[ The Use of “Arabic” Figures.]

It may now be regarded as proved by Bubnov that our present numerals are derived from Greek sources through the so-called Boethian “apices,” which are first found in late tenth century manuscripts. That they were not derived directly from the Arabic seems certain from the different shapes of some of the numerals, especially the 0, which stands for 5 in Arabic. Another Greek form existed, which was introduced into Europe by John of Basingstoke in the thirteenth century, and is figured by Matthew Paris (V. 285); but this form had no success. The date of the introduction of the zero has been hotly debated, but it seems obvious that the twelfth century Latin translators from the Arabic were perfectly well acquainted with the system they met in their Arabic text, while the earliest astronomical tables of the thirteenth century I have seen use numbers of European and not Arabic origin. The fact that Latin writers had a convenient way of writing hundreds and thousands without any cyphers probably delayed the general use of the Arabic notation. Dr. Hill has published a very complete survey of the various forms of numerals in Europe. They began to be common at the middle of the thirteenth century and a very interesting set of family notes concerning births in a British Museum manuscript, Harl. 4350 shows their extension. The first is dated Mijc. lviii., the second Mijc. lxi., the third Mijc. 63, the fourth 1264, and the fifth 1266. Another example is given in a set of astronomical tables for 1269 in a manuscript of Roger Bacon’s works, where the scribe began to write MCC6. and crossed out the figures, substituting the “Arabic” form.

[ The Counting Board.]

The treatise on pp. 52-65 is the only one in English known on the subject. It describes a method of calculation which, with slight modifications, is current in Russia, China, and Japan, to-day, though it went out of use in Western Europe by the seventeenth century. In Germany the method is called “Algorithmus Linealis,” and there are several editions of a tract under this name (with a diagram of the counting board), printed at Leipsic at the end of the fifteenth century and the beginning of the sixteenth. They give the nine rules, but “Capitulum de radicum extractione ad algoritmum integrorum reservato, cujus species per ciffrales figuras ostenduntur ubi ad plenum de hac tractabitur.” The invention of the art is there attributed to Appulegius the philosopher.

The advantage of the counting board, whether permanent or constructed by chalking parallel lines on a table, as shown in some sixteenth-century woodcuts, is that only five counters are needed to indicate the number nine, counters on the lines representing units, and those in the spaces above representing five times those on the line below. The Russian abacus, the “tchatui” or “stchota” has ten beads on the line; the Chinese and Japanese “Swanpan” economises by dividing the line into two parts, the beads on one side representing five times the value of those on the other. The “Swanpan” has usually many more lines than the “stchota,” allowing for more extended calculations, see Tylor, Anthropology (1892), p. 314.

Record’s treatise also mentions another method of counter notation (p. 64) “merchants’ casting” and “auditors’ casting.” These were adapted for the usual English method of reckoning numbers up to 200 by scores. This method seems to have been used in the Exchequer. A counting board for merchants’ use is printed by Halliwell in Rara Mathematica (p. 72) from Sloane MS. 213, and two others are figured in Egerton 2622 f. 82 and f. 83. The latter is said to be “novus modus computandi secundum inventionem Magistri Thome Thorleby,” and is in principle, the same as the “Swanpan.”

The Exchequer table is described in the Dialogus de Scaccario (Oxford, 1902), p. 38.

[1.] Halliwell printed the two sides of his leaf in the wrong order. This and some obvious errors of transcription—‘ferye’ for ‘ferthe,’ ‘lest’ for ‘left,’ etc., have not been corrected in the reprint on pp. 70-71.

[2.] For Egyptian use see Herodotus, ii. 36, Plato, de Legibus, VII.

[3.] See on this Dr. Poole, The Exchequer in the Twelfth Century, Chap. III., and Haskins, Eng. Hist. Review, 27, 101. The hidage of Essex in 1130 was 2364 hides.