Journey through Genius: The Great Theorems of Mathematics

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 AEach 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.

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.

This text was used in a History of Mathematics class that I took. It was one of the few books that had an enormous impact on my life. For the first time I saw the way that mathematics directly influenced and impacted human beings, society and history at large. Each chapter does a fabulous job of bringing to life the mathematicians and their times. Even more telling, the climax of each chapter is a mathematical proof that the author leads you through. I have used this book as a supplementary text as a high school teacher, and though my students could not follow the proofs, they enjoyed reading the stories (advanced high school students could probably follow them). Readers who have a basic background in mathematics should be able to follow most of the proofs with patience. I remember glossing over some of the proofs when I was in college, but when I came back to them later I was genuinely impressed. The writing is great, the choices of mathematicians is fairly diverse, and the math itself is the main feature. Anyone who wants to understand the way that mathematics really fits into our universe should give this book a try. It is arguably one of the best books about mathematics, that includes mathematics, that is out there. (I would highly recommend this for any history of math class and as a supplementary text for appropriate classes...

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.