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DIOPHANTUS
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DIOPHANTUS , of See also:Alexandria, See also:Greek algebraist, probably flourished about the See also:middle of the 3rd See also:century. Not that this date rests on See also:positive See also:evidence. But it seems a See also:fair inference from a passage of See also:Michael See also:Psellus (Diophantus, ed. P. Tannery, ii. p. 38) that he was not later than Anatolius, See also:bishop of See also:Laodicea from A.D. 27o, while he is not quoted by See also:Nicomachus (fl. c. A.D. 1oo), nor by See also:Theon of See also:Smyrna (c. A.D. 130), nor does Greek See also:arithmetic as represented by these authors and by See also:Iamblichus (end of 3rd century) show any trace of his See also: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 See also:hand he is quoted by Theon of Alexandria (who observed an See also:eclipse at Alexandria in A.D. 365); and his See also:work was the subject of a commentary by Theon's daughter See also:Hypatia (d. 415). The Arithmetica, the greatest See also:treatise on which the fame of Diophantus rests, purports to be in thirteen Books, but none of the Greek See also:MSS. which have survived contain more than six (though one has the same See also:text in seven Books). They contain, however, a fragment of a See also:separate See also:tract on Polygonal See also:Numbers. The missing books were apparently lost See also:early, for there is no See also:reason to suppose that the See also:Arabs who translated or commented on Diophantus ever had See also:access to more of the work than we now have. The difference in See also:form and content suggests that the Polygonal Numbers was not See also: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 See also: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 See also:solution of a See also:complete quadratic promised by Diophantus himself (I. def. 11), and really assumed later, was one of the Porisms. Among the See also: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 See also:lead to indeterminate equations of the third and See also:fourth degrees, an easy indeterminate See also:equation of the See also:sixth degree beingalso found. The See also: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, &c. E.g. To find three numbers such that the product of any two added to the sum of those two gives a square (III. 15, ed. Tannery) ; To find four numbers such that, if we take the square of their sum any one of them singly, all the resulting numbers are squares (III. 22) ; To find two numbers such that their product their sum gives a See also:cube (IV. 29) ; To find three squares such that their continued product added to any one of them gives a square (V. 21). Book VI. contains problems of finding rational right-angled triangles such that different functions of their parts (the sides and the See also:area) are squares. A word is necessary on Diophantus' notation. He has only one See also:symbol (written somewhat like a final sigma) for an unknown quantity, which he calls &.pr.Bµbs (defined as " an undefined number of See also:units ") ; the symbol may be a contraction of the initial letters ap, as AY, KY, DYE, &c., are for the See also:powers of the unknown (Suvaµes, square; rsujos, cube; SuvaµoSuvaµcs, fourth See also:power, &c.). The only other algebraical symbol is At for minus; plus being expressed by merely See also:writing terms one after another. With one symbol for an unknown, it will easily be understood what See also:scope there is foradroit assumptions, for the required numbers, of expressions in the one unknown which are at once seen to satisfy some of the conditions, leaving only one or two to be satisfied by the particular value of x to be determined. Often assumptions are made which lead to equations in x which cannot be solved " rationally," i.e. would give negative, surd or imaginary values;'Diophantus then traces how each See also:element of the equation has arisen, and formulates the See also:auxiliary problem of de. termining how the assumptions must be corrected so as to lead to an equation (in See also:place of the " impossible " one) which can be solved rationally. Sometimes his x has to do See also:duty twice, for different unknowns, in one problem. In general his See also:object is to reduce the final equation to a See also:simple one by making such an See also:assumption for the See also:side of the square or cube to which the expression in x is to be equal as will make the necessary number of coefficients vanish. The book is valuable also for the propositions in the theory of numbers, other than the "porisms," stated or assumed in it. Thus Diophantus knew that no number of the form 8n+7 can be the sum of three squares. 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