El Ultimo Teorema de Fermat

by Amir D. Aczel

Amir D. Aczel descubre la belleza de las matemáticas para todo tipo de lectores. Desde los anónimos babilonios y agrimensores egipcios, pasando por Pitágoras y sus seguidores, Arquímedes y Diofanto, hasta llegar al mundo árabe, el autor traza el camino que llevó a la solución del último teorema de Fermat, una ruta llena de intrigas y falsas atribuciones.

Genres: Science, Mathematics, Nonfiction, History, Biography, Popular Science, Physics

175 pages, Paperback

First published January 1, 1996

About the author

Amir Aczel was an Israeli-born American author of popular science and mathematics books. He was a lecturer in mathematics and history of mathematics.

He studied at the University of California, Berkeley. Getting graduating with a BA in mathematics in 1975, received a Master of Science in 1976 and several years later accomplished his Ph.D. in Statistics from the University of Oregon. He died in Nîmes, France in 2015.

Reviews

[Mare · Rating 3 out of 5]

It is an interesting and simple explanation of the complexity of proving one "simple" statement.
But, book is for general public, and as an mathematican it was hard not to cringe on some bad statements.

[Nathan Albright · Rating 5 out of 5]

One of the aspects of this book that is particularly worthwhile is the way it comments on the collaborative effort that it took to solve Fermat's last theorem. On that matter, it is perhaps better to consider it as Fermat's last conjecture, because whatever proof that Fermat had in mind was not the way that the problem was eventually solved, with a whole host of complex theoretical mathematics that was not even remotely considered when Fermat lived. Mathematics is often viewed as an individual endeavor, but a great deal of value comes in collaboration, and those who do contribute to that larger picture are not always given the credit that they deserve. One gets the sense that this author has a lot to say in the revisionist side in that he points out places where mathematicians have not always been candid about where their insights came from and about who gets to share the credit for solving a problem as massive as Fermat's Last Theorem. And although the problem itself might not seem to be a practical one, the mathematics that was developed along the way to solving them ended up opening up some interesting space, to be sure, in theoretical mathematics, even if it is far beyond my own modest understanding of the various fields of topology involved.

This book is a relatively short one at less than 150 pages. The author begins close to the ending with the first announcement of a solution to the book's titular problem, along with the critique and discovery of flaws that led to another lonely race on the part of the solving mathematician to solve the problem before a rival beat him to the solution. Then the author looks back in history at some of the areas of mathematics that went into influencing Fermat to make his conjecture and for people over the course of centuries to solve different problems that ended up playing part of the role in solving the theorem. Of particular interest to the author is the effort of two Japanese mathematicians to connect two disparate fields of mathematics together, which ended up providing some key insight into solving Fermat's last theorem, something that had previously been viewed as impossible. For those who like mathematics puzzles, this book has a strong sense of mystery and excitement.

I am likely speaking for a vast majority of this book's potential readers when I say that the math involved in this book is certainly beyond my understanding. If the problem itself is familiar enough and the equation of the conjecture easy enough to understand, how one goes about proving it is something far beyond my own comprehension. I suspect this will not be an uncommon experience, and yet this book is not aimed merely at theoretical mathematicians but is aimed at a wide audience of people who are interested in mathematics but by no means masters of its far reaches. This is a book and subject matter that is possible to appreciate without knowing all of the levels of mathematical expertise that were required to solve the riddle of Fermat's last theorem. As for this reader, I am still interested in seeing how Fermat thought he had solved the problem with a solution that was too long to fit in the margins of a book. Did he think he had solved his theorem by contradiction or something relatively straightforward? It is impossible to know at this point, but impossible not to wish to know what he was thinking in light of what it took to prove his conjecture.

[Veronica · Rating 5 out of 5]

This is a relatively short book that you can finish in no time. Don’t be intimidated by its title, it reads more like a history book than a math book. While it does contain formulas and some mathematical language, they are presented in an easily understandable way. Even without an in-depth background in mathematics, you can grasp what the author is trying to convey.

Reading this, you’ll see the beauty of mathematics, how discoveries are interconnected and how solving a problem often requires tremendous collective effort, sacrifices, and decades of work. You’ll meet dedicated, passionate people who devote their lives to mathematics, and I find such dedication and love for math truly admirable and beautiful.

***Thanks to the SSM Library for letting me borrow this book.

***
Andrew Wiles himself termed his proof of Fermat's Last Theorem a "twentieth-century proof," based on the fact that it rested on many years of prior work. The solution of the theorem required a ton of work and collaborations from many mathematicians: Ernst Kummer, Barry Mazur, Gerhard Frey, Jean-Pierre Serre, Ken Ribet, Yutaka Taniyama, and Goro Shimura, whose combined efforts drew together diverse areas of mathematics.

Fermat lived many hundreds of years before and therefore could not conceive the contemporary method or techniques required in Wiles' proof, especially those on the Shimura-Taniyama conjecture. Some would argue that he probably did have simpler proof with him, yet there was no evidence at all. His unclear scribbling in the margins and not mentioning the result for nearly 28 years proves that he would have to be wrong in his theorem or just plainly turned towards other things more.

While the theorem’s solution has been verified in meticulous detail, the possibility of an alternative, simpler proof remains open. Fermat’s true intentions and any potential insights he had will likely remain a mystery, as the original text containing his note has never been found.

[strategygamer22 · Rating 2 out of 5]

This is a fascinating book that is quite an entertaining read. The writing style is quick and clear, and simple illustrations are given of complex ideas. I'm not sure that a complete novice would be able to follow along with everything, but overall the author does a good job of "dumbing down" the mathematics without seeming insulting or condescending.

With all this praise, you might wonder why the 2-star rating. Everything one could hope for in a book about mathematics is here: clarity, brevity, simplicity. Everything except accuracy. The author makes multiple statements early in the book that astounded me. I have a hard time believing that this was written by a mathematician. For example, in a section talking about Fermat directly, the author states, "The first few prime numbers are 1, 2, 3, 5, 7, 11, 13, 17..." The number 1 is not prime. Even the ancient Greeks did not consider it prime. In general, the author seems to take a lot of liberty with historical facts, oftentimes presenting folklore as fact and imagining how events might have transpired. Rarely does he say that he is doing this; I was only aware of it because of my own knowledge of ancient mathematics.

As a result of the shaky-at-best ancient history, I found myself unable to fully believe the rest of the book. There are very few sources given. The author states only that he, "drew much of the historical background from a variety of sources" and that "All my important sources are referenced in the endnotes," of which there are only a few. This book has increased my interest in elliptic and modular curves, but I won't be citing it as a source. It is too untrustworthy and will require much cross-referencing to verify what it says.

[RoWoSthlm · Rating 4 out of 5]

This book is a very brief history of a significant part of the mathematics that is presented in the perspective of one of the most difficult mathematical problems - Fermat's Last Theorem. It has been formulated by Pierre de Fermat and challenged many mathematicians for centuries, until it was quite recently solved by Andrew Wiles. Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

As one can imagine, the arrival to the solution was hard and with numerous byways. The search for the solution of the problem prompted the development of some other important number theories. That is presented by the author in a quite clear way and is a good read for everybody with interest in math. I know mathematicians have issues with some of the thesis in this book, but as an overall history of the famous problem it is a valid and interesting text.

[Brenda · Rating 4 out of 5]

Fascinating look into the solution of Fermat's last theorem. First talked about in around 1637 (Fermat hinted at his solution in the margins of a text book by ancient Greek mathematician Diophantus), its roots go back even farther, yet the equation wasn't solved until 1994. There's a section that deals with the intrigue of mathematical proprietorship which feels odd in the context of the rest of the book, but the explanations of mathematical theories & their histories is made almost understandable 😉

[Angelino Desmet · Rating 5 out of 5]

This book fills a significant void in scientific education: making mathematics fascinating and tangible through its historical origins. Despite being exquisitely succinct, Amir takes his readers on a multimillennial voyage as he in turn sets out to prove why Fermat's Last Theorem is proven to be so beautiful, both mathematically and beyond.

[Todd Martin · Rating 4 out of 5]

Whoever believes that math can't be fun has never read Fermat's Last Theoremby Amir D. Aczel. The book provides a lively discussion of, not only the theorem and the story of its eventual solution, but of a number of fundamental concepts in mathematics as well.

Unlike your old college calculus book, this one is a page turner.

[Noloter · Rating 2 out of 5]

Che delusione. Non è altro che un'accozzaglia di pettegolezzi, aneddoti e curiosità su matematici i cui nomi sono buttati lì un po' a caso, con poco rigore e organicità, e senza il minimo approfondimento.
Davvero molto, molto superficiale. Un vero peccato.

[Afonso Bernardes · Rating 4 out of 5]

Depois de ler este livro, apetece-me demonstrar um teorema.

[Giulio Ciacchini · Rating 4 out of 5]

L’autore esplora uno dei problemi matematici più famosi e affascinanti della storia: l'Ultimo Teorema di Fermat. Enunciato dal matematico francese Pierre de Fermat nel XVII secolo, afferma che non esistono soluzioni intere per l'equazione "x^n + y^n = z^n" quando n>2.
Il libro segue l'intricata ricerca per dimostrare il teorema, dall'epoca di Fermat fino ai giorni nostri, evidenziando le diverse strategie e le personalità coinvolte nel tentativo di risolvere questo enigma.
Azcel riesce a fornire al lettore le coordinate matematiche per capire l'essenza del problema, mantenendo un linguaggio poco tecnico, comprensibile ai più.

Uno dei momenti cruciali nella storia del teorema è l'enorme progresso compiuto nel 1994 da Andrew Wiles, un matematico britannico, che è riuscito a dimostrare il teorema utilizzando complesse tecniche matematiche.
Questi progressi non sono stati avulsi da critiche e rivendicazioni tra vari studiosi, aumentando il clamore dietro L'enigma di Fermat.

Il testo segue un approccio storico andando indietro nel tempo per spiegare le fondamenta matematiche della soluzione dell'enigma.
Le pagine finali ci fanno sorridere: sembra che l'affermazione di Fermat, ovvero che era riuscito a dimostrare il teorema, ma che per ragioni di spazio non abbia potuto scriverlo, sia quantomeno bizzarra.
Infatti, molto probabilmente Fermat non conosceva la matematica necessaria a dimostrare il teorema, ma soprattutto l'enigma non poteva essere dimostrato in poche righe: Wiles ha scritto 200 pagine.

[Francesco · Rating 2 out of 5]

Troppa matematica. Cioè, lo so che in un libro che parla del Teorema di Fermat non ci si può aspettare nulla di meno; ma avrei preferito un focus maggiore sulle persone che l'hanno risolto e un po' meno sulla storia della matematica

[Maja]

to jest takie niesamowite dla mnie ile ludzi, lat i obszarów matematyki było potrzebne żeby dowieść tego twierdzenia 😭❤️

[Rodrigo Leonardo · Rating 5 out of 5]

Delighted to read about the intertwined network of fraternal bonds that lead to the most famous proof of the 20th century. Not it's time to actually see the proof after understanding its road-map.

[Alan Davies · Rating 3 out of 5]

Pierre de Fermat (1607-1665) was a French mathematician. Trained as a lawyer / he bought a position as councilor at court which supported him for the rest of his life. Descartes was somewhat his senior and Pascal his junior / but they were all alive at the same time. He corresponded with Pascal. He worked on problems in calculus that were later developed by Newton and Liebniz. The 17th Century was to see the end of the Renaissance / and was on the cusp of modernity.

A book about physics that’s five or at most ten years old is not up-to-date. Advances in both astrophysics and quantum physics occur quickly. New advances in mathematics are being made every day / but not with the rapidity of those in physics.

Fermat’s last theorem says that there aren’t three positive integers (a, b, c) that satisfy the following equation where positive integer n is greater than 2 –

a to the n +b to the n = c to the n

After Fermat’s death his son found the equation written in the margin of one of his father’s books / with a note saying there wasn’t room in the margin to give the solution. The implication was that Fermat had a solution for the equation. Did he? No way to know. He did the same thing with other of his theorems / implying that he had solved them / but not anywhere giving the solution.

Aczel’s writing is cogent and succinct / and written at a time when the theorem was being attacked by Andrew Wiles / an obsessive American mathematician. He had the help of the many who had worked on the theorem over the centuries and failed / and many others whose advances in mathematics contributed to his solution.

The OED defines theorem – A general proposition not self-evident but proved by a chain of reasoning; a truth established by means of accepted truths. The American Heritage Dictionary – An idea that has been demonstrated as true or is assumed to be so demonstrable. And more specifically when used in mathematics – A proposition that has been or is to be proved on the basis of explicit assumptions. A theorem has not necessarily been proven – it is assumed to be so demonstrable and is to be proved on the basis of explicit assumptions. The assumptions were not made explicit by Fermat / at least we have record of them. The only presuppositions that were explicit are the rules of algebra.

I won’t try to summarize the mathematics / because I can’t do it / and even if I could I would not want to confuse the clarity of Aczel’s terrific book. I didn’t study mathematics past algebra / and only a little of that – yet I was able to follow Aczel’s exposition of the solving of Fermat’s Last Theorem. It is worth the work of reading over more difficult sections twice. He begins by describing the occasion in England in 1993 when Professor Wiles / forty years old / presented his solution to Fermat’s Last Theorem to a stunned audience. Aczel then takes us back in time to provide historical underpinnings of number theory / followed by a return to the then present to tell us that flaws were found in Wiles’ proof. He writes about Pythagoras / number worshipers / the Babylonians / Eudoxus / Archimedes / Gauss / imaginary numbers / non-Euclidian geometry / and other mathematicians who did work that would later make possible Wiles’ eventual proof of the theorem. Speaking for example of Poincaré’s Analysis Situs / he notes that Topology—the study of shapes and surfaces and continuous functions— was important in understanding Fermat’s problem in the late twentieth century. He discusses intrigues between competing theories / and competing theorists. He describes such systems of thought as the use of graphs with real numbers on one axis and imaginary ones on the other / which produced very strange mathematical objects. By the 1980s / when another theorem was proved / it could be said that Fermat’s Last Theorem was “almost always” true. But not always true. Neither proven nor disproven. Aczel’s telling of the story is thrillingly engaging / with twists and turns of both thought and emotion. There were fascinating characters involved. It’s an exciting read.

Wiles suffered many setbacks as he continued his work. At times he locked himself in the room at his home in which he was laboring – there were periods of agonizing despair. He enlisted help from a couple other mathematicians / ones he could trust not to make his work public. The news that Ken Ribet had proven the Epsilon Conjecture provided him with new mathematical insights that helped him progress. Mazur’s paper on the Eisenstein Ideal also helped him. Again / instead of publishing his paper in a peer-reviewed journal / he presented it at a conference. His 200-page paper was then sent to noted number theorists for review. His friend Nick Katz / who was one of those who had helped him with aspects of the proof / found a flaw. Wiles could not explain it away. Other mathematicians around the world had found the same problem. Again / failure. Andrew Wiles returned to Princeton in the fall of 1993. He was embarrassed, he was upset, he was angry, frustrated, humiliated. Another year passed. Sitting at his desk in Princeton –

Wiles studied the papers in front of him, concentrating very hard for about twenty minutes. And then he saw exactly why he was unable to make the system work. Finally, he understood what was wrong. “It was the most important moment in my entire working life,” he later described the feeling. “Suddenly, totally unexpectedly, I had this incredible revelation. Nothing I’ll ever do again will…” at that moment tears welled up and Wiles was choking with emotion. This time after correcting his paper / instead of presenting at a conference / he sent it to several peer-reviewed journals. It was published in The Annals of Mathematica … The review process took a few months, but no flaws were found this time. The May 1995, issue of the journal contained Wiles’ original Cambridge paper and the correction by Taylor and Wiles. Fermat’s Last Theorem was finally laid to rest.

From Wiles –

“Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. You go into the first room and it’s dark, completely dark. You stumble around, bumping into the furniture. Gradually, you learn where each piece of furniture is. And finally, after six months or so, you find the light switch and turn it on. Suddenly, it’s all illuminated and you can see exactly where you were. Then you enter the next dark room…”

And from Aczel’s own conclusion –

The profound nature of the theorem is that not only does its history span the length of human civilization, but the final solution of the problem came about by harnessing—and in a sense unifying—the entire breadth of mathematics. It was this unification of seemingly disparate areas of mathematics that finally nailed the theorem.

/ copyright © 2021 Alan Davies

[Jason Furman · Rating 3 out of 5]

This book is much too short to accomplish it's goal--which is to explain Andrew Wiles proof of Fermat's Last Theorem and to ground that explanation in the history of mathematics. Simon Singh's Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem does a much better job but even that falls short. In contrast to Singh's explications that really try to explain the concepts, Amir Aczel presents more of a capsule series of biographies of the many mathematicians over the last several centuries whose work converged in the proof of Fermat's Last Theorem. He tries to explain some concepts with words and figures but the explanations themselves are too elliptical. The individual pieces of the book are interesting and often entertaining enough. And the argument that you need to understand all of mathematics to understand the proof of Fermat's Last Theorem is also driven home with overwhelming force. But I was hoping for something more.

[Erin · Rating 4 out of 5]

Aczel does a thorough job of describing the problems behind Fermat's Last Theorem including the history of mathematical discoveries that lead to the final solution of the proof in 1993. His history is entertaining and completely readable to the layperson, often including simple examples to illustrate principles and the details hardly ever border on tedious. I also enjoyed Aczel's use of brief anecdotes to add depth to the characters featured in what would otherwise be a mathematical name parade.
That said, the final proof of Fermat's Last Theorem involves many different branches of mathematics and some very clever applications of mathematical results spanning the whole history of math, from the ancient Greek Diophantus to the work of Taniyama and Ribet in the 21st century. Grasping even the broad overview of what was accomplished is quite a headful of names and theories. Aczel handles the task quite efficiently, wrapping up the many details into a neat little package of less than 140 pages.
Overall, a great quick read offering a brief and entertaining mathematical history leading up to one of the most dramatic and heart-wrenching stories in mathematics to date.

[Colin Harris · Rating 3 out of 5]

Fermat's Last Theorem runs into the same problem quite a few mathematical narratives I've come across run into - Aczel is tasked with explaining extraordinarily abstract and complex mathematical concepts in ordinary terms.

He does a better-than-average job of making the narrative thread interesting (a better job than Prime Obsession did with the Riemann Hypothesis, for instance) but in 136 pages one shouldn't expect to come away from a book with a greater mathematical understanding of the solution to Fermat's Last Theorem.

I didn't even know it was solved! My mathematical Ghetto Pass has been revoked.

[Ahmed · Rating 5 out of 5]

Simple, elegant, and utterly impossible to prove I think that Fermat's last theorem is one of the most interesting theorems ever created. Amir D. Aczel made this great book to show us how such a simple theorem A^n+B^n=C^n if n is greater than 2.I strongly believe that everyone should read this complex and fascinating book. The author has made the theorm so simple and so hard to solve at the same time. I rate this book a 5/5 stars.

[Linda Isakson · Rating 4 out of 5]

Interesting read. While Fermat's Last Theorem was a mathematical conundrum for hundreds of years, the author presents the quest to proof the theorem in a concise and engaging manner. I particularly enjoyed the behind-the-scenes look at each mathematicians personality and motivations.

[Akrabar · Rating 5 out of 5]

splendid explanation of how the theorem was proved and a beautiful and accessible history of discoveries leading up to it

[Momshad Alvee · Rating 5 out of 5]

First book I ever read out of interest at 7th grade. This probably changed my life...maybe

[Ragdoll Metaphysics · Rating 5 out of 5]

Non technical treatment of the now world famous theorem. Traces the historical development of number theory and really shows how the proof of the theorem is bigger than any one person. I’m glad this is done because Wiles proof has always belonged to the class of problems common in disciplines that make use of logic and symbolic reasoning that are posed and/or have solutions that almost feel like cheating. Something like Zenos Paradoxes or Arrows Impossibility Theorem for eg have a certain form and problems arise or disappear only when they are converted into a totally new form that seems “out of bounds” so to speak. Using infinite series to resolve a paradox seems like moving way past what the original author intended as a valid solution. The authors sketching out the techniques used to resolve the theorem and the background history make this move feel more justified. The author is clear that mathematics is not simply footnotes to Plato or whatever though again I do question the idea that rigor is something invented by the Greeks and every other culture practices math in a way that’s less pure and idealistic and perfectly abstract only to do with its applications to specific problems of their time in land surveying or engineering (this is a common myth in histories of mathematics not just here).

[Christopher Chambers · Rating 2 out of 5]

An easy read - perhaps too easy.

The author takes a classic chronological approach and covers 'all the usual suspects' in the evolution of FLT and its eventual solution. He gives proper emphasis to the crossovers of mathematical areas that informed Wiles's insights. The best parts are perhaps the discussions of the academic spats and the niceties of naming. Aczel is clear in the association of Goro Shimura's name with the (then) conjecture, possibly more of a concern when he was writing in 1996 (this edition 1997 for UK unchanged from earlier US).

There are typos and mis-statements as other reviewers have pointed out. They are annoying but I've sympathy for anyone trying to write lay explanations of science and especially maths. However this book does eventually dissolve - as many popular science tracts do - into a novel-like state where the characters are unknowable constructs rather than fictions: 'Modular forms' appears and the expression gains familiarity but at the end can the reader write down an example?

[Synthetic Vox · Rating 4 out of 5]

I really enjoyed this book. Short and very accessible given that it covers basically the history of number theory from the Babylonian times to the solving of FLT. It uses the people connected to their key mathematical ideas as the thread, which I really appreciated. I knew almost nothing about mathematics or mathematicians and so very much appreciated learning about Gauss’ genius and life, for example.

As the blurb says, “An excellent short history of mathematics, viewed through the lens of one of its great problems—and achievements.” I dropped a star because it didn’t use chapter formats, which I found that a little tedious, and because toward the end the author stopped explaining what certain terms meant.

If you enjoyed this, I would highly recommend reading The Housekeeper and the Professor.

[Heather · Rating 4 out of 5]

This book came to me from my son who has a far deeper understanding of mathematics than I do. Truly I’d give it 4.5 stars because although a bit of the narrative of people through mathematical history became cumbersome, this book has given me a much wider perspective of math that I’ve never considered before. If my sixth graders had an approach similar to the thinkers of this book it would take them much further than our sorry common core math curriculum could.

I loved the deep dedication of the mathematicians throughout history. And although, I’ve never identified myself as a “math” person, I’m lead to the conclusion that the world of math is much, much bigger and I am familiar with very very little of it.

My respect for mathematics has grown significantly after this read.

“Mathematics is the language in which God has written the universe” Galileo

[Linda · Rating 3 out of 5]

Well I did read some of it, seriously when the math gets heavy I exit.
One of Fermat's most stunning achievements was to develop the main ideas of calculus, thirteen years before Sir Isaac Newton. Fermat translated many Greek works into Latin among which was a book called Arithmetica written by the Greek mathematician Diophantus. Next to a problem on breaking down a squared nuber into two squares he wrote
On the other hand, it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. i have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain.
This simple sentence would keep mathematicians busy for centuries!
Fun read.

[Simon Mcleish · Rating 3 out of 5]

A good summary of the work which went into the proof of Fermat's Last Theorem for non-mathematicians. Marred by clear signs of a hurry to get the book out - proof reading errors and small gaffes (there's one place for instance where two adjacent sentences appear to contradict each other which is due to a missing qualification in the second sentence, which only holds for some numbers not globally). Concentrates more on personalities than the mathematics, which makes it less useful for a reader actually trying to get more than a vague understanding of the incredibly difficult ideas which made Wiles' proof possible. But highly readable!

[Gabe · Rating 3 out of 5]

This is a fun little book on the history of Fermat's Last Theorem. Aczel does a good job of telling the story. The mathematics is kept at the level of techno-babble, that is, random sounding jargon that you're not expected to understand. The story is mostly about the strange people who made various mathematical contributions, starting with ancient thinkers and working all the way up to the 1990s when the theorem was proven. There are plenty of stories of mathematicians leading troubled lives and being jerks to each other. It makes me glad not to be a mathematician!

[Roberta · Rating 3 out of 5]

Breve storia della matematica, che sono riuscita a seguire fino a un certo punto. Certo deve essere eccitante, a distanza di secoli, riuscire a risolvere un problema che ha tenuto occupate generazioni di matematici con la sola forza del proprio pensiero.
Inoltre è bellissimo leggere come la soluzione di tale problema si sia creata nel tempo, prendendo da diversi campi e attingendo a una conoscenza diffusa e libera.
Il bello delle scienze, in un libro.