word itself in the sense of 'reckoning-table' is not used for certain in any writer before Polybius (ȧßákιov in v. 26, 13) who belongs only to the 2nd century B. C. It is, however, used in the sense of plain 'board' in many different connexions1. Assuming it to be true, also, that the Semites did generally use a sanded board for their calculations 2, it does not appear that this was called abaq, and the step from Semitic abaq 'sand' to Greek aßağ a 'board' remains practically as wide as before. Lastly, the tradition which connects the aßag with Pythagoras as well as that which connects him with a Semitic people, is so late and belongs to so imaginative authors that no reliance can be placed upon it. Of course, a few lines drawn with a stick in the dust and a handful of stones were as efficient an instrument for calculation as was needed and must always have been used by Greeks upon occasion. Such an impromptu ledger would indeed frequently be preferable to a more elaborate device, since it could be adapted to different fractions, different monetary scales etc., while a permanent machine would probably be restricted to one scale and a few selected fractions. But whether such a scheme of lines drawn on the ground could ever in Greek have been called aẞağ there is no evidence to show.

23. It must be admitted, also, that hardly anything is known of the normal Greek äßağ, using that word in the sense of a reckoning-board with permanent lines drawn on it and possibly permanent balls or pebbles attached to it. Three types

1 The word seems first to occur in the sense of 'trencher' in Cratinus, Kλeoß. 2 (cit. Poll. x. 105). Hesychius says it was a synonym for μáктρа 'trough.' Pollux also cites ἀβάκιον from Lysias, without stating its meaning. It is oddly accented.

2 The evidence adduced by Cantor, Math. Beiträge, p. 141, is not satisfactory on this point, but the fact is hardly worth disputing. A sanded board was certainly used by Greek geometers, but is nowhere attributed to arithmeticians. Cf. Cic. Nat. Deor.

2, 18, 48. Tusc. 5, 23, 64, and other quotations collected by Friedlein, Zahlz. § 76, pp. 52, 3. See also Cantor, Vorl. pp. 109-111. It seems to me not unlikely that äßag was a childish name for the board on which the alphabet was written and from which the children read their Bra ἄλφα βα, βῆτα ελ βε, etc. (Athenaeus, x. 453). "Aẞag would be the 'ABC board,' the termination being chosen by analogy from πίναξ.

3 Iamblichus, for instance, and the pseudo-Boethius, cited post.

at least of such a machine are well known. One of these is the Russian tschotü, in which each wire carries 10 balls1. Some advance is shown in the Chinese suan-pan, where the whole field of the frame is divided by a transverse string: each wire on that part of it which is below this string carries 5 balls : and on the part which is above 2 balls, each of which is worth 5 of those below. On both these machines, apparently, it is possible and usual to remove balls from one wire to another as the case may require. But the third type is the Roman abacus, which, at any rate in its highest development, was closed, so that balls or buttons could not be removed from the wire or groove in which they were originally placed. A few specimens of this sort, constructed with grooves in which buttons (claviculi) slide, are still preserved. One of them which is figured in Daremberg's Dictionnaire des Antiquités (s.v. abacus) and is in the Kircher Museum at Rome, may be roughly represented thus:

Leaving out

of

consideration, for a moment, the two grooves on

the extreme right, it will be seen that the remaining 7 contain buttons representing units, tens, etc. up to millions. The lower

1 The balls are differently coloured, some of the 10 being white and some black. The instrument was introduced into the schools of Eastern France after the great Russian campaign. It is common enough in Pestalozzian schools. See further Cantor, Math. Beitr. pp. 129, 130.

=

2 Suân reckon : phuân=board. Goschkewitsch, an authority quoted by Hankel, Zur Gesch. der Math. p. 54, says that "the practised Chinese reckoner plays with the fingers of the right hand on the suan pan as on a musical instrument and grasps whole numerical chords."

grooves contain 4 buttons each, the higher 1 each, which represents 5 of the same value as those in the lower corresponding groove. The letters indicating the values of the buttons are obscure above C, but are plain enough on another specimen, which once belonged to one Welser, in whose works published at Nuremberg in 1682 there was given a drawing of his abacus (pp. 442 and 819)'. The sign O which distinguishes the penultimate groove on the right, stands for uncia, and as there are 12 unciae to the as, here the lower portion of the groove has 5 buttons for 5 unciae, the upper 1 button for 6 unciae. The signs appended to the last groove on the right are S for semuncia (24th of an as): Ɔ for sicilicus (4th of an as): and Z for sextula (nd of an as). It is not, however, very clear why there should be 4 buttons in this groove or what was the value of each and how, if of different values, they were distinguished from each other. Welser's abacus, which in other respects is exactly similar to this, had three separate grooves for these fractions, the first containing 1 button for the semuncia (4th): the second 1 button for the sicilicus (th): the third 2 buttons, each representing a sextula (nd). These grooves therefore together (and no doubt the last groove of the Kircher abacus) represent 13ths of an uncia. Both abaci are capable of representing all whole numbers from 1 to 9,999,999 and the duodecimal fractions of the as in common use. Since such an abacus could seldom represent more than one number at a time, it is probable that, in calculating with it, the larger of the two numbers to be dealt with would be represented on the table. The smaller would be mentally added or subtracted

1Reproduced by Friedlein in Zeitschr. für Math. u. Physik. Vol. IX. 1864. Plate 5. See also p. 299. A description of this abacus is given also in Friedlein's Zahlzeichen, p. 22, § 32. A figure of it is given in Daremberg, Dict. des Ant. s.v. arithmetica. M. Ruelle, the writer of the article, says that 4 Roman abaci (which he names) are known, but it does not appear that they are all now in existence.

2 This statement, which is taken from Friedlein, seems unlikely. On the analogy of all the preceding grooves, we should expect the table to conclude with 11ths of the uncia, and not 13ths. It will do so if the last two buttons be taken to represent, not sextulae, but dimidiae sextulae, the ordinary sign of which is easily to be confused with that of the sextula. See Friedlein, Zahlz. Plate to § 48.

as the case may be, and the buttons would be successively altered so as to represent the sum or remainder. Multiplication can only have been performed by repeated additions, and division by repeated subtractions 1.

was.

24. It will be seen that the types of abacus now known are not very diverse from one another, and there is no cause to be greatly distressed by our ignorance of what the Greek äßağ A certain table, however, which may be an aẞaş, was discovered in 1846 in the island of Salamis and this, which can be partly explained by reference to the Roman instruments, must serve to assure us that there cannot have been any great superiority in the Greek äßağ at any time. This "Salaminian table" may be figured thus":

It is made of marble and is very large, being about 5 ft. (1.5 metres) long by 2 ft. (75 metre) wide. The letters upon the margin are easily explained. is the customary Attic sign for a drachma. The letters which, in the table, stand on the left of this sign are II for 5 (πévтe), ▲ for 10 (déкa), ♫ for 50, H for 100 (Exaтóv), for 500 and X for 1000 (xico) in the ordinary Attic style. To these are added, in one row, the signs for 5000 and T for тáλavтov or 6000 drachmae. The signs which stand to the right of in the table are the fractions of the drachma, viz. | for th (obol), C forth (obol), T1 for th (τεταρτημορίου of the obol) and X for χαλκοῦς (#th of the obol, th of the drachma). The last three fractions, it will be observed, when added together make ths of an obol, which is the real unit of the table. On the principle of a Roman abacus, this scale would be thus distributed:

But it will be seen that the lines of the Salaminian table do not fall in with this arrangement. Here we have 11 lines, with 10 intervals, in one place: and 5 lines, with 4 intervals, in another. If the table be really an äßağ, the simplest explanation is that

Archéolog. 1846, p. 296, where a very minute description of the stone is given by M. Rangabé. Another Greek abax is also figured on the Darius-vase at Naples. The numerals on it are of the same kind as those on the Salaminian table and it is held by the

reckoner so that the columns are perpendicular to his body. But it is too small and roughly drawn to furnish important information.

1 τεταρτημορίον is Böckh's expla nation: M. Vincent proposed τpitημορίου.