The ``Arithmetica'' of Diophantus. A historical and mathematical reading (Q2449120)

The theory of Diophantine equations is as alive today as it was in the days of Fermat and Euler, and the debt of algebra and number theory to Diophantus' \textit{Arithmetica} is immense. The discovery of the six Greek books by Johannes Regiomontanus in 1462 was an impetus for the development of algebra at the hands of Vieta and others: Diophantus had already symbols for an unknown and its small powers, and he used his symbolic expressions to solve his problems. Mathematicians such as Bombelli and Vieta used Diophantus' problems for displaying their new methods, and it was in the margin of Bachet's edition of these six books where Fermat wrote down his famous conjecture on the unsolvability of \(x^n + y^n = z^n\) in nonzero integers for exponents \(n \geq 3\). In addition to the classical editions of the six ``Greek'' books in various languages, \textit{P. de Fermat} [Varia opera mathematica. Tolosae 1679. Bruxelles: Culture et Civilisation (1969; Zbl 0191.00401)], \textit{T. L. Heath} [Diophantus of Alexandria, a study in the history of Greek algebra. Second edition, with a supplement containing an account of Fermat's theorems and problems connected with Diophantine analysis and some solutions of Diophantine problems by Euler. Cambridge: University Press (1910; JFM 41.0003.02)], \textit{G. Wertheim} [Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Leipzig: B. G. Teubner (1890; JFM 22.0011.02)], and \textit{A. Czwalina} [Arithmetik des Diophantos aus Alexandria. Göttingen: Vandenhoeck \& Ruprecht (1952)], there are four books in Arabic translation that were discoverd by \textit{R. Rashed} [Hist. Sci. (2) 4, No. 1, 39--46 (1994; Zbl 0816.01002)] and edited by \textit{J. Sesiano} [Books IV to VII of Diophantus' Arithmetica in the Arabic translation attributed to Qusta ibn Luqa. New York etc.: Springer-Verlag (1982; Zbl 0501.01004)]. The first few pages of this book introduce the \textit{Arithmetica} and explain the symbolism employed by Diophantus; the authors point out that these problems may be interpreted in an algebraic as well as in a geometric way. Chapter I is dedicated to the ``method of chords'': it introduces the vocabulary of algebraic geometry (algebraic curves, singular points, Bézout's Theorem etc.), and shows how to interpret algebraic equations as curves (in the case at hand as curves of genus \(0\) and \(1\), i.e., conics, reducible cubics, or elliptic curves). In Chapter II it is shown how to apply these tools to a few problems given by Diophantus. Chapter III contains a modern interpretation of all the problems in Books I--VI (Greek) and IV--VII (Arabic) as well as their solutions using the techniques explained in the first two chapters. All these problems are stated and solved ``in general'': the special choices made by Diophantus, who only had a symbol for one unknown, are not mentioned at all, so the readers interested in Diophantus' original formulation still have to consult one of the sources quoted above. The appendix contains a short article on the reception of Diophantus in Antiquity, and the last few pages contain an excellent index and a detailed bibliography. In order to show how the original problems are presented by Rashed and Houzel, let us consider the famous problem II.8: \textit{To decompose a square number into two squares}. Diophantus solves this problem by choosing the square number \(16\) as an example, calls one square \(x^2\) and the other \(16 - x^2\), and for making the second number a square he sets \(16 - x^2 = (2x-4)^2\), where the coefficient \(2\) of \(x\) is arbitrary. This gives him \(x = \frac{16}5\), so the two squares are \(\frac{256}{25}\) and \(\frac{144}{25}\). Rashed and Houzel give II.8 directly as the problem of solving the equation \(x^2 + y^2 = a^2\) for a given number \(a\). They remark that this equation defines a circle, and for solving the problem they intersect this circle with the lines \(y = \lambda x - a\) through the point \((0; -a)\), which gives \(x = \frac{2a\lambda}{\lambda^2+1}\) and \(y = a \frac{\lambda^2-1}{\lambda^2+1}\). Then they observe that ``Diophantus takes \(a = 4\) and chooses \(\lambda = 2\), which gives \(x = \frac{16}5\) and \(y = \frac{12}5\).'' In a similar way, the problems of all the known books by Diophantus are presented and solved in modern algebraic language, which makes the content of the \textit{Arithmetica} accessible to everyone interested in Diophantine analysis. This book is an impressive testimony to the genius of Diophantus, a monument in the literature on Diophantine equations erected by two mathematicians who have devoted much time and effort to their investigation of the history of Diophantine equations.