He reports to the assessor his own age (forty-seven) and the ages of his brother, son and wife. The assessment document also records the size of his fields, from a survey carried out a decade earlier, and their liability for the summer and autumn taxes. It goes into detail about the size of eight plots of land including a paddy field and a patch of marsh, and it closes with a note that the family’s house is constructed of straw and possesses three rooms.

Hong’s summer tax is given as 6 dou 4 sheng 4 ge 1 shao of wheat; the autumn tax 1 shi 6 dou 3 sheng 9 ge of husked rice.

The same scene was played out millions of times each year across Ming China, as tax assessors tried to get a grip on who owed what under the country’s complex system of assessments. Revenues were raised across a land area of 4 million square kilometres, from the Pacific to the Himalayas, and from a population of more than 150 million; the total collected amounted to perhaps 5 or 10 per cent of the country’s agricultural output.

For administrative purposes, the vast country was divided into thirteen provinces, then into prefectures, sub-prefectures and counties. At the lowest level, groups of 110 households were organised into li, each under a rotating lichang (chief, alderman) who performed, among other things, the collection of taxes. The emperor controlled it all from the capital at Beijing, in a system which relied heavily – at least in theory – on his direct control. Some of the questions brought to his attention were very small: ‘the relocation of a particular business tax station, or from which productive area a particular county was to draw its salt supply, or how many rolls of silk were to be awarded to a foreign tributary mission’.

Such matters were passed upwards from the village level and assembled by a pyramid of officials in the prefectures and provinces, headed by the minister of revenue. Those same structures also organised the immense task of moving the goods – after collection – across the country to where they were distributed. For instance,

as late as 1578 the imperial university in Nanking still received 3,500 piculs of husked rice from Ch’ang-chou prefecture, 100 piculs of wheat from Ning-kuo prefecture and 100 piculs of green lentils from Ying-t’ien prefecture, all in South Chihli, and also over 20,000 catties of dried fish from Hukwang province, all these items being charged to the tax quotas of the respective districts.

As well as direct levies on agricultural output, there was an obligation to provide labour services and various specialised kinds of goods – office supplies, military equipment, medical supplies, even brooms to sweep the palace – from time to time. Furthermore, the assessments were adjusted based on a ten-yearly survey of land and an annual census of the number of people and amount of property in each household. Though the system was simple in theory, it inevitably tended to burgeon into increasing complexity, with surcharges and additions to cover the transport of the goods, and an ever-increasing willingness to substitute silver bullion for the set of items nominally due. This led to complex calculations, with outcomes that sometimes had to be specified in thousandths of an ounce of silver. A modern commentator suggests that even the simplest cases might each have taken an hour and a half to calculate, giving the example of ‘a labor service payment of 0.0147445814487 taels of silver on each picul of grain in basic land tax assessment’, a situation which ‘provided a paradise for lower-echelon tax collectors and bookkeepers’. Attempts at consolidation and simplification were made in the sixteenth century, but they were only ever implemented patchily.

Even if there was a tendency for some of this to become a mere formality, with survey and census numbers being carried over from one period to the next unchanged or only mechanically updated, nevertheless the administrative effort involved was huge, as was the volume of tax records being generated. By Hong Gongshou’s time some were complaining that the whole exercise was a waste of paper and writing brushes. Four copies of each tax assessment were made: one each for the county, prefecture, provincial and imperial governments. The last was the subject of an elaborate storage regime on an island on lake Hsuan Wu near Nanjing. By the 1640s there were 1.7 million volumes in seven hundred storage rooms, very probably the largest archive the world had yet seen.

The largest archive, and therefore the largest collection of number symbols. Chinese, then as now, had multiple ways of recording the outcome of a count or a calculation, and the surveys and censuses certainly reflected some of that variety.

One way to write a number was simply to write the logograph for its word. These already appeared in effectively their modern form in inscriptions of the third century BCE, and continue in use to this day. The system, both spoken and written, is a decimal one, with words for the numbers from one to nine and for ‘ten’, ‘hundred’, ‘thousand’ and ‘ten thousand’ which combine as, for instance, two-myriad four-thousand three-hundred six-ten five (24,365). A zero was introduced to the written system from the thirteenth century, if not earlier, but strictly speaking it is redundant. The words ‘ten thousand’, ‘thousand’, ‘hundred’ and ‘ten’ already specify what each unit sign means: there is no need for its position to do so as well.

This system spread throughout East Asia, appearing in Japan by the third century and Korea by the fifteenth. Calligraphic variants of the signs – which did not change the structure of the system – began to appear from the first century BCE. At least two such variants, the so-called accountants’ numerals and the secret numerals, are still in use today, their greater visual complexity providing a guard against forgery.

But what technology was used actually to count and calculate in China and its neighbours? Texts from the first millennium BCE mention sticks being used to count, and by at least the third century BCE they were employed in a system that involved laying them out in groups on a flat surface. The sticks were never completely standardised, and could be made of bamboo, wood, ivory, bone or other materials. In the classical period they ranged between about 9 centimetres and 14 in length and were perhaps 7 millimetres wide: 271 of them could be made into a hexagonal bundle and held in one hand.

The patterns in which the sticks were arranged functioned as number symbols. Counting up to five was done with rods laid down vertically beside one another, just like the most primitive tallies. Numbers from six to nine were shown by the combination of one horizontal with one, two, three or four vertical rods (once again, the eye never needed to deal with a single group larger than four). The same patterns were repeated for the tens, hundreds and so on, with the larger powers placed to the left. The interesting complication was added that for the even-numbered digits – tens, thousands, hundred thousands and so on – the pattern of sticks was rotated by ninety degrees, to help distinguish the digits from one another. A rhyme reported that:

Chinese counting rods

Replica of Han Dynasty Counting rods in National Museum of Natural Science in Taiwan. Wikimedia Commons. CC BY-SA 3.0.

Units are vertical, tens are horizontal,

Hundreds stand, thousands lie down;

Thus thousands and tens look the same,

Ten thousands and hundreds look alike.

Any number of any size – subject to the limits of how much flat space you had and how many rods you possessed – could be denoted in this way.

Literary references seem to place the sticks on tables, beds or the floor. But for the system to work really efficiently, the surface on which they were placed probably needed to be marked in some way: otherwise the gap between one group of rods and the next would have the potential to become ambiguous, 24 looking all too similar to 2004. Large wooden boards would have been cumbersome, and none have turned up as archaeological artefacts. But there is at least one reference to a cloth marked with columns or rows for the sticks to occupy.

Like the Chinese counting words and their equivalent symbols, the counting rods spread to Japan and Korea, and they were much used throughout East Asia for 2,000 years. They were employed not just for counting, but also for calculation. Algorithms for the basic arithmetical operations in the rod system are not at all difficult to devise, and eyewitness reports suggest that they could be performed very rapidly; one expert in the eleventh century ‘could move his counting-rods as if they were flying, so quickly that the eye could not follow their movements before the result was obtained’. Both mental and written calculation seem to have become unusual, displaced by the more efficient rods. By the twelfth century, perhaps earlier, algorithms had been devised not only to extract square and cube roots using the counting rods, but to find numbers satisfying systems of relationships that a Greek would have represented using the geometry of squares and rectangles, a modern European using the algebra of linear and quadratic equations.

The counting rods also prompted their users to write down numbers in the same form: to make sets of ink strokes that matched the layout of the rods. As early as 400 BCE, Chinese coins bore numbers using symbols that seem to be of this kind; and like the rods themselves, these number symbols remained in use until the Ming period. It has sometimes been suggested that this set of number representations in fact travelled as far west as India and was an inspiration for the place-value structure of the Brahmi number symbols. Direct evidence is lacking, and well-informed experts disagree about how plausible the suggestion is. It is certain that by the eighth century CE, the Brahmi numerals themselves had been seen by some in China, and were the subject of comment that compared them unfavourably to the counting rods as a technique of calculation: ‘The method is complicated … and you are lucky if you get it right.’

The rods and their techniques are constantly assumed – and in some cases described in detail – throughout the rich tradition of Chinese mathematical writing: in the set of canonical teaching texts codified in the seventh century, in later learned writings and also in collections of more practical problems. One such collection, compiled in the thirteenth century, ranged across practical subjects such as chronology, surveying, architecture, military problems and trade. Among its sections was one dealing with taxes and service levies of precisely the kind to which Hong Gongshou and millions like him were subject. The problems give a vivid sense of the difficulties taxpayers and tax collectors faced throughout the Ming period. Farmer A makes over 407 mou of paddy fields to farmer B and 516 mou to farmer C; how will this affect the three men’s tax liabilities, in rice and silk? Land in a certain prefecture has emerged from the sea. How should taxes be apportioned among the six villages that claim parts of it across nine different kinds of field? Twelve thousand men are levied to transport military rations to the frontier, in proportion to their field tax liability and in inverse proportion to their distance from the frontier: how many men will have to come from each village?

That last question was to gain a new topicality in the early seventeenth century, as the Ming regime faced the series of crises that led to its downfall. From 1618 onwards there were Manchu invasions; Chinese military spending spiralled and taxes rose commensurately. But the system was not able to mobilise enough of the land’s vast resources with sufficient speed. Although revenues quintupled in just twenty years, the army could not reliably be paid.

To add to the disaster, northern China also saw poor harvests in the mid-1630s, compromising the possibility of raising the sums that were needed, and ultimately triggering local rebellions. Flood, famine and disease arrived to complete the picture. In fact, the year 1641–2 was the last in which the tax survey was drawn up. The Ming state collapsed soon afterwards, in 1644.

The vast collection of documents held in Nanjing was destroyed when the city fell to the Qing army in 1645. One report holds that this, the world’s largest collection of written number symbols, was used as kindling for gunpowder in the last defence of the city. Documents like Hong Gongshou’s tax return are rare survivals today, reminders of a system long since swept away. For the counting rods and the number symbols based on them were also destined for oblivion. Both for counting and for calculation, another device was now on the rise: the suanpan.

Kiyoshi Matsuzaki (and Thomas Wood): Counting with beads

Nobody knows when it was invented, or indeed where. The suanpan in Mandarin, the soroban to the Japanese and the jupan to Koreans. It seems to have been during the sixteenth and seventeenth centuries that the suanpan took over in widespread use from the counting rods as a way of representing and working with numbers, though it made its first appearance at least as early as the thirteenth century, and perhaps earlier still. The idea, like all great ones, is simple enough: a rectangular frame holds a set of parallel wires on which beads are strung. The position of the beads represents a number, and the rapidity with which a few fingers can shift the beads to change or calculate with the number is the evident reason for its success. Its operators became famous far beyond East Asia for the skill and speed with which they worked.

Its name says something about what it is, and indeed about what mathematics is. Historian Joseph Dauben explains:

The character ‘Suan’ is based on a radical meaning cowry for shells or, in a slightly different form, goods … ‘Suan’ originally referred to the counting board, with which bamboo counting rods were used to carry out calculations during transactions, and later came to refer to the abacus or any method of calculation, including mathematics generally. Thus the character ‘Suan’ ideogrammatically embodies both the original objects and methods that became the stock in trade of the court and administrative mathematicians.

The frame is usually of wood, often of bamboo; but a suanpan can be made from many materials. The number of rods never seems to have been standardised, but there is a noticeable tendency over time for larger numbers of rods to be envisaged and for them to become concomitantly smaller, so that the whole device always remained of portable or at least manageable size. By the twentieth century, a typical suanpan for commercial use might leave only a millimetre of clearance between one column of beads and the next, and allow little more than a centimetre of free wire for the sliding of the beads.

Like its predecessor the counting rods, the suanpan represents numbers using a decimal system. Which row represents the number 1 is a matter of choice, depending on the need of the moment; once it has been chosen, its neighbour to the left will represent the tens, the next row the hundreds, and so on. In fact, a beam running across the board divides each wire into an upper and lower section: the beads in the lower section are worth 1, 10, 100 and so on, those in the upper 5, 50, 500 … Beads are brought into play by being pushed against the bar.

The oldest Chinese suanpan had two ‘fives’ beads and five ‘ones’ beads, the second ‘fives’ bead enabling numbers larger than 10 to be briefly represented on a single wire during calculation. In Japan, the ‘fives’ beads were reduced to a single one during the second half of the nineteenth century, and the ‘ones’ beads to four a few decades later, producing the most streamlined decimal soroban possible with just five beads in total on each wire. This Japanese type has since become widespread in Korea although it has – particularly in China – not wholly supplanted the older types.

A Japanese soroban.

Dexter Mikami / Alamy Stock Photo.

It is remarkable that the modern Japanese soroban therefore has the same structure of four-beads-plus-one as some ancient Greek and Roman counting boards. If transmission occurred in either direction – perhaps during the Middle Ages – it seems to have left no direct evidence, and this may instead be a case of two cultures converging on similar solutions to analogous problems of calculation and efficiency. The ‘abacus’ that began to appear in Europe and the USA from the nineteenth century is usually traced to Russian models brought to France during the Napoleonic period; but the ancestry of the Russian schoty, which has ten beads per rod, is itself obscure. The fact that the English word ‘abacus’ can refer to any of these devices – Greek, Roman, medieval, Russian or East Asian – only adds to the confusion.

Some very detailed descriptions of efficient finger technique for the suanpan and soroban exist. The most minimal version has the ‘fives’ bead (or beads) moved up and down by the index finger, while the ‘ones’ beads are moved up by the thumb and down by the index finger. But three-fingered or even two-handed techniques also appear in some books. Arithmetic with the soroban proceeds from left to right, from larger place values to smaller. The addition and multiplication tables are of the same size and complexity as those that have to be learned for calculation with Arabic numerals, or indeed with any decimal representation of numbers. The techniques of arithmetic are closely analogous to those with the number rods, which represent numbers using a similar structure of units, fives, tens, fifties, and so on. At one time, it was also usual to memorise a division table, based ultimately on counting-rod techniques: but it was phased out in the first half of the twentieth century in favour of a method of division that used the multiplication table only. Just as with the rods, it is perfectly possible to do more advanced arithmetic such as the extraction of square and cube roots using a soroban.

The soroban continued to gather interest and expertise, through refinements to the device itself and to its techniques, right up to the end of the twentieth century if not beyond. Its visibility and its operators’ skill probably reached their apex in the second half of that century. By that time it was frequent for children to learn the use of the soroban at home, even before entering school. In both Japan and China, soroban/suanpan techniques were part of the formal curriculum from typically third or fourth grade. Vocational schools preparing students to go into business similarly provided intensive soroban training. And for those wishing to become experts, there were extracurricular programmes usually associated with primary schools, with training that might occupy several hours per week. Patient practice and memorisation could produce results that were impressive by any standard, and national institutes organised exams and competitions: in Japan from the 1930s, and more recently in China. The levels of skill on display on these occasions were startling: in the Japanese system, where most examinees were teenagers, even the lowest grade of examination involved the addition of five columns of fifteen multi-digit numbers in five minutes, with 70 per cent accuracy required. At national level a contestant could expect to add or subtract fifteen numbers of up to nine digits in little more than a minute.

Tokyo, Monday 12 November 1946. The US Army’s Ernie Pyle theatre. An audience of nearly three thousand GIs assembles to witness such classics of entertainment as 759.843 × 57.941 or 4,768,788,098 ÷ 14,593, performed on a pair of calculating machines. According to reliable witnesses, they cheer frenziedly.

By the nineteenth century, it was already a matter of comment among Western observers that soroban calculation was strikingly faster than paper-based alternatives. In the context of the American occupation of Japan after the Second World War, the soroban was famously put to the test against one of the new electric calculating machines. Stars and Stripes magazine organised a head-to-head contest between a US Army private operating a machine made in California, and a Japanese soroban expert.

Barely a year after the end of the war, the Japanese economy and infrastructure remained shattered, the country occupied by foreign troops. Half of Tokyo itself had been destroyed by Allied bombing. The event was planned as an entertainment for American troops, and reports suggest that an easy victory for the electrical machine was expected. There were to be two rounds of addition problems, three each of subtraction, multiplication and division, and finally a round of more complex composite problems requiring several operations. The numbers ranged in size up to twelve digits.