Customer Reviews: Journey through Genius: The Great Theorems of Mathematics
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on December 14, 2000
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.
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on December 26, 2001
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.
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on January 24, 2008
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.
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on March 23, 2000
As a high school math teacher, I found Dunham's book perfectfor filling what is sadly lacking in math textbooks--the idea thatreal people have contributed to the progress of mathematics. Dunham makes it clear that mathematics is not simply calculation, but an exciting journey of discovery. I have included the book in my Advanced Mathematics courses for several years now, and my students always name it as one of the best parts of the class. The other day, one of my calc students corrected an underclassman's pronunciation of Euler, commenting, "he was great enough that we should pronounce his name correctly." Journey Through Genius is where he encountered Euler's greatness.
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on January 5, 2012
This is a wonderful book. People with a basic grasp of math who are open to the idea that math might be beautiful will be rewarded. But I have a PhD in math and thoroughly enjoyed it, and learned some things along the way. (Because math is taught very ahistorically, Chapter 1 was entirely unfamiliar to me).

These are *not* "*The* Great Theorems of Mathematics," as the subtitle suggests, but they certainly are "Great Theorems of Mathematics." Most "Great Theorems" are too technical to be presented in a book of this sort, but Mr. Dunham has done a wonderful job selecting theorems that can be proved with a minimum of prerequisites. In some ways this is a more challenging task than choosing the "greatest" theorems.

My main reservation is the fact that at times the proofs get more ponderous than necessary, and can wind up obscuring the simplicity and elegance of the mathematics. The most glaring example is the already-noted proof of Fermat's Little Theorem (p. 226-9). The proof is incomplete, and presented in a very obscure way. The key fact, that (a+b)^p = a^p + b^p (mod p) follows easily and beautifully from the binomial theorem, so a complete proof could be given quite straightforwardly. I had the sense that some of the other theorems could have been presented somewhat more cleanly as well.

The story behind Bernoulli's proof of the divergence of the harmonic series is enjoyable, but Bernoulli's proof is complex and unmotivated. Happily Mr. Dunham presents the beautiful proof Nicole Oresme from the 14th century. It is superior to Bernoulli's in every way: shorter, more elegant, and more illuminating, since pursuing his line of thinking makes it clear that the series grows as the log of the number of terms. So it's hard to see why Bernoulli is getting high marks for this particular proof, though he is overall a towering figure in the history of mathematics.

Really, all my complaints are nit-picking. This is a wonderful book.

I do want to defend Mr. Dunham from one of the other reviews: Euclid can prove (in modern language) that the area of a circle divided by the radius squared is a constant, and he can prove that the circumference divided by the diameter is a constant. But Euclid didn't show that these are the *same* constant, and that is why Archimedes result can fairly be seen as "greater" than Euclid's. Not that those theorems of Euclid's were slouches by any means.
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on November 6, 1998
I studied mathematics in university but never at any great level. I eventually went to law school and I now practice law. I give you this background so that you can appreciate what I know (and don't know about math). When I feel like reading about mathematics I look for a book that can give me a general idea of the math, but that does not get technical (and therfore boring). I also the lives of mathematicians intereting.
Dunham's book fits the bill for excellent reading in mathematics. It has just enough meat to it so that I can get insights into various mathematical theories. However he never gets so technical that I fall asleep reading the material.
The best parts of the book are the discussions of the various mathematician's and the importance of the mathematical in question. Both form the bulk of the book and are witty and informative. After reading this book, you get the impression that the history of mathematics is filled with a collection of absentminded and colourful men. These parts of the book can be read and enjoyed with absolutely no understanding of the mathematics involved. I would highly recommend this book to anyone who wants to get some basic knowledge of mathematics and its history.
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on November 17, 1998
Absolutely fantastically written. Dunham takes unfamiliar mathematical theorems and makes sense out of the whole works. Plus, he gives detailed (and extremely interesting) accounts of the people and times of certain mathematics. The book has it all, history, humor, interest and most important of all the truth of the universe and the giants who attained those truths. The beauty of it is, you don't even need to be a great mathematician to enjoy and understand this wonderful book.
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on October 28, 1999
The author puts interesting and revolutionary mathematical theorems into their historical context. Perfect book for somebody interested in math (but no significant mathematical education required) and in the history of math (and western civilization in general).
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on March 13, 2014
While this book does have a solid, relatively succinct overview of the most notable theorems in math, unfortunately it is bogged down in incessant, intolerable praise of said theorems and the incredible, amazing, ingenius geniuses who invented them. Seriously, it gets to the point where you can basically see the writer straining for yet more different words to use to describe all this magnificence: "Striking! Remarkable! Uh... dazzling!" He really dove into the thesaurus for this book. While that would be fine here and there, and yes, the theorems and their inventors surely are impressive, ENTIRE PARAGRAPHS are chock-a-block with this stuff. The title is already "Journey Through Genius"; you don't need to repeat the word "genius" over and over!!

I got to the point where I just skipped to the math and ignored all the unbearable encomium. I would have just stopped reading completely, but the explanations of the theorems are good summaries for a lay-person like me.
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on May 24, 2005
There are lots and lots of books written on mathematics claiming to target mass audience and containing none to negligible real "mathematics". Yeah, I'm talking about those funny stupid books which keeps talking about math for 400 pages but shy away from putting one real equation or proof. Well, this book is different and if you ask me, it's the best book on mathematics I've came across so far. It's the collection of some of the cleverest not-too-obvious theorems derived from the scratch with really fluid explanation and plenty of diagrams. One of the coolest thing about this book is that it first gives you a historical preview of the problem which is usually gets really interesting and pretty fun to read, specially all those tid-bits about the people involved. So by the time you reach to the proof, you know why it was a hard to do thing and you can fully appreciate the clever twists and turns in the proof. You can literally enjoy it like some murder mystery thriller. The book is written with loads and loads of infectious passion for mathematics. If this is the way math textbooks are written, there would have been far more people with passion, love and deeper understanding of mathematics.
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