Travelling mathematics. The fate of Diophantos' arithmetic (Q1959760)

``The fate of Diophantos' Arithmetic'' tells us about the reception of Diophantus' work in various ages, and provides us with a solid background and behind-the-scenes stories concerning the various editions of his Arithmetika. In the first chapter, the contributions by pre-Diophantine cultures are discussed: Babylonians, Pythagoreans, and the classical Greek period connected with the names of Euclid and Archimedes. A large part of the content deals with the concept of numbers in Euclid's Elements. Chapter 2 gives informations on the role of Alexandria in the ancient world of science and investigates the role of Heron as a precursor to Diophantus. The manuscripts of Diophantus, their dating, and the use of symbols in this work are the subject of Chapter 3, which also contains problems from the known books. The last part of this chapter is devoted to various geometric interpretations of the algebra in Diophantus' work. The next few chapters sketch the reception of Diophantus by mathematicians from Hypatia over Arab scientists to Stevin, Viete, Bachet and Fermat. There is an extensive bibliography and an index. This book is a very good introduction to the literature available on Diophantus and the interpretations of his work, and provides the reader with solid information on the various editions, their history and their background. From a number theorist's point of view, however, the book is disappointing. The formulas apparently have not been proofread, and there are many mistakes ranging from simple but annoying mistakes such as \(\sum_{i=1}^n (2i+1) = (n+1)^2\) on p.~19 or the equation \((2n+1)^2 - (2n)^2 = 2n+1\) on p. 141 to confusing cubes with squares in the text of problem A IV.1 on p.~64, obviously incorrect equations in footnote 9 on p.~159 and a dozen of similar mistakes in between. The author's opinion on a variety of topics are, at the very least, surprising, and it is not acceptable that neither facts nor any arguments are given that would support these claims. On p.~24 he explains that he does not deal with Euclid's books on number theory because Euclid's number theory is ``geometric, not arithmetic, in nature''. It is true that Euclid's language is geometric, but the nature of the infinitude of primes or aspects of unique factorization is most certainly not geometric. In a similar vein, the author claims that there ``is no doubt in our mind that every single problem we encounter in the Arithmetika had already been solved before Diophantos'', but refuses to provide even a single argument in its favor. Moreover I do not agree with the author's comment on Sesiano's opinion that problem A VI.17 is ``unimaginative''. It most certainly is, and if the problem has gained some prominence then this is because it deals, in modern language, with a curve of genus \(2\). Diophantus did not have any techniques for attacking such problems, and the problem of finding all rational points on the corresponding hyperelliptic curve, which is interesting and was studied in Wetherell's thesis (a simpler proof was given by \textit{E. V. Flynn} and \textit{J. L. Wetherell} [Manuscr. Math. 100, No. 4, 519--533 (1999; Zbl 1029.11024)]), has little if nothing to do with the problem studied by Diophantus. Section 3.8 deals with various geometric interpretations of Diophantus' technique of substitutions, but it is not explained why this technique needs to be interpreted geometrically. Mathematicians studying Diophantus, whether it's Bachet, Fermat, Euler or more recent number theorists, never had difficulties understanding his algebraic technique, and there are absolutely no hints of any geometric interpretation of his substitutions neither as lines in affine or projective space nor as pieces of geometric figures. I also would have preferred had the author omitted the remark that the right angled triangles in Viete's Notae Priores ``may be considered as complex numbers''. There are other places in the book that suggest that the author is not familiar with classical (let alone modern) number theory. His claim that Fermat's techniques rendered Diophantus's problems uninteresting (p.~168), or that Fermat's ``researches into the properties of square numbers brought him to the brink of Gauss's quadratic forms theory'' (p.~168) are as revealing as the fact that the statement of Fermat's Little Theorem (p.~168) is incorrect. The history of the polygonal number theorem given in footnote 24 on p.~164 is almost pure fantasy, and of course Kummer is credited with an incorrect proof of Fermat's Last Theorem on p.~169. Finally, it is claimed on p.~172 that had ``Hilbert's tenth problem been confirmed, then Goldbach's conjecture would have been false''. As for Fermat's contributions to number theory, the author should rather have referred to Weil's excellent book [\textit{A. Weil}, Number Theory. Modern Birkhäuser Classics. Basel: Birkhäuser. (2007; Zbl 1149.01013)] instead of to [\textit{M. S. Mahoney}'s, The mathematical career of Pierre de Fermat. Princeton, NJ: Princeton University Press. (1994; Zbl 0820.01017)] (p.~168), which, by the way, cannot even be found in the bibliography. Finally, the statement (p.~169) of Fermat's marginal note, in which he claimed what became known as his Last Theorem, contains several errors, some of which can be spotted without knowing a single word of Latin. The Arithmetika by Diophantus is still a source of inspiration in our days, and a must-read for anyone studying the number theoretic contributions by mathematicians from Bachet and Fermat to Euler and beyond. A book on the same level as Weil's book cited above, which would explain the number theoretic tools of Diophantus and their development at the hands of subsequent generations of mathematicians, would be much appreciated; the book under review, which claims to ``present a comprehensive study of Diophantos' monumental work known as the \textit{Arithmetika}'', is not what number theorists have been waiting for.