Born: fl. 3rd c. AD
Birthplace: Alexandria, Egypt
Died: fl. 3rd c. AD
Cause of death: unspecified
Race or Ethnicity: White
Nationality: Ancient Rome
Executive summary: Arithmetica
Diophantus, of Alexandria, Greek algebraist, probably flourished about the middle of the 3rd century. Not that this date rests on positive evidence. But it seems a fair inference from a passage of Michael Psellus that he was not later than Anatolius, bishop of Laodicea from AD 270, while he is not quoted by Nicomachus (fl. circa AD 200), nor by Theon of Smyrna (circa AD 130), nor does Greek arithmetic as represented by these authors and by Iamblichus (end of the 3rd century) show any trace of his influence, facts which can only be accounted for by his being later than those arithmeticians at least who would have been capable of understanding him fully. On the other hand he is quoted by Theon of Alexandria (who observed an eclipse at Alexandria in AD 365); and his work was the subject of a commentary by Theon's daughter Hypatia (died in 415). The Arithmetica, the greatest treatise on which the fame of Diophantus rests, purports to be in thirteen Books, but none of the Greek manuscripts which have survived contain more than six (though one has the same text in seven Books). They contain, however, a fragment of a separate tract on Polygonal Numbers. The missing books were apparently lost early, for there is no reason to suppose that the Arabs who translated or commented on Diophantus ever had access to more of the work than we now have. The difference in form and content suggests that the Polygonal Numbers was not part of the larger work. On the other hand the Porisms, to which Diophantus makes three references ("we have it in the Porisms that..."), were probably not a separate book but were embodied in the Arithmetica itself, whether placed all together or, as Tannery thinks, spread over the work in appropriate places. The "Porisms" quoted are interesting propositions in the theory of numbers, one of which was clearly that the difference between two cubes can be resolved into the sum of two cubes. Tannery thinks that the solution of a complete quadratic promised by Diophantus himself, and really assumed later, was one of the Porisms.
Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers. But the bulk of the work consists of problems leading to indeterminate equations of the second degree, and these universally take the form that one or two (and never more) linear or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x. A few problems lead to indeterminate equations of the third and fourth degrees, an easy indeterminate equation of the sixth degree being also found. The general type of problem is to find two, three or four numbers such that different expressions involving them in the first and second, and sometimes the third, degree are squares, cubes, partly squares and partly cubes. Book VI contains problems of finding rational right-angled triangles such that different functions of their parts (the sides and the area) are squares.
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