1911 Encyclopædia Britannica/Diophantus
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 (Diophantus, ed. P. Tannery, ii. p. 38) that he was not later than Anatolius, bishop of Laodicea from A.D. 270, while he is not quoted by Nicomachus (fl. c. A.D. 100), nor by Theon of Smyrna (c. A.D. 130), nor does Greek arithmetic as represented by these authors and by Iamblichus (end of 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 A.D. 365); and his work was the subject of a commentary by Theon’s daughter Hypatia (d. 415). The Arithmetica, the greatest treatise on which the fame of Diophantus rests, purports to be in thirteen Books, but none of the Greek MSS. 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 (I. def. 11), 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, &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 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 area) are squares. A word is necessary on Diophantus’ notation. He has only one symbol (written somewhat like a final sigma) for an unknown quantity, which he calls ἀριθμός (defined as “an undefined number of units”); the symbol may be a contraction of the initial letters αρ, as ΔΥ, ΚΥ, ΔΥΔ, &c., are for the powers of the unknown (δύναμις, square; κύβος, cube; δυναμοδύναμις, fourth power, &c.). The only other algebraical symbol is for minus; plus being expressed by merely writing terms one after another. With one symbol for an unknown, it will easily be understood what scope there is for adroit 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 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. Sometimes his x has to do duty twice, for different unknowns, in one problem. In general his object is to reduce the final equation to a simple one by making such an assumption for the 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. He also says that, if 2n + 1 is to be the sum of two squares, “n must not be odd” (i.e. no number of the form 4n + 3, or 4n - 1, can be the sum of two squares), and goes on to add, practically, the condition stated by Fermat, “and the double of it [n] increased by one, when divided by the greatest square which measures it, must not be divisible by a prime number of the form 4n - 1,” except for the omission of the words “when divided . . . measures it.”