dea 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:

_ __ ___ ____
|, ||, |||, ||||, |||||, |, ||, |||, ||||.

For the sake of clearness, tens, hundreds, etc., were expressed in the
even place by horizontal instead of vertical lines and vice versa; thus
1267 would be formed

__
- || | ||.
-

The rods were arranged on a sort of chessboard called the swan-pan. Much
later the lines were transferred to paper, and a circle used to denote
the vacant square. The use of squares, however, rendered it unnecessary
to arrange the even places differently from the odd, so numbers like
38057 came to be written

+-------+-------+-------+-------+------+
| | ___ | | | __ |
| ||| | ||| | | ||||| | || |
| | | | | |
+-------+-------+-------+-------+------+

instead of

+-------+-------+-------+--