The Encyclopedia Americana (1920)/Ab′acus

AB′ACUS (Greek ἄβάξ, from the Semitic אבק‎‎, abq, dust). In mathematics, a term applied to several forms of reckoning apparatus, and hence for some centuries to arithmetic itself. The primitive form seems to have been a board covered with fine dust, whence the generic name. Among the Hindus this was a wooden tablet covered with pipe clay, upon which was sprinkled purple sand, the numerals being written with a stylus. (Consult Taylor, in the preface to his translation of the ‘Lilawati,’ Bombay 1816, p. 6). That this form was used by the ancient Greeks is evident from Iamblichus, who asserts that Pythagoras taught geometry as well as arithmetic upon an abacus. Its use among the Romans of the classical period is also well attested. Another form of the abacus, having many modifications, is a board with beads sliding in grooved or on wires in a frame. Herodotus tells us that this instrument was used by the Egyptians and the Greeks, and we have evidence that the Romans also knew it, although preferring a form described below. It is at present widely used in India and appears in the form of the swanpan in China, the saroban in Japan, and the tschoty in Russia, the latter being the same as the modern Arabian abacus. In its simplest form, the beads or counters are stored at one end of the frame and the computation is done at the other end by moving the correct number of beads over against that side of the frame. Usually on a decimal scale, the separate wires represent units, tens, hundreds, etc., progressively, but a duodecimal scale is also in use, and among the Chinese there is a separate division horizontally across the frame below which units are counted up to five, and the fives transferred to the upper section where each bead stands for five units. In parts of India where English money is used the wires on the abacus represent pence, shillings, pounds, tens of pounds, hundreds of pounds, etc., there being 11 beads on the first wire, 19 on the second, and 9 on each one above. It is in this type of the abacus that prayer beads have their origin. The third form is a ruled table, upon which counters are placed, somewhat like checkers on a backgammon board, a game derived from this type of abacus. This was the favorite form among the Romans, whose numerals were not at all adapted to calculation, and it maintained its position throughout the Middle Ages and until the latter part of the 16th century. The Hindu-Arabic numerals (see Numericals) having then supplanted the Roman, such an aid to calculation was thought superfluous in western Europe. The counters used were called ψήφοι by the Greeks, calculi (pebbles, whence calculare and our calculate) by the Romans, and in Cicero's time aera because brass discs were used. In mediæval times they were called projectiles because they were thrown upon the table, whence our expression to “cast an account,” and Shakespeare's “counter caster.” The early French translated this as gettons, gectoirs, and jetons, whence our obsolete English jettons and the modern French jeton, meaning a medal, and also a counter for games. The Germans translated the late Latin denarii supputarii (calculating pennies) as Rechenpfennife, the early printed books distinguishing between reckoning on the line (that is, on the ruled table) and with the pen. The Court of the Exchequer (q. v.) derives its name from this form of the abacus, about which the judges of the fiscal court sat. (Hall, ‘The Antiquities and Curiosities of the Exchequer,’ London 1891; Henderson, ‘Select Historical Documents of the Middle Ages,’ London 1892, p. 20.) Another form of the abacus, possible introduced by Gerbert before he became Pope Sylvester II (q. v.), was arranged in columns and employed counters upon which the western Arab forms of the Hindu numerals (see Numerals) were written. The use of the term to designate an instrument of calculation led to its use for arithmetic itself, as in the ‘Liber abaci’ of Leonardo Fibonacci of Pisa (q. v.) and in the works of later writers.

Consult Knott, ‘The Abacus’ (in the ‘Transactions of the Asiatic Society of Japan,’ Vol. XIV); Bayley, in the Journal of the Royal Asiatic Society (N. S., Vol. XV); Chasles, in the Comptes rendus, t. 16, 1843, p. 1409; Woepcke, in the Journal asiatique, 6 ser., t. I. See Finger Notation.

David Eugene Smith, Professor of Mathematics, Teachers College, Columbia University, New York.