Calculating rods were used for arithmetic

In China, calculation was probably first carried out by using the fingers, as in other places. The use of fingers to display numbers when bargaining in a market that may still be occasionally observed is a legacy of this method. There is evidence that numbers were recorded by knots in strings, an example of which was the Pervian *quipu*. It is strange that this method was found on both sides of the wide Pacific, when no cultural connection is known other than a distant Asian origin. Then, counters, perhaps pebbles, bones, or sea shells were used on the counting board, the *suan*, in which each successive column or row was a power of ten, giving rise to the decimal numbers seen throughout the ancient world, and the number-words in many languages, including Chinese.

A distinctive method arose in China, the use of number-rods as counters. These number rods were typically 0.1" in diameter and 6" long, quite plain, with no numbers or characters written on them, and carried in a bag with the person. When these rods were used, no counting-board was necessary, since the successive powers of 10 were separated by the orientation of the rods laid down. Originally, the numbers from 1 to 5 were denoted by the corresponding number of rods laid side-by-side horizontally, and the numbers from 6 to 9 with one rod vertical, the rest horizontal, as shown in the first column of the figure. The ideographs for the numbers from 1 to 3 were clearly taken from the rods, as can be seen. Later, the units appear to have been made with vertical rods, and the single vertical rod below rather than above, the tens as shown for 1-9, and so on. The calculating rods with numbers on them, introduced in the 17th century, were just Napier's Bones. The ideographic word for *calculate*, shown in the title above, is said to show two hands operating an abacus, but I think it more probably shows counting rods on a tablet.

The abacus is familiarly associated with Chinese arithmetic, but the first abacus of the characteristic form, the *suan p'an* or calculating board, is recorded as late as 1593 by Ch'eng Ta-Wei. It is astonishing that it appears in a developed form at this late date, with only rare earlier references to something called *ball arithmetic* around AD 570. Even Needham does not claim that it was known in China from early days and developed there; in fact, he gives no hint as to its evolution. It is probably true that it did not become popular until the 16th century, and was not widely used until then. Most Westerners who talk about the abacus have no idea how one is used, though it takes less than an hour to learn. Addition and subtraction are so easy that anyone who learns the abacus and has no other mechanical help will never give it up. All you have to know are the complements of the numbers from 1 to 9, that is, the number when added to them will give 10, and you can do error-free, rapid arithmetic. Multiplication and division are more difficult, but still possible, as well as the extraction of roots.

The Latin word abacus is borrowed from the Greek abax , tablet, and the word anciently was used only for a gaming or counting board. Later, it was transferred to the portable counting board with captive beads that was known in Roman times, whether a Hellenistic invention or whatever. It is one of those many articles of daily commercial life that scholars have always thought beneath their consideration, so few have been preserved, and even fewer mentioned in writing. Nevertheless, there are examples to show what they were like, and I believe they were very widely used in government and commerce. The differences from the counting board are significant, and make the abacus much more useful. The size of the instrument was reduced until it would fit in a pocket or purse. It was a bronze plate with slots in which metal beads were riveted. The columns were labelled with the power of 10, as I, X, (|), ((|)) and so on. the (|) and so forth are as close as this font can get to the Roman numerals used for 100, 1000 and so on. C and M were not used in classical times, but rather these characters derived from Etruscan. The beads in each column were in two groups, four in the lower, representing units, and one in the upper, representing fives. These five beads did the same work as the 9 or 10 beads in a primitive abacus. Exactly this arrangement of beads appears in the Japanese *soroban*, the most elegant of all abacuses. The appearance of Roman numerals for 5, 50 and so on is surely due to this grouping of beads. A number expressed in Roman numerals, especially as used in classical times, is easily transferred to a abacus as quickly as it can be read. Roman numerals and a Roman abacus make arithmetic much faster than paper calculations with memorised addition tables, and very much more accurate.

The Chinese abacus has an extra bead in each group, which gives alternative settings that are sometimes useful, but has essentially the same groupings, and is used in the same way. The lack of forebears would be explained if the Roman abacus had been transmitted as a finished product to China, and there were many opportunities for this to happen. There is no surprise that the abacus appears in customary Chinese materials, but it would be very enlightening to find a bronze abacus, or any intermediate stage between the captive bead sliding in a slot, and the pierced bead on a rod. The abacus did not reappear in Europe until the 11th or 12th century, it seems. The Chinese abacus was commented upon in the 17th and 18th centuries in Europe, as objects were brought back from the China trade. The Russian abacus, used in monetary transactions, seems to have come from China. It is a more primitive abacus with horizontal rods.

Chinese mathematics consisted almost entirely of numerical calculations, in which considerable skill and ingenuity was shown. Geometrical and theoretical knowledge was surprisingly slight. Needham can only adduce the Pythagorean theorem, which he wrongly attributes to Pythagoras himself, and which was very well known in Egypt and Babylon. The Chinese 'proof' is the one by dissection, and the only proof of any type that he mentions. There is a book containing 750 proofs of the Pythagorean theorem. Needham also seems to regard Euclid's Elements as the epitome of Greek Mathematics. It was called the *Elements*, not the *Hard Bits*, because it was elementary. The Chinese got a very good value for p (although 3 was a popular value) as did Archimedes much earlier. Archimedes' method is general, and was only carried out far enough for practical use and as demonstration. The Chinese had no conic sections, or such like, nor did they worry about irrational numbers. Chinese mathematics was practical, and very good at that, but would never have developed into anything like Greek mathematics, which became modern mathematics in the 18th century.

The I Ching was a book of divination based on figures made from long and short horizontal lines. The eight trigrams are shown to the left; these were stacked to form sixty-four hexagrams. Each had a mystical meaning. The arrangements were combinatorial and binary, but seem to have had no mathematical significance. Ch'ien was associated with maleness, roundness, strength, dragons, heaven, king and head. K'un signified femininity, squareness, docility, mares, earth, people and abdomen. Doubled, as hexagrams, the meaning was the same. The next-to-last hexagram, K'an on Li, called Chi Chi, stood for completeness and order, while the last, Li on K'an, called Wei Chi, stood for incompleteness and disorder. This system was based largely on peasant omen interpretations, liberally embellished, and the symbols were derived from the drawing by lot of plant stalks short and long. Most causes in the natural world were ascribed to the I Ching and mystic influences, not to natural law subject to mathematical interpretation.

Chinese nouns do not have plurals. To express a certain number of things, the number comes first, then a *measure*, which is a word signifying a unit of the class of things counted, and finally the noun. Things in general take the measure *ge*, flat things *jang*, books and bound things *ben*, people (politely) *wei*, and so on. Three men is *san ge ren*, 'three units man'.

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Composed by J. B. Calvert

Created February 2000

Last revised 17 February 2000