Diophantus of Alexandria sits uncomfortably among the Greek-speaking mathematicians of antiquity. Most of the Greek mathematics that survives is thoroughly geometrical. Often the notion of ratio takes a central role, and the only numbers ever considered are positive integers. Not so in Diophantus’ Arithmetic: the topic is solving problems about numbers that strike the modern reader as distinctly algebraic: “given a square, write it as a sum of two squares” or “find two cubes whose sum is a square.” The solutions are usually fractions: 16 is equal to the sum of 256/25 and 144/25, for example.
Diophantus does not prove theorems; instead, he solves problems. Usually, he uses specific numbers. To solve the problem about expressing a square as the sum of two squares, he begins with “I propose to divide 16 into two squares.” Nevertheless, his methods are clearly intended to be general. In some cases, he even specifies conditions for solvability.
Diophantus’ strangeness makes it somewhat surprising that his book has been transmitted to us. As David Fowler once pointed out to me,
Diopantus is remarkably unlike any other Greek writer and, if his manuscript hadn't survived, nobody would ever have dreamed that there had been anything like that. (So what other things might there have been but not been preserved?!)
That we have copies of Diophantus probably testifies to the admiration many have had for his work — that, and a large dose of luck.
Perhaps because the Arithmetic is so unlike everything else, it has had a more complex historical role than, say, Euclid’s Elements. Between the 5th and 17th centuries there were several “rediscoveries” of Diophantus, and each time he has had significant impact. He was translated into Arabic and his methods found their way into the work of several mathematicians of the time. Through them, he seems to have influenced Fibonacci and the Italian abacists. He was rediscovered again during the Renaissance, when the “humanists” were busily investigating every Greek work they could. And, as every mathematician knows, it was when he was reading Diophantus that Fermat wrote that famous marginal note. The main goal of this book is to trace this history.
Reading Diophantus is not easy. The text is a sequence of problems, usually solved using specific numbers, though the solutions are intended to be general (or at least generalizable). The notation is strange, the way of explaining operations is unusual, and there are no explanations for the ordering of problems. One suspects there is an overarching structure, but the reader must discover it, since it is not spelled out.
The only version of Diophantus currently available in English is T. L. Heath’s Diophantus of Alexandria: A Study in the History of Greek Algebra, first published in 1885, with a second edition in 1910. This, however, it is more like a retelling than a translation: the mathematics behind Diophantus’ text is explained using modern notation and operations. Heath even merges problems together and thus changes the numbering of the original, a feature about which L. E. Dickson complained in his 1911 review.
In 2006, Ad Meskens and Nicole van der Auwera published a Dutch translation of Diophantus’ Arithmetic. (Meskens prefers the Greek forms “Diophantos” and “Arithmetika,” but I will use the standard English forms in this review.) Meskens, a historian specializing in Renaissance and Early Modern mathematics, became interested in Diophantus when he was studying the work of Simon Stevin. Classicist van der Auwera did the translation, while Meskens provided “mathematical elucidation in modern notation” on facing pages. This is a smart way of solving the problem of producing a readable Diophantus: the translation provides the actual text, while the elucidation can be done in a more modern style like Heath’s. I wish someone would undertake an English translation in this spirit!
In the introduction to Travelling Mathematics, Meskens tells us that he was constrained, in the 2006 volume, from providing a full introduction. The feeling that more needed to be said has led to the volume under review. This character of the book as the-introduction-that-wasn’t is evident throughout. Many of the chapters are really surveys of the existing literature. Sometimes Meskens just cites the differing opinions of scholars without giving his own conclusions. In some cases, this seems appropriate: for example, barring some spectacular manuscript discovery, there is just no way we can know for sure when Diophantus lived. At other times, one feels that Meskens really does have an opinion even though it is not clearly stated. (On page 116, for example, a footnote reveals that Meskens does not agree with an opinion of Christianides mentioned on page 113 in a more neutral way.)
The book begins with an outline of early arithmetic and algebra (all the while keeping in mind that both terms are anachronistic). This very brief summary of older work is followed by a contextualizing chapter on Alexandria as a center of Hellenistic mathematics. The meat of the book is chapters 3–7. In chapter 3 Meskens gives a very brief survey of the contents of the Arithmetic. The chapters that follow survey the history of the reception of the book. As one would expect given Meskens’ own area of expertise, the richest chapters are 5–7, covering the period between the invention of printing and Fermat’s reading of Bachet’s 17th century edition.
There are many excellent things in the book. For example, I did not know the important role played by Cardinal Basileos Bessarion in the transmission of Greek mathematics and its study by the Renaissance humanists. Meskens tells us that it was he who “in a decisive way, made mathematics an integral part of the studia humanitatis.” He also makes the interesting observation that Bessarion’s private manuscript collection contained texts of almost all the surviving Greek mathematical texts… which makes me wonder to what extent our idea of what the Greeks were about depends on the taste of this one person!
Unfortunately, there are several production glitches that make the reading less pleasant. Meskens is not a native speaker of English, and Birkhäuser does not seem to have made use of a line editor. So we get “destructed” instead of “destroyed”, “syncoptic” instead of “syncopated”, and texts getting “translated into Arab.” Fractions have a “nominator” and a “denominator.” It is always possible to understand what is meant, however. In a few cases, the word choice made me smile, as when Bombelli places Diophantine analysis “in the mathematical footlight again.” (He means “spotlight”, of course.)
The formatting is quite strange as well. Some of the paragraphs in this book have their first line indented, in standard English style. But others do not: there is a line break, but the next line starts flush left. Some paragraphs are preceded by a blank line, others are not. I tried to figure out whether some kind of hierarchy was intended, say blank lines being more significant than new line/indent, which is more significant than new line/no indent, but I don’t think that’s right. I was also bothered by large numbers such as 300000 being printed without commas or spacing.
There are occasional minor errors. The standard story about Archimedes’ death does not say that “he was killed as he tried to approach a Roman camp with some of his inventions after the fall of Syracuse” (p. 14). It is parchment, not papyrus (p. 15), that is valuable and therefore was sometimes reused to create a palimpsest. If Diophantus lived “somewhere during the third century” it seems unlikely that he would have had to pay “the high taxes imposed by Augustus” (p. 32). On page 64, “we wish to find two square numbers…” should have been “two cube numbers,” as the solution immediately following indicates. On page 168, “p is a divisor of a” should be “p is not a divisor of a.”
(And since I'm picking nits: the cover is not featureless green, but has some writing in the background. But see the snippet on the right, in which I have increased the contrast and the brightness to bring out the image. Why is there an integral on the cover of a book about Diophantus? It's even a complex integral! The copyright page says that it comes from a manuscript by Augustus Beer, Über die Correction des Cosinusgesetzes bei der Anwendlung des Nicol'schen Prismas in der Photometrie. Why, it sayeth not. Every time I picked up the book, this integral bothered me. Do I need to “get a life”?)
All of this is annoying but minor; one only regrets that the publisher did not see fit to provide some copyediting.
Travelling Mathematics provides a wonderful overview of the Arithmetic and a valuable account of its influence. Anyone interested in Greek mathematics and in the history of algebra and number theory will want to read Meskens’ book and follow the many references provided. Indeed it would have been an excellent introduction to that Dutch edition of Diophantus.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.