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Book DetailsSpectrumVolume: 64; 2010; 287 ppRecipient of the Mathematical Association of America's Beckenbach Book Prize in 2014!
Judith Grabiner, the author of A Historian Looks Back, has long been interested in investigating what mathematicians actually do, and how mathematics actually has developed. She addresses the results of her investigations not principally to other historians, but to mathematicians and teachers of mathematics. This book brings together much of what she has had to say to this audience. The centerpiece of the book is The Calculus as Algebra: J.L. Lagrange, 1736–1813. The book describes the achievements, setbacks, and influence of Lagrange's pioneering attempt to reduce the calculus to algebra. Nine additional articles round out the book describing the history of the derivative; the origin of deltaepsilon proofs; Descartes and problem solving; the contrast between the calculus of Newton and Maclaurin, and that of Lagrange; Maclaurin's way of doing mathematics and science and his surprisingly important influence; some widely held myths about the history of mathematics; Lagrange's attempt to prove Euclid's parallel postulate; and the central role that mathematics has played throughout the history of western civilization. The development of mathematics cannot be programmed or predicted. Still, seeing how ideas have been formed over time and what the difficulties were can help teachers find new ways to explain mathematics. Appreciating its cultural background can humanize mathematics for students. And famous mathematicians' struggles and successes should interest—and perhaps inspire—researchers. Readers will see not only what the mathematical past was like, but also how important parts of the mathematical present came to be.

Table of Contents

The Calculus as Algebra

Preface to the Garland Edition

Acknowledgements

Introduction

Chapter 1. The Development of Lagrange’s Ideas on the Calculus: 1754–1797

Chapter 2. The Algebraic Background of the Theory of Analytic Functions

Chapter 3. The Contents of the Fonctions Analytiques

Chapter 4. From ProofTechnique to Definition: The PreHistory of DeltaEpsilon Methods

Conclusion

Appendix

Bibliography

Selected Writings

Chapter 1. The Mathematician, the Historian, and the History of Mathematics

Chapter 2. Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus

Chapter 3. The Changing Concept of Change: The Derivative from Fermat to Weierstrass

Chapter 4. The Centrality of Mathematics in the History of Western Thought

Chapter 5. Descartes and ProblemSolving

Chapter 6. The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and Their Legacy

Chapter 7. Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions

Chapter 8. Newton, Maclaurin, and the Authority of Mathematics

Chapter 9. Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while Promoting Maths

Chapter 10. Why Did Lagrange “Prove” the Parallel Postulate?


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Judith Grabiner, the author of A Historian Looks Back, has long been interested in investigating what mathematicians actually do, and how mathematics actually has developed. She addresses the results of her investigations not principally to other historians, but to mathematicians and teachers of mathematics. This book brings together much of what she has had to say to this audience. The centerpiece of the book is The Calculus as Algebra: J.L. Lagrange, 1736–1813. The book describes the achievements, setbacks, and influence of Lagrange's pioneering attempt to reduce the calculus to algebra. Nine additional articles round out the book describing the history of the derivative; the origin of deltaepsilon proofs; Descartes and problem solving; the contrast between the calculus of Newton and Maclaurin, and that of Lagrange; Maclaurin's way of doing mathematics and science and his surprisingly important influence; some widely held myths about the history of mathematics; Lagrange's attempt to prove Euclid's parallel postulate; and the central role that mathematics has played throughout the history of western civilization. The development of mathematics cannot be programmed or predicted. Still, seeing how ideas have been formed over time and what the difficulties were can help teachers find new ways to explain mathematics. Appreciating its cultural background can humanize mathematics for students. And famous mathematicians' struggles and successes should interest—and perhaps inspire—researchers. Readers will see not only what the mathematical past was like, but also how important parts of the mathematical present came to be.

The Calculus as Algebra

Preface to the Garland Edition

Acknowledgements

Introduction

Chapter 1. The Development of Lagrange’s Ideas on the Calculus: 1754–1797

Chapter 2. The Algebraic Background of the Theory of Analytic Functions

Chapter 3. The Contents of the Fonctions Analytiques

Chapter 4. From ProofTechnique to Definition: The PreHistory of DeltaEpsilon Methods

Conclusion

Appendix

Bibliography

Selected Writings

Chapter 1. The Mathematician, the Historian, and the History of Mathematics

Chapter 2. Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus

Chapter 3. The Changing Concept of Change: The Derivative from Fermat to Weierstrass

Chapter 4. The Centrality of Mathematics in the History of Western Thought

Chapter 5. Descartes and ProblemSolving

Chapter 6. The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and Their Legacy

Chapter 7. Was Newton’s Calculus a Dead End? The Continental Influence of Maclaurin’s Treatise of Fluxions

Chapter 8. Newton, Maclaurin, and the Authority of Mathematics

Chapter 9. Why Should Historical Truth Matter to Mathematicians? Dispelling Myths while Promoting Maths

Chapter 10. Why Did Lagrange “Prove” the Parallel Postulate?