Book Review: Journey Through Genius by William Dunham

"This, therefore, is mathematics; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes the oblivion and ignorance which are ours by birth. "-Proclus 

William Dunham's Journey Through Genius, upon first glance, seems daunting. If you were to flip through it, you would encounter many diagrams and several proofs. To people who have encountered concepts such as harmonic series or non-Euclidean geometry, these may seem like child's play. But to those who aren't as versed in mathematics (who Dunham writes in favor of), the book seems impossible. However, don't be so quick to judge the book by a quick skim: Dunham's intertwining of history, philosophy, logic, and mathematics is easy to read, interesting, and shows the interesting history of mathematics in a way I never thought possible.
As a brief outline of the contents of Dunham's book (without giving away some of the wonderful content he provides), the listing of each chapter is below:

1. Hippocrates' Quadrature of the Lune
2. Euclid's Proof of the Pythagorean Theorem 
3. Euclid and the Infinitude of Primes 
4. Archimedes' Determination of Circular Area 
5. Heron's Formula for Triangular Area 
6. Cardano and the Solution of the Cubic 
7. A Gem from Isaac Newton 
8. The Bernoullis and the Harmonic Series 
9. The Extraordinary Sums of Leonhard Euler
10. A Sampler of Euler's Number Theory 
11. The Non-Denumerability of the Continuum 
12. Cantor and the Transfinite Realm 

Starting from the 400s BCE and ending with thinkers of the twentieth century, Dunham goes into detail about twelve of the proofs that define modern mathematics. Sometimes these proofs are challenging, especially since several of these great mathematicians didn't have the same technological benefits that we do today. However, every chapter remains informative, and it is far more enjoyable to read than a section of a textbook.
For anyone who wants to know more about the lives of the mathematicians who proved these theorems that helped formulate modern math, Dunham also does a wonderful job of detailing the lives of these mathematicians. For example, Dunham discusses the competition between brothers in his history of Johann and Jakob Bernoulli, and also describes the devastation that mental illness can cause in his summary of Cantor's life.  In this way, Dunham brings to life the lives of these mathematicians, and shows us that these people were more than just names in a history book. Then, Journey Through Genius is a fantastic glimpse not only at mathematics' greatest theorems, but also at the lives of those historical figures who made math what it is today.
You may be asking, "There are many 'great theorems' in the mathematical world. Why did Dunham only pick twelve?" While Dunham never fully addresses why he chooses the theorems he does anywhere in his book, his intent is clear: every theorem he discusses in this book is an art. Consider Heron, who proved the area of a triangle: his argument seems disorganized and scattered, divided into three parts that don't seem to go together. However, once the proof is complete, the reader is left in awe, thinking, "How did he even come up with that?" Every proof in this book makes the reader feel that way in one way or another. Then, Dunham's point in choosing these theorems is to show everyone that math can be beautiful and artistic, even in its own unique way. Additionally, I believe Dunham chose the theorems he did to take us on a historical journey, picking a mathematical phenomenon from each time period. Then, Dunham appeals to the lovers of history, making his book all the more readable.
This book would appeal to all audiences, in my opinion. I am of the belief that anyone has a capability to understand mathematics, and I think this book completely shows that. I am in no way, shape, or form an expert in math, yet I was able to follow Dunham's logic as he explained the proofs of each mathematical figure. Additionally, this book provides a great learning process for anyone who is willing to read about something new and interesting. However, if you know any aspiring math buffs, anyone who loves to read about "weird history," or anyone who just likes to read about something new, then I would definitely recommend this book.
One thing I loved about the book was how it was organized. Dunham begins each chapter with a historical note about the time period and the key mathematicians involved in this time period, eventually leading to the mathematician responsible for the "great theorem" that he presents. Dunham lays out several key "problems" that were in need of solving as well, showing how these problems were solved and how logic evolved over time. Dunham then moves on to his "great theorem," and without losing the universal voice of a patient teacher, guides the reader through the theorem in understandable language while still making the proof complex and interesting. Finally, Dunahm ends with an epilogue that transitions to the next chapter. This element of the book made it flow wonderfully, and made it incredibly interesting and easy to read. While the sections detailing the various mathematical proofs do take some critical thinking to get through, they are very interesting and leave the reader feeling incredibly appreciative of the various accomplishments of mathematicians throughout history.
One thing, however, that I didn't like about the book was the occasional assumptions of further knowledge than I had. One example of this appears in the last chapter, where Dunham discusses the infinitesimalism of the power series, leading to a discussion of Cantor's development of transcendental numbers and infinity as a whole in set theory. While I am mildly familiar with set theory, I was not at all familiar with the vast concept of infinity, its role in set theory, or with the definition of transcendental numbers. For that reason, this is one of several instances where I wish Dunham would have allotted a bit more time to explain some of these complicated concepts, instead of discussing some of the minute historical details.
Despite my small complaints, I give Journey Through Genius a 4.5/5. This is by far one of the most interesting books I have ever read, and I vastly enjoyed learning new things about mathematics. If you enjoy learning as well, then I think this might be the book for you!