You are here

Real Analysis - A Historical Approach

Saul Stahl
John Wiley
Publication Date: 
Number of Pages: 
[Reviewed by
Ioana Mihaila
, on

As one of the people that learned analysis as a beautiful weave of proofs devoid of any history, I was extremely curious about Stahl's book. Yes, of course, I knew that Newton and Leibniz were the "parents" of calculus, that Archimedes must have had something to do with the Archimedean Property, but I never took the time to find out what each of these people actually did. I wondered how they actually reasoned, what their mathematical statements sounded like, how rigorous their arguments were, given their knowledge at the time.

Just reading the title of this new analysis book, I didn't know whether I would find any of the information that I was hoping for, or merely just a careful encyclopedic review of who discovered what first. And so the first five chapter of the book came as a wonderful surprise to me. Not only did I find out that Archimedes did discover his namesake property, but also what he was doing when he stumbled onto it, and how he felt the need for "epsilon-proofs" centuries before they were developed. I learned how Newton found not only the binomial formula, but more general series expansions by performing long divisions and taking roots of polynomials. And the list could go on and on.

All this reading was so exciting that I began to fear that normal undergraduate students will never make it through a full course of real analysis laid over such a thick slice of history. However, after setting this motivating foundation, Dr. Stahl proceeds to write the rest of the book in a more classical fashion. His sequencing of the notions is incredibly smooth, even though he manages to include trigonometric series right next to the traditional power series. The proofs are clear and concise, and the exposition is really beautiful. To top that, each section has an abundance of exercises.

Since I wasn't reading this book with the eyes of an undergraduate student, so that most of the facts (though perhaps not the proofs), were already familiar to me, I was a bit disappointed that after a while the history part faded. But I would have to admit that this sacrifice had to be made in favor of keeping the exposition of the analysis part flowing smoothly. And as a bonus for the persistent reader, the six appendices at the end of the book present excerpts from the original works of Archimedes, Fermat, Newton, and Euler - what a treat!

Overall, a great book to read. And now it's time to go to the library. After all, how did Cauchy come up with all of the "Cauchy's theorems?" I am curious...

Ioana Mihaila ( is assistant professor of mathematics at Coastal Carolina University, SC. Her research area is analysis and she has a special interest in student math contests at all grade levels.

Archimedes and the Parabola.

Fermat, Differentiation, and Integration.

Newton's Calculus (Part 1).

Newton's Calculus (Part 2).


The Real Numbers.

Sequences and Their Limits.

The Cauchy Property.

The Convergence of Infinite Series.

Series of Functions.



Uniform Convergence.

The Vindication.


Solutions to Selected Exercises.