Part of the Rhind Mathematical Papyrus showing how to calculate the area of a triangle
Papyrus found at Thebes (near Luxor), Egypt
AROUND 1550 BC
In seven houses there are seven cats. Each cat catches seven mice. If each mouse were to eat seven ears of corn and each ear of corn, if sown, were to produce seven gallons of grain, how many things are mentioned in total?
This is just one of dozens of similar problems, all equally complicated, all carefully written out – with the answers and showing the working in best schoolbook manner – that are recorded in the Rhind Mathematical Papyrus. This object is 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.
The Rhind Papyrus 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’re paying for has been adulterated.
The papyrus contains eighty-four mathematical problems. Red ink indicates the name of or answer to a problem
The whole Rhind Papyrus contains eighty-four different problems – calculations that would have been used in different scenarios to solve the practical difficulties of administrative life, for instance how to calculate the slope of a pyramid, or the amount of food necessary for different kinds of domesticated birds. It’s mostly written in black, but red is used for each problem’s title and solution. And, interestingly, it is written not in hieroglyphs but in a particular kind of scribbly administrative shorthand that’s much quicker, much simpler, to write.
The papyrus owes its name to an Aberdeen lawyer, Alexander Rhind, who took to wintering in Egypt in the 1850s because the dry heat helped his tuberculosis. There, in Luxor, he bought this papyrus, which turned out to be the largest ancient mathematical text we know, not just from Egypt but from anywhere in the ancient world.
Because it is extremely sensitive to humidity and to light, we keep it in the Papyrus Room of the British Museum. It’s pretty dry and stuffy there, and above all it’s dark, all of which suits the papyrus, which rots in the damp and fades in bright light. It’s the nearest we can get in Bloomsbury to conditions in an ancient Egyptian tomb, where the papyrus presumably spent most of its existence. The whole papyrus would originally have been about 5 metres (17 feet) long and would normally have been rolled up in a scroll. Today it’s in three pieces. The two largest ones are in the British Museum, framed under glass to protect them (the third is in the Brooklyn Museum, New York). The papyrus is about 30 centimetres (roughly a foot) high, and if you look closely you can see the fibres of the papyrus plant.
Making papyrus is laborious but quite straightforward. The plant itself – a kind of reed which can grow to about 4.5 metres (15 feet) 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, and the sheets are then dried and rubbed smooth with a stone. Conveniently, the organic fibres of papyrus mesh together without the need for glue. 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 papyrus was expensive – a 5-metre roll like the Rhind Papyrus would have cost two copper deben, about the same as a small goat. So this is an object for the well off.
Why would you spend so much money on a book of mathematical puzzles? I think because to own this scroll would have been a 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 theirs needed people who could supervise building works, organize payments, manage food supplies, plan troop movements, compute the flood levels of the Nile – and 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.
The Rhind Papyrus teaches you all you need to know for a dazzling administrative career. It is effectively a crammer for the Egyptian civil service exams around 1550 BC. 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: ‘All the maths you need to know. Buy me, and you buy success.’
The numeracy of the Egyptians, honed by works like the Rhind Papyrus, was widely admired across the ancient world. Plato, for example, urged the Greeks to copy the Egyptians, for whom
The teachers, by applying the rules and practices of arithmetic to play, prepare their pupils for the tasks of marshalling and leading armies and organizing military expeditions and all together form them into persons more useful to themselves and to others and a great deal wider awake.
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. 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 do not have the evidence for it. Professor Clive Rix, of the University of Leicester, emphasizes the significance of the Rhind Papyrus:
The traditional view has always been that the Greeks learnt their geometry from the Egyptians. Greek writers such as Herodotus, Plato and Aristotle all refer to the outstanding skills of the Egyptians in geometry.
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 of what we call arithmetical progressions, which are a little bit more sophisticated. The geometry’s a very basic kind as well. Ahmose [the original copyist of the papyrus] 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 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 10 gallons 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 energized. Eleanor Robson, a specialist in ancient mathematics from Cambridge University, explains:
Everyone who was writing mathematics was doing it because they were learning how to be a literate, numerate manager, a bureaucrat, a scribe – and they were learning both 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 workforces 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 that it tends to crumble, it rots in damp conditions, and it burns so easily. We don’t even know where the Rhind 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 with its neighbours and to understand exactly how representative Egyptian mathematics is around 1550 BC. Eleanor Robson tells us:
The only evidence from the same time we’ve got to compare it with is from Babylonia, in southern Iraq, because they were the only two civilizations at that point that actually used writing. I’m sure that lots of other 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.
For me, the most remarkable thing about this papyrus is how close it lets us get to the quirky details of daily life under the pharaohs, not least the culinary aspects. From it we learn that if you force-feed a goose it needs five times as much grain as a free-range goose will eat. So did the Egyptians eat foie gras? Ancient Egypt also seems to have had battery-farming, because we’re told that geese kept in a coop – 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.
‘In seven houses there are seven cats …’
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 mobilize 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 puzzling over the cats, and the mice, and the ears of grain in the puzzle that I began with, the answer is … 19,607.