Diophantus, one of the last of the great Greek mathematicians, lived at Alexandria, most probably in the second half of the 3d century of our era. He died at the age of eighty-four. The titles of three of his works are Arithmetics, Polygonal Numbers, and Porisms. Of the first, which consisted of thirteen books, only six remain; of the second we possess merely a fragment; and the third has been entirely lost. The Arithmetics is the earliest extant treatise on algebra, but it would be rash to say that Diophantus was the inventor of algebra, though to what extent he was indebted to his predecessors cannot now be decided. The first book of the Arithmetics is occupied with problems leading to determinate equations of the first degree, the rest of the books with problems leading to indeterminate equations of the second degree, the sixth book in particular being devoted to the finding of right-angled triangles where some linear or quadratic function of the sides is to be a square or a cube. The treatise on Polygonal Numbers is not analytical but synthetical—i.e. in the manner of Euclid's arithmetical books—and in it numbers are represented by lines. The Porisms were probably a collection of propositions on the properties of certain numbers. The first translation of Diophantus was into Latin by Xylander (Wilhelm Holzmann) in 1575. The only edition of the Greek text is that by Bachet, published along with a Latin translation in 1621, and reprinted with the addition of Fermat's notes and many misprints in 1670. A translation into German by Otto Schulz appeared in 1822. See T. L. Heath's Diophantos of Alexandria (1885).
DIOPHANTINE ANALYSIS, so called from Diophantus, is that part of algebra which treats of the finding of particular rational values for general expressions under a surd form. A simple example of a diophantine problem is to find a right-angled triangle whose three sides are expressible by rational numbers, or in other words, to divide a square number into two squares (Diophantus, Arithmetics, ii. 9). A diophantine theorem less simple is the statement of Fermat, which even yet has only been partially proved, that the equation is impossible for every integral value of greater than 2. The diophantine analysis is really a part of what is now called the theory of numbers, and its development is to be sought in the writings of those mathematicians, from Fermat and Euler downwards, who have cultivated this subject. Much information regarding it will be found in the second part of Euler's Algebra.