Journey Through Genius: The Great Theorems of Mathematics [Brossura]

Some books, such as Ball's and Beiler's seem to have sparked a life-long love of mathematics in practically everyone who reads them. "Journey Through Genius" should be another such book.

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.

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.

Dunham has done an excellent job of taking us through the history of mathematics providing a context with the civilization of the time. He shapes his production around what he considers to be the great theorems of mathematics.

The order of presentation is chronological. Early on we see great admiration for Euclid and his "Elements" as two of Euclid's theorems appear on the list, a proof of the Pythagorean theorem and the proof that there are infinitely many primes. Euler and Cantor are also honored with two theorems included among the collection.

However there is more to Dunham's presentation than just the proofs. We find other related results by these masters and other great mathematicians that were their contemporaries. He shows reverence for Newton. Gauss and Weierstrass and others are mentioned but none of their theorems are highlighted.

It is not his intention to slight these great mathematicians. Rather, Dunham's criteria seems to be to present the theorems that have simple and elegant proofs but often surprising results. His coverage of Cantor is particularly good. It seems that he is most knowledgeable about Cantor's mathematics of transfinite numbers and the related axiomatic set theory.

For a detailed description of the chapters in this work, look at the detailed review by Shard here at Amazon. I found this book to be well written and authoritative and learned a few things about Euler and number theory that I hadn't known from my undergraduate and graduate training in mathematics. Yet I did not give the book five stars.

There are a couple of omissions that I find reduce it to a four star rating. My main objection is the slighting of Evariste Galois. Galois was the great French mathematician who died in a duel at the early age of 21 in the year 1832. Yet, in his short life he developed a theory of abstract algebra seemingly unrelated to the great unsolved questions about constructions with straight edge and compass due to the Greeks and yet his theory resolved many of these questions. I was very impressed in graduate school when I learned the Galois theory and came to realize that problems such as a solution to the general 5th degree equation by radicals and the trisection of an arbitrary angle with straight edge and compass were impossible.

Now, Galois theory is certainly beyond the scope of this book but so is non-Euclidean geometry and aspects of number theory and set theory that Dunham chooses to mention. He spends a great deal of time on Euclid's work and the various possible constructions with straight edge and compass.

Also, in the chapter on Cardano's proof of the general solution to the cubic, he also presents the solution to the quartic and refers to Abel's result on the impossibility of the general solution to the quintic equation. This would have been the perfect place to introduce Galois who independently and at the same time in history proved the impossibility of solving the general quintic equation by radicals. Oddly Galois is never once mentioned in the entire book.

In discussing number theory and Euler's contributions, the theorems and conjectures of Fermat are mentioned. This book was written in 1991 and it presents Fermat's last theorem as an unproven conjecture.

Andrew Wiles presented a proof of Fermat's last theorem to the mathematical community in 1993 and after some needed patchwork to the proof, it is now agreed that Fermat's last theorem is true. There are a number of books written on Fermat's last theorem including an excellent book by Simon Singh. It seems that Dunham's book is popular and has been reprinted at least 10 times since the original printing in 1991. It would have been appropriate to modify the discussion of Fermat's last theorem in one of these reprintings.