History of classical Diophantine analysis. From Abū Kāmil to Fermat (Q2439840)

In this book, the author presents a carefully researched history of Diophantine analysis from Abu Kamil to Fermat. The first half of the book is dedicated to the contributions of Islamic mathematicians: Abu Kamil, Al-Karaji, Al-Khazin, Al-Sijzi, Abu al-Jud, Ibn al-Haytham and Ibn al-Khawwam. The author explains the development from problems originating in the reading of Diophantus to problems requiring solutions in integers, to ``congruence conditions'', and to impossibility results, such as the first attempts to prove the unsolvability of the ``Fermat equation'' \(x^4 + y^4 = z^2\). The second part deals with the development that has taken place between Fibonacci and Fermat. Here the contributions of Fibonacci (\textit{Liber abaci}, \textit{Liber quadratorum}), Rafael Bombelli, Simon Stevin, François Viète and Bachet de Méziriac are discussed. Whereas Fibonacci studied problems imported from Arabic sources, Bombelli and Viète incorporated problems and methods from Diophantus, and Bachet published a very successful edition of the six books of Diophantus's \textit{Arithmetica}. The last chapter covers Fermat's work in Diophantine analysis: double and triple equations, cubic and quartic equations, infinite descent, and the Pell equation. The book also contains a carefully prepared author index, a subject index, as well as a valuable bibliography. The fact that the author presents the results in modern algebraic notation makes this book accessible to everyone interested in the history of Diophantine analysis, and specialists will have no problems distinguishing the original contribution from the author's modern interpretation. There are some places where the author's interpretation is perhaps too bold; on p.~7, for example, he discusses Abu Kamil's solution of Diophantine equations of the type \(ax - x^2 + b = y^2\). Abu Kamil solves the equation \(8x - x^2 + 109 = y^2\) and then explains (in words) how to proceed for ``general'' values of the coefficients. His method boils down to writing the equation in the form \(y^2 + (\frac a2-x)^2 = b + (\frac a2)^2\), at which point he remarks that if the sum on the right can be written as a sum of two squares, then there will be innumerable rational solutions, whereas otherwise there will not be any. The author carries this statement further and finally remarks: ``In other words, Abu Kamil expresses the following condition: if one of the variables may be expressed as a rational function of the other, or equivalently, if there is a rational parametrization, then we have all solutions''. This statement would be less controversial if Abu Kamil's name had been omitted; in addition, Abu Kamil talks about having found infinitely many solutions and not about all of them (the same confusion between infinitely many solutions and all solutions can be found in the discussion of de Billy on pp.~242--243). The other point on which this reviewer disagrees with the author concerns the dating of some of Fermat's discoveries, in particular infinite descent. On p.~231 he is still cautious, and writes that in the middle of 1640, Fermat seems to have been in possession of his technique of infinite descent and of his ``Little Theorem''. Fermat had stated the latter explicitly in his letters together with the claim of having a proof, but he never mentioned ``his method'' in this early period. On pp.~240, 263, 274, 290, 305 and 308, this speculation concerning Fermat's discovery of infinite descent in 1640 is presented as a fact. The single argument supporting this claim, and an argument that already Itard has used extensively, is the fact that in 1640, Fermat asked Frenicle to solve four problems, among them the Diophantine equations \(x^3 + y^3 = z^3\) and \(x^4 + y^4 = z^4\). Fermat did not claim that there are no solutions, nor does he claim to be able to demonstrate the statement he did not make. Another problem with the chronology presented by the author concerns Fermat's supplement discovered by \textit{J. E. Hofmann} [Abh. Preuß. Akad. Wiss., Math.-Naturw. Kl. 1943, No. 9, 52 S. (1944; Zbl 0060.01003)]. In this appendix to Frenicle's work on one of the problems that Fermat had posed in particular to the British mathematicians in 1657, Fermat explains how to prove his results on integers represented by the quadratic form \(x^2 - 2y^2\). In a footnote, the author writes ``This text, probably from 1643 if not earlier, [\dots]''. Even if we are willing to believe that Fermat composed, in 1643, a supplement to Frenicle's article from 1657, no reason is given as to why this must have been in 1643 or even earlier. These remarks must not divert from the fact that this book is an excellent source of information on the history of Diophantine analysis in Islamic countries and in Western Europe up to Fermat.