Les "Arithmétiques" de Diophante

Diophantus is a unique figure in the Ancient tradition of mathematics in Greek. His Arithmetica is concerned with problems about numbers rather than theorems. There are no diagrams at all, and we seem closer to the spirit of what was later to become known as algebra, though of course it is anachronistic to use that word for something written (probably) in the 3rd century. It is organized in order of increasing complexity, and it gets quite challenging. For example, consider this problem:

To find three numbers such that the difference of the greatest and the middle has a given ratio to the difference of the middle and the least, and further such that the sum of any two is a square. 

[Problem IV.39, from Ivor Thomas, Greek Mathematical Works II, Loeb Classical Library]

Like many of Diophantus’s problems, this is indeterminate: it has many solutions. As we learn by reading through his solution, the kind of “number” he wants is a positive rational number. The solution he finds is \[ \frac{87}{726}, \frac{2817}{726}, \frac{11007}{726}.\] I’ll leave it as a challenge to the reader.

A contemporary mathematician attempting to understand the Arithmetica will inevitably want to use modern algebraic symbolism and methods to follow what is going on. Indeed, intrepid translator T. L. Heath never published a literal translation, producing instead a version rewritten (“for the sake of brevity and perspicacity”) in modern algebraic notation. (I have heard that Heath produced a literal translation then discarded it, but I do not know whether that is true.) The only close-to-literal translations we have are the few selections that appear in Ivor Thomas’s Loeb volumes.

Heath’s “study,” published in 1910, covers the six chapters of the Arithmetica that survived in Greek. Around 1970, however, an Arabic translation of Diophantus was discovered. This turned out to contain four chapters that did not survive in Greek. (An English translation of these four chapters was done by Jacques Sesiano and published by Springer in their Sources and Studies series.) Surprisingly, scholars have concluded that these four chapters fit in the middle of the six that survived in Greek, which tends to wreak havoc with earlier analyses of the book.

French speakers are a little luckier on the translation front: there are two translations of the Greek text available, by Paul Ver Eecke and by André Allard, and Roshdi Rashed has translated the Arabic chapters. There being no need for another translation, what Rashed and Houzel present in this volume is more along the lines of what Heath did: a modern “reading” of Diophantus that attempts to explain his goals and methods.

Rashed and Houzel have a huge advantage over Heath in that they work with the ten surviving chapters rather than only the six chapters available in Greek. They work through the material, problem by problem, analyzing in modern terms what Diophantus was doing.

“In modern terms” is an important modifier here: Rashed and Houzel do not hesitate to use the language and ideas of arithmetical algebraic geometry, something that is heresy to many historians. They are careful not to claim that Diophantus was actually thinking in such terms. In fact, they do not try to reconstruct what Diophantus was actually thinking. Rather, they are skeptical that it is even possible to do that.

What we have here, then, is a modern mathematical commentary on a classical text. But while one might prefer to read this book cum grano salis when it comes to the history, it is an invaluable help to the mathematical reader who simply wants to see what is there.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College. He still hopes to someday read an English translation of Diophantus.