Book title: Calculus Reordered: A History of the Big Ideas
Author : David M. Bressoud

Princeton University Press
Link to the book: Calculus Reordered: A History of the Big Ideas

Discussions on the history of different fields are usually dry, wordy and generally, when you are studying the field, hard to read. This is because they are usually geared towards the general audience, and in doing so most authors tend to strip away the very exciting technical details. I expected the same treatment from the author, but I was pleasantly surprised.

The book contains $5$ chapters, which are the following:

1) Accumulations
2) Ratios of Change
3) Sequences of Partial Sums
4) The Algebra of Inequalities
5) Analysis

Each of these chapters has a central theme that is being covered, but they are not at all disjoint. For instance, the last three contain the history of concepts that would normally be found in a first course for Real Analysis, while the first two are essentially the more applied spectrum to serve as some form of motivation for going through all this trouble, although they can certainly stand on their own.

The main structure of the text in the book is as follows:
In an orderly sense, the writer introduces the context and the proceeds to state exactly what was the major draw back in the context during the relevant period of time. He then proceeds to start with the most basic step that was taken, continues to build up on this until there is enough to talk about really fascinating results all of which should be familiar to someone who is pursuing a degree in Mathematics, but nonetheless are quite exciting to see. The progression, as well as the way in which he uses simple techniques to demolish towers of problems in the same sense as it was done back in the day is certainly worth appreciation.

I found reading the book as a very pleasant exercise. It excites me that someone managed to give a very well written history of the relevant big ideas. In some cases, known theorems are given with proofs (sometimes). For example: in the $4$th chapter he writes the well known Intermediate Value Theorem, then proceeds to give a proof. The book goes as far as having discussions about measures on the real line, which almost never happens without the Cantor set. The formality is significantly dailed down, but the book is more of a narrative, and hence can be read like a novel and simply appreciated for exactly what it is. In fact, I wish I had read this book in my first year. It offers a great scope into what comes after one’s first calculus class. I did find that there is a strong emphasis on the analysis aspect of mathematics, and the algebra is lacking, but even with that it was well worth a read.

I certainly recommend this book to anyone who is interested in Mathematics (and or its development).

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