Ancient and medieval writers observed a distinction between the Science and the Art of Arithmetic. The classical treatises on the subject, those of Euclid among the Greeks and Boethius among the Latins, are devoted to the Science of Arithmetic, but it is obvious that coeval with practical Astronomy the Art of Calculation must have existed and have made considerable progress. If early treatises on this art existed at all they must, almost of necessity, have been in Greek, which was the language of science for the Romans as long as Latin civilisation existed. But in their absence it is safe to say that no involved operations were or could have been carried out by means of the alphabetic notation of the Greeks and Romans. Specimen sums have indeed been constructed by moderns which show its possibility, but it is absurd to think that men of science, acquainted with Egyptian methods and in possession of the abacus,[2] were unable to devise methods for its use.
[ The Pre-Medieval Instruments Used in Calculation.]
The following are known:—
(1) A flat polished surface or tablets, strewn with sand, on which figures were inscribed with a stylus.
(2) A polished tablet divided longitudinally into nine columns (or more) grouped in threes, with which counters were used, either plain or marked with signs denoting the nine numerals, etc.
(3) Tablets or boxes containing nine grooves or wires, in or on which ran beads.
(4) Tablets on which nine (or more) horizontal lines were marked, each third being marked off.
The only Greek counting board we have is of the fourth class and was discovered at Salamis. It was engraved on a block of marble, and measures 5 feet by 2½. Its chief part consists of eleven parallel lines, the 3rd, 6th, and 9th being marked with a cross. Another section consists of five parallel lines, and there are three rows of arithmetical symbols. This board could only have been used with counters (calculi), preferably unmarked, as in our treatise of Accomptynge by Counters.
[ Classical Roman Methods of Calculation.]
We have proof of two methods of calculation in ancient Rome, one by the first method, in which the surface of sand was divided into columns by a stylus or the hand. Counters (calculi, or lapilli), which were kept in boxes (loculi), were used in calculation, as we learn from Horace’s schoolboys (Sat. 1. vi. 74). For the sand see Persius I. 131, “Nec qui abaco numeros et secto in pulvere metas scit risisse,” Apul. Apolog. 16 (pulvisculo), Mart. Capella, lib. vii. 3, 4, etc. Cicero says of an expert calculator “eruditum attigisse pulverem,” (de nat. Deorum, ii. 18). Tertullian calls a teacher of arithmetic “primus numerorum arenarius” (de Pallio, in fine). The counters were made of various materials, ivory principally, “Adeo nulla uncia nobis est eboris, etc.” (Juv. XI. 131), sometimes of precious metals, “Pro calculis albis et nigris aureos argenteosque habebat denarios” (Pet. Arb. Satyricon, 33).