A History of Mathematics From Mesopotamia to Modernity

ChineseMathematics 85

5. Counting rods—who needs them?

It follows that the Hindu–Arabic numeral system originated from the rod numeral system, which was developed

centuries earlier. (Lam and Ang 1992, p. 148)

Most authors believe that counting-rods were manipulated on a special surface called the counting-board or chess-

board, which would have been to rods what the frame and the bars are to the abacus. However there is no proof that

such boards existed. (Martzloff 1995, p. 209)

What (in ancient Chinese mathematics) was done with counting rods was considered fairly well

established before the doubts raised by Martzloff,^7 and the important claims for the numeral system

made by Lam and Ang, following Needham and others, make it desirable to establish what we

can. Not quite a way of writing numbers, nor simply a calculating tool, the rods were used to

combine the two in a unique way which some specialists at least see as providing an approach to

number-manipulation which was better than anything used before or since. Indeed, it seems that

‘difficult’ mathematics declined when, around the sixteenth century, the abacus replaced the rods

as the instrument of calculation.

Numbers have been written in the Chinese script at least since the Zhou dynasty in a form which

corresponds exactly to the words:

san bai ba shi qi

three hundred eight ten seven

which means: ‘387’ (as is obvious). In particular, it is in this form that they are written in the

classics such as theNine Chapters. As Lam and Ang (1992, p. 14) point out, this means that, given

the particular nature of Chinese writing, the usual distinction between writing numbers in words

(e.g. ‘three hundred and eighty-seven’) and figures (e.g. ‘387’) disappears.

This is quite convenient in itself. However, at some time in the Zhou dynasty the counting rods

were developed as an aid to actually doing sums—one could conjecture, sums of the kinds needed

by merchants or bureaucrats. A rod was

a round bamboo stick 1fen(about 2.5 mm) in diameter and 6cun(about 25 cm) in length...(Shen et al. 1999, p. 12)

(The exact dimensions, and the materials were more variable than this description suggests.)
Placed in patterns, they could symbolize the numbers 1–9, in one of two forms or ‘series’—either
horizontal or vertical (Fig. 1).
There are references to how they were used in early classics, for example, theSunzi suanjing,^8 but
(presumably because the texts were supplemented by a teacher’s instructions) they are not explicit.
The usual explanation of their use, which follows the way in which they were used in Japan in the
eighteenth century, is that:

1. Laid out along a single row of the counting board, the rods gave the decimal representation of
   a number, with an empty space denoting a ‘zero’: so ‘60390’ was represented by Fig. 2 below.


2. By convention, vertical and horizontal types alternated, so that there was less room for con-
   fusion about where an empty space had occurred (‘84’ looked different from ‘804’—try to
   see why).


3. Who would concede that we know a great deal, but argues that it is not quite as much as we think.


4. This is the text, of very uncertain date (between the first and fifth centuriesce?) edited and translated by Lam and Ang (1992).