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Rhind Mathematical Papyrus (made around 3,500 years ago) found in Thebes, Egypt

In seven houses there are seven cats. Each cat catches seven mice. Each mouse would have eaten seven ears of corn and each ear of corn, if sown, would have produced seven gallons of grain. How many things are mentioned in total?

While you're counting, I'll tell you that this is just one of nearly a hundred similar problems, all equally complicated, all carefully written out, with the answers, and showing the workings in best schoolbook manner, that are recorded in the Rhind Mathematical Papyrus - the most famous mathematical papyrus to have survived from Ancient Egypt, and the major source for our understanding of how the Egyptians thought about numbers.

"Some of the maths is very very practical, other problems are more abstract." (Eleanor Robson)

"I think we see the beginnings of a realisation that mathematics is not about specific numbers, about specific problems; that you can extract from it general procedures, general rules, that you can follow in lots of cases." (Clive Rix)

It owes its name to an Aberdeen lawyer, Alexander Rhind, who in the 1850s took to wintering in Egypt because the dry heat helped his tuberculosis. There, in Luxor, he bought this papyrus, which turns out to be the largest mathematical text we know, not just from Egypt but from anywhere in the ancient world.

As it is extremely sensitive to humidity and to light, we keep it here in the British Museum, in the Papyrus Room, which I am just going to go into now ... It's pretty dry and pretty stuffy in here, in fact I imagine rather like the conditions in an Ancient Egyptian tomb, which suits the papyrus - above all because of course it's dark, and therefore the writing doesn't fade. The whole papyrus would originally have been about 17 feet (or 5m) long and would normally have been rolled up in a scroll. Today it's in three pieces - the two largest ones in the British Museum. It's simply framed under glass to protect it. The papyrus is about a foot (or 30cm) high, and if you look closely you can see the fibres of the papyrus plant.

Making papyrus is laborious but in fact, quite straightforward. The plant itself - a kind of reed that can grow to about 15 feet (or 4.5m) high - was plentiful in the Nile Delta. The pith of the plant is sliced into strips, which are soaked and pressed together to form sheets - very conveniently, the organic fibres of papyrus were meshed together without needing glue, and the sheets are then dried and rubbed smooth with a stone.

The result is a wonderful surface for writing on - papyrus went on being used across the Mediterranean until about a thousand years ago, and indeed gave most European languages their very word for paper.

But it was expensive - a 17-foot roll like the Rhind Mathematical Papyrus would have cost two copper deben, about the same as a small goat. So this is an object for the well-off.

But why would you spend so much money on a book of mathematical puzzles? Is this the Ancient Egyptian version of our craze for Sudoku? The answer is ... not quite. Because to own this scroll would, in fact, have been a very good career move. If you wanted to play any serious part in the Egyptian state, you had to be numerate. A society as complex as this needed people who could supervise building works, organise payments, manage food supplies, plan troop movements, compute the flood levels of the Nile - and much, much more. To be a scribe, a member of the civil service of the pharaohs, you had to demonstrate your mathematical competence. As one contemporary writer put it:

"So that you may open treasuries and granaries, so that you may take delivery from one corn-bearing ship at the entrance to the granary, so that on feast days you may measure out the gods' offerings." [Papyrus Lausing BM 9994]

The whole Rhind Mathematical Papyrus contains 84 different problems - calculations that would have been used in different scenarios to solve practical difficulties of life; for instance, how to calculate the slope of a pyramid, or the amount of food necessary for different kinds of birds. It's mostly written in black, but red is used for each problem's title and to explain the solution. And, interestingly, it's not written in hieroglyphs, but in a particular kind of administrative short-hand that's much quicker, much simpler, to write - and looks a bit like scribbly hand-writing.

In short, the Rhind Mathematical Papyrus teaches you all you need to know for a dazzling administrative career. It's a crammer for the Civil Service exams around 1550 BC and, like self-help publications today that promise instant success, it has a wonderful title, written boldly in red on the front page:

"The correct method of reckoning, for grasping the meaning of things, and knowing everything - obscurities and all secrets."

In other words: "Buy me, and you can't go wrong". The numeracy of the Egyptians, honed by works like the Rhind Mathematical Papyrus, was widely admired across the ancient world. Plato, for example, urged the Greeks to copy the Egyptians, where ...

" ... The teachers, by applying the rules and practices of arithmetic to play, prepare their pupils for the tasks of marshalling and leading armies and organising military expeditions and all together form them into persons more useful to themselves and to others and a great deal wider awake." [Laws 7,819]

But if everybody agreed that training like this produced a formidable state machine, the question of what mathematics the Greeks actually did learn from the Egyptians, remains a matter of debate. Clive Rix, of the University of Leicester:

"Well, the interesting thing is, that the traditional view has always been that the Greeks learned of their geometry from the Egyptians. Greek writers such as Herodotus, Plato, Aristotle, all refer to the outstanding skills of the Egyptians in geometry."

The problem is that we have only a very few surviving Egyptian mathematical documents - many others must have perished. So, although we have to assume that there was a flourishing higher mathematics, we just don't have the evidence for it. Clive Rix again:

"If we didn't have the Rhind Mathematical Papyrus, we'd actually know very little indeed about how the Egyptians did mathematics. The algebra is entirely what we would call linear algebra, straight line equations. There are some what now we would call arithmetical progressions, which are a little bit more sophisticated. The geometry's a very basic kind as well. Ahmose tells us how to calculate the area of a circle, and how to calculate the area of a triangle. There is nothing in this papyrus that would trouble your average GCSE student, and most of the stuff is rather less advanced than that."

But this is, of course, what you'd expect, because the person using the Rhind Mathematical Papyrus is not training to be a mathematician. He just needs to know enough to handle tricky practical problems - like how to divide up rations among workmen. If, for instance, you have ten gallons (or 45 litres) of animal fat to get you through the year, how much can you consume every day? Dividing 10 by 365 was as tricky then as it is now, but it was essential if you were going to keep a workforce properly supplied and energised. As Eleanor Robson, a specialist in ancient mathematics from Cambridge University explains:

"Everyone who's writing mathematics is doing it because they're learning how to be a literate, numerate manager, a bureaucrat, a scribe - and they're both learning the technical skills and how to manage numbers and weights and measures, in order to help palaces and temples manage their large economies. There must have been a whole lot of discussion of mathematics and how to solve the problems of managing huge building projects like the pyramids and the temples, and managing the huge work-forces that went with it, and feeding them all."

How that more sophisticated discussion of mathematics was conducted, or transmitted, we can only guess. The evidence that has come down to us is maddeningly fragmentary, because papyrus is so fragile, because it rots in the damp, and it burns so easily. We don't know where the Rhind Mathematical Papyrus came from, but we presume that it must have been a tomb. There are some examples of private libraries being buried with their owners - presumably to establish their educational and administrative credentials in the afterlife.

This loss of evidence makes it very hard to form a view of how Egypt stood in comparison to its neighbours. Eleanor Robson again:

"It's quite difficult to tell exactly how representative Egyptian mathematics is in the early second millennium BC. The only evidence we've got to compare it with at the same time is from Babylonia, southern Iraq. Because they were the only two civilisations at that point that actually used writing. So I'm sure that lots of cultures were counting and managing with numbers, but they all did it - as far as we know - without ever writing things down. The Babylonians we know a lot more about, because they wrote on clay tablets and, unlike papyrus, clay survives very well in the ground over thousands of years. So for Egyptian mathematics we have perhaps six, maximum ten, pieces of writing about mathematics, and the biggest of course is the Rhind Papyrus."

The Rhind Papyrus certainly gives us no sense of maths as an abstract discipline through which the world can be conceived and contemplated anew. But it does let us glimpse - and share - the daily headaches of an Egyptian administrator. Like all civil servants, he seems to be looking anxiously over his shoulder at the National Audit Office, eager to ensure that he is getting value for money. So there are calculations about how many gallons of beer, or how many loaves of bread, you should be able to get from a given amount of grain, and how to calculate whether the beer or the bread that you've been paying for has been adulterated.

For me, the most remarkable thing about this papyrus is how close it lets me get to the fascinating quirky details of aspects of daily life under the pharaohs, not least culinary. So did the Egyptians really eat foie gras? From the papyrus, you can learn that if you force-feed a goose, it needs five times as much grain as a free-range goose will eat. But Ancient Egypt also seems to have had battery-farming, because we're told that geese kept in a coop - and so presumably unable to move - will need only a quarter of the food consumed by their free-range counterparts, and so would be much cheaper to fatten for market. Whether there were also champions of animal rights in Ancient Egypt at this date we just don't know.

But in between the beer and the bread, and the hypothetical foie gras, you can see the logistical infrastructure of an enduring and powerful state, able to mobilise vast human and economic resources for public works and military campaigns. The Egypt of the pharaohs was, to its contemporaries, a land of superlatives - astonishing visitors from all over the Middle East by the colossal scale of its buildings and sculptures, as it still does us today. Like all successful states, then as now, it needed people who could do the maths.

And if you're still counting the cats, and the mice, and the ears of grain in the puzzle that I began with, the answer is, of course ...19,607.

The transcript for this programme will be published when the programme is broadcast.

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