A Historian Looks Back: The Calculus as Algebra and Selected Writings, by Judith V. Grabiner, The Mathematical Association of America, Washington, D. C., 2010. 287 pp., $62.95 (hardbound). ISBN 9780883855720.
Five out of five stars
This is literally the reprint of a previously published book followed by some additional articles written by the author. The first section is the book “The Calculus as Algebra: J. L. Lagrange, 1736-1813,” that was first published by Garland in 1990. The content is largely as the name implies, it is a description of how Lagrange worked to treat functions as analytic, so they and their derivatives could be expressed in the form of an infinite series (algebraic expressions). As all students of calculus learn very quickly, a function expressed in this form is very easy to integrate and differentiate. Of course, this was before all was made as rigorous as it is now, so there are some deficiencies in Lagrange’s work that are pointed out. Yet, the reader clearly understands the extent of the progress he made.
The remainder of the book contains ten papers by Grabiner, there is no question as to the best, for it answers the question asked of so many calculus students, “Whom do I cuss for this?” That paper is “Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus.” It seems a rite of passage for calculus students that they dislike and complain about epsilon-delta proofs. After reading this article, they know who to “blame” for their misery.
In the first paper in the collection of ten, “The Mathematician, the Historian and the History of Mathematics,” Grabiner explores the relationship between mathematics, general history and the people that write about their interconnectedness. Most of the major mathematical figures were buffeted by the context within which they lived, so no history of their lives can ever be effectively written without including some understanding of social and political history. This is a thought-provoking paper.
If you are interested in or are teaching a course in the history of mathematics, this is a book you will find valuable.