The Calculus Gallery: Masterpieces from Newton to Lebesgue Paperback – July 21, 2008

Begin with a quote: "mathematicians have gone from curves to functions, from geometry to algebra, and from intuition to cold, clear logic." (page 220). That line, from the Afterword, sums up the content of this exposition. I highlight Dunham's expository skills, almost second-to-none, his book is a delight to peruse. Read: "by 1880, integration had come to be regarded primarily as the inverse of differentiation, occupying a secondary position in the pantheon of mathematical concepts." (page 86). Now, Cauchy disagreed: "he believed the integral must have an independent existence." Thus, if you never get anything else from calculus, Cauchy is the name to remember ! Pay attention, physicists !(1) Of late, there has been a happy increase in the quality and quantity of mathematics books which pay some attention to the evolution of the subject. Dunham was instrumental in getting us there ! Textbooks advocating this approach now abound: Bressoud's Radical Approach to Real Analysis, Hairer and Wanner's Analysis By Its History and Stahl's Real Analysis. Those textbooks are for the classroom. Dunham does not claim to be such a classroom text (for example, there are no exercises). Dunham is motivator, an instance, motivating the oft-utilized "integration-by-parts" formula, here Leibniz is guide (page 29).(2) An overarching guidepost in his evolutionary reading of calculus (and one I find utterly fascinating) is a reinforcing of the "separation of calculus from geometry." Learn of Weierstrass' function: "lies somewhere beyond the intuition, far removed from geometrical diagrams..." (page 148). A parallel thread, yet another guide, is an escape from curves and simple formulas to arbitrary functions. Read: "generality lies at the heart of modern analysis." (page 116).(3) About proofs: As is well-known, you won't get far beyond elementary mathematics (manipulative skill) without understanding mathematical proof. It is here that Dunham's expository skills excel. Find Cauchy's verbal description (of limits, page 129) followed by Weierstrass' uniform convergence, learn: "these ideas appear throughout the remainder of the book." (page 137). Seeing, how "intuition misleads." You utilize the "triangle inequality" (for example, page 144). Baire is the most difficult encounter herein (page 191).(4) Consider that Cantor and Dedekind's approach was "the final step in the separation of calculus from geometry." (page 161). It is well worth revisiting Dunham's recapitulation of "completeness," that is, four incarnations found here (pages 159 & 160). Finally, Lebesgue recapitulates the "fundamental theorem of calculus" (Dunham: "back in all its glory" page 218) which is met in different guise throughout. Learn to appreciate inequalities (page 130). Again: "lacks the charm of intuition and the immediacy of geometry." (5) In conclusion: Dunham has written a thoughtful book detailing an evolution of calculus. There are end notes to lead the reader elsewhere (yet, no bibliography). The index is three pages (you will not find the word "sequence"). Those minor quibbles aside, you get a lovely excursion into analysis, worth the effort.

This is one of the most interesting books on the history of Calculus that I have ever read. It does require a moderate amount of mathematical knowledge (although not more than the standard first year undergraduate Analysis courses), but it is written with such a brilliance that one reads it with the eagerness more frequently experienced when reading a good thriller. But then, the history of Mathematical Analysis is, when we look at it in the proper way, one of the most fascinating and thrilling episodes in the intellectual history of mankind. This book is but one of the different stories that can be written: not being the history of Calculus, not even a history, it is, as the title indicates, a gallery, like an art gallery: reading along it we travel from the founding fathers Newton and Leibnitz, until the pinnacle of rigour and generality (and beauty!!) attained in the beginning of the 20th Century by Baire and Lebesgue. Along the way we visit some of the brilliant ideas of the Bernoulli brothers, Euler, Cauchy, Riemann, Liouville, Weierstrass, Cantor, and Volterra, and we see how, in two and a half centuries, the combined work of these (and others) outstanding minds shaped one of the most beautiful and powerful of all human creations. Like in any art gallery, a lot of names, some of then genius, are missing, but what is there is enough to tell a story, to disquiet and to awe the visitor.All in all, this is a magnificent book that all teachers and students of mathematics should read. It is also a work that should sadden us for the beauty herein is not likely to be appreciated by many more. It comes to mind the following famous poem by Fernando Pessoa, one of the most celebrated of all Portuguese poets (in my loose translation): Newton's binomial is as beautiful as the Venus of Milo. The trouble is that few people can be aware of this. And the (generalized) Newton's binomial expansion is just the beginning: it is the very first section of the first chapter in this book...

Shows the power of counterexamples in mathematical development and a great reminder of the key issues calculus learners would have encountered if they were lucky enough to have had good teachers. If you didn't, this book would open a whole new world of mathematical thinking to you. A great book for teachers of calculus at all levels and an exciting story of the calculus for all exposed to the ideas of the calculus now or in their past. For me, as a former researcher of the role of counterexamples in the development of mathematical thinking, this book highlights the role counterexamples played in mathematicians thinking about the key problems and solutions in moving from quick computations to the real subject of real analysis. It is a great story of mystery and resolution, missing in the exposure to the calculus for many learners. If you are not so mathematically inclined, you can gloss over some of the details and see the key ideas and mysteries. If you have a mathematical bent, it is also exciting to see how great mathematicians thought as they grappled with the big ideas of calculus and analysis. It is a very hard book to put down, even to eat or sleep — a great read!

Pequeñas obras de arte cuya lectura te pone en la piel de algunos de los grandes matemáticos que hicieron la historia del ''nuevo'' Cálculo (Newton, Leibniz,Cauchy, Riemann, Lebesgue.....)pinceladas geniales para los amantes de la historia de la matemática. Por qué nadie tradujo este libro al español?

Dunham is a skilled and knowledgable author who chooses interesting parts of the developments of Calculus as it evolves from dubious foundations to the firm foundations used at the present, The reader encounters several surprises.

If you are reading this review you probably have an interest in Mathematics and Calculus. If you do, then buy this book. It gives an interesting account of the history of Calculus and of the characters that have refined and developed it over the course of centuries. The book glosses over some aspects with the odd note implying that modern mathematicians would be horrified if anyone used such a technique today, but it does not offer any insight into these developments (e.g. the handling of discontinuities) in the modern age.If however you are a mathematician looking for a good overview of Calculus techniques then this is definitely not it, with the emphasis very much on the personalities and the context of their work.

英文は読みやすく、また他のダンハムの本と同様に彼の数学史への情熱が伝わってきます。読者としては大学教養程度の数学的知識を必要としますが、数学と英文への熱心さがあれば十分に読みこなすことができるのではないかと思います。この本は微積分学の通史ではなく、彼が意図している数学史を美術館のギャラリーのように見て歩くというスタイルです。ある程度のバックグランド的な知識を持っていることが要求されるように思えますが、内容的にうまく読者を引き込んで読ませてしまいます。ダンハムのストーリーテラーとしての力が発揮された一冊だと思いました。

Both the book itself and the timeliness of delivery are superb.