Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/584

Rh 524 A R I A R I APISTOXENUS, of Tarentum, a celebrated Greek philosopher and writer on music, was the son of Spintharus or Mnesias. He was at first a pupil of the Pythagorean school, and received instruction from Xenophilus; afterwards re moving to Athens, he studied under Aristotle. He became one of the most distinguished pupils of the Peripatetic school, and is said to have been much disappointed when, after the death of Aristotle, Theophrastus was preferred to him as scholarch. His writings, which, according to Suidas, were 453 in number, have been almost entirely lost. The titles of some of them have been preserved, and show that his activity had been directed to a great variety of topics. With the exception of a few fragments quoted by other authors, there is extant of all his writings only one treatise on Harmony, in three books, which is probably not a complete work, but made up of portions of one or more separate writings. His doctrine of harmony is directly opposed to that of the Pythagoreans, according to whom musical concord depended upon certain numerical ratios, and who were obliged to reject some combinations as dissonant, only because there was no ratio corresponding to them. This theory Aristoxenus considered to Le an attempt to force a priori determinations upon nature, and he sought to develop a theory of harmony from an empirical basis. According to him, the ear is the true judge of concord, and its impressions can be generalised into rules. His followers were called /AOVCTIKOI, or musicians by ear, in opposition to the Pythagoreans, who were Ka.vovi.Kot, or musicians by rule. Another doctrine attributed to Aris toxenus brings out forcibly the strong empirical tendency of his mind. He is said to have held that the soul stood in the same relation to the parts of the body as harmony stands to the parts of a musical instrument; it was the result of organisation. What proofs he advanced in favour of this view, and how the opinion was connected with his general system of thought, we have not now the means of determining. The best edition of Aristoxenus is by Marquard, with German translation, and full commentary, Arisfoxenus harmoniscJie Frag- mcnte, 1868. The fragments are also given in Miiller, Frag. Hist. Grcec., ii. 2(39, sqq. See also Mahne s work, Diatribe de Aristoxeno, 1793; and that of Brill, Aristoxenus rhythmische und metrische Messungcn, 1871. A RITHMETIC is the science that treats of numbers, JLJL and of the methods of computing by means of them. In introducing the subject, and endeavouring to trace the progress of the science, there appear to be three points that call for particular notice, viz., the conception of number, the representation of numbers either by words or graphically by characters, and the principles and modes of computation. 1. The primary conceptions of numbers are necessarily of a very crude kind. The child attains the notion slowly by experience, and the ability that even adults have to apprehend the significance of numbers with precision is re stricted to an extremely narrow area. This is still more the case among uncivilised races, some of which do not appear to be able to count beyond 3, or 4, or 5, or are at least believed to have no words in their vocabularies that express larger numbers. It is to be remembered that the knowledge that is acquired regarding numbers through experience and culture is not of numbers absolutely or in the abstract, but rests almost entirely on a perception of the relations which numbers bear to each other. The power to form a direct and immediate conception of numbers is very limited; but the relative magnitudes of numbers, large as well as small, can be expressed with the utmost accuracy, and so as to .be clearly understood. The system of notation in common use, whereby we express not merely numbers, but parts of numbers, supplies us with means of comparing arithmetically together either the greatest or the minutest magnitudes, to which there is absolutely no limit. The proportion, for instance, that the circumference of a circle bears to the diameter, though it cannot be stated with arithmetical exactness, has been cal culated to upwards of 200 decimal places, a nicety for which the vast dimensions that science discloses in the physical universe furnish no means of comparison what ever. For let it be supposed that a circle were described, with a point on the earth s surface as centre, so as to extend beyond the most distant star that can be discerned by the most powerful existing telescope, that the radius of the circle were known, and that the circumference were computed from it, it does not appear that an error in the thirtieth decimal place would be of such magnitude that the keenest vision, aided by the most powerful microscope, could detect it. In all systems of number, with the exception, perhaps, of the very rudest, numerical conceptions are aided by the introduction, usually at a very early stage, of methods of grouping. In nearly every case the methods adopted con nect themselves with the number of the fingers, either of one or, more usually, of the two hands. Having reached 5 or 10, the reckoner proceeds by adding to these the prior numbers; and when a second 5 or 10 is reached, a new word or sign is employed, the significance or derivation of which is generally well marked in the name or form it bears. Similarly when (say) five fives or ten tens are reached, a fresh start is made. These processes of grouping are of great importance, conveying clearer conceptions than could otherwise be obtained of the relative magnitudes of numbers. An additional evidence of the value of the prin ciple is to be found in certain numerical combinations which are not additive, as such combinations mostly are, but subtract! ve. Thus, if a conception be formed of 10, and again of 20, 19 and even 18 will connect themselves more readily with the latter than with the former, and so we have such forms as duodeviginti and undeviyinti; and there is little doubt that similar considerations, in combina tion with a regard for brevity of expression, have led to the use of the subtractive forms IX., XL., &c. 2. There are two ways in which numbers are represented, either in words or by particular characters or symbols. It is with the latter that arithmetic has more especially to do, the numerous important and interesting questions that relate to word-numerals falling rather within the domain of language. It is only in so far as they arise out of the systems of grouping already referred to, suggesting in their formation such processes as addition or multiplication, that these call for any notice here. Dr Tylor (Primitive Cul ture, chap, vii.) gives a variety of remarkable collocations of this sort, as well as of descriptive numerals, that are in use among different races and tribes. There is also a common use of characters to represent numbers which docs not properly belong to this subject. The letters A, B, C, &c., or a, /?, y, or the like, the order of whose succession is fixed, are often employed to indicate numerical order, either singly or in combination with numerals proper. This is not, however, an arithmetical use of the characters; they are merely ordinals, and cannot furnish a basis for calculation. The origin of the various characters which have been employed to indicate numbers proper, and by mean of