Book Review: The History of the Calculus and its Conceptual Development, by Carl B. Boyer.

Dover published this book in 1959, although Boyer no doubt wrote it earlier.

I picked this book up mainly because I wanted to know as much as possible about any subject I want to teach, calculus being very high on that list. This book was definitely an eye-opener. There were several main points Boyer wanted to get across, namely the following:

1. There are many instances in which historians ascribe some complicated, nuanced concept such as the limit to an early thinker like Archimedes, when in fact said early thinker did not have the concept fully in his head (or so we judge by his writings).

2. The founders of the calculus, and even some who came after, did not invent calculus in a vacuum. Although Newton and Leibniz may have bundled the existing algorithms into a nice package and thus deserve to be called the inventors of the calculus, the story was not over then, and even so bold a thinker as Lagrange didn't get some things right (at least, as we think of "right" these days.)

3. The development of calculus did not happen overnight (more like over the span of 2500 years!).

There are some interesting criticisms of Boyer's book on Amazon. I'd encourage you to read them. I found a few of them to be right on. For though there are few books which collect all of the information in one place like Boyer's, he does have some hidden assumptions, especially modernism.

Another criticism I would have of the book is the disassociation of calculus from physics. Physics provided a good deal of the motivation for developing the calculus in the first place, but Boyer barely mentions it.

Is it worth reading? Yes, if you want to teach calculus or write about the philosophy of math or the history of math. It would even be useful for students to read so that they don't get the impression that calculus fell out of the sky in the "perfect" delta-epsilon form we have today.

For that matter, my Dad wrote a calculus book in which in put forth the Third Derivative Test, which is not in any textbook of which I am aware. Calculus, while a full, rich body of knowledge, cannot possibly be complete. (See Goedel's proof of the incompleteness of first-order logic.)