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Zentralblatt MATH DatabaseA fascinating, competent visit too the calculus gallery.
— Eberhard Knobloch
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.
"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
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