Abacus

ABACUS, an early device for tracing numerals on a dust covered board. There have been three standard types: (1) a board covered with dust, whence the derivation, or, at a later period. with wax, and serving the same purpose as a slate in the 19th century or as the modern pad of paper; (2) a table marked with lines upon or between which were placed loose counters in the form of small discs of metal, bone, glass or other material, or (in China) in the form of rods; and (3) a table on which the counters were fastened by means of grooves.

The early counter abacus was a table with lines to represent units, tens, hundreds, etc., or to represent different units of value like pounds, shillings and pence. Addition on such an abacus was doubtless performed as represented in fig. 1, which shows the successive steps in the addition of 64 and 239.

FROM SMITH. "HISTORY OF MATHEMATICS" Fig 1--A PROBABLE FORM OF EARLY COMPUTATION The lines represent hundreds, tens and units. The computation represented is: 239+64=29[13]=2[10]3=303

Subtraction was simply the process of taking away counters; multiplication was considered as repeated addition, and division as repeated subtraction. This type of abacus was known in the Mediterranean countries, and Herodotus (c. 450 B.C.) is authority for the statement that the Egyptians wrote their figures and reckoned with pebbles "bringing the hand from right to left," while the Greeks proceeded in the opposite direction. There are several references to the abacus in Roman literature, and what is apparently a Greek computing table was found in the 19th century on the island of Salamis.

Latin writers tell of three types of abacus in use in Rome, namely: (1) the sand board or the wax tablet, (2) a marked table for counters, and (3) a table with grooves in which the counters were free to slide. Fig. 2 shows a late Roman abacus now in the British museum, each upper button representing five units of the order in which the column stands, and each lower button representing one unit of the same order. Cicero (Phil. Frag. V. 59 speaks of the counters as aera (bronzes), but the common name was calculi (pebbles) or abaculi the pieces being made of stone ivory (Juvenal xi, 131) metal or coloured glass (Pliny. Hist Nat. xxxvi 26, 67)

FROM SMITH. "HISTORY OF MATHEMATICS" Fig. 2--A late Roman abacus, whose source is unknown. This bronze piece of uncertain date has grooves in which the counters slide. Each upper button represents five units and each lower button one unit of the order in which the column stands

The earliest type of abacus in China seems to have been the "bamboo rods" that served instead of counters. These were known as early as the 6th century B.C. and they survived in Korea until the close of the 19th century (fig. 3).

FROM SMITH. "HISTORY OF MATHEMATICS" Fig. 3.--Korean computing rods. Made of bone, these rods were used in the native Korean schools until the end of the 19th century

They found their way into Japan about the year 600 and were known as sangi or sanchu Until recent times they were used to represent algebraic coefficients, being placed on a board ruled as shown in fig. 4.

FROM SMITH. "HISTORY OF MATHEMATICS" Fig. 4--A Japanese ruled "Sangi" board. Upon this board the computing rods ("Sangi") were placed to represent algebraic coefficients

Since the 12th century the suan-pan (computing tray; see fig. 5) has been generally used throughout China. The chief difference between this and the Roman abacus lies in the fact that it has one more bead in each section.

FROM SMITH. "HISTORY OF MATHEMATICS" Fig. 5.--A modern Chinese abacus. The "suan-pan" was used as early as the 12th century. The number represented is 27091. Each upper bead has a value of five of its order and each lower bead the value of one of its order. Often employed in Chinese laundries

In the 16th century this type, slightly changed and bearing the kindred name of soroban (fig. 6), found its way into Japan, where it is still in use.

FROM SMITH. "HISTORY OF MATHEMATICS" Fig. 6.--A Japanese abacus. The "soroban" was in use as early as the 16th century. The number represented in the central part is 90278

An abacus differing considerably from the Roman or oriental types is found in the middle eastern countries. The Turks call it the coulba the Armenians, the choreb and the Russians, the s'choty (fig. 7) As in the case of the suan-pan and the sorohan, this permits of rapid computation and serves a purpose similar to that of the modern calculating machines ( q.v. ).

FROM SMITH. "HISTORY OF MATHEMATICS" FIG. 7.--A RUSSIAN ABACUS The "s'choty" of the R u s s i a n s "choreb" of the Armenians or "coulba" of the Turks. The wires are slightly arched, a feature which is not shown in the illustration

There appeared in Europe in the middle ages the line abacus. This type of abacus consisted of a table ruled with horizontal lines representing the successive powers of ten, each space representing half the value of the line immediately above it (fig. 8).

FROM SMITH. "HISTORY OF MATHEMATICS" FIG. 8.-- MEDIAEVAL COMPUTING TABLE This method of ruling the computing board was extensively used in the middle ages

This type was in use until well into the 18th century. A pupil who could compute on such a table was said to "know the lines," and the verb "to abacus" was used as the equivalent of the verb "to compute." The method employed may be inferred from a passage in Robert Recorde's Ground of Artes (about 1542) as shown in fig 9.

FIG. 9 --A FACSIMILE PAGE FROM ROBERT RECORDE. ''GROUND OF ARTES', This reduced gage, from the edition of 1558, shows the first step in the addition of 2659 arid 8342

The discs that were used in computing were commonly known in Great Britain as "counters' (countures cowntouris); in the Latin books as projectiles (pro, forward, + jacere to throw) being thrown or cast upon the counting board, or as denarii supputarii (computing pennies); and in France as jetons (with many such variants as gects, jectoirs gietons and jettons, from jacere, to throw). In Germany a counter was called a Reckenpfennig or zalllpfennig (number penny). Such later expressions as "cast an account," "borrow one," "carry two," and possibly "lay a wager" have their origin in this kind of computation. The Court of the Exchequer. the "counting house," the "counter" in the modern shop, billiard counters, poker "chips," and various games trace their origin to the counting board.

Gerbert (c. 1000), who became Pope Sylvester II, invented an "arc abacus" in which each counter bore a certain Indo-Arabic numeral excepting the zero, the periods in groups of three being united by an arc. It had little recognition, however, for as soon as the significance of the zero became apparent (see NUMERALS) it was manifestly of no value. See CALCULATING MACHINES, FINGER NUMERALS. (D. E. S.)

BIBLIOGRAPHY.--F. P. Barnard, The Casting-counter and the Counting- board (Oxford, 1916); C. G. Knott, "The Abacus in its Historic and Scientific Aspects," Trans. Asiat. Soc. Japan 14:18 (1886); F. C. Scesney, The Chinese Abacus (I944); D. E. Smith, History of Mathematics, 2:156-95 (1925); Y. Yoshimo, The Japanese Abacus Explained (Tokyo 1937).