The Calculus Gallery: Masterpieces from Newton to Lebesgue by William Dunham (PDF)

Description

More than three centuries after its creation, calculus remains a dazzling intellectual achievement and the gateway into higher mathematics. This book charts its growth and development by sampling from the work of some of its foremost practitioners, beginning with Isaac Newton and Gottfried Wilhelm Leibniz in the late seventeenth century and continuing to Henri Lebesgue at the dawn of the twentieth–mathematicians whose achievements are comparable to those of Bach in music or Shakespeare in literature. William Dunham lucidly presents the definitions, theorems, and proofs. “Students of literature read Shakespeare; students of music listen to Bach,” he writes. But this tradition of studying the major works of the “masters” is, if not wholly absent, certainly uncommon in mathematics. This book seeks to redress that situation. Like a great museum, The Calculus Gallery is filled with masterpieces, among which are Bernoulli’s early attack upon the harmonic series (1689), Euler’s brilliant approximation of pi (1779), Cauchy’s classic proof of the fundamental theorem of calculus (1823), Weierstrass’s mind-boggling counterexample (1872), and Baire’s original “category theorem” (1899). Collectively, these selections document the evolution of calculus from a powerful but logically chaotic subject into one whose foundations are thorough, rigorous, and unflinching–a story of genius triumphing over some of the toughest, most subtle problems imaginable. Anyone who has studied and enjoyed calculus will discover in these pages the sheer excitement each mathematician must have felt when pushing into the unknown. In touring The Calculus Gallery, we can see how it all came to be.

User’s Reviews

Editorial Reviews: Review “William Dunham, Winner of the 2006 Lester R. Ford Award for his expository article on “Touring the Calculus Gallery” in The American Mathematical Monthly””One of Choice’s Outstanding Academic Titles for 2005″”The Calculus Gallery is a wonderful book. The style is inviting; the explanations are clear and accessible. . . . Mathematicians, scientists, and historians alike can learn much that is interesting, much that is mathematically significant, and a good deal that is both.”—Judith V. Grabiner, Science”[A] brilliant book. . . . I predict that Dunham’s book will itself come to be considered a masterpiece in its field.”—Victor J. Katz, American Scientist”What distinguishes this selection is it truly provides a history of mathematics, not just a history of mathematicians. . . . If a better historical treatment of the development of the calculus is available, this reviewer has yet to see it. . . . Essential.” ― Choice”A joy to read, The Calculus Gallery showcases one of the great intellectual pursuits of all time and, in the words of John von Neumann, ‘the first achievement of modern mathematics.’ Thirteen scholars, beginning with Newton and Leibniz, who gave birth to calculus in the seventeenth century, are featured in this sequential development of the important ideas that shaped calculus as we know it and gave rise to modern analysis. . . . [I]t is a lovely and engaging gallery of the ‘masters’ that belongs in the library of everyone who seriously teaches or studies the subject.”—Diane M. Spresser, Mathematics Teacher”A fascinating, competent visit too the calculus gallery.”—Eberhard Knobloch, Zentralblatt MATH Review “The Calculus Gallery is one of the best efforts at mathematical exposition I have ever read! Dunham presents in detail and in his own words the sequence of ideas of classical giants of mathematics, but each new idea is described in modern terms and notation, so I had absolutely no trouble following along. Furthermore―and this is an astounding achievement―the entire work has a tightly woven development. If it were a detective story I would say it had a plot with no loose ends. An amazing feat. I wish I could plan a single lecture, never mind a course or a book, that well!”―Henry Pollak, Teachers College, Columbia University”What a fine resource! All of the famous functions that have shaped calculus and analysis parade before the reader in the original words of their creators. Bill Dunham has produced an excellent volume that teachers and students will enjoy and appreciate.”―Thomas Banchoff, Brown University”Bill Dunham has done it again. The Calculus Gallery is a masterly journey through the works of thirteen mathematicians who formulated, formalised, and reformed the calculus into the modern analysis we learn today. Readers of his earlier books have learned to expect a clarity of exposition that few others can attain: they will not be disappointed.”―Robin Wilson, author of Four Colors Suffice”This is an excellent book―an amazing mathematical page-turner. William Dunham has done the seemingly impossible: he has taken some difficult, advanced mathematics and, without sacrificing the technical details, written a lively, readable book about it.”―Barry Cipra, author of Misteaks . . . and How to Find Them Before the Teacher Does”Pedagogically excellent and extremely well written, The Calculus Gallery bridges the gap between general histories and detailed studies of individual mathematicians. Dunham has described mathematical developments in an engaging style rarely found in literature of this kind.”―Annette Imhausen, Trinity Hall, Cambridge”A welcome addition to the literature. The idea of presenting a ‘museum of mathematics’ is new. It allows the author to present a nonstandard selection of theorems, so that even mathematicians with a strong historical background will learn a few things.”―Franz Lemmermeyer, Bilkent University, author of Reciprocity Laws: From Euler to Eisenstein From the Back Cover “The Calculus Gallery is one of the best efforts at mathematical exposition I have ever read! Dunham presents in detail and in his own words the sequence of ideas of classical giants of mathematics, but each new idea is described in modern terms and notation, so I had absolutely no trouble following along. Furthermore–and this is an astounding achievement–the entire work has a tightly woven development. If it were a detective story I would say it had a plot with no loose ends. An amazing feat. I wish I could plan a single lecture, never mind a course or a book, that well!”–Henry Pollak, Teachers College, Columbia University”What a fine resource! All of the famous functions that have shaped calculus and analysis parade before the reader in the original words of their creators. Bill Dunham has produced an excellent volume that teachers and students will enjoy and appreciate.”–Thomas Banchoff, Brown University”Bill Dunham has done it again. The Calculus Gallery is a masterly journey through the works of thirteen mathematicians who formulated, formalised, and reformed the calculus into the modern analysis we learn today. Readers of his earlier books have learned to expect a clarity of exposition that few others can attain: they will not be disappointed.”–Robin Wilson, author of Four Colors Suffice”This is an excellent book–an amazing mathematical page-turner. William Dunham has done the seemingly impossible: he has taken some difficult, advanced mathematics and, without sacrificing the technical details, written a lively, readable book about it.”–Barry Cipra, author of Misteaks . . . and How to Find Them Before the Teacher Does”Pedagogically excellent and extremely well written, The Calculus Gallery bridges the gap between general histories and detailed studies of individual mathematicians. Dunham has described mathematical developments in an engaging style rarely found in literature of this kind.”–Annette Imhausen, Trinity Hall, Cambridge”A welcome addition to the literature. The idea of presenting a ‘museum of mathematics’ is new. It allows the author to present a nonstandard selection of theorems, so that even mathematicians with a strong historical background will learn a few things.”–Franz Lemmermeyer, Bilkent University, author of Reciprocity Laws: From Euler to Eisenstein About the Author William Dunham, Koehler Professor of Mathematics at Muhlenberg College, is the author of Journey Through Genius: The Great Theorems of Mathematics; The Mathematical Universe; and Euler: The Master of Us All. He has received the Mathematical Association of America’s George Pólya, Trevor Evans, and Lester R. Ford awards, as well as its Beckenbach Prize for expository writing. Read more

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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!

⭐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.

⭐By chance, the proofs I was most interested in didn’t show up in the book. But it is an excellent book.

⭐Buen libro para leer si te gustan las matemáticas y estás interesado en la historia. Para mi gusto, peca algo de excesivamente teórico (aunque para seguramente para un matemática sea una delicia). Muy interesante.

⭐

⭐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?

⭐

⭐The book I had been looking for a long time, entertaining like Agatha Christie’s novels. A great deal of math if course.

Keywords

Free Download The Calculus Gallery: Masterpieces from Newton to Lebesgue in PDF format
The Calculus Gallery: Masterpieces from Newton to Lebesgue PDF Free Download
Download The Calculus Gallery: Masterpieces from Newton to Lebesgue 2008 PDF Free
The Calculus Gallery: Masterpieces from Newton to Lebesgue 2008 PDF Free Download
Download The Calculus Gallery: Masterpieces from Newton to Lebesgue PDF
Free Download Ebook The Calculus Gallery: Masterpieces from Newton to Lebesgue