Arithmetica


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 and indeterminate equations.

Summary

Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the Arithmetica problems lead to quadratic equations.
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 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, his doing so would be truly remarkable: 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 was originally written in thirteen books, but the Greek manuscripts that survived to the present contain no more than six books. In 1968, Fuat Sezgin found four previously unknown books of Arithmetica at the shrine of Imam Rezā in the holy Islamic city of Mashhad in northeastern Iran. The four books are thought to have been translated from Greek to Arabic by Qusta ibn Luqa. Norbert Schappacher has written:

resurfaced around 1971 in the Astan Quds Library in Meshed in a copy from 1198 AD. It was not catalogued under the name of Diophantus because the librarian was apparently not able to read the main line of the cover page where Diophantus’s name appears in geometric Kufi calligraphy.

Arithmetica became known to mathematicians in the Islamic world in the tenth century when Abu'l-Wefa translated it into Arabic.