Newmarket contained in the British Museum MS. Reg. 12, E. 1. A fragment of another translation of the same gloss was printed by Halliwell in his Rara Mathematica (1835) p. 29.1 It corresponds, as far as p. 71, l. 2, roughly to p. 3 of our version, and from thence to the end p. 2, ll. 16-40.

The Art of Nombryng is one of the treatises bound up in the Bodleian MS. Ashmole 396. It measures 11½” × 17¾”, and is written with thirty-three lines to the page in a fifteenth century hand. It is a translation, rather literal, with amplifications of the de arte numerandi attributed to John of Holywood (Sacrobosco) and the translator had obviously a poor MS. before him. The de arte numerandi was printed in 1488, 1490 (s.n.), 1501, 1503, 1510, 1517, 1521, 1522, 1523, 1582, and by Halliwell separately and in his two editions of Rara Mathematica, 1839 and 1841, and reprinted by Curze in 1897.

Both these tracts are here printed for the first time, but the first having been circulated in proof a number of years ago, in an endeavour to discover other manuscripts or parts of manuscripts of it, Dr. David Eugene Smith, misunderstanding the position, printed some pages in a curious transcript with four facsimiles in the Archiv für die Geschichte der Naturwissenschaften und der Technik, 1909, and invited the scientific world to take up the “not unpleasant task” of editing it.

Accomptynge by Counters is reprinted from the 1543 edition of Robert Record’s Arithmetic, printed by R. Wolfe. It has been reprinted within the last few years by Mr. F. P. Barnard, in his work on Casting Counters. It is the earliest English treatise we have on this variety of the Abacus (there are Latin ones of the end of the fifteenth century), but there is little doubt in my mind that this method of performing the simple operations of arithmetic is much older than any of the pen methods. At the end of the treatise there follows a note on merchants’ and auditors’ ways of setting down sums, and lastly, a system of digital numeration which seems of great antiquity and almost world-wide extension.

After the fragment already referred to, I print as an appendix the ‘Carmen de Algorismo’ of Alexander de Villa Dei in an enlarged and corrected form. It was printed for the first time by Halliwell in Rara Mathemathica, but I have added a number of stanzas from various manuscripts, selecting various readings on the principle that the verses were made to scan, aided by the advice of my friend Mr. Vernon Rendall, who is not responsible for the few doubtful lines I have conserved. This poem is at the base of all other treatises on the subject in medieval times, but I am unable to indicate its sources.

The Subject Matter.

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).

There are, however, still in existence four Roman counting boards of a kind which does not appear to come into literature. A typical one is of the third class. It consists of a number of transverse wires, broken at the middle. On the left hand portion four beads are strung, on the right one (or two). The left hand beads signify units, the right hand one five units. Thus any number up to nine can be represented. This instrument is in all essentials the same as the Swanpan or Abacus in use throughout the Far East. The Russian stchota in use throughout Eastern Europe is simpler still. The method of using this system is exactly the same as that of Accomptynge by Counters, the right-hand five bead replacing the counter between the lines.

The Boethian Abacus.

Between classical times and the tenth century we have little or no guidance as to the art of calculation. Boethius (fifth century), at the end of lib. II. of his Geometria gives us a figure of an abacus of the second class with a set of counters arranged within it. It has, however, been contended with great probability that the whole passage is a tenth century interpolation. As no rules are given for its use, the chief value of the figure is that it gives the signs of the nine numbers, known as the Boethian “apices” or “notae” (from whence our word “notation”). To these we shall return later on.