It took 17 years to piece together the 190 bamboo strips that make up Chinaâs earliest known maths test
âA fox, a wildcat and a dog go through a customs-post. They are taxed 111 cash. The dog says to the wildcat, and the wildcat says to the fox, âYour skin is worth twice mine; you should pay twice as much tax!'â Who pays what?
In 186 BC, a civil servant employed by the Emperor of China was buried in a tomb in what is now HĂșbei Province, along with a few books he might want to read in the afterlife. Among them was Chinaâs earliest known work on mathematics, a collection of problems, including the one about the fox, complete with answers and methods of calculation. Against expectations, not all the maths was practical, its purpose to keep the empire running smoothly. It seems even then some people did maths just for fun.
IN DECEMBER 1983, archaeologists opened up one of the many tombs at an ancient burial ground near Zhangjiashan, in Chinaâs HĂșbei Province. The anonymous man in tomb 247 had been buried around 186 BC. He was neither rich nor famous: he was a civil servant, one of a great army of officials employed to run the new Chinese empire. His career started under QĂn Shi HuĂĄng DĂŹ, âThe First Sovereign Lord of QĂnâ, who in 221 BC destroyed all his rivals and established a unified empire that extended over much of what is now modern China.
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When the First Emperor died in 210 BC, he was buried in a vast mausoleum along with his now famous army of terracotta warriors. Although the QĂn dynasty didnât last long, the bureaucracy he set up to administer the state lived on, and the man destined for tomb 247 continued his official duties under the new HĂ n dynasty. When he died, he too was buried with the things he thought he would need most in the next world: his books. One of them, the SuĂ n shĂč shu â Writings on reckoning â is the oldest work on mathematics to emerge from China.
Ancient Chinese books were written in ink on strips of bamboo strung together to form a scroll. Unfortunately for the archaeologists, the strings had long since rotted, so instead of a neat stack of scrolls they found a jumble of more than 1200 bamboo strips. Painstaking reconstruction produced a set of government statutes, law reports and writings on therapeutic gymnastics. Most exciting of all were the 190 strips that made up the SuĂ n shĂč shu. It took Chinese scholars 17 years to fit them together in a way that made sense.
Over the past six years, Christopher Cullen, director of the Needham Research Institute in Cambridge, UK, and a historian of Chinese mathematics, has translated and studied the SuĂ n shĂč shu It is not a book in the usual sense, he says, but a compilation of problems put together by a mathematical magpie (Historia Mathematica, in press). Someone, perhaps the dead civil servant himself, had collected 69 problems from a variety of sources and stitched them into a sort of album. Each problem has a question, an answer and a method for reaching it, but the varied styles of working suggest they were written by a number of people. At least two â Mr WĂĄng and Mr YĂĄng â were proud enough of their creations to put their names to them.
âIf he had spare time he could relax with a little mathematicsâ
The SuĂ n shĂč shu extends what we know about the development of Chinese maths by three centuries. It has also produced surprises. Chinese mathematics has often been stereotyped by westerners as relentlessly practical, developed as a tool in the service of those who ran the empire, yet the SuĂ n shĂč shu suggests that the early imperial age was also a time of mathematical creativity.
Mathematics in China developed along different lines from that in early western civilisations. The westâs most influential ancient mathematician was Euclid, a Greek who lived in the 4th century BC. His approach, as laid out in his treatise on geometry, Elements, was deductive: he began with a small number of apparently indisputable truths, such as âthe whole is greater than the partâ, and on these foundations proved many less obvious things by logical deduction. As a result, the western tradition of maths is based on theorems and proofs.
While the Greeks dazzled with their abstract concepts and elegant proofs, the Chinese approach was fundamentally different. Chinese mathematicians began by providing workaday tools for people who needed to perform a raft of calculations in the highly organised new society they lived in. These generalised methods â algorithms â could be applied widely to many problems, and the aim was to generalise them still further to solve even more types of problem. âThe Chinese were the first people to talk explicitly about how to generate âsoftwareâ that can do a lot of jobs,â says Cullen.
Before the opening of tomb 247, the earliest known Chinese maths text was the Nine Chapters on Mathematical Procedures, which dates from around AD 100. Like Elements, this was a treatise on maths that became well known and was in continuous circulation for many centuries. âWeâve always wondered how Chinese maths got started,â says Cullen. âJust as Euclid must have built on the work of earlier scholars, the material in the Nine Chapters must have come from somewhere. The SuĂ n shĂč shu is as big a discovery as finding a book by one of Euclidâs predecessors.â
Some of the maths in the Nine Chapters was in use early in the imperial age: by 186 BC, the distinctive Chinese way of doing things was already established. The scroll includes problems ranging from simple calculations with fractions to methods for working out the volumes of solid shapes. There are techniques for calculating tax rates, the productivity of workers and much else of use to the budding bureaucrat. What is not in the collection is also interesting. There is no method for extracting square roots, although there is a technique for getting good approximations, and the Chinese equivalents of simultaneous equations and Pythagorasâs theorem â both in the Nine Chapters â are missing.
Clearly, there were people who had to perform a range of calculations of varying degrees of difficulty, and there were people who could provide methods that enabled them to do the job efficiently. So was the SuĂ n shĂč shu a handbook of reckoning that enabled the man in tomb 247 to fulfil his daily duties? âOfficials were expected to be able to do accounts. They had to add and subtract, multiply and divide. But this goes way beyond that,â says Cullen. âWithout being a QĂn official itâs impossible to know whether some of the more complex problems reflected any practical needs in the course of oneâs work. But some of the problems are clearly designed to make the reader rack their brains.â At least two have absolutely no practical use, he says. âThey are intended to show off the authorâs mathematical virtuosity.â So early Chinese maths wasnât entirely mundane and useful. âRight from the beginning there were people interested in maths for its own sake.â
To a keen mathematician like Cullen, thatâs no big surprise. What he did find surprising was how people learned about maths. In the early days of empire there was no well-ordered textbook like the Nine Chapters to refer to. People acquired knowledge piecemeal. When a mathematician came up with a new method, he would write it down on a strip or two of bamboo, and such snippets became the currency of learning. When you heard about a new method, you could ask for a copy and add it to your collection. âIt was a bit like surfing the net and downloading bits you found interesting into a file,â says Cullen.
So why did our anonymous official need his books after he was dead? In pre-Buddhist China, people expected to go to a shadowy place after death where they would carry on much as before. âHis was a high-status job that made him a person of importance. He would probably hope to continue doing it in the next world and so heâd want to have his equipment with him â in this case, his books.â And if he had a little spare time after taxing and judging his shadowy peers, he might relax with a little mathematics.
