ARISTOXENUS, of Tarentum (4th century B.C.), a Greek peripatetic philosopher, and writer on music and rhythm. He was taught first by his father Spintharus, a pupil of Socrates, and later by the Pythagoreans, Lamprus of Erythrae and Xenophilus, from whom he learned the theory of music. Finally he studied under Aristotle at Athens, and was deeply annoyed, it is said, when Theophrastus was appointed head of the school on Aristotle’s death. His writings, said to have numbered four hundred and fifty-three, were in the style of Aristotle, and dealt with philosophy, ethics and music. The empirical tendency of his thought is shown in his theory that the soul is related to the body as harmony to the parts of a musical instrument. We have no evidence as to the method by which he deduced this theory (cf. T. Gomperz, Greek Thinkers, Eng. trans. 1905, vol. iii. p. 43). In music he held that the notes of the scale are to be judged, not as the Pythagoreans held, by mathematical ratio, but by the ear. The only work of his that has come down to us is the three books of the Elements of Harmony (ῥυθμικὰ στοιχεῖα), an incomplete musical treatise. Grenfell and Hunt’s Oxyrhynchus Papyri (vol. i., 1898) contains a five-column fragment of a treatise on metre, probably this treatise of Aristoxenus.
The best edition is by Paul Marquard, with German translation and full commentary, Die harmonischen Fragmente des Aristoxenus (Berlin, 1868). The fragments are also given in C.W. Müller, Frag. Hist. Graec., ii. 269 sqq.; and R. Westphal, Melik und Rhythmik d. klass. Hellenenthums (2nd vol. edited by F. Saran, Leipzig, 1893). Eng. trans. by H.S. Macran (Oxford, 1902). See also W.L. Mahne, Diatribe de Aristoxeno (Amsterdam, 1793); B. Brill, Aristoxenus’ rhythmische und metrische Messungen (1871); R. Westphal, Griechische Rhythmik und Harmonik (Leipzig, 1867); L. Laloy, Aristoxène de Tarente et la musique de l’antiquité (Paris, 1904); See [Peripatetics], [Pythagoras] (Music) and art. “Greek Music” in Grove’s Dict. of Music (1904). For the Oxyrhynchus fragment see Classical Review (January 1898), and C. van Jan in Bursian’s Jahresbericht, civ. (1901).
ARISUGAWA, the name of one of the royal families of Japan, going back to the seventh son of the mikado Go-Yozei (d. 1638). After the revolution of 1868, when the mikado Mutsu-hito was restored, his uncle, Prince Taruhito Arisugawa (1835-1895), became commander-in-chief, and in 1875 president of the senate. After his suppression of the Satsuma rebellion he was made a field-marshal, and he was chief of the staff in the war with China (1894-95). His younger brother, Prince Takehito Arisugawa (b. 1862), was from 1879 to 1882 in the British navy, serving in the Channel Squadron, and studied at the Naval College, Greenwich. In the Chino-Japanese War of 1894-95 he was in command of a cruiser, and subsequently became admiral-superintendent at Yokosuka. Prince Arisugawa represented Japan in England together with Marquis Ito at the Diamond Jubilee (1897), and in 1905 was again received there as the king’s guest.
ARITHMETIC (Gr. ἀριθμητική, sc. τέχνη, the art of counting, from ἀριθμός, number), the art of dealing with numerical quantities in their numerical relations.
1. Arithmetic is usually divided into Abstract Arithmetic and Concrete Arithmetic, the former dealing with numbers and the latter with concrete objects. This distinction, however, might be misleading. In stating that the sum of 11d. and 9d. is 1s. 8d. we do not mean that nine pennies when added to eleven pennies produce a shilling and eight pennies. The sum of money corresponding to 11d. may in fact be made up of coins in several different ways, so that the symbol “11d.” cannot be taken as denoting any definite concrete objects. The arithmetical fact is that 11 and 9 may be regrouped as 12 and 8, and the statement “11d. + 9d. = 1s. 8d.” is only an arithmetical statement in so far as each of the three expressions denotes a numerical quantity (§ 11).
2. The various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view. There are three main purposes, the practical, the educational, and the scientific; i.e. the subject may be studied with a view to technical skill in dealing with the arithmetical problems that arise in actual life, or for the sake of its general influence on mental development, or as an elementary stage in mathematical study.
3. The practical aspect is an important one. The daily activities of the great mass of the adult population, in countries where commodities are sold at definite prices for definite quantities, include calculations which have often to be performed rapidly, on data orally given, and leading in general to results which can only be approximate; and almost every branch of manufacture or commerce has its own range of applications of arithmetic. Arithmetic as a school subject has been largely regarded from this point of view.
4. From the educational point of view, the value of arithmetic has usually been regarded as consisting in the stress it lays on accuracy. This aspect of the matter, however, belongs mainly to the period when arithmetic was studied almost entirely for commercial purposes; and even then accuracy was not found always to harmonize with actuality. The development of physical science has tended to emphasize an exactly opposite aspect, viz. the impossibility, outside a certain limited range of subjects, of ever obtaining absolute accuracy, and the consequent importance of not wasting time in attempting to obtain results beyond a certain degree of approximation.