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A Historian Looks Back: The Calculus as Algebra and Selected Writings

A Historian Looks Back: The Calculus as Algebra and Selected Writings

By Judith V. Grabiner

Print ISBN: 978-0-88385-572-0
282 pp., Hardbound, 2010
List Price: $64.00
Member Price: $48.00
Series: Spectrum

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Judith Grabiner has written extensively on the history of mathematics. This collection, representing some of Grabiner's finest work, highlights the benefits of studying the development of mathematical ideas and the relationship between culture and mathematics.

A large part of the book (Part I) is a welcome reprinting of Grabiner's The Calculus as Algebra: J.-L. Lagrange, 1736-1813 (1990), which focuses on Lagrange's pioneering effort to reduce the calculus to algebra.

Ten articles (Part II) span a range of other mathematical topics, including widely held myths about the history of mathematics and the work of such mathematicians as Descartes, Newton, and Maclaurin. Six of these articles won awards from the MAA for expository excellence.

This collection is an inspiring resource for courses on the history of mathematics. It reveals the creativity that has produced the mathematics we see as the finished product in textbooks.

Table of Contents

Part I. The Calculus as Algebra
Preface to the Garland Edition
1. The Development of Lagrange's Ideas on the Calculus: 1754’1797
2. The Algebraic Background of the Theory of Analytic Functions
3. The Contents of the Functions Analytiques
4. From Proof-technique to Definition: The Pre-history of Delta’Epsilon Methods
Part II. Selected Writings
1. The Mathematician, the Historian, and the History of Mathematics
2. Who gave you the Epsilon? Cauchy and the Origins of Rigorous Calculus
3. The Changing Concept of Change: The Derivative from Fermat to Weierstrass
4. The Centrality of Mathematics in the History of Western Thought
5. Descartes and Problem-solving
6. The Calculus as Algebra, the Calculus as Geometry: Lagrange, Maclaurin, and Their Legacy
7. Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions
8. Newton, Maclaurin, and the Authority of Mathematics
10. Why did Lagrange "Prove" the Parallel Postulate?

Excerpt: Lagrange's Critique of Earlier Methods (p. 64)

Newton and the English school had worked out a calculus of fluxions, which Lagrange viewed as leading to the same operations as the differential calculus. But the conceptions were different. Newton "considered mathematical quantities as engendered by motion," and the method of fluxions sought "the ratio of the variable velocities with which the quantities are produced." Lagrange recognized that Newton's view had a deceptive plausibility, saying that "everyone has or believes to have an idea of velocity." But Lagrange held that we do not have a clear enough idea of an instantaneous velocity when that velocity is variable. And he had a more fundamental objection to this view. The calculus has only "algebraic quantities" as its object. Velocity is thus, in Lagrange's view, "a foreign idea," and its introduction into the calculus would force us to regard quantities properly algebraic "as lines covered by a moving body."

About the Author

Judith V. Grabiner, who received a Ph.D. in the History of Science from Harvard, has garnered numerous awards from the MAA. She received three Carl Allendoerfer Awards (for the best article in Mathematics Magazine, 1984, 1988, and 1996); four Lester R. Ford Awards (for the best article in the American Mathematical Monthly, 1984, 1998, 2005, and 2009); and the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching (2003).


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