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Diophantus and Diophantine Equations

Isabella Grigoryevna Bashmakova
Mathematical Association of America
Publication Date: 
Number of Pages: 
Dolciani Mathematical Expositions
[Reviewed by
David Graves
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This brief but fascinating book is a recent translation by Abe Shenitzer of a book first published in Russian in 1972. The author, whose entire teaching career has been at Moscow University, presents a blend of history and mathematics centered on the work of Diophantus and its subsequent development up through the early twentieth century. A substantial amount of information is packed into 85 pages of text supplemented by a brief bibliography and an index of more than a hundred names (mainly mathematicians). A two-page addendum by J. H. Silverman skillfully surveys results in algebraic geometry since the book was written.

The book's introduction delimits the main topics to be discussed and presents (for the first of several times) the suggestion that quite general methods are discernible in Diophantus' particular solutions of problems. The first chapter then tells us what little is known of Diophantus personally (practically nothing), reviews Tannery's close-to-convincing argument placing Diophantus in the third century, and surveys some other famous figures from Alexandrian mathematics. In discussing the 189 problems of Diophantus' "Arithmetica" the author presents again her claim that the problems selected "illustrate definite, rigorously thought-out methods." The second chapter comments that Diophantus extended the notion of number to include negatives and rationals, describes his symbols for exponents from -6 to 6, and notes that he moved beyond Greek traditions in permitting addition of non-homogeneous magnitudes. The third chapter raises the mathematical ante a bit by discussing order, genus, and birational equivalence of algebraic curves, material that is needed to understand much of the remainder of the book.

In Chapter 4 we find Bashmakova's claim that "Diophantus had a general method for the determination of rational points on quadratic curves" juxtaposed with opinions to the contrary of several historians of mathematics. In Chapter 5, using the famous Fermat-related problem 8 of Book II as an example, the author further elucidates her claim by inferring, from Diophantus' particular solutions of indeterminate quadratics, that he knew there were infinitely many solutions, and that they could be expressed as rational functions of one parameter. And in Chapter 6 she sees Diophantus' solutions of indeterminate cubics yielding a general method of finding additional rational points on a curve using tangent or secant lines.

The seventh chapter argues that Diophantus was aware of conditions for an integer to be a sum of two square integers, and knew how many ways such an integer could be written as a sum of two squares. The author marshals support for this view from Jacobi's reconstruction of a distorted passage in problem 9 of Book V of the "Arithmetica." The next three chapters trace the influence of Diophantus after Hypatia through the nineteenth century. The algebraic ideas are followed through Byzantine commentators and Arab mathematicians, and into Europe via Fibonacci. The arithmetical ideas only surfaced later, from Regiomontanus' decision to translate the "Arithmetica" (abandoned when only 6 of the 13 books could be found) to an inclusion of 143 problems of Diophantus in Bombelli's "Algebra" in 1572, followed three years later by the first Latin translation of the "Arithmetica." There is a description of Viète's solution of problems involving sums and differences of cubes using Diophantus' "method of tangents" discussed in Chapter 6, then of Fermat's advances on such problems through iteration of that method. Mention of the 1621 Bachet translation of Diophantus triggers a digression on the career and work of Fermat. We then have a description of Euler's systematic analysis of rational solutions of indeterminate quadratics and cubics, and of Jacobi's observation that other work of Euler on elliptic integrals has some bearing on Diophantine analysis. At the end of Chapter 10, Bashmakova remarks that Jacobi's work in this context can be described in modern terms as examining the structure of the group of rational points on an elliptic curve.

Chapter 11 gives a geometric interpretation of the group operation introduced in the previous chapter, and connects the work of Euler and Jacobi with the tangent and secant methods of Diophantus. In Chapter 12, the author briefly notes contributions of both Abel and Riemann to the theory of algebraic curves, then gives an extensive discussion of Poincaré's contribution to the subject: determination of birationally equivalent curves, an explicit articulation of the fact that the set of rational points on an elliptic curve forms an abelian group, and the investigation of generators of the group (including the conjecture - proven by Mordell in 1922 - that the group is finitely generated).

In the concluding chapter, the author's remarks about generalizations and conjectures from the early part of this century by Poincaré, Weil, Mordell, and Siegel are followed by Silverman's two-page summary of developments from Mordell's conjecture (settled in 1983 by Faltings) to the Taniyama-Shimura conjecture and its FLT-related resolution by Wiles and Taylor. In connection with that result, the concise summary manages to include references to the usual list of Frey, Ribet, Rubin, and Serre, and to another dozen and a half mathematicians as well. In a short "supplement" Bashmakova puts in a final word for her theme that in the works of Diophantus, " ... the concrete number symbols ... double as parameters."

One major point the author repeats several times - that behind the particular solutions of Diophantus lie general algorithms and even proofs of which Diophantus himself was well aware - is bound to be controversial. Bashmakova's recognition that well-chosen examples can contain seeds of general proof is a valuable insight, and presumably many of us focus on such examples in our teaching as we try to guide our students through a rigorous proof. But the next big step - concluding that Diophantus was consciously doing this - is a step that many will find difficult to take.

Bashmakova argues that Diophantus must have had a general method in mind, rather than simply a bag of tricks. She argues, in fact, that his method was an algebraic version of modern geometric methods, such as the "method of secants and tangents" for finding rational points on cubic curves. Her evidence consists of showing that in many cases it is possible to interpret what Diophantus does in geometric terms. This interpretation is certainly illuminating to us, but one wonders what it really tells us about how Diophantus understood his method.

Despite these qualms about Bashmakova's historical claims, this book can be recommended because it teaches us so much interesting mathematics and because it opens us to further study of Diophantus. A wealth of material, both mathematical and historical, is crammed into this remarkable little book, which should find a place on the shelves of every college library and of everyone who teaches number theory or history of mathematics.

Diophantus and Diophantine Equations was named a CHOICE Outstanding Academic Book for 1998.

 David Graves ( is Associate Professor of Mathematics at Elmira College, where he is active as a pianist, and has taught courses in opera and history of astronomy as well as the usual run of mathematics courses.

Introduction; Diophantus; Numbers and symbols; Diophantine equations; Evaluation of Diophantus' methods by historians of science; Indeterminate quadratic equations; Indeterminate cubic equations; Diophantus and number theory; Diophantus and the mathematicians of the 15th and 16th centuries; Diophantus' methods in the works of Viéte and Fermat, Diophantine equations in the works of Euler and Jacobi; The geometric meaning of the operation of addition of points; The arithmetic of algebraic curves; Conclusion; Supplement: the role of concrete numbers in Diophantus' Arithmetic; Bibliography.