Diophantus

Greek mathematician

• Born: fl. c. 250

Diophantus wrote a treatise on arithmetic that represents the most complete collection of problems dating from Greek times involving solutions of determinate and indeterminate equations. This work was the basis of much medieval Arabic and European Renaissance algebra.

Early Life

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.

Life’s Work

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 Qusṭā ibn Lūqā al-Ba՚labakkī (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 Diophantus’s work the term “arithmetic” takes a whole new meaning. The Greek tradition sharply distinguished between arithmetic and logistics. Arithmetic dealt with abstract properties of numbers, whereas logistics meant the computational techniques of reckoning. Diophantus dropped this distinction because he realized that although he was working with numerical examples, the techniques he used were quite general. Diophantus has often been called “the father of algebra,” but this is inaccurate: Diophantus merely uses definitional abbreviations and not a system of notation that is completely symbolic. At the outset of the Arithmetica, Diophantus gives his notation for powers of the unknown x, called arithmoi (and indicated by the symbol σ), and for their reciprocals. (For example, x² is denoted by Δ^ν and x³ by K^ν.) Diophantus has no signs for addition and multiplication, although he has a special sign for minus and a special word for “divided by.”

It is impossible to summarize the rich content of the 290 problems of the Arithmetica (189 in the Greek and 101 in the Arabic books), but from the technical point of view a very rough description of the books can be given as follows: IG deals mainly with determinate equations of the first and second degree; IIG and IIIG address many problems that involve determinate and indeterminate equations of degree no higher than two; IVA to VIIA are mainly devoted to consolidating the knowledge acquired in IG-IIIG; and IVG to VIG address problems involving the use of indeterminate equations of degree higher than two.

Throughout the Arithmetica, Diophantus admits only positive rational solutions (that is, solutions of the form p/q where p and q are natural numbers). Although negative numbers are used in his work, he seems to make sense of them only with respect to some positive quantity and not as having a meaning on their own. For example, in VG.2 (where 2 refers to problem 2 of VG), the equation 4 = 4x + 20 is considered absurd because the only solution is -4.

In IG are found many problems involving pure determinate equations, such as equations in which the unknown is present only in one power. The solution to IG.30, for example, requires solution of the equation 100 - x² = 96, which gives x = 2. Note that Diophantus is not interested in the solution x = -2. Diophantus gives a general rule for solving pure equations:

Next, if there results from a problem an equation in which certain terms are equal to terms of the same species, but with different coefficients, it will be necessary to subtract like from like on both sides until one term is found equal to one term. If perchance there be on either side or on both sides any negative terms, it will be necessary to add the negative terms on both sides, until the terms on both sides become positive, and again to subtract like from like until on each side only one term is left.

In other words, Diophantus reduces the equation to the normal form ax^m = c. If the result were a mixed quadratic, however, such as ax² + bx + c = 0, Diophantus might have solved it by using a general method of solution similar to the one commonly learned in high school. As an example, problem VIG.9 can be reduced to finding the solution of 630x² - 73x = 6, for which Diophantus merely states the solution to be x = 6/35. Although the possibility that Diophantus might have solved these problems by trial and error is open, internal evidence strongly suggests that he knew more than is relayed in the Arithmetica. In fact, the passage immediately following the above quote reads, “we will show you afterwards how, in the case also when two terms are left equal to a single term, such an equation can be solved.” The promised solution may be in the lost three books.

Diophantus also solves problems involving equations (or systems of equations) of the form

(a) a_nxⁿ + a_{n - 1}x^{n - 1} + . . . + a₁x - a₀ = y²(where n is at most 6)(b) a_nxⁿ + a_{n - 1}x^{n - 1} + . . . + a₁x - a₀ = y²(where n is at most 3)

The methods are seldom general, however, and rely on special cases of the above equations as found in VIG.19, where one finds the system given by the two equations 4x + 2 = y³ and 2x + 1 = z². (The reader is reminded that Diophantus always works with numerical cases and so equations in abstract form are not to be found in his work.)

In many problems, Diophantus needs to find solutions that are subject to certain limits imposed by a condition of the problem at hand. He often uses some very interesting techniques to deal with such situations (so-called methods of limits and approximation to limits).

The tract on polygonal numbers has been transmitted in incomplete form. Whereas the Arithmetica used methods that could be called algebraic, the treatise on polygonal numbers follows the geometrical method, in which numbers are represented by geometrical objects.

Of the other two works, Porismata and Moriastica, virtually nothing is known. The Moriastica was mentioned by Iamblichus (fourth century c.e.) and seems to have been merely a compendium of rules for computing with fractions similar (or identical) to the one found in IG. The Porismata is referred to often by Diophantus himself. In the Arithmetica, he often appeals to some results of number theoretic nature and refers to the Porismata for their proofs. It is unclear, as in the case of the Moriastica, whether the Porismata was part of the Arithmetica or a different work. There are other number theoretic statements that are used by Diophantus in the Arithmetica and that might have been part of the Porismata. They concern the expressibility of numbers as sums of two, three, or four squares. For example, Diophantus certainly knew that numbers of the form 4n + 3 cannot be odd and that numbers of the form 8n + 7 cannot be written as sums of three squares. It was in commenting on these insights of Diophantus that the distinguished mathematician Pierre de Fermat (1601-1665) gave some of his most famous number theoretic statements.

Significance

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.

Bibliography

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.