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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.… (plus d'informations)
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 Leibitz, until the pinacle of rigor and generality (and beauy!!) 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... ( )
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.
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