One of my all-time favorite math books for lay readers. Dunham takes us on a tour through a dozen of the greatest theorems in the history of mathematics, from Hippocrates to Cantor.
[Historical tidbits: The pharaoh’s treasure -- 5/24/07]

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Product Description

Praise for William Dunhams Journey Through Genius The Great Theorems of Mathematics "Dunham deftly guides the reader through the verbal and logical intricacies of major mathematical questions and proofs, conveying a splendid sense of how the greatest mathematicians from ancient to modern times presented their arguments." —Ivars Peterson Author, The Mathematical Tourist Mathematics and Physics Editor, Science News

"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 ideas—ideas 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

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Product Details

• Amazon Sales Rank: #5733 in Books
• Published on: 1991-08-01
• Original language: English
• Number of items: 1
• Binding: Paperback
• 320 pages

Amazon.com Review
In Journey through Genius, author William Dunham strikes an extraordinary balance between the historical and technical. He devotes each chapter to a principal result of mathematics, such as the solution of the cubic series and the divergence of the harmonic series. Not only does this book tell the stories of the people behind the math, but it also includes discussions and rigorous proofs of the relevant mathematical results.

From the Inside Flap
There is a remarkable permanence about mathematical ideas. Whereas other scientific disciplines regularly discard the old and outmoded, in mathematics new results build upon their predecessors without rendering them obsolete. The astronomical theories and medical practices of the Alexandrian Greeks, works of undisputed genius in their day, have long since become archaic curios. Yet Euclid’s proof of the Pythagorean theorem, set forth in 300 B.C., has lost none of its beauty or validity with the passage of time. A theorem, correctly proved within the rigors of logic, is a theorem forever. Journey Through Genius explores some of the most significant and enduring ideas in mathematics: the great theorems, discoveries of beauty and insight that stand today as monuments to the human intellect. Writing with extraordinary clarity, wit, and enthusiasm, Professor William Dunham takes us on a fascinating journey through the intricate reasoning of these masterworks and the often turbulent lives and times of their creators. Along with the essential mathematics, Professor Dunham uniquely captures the humanity of these great mathematicians. You’ll meet Archimedes of Syracuse, who pushed mathematics to frontiers that would stand some 1,500 years. Unchallenged as the greatest mathematician of antiquity, Archimedes was the stereotypically "absent minded" mathematician, capable of forgetting to eat or bathe while at work on a problem. From the sixteenth century you’ll encounter Gerolamo Cardano, whose mathematical accomplishments provide a fascinating counterpoint to his extraordinary misadventures. In the next century, there appeared the competitive, bickering Bernoulli brothers, who explored the arcane world of infinite series when not engaged in contentious wrangling with one another. And from more modern times you’ll read of the paranoid genius of Georg Cantor, who had the ability and courage to make a frontal assault on that most challenging of mathematical ideas—the infinite. Journey Through Genius is a rare combination of the historical, biographical, and mathematical. Readers will find the history engaging and fast-paced, the mathematics presented in careful steps. Indeed, those who keep paper, pencil, and straightedge nearby will find themselves rewarded by a deeper understanding and appreciation of these powerful discoveries. Regardless of one’s mathematical facility, all readers will come away from this exhilarating book with a keen sense of the majesty and power, the creativity and genius of these mathematical masterpieces.

From the Back Cover
Praise for William Dunhams Journey Through Genius The Great Theorems of Mathematics "Dunham deftly guides the reader through the verbal and logical intricacies of major mathematical questions and proofs, conveying a splendid sense of how the greatest mathematicians from ancient to modern times presented their arguments." Ivars Peterson Author, The Mathematical Tourist Mathematics and Physics Editor, Science News

"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

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Customer Reviews

Brilliant book. Belongs on the top shelf.
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.

Brings Mathematics To Life!
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.

math history through great theorems
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 chpaters in this work, look at the detailed review by Shard here at Amazon. I found this book 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.