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Journey through Genius: The Great Theorems of Mathematics
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
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
320 pages, Paperback
First published January 1, 1990
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9,140 people want to read
About the author
William Dunham
20 books80 followersAn American writer who was originally trained in topology but became interested in the history of mathematics and specializes in Leonhard Euler. He has received several awards for writing and teaching on this subject.
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Displaying 1 - 30 of 161 reviews
July 15, 2012
The preface to this book contains the following explanation, which I think suffices to explain its reason for being:
"For disciplines as diverse as literature, music, and art, there is a tradition of examining masterpieces -- the "great novels", the "great symphonies", the "great paintings" -- as the fittest and most illuminating objects of study. Books are written and courses are taught on precisely these topics in order to acquaint us with some of the creative milestones of the discipline and with the men and women who produced them.
The present book offers an analogous approach to mathematics, where the creative unit is not the novel or symphony, but the theorem. ... "
On the whole, I think the author does a commendable job in fulfilling this promise. There are, however, some shortcomings: The book states that Fermat's Last Theorem is still unsolved, when it was in fact proved by Andrew Wiles and Richard Taylor in 1995. Of course, the book was published in 1990, so what it stated was true at the time of writing. A second shortcoming, in my estimation, is the total absence of any mention of Group Theory, its origins during the French Revolution (Evariste Galois and subsequent enhancements by Sophus Lie, both of whom have fascinating histories, albeit tragic ones). Perhaps the importance of Group Theory was not apparent in 1990, but I doubt that. Bottom line: The book needs a new edition.
Just as I would have a hard time going through one of the standard "great masterpieces of fiction" tomes because of my lack of appreciation of many so-called classics, so a typically non-mathematical reader would probably have a difficult time reading this book, at least in the sections that deal with the proofs of the theorems. Even I, with a strong mathematical background, found my eyes glazing over during some of the proofs -- especially those based on Euclidean style geometry (all the ancient Greek and other ancient cultures were based on this kind of geometry). You know, the kind of geometry where, in school, you were asked to prove that this angle is equal to that other angle. I was never very good at that -- perhaps I lacked the spatial aptitude, and there never seemed to be any real rules to follow; it was basically trial and error. I *was* good at what we called Coordinate Geometry (I think they call it Analytic Geometry now), probably because there were more easily identified rules.
I found the last two chapters, "The Non-Denumerability of the Continuum" and "Cantor and the Transfinite Realm", to be particularly interesting, because I had not previously learned about those areas.
One final comment, which applies not only to this book but to most if not all mathematics books I've read or studied: I feel that one of the reasons why so many people stop paying detailed attention (glazed eye onset) to a mathematical proof is not only that it's often difficult to understand, but that it's often presented in a very dense manner. Essential steps are subsumed into a single paragraph with no attempt to identify each step. I believe that, if there was an express attempt to present the proof using bulleted or numbered lists, with relatively short explanations in each item, more people could stay awake longer and be more likely to understand the proof. Perhaps this is because it would provide more of a visual aid than densely written paragraphs. Mathematicians often forget that they do this for a living, so it becomes second nature to them, while other mere mortals might benefit from a different approach and/or presentation.
"For disciplines as diverse as literature, music, and art, there is a tradition of examining masterpieces -- the "great novels", the "great symphonies", the "great paintings" -- as the fittest and most illuminating objects of study. Books are written and courses are taught on precisely these topics in order to acquaint us with some of the creative milestones of the discipline and with the men and women who produced them.
The present book offers an analogous approach to mathematics, where the creative unit is not the novel or symphony, but the theorem. ... "
On the whole, I think the author does a commendable job in fulfilling this promise. There are, however, some shortcomings: The book states that Fermat's Last Theorem is still unsolved, when it was in fact proved by Andrew Wiles and Richard Taylor in 1995. Of course, the book was published in 1990, so what it stated was true at the time of writing. A second shortcoming, in my estimation, is the total absence of any mention of Group Theory, its origins during the French Revolution (Evariste Galois and subsequent enhancements by Sophus Lie, both of whom have fascinating histories, albeit tragic ones). Perhaps the importance of Group Theory was not apparent in 1990, but I doubt that. Bottom line: The book needs a new edition.
Just as I would have a hard time going through one of the standard "great masterpieces of fiction" tomes because of my lack of appreciation of many so-called classics, so a typically non-mathematical reader would probably have a difficult time reading this book, at least in the sections that deal with the proofs of the theorems. Even I, with a strong mathematical background, found my eyes glazing over during some of the proofs -- especially those based on Euclidean style geometry (all the ancient Greek and other ancient cultures were based on this kind of geometry). You know, the kind of geometry where, in school, you were asked to prove that this angle is equal to that other angle. I was never very good at that -- perhaps I lacked the spatial aptitude, and there never seemed to be any real rules to follow; it was basically trial and error. I *was* good at what we called Coordinate Geometry (I think they call it Analytic Geometry now), probably because there were more easily identified rules.
I found the last two chapters, "The Non-Denumerability of the Continuum" and "Cantor and the Transfinite Realm", to be particularly interesting, because I had not previously learned about those areas.
One final comment, which applies not only to this book but to most if not all mathematics books I've read or studied: I feel that one of the reasons why so many people stop paying detailed attention (glazed eye onset) to a mathematical proof is not only that it's often difficult to understand, but that it's often presented in a very dense manner. Essential steps are subsumed into a single paragraph with no attempt to identify each step. I believe that, if there was an express attempt to present the proof using bulleted or numbered lists, with relatively short explanations in each item, more people could stay awake longer and be more likely to understand the proof. Perhaps this is because it would provide more of a visual aid than densely written paragraphs. Mathematicians often forget that they do this for a living, so it becomes second nature to them, while other mere mortals might benefit from a different approach and/or presentation.
July 10, 2016
Not too bad - but way too easy: I was expecting something at a higher level. I was misled (by comments by other reviewers) into believing that the level of sophistication of the items treated in this book was reasonably good.
June 7, 2010
I finally finished this book! It's been a long time coming. I've owned it for almost ten years. I finally picked it up to read a few months ago. I don't know why I waited so long. It's a real gem. The main reason it took me so long to get through is the format. You can read it a chapter at a time, as you have time, and read other books in between, etc, and it really doesn't matter. I'd read a chapter, then read other books, then read another chapter, etc. Each chapter is about one of the more important theorems in mathematics. I loved almost every chapter. I was a little bored by the two Cantor chapters, but I think that was primarily because I don't love set theory and didn't study it a whole lot in college. I'd give the book five stars if not for those two chapters.
The chapter dealing with Sir Isaac Newton nearly had me in tears. His accomplishments and understanding are just mind boggling.
I took a course on the history of mathematics in college. It was horribly boring. I had to fight sleep off. This book would have been a brilliant text for that class. Sure it doesn't cover every blasted mathematical discovery since the beginning of time, but the approach this book takes seems so much more practical and well rounded. It has just the right mix of history and proofs, and was almost always very engaging and interesting. It covers so many of the really really important things. My class would have been much better if it had used this sort of approach verses the linear, this happened then this happened then this happened... approach. History can be so boring, but this book was very fun.
People who don't like math, and wouldn't care to follow a proof while reading won't like this one bit.
One final thought: Newton once explained that the reason he was so successful at solving problems was "by thinking on them continuously." I wish I could harness a small bit of that focus. Sure, I've got to be able to multitask a little, but wouldn't it be nice to use a bit more focus in our lives? I think I could get more done, and do it better, if I'd procrastinate less, and not do things in pieces, but just set about doing a task, and get it done. If that makes sense. Isn't Newton inspiring in so many ways? Don't we all wish we could be like him? Except maybe a little more well adjusted and happy?
The chapter dealing with Sir Isaac Newton nearly had me in tears. His accomplishments and understanding are just mind boggling.
I took a course on the history of mathematics in college. It was horribly boring. I had to fight sleep off. This book would have been a brilliant text for that class. Sure it doesn't cover every blasted mathematical discovery since the beginning of time, but the approach this book takes seems so much more practical and well rounded. It has just the right mix of history and proofs, and was almost always very engaging and interesting. It covers so many of the really really important things. My class would have been much better if it had used this sort of approach verses the linear, this happened then this happened then this happened... approach. History can be so boring, but this book was very fun.
People who don't like math, and wouldn't care to follow a proof while reading won't like this one bit.
One final thought: Newton once explained that the reason he was so successful at solving problems was "by thinking on them continuously." I wish I could harness a small bit of that focus. Sure, I've got to be able to multitask a little, but wouldn't it be nice to use a bit more focus in our lives? I think I could get more done, and do it better, if I'd procrastinate less, and not do things in pieces, but just set about doing a task, and get it done. If that makes sense. Isn't Newton inspiring in so many ways? Don't we all wish we could be like him? Except maybe a little more well adjusted and happy?
July 20, 2014
What a merry walkthrough over the work of History’s mathematical geniuses!, faith in Humanity: Restored!
And in Bertrand Russels's words: Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
And in Bertrand Russels's words: Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
May 29, 2011
The math history presented is very good. The mathematical exposition is uneven. Some of it is good and some not so good.
The chapter I had the most difficulty with was the one on Heron's formula. Theorems are presented without any indicator of where they are headed. Dunham keeps promising that the formula will eventually be derived, but I gave up beforehand.
The other chapter I would criticize is the one on Euler's number theory, but for different reasons. In developing Fermat's Little Theorem, Dunham derives the result that (a+1)^p - a^p -1 is divisible by p, by asking us to assume that the expression for 1/p((a+1)^p - a^p -1) is a whole number. I don't know why this was done, since it is easy to prove the divisibility directly, using the Binomial Equation discussed at length in the book. Later in the same chapter, mentioning the formula for geometric series would have made much clearer Euler's work on factoring Fermat numbers.
Overall I would recommend the book. If you are not that much into proofs, there is much of interest in the biographical and historical material. If you are willing to do a little work in following the proofs, there is much to appreciate.
The chapter I had the most difficulty with was the one on Heron's formula. Theorems are presented without any indicator of where they are headed. Dunham keeps promising that the formula will eventually be derived, but I gave up beforehand.
The other chapter I would criticize is the one on Euler's number theory, but for different reasons. In developing Fermat's Little Theorem, Dunham derives the result that (a+1)^p - a^p -1 is divisible by p, by asking us to assume that the expression for 1/p((a+1)^p - a^p -1) is a whole number. I don't know why this was done, since it is easy to prove the divisibility directly, using the Binomial Equation discussed at length in the book. Later in the same chapter, mentioning the formula for geometric series would have made much clearer Euler's work on factoring Fermat numbers.
Overall I would recommend the book. If you are not that much into proofs, there is much of interest in the biographical and historical material. If you are willing to do a little work in following the proofs, there is much to appreciate.
May 12, 2019
I enjoyed this tour through the history of the development of mathematics, from Euclidean geometry to the more abstract challenges of modern theory. The author tells a good story, and there is plenty of mathematical detail for those who are interested. In particular, the explanation of Cantor’s work on infinity is the clearest I have seen.
Sometimes the story may be too good. My confidence in the history I did not know was reduced by two errors (both on page 129) in the history that I do know. There is no evidence that the Arabs burned the Library of Alexandria in 621, as it was probably long gone by then. And Muhammad did not die in Jerusalem. How many other old myths are being recycled here?
However, I am sure the math is good, even if the story behind it might occasionally be questionable. I recommend this book for those with an interest in mathematics, but do not expect an advanced treatment of the ideas in question.
Sometimes the story may be too good. My confidence in the history I did not know was reduced by two errors (both on page 129) in the history that I do know. There is no evidence that the Arabs burned the Library of Alexandria in 621, as it was probably long gone by then. And Muhammad did not die in Jerusalem. How many other old myths are being recycled here?
However, I am sure the math is good, even if the story behind it might occasionally be questionable. I recommend this book for those with an interest in mathematics, but do not expect an advanced treatment of the ideas in question.
May 13, 2013
The title is a fair description: Dunham presents highlights from math history as great works of art. He carries this analogy through the book consistently, for example identifying Georg Cantor (-1918) as the mathematical parallel of his contemporary Vincent van Gogh.
Dunham has done an excellent job of selecting exemplary theorems that can be explained to an interested reader having no special mathematical training, that are associated with the most greatest mathematicians of all time, and that influenced the future of mathematics. In addition to the "great theorems," he finds time to describe many other masterpieces.
Of Dunham's twelve "Great Theorems," Heron's formula and Newton's general binomial theorem have by far the weakest credentials. Looking for candidates to replace them, though, gives a good idea of why they were chosen. Dunham certainly had a clear idea of his intended audience, and decided on that basis not to include any calculus beyond the Cauchy/Weierstrass definition of the limit -- this eliminates some clearly Great Theorems, most notably the Fundamental Theorem of Calculus, and explains the inclusion of the binomial theorem: "It was a must, for example, to have selections from ...Newton." (p. vii) On the other hand, the beautiful results of graph theory (particularly Euler's polyhedron formula) could hardly have been included without requiring a third chapter about Euler.
I recommend this book to a general reader interested in the history of mathematics, and particularly to undergraduate students of mathematics: it's a great chance to pick up the folklore that "everyone knows" but that isn't always taught explicitly.
Dunham has done an excellent job of selecting exemplary theorems that can be explained to an interested reader having no special mathematical training, that are associated with the most greatest mathematicians of all time, and that influenced the future of mathematics. In addition to the "great theorems," he finds time to describe many other masterpieces.
Of Dunham's twelve "Great Theorems," Heron's formula and Newton's general binomial theorem have by far the weakest credentials. Looking for candidates to replace them, though, gives a good idea of why they were chosen. Dunham certainly had a clear idea of his intended audience, and decided on that basis not to include any calculus beyond the Cauchy/Weierstrass definition of the limit -- this eliminates some clearly Great Theorems, most notably the Fundamental Theorem of Calculus, and explains the inclusion of the binomial theorem: "It was a must, for example, to have selections from ...Newton." (p. vii) On the other hand, the beautiful results of graph theory (particularly Euler's polyhedron formula) could hardly have been included without requiring a third chapter about Euler.
I recommend this book to a general reader interested in the history of mathematics, and particularly to undergraduate students of mathematics: it's a great chance to pick up the folklore that "everyone knows" but that isn't always taught explicitly.
October 28, 2011
This was a class book for a 'History of Math' course I took during my undergrad and it remains one of the few books from that era in my life that I actually return to now and then... Geeky, sure. Dorky, definitely, but this book provides a fascinating account of how advances in mathematics follows progress in civilization and vice versa.
From everybody's favorite theorem (the Pythagorean theorem that is) to the dreaded nightmare-inducing calculus (thank you, Sir Isaac Newton!) and beyond this little book packs quite the punch. Even if math isn't your thing, you can skip the proofs and just read the blurbs about the life, times and genius-inspiring circumstances of great historical figures.
From everybody's favorite theorem (the Pythagorean theorem that is) to the dreaded nightmare-inducing calculus (thank you, Sir Isaac Newton!) and beyond this little book packs quite the punch. Even if math isn't your thing, you can skip the proofs and just read the blurbs about the life, times and genius-inspiring circumstances of great historical figures.
July 24, 2020
The historical and biographical aspects of this text are fascinating in themselves. One certainly has to have a strong foundation, however, to understand the mathematical components fully and in context.
July 25, 2023
3. 5
Rayillos, si que me tardé mucho en leerlo, espero que la persona que me lo prestó no piense que me lo robé.
El libro es un estudio de 1990 sobre la historia de las matemáticas por el escritor y matemático William Dunham. Se expande más allá del alcance de la vida de Euler, perfilando sobre una docena de matemáticos. Sin embargo, en lugar de utilizar a estas personas como sus puntos focales, todo se organiza en torno a doce resultados importantes de la investigación matemática. Como resultado, se demuestra cómo los frutos de la creación de conocimiento se multiplican más allá de los límites de cualquier vida individual.
Parece que Dunham cuando buscó teoremas para su libro, no sólo seleccionó los que son ya exaltados en la academia o en la divulgación científica. Eligió sus doce teoremas preguntándose también si son herramientas efectivas para transmitir cómo era la vida intelectual en el momento en que se formaron.
Otra cosa que también buscaba lograr este señor era tratar las matemáticas como una forma de arte: así como las grandes novelas, las grandes sinfonías y las grandes pinturas son objetos de estudio para apreciar la literatura, la música y el arte, los teoremas son los objetos de estudio apropiados para apreciar las matemáticas.
Creo que la historia de las matemáticas presentada es muy buena, mientras que la exposición matemática es un poco desigual. Algunas de ellas son buenas y otras no tan buenas. Mi principal incomodidad es el hecho de que a veces las demostraciones se vuelven más pesadas de lo necesario y pueden terminar oscureciendo la simplicidad y elegancia de las matemáticas.
Eso sí, el libro es para lectores interesados, no especialistas, preferiblemente aquellos que disfrutaron de las matemáticas en la preparatoria y recuerdan algo de ellas, y si es que no les gustan mucho las demostraciones, hay mucho de interés en el material biográfico e histórico.
Rayillos, si que me tardé mucho en leerlo, espero que la persona que me lo prestó no piense que me lo robé.
El libro es un estudio de 1990 sobre la historia de las matemáticas por el escritor y matemático William Dunham. Se expande más allá del alcance de la vida de Euler, perfilando sobre una docena de matemáticos. Sin embargo, en lugar de utilizar a estas personas como sus puntos focales, todo se organiza en torno a doce resultados importantes de la investigación matemática. Como resultado, se demuestra cómo los frutos de la creación de conocimiento se multiplican más allá de los límites de cualquier vida individual.
Parece que Dunham cuando buscó teoremas para su libro, no sólo seleccionó los que son ya exaltados en la academia o en la divulgación científica. Eligió sus doce teoremas preguntándose también si son herramientas efectivas para transmitir cómo era la vida intelectual en el momento en que se formaron.
Otra cosa que también buscaba lograr este señor era tratar las matemáticas como una forma de arte: así como las grandes novelas, las grandes sinfonías y las grandes pinturas son objetos de estudio para apreciar la literatura, la música y el arte, los teoremas son los objetos de estudio apropiados para apreciar las matemáticas.
Creo que la historia de las matemáticas presentada es muy buena, mientras que la exposición matemática es un poco desigual. Algunas de ellas son buenas y otras no tan buenas. Mi principal incomodidad es el hecho de que a veces las demostraciones se vuelven más pesadas de lo necesario y pueden terminar oscureciendo la simplicidad y elegancia de las matemáticas.
Eso sí, el libro es para lectores interesados, no especialistas, preferiblemente aquellos que disfrutaron de las matemáticas en la preparatoria y recuerdan algo de ellas, y si es que no les gustan mucho las demostraciones, hay mucho de interés en el material biográfico e histórico.
June 13, 2019
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.
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.
July 22, 2015
Bad habits die hard, so let's start with a quotation, shall we? Make it a double one, since in the book, it originally is a quotation already. (And, like I said, bad habits die hard, so this is actually the conclusion of the book.)
If you ask me, this quite sums up the idea of the book in the sense that, by going through the history behind some of mathematics' greatest theorems, it shows the beauty of mathematics... but it should be noted that it's quite difficult to go through the book if you don't have some basics (and maybe more than basics) in the domains of mathematics explored in there. In other words, I don't think it suits people who would only be interested in the historical side of mathematics, you need to know a bit of the mathematics themselves. Well, maybe not for all chapters, because some of these theorems are easy concepts to grasp and/or are explained quite simply-ish (and Dunham even writes at times that such theorem goes beyond the level of the book)... but other chapters, not so much. Or maybe you can just skip over the proof and technical details, but I'm not sure if it's enjoyable to read this book that way.
That aside, from a (kinda) mathematics student POV, it's a very enjoyable reading. When you go through years of studies, you usually only get the technical side of stuffs (unless you have special profs who like to add anecdotes to their classes --those are the best, it makes it more interesting imo) and all history is swept under the rug. With this book, I got to catch up a bit on history. I think there's only Cardano's solution of the cubic (chapter 6) and Cantor's infinites (chapters 11-12) for which I knew most of the story already. In any case, all through the book, I was reminded of my many many classes, a small trip down memory lane so to say.
It covers a variety of domains: geometry (that I feel like I haven't been in touch with for a couple of years), algebra, number theory (the one I've, most likely, been studying more thoroughly), calculus (the one I've never really got along with)... And although the table contents seems to indicate that only a couple of great mathematicians are included in the book, there are actually so many more that come into the picture as you go through the chapters (a couple of them I've never even heard of). Another interesting point was to be confronted to those long-ass sentences they used to use. Maybe it's because my classes were all taught in French and the phrasing is slightly different from the modern English one, but sometimes I had to read it over a couple times to grasp the idea.
The only bad point I have in mind is that it was too short (although I do admit that a longer book might have been harder to digest). I feel like there's a good amount of other interesting theorems/theories that could have fitted in the book (not that I have specific ones in mind right now though), but maybe the history behind them isn't as fascinating as those included...
Mathematics, rightly viewed, possesses not only truth, but supreme beauty -a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show.
If you ask me, this quite sums up the idea of the book in the sense that, by going through the history behind some of mathematics' greatest theorems, it shows the beauty of mathematics... but it should be noted that it's quite difficult to go through the book if you don't have some basics (and maybe more than basics) in the domains of mathematics explored in there. In other words, I don't think it suits people who would only be interested in the historical side of mathematics, you need to know a bit of the mathematics themselves. Well, maybe not for all chapters, because some of these theorems are easy concepts to grasp and/or are explained quite simply-ish (and Dunham even writes at times that such theorem goes beyond the level of the book)... but other chapters, not so much. Or maybe you can just skip over the proof and technical details, but I'm not sure if it's enjoyable to read this book that way.
That aside, from a (kinda) mathematics student POV, it's a very enjoyable reading. When you go through years of studies, you usually only get the technical side of stuffs (unless you have special profs who like to add anecdotes to their classes --those are the best, it makes it more interesting imo) and all history is swept under the rug. With this book, I got to catch up a bit on history. I think there's only Cardano's solution of the cubic (chapter 6) and Cantor's infinites (chapters 11-12) for which I knew most of the story already. In any case, all through the book, I was reminded of my many many classes, a small trip down memory lane so to say.
It covers a variety of domains: geometry (that I feel like I haven't been in touch with for a couple of years), algebra, number theory (the one I've, most likely, been studying more thoroughly), calculus (the one I've never really got along with)... And although the table contents seems to indicate that only a couple of great mathematicians are included in the book, there are actually so many more that come into the picture as you go through the chapters (a couple of them I've never even heard of). Another interesting point was to be confronted to those long-ass sentences they used to use. Maybe it's because my classes were all taught in French and the phrasing is slightly different from the modern English one, but sometimes I had to read it over a couple times to grasp the idea.
The only bad point I have in mind is that it was too short (although I do admit that a longer book might have been harder to digest). I feel like there's a good amount of other interesting theorems/theories that could have fitted in the book (not that I have specific ones in mind right now though), but maybe the history behind them isn't as fascinating as those included...
February 17, 2008
At times the proofs can be a little hard to follow, but the book was definitely written for the layman with some calculus background. However, since the book covers such diverse mathematical topics, it is difficult to fully appreciate every theorem. The author does try to present every theorem in its historical context and give background on the great minds of the discoverers.
The most striking point of the entire book to me was how miserable the vast majority of the featured mathematicians lives were. They lived lonely lives and suffered from bouts of mental illness. I wonder if people like Cardano, Cantor, Newton, and Turing succeeded because of their faults or in spite of them. Their work was a great service to humanity, but was it worth it in the end for them. Would they have traded their greatness for a normal, happy life? Reading about their lives is enough to make one hope that they never accomplish anything great. Of course, there are plenty of people who lead miserable lives without accomplishing anything too.
The example that really sits with me was the final one illustrated by Cantor. He was a brilliant man who tackled the problem of the infinite. He spent much of his life struggling to prove or disprove his continuum hypothesis. It is believed that his obsession over this problem contributed to his bouts of mental illness and the complete breakdowns that he suffered. It may have even expedited his death. Twenty years after his death it was proven that this hypothesis could not be disproved. Another twenty years later it was also proven that the hypothesis could not be proven. This meant that he spent his life obsessing over a problem that could not be solved.
The most striking point of the entire book to me was how miserable the vast majority of the featured mathematicians lives were. They lived lonely lives and suffered from bouts of mental illness. I wonder if people like Cardano, Cantor, Newton, and Turing succeeded because of their faults or in spite of them. Their work was a great service to humanity, but was it worth it in the end for them. Would they have traded their greatness for a normal, happy life? Reading about their lives is enough to make one hope that they never accomplish anything great. Of course, there are plenty of people who lead miserable lives without accomplishing anything too.
The example that really sits with me was the final one illustrated by Cantor. He was a brilliant man who tackled the problem of the infinite. He spent much of his life struggling to prove or disprove his continuum hypothesis. It is believed that his obsession over this problem contributed to his bouts of mental illness and the complete breakdowns that he suffered. It may have even expedited his death. Twenty years after his death it was proven that this hypothesis could not be disproved. Another twenty years later it was also proven that the hypothesis could not be proven. This meant that he spent his life obsessing over a problem that could not be solved.
June 26, 2019
I read this book for a history of mathematics course I took in college. It was an excellent introduction to the history of mathematics. Talking to William Dunham was also an interesting experience. He mentioned that the publishers didn't want anything to be included that required more knowledge than some basic calculus, and he also didn't like the title "Journey Through Genius" at first though it grew on him as the book gained popularity.
Dunham teaches (or taught, I don't know) a history of math course himself, and is obviously an expert on the topic. It would have been nice to have included more advanced material, but the math covered is both important to the history of math and engaging enough that you will not forget it. Aside from the math, Dunham also includes historical bits about the mathematicians themselves. From the outrageous Pythagorean response to the discovery of irrational numbers to the battle of Tartaglia vs Cardano, the bizarre history of mathematicians lightens the more rigorous concepts introduced. A must-read for anyone unfamiliar with the history of math.
Dunham teaches (or taught, I don't know) a history of math course himself, and is obviously an expert on the topic. It would have been nice to have included more advanced material, but the math covered is both important to the history of math and engaging enough that you will not forget it. Aside from the math, Dunham also includes historical bits about the mathematicians themselves. From the outrageous Pythagorean response to the discovery of irrational numbers to the battle of Tartaglia vs Cardano, the bizarre history of mathematicians lightens the more rigorous concepts introduced. A must-read for anyone unfamiliar with the history of math.
June 10, 2015
A book about mathematics, written for the layman, but with some pretty deep math in there. As someone who likes math, this book was fascinating. A lot of it was about famous proofs I was already familiar with (Euclid's infinite primes, Cantor's diagonalization) but it was still really cool to read about them again. Dunham's tone is casual and fun, and even if the historical/biographical bits didn't seem very rigorous, the book was still very fun to read. I would definitely recommend it to anyone who thinks math is fun.
February 26, 2020
A quote from Bertrand Russel, which is offered in the afterword, perfectly sums up the contents of this book:
"Mathematics, rightly viewed, possesses not only thruth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure and capable of a stern perfection such as only the greatest art can show."
William Dunham perfectly transports this feeling. 5/5.
"Mathematics, rightly viewed, possesses not only thruth, but supreme beauty - a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure and capable of a stern perfection such as only the greatest art can show."
William Dunham perfectly transports this feeling. 5/5.
July 24, 2017
The author had an interesting idea for a book. Unfortunately, by only scratching a tiny portion of the surface, he executed that idea very poorly. I know ... I should have expected that from reading the Contents.
The real detraction for me is Dunham's overt prejudices: anti-Oriental; anti-Islamic; anti-German; and, anti-Catholic.
Leave this book on the shelf.
The real detraction for me is Dunham's overt prejudices: anti-Oriental; anti-Islamic; anti-German; and, anti-Catholic.
Leave this book on the shelf.
June 29, 2019
quite an interesting book，and the explanation is very clear. The only thing I feel a bit not very good is that the gap between math before 1750 and after 1850 is quite large. after euler the book jumps to cantor,which is a huge jump.
April 3, 2020
Wonderful, amazing and great 😊 this book is in itself a genius way to present maths...Math is an art and I am everyday fascinated by its history and its answers ....strongly recommend this book for all lovers of Mathematics
June 15, 2020
A good read about math history!
October 9, 2022
To hear honest, I don’t understand math. After reading A Mathematician’s Lament I realized there was something more to math than my education led me to believe. I read this book in an attempt to further understand and (hopefully) love math. The 3 stars are not a proper reflection of this book, but is due to my utter lack of understanding of 2/3 of the book. However, I did come out of it with an admiration for the beauty of math and I really enjoyed the historical and biographical backgrounds.
August 13, 2023
Pra quem gosta de matemática é perfeito. O autor faz um resumo de como a matemática evoluiu brilhantemente. Hoje tenho uma visão geral muito boa sobre cada período da matemática e pude apreciar a beleza dos belos teoremas escolhido a dedo pelo autor.
March 24, 2021
The geometric proofs described are fantastic (5/5)! Eye-opening and intriguing. The algebraic ones - not so much (2/5).
December 13, 2024
- Really great read. Maybe needs more chapters, org and a better title. Overall great 4000 year history. With great insights to key theorems as well as personalities. Thanks KD for the reco;).
- journey of Geometry & Algebra. Egypt --2000; Mezopotamya/Babil --1900-1600; Ege/Greece --1000; Thales of Milet --600; Pythagoras of Ege --572; Geometry over arithmetic drives math concepts these days for 1000years those days (since irrational numbers like sqrt(2) are not well defined yet); Hippocrates of Chios --440 (not the same as medicine Hippocrates of Kos:)): quadrature of lune; Euclid --300: Archimedes of Syracuse --287;
- Euclid
* date --300; came to alexandria
* wrote the "elements", 13 books, superbly organized treatise. bible of math. 2000 editions.
* Beautiful Proof of pythagorean theorem.
* Euclid's number theory. Oldest dev of subject.
* Euid method for (GCD) OBEB.
* Prime numbers. Bunch of theos. Infinitude of primes!
* Perfect numbers. Odd perfect numbers still unsolved today. wow.
- Real Nums : algebraic nums (rationals, sqrt(2)) + transcendental nums (pi).
- Goldbach Conjecture: Any even num gt 4 can be written as the sum of two primes. Euler agrees, but did not solve. not solved to date. wow.
- Archimedes
- Greek. born Syracuse in Sicily --287. Studied in Egypt.
- Invented archimedan screw.
- deep focus, single minded. "eureka" story.
- Floating bodies, optics, mechanics, pulleys, military defense devices when under attack by romans.
- Killed by roman soldier :(
- Great Theorem: area of circle.
- reductio ad absurdum
- Pi approximation using polygons. (this geometric approach lasted for ~2000 years. Then there was Newton/Leibniz and he crushed it w ease via binomial expansion and fluxions). Ramanujan did work in this 1912. 1948 pi got to 808 decimal places. Then computers came. ENIAC dis 2037 places. 1980s: 500M places.
- Sphere area.
- ratio of solid volumes.
- tombstone sphere in cylindr
- Eratosthenes
- --284. Chief librarian in alexandria
- Built "sieve" for listing primes.
- apprximated earth diameter via shadows.
- Heron
- ++75. of Alexandria
- Area of triangle using the edges: s=(a+b+c)/2 => A=sqrt(s(s-a)(s-b)(s-c)). Impressive proof.
- Alexandria library:
- founded --300 and closed by Christians ++529. burnt 641.
- Produced Euclid, Archimedes, Eratosthenes, Heron, et al.
- Cardano
- Pacioli (1445, Italian)considered cubic (ax^3+...=0) eqn a challenge in Summa de Arithmetica.
- del Ferro (1465, U of Bologna) solved depressed (wo x^2 term) cubic. Kept secret due to academic politics of time. Passed to student Fior (1506) at death.
- Niccolo Fontana (Tartaglia = stammerer, disfigured by sword as child). Fior challenged Tartaglia. Tartaglia solved depressed cubic on 1535.
- Cardano of Milan (1501). Physically fragile. Was not allowed tonpractice medicine. Asked tartaglia for the solution. Tartaglia eventually shared the secret to Cardano on 1539. Cardano sweared not to publish the secret. Cardano and protege Ferrari used this info to solve the generic cubic eqn. Ferrari also solved x^4 poly eqn. They did not publish due to their oath to Tartaglia.
- Then 1543, they discovered del Ferro's soln. This gave them a way to publish. 1545 Cardano published the cubic soln in his "Ars Magna" book.
- At this point they still thought of physical cubes and positive numbers. Negative numbers were not as common.
- 1824 Abel proved quintic (x^5) polynomials have no solution by radicals. There exists no algebraic formula involving only the coeffs of the quintic eq-n that can generate the solutions. WOW. unlike 2,3,4 degree polynoms. Wow. limits of algebra.
- odd degree polynomials have at least one real solution.
- Heroic century: 16th century
- Focus shifted from Italians to French, German, British.
- 1590: French Viete introduced algebraic notation
- 1600s: Napier and Briggs of British Isles introduced logarithm.
- Descartes, french, 1596, was also mathematician. Introduced algebraic geometry.
- Pascal, 1623, gifted, impressed descartes when 16. Probability theory. Also big into theology. Cycloid curve. Pascal's triangle!
- Fermat, 1601, french. Also analytic geometry. Collab w Pascal on Probabiliy theory. Differential calculus. Biggest Impact on Number theory. Fermat had many theorems. Not as big on proofs. Euler proved a whole bunch of them later.
- Fermat famous theorem. a^2+b^2=c^2 works. But no such solution exists for cubes, 4th power, etc. for a^n+b^n=c^n. Margin narrow for proof :). Eventually proved in 1900s:). Euler proved for n=3 and n=4.
- Newton
- dob 1642. premature, frail, lived. single mother. remarried. left to grandma. Neurotic, misanthropic adult. Went to Trinity College (Cambridge) 1661. More political, less academic env. Read Euclid's Elements. Descartes's La Geometrie. Prof Isaac Barrow was one who offered some direction, and later recognition.
- Ability to hold a problem in mind continuoisly. "by thinking on them continuously"..
- Plague years of 1665, 1666 were super productive. With all these discoveries, he was still an unknown, anon Cambridge student of age 24.
- 1665 discovered generalized binomial theorem
- Fluxions (differential calculus)
- 1666: inverse fluxions (integral calculus)
- Theory of colors
- Gravity
- Binomial Theorem:
- (a+b)^n expansion -> a^n + c1.a^(n-1).b + … + b^n
- Pascal Triangle
- Newton found expression of coeffs w/o iterating thru each stage, shared w Leibniz 1676
- Newton’s Pi Approximation: using geometry, fluxions and binomial theorem.
- Did innovation in astronomy. Did not publish much due too dealing w criticism.
- Principia
- 1684, Halley (of Comet) urged Newton to publish some his phys/astro work. From this came “Philosipia Naturalis Principia Mathematica”, laws of motion and gravtity (1687).
- Unleashed Newtonian physics.
- 1696: Moved to govt job. British Mint.
- 1704: Published “Opticks” from his 40 y/o ideas.
- 1705: became Sir Isaac.
- Leibniz
- dob 1646, child prodigy, completed phd at 20, went to govt/legal/diplomat job, built calculating machine
- Sent to Paris 1676, met Huygens (Dutch), Huygens: work on cycloid curve, used in pendulums; Saturn rings,
- Huygens suggested “sum of reciprocal oftriangular numbers” problem:
- S= 1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + ... => Leibniz solved S=2
- Saw Newton’s docs on fluxions before they were widely published; corresponded w Newton;
- Then published his work on maxima/minima. The world learned calculus from Leaibniz, before Newton published. Led to some controversy between them.
- Bernoillis
- Jakob (1654), Johann (1667) followed in steps of Leibniz to disseminate calculus. Big contrib to probability.
- Johann worked w L’Hospital.
- Catenary curve: chain hanging from two ends.
- Geo Series: a + a^2 + a^3 + … => converges to a/(1-a), where -1 - Harmonic Series:
- Sh = 1 + 1/2 + 1/3 + 1/4 + …
- Bernoillis proved Sh is divergent. Very slow growing (250M terms to get to a sum of 20), but divergent.
- Brachistochrone Curve Challenge:
- Johann posed challenge in 1696-1697
- Leibniz had one solution
- 1697, Johann sent the problem to Newton, to antagonize
- Newton, while dealing w Mint those days, solved it in a day: “I dont love to be teased by foreigners on Math things…”
- 5 total solutions: Johann, Leibniz, Jakob, L’Hospital, and a letter w ENglish postmark (Newton:))
- Johann: “I recognize the lion by his paw!” (speculative)
- The soluion was: Upside down Cycloid of Huygens :o
- Euler
- dob 1707, Basel, Swiss. Super prolific. 73 volumes of work “Opera Omnia”. Pub backlog lasted 47 years after death.
- Lost right eyesight 1730. Lost most vision 1771. Warmer personality. 13 children. Passionate about teaching.
- STudied w Bernoulli.
- Solved optimum placement of Masts on ship at 19. Won French prize.
- 1727, went to St Petersburg Academy, Russia. Via Daniel Bernoulli, son of Johann.
- 1741 back to Berlin Academy. Back to St Petersburg after some time.
- Seminal works: 1748 - Intro to “Analysin Infinitorum”; 1755 - Differential Calc; 1768-74 Integral calc;
- Insane breadth: Euler Triangle in Geo; Euler characteristic in Topology; Euler Circuit in Graph; Euler constant, polynomials, integrals, etc. Wah.
- S=1+1/4+1/9+1/25+...+1/k^2+... (I believe this is Riemann Zeta func-n where power is 2)
- Bernoullis and Leibniz did not solve but knew <2.
- Euler found a solution from expansion of Sin(x) = x - x^3/3! + x^5/5! - …
- Found s = pi^2 / 6. Wow.
- He computed bunch of other power sums.
- Wow. Sum of reciprocals of odd powers is still unsolved!
- 1 + 1/2^3 + 1/3^3 + … + 1/k^3 + …
- Goldbach (of Goldbach conjecture) drove Euler’s interest in Number thery and Fermat’s theorems.
- Fermats’s theorem on primes: “Primes are either of form A: 4k+1 or form B: 4k+3. Form A primes can be written as sum of two perfect squares in one unique way. Form B primes cannot be written as sum of two perfect squares at all.”
- Euler proved 1747
- Euler proved bunch of Fermat theorems and debunked Euler’s prime conjecture.
- Number Theory vs Calculus
- Euclid’s proof ot infinite primes
- Fermat’s work
- Little progress in 17th century becoz calculus was all the hype. Not much appln of number theory.
- Gauss
- dob 1777. 6y/o when Euler died. Became the next big mathematician. phd 1799.
- Gauss Theorem: 1+2+…+N = N(N+1)/2.
- Solved at school when teacher gave to keep the students busy.
- Fundamental theorem of algebra.
- At 30 became director of Gottingen observatory
- Ceres’s orbit.
- Magnetic field => Unit Gauss.
- Supported female mathematician Sophie Germain
- 18th century recap:
- Euler, Newton, Leibniz
- Newton's fluxions and Leibniz's Calculus.
- 19th century: both art and math started deviating from being closely coupled w physical phenomena
- eg, Cezanne, Gaugin, Van Gogh,
- Math also shifted to abstract concepts.
- Maybe both ventured too far?
- Until 19th century, concept of limits was still poorly defined by Newton/Leibniz.
- Cauchy (dob 1789) defined limits proper.
- Weierstrass made this in the symbolic form we undertand today.
- Cantor tackled the challenge of infinites.
- Cantor:
- dob 1845 Russia -> Germany. Also had erratic, bipolar.
- denumerable: countably infinite; eg Natural nums, whole nums, rational nums.
- Natural numbers, whole numbers, integers, even numbers, rational numbers all set equivalent (N_0).; same cardinality.
- Non-denumerability of continuum: Real numbers in [0,1] not denumerable.
- Wow: cardinality of set Irrational numbers > cardinality of set of rational numbers (c).
- WOW: cardinality of Set of transcendal numbers (eg PI) > cardinality of set of algebraic (constructable) numbers (eg all rationals and subset of irrationals a la sqrt(2). (algebraics are like stars in space, transcendentals are like the space in space).
- Continuum Hypothesis (neither can be proven nor disporven): No cardinality exists betw N_0 (denumerable, e.g., natural nums) and c (no -debumerable, e.g., real nums). Like 0 and 1 of whole nums.
- About Pi
- Pi is our first found transcendental \o/
- Quadrature of circle is not possible.
- Becoz Area(circle)=pi.r^2. Square side would be sqrt(pi).r. We cannot construct sqrt(pi). :)
- Pi approximations
- Egyptians did empirical estimates: 3.16...
- Archimedes: Estimated 2 digits by drawing 96-gons inside/outside circle.
- Ptolemy of Alexandria (++150) did 360-gon. Got to 4 digits.
- Viete (dob 1540, French) got to 9 digits w 393216-gon. :)
- Van Ceulen (1600s, dutch) got to 35 digits. Still w geo approach.
- Newton (late 1600s) changed the game w binomial theorems and fluxions to get accurate estimates w ease
- Leibniz (1674) got the series: 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... = pi/4. But This approached pi very slowly. Zillions of terms needed to get to 100 places.
- Sharp and Machin (late 1600-early 1700) built faster converging series. Got to 71 and 100 decima places respectively.
- Lambert (1767) proved pi is irrational.
- Lindeman (1882) proved pi is transcendental.
- Ramanujan (1914) built eff, rapid estimation formulas for pi.
- Ferguson (1946, English) got to 710 places.
- Wrench (1947, American) got to 808.
- ENIAC (1949) got to 2037 places.
- Supercomputers (1980) got to 500M places.
- QED. :)
- journey of Geometry & Algebra. Egypt --2000; Mezopotamya/Babil --1900-1600; Ege/Greece --1000; Thales of Milet --600; Pythagoras of Ege --572; Geometry over arithmetic drives math concepts these days for 1000years those days (since irrational numbers like sqrt(2) are not well defined yet); Hippocrates of Chios --440 (not the same as medicine Hippocrates of Kos:)): quadrature of lune; Euclid --300: Archimedes of Syracuse --287;
- Euclid
* date --300; came to alexandria
* wrote the "elements", 13 books, superbly organized treatise. bible of math. 2000 editions.
* Beautiful Proof of pythagorean theorem.
* Euclid's number theory. Oldest dev of subject.
* Euid method for (GCD) OBEB.
* Prime numbers. Bunch of theos. Infinitude of primes!
* Perfect numbers. Odd perfect numbers still unsolved today. wow.
- Real Nums : algebraic nums (rationals, sqrt(2)) + transcendental nums (pi).
- Goldbach Conjecture: Any even num gt 4 can be written as the sum of two primes. Euler agrees, but did not solve. not solved to date. wow.
- Archimedes
- Greek. born Syracuse in Sicily --287. Studied in Egypt.
- Invented archimedan screw.
- deep focus, single minded. "eureka" story.
- Floating bodies, optics, mechanics, pulleys, military defense devices when under attack by romans.
- Killed by roman soldier :(
- Great Theorem: area of circle.
- reductio ad absurdum
- Pi approximation using polygons. (this geometric approach lasted for ~2000 years. Then there was Newton/Leibniz and he crushed it w ease via binomial expansion and fluxions). Ramanujan did work in this 1912. 1948 pi got to 808 decimal places. Then computers came. ENIAC dis 2037 places. 1980s: 500M places.
- Sphere area.
- ratio of solid volumes.
- tombstone sphere in cylindr
- Eratosthenes
- --284. Chief librarian in alexandria
- Built "sieve" for listing primes.
- apprximated earth diameter via shadows.
- Heron
- ++75. of Alexandria
- Area of triangle using the edges: s=(a+b+c)/2 => A=sqrt(s(s-a)(s-b)(s-c)). Impressive proof.
- Alexandria library:
- founded --300 and closed by Christians ++529. burnt 641.
- Produced Euclid, Archimedes, Eratosthenes, Heron, et al.
- Cardano
- Pacioli (1445, Italian)considered cubic (ax^3+...=0) eqn a challenge in Summa de Arithmetica.
- del Ferro (1465, U of Bologna) solved depressed (wo x^2 term) cubic. Kept secret due to academic politics of time. Passed to student Fior (1506) at death.
- Niccolo Fontana (Tartaglia = stammerer, disfigured by sword as child). Fior challenged Tartaglia. Tartaglia solved depressed cubic on 1535.
- Cardano of Milan (1501). Physically fragile. Was not allowed tonpractice medicine. Asked tartaglia for the solution. Tartaglia eventually shared the secret to Cardano on 1539. Cardano sweared not to publish the secret. Cardano and protege Ferrari used this info to solve the generic cubic eqn. Ferrari also solved x^4 poly eqn. They did not publish due to their oath to Tartaglia.
- Then 1543, they discovered del Ferro's soln. This gave them a way to publish. 1545 Cardano published the cubic soln in his "Ars Magna" book.
- At this point they still thought of physical cubes and positive numbers. Negative numbers were not as common.
- 1824 Abel proved quintic (x^5) polynomials have no solution by radicals. There exists no algebraic formula involving only the coeffs of the quintic eq-n that can generate the solutions. WOW. unlike 2,3,4 degree polynoms. Wow. limits of algebra.
- odd degree polynomials have at least one real solution.
- Heroic century: 16th century
- Focus shifted from Italians to French, German, British.
- 1590: French Viete introduced algebraic notation
- 1600s: Napier and Briggs of British Isles introduced logarithm.
- Descartes, french, 1596, was also mathematician. Introduced algebraic geometry.
- Pascal, 1623, gifted, impressed descartes when 16. Probability theory. Also big into theology. Cycloid curve. Pascal's triangle!
- Fermat, 1601, french. Also analytic geometry. Collab w Pascal on Probabiliy theory. Differential calculus. Biggest Impact on Number theory. Fermat had many theorems. Not as big on proofs. Euler proved a whole bunch of them later.
- Fermat famous theorem. a^2+b^2=c^2 works. But no such solution exists for cubes, 4th power, etc. for a^n+b^n=c^n. Margin narrow for proof :). Eventually proved in 1900s:). Euler proved for n=3 and n=4.
- Newton
- dob 1642. premature, frail, lived. single mother. remarried. left to grandma. Neurotic, misanthropic adult. Went to Trinity College (Cambridge) 1661. More political, less academic env. Read Euclid's Elements. Descartes's La Geometrie. Prof Isaac Barrow was one who offered some direction, and later recognition.
- Ability to hold a problem in mind continuoisly. "by thinking on them continuously"..
- Plague years of 1665, 1666 were super productive. With all these discoveries, he was still an unknown, anon Cambridge student of age 24.
- 1665 discovered generalized binomial theorem
- Fluxions (differential calculus)
- 1666: inverse fluxions (integral calculus)
- Theory of colors
- Gravity
- Binomial Theorem:
- (a+b)^n expansion -> a^n + c1.a^(n-1).b + … + b^n
- Pascal Triangle
- Newton found expression of coeffs w/o iterating thru each stage, shared w Leibniz 1676
- Newton’s Pi Approximation: using geometry, fluxions and binomial theorem.
- Did innovation in astronomy. Did not publish much due too dealing w criticism.
- Principia
- 1684, Halley (of Comet) urged Newton to publish some his phys/astro work. From this came “Philosipia Naturalis Principia Mathematica”, laws of motion and gravtity (1687).
- Unleashed Newtonian physics.
- 1696: Moved to govt job. British Mint.
- 1704: Published “Opticks” from his 40 y/o ideas.
- 1705: became Sir Isaac.
- Leibniz
- dob 1646, child prodigy, completed phd at 20, went to govt/legal/diplomat job, built calculating machine
- Sent to Paris 1676, met Huygens (Dutch), Huygens: work on cycloid curve, used in pendulums; Saturn rings,
- Huygens suggested “sum of reciprocal oftriangular numbers” problem:
- S= 1 + 1/3 + 1/6 + 1/10 + 1/15 + 1/21 + ... => Leibniz solved S=2
- Saw Newton’s docs on fluxions before they were widely published; corresponded w Newton;
- Then published his work on maxima/minima. The world learned calculus from Leaibniz, before Newton published. Led to some controversy between them.
- Bernoillis
- Jakob (1654), Johann (1667) followed in steps of Leibniz to disseminate calculus. Big contrib to probability.
- Johann worked w L’Hospital.
- Catenary curve: chain hanging from two ends.
- Geo Series: a + a^2 + a^3 + … => converges to a/(1-a), where -1 - Harmonic Series:
- Sh = 1 + 1/2 + 1/3 + 1/4 + …
- Bernoillis proved Sh is divergent. Very slow growing (250M terms to get to a sum of 20), but divergent.
- Brachistochrone Curve Challenge:
- Johann posed challenge in 1696-1697
- Leibniz had one solution
- 1697, Johann sent the problem to Newton, to antagonize
- Newton, while dealing w Mint those days, solved it in a day: “I dont love to be teased by foreigners on Math things…”
- 5 total solutions: Johann, Leibniz, Jakob, L’Hospital, and a letter w ENglish postmark (Newton:))
- Johann: “I recognize the lion by his paw!” (speculative)
- The soluion was: Upside down Cycloid of Huygens :o
- Euler
- dob 1707, Basel, Swiss. Super prolific. 73 volumes of work “Opera Omnia”. Pub backlog lasted 47 years after death.
- Lost right eyesight 1730. Lost most vision 1771. Warmer personality. 13 children. Passionate about teaching.
- STudied w Bernoulli.
- Solved optimum placement of Masts on ship at 19. Won French prize.
- 1727, went to St Petersburg Academy, Russia. Via Daniel Bernoulli, son of Johann.
- 1741 back to Berlin Academy. Back to St Petersburg after some time.
- Seminal works: 1748 - Intro to “Analysin Infinitorum”; 1755 - Differential Calc; 1768-74 Integral calc;
- Insane breadth: Euler Triangle in Geo; Euler characteristic in Topology; Euler Circuit in Graph; Euler constant, polynomials, integrals, etc. Wah.
- S=1+1/4+1/9+1/25+...+1/k^2+... (I believe this is Riemann Zeta func-n where power is 2)
- Bernoullis and Leibniz did not solve but knew <2.
- Euler found a solution from expansion of Sin(x) = x - x^3/3! + x^5/5! - …
- Found s = pi^2 / 6. Wow.
- He computed bunch of other power sums.
- Wow. Sum of reciprocals of odd powers is still unsolved!
- 1 + 1/2^3 + 1/3^3 + … + 1/k^3 + …
- Goldbach (of Goldbach conjecture) drove Euler’s interest in Number thery and Fermat’s theorems.
- Fermats’s theorem on primes: “Primes are either of form A: 4k+1 or form B: 4k+3. Form A primes can be written as sum of two perfect squares in one unique way. Form B primes cannot be written as sum of two perfect squares at all.”
- Euler proved 1747
- Euler proved bunch of Fermat theorems and debunked Euler’s prime conjecture.
- Number Theory vs Calculus
- Euclid’s proof ot infinite primes
- Fermat’s work
- Little progress in 17th century becoz calculus was all the hype. Not much appln of number theory.
- Gauss
- dob 1777. 6y/o when Euler died. Became the next big mathematician. phd 1799.
- Gauss Theorem: 1+2+…+N = N(N+1)/2.
- Solved at school when teacher gave to keep the students busy.
- Fundamental theorem of algebra.
- At 30 became director of Gottingen observatory
- Ceres’s orbit.
- Magnetic field => Unit Gauss.
- Supported female mathematician Sophie Germain
- 18th century recap:
- Euler, Newton, Leibniz
- Newton's fluxions and Leibniz's Calculus.
- 19th century: both art and math started deviating from being closely coupled w physical phenomena
- eg, Cezanne, Gaugin, Van Gogh,
- Math also shifted to abstract concepts.
- Maybe both ventured too far?
- Until 19th century, concept of limits was still poorly defined by Newton/Leibniz.
- Cauchy (dob 1789) defined limits proper.
- Weierstrass made this in the symbolic form we undertand today.
- Cantor tackled the challenge of infinites.
- Cantor:
- dob 1845 Russia -> Germany. Also had erratic, bipolar.
- denumerable: countably infinite; eg Natural nums, whole nums, rational nums.
- Natural numbers, whole numbers, integers, even numbers, rational numbers all set equivalent (N_0).; same cardinality.
- Non-denumerability of continuum: Real numbers in [0,1] not denumerable.
- Wow: cardinality of set Irrational numbers > cardinality of set of rational numbers (c).
- WOW: cardinality of Set of transcendal numbers (eg PI) > cardinality of set of algebraic (constructable) numbers (eg all rationals and subset of irrationals a la sqrt(2). (algebraics are like stars in space, transcendentals are like the space in space).
- Continuum Hypothesis (neither can be proven nor disporven): No cardinality exists betw N_0 (denumerable, e.g., natural nums) and c (no -debumerable, e.g., real nums). Like 0 and 1 of whole nums.
- About Pi
- Pi is our first found transcendental \o/
- Quadrature of circle is not possible.
- Becoz Area(circle)=pi.r^2. Square side would be sqrt(pi).r. We cannot construct sqrt(pi). :)
- Pi approximations
- Egyptians did empirical estimates: 3.16...
- Archimedes: Estimated 2 digits by drawing 96-gons inside/outside circle.
- Ptolemy of Alexandria (++150) did 360-gon. Got to 4 digits.
- Viete (dob 1540, French) got to 9 digits w 393216-gon. :)
- Van Ceulen (1600s, dutch) got to 35 digits. Still w geo approach.
- Newton (late 1600s) changed the game w binomial theorems and fluxions to get accurate estimates w ease
- Leibniz (1674) got the series: 1 - 1/3 + 1/5 - 1/7 + 1/9 - ... = pi/4. But This approached pi very slowly. Zillions of terms needed to get to 100 places.
- Sharp and Machin (late 1600-early 1700) built faster converging series. Got to 71 and 100 decima places respectively.
- Lambert (1767) proved pi is irrational.
- Lindeman (1882) proved pi is transcendental.
- Ramanujan (1914) built eff, rapid estimation formulas for pi.
- Ferguson (1946, English) got to 710 places.
- Wrench (1947, American) got to 808.
- ENIAC (1949) got to 2037 places.
- Supercomputers (1980) got to 500M places.
- QED. :)
January 3, 2015
Loved it. Great exposition of some of the best ideas in the history of mathematics. I've seen some other books like this, but this is the first one that really explains the way the Greeks did their mathematics. I've seen Euclid's proof of the Pythagorean Theorem before, but this is the first time it really made sense, and the first time I understood why Euclid had to prove it that way: he had no concept of what we would call algebra. For him a square was just that, and the theorem isn't a^2 + b^2 = c^2, but "if you make each side of a triangle into a square, the total area of the squares on the shorter sides is the same as the area of the square on the long side."
I have a couple of quibbles. He uses "awesome" too much; that's more of an obnoxious mannerism now than it was when Dunham wrote, and it grates. More important: One of the virtues of this book is that it's brief, and Dunham limits himself to math that is important, elegant, and easily explained. Selection is very important, and Dunham's excellent sense of what to leave out is why this book works so well. With that in mind, maybe it's unfair of me to complain about what was left out. But this book ends with the 19th century. There are brief mentions of Hilbert and Gödel, and that's about it. I would have liked to see it push a bit farther: Hilbert's reformulation of mathematics, Gödel's incompleteness theorem, and maybe even Turing's undecidability theorem. Surely that's not too much to ask; it's just following the story he was telling in his last chapter through until the end.
I have a couple of quibbles. He uses "awesome" too much; that's more of an obnoxious mannerism now than it was when Dunham wrote, and it grates. More important: One of the virtues of this book is that it's brief, and Dunham limits himself to math that is important, elegant, and easily explained. Selection is very important, and Dunham's excellent sense of what to leave out is why this book works so well. With that in mind, maybe it's unfair of me to complain about what was left out. But this book ends with the 19th century. There are brief mentions of Hilbert and Gödel, and that's about it. I would have liked to see it push a bit farther: Hilbert's reformulation of mathematics, Gödel's incompleteness theorem, and maybe even Turing's undecidability theorem. Surely that's not too much to ask; it's just following the story he was telling in his last chapter through until the end.
February 13, 2019
As a Calculus teacher, I am always baffled at students knowledge of history especially in regard to the mathematics. This of course is the fault of stale 21st century curriculum that teaches math as mastery of procedure than as the art of problem solving. It pains when when I ask students which mathematical mastermind derived such a beautiful argument, and of a sample size of 100 only 2 to 3 have heard of Archimedes let alone Gauss or Euler. As such, I began intentionally adding historical and small bios of history's finest in to as many lessons as I could. I feel that this shift has brought mathematics to more life in the classroom, as the focus is now viewing math as a body of knowledge that humanity has building upon and exploring together for thousands of years. It also helps build connections among their other classroom studies building stronger academic awareness. After reading Journey through Genius, I have learned so much more history and beautiful arguments to bring into the classroom. The author's ability to tell a story was gripping, his ability to present mathematical proofs coherently and seamlessly was incredible, and the wide array of tangential tidbits of historical figures and factoids was masterfully chosen. I am truly inspired by this book, and believe that my students will love to hear these stories in the classroom.
June 18, 2017
A decent trawl through a few millennia of mathematics, focusing on specific theorems that the author describes as "the great theorems of mathematics". The pen pictures of the mathematicians are good but what sets the book apart from the large number of similar books is the focus on specific theorems and their proofs. So there's quite a lot of actual mathematics in the book. For me this is both a strength and a weakness - the former because it makes the achievements more real but the latter because it's been many years since I had the patience to go through a mathematical proof properly.
I wasn't entirely enamoured of the choice of theorems but I understand the author's dilemma in wanting to choose theorems whose proofs could be understood by non-mathematicians and also wanting to include contributions from the most important mathematicians in his list. Ultimately, I felt that the book was only partially successful in convincing me of the beauty of the proofs and I suspect that this was partly down to the choice of theorems covered. Cantor's diagonal arguments being some of the more successful examples in this regard.
August 16, 2018
It took me a while to finally read this book but I believe it to be an all time classic. This sets the example for great math history books. Of course there is an inherent risk at choosing to tell certain stories (in this case, certain theorems) above others, Dunham really does make a great selection of great, unexpected, brilliant theorems that are easy to explain, easy to understand, that had a great impact and with solutions that were truly a work of genius.
Dunham does a great job at linking each theorem of his choosing to the next, telling the little anecdotes that happened in between, keeps us updated with important characters in the story that were incredibly decisive even if their work was not chosen as the protagonist.
Along with the story, we get to read a simplified and modern version of both the theorem and its proof, so it is not only history but some great math in as well. I would absolutely recommend this book to every math aficionado.
Dunham does a great job at linking each theorem of his choosing to the next, telling the little anecdotes that happened in between, keeps us updated with important characters in the story that were incredibly decisive even if their work was not chosen as the protagonist.
Along with the story, we get to read a simplified and modern version of both the theorem and its proof, so it is not only history but some great math in as well. I would absolutely recommend this book to every math aficionado.
June 21, 2020
I read this for class, but I actually really enjoyed it and finished the whole thing even though the last few chapters were cut from our curriculum. The book focuses more on the history of mathematics than the mathematics itself; I think most of it is written for a reader with a pre-calculus background, with a few chapters needing a calculus background to understand the math more clearly (though not understanding the details of the proofs isn't necessary to understand the larger significance or history). The way the history is told is really engaging and interesting. It's a bit dated; for example, it states that Fermat's Last Theorem remains unproven, though it was proven in 1995, but for the most part it is a fairly expansive coverage of mathematical history.
May 1, 2019
Journey Through Genius contains mathematics histories and technical details on important subjects, written for layman. With this huge subject, Dunham probably put a lot of thoughts into selecting what to include, and how.
Readers will get some sense on the broad pattern of mathematics development and how it fits into human civilization progression. Readers will also find detailed stories such as how Issac Newton left his work on Calculus unpublished for years.
Dunham shares with readers several scenes that demonstrate the beauty of mathematics. These scenes vary in the level of details from almost-complete technical proofs to hand waving description of critical ideas.
Readers will get some sense on the broad pattern of mathematics development and how it fits into human civilization progression. Readers will also find detailed stories such as how Issac Newton left his work on Calculus unpublished for years.
Dunham shares with readers several scenes that demonstrate the beauty of mathematics. These scenes vary in the level of details from almost-complete technical proofs to hand waving description of critical ideas.
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