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2020 | OriginalPaper | Chapter

1. Diophantus of Alexandria

Author : Daniel Coray

Published in: Notes on Geometry and Arithmetic

Publisher: Springer International Publishing

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Abstract

Diophantus is like an island in the history of mathematics. He lived in Alexandria around 250 C.E. Nobody before him had ever tackled a study of arithmetic over the field of rational numbers. It was 1,300 years before Western mathematicians became interested in this type of problem (Bombelli, Viète, Bachet, Fermat), … on reading Diophantus to be precise. He also introduced new methods and a special symbol to express an unknown, which makes him an essential precursor of algebraic notation.

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next chapter Algebraic Closure; Affine Space

But we know that in the tenth century Arab mathematicians (among others abu’l-Wafa and al-Karajı̄) were already studying and commenting on Diophantus.

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(in the resolution of Problem V.16: “but we have in the Porisms that…”), but nobody knows if the word “Porisms” refers to statements proved elsewhere, in Diophantus’ books that have not survived, or if it is for instance the title of a book.

This approach led many commentators to say that Diophantus treated only particular cases. However, it is obvious that the method he developed is extremely general. Diophantus did not really claim that his method works without changes for all positive rational numbers, but the problem is formulated for a general enough number, not only for a = 6.

A very original contribution of Diophantus is the introduction of a symbol for an unknown x. However, as he lacked a symbol for a second variable, he had to use circumlocutions to name the other unknowns. Here he says: “we put the first number to be x; the second is then 6 − x”.

This method, called “of the false position”, has been found in mathematical texts since the highest antiquity. It serves to address the lack of notation for a variable. Here we would write y = mx − 1 and would show that it would better to take m = a∕2, but Diophantus lacked a second letter for the variable m. We do not keep y = 2x − 1 at the end of the solution. See the discussion that follows to understand the geometric meaning of this working hypothesis.

We can solve the equation, but this introduces \(\sqrt {185}\), which is not a rational number. Diophantus actually says “if the coefficients of x were the same on both sides of the equality…”, then he looks for where the 4 in 4x comes from.

Diophantus worked very well with polynomials, even if he constantly wrote (27x ³ + 6x) − (27x ²) rather than 27x ³ − 27x ² + 6x, which allowed him to always subtract a positive number from another greater positive number.

[Ba]

Bashmakova, I.G.: Diophantus and Diophantine Equations. Trans. from Russian (Nauka, Moscow, 1972). The Dolciani Mathematical Expositions, vol. 20. Math. Assoc. America, Washington, DC (1997)

[BaSl]

Bashmakova, I.G., Slavutin, E.I.: La méthode des approximations successives dans l’Arithmétique de Diophante. Istor. Metodol. Estestv. Nauk 16, 25–35 (1974, in Russian)

[Cas]

Cassels, J.W.S.: The rational solutions of the diophantine equation Y ² = X ³ − D. Acta Math. 82, 243–273 (1950)

[Di]

Diophantus Alexandrinus: Opera Omnia, edition P. Tannery. Teubner, Stuttgart (1893)

[Fe]

Fermat, P.: Œuvres, edition P. Tannery. Gauthier-Villars, Paris (1892–1922)

[Ra]

Rashed, R.: Les travaux perdus de Diophante. I, II. Rev. Hist. Sci. 27, 97–122 (1974); 28, 3–30 (1975)

[Ses]

Sesiano, J.: Books IV to VII of Diophantus’ Arithmetica: in the Arabic Translation Attributed to Qusṭā ibn Lūqā. Springer, New York (1982)CrossRef

[Vi]

Viète, F.: Zeteticorum Libri Quinque (publié en 1591). In: Opera Mathematica, pp. 42–81. G. Olms Verlag, Hildesheim (1970)

Title: Diophantus of Alexandria
Author: Daniel Coray
Publisher: Springer International Publishing
Book: Notes on Geometry and Arithmetic
Print ISBN: 978-3-030-43780-0

Electronic ISBN: 978-3-030-43781-7

Copyright Year: 2020
DOI: https://doi.org/10.1007/978-3-030-43781-7_1

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