Judy Grabiner's The Origins of Cauchy's Rigorous Calculus is back in print, and at a friendly price. This is a classic study of the rigorization of analysis that is a "must have" for anyone who is serious about understanding the history of the subject.
Grabiner's book, which was originally published by MIT Press in 1981, is well-described by its title. Though the first chapter argues for Cauchy's central role in creating a rigorous calculus, the focus is really on "origins." In other words, Grabiner's question is about the context of Cauchy's work, the authors from which he learned and with which he interacted, and the early development of Cauchy's thought: "Since no great work arises in a vacuum, what, in the thought of his predecessors, made Cauchy's achievement possible?"
It's an interesting historical question, but it also has implications for how we teach calculus. Here's how Grabiner herself puts it:
The history of the foundations of the calculus provides the real motivation for the basic ideas, and also helps us to see which ideas were — and thus are — really hard.
In order to keep the book accessible to those mathematicians and teachers who might not be specialists in history, Grabiner has been careful to cite her sources in translation. She even provides her own translation of a few key passages in an appendix.
Like most Dover reprints, this is an unchanged photographic reproduction of the original edition. As such, it preserves the original's somewhat strange page layout, with narrow columns of text and section headings in the margins. (But always in the left margin, so sometimes in the inside margin and sometimes in the outside margin, which seems particularly strange.) So ignore the layout, and enjoy the book.
Fernando Q. Gouvêa is Professor of Mathematics at Colby College; in his free time, he is the editor of MAA Reviews.