A History of the World in 100 Objects (17 page)

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.

A small bronze sculpture of a bull with a figure leaping over it is now one of the highlights of the British Museum’s Minoan collection. It comes from the Mediterranean island of Crete, where it was made around 3,700 years ago.

The bull and the leaper are both made of bronze, and together they’re about 5 centimetres (2 inches) long and between 10 and 13 centimetres (4 or 5 inches) high. The bull is in full gallop – legs outstretched and head raised – and the figure is leaping over it in a great arching somersault. It’s probably a young man. He’s seized the bull’s horns and thrown his body right over, so that we see him at the point where his body has completely flipped. The two arching figures echo each other – the outward curve of the boy’s body being answered by the inward curve of the bull’s spine. It’s a most dynamic and beautiful piece of sculpture, and it carries us at once into the reality – and, no less important, the myth – of the history of Crete.

The image is a literal representation of something that to most people today is just a metaphor – ‘taking the bull by the horns’ is what we’re all meant to do when confronted with the big moral problems of life. But archaeology suggests that about 4,000 years ago a whole civilization seems to have been collectively fascinated by both the idea and the act of confronting the bull. Just why they were is one of the many mysteries of a society at the crossroads of Africa, Asia and Europe that played a key role in shaping what we now call the Middle East. It was a society that Homer described in lyric terms:

Out in the middle of the wine-dark sea, there is a land called Crete, a rich and lovely land washed by the sea on every side; and in it are many peoples and ninety cities. There, one language mingles with another … Among the cities is Knossos, a great city; and there Minos was nine years king, the boon companion of mighty Zeus.

In Greek myth, Minos, ruler of Crete, had a complex relationship with bulls. He was the son of the beautiful Europa by Zeus, king of the gods, but in order to father him and abduct Europa, Zeus had turned himself into a bull. Minos’s wife in turn had conceived an unnatural passion for a very beautiful bull, and the fruit of that obsession was the Minotaur, half-man, half-bull. Minos was so ashamed of his monstrous stepson that he had him imprisoned in an underground labyrinth, and there the Minotaur devoured a regular supply of maidens and youths sent every year by Athens – until, that is, the Greek hero Theseus succeeded in killing him. The story of Theseus and the Minotaur, of man first burying then confronting and slaying his monstrous demons, has been told and retold for centuries, by Ovid, Plutarch, Virgil and others. It’s part of the high canon of Greek myth, of Freudian psychology and of European art.

Archaeologists were captivated by these tales. Just over a hundred years ago, when Arthur Evans explored the island and decided to dig at Knossos, the bulls and monsters, palaces and labyrinths of Crete were very much in his mind. So although we have no idea what the people of this rich civilization around 1700
BC
actually called themselves, Evans, believing he was uncovering the world of Minos, called them quite simply Minoans, and they’ve remained Minoans to archaeologists ever since. In his extensive excavations, Evans uncovered the remains of a vast building complex, finding pottery and jewellery, carved stone seals, ivory, gold and bronze, and colourful frescoes, often depicting bulls; and he sought to interpret these finds in the light of the familiar myths. He was eager to reconstruct the role that the bulls might have played in the island’s economic and ceremonial life, so he was particularly interested in the discovery, some distance from Knossos, of the ‘Minoan’ bull-leaper.

It’s thought to have come from Rethymnon, a town on the north coast of the island, and it was probably originally deposited as an offering in a mountain shrine or in a cave sanctuary. Objects like this are often found in these holy places of Crete, suggesting that cattle played an important role in local religious rituals. Many scholars since Evans have tried to explain why these images were so important. They’ve asked what bull-leaping was for, and even if it was actually possible. Evans thought it was part of a festival in honour of a mother goddess. Others disagree, but bull-leaping has often been seen as a religious performance, possibly involving the sacrifice of the animal, and even occasionally the death of the leaper. Certainly, in this sculpture, both bull and human are engaged in a highly dangerous exercise. Being able to vault the animals would have taken months of training. We can say this with confidence, because the sport still survives today in parts of France and Spain. Sergio Delgado, a leading modern-day bull-leaper – or, to use the proper Spanish term,
recortador
– explains:

There has always been a kind of game between men and bulls, always. There is not a proper school for
recortadores
. You just learn how to understand the animal and how he will react to the arena. You only get this knowledge with experience.

There are three main techniques we had to learn: first the
recorte de riñón
[the ‘kidney cut’]; second it’s the
quiebro
[the ‘break’ or the ‘swing’]; the third one is the
salto
[or ‘leap’], which is mainly jumping right over the bull in a different variety of styles.

The bulls are not injured before the match, like in bullfighting. The bull never dies in the arena. We are risking our lives here, we get butted and gored as frequently as bullfighters. The bull is unpredictable. He is the one in charge. We never lose respect for the bull.

This continuing reverence for the bull is a fascinating contemporary echo of the suggestion made by some scholars that bull-leaping on Crete at the time of this little statue probably had a religious significance. Even the valuable bronze it’s made of suggests an offering to the gods.

The sculpture was made around 1700
BC
, in the middle of what archaeologists call the Bronze Age, when huge advances in making metals transformed the way humans could shape the world. Bronze, an alloy of copper and tin, is much harder and cuts much better than copper or gold; once discovered, it was widely used to make tools and weapons for more than a thousand years. But it also makes very beautiful sculpture, so it was frequently used for precious, probably devotional objects.