Customer Reviews

Analysis by Its History (Undergraduate Texts in Mathematics)

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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.

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.

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.

This very interesting book contains very good historical perspectives on analysis. If you want to know how things like trigonometric functions, logarithms, infinite series, differential and integral calculus and differential equations come about (but written from a modern viewpoint), then this is the book for you. It is not a book for casual reading like E T Bell's Men of Mathematics, but the reader will learn a lot of college and undergraduate mathematics along the way.

Starting with the bottom line, THIS IS NOT a first-year undergraduate course in Mathematical Analysis and should never be used as such (you wouldn't use Hardy's A Course in Pure Mathematics, either, for all its venerable merits).

If you ARE doing first-course Analysis, you'll have to stick to whatever textbook your professor chooses (hopefully NOT Rudin's masterpiece). If you get stuck with that, you will do well to search for a more user friendly alternative or supplement in the line of BINMORE or BRANNAN's book (there are others, for the market is flooded in this area), some good text that emphasizes the fundamentals taking his time, that emphasizes the practical connection between Analysis and (high-school) Calculus, and that uses a lot of visual illustration for good. If you remain stuck still, then you will have 1) to rethink the wiseness in choosing a career that requires advanced mathematics, and/or 2) as a last resource there is the "Yet Another Introduction to Analysis", which isn't the standard textbook either, but it can get you unstuck and on safer ground.

Then, what on Earth do we need HAIRER and WANNER's Analysis by it's history for?
For starters, it's a VERY BEAUTIFUL book (in the line of Proofs by the Book, if you know what I mean, but a step lower). The explanations are right and straighrforward, at an accessible level that reminds me of that other gem of a book, DUNHAM's Euler: The Master of Us All. It has a lot of examples and graphic elements and takes you into a wonderful journey into what is rather more Calculus than Analysis, alohg the path taken by the great classics (without being a book on the History of Calculus -try EDWARDS for that, and forget Boyer's-). In addition, if you just don't read casually the book for the sheer pleasure of it, but you DO WORK through it and do the exercises, you will end with a sure grasp of the fundamentals of Calculus (and thet is what Analysis is supposed to be), a surer grasp than you would have memorizing the whole of Rudin's consecrated and masterful text, Another reviewer takes issue with the many short quotations that decorate the book, as epigraphs to every little section, that are intended to maintain a historic flavor. Some of them make good sense, and a few others are rather quizzical or downright cryptic: all in all, it's a rather idiosyncratic and funny feature of the book, but it's not a matter to take issue with.

IN SUMMARY: it's NOT supposed to be a standard textbook nor should it be used as one. But if you think that Calculus (and Analysis) is the greatest endeavor mankind has engaged in -take that "com grano salis"- and you remember still your high-school Calculus although you're not a working mathematician, but something more on the amateur side of it... if you appreciate mathematical books looking the beautiful way they should all look like (and remember -or get outright- Proofs from the Book)...
if you'd love getting a refresher on Calculus cum Analysis, THEN PLEASE DON'T MISS that book, so very clearly and beautifully conceived and written. One book like this occurs only a few times in a century, unfortunately you and I haven't written this one, at the very least let's share the joy of reading it, at whatever depth level we feel like!

This book was my secret weapon in analysis class. It belongs on the shelf of every student and professional. I just wish the authors would provide solutions to problems so we can check our work. I recommend this book head and shoulders over Rudin's Principles of Math. Analysis.

I return to this book again and again just out of sheer pleasure. The depth of scholarship of the authors shines through on every page and the choice of historical material is fascinating.Topics like compactness and uniform convergence can here be seen to have arisen out of genuine necessity-they are not (as would seem from other books)mere names in a standard syllabus. If you have any mathematica interest at all, take this book on holiday and sleep with it under the pillow to extract more from it by osmotic pressure overnight.

Chapters 1 and 2 treat classical differential and integral calculus. This is a disorganised mess of historical and mathematical tidbits. It's not a great place to learn calculus, but it's good side reading since there are many interesting topics, some of which are often neglected in today's books: continued fractions (!), complex functions already on page 56, an interesting section on differential geometry, Euler-Maclaurin summation, etc. The authors also have the very commendable habit of including charming facsimiles of figures from original works.

Chapter 3 "Foundations of Classical Analysis" and chapter 4 "Calculus in Several Variables" are almost completely ahistorical. The "by its history" part of the exposition is restricted to some scattered superficial remarks, including silly nonsense such as that if Leibniz had know of the intricate progression of theorems needed for a modern proof of the "fundamental theorem" then "he might not have had the courage to state and use this theorem" (p. 239). And in another parodic misuse of the historical perspective, the authors introduce Descartes's folium merely for the purpose of practising the determination of stationary points (p. 322)---of course, Descartes introduced the folium for a much more interesting purpose, but to learn that story we must look for an "Analysis by Its History" book worthy of its name.

The idea behind this book, of presenting the topics involved in calculus in the order in which they were developed in history, is commendable. However, the explanations, at least in the first few pages, are far too concise and confusing for my liking.