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* DIOPHANTINE ANALYSIS. 268 DIOSCOREACE^. the use of syiitlu'sis (q.v.). (8) He showed skill in the iiitroduelion of urhitrary eoiuiilions: e. g. * two numbers bein},' sought such that the cube of one is greater by 2 tluin the square of the other. Diophantus arbitrarily assumes that the numbers are x + 1, j; — I. The strength of this analysis does not consist in the elegance and deli- niteness of the mclliod, but in the consummate skill with which Diophantus makes usf of the above devices. Diophantus's name is also associ- ated with the following theorem: The sum of the squares of any three integers can never be ex- pressed as ihcsum of two such squares. Fermat first proved this theorem and added the corol- lary: It is impossible that any multii)le of a prime of the form 4»i — 1, by a number prime to it, can either be a square or the sum of two squares, integral or fractional. DIOPHANTINE THEOREM, See Dio- IMIA.NIIM; An.m.vms. DIOPHAN'TUS (Lat., from Gk. Ai6<pav-ot). A distinguished Greek mathematician, who lived at Alexandria in all probability in the second half of the third century. He is called the Father of Algebra, although he speaks of that science as known before his time. It is not probable that he knew anything of the Hindu algebra, although the Orientals were quite early familiar with his work. He occupied himself chielly with a class of problems discussed under DioriiANTiKE Analysis (q.v,). Of his three works, (1) on arUhmetic, (2) on polygonal numberx, (3) the porisms, the Arilhmclicii consisted origi- nally of thirteen books, only si. of which are extant. The polygonal numbers may have been one book of the Arithmetica. and the Poris-mata, which is lost, may also have been part of this greater work. The best-known editions of Dio- phantus's v.orks are those of Bachet de iU'zeriac (Paris, 1021) : Scliultz (Berlin, 1822) : and Tan- nery (Leipzig, ]8!»;M)5), Consult: Gow, Iliitory of (Ireck Mathematirx (Cambridge, 1884); Nes- selmann, (Irschichtc der Algchra der Oriechen (Berlin, 1842): Heath, Diophantos of Alexan- dria (London, 1885), DIOPSIDE (Gk. J/oi^ir, rfiop.sis, view through, from Sid, <iin. through + f^pic, opsis. view). A variety of pyro.xene that crystallizes in the mono- clinic "system, and is essentially a calcium magne- sium silicate. It is usually white or light gray, yellow, or green in color, although sometimes, owing to the presence of iron, it is dark green and nearly black. Diopside is a common mineral occurring in crystalline limestone and dolomite, in serpentine or igneous rocks. Transparent, oily, green crystals, especially those from Saint Lawrence County, N. Y., have been found sulTi- cientlv large and clear to be cut into gems weigh- ing from (i to 8 carats each. A variety contain- ing chromium that is deep emerald green in color has been foniiil of suOicient size for small gems. DIOPTOMETER (Lat. dioplrn. Gk. SilnrTpa, leveling instrument, from 5i<£. 'liii. through + dxT--, opt-, visual + pirpnv. mrlrnn. measure). An instrument used by oculists and opticians in measuring directly (he curature and focal length of glasses used to correct defective vision. It is based on the principle that the focal length of lenses of the glass usiinlly employed is very nearly equivalent to one half the sums of the radii of curvature. This follows from the for- mula that -.= (ii — 1) ~ -- - .where / is the prin- cipal focus, IX is the index of refraction, and r and s arc the radii of curvature of the two sur- faces. As fi is approximately 1.5. it will be seen readily how easily the focal length can be obtained wlicn once the two radii of curvature are known. The dioplomcter is a simple arrange- ment of levers by means of which the curvature of any spherical surface is indicated, and the ■ result shown in diopters. If the lens is plane, convex, or concave, the focal length is obtained directly by inspection, but if both surfaces of the lens are curved, the algebraical sum of the two readings must be taken. DIOPTER, DIOPTRIC SCALE, The unit ado]ited by ophthalmologists in measuring the strengtli of lenses used in the correction of de- fective vision, A concave or convex lens with a focal length of one meter is taken as the unit, and the reciprocals of the focal lengths supply the numbers which are assigned to the len>es. Thus a lens with a focal length of .5 meter would be designated as Number 2, and one with a focal length of .1 would be a Number 10. The measures are the same whether the lenses are positive (convex) or negative (concave), luit in the former case the + sign is prefixed to the number, while in the latter it is the — sign. DIOPTRICS. That branch of optics which deals uitli tlic refraction of light. See Light. DI'ORITE (from Gk. Siopfjtic, diorizein. to distinguish, from Sid, din. llirough 4- ApffttK horizcin, to divide, from ipoc . hurus, boundary). An igneous rock of granitic texture, the essential constituent minerals of which are a lime-soda feldspar and either hornblende or dark mica, or both of these minerals. Tiic average clu'niical composition of diorite is about as follows: Silica, Gl per cent.; alumina. Hi per cent.; ferric oxide, 3 per cent.; ferrous oxide, 5 per cent.; magnesia, 4 per cent.; lime, G per cent.: soda, 3 per cent.; potash, 2 per cent. Many rocks formerly classi- fied as diorites arc now by many writers sepa- rated because of i)eculiarities of texture and of manner of occurrence; e.g. kersanfite, camptim- tite, aphanite, etc. In addition to lime-soda feld- spar, hornblende, or mica, diorites m.ay contain quartz, alkali feldspar, or augite, and other minerals in smaller quantities, .ccording to the presence or absence of quartz, diorites are classi- fied as quartz diorites or diorites proper. Each of these groups is subdivided into mica diorites, hornblende diorites, or augite diorites, according as mica, hornblende, or augite predominates among the darker minerals. Peculiarities of tex- ture alone serve to di-tingiiish quartz diorite4 from dacites (q.v.), and diorites from andesltes (q.v.), the composition being essentially the same. Chemically, diorites dilTer from granites chiefly in containing less silica and potash and more lime, magnesia, and iron. Mincralogically, they are distingtiished by the greater alomdance of lime-soda feldspar and by the darker color, due to the greater abundance of mica and horn- blende, lliorite is often quarried for use as a buildinL' stone. DI OSCO'REA'CEiE (Neo-T,at. nom. pi. named after the Greek physician and botanint Dio.seorides). . order of monoeotyledonous