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DIOPHANTUS, of Alexandria, Greek algebraist, probably flourished about the middle of the 3rd century.

00The Arithmetica, the greatest treatise on which the fame of Diophantus rests, purports to be in thirteen Books, but none of the Greek MSS.

00The 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.

00On the other hand the Porisms, to which Diophantus makes three references ("we have it in the Porisms that.

00Among 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.

00A word is necessary on Diophantus' notation.

00would give negative, surd or imaginary values; Diophantus then traces how each element of the equation has arisen, and formulates the auxiliary problem of determining how the assumptions must be corrected so as to lead to an equation (in place of the "impossible" one) which can be solved rationally.

00Thus Diophantus knew that no number of the form 8n-1-7 can be the sum of three squares.

00General accounts of Diophantus' work are to be found in H.

00For many centuries algebra was confined almost entirely to the solution of equations; one of the most important steps being the enunciation by Diophantus of Alexandria of the laws governing the use of the minus sign.

00The first extant work which approaches to a treatise on algebra is by Diophantus, an Alexandrian mathematician, who flourished about A.D.

00In the preface to this work, which is dedicated to one Dionysius, Diophantus explains his notation, naming the square, cube and fourth powers, dynamis, cubus, dynamodinimus, and so on, according to the sum in the indices.

00It is difficult to believe that this work of Diophantus arose spontaneously in a period of general stagnation.

00We find that geometry was neglected except in so far as it was of service to astronomy; trigonometry was advanced, and algebra improved far beyond the attainments of Diophantus.

00However this may be, it is certain that the Hindu algebraists were far in advance of Diophantus.

00A notable improvement on the ideas of Diophantus is to be found in the fact that the Hindus recognized the existence of two roots of a quadratic equation, but the negative roots were considered to be inadequate, since no interpretation could be found for them.

00Great advances were made in the study of indeterminate equations, a branch of analysis in which Diophantus excelled.

00But whereas Diophantus aimed at obtaining a single solution, the Hindus strove for a general method by which any indeterminate problem could be resolved.

00Ptolemy's Almagest, the works of Apollonius, Archimedes, Diophantus and portions of the Brahmasiddhanta, were also translated.

00He follows the methods of Diophantus; his work on indeterminate equations has no resemblance to the Indian methods, and contains nothing that cannot be gathered from Diophantus.

00Heath's Diophantus.

00That principle had been made use of by the Greek authors of the classic age; but of later mathematicians only Hero, Diophantus, &c., ventured to regard lines and surfaces as mere numbers that could be joined to give a new number, their sum.

00It may be that the study of such sums, which he found in the works of Diophantus, prompted him to lay it down as a principle that quantities occurring in an equation ought to be homogeneous, all of them lines, or surfaces, or solids, or supersolidsan equation between mere numbers being inadmissible.

00In this respect the fate of Pappus strikingly resembles that of Diophantus.

00by Diophantus, the general of Mithradates the Great, king of Pontus.

00Hypatia, according to Suidas, was the author of commentaries on the Arithmetica of Diophantus of Alexandria, on the Conics of Apollonius of Perga and on the astronomical canon (of Ptolemy).

00and hi; general Diophantus, c. 110 B.e., and submitted to the Pontic dynasty.

00We know only that the last of them, a Paerisades, unable to make headway against the power of the natives, called in the help of Diophantus, general of Mithradates VI.

00It is addressed to Diophantus and conveys a moral, that one should work and not dream, illustrated by the story of an old fisherman who dreams that he has caught a fish of gold and narrates his vision to his mate.

00As Leonidas of Tarentum wrote epigrams on fishermen, and one of them is a dedication of his tackle to Poseidon by Diophantus, the fisher, 8 is likely that the author of this poem was an imitator of Leonidas.

00Diophantus >>

00A different method was used by Diophantus, accents being omitted, and the denominator being written above and to the right of the numerator.

00They originally took the form of marginal notes in a copy of Bachet's Diophantus, and were published in 1670 by his son Samuel, who incorporated them in a new edition of this Greek writer.

00The first contains the "Arithmetic of Diophantus," with notes and additions.

00We do know three lemmas contained in The Porisms since Diophantus refers to them in the Arithmetica.

00But it seems a fair inference from a passage of Michael Psellus (Diophantus, ed.

00Tannery thinks that the solution of a complete quadratic promised by Diophantus himself (I.

00- The first to publish anything on Diophantus in Europe was Rafael Bombelli, who embodied in his Algebra (1572) all the problems of Books I.

00

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