errors, very well edited and excellent pictures explaining some concepts. The gradual progression to difficult topics is also well thought out. I agree that it is a hard read getting to understand the concepts - but it is not the book - it is the topic which is that way.
Things sorely missing - Lebesgue integrals .....
A must read for those who will get swayed by Landau's quote about his doubts that his daughters really understand what it means to add and multiply two real numbers - even if they have done it efficiently so many times.

### About the author (2000)

Ernst Hairer is a Professor of Mathematics at the University of Geneva and has been awarded the Henrici Prize by the Society of Industrial and Applied Mathematics.