Analysis by Its History (Undergraduate Texts in Mathematics / Readings in Mathematics) [Hardcover]

Ernst Hairer (Author), Gerhard Wanner (Author)

Book Description

ISBN-10: 0387945512 | ISBN-13: 978-0387945514 | Publication Date: 1996

This book presents first-year calculus roughly in the order in which it was first discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. Many quotations are included to give the flavor of the history. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, as well as researchers.

Editorial Reviews

Review

"...well done, attractively designed...And above all, it proposes an interesting approach to teaching analysis." Internationale Mathematische Nachrichten

From the Back Cover

This book presents first-year calculus roughly in the order in which it first was discovered. The first two chapters show how the ancient calculations of practical problems led to infinite series, differential and integral calculus and to differential equations. The establishment of mathematical rigour for these subjects in the 19th century for one and several variables is treated in chapters III and IV. The text is complemented by a large number of examples, calculations and mathematical pictures and will provide stimulating and enjoyable reading for students, teachers, as well as researchers.   From the reviews: The aim of this interesting new contribution to the series Readings in Mathematics is an attempt to restore the historical order in the presentation of basic mathematical analysis...such a historical approach can provide a very fruitful and interesting approach to mathematical analysis. - Jean Mawhin, Zentralblatt The authors include a large number of once-traditional subjects which have now vanished from the analysis curriculum, at least in the standard American courses. Thus we find continued fractions, elliptic integrals, the Euler-MacLaurin summation formula, etc., most of which are found only in more compendious works. Many of the exercises are inspired by original papers, with the bibliographic references sometimes given. The work is very well illustrated. The book is definitely an analysis text, rather than a history, but a great deal of reliable historical material is included. For those seeking an alternative to the traditional approach, it seems to me to be of great interest. - Thomas Archibald, Mathematical Reviews The authors...have assembled an impressive array of annotated results, quotations, tables, charts, figures and drawings, many copied from original documents....they write with great enthusiasm and with evident affection for both analysis and history. - John Troutman, American Mathematical Monthly

Product Details

• Hardcover: 387 pages
• Publisher: Springer (1996)
• Language: English
• ISBN-10: 0387945512
• ISBN-13: 978-0387945514
• Product Dimensions: 9.3 x 6.3 x 1.1 inches
• Shipping Weight: 1.5 pounds (View shipping rates and policies)
• Average Customer Review: 4.2 out of 5 stars  See all reviews (9 customer reviews)
• Amazon Best Sellers Rank: #1,488,821 in Books (See Top 100 in Books)

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34 of 37 people found the following review helpful:

5.0 out of 5 stars Brilliant, unorthodox, a very commendable approach., September 1, 2000

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R. Ball (London W14, England United Kingdom) - See all my reviews
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This review is from: Analysis by Its History (Undergraduate Texts in Mathematics / Readings in Mathematics) (Hardcover)

I wish there had been books like this when I was at (high)school! It is one of those rare books that bridge the yawning gap between the popular personalised history books that are so inspiring to the young mind, (eg. E.T.Bell's "Men of Mathematics", Kasner & Newman's "Mathematics & the Imagination" or Kak & Ulam's "Logic and(?) Mathematics") and the terse, somewhat desiccated university text books. This can leave the undergraduate not fully appreciating the motivation for exhaustive rigor and also losing any perspective of where the abstract theorems and lemmas are ultimately distilled from. This book links the historical characters, controversies and challenges with the modern techniques that gradually emerged to deal with the pathological behaviour of sets, series and functions. It would be a mistake to confuse this book, as some of your reviewers have done, with the many first-year undergraduate texts that are available. It could be regarded as a sophisticated high school book that gives a real flavour of how the classical problems are treated in modern rigorous style, or alternatively as a colourful motivational aid to early undergraduate analysis courses. I hope that the publishers encourage similar ventures in other branches of the subject, for instance algebra, differential & integral equations, probability and perhaps even quantum theory.

12 of 13 people found the following review helpful:

5.0 out of 5 stars A Good Mix of Calculus and its History, January 9, 2007

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J. Keesling "Jed" (Gainesville, FL USA) - See all my reviews
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This review is from: Analysis by Its History (Undergraduate Texts in Mathematics / Readings in Mathematics) (Hardcover)

This books gives a unique approach to Calculus using its historical development. The most notable feature of the book is that the order of topics is reversed from what has become standard in current textbooks. It begins with the analysis of areas and volumes. This is followed by derivatives, continuity, and the notion of function. This is the order in which analysis developed, but not the order one would follow if building understanding of the subject from a foundation upward. Historically, the foundations were laid last.

The book is not intended as a history of analysis. It is rather intended as a textbook or reference in which the topics are presented in historical order. The historical background is intended to give insight into a modern view of the subject. It accomplishes this admirably.

The book is filled with examples, quotes, vignettes, historical background, computer graphics, and copies of original documents. Special topics are interspersed throughout. The book gives us a fresh and envigorating view of Calculus. It is an invaluable resource.

26 of 35 people found the following review helpful:

3.0 out of 5 stars Mathematics made concrete, December 24, 1999

By 

Fabrice P. Laussy (Southampton (UK)) - See all my reviews
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This review is from: Analysis by Its History (Undergraduate Texts in Mathematics / Readings in Mathematics) (Hardcover)

This book's aim is really to teach analysis. It is not a book on history of science, the kind you read like a novel. The difference with a standard text is it proceeds after the historic evolution. It's quite an audacious approach, for mathematical rigor flowing from axioms towards theorems through lemmas and hypothesis doesn't fit well with historical connections which are chaotic, incomplete and abstruse. It's really not like the (many) books which have great concern for historical context discussed in appendices or footnotes. Here the history is underlying everything, but--once again--it isn't an history book anyway. Theorems are proved.

I do not recommend it, not even to beginners, though it can be a good introductory book. It indeed is much less abstract than a classic text of the same level, with many illustrations, and in depth detailed explanations (for beginners serious after the idea of doing Mathematics, I suggest Rudin's "Principles of Mathematical Analysis"). It has many things at its advantage anyway. It shows for instance how many astoundingly insecure results were granted, and thus illustrates well the experimental aspect of mathematics, often denied. It comes with false proof (for instance Euler's taking limit of series or Ampere's theorem about derivatives of continuous functions), and reveal the difficulty of such giants like the Bernoulli, Cauchy or Weierstrass with the problems of convergence. It sure helps understand how mathematics are partly a science of discovery, not a science of just invention. It shows mathematicians are mere people, after all, and that one's difficulties have little significance. In the overall, it sheds light on the genuine mathematical world, which is often seen as a cold topic where one makes its way to the solution through lengthy linear computations. This a book that can definitely make you love mathematics, and ask once you caught the hint for more abstract, deeper texts (Rudin for instance).

Thus while the merging (once more not the simple association) of the theory with its development's history was not necessary, it has been _very well_ done. If this approach pleases you, this book is for you.