Almost contemporaneously with the beginnings of counting various auxiliary devices were introduced to help out the difficult task. In place of many men, notched sticks, knotted strings, pebbles, or finger pantomime were used. In the best form, these devices resulted in the abacus; indeed, it was not until after the introduction of arabic numerals and well into the Renaissance period that instrumental arithmetic gave way to graphical in Europe (D. E. Smith, _Rara Arithmetica_, under "Counters"). "In eastern Europe," say Smith and Mikami (_Japanese Mathematics_, pp. 18-19), "it"--the abacus--"has never been replaced, for the tschotü is used everywhere in Russia to-day, and when one passes over into Persia the same type of abacus is common in all the bazaars. In China the swan-pan is universally used for the purposes of computation, and in Japan the soroban is as strongly entrenched as it was before the invasion of western ideas."
Given, then, the idea of counting, and a mechanical device to aid computation, it still remains necessary to obtain some notation in which to record results. At the early dawn of history the Egyptians seem to have been already possessed of number signs (cf. Cantor, _Gesch. de. Math._, p. 44) and the Phoenicians either wrote out their number words or used a few simple signs, vertical, horizontal, and oblique lines, a process which the Arabians perpetuated up to the beginning of the eleventh century (Fink, p. 15); the Greeks, as early as 600 B. C., used the initial letters of words for numbers. But speaking generally, historical beginnings of European number signs are too obscure to furnish us good material.
Our Indians have few number symbols other than words, but when they occur (cf. Eells, _loc. cit._) they usually take the form of pictorial presentation of some counting device such as strokes, lines dotted to suggest a knotted cord, etc. Indeed, the smaller Roman numerals were probably but a pictorial representation of finger symbols. However, a beautiful concrete instance is furnished us in the Japanese mathematics (cf. Smith and Mikami, Ch. III). The earliest instrument of reckoning in Japan seems to have been the rod, Ch'eou, adapted from the Chinese under the name of Chikusaku (bamboo rods) about 600 A. D. At first relatively large (measuring rods?), they became reduced to about 12 cm., but from their tendency to roll were quickly replaced by the sangi (square prisms, about 7 mm. thick and 5 cm. long) and the number symbols were evidently derived from the use of these rods: