Bamboo strips of the Suan shu shu, "Writings on Reckoning"

China’s rich history of scientific and technical contributions to human culture stretches back to antiquity, as Christopher Cullen has been discovering when it comes to numbers.

Numbers mattered vitally to the new centralised state’s view of what was needed for order to be maintained, both in relation to the universe and in everyday life.

The idea that the material world can be understood through numbers, patterns and mathematical laws was one of the most fruitful departures in the intellectual history of mankind. But when and how did that departure take place? For some, the answer goes back to the ancient Greeks – and amongst the Greeks themselves, Aristotle looked back to the disciples of Pythagoras who, he says, ‘thought the principles [of mathematics] were the principles of all things’ (Metaphysics, Book 1). The research interests of Professor Christopher Cullen, of the Needham Research Institute in Cambridge and the Faculty of Asian and Middle Eastern Studies, look in a different direction – to ancient China – to discover other evidence of humanity’s early efforts to make quantitative sense of the cosmos.

Preserving harmony and order

The Chinese Empire began with the unification of a collection of culturally related but often quarrelsome states on the East Asian mainland by the First Emperor of the short-lived Qin dynasty (BC 221–207), whose rule was succeeded by the four centuries of the Han dynasty (BC 206–AD 220). Numbers mattered vitally to the new centralised state’s view of what was needed for order to be maintained, both in relation to the universe and in everyday life.

When it came to relations between the human world and the impersonal powers that underpinned the regularities of the cosmos, the lead human actor was the Emperor himself, who performed annual rituals to ensure that the harmony of the cosmos was preserved. To ensure he did the right thing at exactly the right time, a permanent staff of sky-watchers and calculators, equipped with the best instruments available, were tasked by the Imperial Chinese government to set up systems to predict all predictable celestial events – from new moons, solstices and equinoxes, to the motions of the planets.

But it was not only at the top of the Chinese state that numbers were supremely important. Much of the life of an ordinary official consisted of quantitative management of physical resources such as grain, labour and time. All of these required the use of calculation techniques that were often considerably more than simple addition and subtraction. Our understanding of the importance of these techniques in official life has been immensely deepened by the discovery of a collection of material written on 190 bamboo strips in the tomb of a provincial official who died in BC 186. This collection bears the label Suan shu shu ‘Writings on Reckoning’ and is the oldest known Chinese text on mathematics.

Professor Cullen’s translation of this collection has shown that its 7000 characters explain mathematical techniques ranging from elementary fractions through to methods for solving problems that would nowadays be formulated as linear algebraic equations. One obsession of the ancient official mind that pervades this collection is the need to control every activity in the minutest detail. A fascinating example describes the most mathematically efficient use of time for a man working in the forest making charcoal. In one day he produces seven bushels of charcoal, but if the next day he carts it all to the Charcoal Office it would only take him 7/10 of a day’s work, leaving him with 3/10 of a day free – clearly an intolerable situation. The calculations recommend that he should work for 10 full days producing charcoal and for seven full days carting it. Just as there was to be order and predictability in the heavens, so it was to be on Earth.

Celestial software

Detailed contemporary accounts of the predictive systems constructed by early Chinese astronomers – in effect, ancient computing programs – are preserved in historical sources. These include the Han astronomical system (Han li) officially adopted in AD 85, whose procedures Professor Cullen has translated into modern-day Excel spreadsheets. At the tap of a key, these astronomical systems, made nearly 2000 years ago, will calculate all the solar lunar and planetary data for any given year since.

Impressive as this is, the records of the Chinese astronomers go further than simple predictions. Often these historical accounts also furnish the evidence and reasoning that led to the adoption of new (and hopefully improved) systems from time to time. They provide fascinating evidence of such fundamental innovations as the adoption of the ecliptic (the path that the Sun traces out in the sky) as a reference system and the discovery of the varying apparent speeds of the Sun and the Moon. Remarkably, these full Chinese records begin at an earlier stage of astronomical sophistication than found anywhere else in the world.

A ‘Theory of Everything’

The first Chinese astronomical system for which full documentation has been found is the San tong li ‘Triple Concordance System’, which was elaborated by the scholar and statesman Liu Xin around AD 10. As well as specifying his ‘computing program’, Liu Xin wrote a lengthy treatise explaining the rationale behind it. His celestial predictions were derived from numbers that were (in his view) basic to the cosmic order. The ambition was clear: a Grand Universal Theory of Everything. In fact, Liu Xin’s predictions are strikingly accurate. His theoretical value for the mean length of a lunar month was 29.5309 days, not far from the modern value of 29.5306 days; the error amounts to no more than one hour in 139 months. At the other end of the world, the Pythagoreans would certainly have agreed that Liu Xin was on to something!

The place of numbers

Professor Cullen has recently been looking into the broader question of who studied methods of calculation in early Imperial China – who taught whom, and why? One fascinating point that emerges from this enquiry is that the first historically identifiable teacher–pupil pair in this area was female. The teacher was the great scholar Ban Zhao (c. AD 45–114) and the pupil was the Empress Deng (AD 80–121), who as a girl was so obsessed with books, to the exclusion of all interest in ‘women's work’, that her exasperated mother once exclaimed ‘Are you going in for a doctorate?’

It is around the time that the Empress took her supervisions from Ban Zhao that some scholars date the compilation of the first systematic Chinese book on mathematics extant today, the Jiu zhang suan shu ‘Mathematical Methods in Nine Sections’. The compiler of this book is unknown. Could it have been a teaching aid designed by Ban Zhao for her Imperial pupil? The speculation is an intriguing one.

There is certainly much more to be settled about the place of numbers in ancient Chinese thought and practice. The answers should help us to understand an important part of the intellectual history of the whole of humankind.

For more information, please contact the author Professor Christopher Cullen ( at the Needham Research Institute(

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