protection of trees generally (according to Pherecydes in C. W. Müller, Frag. Hist. Graec. iv. p. 637, the word νῦσα signified “tree”). It is suggested that the cult of Dionysus absorbed that of an old tree-spirit. He was figured also, like Hermes, in the form of a pillar or term surmounted by his head. For the connexion of Dionysus with Greek tragedy see Drama.
See Farnell, Cults of the Greek States, v. (1910); also O. Rapp, Beziehungen des Dionysuskultus zu Thrakien (1882); O. Ribbeck, Anfänge und Entwickelung des Dionysuskultes in Attica (1869); A. Lang, Myth, Ritual and Religion, ii. p. 241; L. Dyer, The Gods in Greece (1891); J. E. Harrison, Prolegomena to the Study of Greek Religion (1903); J. G. Frazer, The Golden Bough, ii (1900), pp. 160, 291, who regards the bull and goat form of Dionysus as expressions of his proper character as a deity of vegetation; F. A. Voigt in Roscher’s Lexikon der Mythologie; L. Preller, Griechische Mythologie (4th ed. by C. Robert); F. Lenormant (s.v. “Bacchus”) in Daremberg and Saglio’s Dictionnaire des antiquités; O. Kern in Pauly-Wissowa’s Realencyclopädie (with list of cult titles); W. Pater, Greek Studies (1895); E. Rohde, Psyche, ii., who finds the origin of the Hellenic belief in the immortality of the soul in the “enthusiastic” rites of the Thracian Dionysus, which lifted persons out of themselves, and exalted them to a fancied equality with the gods; O. Gruppe, Griechische Mythologie und Religionsgeschichte, ii. (1907), who considers Boeotia, not Thrace, to have been the original home of Dionysus; P. Foucart, “Le Culte de Dionysos en Attique” in Mémoires de l’Institut national de France, xxxvii. (1906), who finds the prototype of Dionysus in Egypt. The Great Dionysiak Myth (1877–1878) by R. Brown contains a wealth of material, but is weak in scholarship. For a striking survival of Dionysiac rites in Thrace (Bizye), see Dawkins, in J.H.S. (1906), p. 191.
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.”
Authorities.—The first to publish anything on Diophantus in Europe was Rafael Bombelli, who embodied in his Algebra (1572) all the problems of Books I.–IV. and some of Book V. interspersing them with his own problems. Next Xylander (Wilhelm Holzmann) published a Latin translation (Basel, 1575), an altogether meritorious work, especially having regard to the difficulties he had with the text of his MS. The Greek text was first edited by C. G. Bachet (Diophanti Alexandrini arithmeticorum libri sex, et de numeris multangulis liber unus, nunc primum graece et latine editi atque absolutissimis commentariis illustrati . . . Lutetiae Parisiorum . . . MDCXXI.). A reprint of 1670 is only valuable because it contains P. de Fermat’s notes; as far as the Greek text is concerned it is much inferior to the other. There are two German translations, one by Otto Schulz (1822) and the other by G. Wertheim (Leipzig, 1890), and an English edition in modern notation (T. L. Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra (Cambridge, 1885)). The Greek text has now been definitively edited (with Latin translation, Scholia, &c.) by P. Tannery (Teubner, vol. i., 1893; vol. ii., 1895). General accounts of Diophantus’ work are to be found in H. Hankel and M. Cantor’s histories of mathematics, and more elaborate analyses are those of Nesselmann (Die Algebra der Griechen, Berlin, 1842) and G. Loria (Le Scienze esatte nell’ antica Grecia, libro v., Modena, 1902, pp. 95-158). (T. L. H.)
DIOPSIDE, an important member of the pyroxene group of rock-forming minerals. It is a calcium-magnesium metasilicate, CaMg(SiO3)2, and crystallizes in the monoclinic system. Usually some iron is present replacing magnesium, and when this predominates there is a passage to hedenbergite, CaFe(SiO3)2, a closely allied variety of monoclinic pyroxene. These are distinguished from augite by containing little or no aluminium. Diopside is colourless, white, pale green to dark green or nearly black in colour, the depth of the colour depending on the amount of iron present. The specific gravity and optical constants also vary with the chemical composition; the sp. gr. of diopside is 3·2, increasing to 3·6 in hedenbergite, and the angle of optical extinction in the plane of symmetry varies between 38° and 47° in the two extremes of the series. Crystals are usually prismatic in habit with a rectangular cross-section as shown in the figure: the angle between the prism faces m, parallel to which there are perfect cleavages, is 92° 50′.
