Page:EB1911 - Volume 08.djvu/305

 protection of trees generally (according to Pherecydes in C. W. Müller, Frag. Hist. Graec. iv. p. 637, the word  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.

 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 270, while he is not quoted by Nicomachus (fl. c.  100), nor by Theon of Smyrna (c.  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 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.

 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′.