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The Calculus Gallery


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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"

Table of Contents

Illustrations ix Acknowledgments xiii INTRODUCTION 1 CHAPTER 1: Newton 5 CHAPTER 2: Leibniz 20 CHAPTER 3: The Bernoullis 35 CHAPTER 4: Euler 52 CHAPTER 5: First Interlude 69 CHAPTER 6: Cauchy 76 CHAPTER 7: Riemann 96 CHAPTER 8: Liouville 116 CHAPTER 9: Weierstrass 128 CHAPTER 10: Second Interlude 149 CHAPTER 11: Cantor 158 CHAPTER 12: Volterra 170 CHAPTER 13: Baire 183 CHAPTER 14: Lebesgue 200 Afterword 220 Notes 223 Index 233

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 Polya, Trevor Evans, and Lester R. Ford awards, as well as its Beckenbach Prize for expository writing.


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

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