Journey through Genius: The Great Theorems of Mathematics

"It is mathematics presented as a series of works of art; a fascinating lingering over individual examples of ingenuity and insight. It is mathematics by lightning flash." Isaac Asimov

"It is a captivating collection of essays of major mathematical achievements brought to life by the personal and historical anecdotes which the author has skillfully woven into the text. This is a book which should find its place on the bookshelf of anyone interested in science and the scientists who create it." R. L. Graham, AT&T Bell Laboratories

"Come on a time-machine tour through 2,300 years in which Dunham drops in on some of the greatest mathematicians in history. Almost as if we chat over tea and crumpets, we get to know them and their ideasideas that ring with eternity and that offer glimpses into the often veiled beauty of mathematics and logic. And all the while we marvel, hoping that the tour will not stop." Jearl Walker, Physics Department, Cleveland State University Author of The Flying Circus of Physics

In the Preface, the author comments that it is common practice to teach appreciation for art through a study of the great masterpieces. Art history students study not only the great works, but also the lives of the great artists, and it is hard to imagine how one could learn the subject any other way. Why then do we neglect to teach the Great Theorems of mathematics, and the lives of their creators? Dunham sets out to do just this, and succeeds beyond all expectations.

Each chapter consists of a biography of the main character interwoven with an exposition of one of the Great Theorems. Also included are enough additional theorems and proofs to support each of the main topics so that Dunham essentially moves from the origins of mathematical proof to modern axiomatic set theory with no prerequisites. Admittedly it will help if the reader has taken a couple of high school algebra classes, but if not, it should not be a barrier to appreciating the book. Each chapter concludes with an epilogue that traces the evolution of the central ideas forward in time through the history of mathematics, placing each theorem in context.

The journey begins with Hippocrates of Chios who demonstrated how to construct a square with area equal to a particular curved shape called a Lune. This "Quadrature of the Lune" is believed to be the earliest proof in mathematics, and in Dunham's capable hands, we see it for the gem of mathematics that it is. The epilogue discusses the infamous problem of "squaring the circle", which mathematicians tried to solve for over 2000 years before Lindeman proved that it is impossible.

In chapters 2 and 3 we get a healthy dose of Euclid. Dunham briefly covers all 13 books of "The Elements", discussing the general contents and importance of each. He selects several propositions directly from Euclid and proves them in full using Euclid's arguments paraphrased in modern language. The diagrams are excellent, and very helpful in understanding the proofs. If you've ever tried to read Euclid in a direct translation, you should truly appreciate Dunham's exposition: the mathematics is at once elementary, intricate, and beautiful, but Dunham is vastly easier to read than Euclid. The Great Theorems of these chapters are Euclid's proof of the Pythagorean theorem and The Infinitude of Primes, which rests at the heart of modern number theory. Dunham obviously loves Euclid, and his enthusiasm is infectious. After reading this, it is easy to see why "The Elements" is the second most analyzed text in history (after The Bible).

Archimedes is the subject of chapter 4, and he was a true Greek Hero. Even if most of the stories of Archimedes' life are apocryphal, they still make very interesting reading. However the core of the chapter is the Great Theorem, Archimedes' Determination of Circular Area. His method anticipated the integral calculus by some 1800 years, and also introduced the world to the wonderful and ubiquitous number pi. The epilogue traces attempts to approximate pi all the way up to the incomparable Indian mathematician of the 20th century, Ramanujan.

Chapter 5 concerns Heron's formula for the area of a triangle. The proof is extremely convoluted and intricate, with a great surprise ending. It is well worth the effort to follow it through to the end. Chapter 6 is about Cardano's solution to the general cubic equation of algebra. Cardono is certainly one of the strangest characters in the history of mathematics, and Dunham does a great job telling the story. The epilogue discusses the problem of solving the general quintic or higher degree equation, and Neils Abel's shocking 1824 proof that such a solution is impossible.

Sir Isaac Newton is the topic of chapter 7. Rather than go into the calculus deeply, Dunham gives us Newton's Binomial Theorem, which he didn't really prove, but nevertheless showed how it could be put to great use in the Great Theorem of this chapter, namely the approximation of pi. Chapter 8 breezes through the Bernoulli brothers' proof that the Harmonic Series does not converge, with lots of very interesting historical biography thrown in for good measure.

Chapters 9 and 10 discuss the incredible genius of Leonard Euler, who contributed very significant results to virtually every field of mathematics, and seems to have been a decent human being to boot. Chapter 10, "A Sampler of Euler's Number Theory", is my favorite in the book. A large portion of his work in number theory came from proving (or disproving) propositions due to Fermat, which were passed on to him by his friend Goldbach. This chapter gives complete proofs of several of these wonderful theorems including Fermat's Little Theorem, all of which lead up to the gem of the chapter. Taken as a whole it is the kind of number theory detective work that has lured so many people into the field over the years. Chapter 10 is a mathematical tour de force.

The last 2 chapters handle Cantor's work in the "transfinite realm", and should certainly serve to expand the mind of any reader. By the time you finish, you'll have an idea about the twentieth century crisis in mathematics, and its resolution, and what sorts of concepts are capable of making modern mathematicians squirm in their seats. Dunham does a beautiful job of demonstrating Cantor's proof of the non-denumerability of the continuum. At this altitude of intellectual mountain-climbing the air is thin, but it is well worth the climb!

In brief, "Journey Through Genius" might almost be considered a genius work of mathematical exposition. I can think of few authors more capable of conveying the excitement and beauty of mathematics, as well as an appreciation for the sheer enormity of the achievements of the human mind and spirit.


Brings Mathematics To Life!   December 26, 2001
 51 out of 53 found this review helpful

William Dunham has brought life to a subject that almost everyone considers dull, boring and dead. Dunham investigates and explains, in easy-to-understand language and simple algebra, some of the most famous theorems of mathematics. But what sets this book apart is his descriptions of the mathemeticians themselves, and their lives. It becomes easier to understand their thinking process, and thus to understand their theorems.
I am a layman with a computer science degree, and a layman's understanding of mathematics, so I am no expert! But I loved this book.
I found Dunham's description of Archimedes' life and his reasoning for finding the area of a circle and volume of a cylinder to be (almost!) riveting.
Dunham's decription of Cantor and his reasoning regarding the cardinality of infinite sets was fascinating to me. But most of all, I loved his chapter on Leonhard Euler. Having in high school been fascinated by Euler's derivation of e^(i*PI) = -1, I was even more amazed at the scope of this man's genius, and Dunham's description of his life.
The chapter on Isaac Newton is an especially good one as well.
Dunham smartly weaves these important theorems of mathematics into the history of mathematics, making this book even more understandable, and, dare I say it, actually entertaining!
This book is a gem, and for anyone interested in mathematics, it is not to be missed.


A delicious book and mathematical wonderments   July 12, 2003
 4 out of 4 found this review helpful

This is sumptuous and beautiful writing. Years ago, I took 4 years of math in college, and had forgotten how lovely and surprising these ideas can be, especially when elegantly explained. Would that I had had a professor then, who could have walked through this history so well.

With regard to the theorems themselves, how is it possible that our scientific ancestors were so clever and insightful? How can we explain the genius of men like Euclid, Newton, and Euler?

This book dazzled me, and I predict the same effect for any readers with even a smattering of math education and a taste for scientific wonderments.


a math history you can play with   November 23, 2000
 14 out of 14 found this review helpful

This book is an important read for a layman trying to get a better grasp on the actual historical building blocks of math. Because Dunham goes through the actual problems and solutions of great mathemeticians the book moves beyond a simple narrative. A deceptively relevant book for anyone trying to understand intellectual history.

Its nice to finally see the beauty of what the dullards back in math class were trying to teach us. The reviewers are right that the personalities of some of the mathemeticians do come alive here. But that has been done elsewhere (in more detail as well).

The discovery and refinement of mathematics is a central component of civilization, and this is a wonderful way to see its actual historical footings.


Sublime beauty   August 25, 2003
 9 out of 9 found this review helpful

Rarely is it properly appreciated that mathematics is one of the arts, and --- like all the other arts --- has created monuments of surpassing beauty through the centuries. Dunham does a wonderful job in this whirlwind tour of the past two thousand years of mathematics. He presents math as a story of triumph after triumph. Each chapter highlights one "great" theorem, and in every chapter he makes clear the context of the theorem by discussing preceding work, the life of the mathematician who proved the theorem, and the applications it opened up. He is masterful at mentioning tidbits in historical context that will be logically necessary to understand a few chapters further. No advanced knowledge of math is necessary, but I will caution: one must be at least reasonably fluent in both geometry and second year algebra in order to get the most out of this book. The more rusty one's algebra skills are, the more burdensome the proofs will be. For someone comfortable with that level of math, the book is breathtaking in the panoply of intellectual vistas it opens up. For anyone doing any kind of work in any technical field, I simply cannot recommend this book highly enough.