Arithmetica

Cover of the 1621 edition, translated into Latin by Claude Gaspard Bachet de Méziriac.

Arithmetica is an ancient Greek text on mathematics written by the mathematician Diophantus in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

Equations in the book are called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the Arithmetica problems lead to quadratic equations. It was these equations which inspired Pierre de Fermat to propose Fermat's Last Theorem, scrawled in the margins of Fermat's copy of 'Arithmetica', which states that the equation $x^{n}+y^{n}=z^{n}$ , where $x$ , $y$ , $z$ and $n$ are non-zero integers, has no solution with $n$ greater than 2.

In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form ( $4n+3$ ) cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result it would be truly remarkable for even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph Louis Lagrange proved it using results due to Leonhard Euler.

Arithmetica became known to Muslim mathematicians in the tenth century^[1] when Abu'l-Wefa translated it into Arabic.^[2]

References

^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 234. ISBN 0-471-54397-7. Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century. {{cite book}}: |edition= has extra text (help)
^ Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 239. ISBN 0-471-54397-7. Abu'l-Wefa was a capable algebraist as well as a trigonometer. He commented on al-Khwarizmi's Algebra and translated from Greek one of the last great classics, the Arithmetica of Diophantus. {{cite book}}: |edition= has extra text (help)

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[1] Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 234. ISBN 0-471-54397-7. Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century. {{cite book}}: |edition= has extra text (help)

[2] Boyer, Carl B. (1991). "The Arabic Hegemony". A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. p. 239. ISBN 0-471-54397-7. Abu'l-Wefa was a capable algebraist as well as a trigonometer. He commented on al-Khwarizmi's Algebra and translated from Greek one of the last great classics, the Arithmetica of Diophantus. {{cite book}}: |edition= has extra text (help)

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See also

References