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* DIONYSItrS. •2<i7 DIOPHANTINE ANALYSIS. improving him by putting him under the instruc- tion of Plato, but the influence of the liistorian Philistus proved to !» stronger than that of Plato; Dion was sent into exile, Plato was dis- missed, and Dionysius encouraged to pursue a course of pleasure. Afterwards Plato was re- called to Syracuse, with even worse results. Jn B.C. 3.5t; Dion returned to Syracuse at the head of a force of mercenaries, and Dionysius was obliged to leave Sicily. He retired to Locri in, Italy, the birthplace of his mother, Doris, and ruled there despotically until n.c. 34, when he once more obtained possession of the Government at Syracuse. His rule was so oppressive that the Syraeusans finally sought aid of Corinth, and Timoleon was sent to their assistance. Diony- sius was shut up in the citadel of Syracuse and compelled to surrender B.C. 343. He was taken to Corinth, where he remained till his death. DIONYSIUS, E.<.R OF, The name given to one of the Lantumiae or quarries of Syracuse, said to have been constructed by Dionysius the Elder. The cavern is so arranged that all sounds in it converged to one point, where Dionysius was said to have a secret chamber, in which he could overhear the couAersation of the prisoners confijicd in the quarries. Though the cavern is partially tilled up. it still possesses remarkable acoustic properties. DIONYSIUS. Saixt, Pope, 2.59-69, He was a (Jrcck liy l>irth, and must have been among the most distinguished members of the Church in Rome, even before his elevation, as his dis- tinguished namesake of Alexandria addressed to him the fourth of his letters on baptism: after- wards the teaching of the latter on the Trinity was reviewed by him in a synod held in 262, and was the occasion of a notable letter to the Egyptian churches. His day in the calendar is December 26. DIONYSIUS EXIG'UUS (Lat.. Dionysius the I.ittlc I. A learned monastic who lived in the fifth and sixth centuries. The epithet of Exiijuus is merely a term of humility, used by Dionysius himself. Cassiodorus (q.v. ), who was his fellow pupil, says that Dionysius was "a ScMhian by birth, but wholly Roman in other respects." He made a collection containing ( 1 ) the first fifty canons of the Apostles; (2) the canons of ten councils ; ( 3 ) thirty-eight decretals of the popes, from 384 to 498. This collection, of which there were two redactions, had great au- thority in the West. Somewhat changed, it was solemnly promulgated by Charles the Great in "02. (See Caxox Law.") Dionysius completed the paschal cycle and introduced the custom of counting the years from the birth of Christ. (See CirRoxoLoCY. ) He died before aon. Various works of Dionysius are printed in Migne's Pntro- loffiw iMtinae Cursus Completus. vol. Ixvii. But this, as well as the earlier editions, is uncritical and incomplete. Consult Tardif. Ilintoire des sourriH liii droit canoniqup (Paris, 1887). DIONYSIUS PERIEGETES (Lat., from Gk. AionViof IleptiryTTTijf ). A Greek geographer, .luthor of a (Jreck poem in 11S7 hexameters, en- titled T^f T^r OUnvfihijc Jlipiriyriaic. A Dritcrip- tion of the Bahilnhle Worhl. The time and place of his birth are not known. It is probable, however, that he was a native of Vol. vr — 18. Charax, in Susiana, and lived in the latter part of the third or the beginning of the fourth cen- tury, A.D. llis work enjoyed a wide popularity in ancient times, and was translated into Latin by Rufus Kestus Avienus and by Priscianus. A (ireek i)araphrase and scholia and a valuable com- mentary upon it by Eust.ichius are still extjint. I lie best editions of the Perivgesis are those hy Bcrnhardy in Georjraphi Vitrei Minores vol i (Leipzig, 1828); and bv Miiller, in Gcographi Gidci M mores, vol. ii. (Paris, 1861). DIONY'SUS. See Bacchus. DIONYSUS, Theatre of. The oldest known theatre in the world, on the slope of the Acropolis at Athens. It has been the scene of excavations by the German Archteologieal Institute since 1886. The results of the researches show that, contrary to previously accepted belief, no perma- nent building for dramatic representations had existed on the spot until about B.C. 330, when Lycurgus erected a wall with wings to support temporary wooden decorations, and stone seats were jirovided for the audience. Later changes were made in the stage, but the existing ruTna still leave the vexed question of its arrangement in doubt. DI'ONY'ZA. The wife of Cleon, Governor of Tharsus, in Shakespeare's Pericles. In the epilogue, it is announced that she, her husband, and her house have been burned by the populace, angry at her attempt upon the child of Pericles. DIOON, dl-f)'6n. See Ctcadace.e. DIOPHAN'TINE ANALYSIS. That por- tion of the theory of indeterminate problems which seeks rational and commensurable roots of equations involving the squares and cubes of the unknown quautitj'. This class of problems was first and chiefly treated by Diophantus, whose name the method bears. Although Dio- phantus gave problems involving the indetermi- nate linear equations, he was chiefly interested in solving equations of the second degree: e. g. to find two whole numbers, the sum of whose squares is a square; or to find three square numbers in arithmetic progression. The method of Diophantus, being neither concise nor definite, does not admit of a brief exposition. The follow- ing are, however, some of its characteristics: (1) His quantitative sj-mbolism was limited to that of one unknown. (2) His rules of operation are the common axioms of adding to and sub- tracting from both members of an equation. (3) He showed much adroitness in selecting the un- known ; e.g. so as to avoid affected quadratics or complete cubies. (4) He also showed skill in his tentative assumptions, in assigning sepa- rately to the unknown preliminary values which satisfy one or two only of the necessarv' condi- tions: then, from its failure to satisfy" the re- maining conditions, he discovered the' required number, a method somewhat analogous to the regiila falsi (q.v.). (5) He used the symbol for the unknown in different senses, which amounted to the same thing as substituting one unknown for a fimction of another. (6) He made some use of limits: e. g. to find a square between 10 and 11, Diophantus proceeds thus: Ko square lies between 40 and 44. nor between 90 and 99, but 169 lies between 160 and 176. These numbers being 16 times 10 and 11, respectivelv, the square between 10 and 11 is Ye"- (7) He developed