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The King of Infinite Space: Euclid and His Elements
Geometry defines the world around us, helping us make sense of everything from architecture to military science to fashion. And for over two thousand years, geometry has been equated with Euclid’s Elements, arguably the most influential book in the history of mathematics In The King of Infinite Space, renowned mathematics writer David Berlinski provides a concise homage to this elusive mathematician and his staggering achievements. Berlinski shows that, for centuries, scientists and thinkers from Copernicus to Newton to Einstein have relied on Euclid’s axiomatic system, a method of proof still taught in classrooms around the world. Euclid’s use of elemental logic—and the mathematical statements he and others built from it—have dramatically expanded the frontiers of human knowledge.
The King of Infinite Space presents a rich, accessible treatment of Euclid and his beautifully simple geometric system, which continues to shape the way we see the world.
The King of Infinite Space presents a rich, accessible treatment of Euclid and his beautifully simple geometric system, which continues to shape the way we see the world.
172 pages, Hardcover
First published January 8, 2013
35 people are currently reading
456 people want to read
About the author
David Berlinski
31 books265 followersDavid Berlinski is a senior fellow in the Discovery Institute’s Center for Science and Culture.
Recent articles by Berlinski have been prominently featured in Commentary, Forbes ASAP, and the Boston Review. Two of his articles, “On the Origins of the Mind” (November 2004) and “What Brings a World into Being” (March 2001), have been anthologized in The Best American Science Writing 2005, edited by Alan Lightman (Harper Perennial), and The Best American Science Writing 2002, edited by Jesse Cohen, respectively.
Berlinski received his Ph.D. in philosophy from Princeton University and was later a postdoctoral fellow in mathematics and molecular biology at Columbia University. He has authored works on systems analysis, differential topology, theoretical biology, analytic philosophy, and the philosophy of mathematics, as well as three novels. He has also taught philosophy, mathematics and English at Stanford, Rutgers, the City University of New York and the Université de Paris. In addition, he has held research fellowships at the International Institute for Applied Systems Analysis in Austria and the Institut des Hautes Études Scientifiques. He lives in Paris.
Recent articles by Berlinski have been prominently featured in Commentary, Forbes ASAP, and the Boston Review. Two of his articles, “On the Origins of the Mind” (November 2004) and “What Brings a World into Being” (March 2001), have been anthologized in The Best American Science Writing 2005, edited by Alan Lightman (Harper Perennial), and The Best American Science Writing 2002, edited by Jesse Cohen, respectively.
Berlinski received his Ph.D. in philosophy from Princeton University and was later a postdoctoral fellow in mathematics and molecular biology at Columbia University. He has authored works on systems analysis, differential topology, theoretical biology, analytic philosophy, and the philosophy of mathematics, as well as three novels. He has also taught philosophy, mathematics and English at Stanford, Rutgers, the City University of New York and the Université de Paris. In addition, he has held research fellowships at the International Institute for Applied Systems Analysis in Austria and the Institut des Hautes Études Scientifiques. He lives in Paris.
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Displaying 1 - 30 of 51 reviews
April 18, 2013
Though the title suggests we might learn a little about Euclid, we instead find that there's little to know about Euclid. Greek, geometer, wrote The Elements. And . . . That's about it. Oh, and he lived--er, flourished, we're reminded by the author--about 300 BC.
Berlinski occasionally attempts to weave a bit of story from the spare bits known, but we leave the book as ignorant of Euclid the man as when we started, assuming we're members of the mature audience Berlinski is writing for.
Berlinski in the mid-90s produced the successful "Tour of the Calculus", a work which succeeded in spite of an air of being occasionally too clever by half. The same malady, which looks more like a permanent affliction now, is seen throughout his work on Euclid and Elements.
When discussing Euclid's work, Berlinski proves very capable and often entertaining, even humorous, while teaching the reader about geometry's foundations. Just a bit too often, though, the reader is distracted by a writer's overreach for the unusual word or striking phrase. It sometimes works, but sometimes not. And if a writer's going to show off, he must be certain not to be seen as doing so, or, if so brazen as to display even the showboating, he better get it right. I twice reached for the OED, thinking a word a poor choice, and twice was proved correct.
A word of kindness, though, is due. Berlinski occasionally sprinkles a joke or humorous bon mot across the top of things, and when he does, he often succeeds in producing a chuckle or even an outright laugh from the serious reader of what my aunt would call a math book.
He does a generally good job of discussing proofs and propositions, including some of the weak spots. In a couple places, anecdotes or metaphors are of a sufficiently grown-up nature as to warrant a recommendation that teachers and parents read it through carefully before recommending it to students of in the early high school years or younger. We're not talking NSFW labeling, but to a youngster one might oneself explaining a remark one wishes hadn't come up. All things considered, though, recommended, with only modest reservations.
Berlinski occasionally attempts to weave a bit of story from the spare bits known, but we leave the book as ignorant of Euclid the man as when we started, assuming we're members of the mature audience Berlinski is writing for.
Berlinski in the mid-90s produced the successful "Tour of the Calculus", a work which succeeded in spite of an air of being occasionally too clever by half. The same malady, which looks more like a permanent affliction now, is seen throughout his work on Euclid and Elements.
When discussing Euclid's work, Berlinski proves very capable and often entertaining, even humorous, while teaching the reader about geometry's foundations. Just a bit too often, though, the reader is distracted by a writer's overreach for the unusual word or striking phrase. It sometimes works, but sometimes not. And if a writer's going to show off, he must be certain not to be seen as doing so, or, if so brazen as to display even the showboating, he better get it right. I twice reached for the OED, thinking a word a poor choice, and twice was proved correct.
A word of kindness, though, is due. Berlinski occasionally sprinkles a joke or humorous bon mot across the top of things, and when he does, he often succeeds in producing a chuckle or even an outright laugh from the serious reader of what my aunt would call a math book.
He does a generally good job of discussing proofs and propositions, including some of the weak spots. In a couple places, anecdotes or metaphors are of a sufficiently grown-up nature as to warrant a recommendation that teachers and parents read it through carefully before recommending it to students of in the early high school years or younger. We're not talking NSFW labeling, but to a youngster one might oneself explaining a remark one wishes hadn't come up. All things considered, though, recommended, with only modest reservations.
February 5, 2014
I apologize in advance for the fact that even after reading this, I still don't really know who the intended audience is or why the author felt the need to write it. But I did enjoy reading it. I really think I should get credit for reading it 3 or 4 times because I read all the chapters save the last about 3 or 4 times.
I liked the way the author presented the information. There was enough humor to make the subject matter seem light-hearted and breezy, which made you feel a great deal duller and far thicker-skulled for not understanding it than if it had been bone dry and dead serious. I think it is good to be slapped upside the head by the limits of your knowledge and if it can be done with a rubber chicken...even better.
I am studying Euclid right now because I don't know. It isn't really my thing. Math does not interest me. In fact, I don't think it is going to far to say that it repels me. But, I have undertaken the project of re-educating myself in the classical tradition and it occurred to me recently that my plan did not include much science or any math. And here I am puttering about ancient Hellas just ignoring that some of the most important Greek contributions to Western Thought have been in the areas of science and math.
Additionally, I'm hoping that my study of Euclidean geometry will do two things: force my brain to work in ways that challenge it in order to keep those neurons firing and carving out new pathways well into my dotage. And allay, once and forever, the idea that I am not good at math. Because I have suspected for a long time now that I actually could understand math concepts with competence, if not mastery, and that years of being told otherwise gave me a ready excuse for giving up on it.
That said, merely sitting down with Euclid's "Elements" was daunting. I did it anyway but felt I wanted some context to go with the proofs. Along comes Berlinski and I don't know that he wrote this book for the purpose to which I applied it but I don't think he'd mind. I did not understand everything he said even after 4 readings (Poincare's hyperbolic disk? Yeah, that wasn't happening. I asked my husband, "Is it shaped like an M&M?" I was hungry, maybe) but I tried and I think I got the basic sentiment and it did help to have math concepts run through the filter of literature.
I can recommend the book but to whom or for what purpose I can't say. Again, ever so sorry about that. I can tell you that if you're looking for a juicy bio of Euclid the man, this isn't the book you're looking for. That book does not exist.
I liked the way the author presented the information. There was enough humor to make the subject matter seem light-hearted and breezy, which made you feel a great deal duller and far thicker-skulled for not understanding it than if it had been bone dry and dead serious. I think it is good to be slapped upside the head by the limits of your knowledge and if it can be done with a rubber chicken...even better.
I am studying Euclid right now because I don't know. It isn't really my thing. Math does not interest me. In fact, I don't think it is going to far to say that it repels me. But, I have undertaken the project of re-educating myself in the classical tradition and it occurred to me recently that my plan did not include much science or any math. And here I am puttering about ancient Hellas just ignoring that some of the most important Greek contributions to Western Thought have been in the areas of science and math.
Additionally, I'm hoping that my study of Euclidean geometry will do two things: force my brain to work in ways that challenge it in order to keep those neurons firing and carving out new pathways well into my dotage. And allay, once and forever, the idea that I am not good at math. Because I have suspected for a long time now that I actually could understand math concepts with competence, if not mastery, and that years of being told otherwise gave me a ready excuse for giving up on it.
That said, merely sitting down with Euclid's "Elements" was daunting. I did it anyway but felt I wanted some context to go with the proofs. Along comes Berlinski and I don't know that he wrote this book for the purpose to which I applied it but I don't think he'd mind. I did not understand everything he said even after 4 readings (Poincare's hyperbolic disk? Yeah, that wasn't happening. I asked my husband, "Is it shaped like an M&M?" I was hungry, maybe) but I tried and I think I got the basic sentiment and it did help to have math concepts run through the filter of literature.
I can recommend the book but to whom or for what purpose I can't say. Again, ever so sorry about that. I can tell you that if you're looking for a juicy bio of Euclid the man, this isn't the book you're looking for. That book does not exist.
October 25, 2016
That David Berlinski's books exists is something of a mystery in need of explanation. I have by no means read all of his work, but the ones that I have read, should not rightfully exist in our modern, math-averse reading populace. I have read four books of his prior to this one, and they were on the topics of:
1) the integers
2) the algorithm
3) a short history of mathematics
4) the calculus
These books are, you understand, not textbooks. They are not the sort of thing you read because you must, or because someone will pay you a greater salary if you do. These are books read for their entertainment value only. I have no idea what to make of this, but somehow even though I am not by talent or inclination a mathematician, I have greatly enjoyed all of these, so when I found that he had written a book on Euclid's "The Elements", the ancient Greek work which both epitomizes and very nearly defines geometry, I did not hesitate. This is not because I am the sort of person who does geometry for fun (would that I were), but rather because I know that Berlinski is able to make this topic enjoyable, if he so chooses.
Some authors use a narrative voice which is self-effacing, fading into the background, with the content and the topic ever in the foreground. One almost forgets that there is a narrator. Not so, David Berlinski. He has a distinctive character which is ever present, in an analogous way to how Mary Roach writes about science. Of course, one should not take the comparison too far; Roach is smiling and laughing as she tells you about the science of digestion, sex, or death. Berlinski rarely smiles, and very rarely laughs, or so one imagines, but his dry wit and arched eyebrows are present in every page. This could drive you nuts, if you prefer your narrator to fade into the woodwork. For me, this is a rare treat.
Unlike Berlinski, Euclid is for us a name on the spine of a book, and virtually nothing else. For an author whose work has never been out of print in two millennia (far longer than the printing press has existed, which is an even greater accomplishment), we know nearly nothing about Euclid, the person. Whether he arched his eyebrows sardonically with the faintest of smirks, or guffawed like a drunken donkey, we have no clue. But we do know somewhat more about what "The Elements" has meant to the history of mathematics since then, and Berlinski takes us through Euclid's work and the history of how it has been used and interpreted. It is nothing less than the history of what it means to prove something; what it means to not only use a thesis as a working hypothesis, but to regard it as beyond question, without any reference to religion or mysticism. Can anything be proved? What does it matter? Who does it matter to, and why? The answers to these kinds of questions change from century to century, and they tell us more about the culture of those centuries than they do about Euclid and his work.
What will people think of Berlinski's work, centuries hence, if they still know of it? Will his dry humor and the clarity of his explanations still impress readers from the perspective of later generations? Will they get the (almost wholly mistaken) impression that the typical reader of the late 20th and early 21st century was given to thoughts on the nature of the fundamentals of math? I cannot say. But I can say that most centuries do not have a writer such as Berlinski to draw our attention to Euclid, and we are fortunate to have him in our time.
1) the integers
2) the algorithm
3) a short history of mathematics
4) the calculus
These books are, you understand, not textbooks. They are not the sort of thing you read because you must, or because someone will pay you a greater salary if you do. These are books read for their entertainment value only. I have no idea what to make of this, but somehow even though I am not by talent or inclination a mathematician, I have greatly enjoyed all of these, so when I found that he had written a book on Euclid's "The Elements", the ancient Greek work which both epitomizes and very nearly defines geometry, I did not hesitate. This is not because I am the sort of person who does geometry for fun (would that I were), but rather because I know that Berlinski is able to make this topic enjoyable, if he so chooses.
Some authors use a narrative voice which is self-effacing, fading into the background, with the content and the topic ever in the foreground. One almost forgets that there is a narrator. Not so, David Berlinski. He has a distinctive character which is ever present, in an analogous way to how Mary Roach writes about science. Of course, one should not take the comparison too far; Roach is smiling and laughing as she tells you about the science of digestion, sex, or death. Berlinski rarely smiles, and very rarely laughs, or so one imagines, but his dry wit and arched eyebrows are present in every page. This could drive you nuts, if you prefer your narrator to fade into the woodwork. For me, this is a rare treat.
Unlike Berlinski, Euclid is for us a name on the spine of a book, and virtually nothing else. For an author whose work has never been out of print in two millennia (far longer than the printing press has existed, which is an even greater accomplishment), we know nearly nothing about Euclid, the person. Whether he arched his eyebrows sardonically with the faintest of smirks, or guffawed like a drunken donkey, we have no clue. But we do know somewhat more about what "The Elements" has meant to the history of mathematics since then, and Berlinski takes us through Euclid's work and the history of how it has been used and interpreted. It is nothing less than the history of what it means to prove something; what it means to not only use a thesis as a working hypothesis, but to regard it as beyond question, without any reference to religion or mysticism. Can anything be proved? What does it matter? Who does it matter to, and why? The answers to these kinds of questions change from century to century, and they tell us more about the culture of those centuries than they do about Euclid and his work.
What will people think of Berlinski's work, centuries hence, if they still know of it? Will his dry humor and the clarity of his explanations still impress readers from the perspective of later generations? Will they get the (almost wholly mistaken) impression that the typical reader of the late 20th and early 21st century was given to thoughts on the nature of the fundamentals of math? I cannot say. But I can say that most centuries do not have a writer such as Berlinski to draw our attention to Euclid, and we are fortunate to have him in our time.
November 14, 2025
i’m confused but i think maybe i’m just stupid
March 25, 2019
Os elementos de Euclides: Uma história da geometria e o poder das idéias é uma análise do matemático David Berlinski da obra do geômetra Euclides, o fundador de um sistema de axiomas que perdura e é ensinado em escolas até hoje. "O homem não é nada; sua obra, tudo", como se diz no livro, o que é verdade; Euclides viveu na Antiguidade, mas continua imortal através de sua obra.
O livro é uma mistura de explicações de geometria, História da Matemática e análise do que o autor chama de "estilo de vida" de Euclides. Isso é bom e ruim ao mesmo tempo.
Vi o livro como uma alternativa à obra original, que conta com 13 livros com demonstrações extensas. Sou curioso pelo tema e não atuo em matemática; quis ler o livro dado que tenho a filosofia de vida de ler os clássicos pelo impacto que tiveram.
O livro começa bem e o autor dá o contexto da época em que Euclides viveu. Apresenta conceitos importantes para o entendimento de sua obra.
Parte para demonstrações e discussões dos axiomas e teoremas de Euclides. Começa bem, mas começa a eventualmente misturar com uma análise de quem era Euclides com detalhes de como matemáticos posteriores viam sua obra. Começou a ficar misturado e um pouco confuso, o que me desagradou de certa forma. O livro acaba não sendo nem tanto de geometria, nem tanto de História da Matemática, nem tanto de análise da vida de Euclides; é uma grande mistura sem um foco claro.
Em direção ao fim, o livro fica mais difícil de entender (para um leigo) e de certa forma maçante. Faltam diagramas que demonstrem o que o autor está falando (Euclides ficaria bravo!), principalmente na parte de geometria hiperbólica, que é de mais difícil digestão do que a euclidiana.
O que me marcou mais foram os questionamentos do autor quanto à geometria euclidiana (que aprendemos com tanta certeza nas escolas que parece uma verdade absoluta e inquestionável) e suas explicações do processo de investigação e demonstração matemáticos.
Vale a pena ler o livro, porém, se você está esperando um foco grande na geometria de fato, se desapontará. Por outro lado, também é divertido pela forma como o autor apresenta suas idéias e fala sobre a relação de outros matemáticos com a obra de Euclides, O Grande.
O livro é uma mistura de explicações de geometria, História da Matemática e análise do que o autor chama de "estilo de vida" de Euclides. Isso é bom e ruim ao mesmo tempo.
Vi o livro como uma alternativa à obra original, que conta com 13 livros com demonstrações extensas. Sou curioso pelo tema e não atuo em matemática; quis ler o livro dado que tenho a filosofia de vida de ler os clássicos pelo impacto que tiveram.
O livro começa bem e o autor dá o contexto da época em que Euclides viveu. Apresenta conceitos importantes para o entendimento de sua obra.
Parte para demonstrações e discussões dos axiomas e teoremas de Euclides. Começa bem, mas começa a eventualmente misturar com uma análise de quem era Euclides com detalhes de como matemáticos posteriores viam sua obra. Começou a ficar misturado e um pouco confuso, o que me desagradou de certa forma. O livro acaba não sendo nem tanto de geometria, nem tanto de História da Matemática, nem tanto de análise da vida de Euclides; é uma grande mistura sem um foco claro.
Em direção ao fim, o livro fica mais difícil de entender (para um leigo) e de certa forma maçante. Faltam diagramas que demonstrem o que o autor está falando (Euclides ficaria bravo!), principalmente na parte de geometria hiperbólica, que é de mais difícil digestão do que a euclidiana.
O que me marcou mais foram os questionamentos do autor quanto à geometria euclidiana (que aprendemos com tanta certeza nas escolas que parece uma verdade absoluta e inquestionável) e suas explicações do processo de investigação e demonstração matemáticos.
Vale a pena ler o livro, porém, se você está esperando um foco grande na geometria de fato, se desapontará. Por outro lado, também é divertido pela forma como o autor apresenta suas idéias e fala sobre a relação de outros matemáticos com a obra de Euclides, O Grande.
July 4, 2024
All the way around the historical solar system
January 5, 2018
I found this read to be a bit odd. We basically know nothing about Euclid, himself, yet here we have nearly 200 pages to a book claiming to be all about him. What this book should have been called was, “How The Elements Inspired and Frustrated Other Mathematicians” (or something to that effect). While Berlinski does his best to try to see who Euclid was as a person, through his writing, Euclid was not any sort of flamboyant writer and did not fill The Elements ramblings or frills. In fact, according to this piece, a lot of mathematicians have complained how little is actually written by Euclid and the frustrations they have felt towards his work due to how it comes across as vague and incomplete by modern standards. Berlinski went back and forth between, essentially, cursing Euclid for his lack of words and praising him for what he had done. It comes across as a very love-hate relationship as he desperately reads between the lines of a text book to the tell the tale of man we ultimately will never know.
December 25, 2013
Even though I feel that I should know more about the Euclid and his Elements that are discussed in Euclid and His Elements, I had a difficult time with this book. In general, it was (or at least I think it was) about how great Euclid was, and how his Elements underlies much (or completely
all
?) of modern mathematics. But I found it too difficult to follow what Berlinski was writing about.
On the other hand, I recently saw a bit of the 2012 Lincoln movie for the first time. In it, the title character discussed Euclid's first Common Notion. I must have missed that it school, but I was able to think, "I just read about that in Berlinski's book!"
But, in the end, I must confess that I "did not like" this book.
On the other hand, I recently saw a bit of the 2012 Lincoln movie for the first time. In it, the title character discussed Euclid's first Common Notion. I must have missed that it school, but I was able to think, "I just read about that in Berlinski's book!"
But, in the end, I must confess that I "did not like" this book.
March 8, 2013
I was hoping to learn more about Euclid or get an introduction to Elements, but this book was much more about how clever the author thought he could be. The writing is choppy and disconnected, and the sections rarely go on for more than two (small) pages, making it difficult to develop a complex or interesting point about the subject. I read half, but could not force myself through the rest. If the text was originally a series of oral presentations, it should have been rewritten or edited before being printed as a book.
July 22, 2021
I was surprised by this title, I went in thinking that it was going to be a shallow overview of Euclid and his geometry, but what Berlinski’s book ends up being is a very good introductory/preface to the Elements. In fact, this book would be an excellent preparatory read for anyone attempting to go through the Elements for self-study or within the guidance of a formal course.
The first few dozen pages of the book is an obligatory quick history of the Elements and its impact on Western society. There’s a bit of pomposity here as Berlinski makes note to inform the reader that no other civilization generated a formal axiomatic approach towards geometry like the Greeks, and that the axiom system is the superior one. This later statement is not fact, it’s opinion. It’s also unclear whether ‘formalism’ did not exist in other regions / civilizations on Earth. Of the literate societies, there’s evidence to suggest that the logic of the Greeks at least may have been replicated independently in other regions, including in China and India at least (possibly more). The Arabs traded too frequently with the Mediterranian to be considered truly independent of the Western system of thought, but even here, Arab innovations in mathematics beyond “rote” equational constructions are noteworthy.
Still, outside of a purely aesthetic argument, it’s not even clear that axiom systems are more efficacious when it comes to discovering more useful mathematical facts/information vis-a-vis ‘mere’ human observation/curve-fitting (which let’s be clear, almost all of the quantitative elements of science essentially is in the natural sciences). In fact, if we take note of recent history of physics, we do not see axioms taking lead in the development of a subject, like quantum mechanics, but being part of a “follow-on’ process, whereby a naive-theory was constructed mostly via intuitive construction by the physicists to explain observed facts. Those were refined, and only after many years of refinement, was an axiom system “grafted” on top of it (i.e. the essential facts observed in QM were put in their proper context/box within the formalism to ensure everything was consistent). I wouldn’t be surprised if Newton came about his ‘system of the world’ in Principia in a similar manner.
Getting back to the book-proper, Berlinski does a great job tying the Elements to the notion of formalism and extending this conversation to include a more general discussion on basic logic/analytic philosophy as well as the nature of broader mathematics. This is natural as in many curriculum Geometry is often used as the “model” subject to tackle a first course on formal proof methods for mathematics students, exactly because the subject lends itself to these kinds of connections.
Interestingly unlike other “pop sci” nonfiction books Berlinski actually goes into a bit of detail on the nature of the 20 odd axioms that make up Euclid's geometry (as well as make commentary on Hilbert et. al. program to reduce this to less than half a dozen in the early 20th century). He goes through some of these within the book, and although having read about half of the Elements (Heath’s translation) many years ago, listening to the constructions does not lend well to indstruction. Definitely need to follow with the PDF, or Kindle, or better yet, a copy of the Elements. Speaking of which, for those who will use this book as a preface to a personal reading of the Elements, Berlinski’s scope of coverage is effectively the first Dover book partition of the Elements, much of the tricks you see in the 2nd (which covers more intricate constructions that reveal relationships between areas and sides of various segments of a shape, often a triangle and is critically important to begin to understand conics and the kind of constructions performed in the Principia) or the 3rd Dover book (which is mostly on solids and their relationships / constructions /properties) are not covered here. It will give you a great start however, so don’t let that dissuade you from purchasing/reading it. It really is worth it.
Berlinski ends the book with some discussion on Geometry’s later developments in the mid-to-late 1800s, which saw the formation of various non-Euclidean geometries (the kind of which are often mention in Lovecraft novels say), and here Berlniski’s comments on the nature of the parallel postulate and how simple changes at the definitional level can still yield a self-consistent geometry was insightful. Overall, I am pleased with this book, it’s definitely a great companion to reading the Elements, or maybe even a standard UG geometry course. Alone, it’s probably also good, though I think the reader only gets the full effect by delving further at least the entire “Book 1” of the Dover edition of Euclid (the first 3 books or so). Recommended.
The first few dozen pages of the book is an obligatory quick history of the Elements and its impact on Western society. There’s a bit of pomposity here as Berlinski makes note to inform the reader that no other civilization generated a formal axiomatic approach towards geometry like the Greeks, and that the axiom system is the superior one. This later statement is not fact, it’s opinion. It’s also unclear whether ‘formalism’ did not exist in other regions / civilizations on Earth. Of the literate societies, there’s evidence to suggest that the logic of the Greeks at least may have been replicated independently in other regions, including in China and India at least (possibly more). The Arabs traded too frequently with the Mediterranian to be considered truly independent of the Western system of thought, but even here, Arab innovations in mathematics beyond “rote” equational constructions are noteworthy.
Still, outside of a purely aesthetic argument, it’s not even clear that axiom systems are more efficacious when it comes to discovering more useful mathematical facts/information vis-a-vis ‘mere’ human observation/curve-fitting (which let’s be clear, almost all of the quantitative elements of science essentially is in the natural sciences). In fact, if we take note of recent history of physics, we do not see axioms taking lead in the development of a subject, like quantum mechanics, but being part of a “follow-on’ process, whereby a naive-theory was constructed mostly via intuitive construction by the physicists to explain observed facts. Those were refined, and only after many years of refinement, was an axiom system “grafted” on top of it (i.e. the essential facts observed in QM were put in their proper context/box within the formalism to ensure everything was consistent). I wouldn’t be surprised if Newton came about his ‘system of the world’ in Principia in a similar manner.
Getting back to the book-proper, Berlinski does a great job tying the Elements to the notion of formalism and extending this conversation to include a more general discussion on basic logic/analytic philosophy as well as the nature of broader mathematics. This is natural as in many curriculum Geometry is often used as the “model” subject to tackle a first course on formal proof methods for mathematics students, exactly because the subject lends itself to these kinds of connections.
Interestingly unlike other “pop sci” nonfiction books Berlinski actually goes into a bit of detail on the nature of the 20 odd axioms that make up Euclid's geometry (as well as make commentary on Hilbert et. al. program to reduce this to less than half a dozen in the early 20th century). He goes through some of these within the book, and although having read about half of the Elements (Heath’s translation) many years ago, listening to the constructions does not lend well to indstruction. Definitely need to follow with the PDF, or Kindle, or better yet, a copy of the Elements. Speaking of which, for those who will use this book as a preface to a personal reading of the Elements, Berlinski’s scope of coverage is effectively the first Dover book partition of the Elements, much of the tricks you see in the 2nd (which covers more intricate constructions that reveal relationships between areas and sides of various segments of a shape, often a triangle and is critically important to begin to understand conics and the kind of constructions performed in the Principia) or the 3rd Dover book (which is mostly on solids and their relationships / constructions /properties) are not covered here. It will give you a great start however, so don’t let that dissuade you from purchasing/reading it. It really is worth it.
Berlinski ends the book with some discussion on Geometry’s later developments in the mid-to-late 1800s, which saw the formation of various non-Euclidean geometries (the kind of which are often mention in Lovecraft novels say), and here Berlniski’s comments on the nature of the parallel postulate and how simple changes at the definitional level can still yield a self-consistent geometry was insightful. Overall, I am pleased with this book, it’s definitely a great companion to reading the Elements, or maybe even a standard UG geometry course. Alone, it’s probably also good, though I think the reader only gets the full effect by delving further at least the entire “Book 1” of the Dover edition of Euclid (the first 3 books or so). Recommended.
June 22, 2021
Though I find math interesting, I wasn’t too enthusiastic throughout the book. Not sure why. The author, David Berlinski did a fantastic job of explaining everything, I just wasn’t into it. I should mention that reading this book was printed by the idea of preparing my mind for two books that will follow. 1.) “The Mathematical Principles of Natural Philosophy”, by Sir Isaac Newton, and 2.) “The Devil’s Delusion”, by David Berlinski. This book gave me insight into what I can expect when reading a book that involves heavy mathematics in one, and also the writing style of the author for the other. Euclid, while know as the father of geometry, wasn’t very interesting. I was much more fascinated in learning that Euclid wasn’t the only Euclid and is now know as Euclid of Alexandria to be juxtaposed with Euclid of Megara, who was a philosopher instead of a mathematician, and also born about 135 years earlier in 435 BC.
March 27, 2017
In some ways, I loved this book. The author is so enthusiastic, and approaches each topic thoughtfully and with humor, which endears me to him. This said, the book, despite how short it was, took me a while to get through, since I found my mind wandering quite a bit. I think that sections that rely on words to describe geometric relationships would have done better with illustrations (there are illustrations and figures in the book, but not enough). Maybe the expectation is that the only kind of person who would be reading a recreational book about mathematics would instantly recognize the principles and proofs being described (and in cases where I did know the idea being referred to, I could follow easily.) However, I am glad that I read it.
January 3, 2024
The author's conclusion is made quite obvious by the rest of the book, so I would say that the information was the main thing I got out of the book, rather than the conclusion.
The information was, for the most part, interesting and not too complicated to understand. It was the author's attempts at being poetic and artful in his writing that made the book very difficult and unenjoyable to get through. The constant use of metaphor made the concepts less, rather than more, clear. I heavily considered giving up on the book at many points, and I am still unsure whether to regret not doing so - I wish I could have gotten the same information from a source that was easier for me to read.
The information was, for the most part, interesting and not too complicated to understand. It was the author's attempts at being poetic and artful in his writing that made the book very difficult and unenjoyable to get through. The constant use of metaphor made the concepts less, rather than more, clear. I heavily considered giving up on the book at many points, and I am still unsure whether to regret not doing so - I wish I could have gotten the same information from a source that was easier for me to read.
March 19, 2020
Not so much about Euclid (because really what do we know?) as it's about the ideas he presented and how notable mathematicians interacted with them. I'm sort of intrigued that the author was so heavy on the Hilbert and just didn't mention his female colleagues at all. I know that searching the margins for women in math isnt easy, but it's kind of necessary. Also, he leans too heavily on the old algebra vs geometry thing and it's just so lame.
June 13, 2018
I appreciated the effort to make Euclid more accessible. I love the aesthetic value of geometry as well as its importance in architectural soundness. Also, the author talks about the idea of proof, and made me appreciate the slow build up of theorems. This is a good read for anybody that wants to appreciate geometry more.
February 26, 2020
A vetting process for the common notions and postulates in Euclid's Elements. Why are they there? What might have been missing. A bridge of 2000 years of time from Euclid to Hilbert. It is a good companion if you are studying the Elements.
January 8, 2018
Berlinski's prose can be florid, but his writing about math never disappoints.
April 3, 2018
Nice short background for anyone who’s attempting to read Euclid for the first time.
December 23, 2018
An erudite author takes the reader through various and sundry related to Euclid's immortal work.
July 23, 2020
Fascinating mix of mathematical postulate, philosophy, history, and biography. I'm looking forward to reading more from this author.
June 15, 2021
Must read for any aspirant mathematician...
October 13, 2021
I did not understand this all because I did not offer the book the attention Euclidean studies demand, but I was still delighted by this quirky act of devotion.
April 5, 2022
Gets a bit thick at the end, might be for more advanced students than myself, perhaps. Overall a wonderful history of the man and his science.
June 22, 2022
A stirring, imaginative overview of the history of footnotes to Euclid.
December 28, 2023
Reads like a philosophical and historical dream.
June 30, 2024
A digestible if uneven account of how the analysis and influence Euclids work has evolved over time.
July 8, 2024
good introduction to the field. too many opinions. too much french and latin
May 31, 2025
Would rather read a textbook.
June 25, 2025
While reading this book made me feel stupid, it did succeed in giving me the urge to learn geometry
October 20, 2025
It's been years since I read this but I especially loved the Bridge of Asses.
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