Euler, Master of Us All

I recently finished reading Euler, Master of us All, by William Dunham. I was most interested in this book due to my fascination of the natural number. Before reading the book I had seen Euler’s identity, and of course Euler’s equation. I also remember reading somewhere that we denote the natural number with an e in honor of Euler himself. I’m not sure if this is true, but after reading Euler, Master of us All (EMA), I’ve been convinced that he deserves the honor regardless.

The book started off with a biography of Euler to give the reader an idea of what he did when, mostly noting where he was working at the time. I liked this a lot because I honestly hadn’t a clue when Euler lived at all (he lived at about the same time as Ben Franklin). The rest of the book didn’t follow any chronological order, but was organized by different types of mathematics.

I felt that the style of the book was a bit repetitive and predictable at times. The author used the same formatting for each chapter, but this was only minutely annoying. It didn’t detract from my amazement at some of the things Euler did. He truly was a master of mathematics. There were a few trends that I noticed throughout the text. Euler loved logarithms. The second chapter of the book is dedicated to his work on logarithms and points out that Euler was the first to recognize the relationship between logarithms and exponentials. He discovered the “golden rule of logarithms” which is used to change base of a logarithm with a given base. Though the second chapter is the only one with logarithms in its title, by no means is it the only place they show up. Euler used logarithms in many of his “proofs” even when they didn’t seem applicable at all. One of my favorites was to show the sum of reciprocals of all primes diverges. In this particular proof Euler not only used the natural logarithm, but also expressed an infinite series of natural logs as an infinite sum of infinite series.

This brings me to another theme I picked up on, namely that Euler loved infinite series. They were another tool on his proof tool belt. As I was reading it felt like every other proof involved an infinite series. One of my favorite excerpts from the book was how Euler calculated e to 20 digits. His method seemed to me to be very loosely related to taking derivatives. It involved exponentiation by an infinitesimal number. He realized that this would be very near to one, and wrote it as a^w=1+kw where w was the infinitesimal, a was some number, and k was a constant he found to be dependent on a. After some change of variables and a series expansion, he set k to be one, and found that a must be e, the natural number! I thought this was brilliant, as the set up reminded me vaguely of Newton’s method.

I would recommend this book to a fellow math fanatic. There is so much great math history and so many cool math tricks, though “hand wavy” they might be. I would warn a potential reader of the amount proofs included in the text. It’s totally possible to read the text without carefully following the proofs (as I did for some sections), however I found it much more interesting to try to at least see the logic behind the proofs. This slowed me down a bit while reading, but I was ok with it. I’ve always been one of those who oddly enjoys math proofs. In short, I would definitely recommend this book to any math major with the cautioning of light to mild proofs involved. It was a good read and I learned a lot about a great mathematician.

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