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William W. Dunham
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Journey through Genius: The Great Theorems of Mathematics Paperback – August 1, 1991

by William Dunham (Author)
4.6 4.6 out of 5 stars 441 ratings
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Like masterpieces of art, music, and literature, great mathematical theorems are creative milestones, works of genius destined to last forever. Now William Dunham gives them the attention they deserve.

Dunham places each theorem within its historical context and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics. A rare combination of the historical, biographical, and mathematical, Journey Through Genius is a fascinating introduction to a neglected field of human creativity.

“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
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  1. Print length
    320 pages
  2. Language
    English
  3. Publisher
    Penguin Books
  4. Publication date
    August 1, 1991
  5. Dimensions
    0.3 x 5.43 x 8.5 inches
  6. ISBN-10
    9780140147391
  7. ISBN-13
    978-0140147391
  8. See all details
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Editorial Reviews

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.

Review

"An inspired piece of intellectual history."— Los Angeles Times

“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

“Dunham deftly guides the reader through the verbal and logical intricacies of major mathematical questions, conveying a splendid sense of how the greatest mathematicians from ancient to modern times presented their arguments.”—Ivars Peterson, author of The Mathematical Tourist

Product details

  • ASIN ‏ : ‎ 014014739X
  • Publisher ‏ : ‎ Penguin Books; First Edition (August 1, 1991)
  • Language ‏ : ‎ English
  • Paperback ‏ : ‎ 320 pages
  • ISBN-10 ‏ : ‎ 9780140147391
  • ISBN-13 ‏ : ‎ 978-0140147391
  • Item Weight ‏ : ‎ 9.2 ounces
  • Dimensions ‏ : ‎ 0.3 x 5.43 x 8.5 inches
  • Customer Reviews:
    4.6 4.6 out of 5 stars 441 ratings

About the author

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William W. Dunham

William W. Dunham

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William Dunham, Koehler Professor of Mathematics at Muhlenberg College, is the author of "Journey Through Genius: The Great Theorems of Mathematics"; "The Mathematical Universe"; and "Euler: The Master of Us All". He has received the Mathematical Association of America's George Polya, Trevor Evans, and Lester R. Ford awards, as well as its Beckenbach Prize for expository writing.

Customer reviews

4.6 out of 5 stars
441 global ratings

Customers say

Customers find the book readable and joyous. They appreciate the historical context and interesting back stories. Readers describe the book as well-explained, fluid, and easy to understand. They also find it fascinating, informative, and challenging enough to inspire inquiry. Additionally, they appreciate the sublime beauty and works of art.

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ReadabilityHistoryClarityInterestBeauty
57 customers mention "Readability"57 positive0 negative

Customers find the book readable, enjoyable, and a must-read for anyone curious about the significance of mathematics. They say the content is amazing and well-thought-out. Readers also mention it's fun to work through and interesting to see how masters experimented.

"...And to top it off, well-written and engaging. I was first exposed to Dr. Dunham through The Great Courses...." Read more

"...It is useful for the student of mathematics to understand that Cantor's work on the transfinite was resisted by the mathematicians of his day just..." Read more

"...The book itself is wonderful. The whole experience was great." Read more

"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...." Read more

40 customers mention "History"36 positive4 negative

Customers find the book excellent for relating the historical context of mathematics. They appreciate the interesting back stories and biographical information. Readers also mention that the proofs are interesting, historical, and often comedic.

"...A combination of math history, biographies, and proofs. And to top it off, well-written and engaging...." Read more

"...Each of these theorems is surrounded by the historical discussion that makes this book a triumph not merely of teaching a dozen results to students..." Read more

"...the choices of mathematicians is fairly diverse, and the math itself is the main feature...." Read more

"...The book is written with loads and loads of infectious passion for mathematics...." Read more

37 customers mention "Clarity"33 positive4 negative

Customers find the book very well-explained, with fluid explanations and plenty of diagrams. They say it's easy to understand, and a great introduction to many theorems. Readers also mention the style is easy to follow and the structure is simple.

"...And to top it off, well-written and engaging. I was first exposed to Dr. Dunham through The Great Courses...." Read more

"...from combinatorics--but no one can deny that these theorems are remarkable in their elegance and in their importance in the development of..." Read more

"...The writing is great, the choices of mathematicians is fairly diverse, and the math itself is the main feature...." Read more

"...not-too-obvious theorems derived from the scratch with really fluid explanation and plenty of diagrams...." Read more

15 customers mention "Interest"15 positive0 negative

Customers find the book fascinating, informative, and challenging. They say it provides a very interesting description of the highlights. Readers also mention the human interest flows throughout.

"...This book, though, goes into much more depth. A big thank you to Dr. Dunham." Read more

"...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..." Read more

"For those who enjoy Math history, this book is fascinating...." Read more

"...My hat to Mr Dunham to make it approachable and fascinating." Read more

10 customers mention "Beauty"7 positive3 negative

Customers find the book sublime, saying it's a work of art. They also appreciate the beauty of its methods. Readers mention it's the product of very creative and interesting people.

"...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..." Read more

"...It is pure. Euclid's proof of the Pythagorean Theorem truly humbled me...." Read more

"...He does a great job explaining complicated topics, but it was very dull." Read more

"...This book is for math lovers who appreciate the beauty of its methods...." Read more

A lesson oft forgotten
4 out of 5 stars
A lesson oft forgotten
Who knew such great life advice in reality would be in here! A lot of people ought to read this passage.
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Top reviews from the United States

Carl Smith
5.0 out of 5 stars What a great book on the history of mathematics
Reviewed in the United States on November 12, 2024
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What a fun read. I love mathematics, but I don't remember ever enjoying a math book so much. A combination of math history, biographies, and proofs. And to top it off, well-written and engaging. I was first exposed to Dr. Dunham through The Great Courses. He offers a class there with the same format as this book. This book, though, goes into much more depth. A big thank you to Dr. Dunham.
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Bob Lewis
5.0 out of 5 stars A personal favorite
Reviewed in the United States on June 13, 2019
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In a phrase, this is one of my favorite books on mathematics. I read it first when it was recommended by my Calculus I professor and thought it was great. I read it again when I took a course in the history of mathematics and thought it was brilliant. Now it remains one of my favorites and I return to it regularly for discussion of some remarkable theorems and the great minds who produced them.

One of the first questions anyone might have before reading a book about mathematics is what level of mathematical sophistication is required on the part of the reader. In this case, the reader can feel pretty safe. While these are real and deep mathematical theorems, their proofs only require high-school level mathematics. In the vast majority of cases, the reader familiar with basic algebra and a little bit of geometry will have no trouble following the discussions. One theorem (Newton's approximation of pi) requires a little bit of integral calculus and another (the discussion of some of Euler's sums) requires a smidge of elementary trigonometry. In both cases, the author holds the reader's hand through the discussion so even if you haven't taken a course in trigonometry or calculus, you'll still be able to follow most of the conversation.

In fact, even if you don't really have a lot of algebra and geometry, the bulk of the book will still be accessible to you. The majority of the text is a history of mathematics wherein the author discusses the context and importance of the theorems and some biographical details of their discoverers. While I find the recreations of the proofs themselves to be perhaps the most interesting part, the reader with a general interest (even if that interest is not supported by mathematical skill) will find the book fascinating. For those of us who do have some knowledge of mathematics, though, the recreations of the theorems presented in their historical context provides a rich and inspiring series of vignettes from the history of mathematics.

This brings us to another important point. While this is a book about the history of mathematics. it is not *a* history of mathematics, and the theorems selected are not the only "great" theorems of mathematics, but a cross-section thereof. Many readers of sufficient mathematical background may quibble over the inclusion of some theorems at the expense of others--personally I would like to have seen more from combinatorics--but no one can deny that these theorems are remarkable in their elegance and in their importance in the development of mathematics from the Ancient Greeks to the very end of the nineteenth century.

It might be helpful to know what theorems are actually included in the book. Aside from a handful of lemmas and minor results presented before or after each of the "Great Theorems," the book consists of a single major result per chapter. They are:

*Hippocrates' quadrature of the lune
*Euclid's proof of the Pythagorean Theorem
*Euclid's proof of the infinitude of primes
*Archimedes' determination of a formula for circular area
*Heron's formula for triangular area
*Cardano's solution of the cubic
*Netwon's approximation of pi
*Bernoulli's proof of the divergence of the harmonic series
*Euler's evaluation of the infinite series 1+1/4+1/9+1/16+...
*Euler's refutation of Fermat's conjecture
*Cantor's proof that the interval (0,1) is not countable
*Cantor's theorem that the power set of A has strictly greater cardinality than A

Each of these theorems is surrounded by the historical discussion that makes this book a triumph not merely of teaching a dozen results to students but of actually educating students on the human enterprise of mathematics. It is not only interesting but, I think, important to be reminded of the human side of a field as abstract as mathematics, and Dunham bridges the mathematical and the biographical with remarkable dexterity. It is useful for the student of mathematics to understand that Cantor's work on the transfinite was resisted by the mathematicians of his day just as much as students struggle with it when they're exposed to it in today's lecture halls. It might further be useful to know that, perhaps partly due to his demeanor and perhaps partly due to the attacks on his work, Cantor spent much of his life in mental hospitals--and yet, despite his unhappy life his work has achieved immortality as one of the great developments in mathematical history.

I can't recommend this book highly enough for the mathematician, the math student, or the merely curious. In fact, I recommend reading it twice. First, just read it straight through and enjoy the story of mathematics told through these vignettes. Then read it again with pencil and paper in hand and work through the theorems and proofs with the author as your guide. You'll come away with a much deeper understanding of and appreciation for these great theorems in particular and mathematics in general.
35 people found this helpful
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5.0 out of 5 stars Pristine condition and prompt delivery.
Reviewed in the United States on October 23, 2024
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The book was delivered in pristine condition and the delivery was prompt. The book itself is wonderful. The whole experience was great.
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N. Kuhn
4.0 out of 5 stars Highly recommended, though imperfect
Reviewed in the United States on January 6, 2012
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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.
32 people found this helpful
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Art Weaver
5.0 out of 5 stars Prompt and precise.
Reviewed in Canada on August 8, 2022
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This book arrived quickly and in 'like new' condition
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Daniel
4.0 out of 5 stars Freshly printed book delivered, but with slightly warped hard cover and stain on bottom edge
Reviewed in Mexico on June 19, 2022
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I ordered this book which was very influential to me the first time I read a copy from my local library back in 1992; this is one of my favorite books, and I decided to buy a hard cover copy from Amazon. The book came mostly in good condition, freshly printed, just 5 days before it was delivered to my home. I rate the Amazon service with 4 stars because the covers are slightly warped, and there is a stain (fingerprint?) at the bottom edge of the book. This was an unpleasant experience for me, I can't help but feeling a little disappointed and frustrated, but at least now I have my own copy of this great book (don't judge a book by its cover). All pages appear to be properly printed. Ordered on May 13th, 2022 from Amazon website, delivered to my home on June 2nd, 2022; printed in Monee, IL, USA, on May 28th, 2022.
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Daniel
4.0 out of 5 stars Freshly printed book delivered, but with slightly warped hard cover and stain on bottom edge
Reviewed in Mexico on June 19, 2022
I ordered this book which was very influential to me the first time I read a copy from my local library back in 1992; this is one of my favorite books, and I decided to buy a hard cover copy from Amazon. The book came mostly in good condition, freshly printed, just 5 days before it was delivered to my home. I rate the Amazon service with 4 stars because the covers are slightly warped, and there is a stain (fingerprint?) at the bottom edge of the book. This was an unpleasant experience for me, I can't help but feeling a little disappointed and frustrated, but at least now I have my own copy of this great book (don't judge a book by its cover). All pages appear to be properly printed. Ordered on May 13th, 2022 from Amazon website, delivered to my home on June 2nd, 2022; printed in Monee, IL, USA, on May 28th, 2022.
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Aurelius M
5.0 out of 5 stars Très bien
Reviewed in France on February 23, 2024
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Parfait
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Cliente Amazon
5.0 out of 5 stars Journey Through Genius: The Great Theorems of...
Reviewed in Brazil on April 9, 2018
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Aborda o assunto de maneira clara, sem se afastar muito do embasamento teórico. O escopo do livro é bem estruturado.
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Siddharth Pathak
5.0 out of 5 stars School Teachers don't want you to know this one little trick that makes math fun. Buy this book to know it.
Reviewed in India on November 7, 2015
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Phew! What a journey that was. From the Hippocrates of Chios to Georg Cantor.

Dunham is a fantastic storyteller as well as a great mathematician. The book really brings out the true beauty and awe hidden in mathematics.
Dunham starts by telling us about the life of the mathematician and the circumstances he lived in. This is a sure-fire way to get your readers interested. And boy does he do it well. His facts are sound and the biographies are well sourced.

For the mathematics, well that is a very good aspect too! He has done brilliant work in presenting proofs in an organized and understandable way. Some have been dumbed down but I can live with that.

These are not THE great theorems of math, some are more simple and elegant yet inconceivably difficult. Notwithstanding, the theorems that Dunham chose were excellent since it wasn't an easy task to find math that is common to every reader.

Really the greatest book on math that I have ever read. I would highly recommend it to anyone who is bored by mathematics. Believe me, Math is truly understood can surprise you and give immense pleasure. This book has done that and I thank William Dunham for that.
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