Aristoxenus
Collier's New Encyclopedia (1921)
an ancient Greek musician and philosopher of Tarentum, born about b. c. 324. He studied music under his father Mnesias, and philosophy under Aristotle, whose successor he aspired to be. He endeavored to apply his musical knowledge to philosophy, and especially to the science of mind. We have a work on the "Elements of Harmony" by him. ARITHMETIC. Viewed as a science, arithmetic is a branch of mathematics; looked on as an art, its object is to carry out for practical purposes certain rules regarding numbers, without troubling itself to investigate the foundation on which those rules are based. It is variously divided, as into integral and fractional arithmetic, the former treating of integers and the latter of fractions. Integral arithmetic is sometimes called vulgar or common arithmetic; and from fractional arithmetic is sometimes separated decimal arithmetic, treating, as the name implies, of decimals. There are also logarithmic arithmetic for computation by logarithms, and instrulog mental arithmetic for calculation by means of instruments or machines. Another division is into theoretical arithmetic, treating of the science of numbers, and practical arithmetic, which points out the best method of practically working questions or sums. Political arithmetic is arithmetic applied to political economy, as is done in the statistical re- ARITHMETICAL COMPLEMENT 250 turns so continually presented to Parliament or Congress. Finally, universal arithmetic is a name sometimes applied to algebra. The chief subjects generally treated under the science or art of arithmetic are: (1) numeration and notation; (2) addition; n; (3) subtraction; (4) mul- (1067-1148). He was the first Icelander tiplication; (5) division; (6) reduction; to use his mother tongue as a literary compound addition (8) (8) compound compound ; (9) compound multiplicasubtraction tion; (10) compound division; (11) simple proportion (rule of three); (12) compound proportion; (13) vulgar fractions; (14) decimal fractions; (15) duodecimals; (16) involution; (17) evolution; (18) ratios, proportions, and progressions; (19) fellowship or partnership; (20) simple interest; (21) compound interest, and (22) position. Of these, the most important are the simple processes of addition, subtraction, multiplication and division, the judicious use of which, singly or in combination, will solve the most complex arithmetical questions.
What Has Changed in the Last Century
Collier's date of "about b.c. 324" for Aristoxenus's birth is now read as a garbled floruit: scholars place his birth at Tarentum somewhere around 375-360 BC, which is what allows him to have studied under Aristotle at all. The Elements of Harmony is still his surviving monument, but modern readers also have substantial fragments of his Elements of Rhythm and hundreds of scraps of his lost biographical and ethical works, gathered by Fritz Wehrli in Die Schule des Aristoteles (1945) and edited afresh since. The largest shift is in how he is valued: where he was once a curiosity, his insistence that the trained ear, not Pythagorean number-ratios, is the judge of intervals now makes him the central figure in the history of Greek music theory, helped by the recovery of actual notated Greek music such as the Delphic hymns found in the 1893 excavations. Collier's remark that he "aspired to be" Aristotle's successor and was passed over for Theophrastus still stands, as does the account of his attempt to bring music to bear on the theory of mind.
Written for this edition, 2026.