Almost nothing is known about the life of Diophantus (di-oh-FAHN-tuhs), and there is no mention of him by any of his contemporaries. A reference to the mathematician by Hypsicles (active around 170 b.c.e.) in his tract on polygonal numbers and a mention of him by Theon of Alexandria (fl. 365-390 c.e.) give respectively a lower and an upper bound for the period in which Diophantus lived. There is also evidence that points to the middle of the third century c.e. as the flourishing period of Diophantus. Indeed, the Byzantine Michael Psellus (latter part of the eleventh century) asserts in a letter that Anatolius, bishop of Laodicea around 280 c.e., wrote a brief work on the Diophantine art of reckoning. Psellus’s remark seems to fit well with the dedication of Diophantus’s masterpiece Arithmētika (Arithmetica, 1885) to a certain Dionysius, who might possibly be identified with Saint Dionysius, bishop of Alexandria after 247. The only dates known about Diophantus’s life are obtained as a solution to an arithmetical riddle contained in the Greek Anthology, which gives thirty-three for his wedding age, thirty-eight for when he became a father, and eighty-four for the age of his death. The trustworthiness of the riddle is hard to determine. During his life, Diophantus wrote the Arithmetica, the Porismata, the Moriastica, and the tract on polygonal numbers.
Diophantus’s main achievement was the Arithmetica, a collection of arithmetical problems involving the solution of determinate and indeterminate equations. A determinate equation is an equation with a fixed number of solutions, such as the equation x2 - 2x + 1 = 0, which admits only 1 as a solution. An indeterminate equation usually contains more than one variable, as for example the equation x + 2y = 8. The name indeterminate is motivated by the fact that such equations often admit an infinite number of solutions. The degree of an equation is the degree of its highest degree term; a term in several variables has degree equal to the sum of the exponents of its variables. For example, x2 + x = 0 is of degree two, and x3 + x2y4 + 3 = 0 is of degree six but of degree three in x and degree four in y.
Although Diophantus presents solutions to arithmetic problems employing methods of varying degrees of generality, his work cannot be fairly described as a systematic exposition of the theory of solution of determinate and indeterminate equations. The Arithmetica is in fact merely a collection of problems and lacks any deductive structure whatsoever. Moreover, it is extremely hard to pinpoint exactly which general methods may constitute a key for reading the Arithmetica. This observation, however, by no means diminishes Diophantus’s achievements. The Arithmetica represents the first systematic collection of such problems in Greek mathematics and thus by itself must be considered a major step toward recognizing the unity of the field of mathematics dealing with determinate and indeterminate equations and their solutions, in short, the field of Diophantine problems.
The Arithmetica was originally divided into thirteen books. Only six of them were known until 1971, when the discovery of four lost books in Arabic translation greatly increased knowledge of the work. The six books that were known before that discovery were transmitted to the West through Greek manuscripts dating from the thirteenth century (these will be referred to as books IG-VIG). The four books in Arabic translation (henceforth IVA-VIIA) represent a translation from the Greek attributed to Qustā ibn Lūqā al-Balabakkī (fl. mid-ninth century). The Arabic books present themselves as books 4 through 7 of the Arithmetica. Because none of the Greek books overlaps with the Arabic books, a reorganization of the Diophantine corpus is necessary.
Scholars agree that the four Arabic books should probably be spliced between IIIG and IVG on grounds of internal coherence: The techniques used to solve the problems in IVA-VIIA presuppose only the knowledge of IG-IIIG, whereas the techniques used in IVG through VIG are radically different and more complicated than those found in IVA-VIIA. There is also compelling external evidence that this is the right order. The organization of problems in al-Karaji’s al-Fakhri (c. 1010), an Islamic textbook of algebra heavily dependent on Diophantus, shows that the problems taken from IG-IIIG are immediately followed by problems found in IVA. The most interesting difference between IG-VIG and IVA-VIIA consists in the fact that in the Greek books, after having found the sought solutions (analysis), Diophantus never checks the correctness of the results obtained; in the Arabic books, the analysis is always followed by a computation establishing the correctness of the solution obtained (synthesis).
Before delving into some of the contents of the Arithmetica, the reader must remember that in...