Diophantus Analysis


Diophantus’s Arithmetica represents the most extensive treatment of arithmetic problems involving determinate and indeterminate equations from Greek times. It is clear from the sources that Diophantus did not create the field anew but was heavily dependent on the older Greek tradition. Although it is difficult to assess how much he improved on his predecessors’ results, his creativeness in solving so many problems by exploiting new stratagems to supplement the few general techniques at his disposal was impressive.

The Arithmetica was instrumental in the development of algebra in the medieval Islamic world and Renaissance Europe. The Arabic writers al-Khazin (fl. c. 940), Abul Wefa (940-998), and al-Karaji (fl. c. 1010), among others, were deeply influenced by Diophantus’s work and incorporated many of his problems in their algebra textbooks. The Greek books have come to the West through Byzantium. The Byzantine monk Maximus Planudes (c. 1260-c. 1310) wrote a commentary on the first two Greek books and collected several extant manuscripts of Diophantus that were brought to Italy by Cardinal Bessarion. Apart from a few sporadic quotations, there was no extensive work on the Arithmetica until the Italian algebraist Rafael Bombelli ventured into a translation (with Antonio Maria Pazzi), which was never published, and used most of the problems found in IG-VIG in his Algebra, published in 1572. François Viète, the famous French algebraist, also made use of several problems from Diophantus in his Zetetica (1593). In 1575, the first Latin translation, by Wilhelm Holtzmann (who grecized his name as Xylander), appeared with a commentary. In 1621, the Greek text was published with a Latin translation by Claude-Gaspar Bachet. This volume became the standard edition until the end of the nineteenth century, when Paul Tannery’s edition became available. A new French-Greek edition of the Greek books is planned since the Tannery edition is long outdated.

Further Reading

Bashmakova, Isabella G. Diophantus and Diophantine Equations. Translated by Abe Shenitzer. Washington, D.C.: Mathematical Association of America, 1997. A discussion of the methods of Diophantus, accessible to readers who have taken some university mathematics. It includes the elementary facts of algebraic geometry indispensable for its understanding. Examines the development of Diophantine methods during the Renaissance and in the work of Pierre de Fermat.

Heath, Thomas L. Diophantos of Alexandria: A Study in the History of Greek Algebra. Cambridge, England: Cambridge University Press, 1885. This volume is still the major reference work on Diophantus in English. It gives an extensive treatment of the sources, the works, and the influence of Diophantus. The appendix contains translations and a good sample of problems from IG-VIG of the Arithmetica and translations from the tract on polygonal numbers.

Heath, Thomas L. A History of Greek Mathematics. 2 vols. 1921. Reprint. New York: Dover Press, 1981. The second volume of this classic study contains a thorough exposition of Diophantus’s work with a rich analysis of types of problems from the Arithmetica.

Sesiano, Jacques. Books IV to VII of Diophantus’s Arithmetica: In the Arabic Translation Attributed to Qusta Ibn Luqa. New York: Springer-Verlag, 1982. A detailed analysis of the Arabic books with a translation and a commentary on the text. The introduction presents a summary of the textual history of arithmetic theory in Greek and Arabic. The English translation and the commentary are followed by an edition of the Arabic text. Other features include an Arabic index, an appendix that gives a conspectus of the problems in the Arithmetica, and an extensive bibliography.

Thomas, Ivor, ed. Greek Mathematical Works. 2 vols. Cambridge, Mass.: Harvard University Press, 1980. Volume 2 of this work contains selections from the Arithmetica and the quotations from the Greek Anthology, Psellus, and Theon of Alexandria that are relevant for Diophantus’s dates. Greek texts with English translation.

Vogel, Kurt. “Diophantus of Alexandria.” In Concise Dictionary of Scientific Biography, vol. 4. New York: Scribner’s, 2000. A survey of Diophantus’s life and works, with an extensive selection of types of problems and solutions found in the Arithmetica.