The Evolution of Mathematics through History

Throughout history humankind has struggled to understand the fundamental workings of the material world. We've endeavored to discover the rules and patterns that determine the qualities of the objects that surround us, and their complex relationship to us and each other. Over thousands of years, societies all over the world have found that one discipline above all others yields certain knowledge about the underlying realities of the physical world. That discipline is mathematics. I'm Marcus de Sotois, and I'm a mathematician. I see myself as a pattern searcher, hunting down the hidden structures that lie behind the apparent chaos and complexity of the world around us. In my search for pattern and order, I draw upon the work of the great mathematicians who've gone before me. People belonging to cultures across the globe, whose innovations created the language the universe is written in. I want to take you on a journey through time and space, and track the growth of mathematics, from its awakening to the sophisticated subject we know today. Using computer generated imagery, we'll explore the trailblazing discoveries that allowed the earliest civilizations to understand the world mathematically. This is the story of maths. Our world is made up of patterns and sequences. They're all around us. Day becomes night. Animals travel across the Earth in ever-changing formations. Landscapes are constantly altering. One of the reasons mathematics began was because we needed to find a way of making sense of these natural patterns. The most basic concepts of maths, space and quantity, are hardwired into our brains. Even animals have a sense of distance and number, assessing when their pack is outnumbered and whether to fight or fly. Calculating whether their prey is within striking distance. Understanding maths is the difference between life and death. But it was man who took these basic concepts and started to build upon these foundations. At some point, humans started to spot patterns, to make connections, to count and to order the world around them. And with this, a whole new mathematical universe began to emerge. This is the River Nile. It's been the lifeline of Egypt for millennia. I've come here because it's where some of the first signs of mathematics as we know it today emerged. People abandoned nomadic life and began settling here as early as 6000 BC. The conditions were perfect for farming. The most important event for Egyptian agriculture each year was the flooding of the Nile. So this was used as a marker to start each new year. Egyptians did record what was going on over periods of time. So in order to establish a calendar like this, you need to count how many days, for example, happened in between lunar phases, or how many days happened in between two floodings of the Nile. Recording the patterns for seasons was essential, not only to their management of the land, but also their religion. The ancient Egyptians who settled on the Nile banks believed it was the river god Happy who flooded the river each year. And in return for the life-giving water, the citizens offered a portion of the yield as a thanksgiving. As settlements grew larger, it became necessary to find ways to administer them. Areas of land needed to be calculated, crop yields predicted, taxes charged and collated. In short, people needed to count and measure. The Egyptians used their bodies to measure the world, and it's how their units of measurement evolved. A palm was the width of a hand, a cubit an arm. from elbow to fingertips. Land cubits, strips of land measuring a cubit by a hundred, were used by the Faroes'surveyors to calculate areas. There's a very strong link between bureaucracy and the development of mathematics in ancient Egypt and we can see this link right from the beginning from the invention of the number system throughout Egyptian history really. For the Old Kingdom the only evidence we have are metrological systems, that is measurements for areas for length. This points to a bureaucratic need to develop such things. It was vital to know the area of a farmer's land so he could be taxed accordingly. Or the Nile robbed him of part of his land so he could request a rebate. It meant that the pharaoh's surveyors were often calculating the area of irregular parcels of land. And it was the need to solve such practical problems that made them the earliest mathematical innovators. The Egyptians needed some way to record the results of their calculations. Amongst all the hieroglyphs that cover the tourist souvenirs scattered around the city of Cairo, I was on the hunt for those that recorded some of the first numbers in history. They were difficult to track down. But I did find them in the end. The Egyptians were using a decimal system, motivated by the ten fingers on our hands. The sign for one was a stroke, ten a heel bump. 100 a coil of rope and a thousand and Lotus plans yeah how much is this t-shirt 25 so that'd be two neighbors and five strokes so you So you're not going to charge me anything up here? Here, one million. One million? Oh, my God. No, no, no. Is this one million? One million. Yeah, that's pretty big. The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed. They had no concept of a place value. So one stroke could only represent one unit, not a hundred or a thousand. Although you can write a million with just one character, rather than the seven that we use, if you want to write a million minus one, then the poor old Egyptian scribe has got to write nine strokes, nine heel bones, nine coils of rope, and so on, a total of 54 characters. Despite the drawback of this number system, the Egyptians were brilliant problem solvers. We know this because of the few records that have survived. The Egyptian scribes used sheets of papyrus to record their mathematical discoveries. This delicate material made from reeds decayed over time, and many secrets perished with it. But there's one revealing document that has survived. The Rhind Mathematical Papyrus is the most important document we have today for Egyptian mathematics. We get a good overview of what types of problems the Egyptians would have dealt with in their mathematics. We also get explicitly stated how multiplications and divisions were carried out. The papyri show how to multiply two large numbers together. But to illustrate the method, let's take two smaller numbers. Let's do 3 times 6. The scribe would take the first number 3 and put it in one column. In the second column, he would place the number 1. Then he would double the numbers in each column. So 3 becomes 6. And 6 would become 12. And then in the second column, 1 would become 2, and 2 becomes 4. Now here's the really clever bit. The scribe wants to multiply 3 by 6. So he takes the powers of 2 in the second column, which add up to 6. So that's 2 plus 4. Then he moves back to the first column and just takes those rows corresponding to the 2 and the 4. So that's 6 and the 12. He adds those together to get the answer of 18. But for me, the most striking thing about this method is that the scribe has effectively written that second number in binary. 6 is 1 lot of 4, 1 lot of 2, and no units, which is 1, 1, 0. The Egyptians have understood the power of binary over 3,000 years before the mathematician and philosopher Leibniz would reveal their potential. Today, the whole technological world depends on the same principles that were used in ancient Egypt. The Rhine Papyrus was recorded by a scribe called Achmes around 1650 BC and his problems are concerned with finding solutions to everyday situations. Several of the problems mention bread and beer, which isn't surprising as Egyptian workers were paid in food and drink. One is concerned with how to divide nine loaves equally between ten people without a fight breaking out. I've got nine loaves of bread here. I'm going to take five of them. and cut them into halves. Of course, nine people could shave a tenth off their loaf and give the pile of crumbs to the tenth person, but the Egyptians developed a far more elegant solution. Take the next four and divide those into thirds. For two of the thirds, I'm now going to cut into fifths, so each piece will be one-fifteenth. Each person then gets one half, and one third, and one fifteenth. It's through such seemingly practical problems that we start to see a more abstract mathematics developing. Suddenly new numbers are on the scene, fractions. And it isn't too long before the Egyptians are exploring the mathematics of these numbers. Fractions are clearly of practical importance to anyone dividing quantities for trade in the market. To log these transactions, the Egyptians developed notation which recorded these new numbers. One of the earliest representations of these fractions came from a hieroglyph which had great mystical significance. It's called the Eye of Horus. Horus was an old kingdom god, depicted as half man, half falcon. According to legend, Horus's father was killed by his other son, Seth. Horus was determined to avenge the murder. During one particularly fierce battle, Seth ripped out Horus's eye, tore it up and scattered it over Egypt. But the gods were looking favourably on Horus. They gathered up the scattered pieces and reassembled the eye. Each part of the eye represented a different fraction, each one half the fraction before. Although the original eye represented a whole unit, the reassembled eye is one sixty-fourth short. Although the Egyptians stopped at one over sixty-four, implicit in this picture is the possibility of adding more fractions, halving them each time, the sum getting closer and closer to one, but never quite reaching it. This is the first hint of something called a geometric series, and it appears at a number of points in the Rhine papyrus. But the concept of infinite series would remain hidden until the mathematicians of Asia discovered it centuries later. Having worked out a system of numbers, including these new fractions, it was time for the Egyptians to apply their knowledge to understanding shapes that they encountered day to day. These shapes were rarely regular squares or rectangles. And in the Rhine papyrus we find the area of a more organic form, the circle. What is astounding in the calculation of the area of a circle is its exactness, really. How they would have found their method is open to speculation because the texts we have do not show us the methods how they were found. This calculation is particularly striking because it depends on seeing how the shape of the circle can be approximated by shapes the Egyptians already understood. The Rhine Papyrus states that a circular field with a diameter of 9 units is close in area to a square with sides of 8. But how would this relationship have been discovered? My favorite theory sees the answer in the ancient game of Mancala. Mancala boards were found carved on the roofs of temples. Each player starts with an equal number of stones, and the object of the game is to move them around the board, capturing your opponent's counters on the way. As the players sat around waiting to make their next move, perhaps one of them realized that sometimes the balls fill the circular holes of the Mancala board in a rather nice way. He might have gone on to experiment with trying to make larger circles. Perhaps he noticed that 64 stones, the square of 8, can be used to make a circle with diameter 9 stones. By rearranging the stones, the circle has been approximated by a square. And because the area of a circle is pi times the radius squared, the Egyptian calculation gives us the first accurate value for pi. The area of the circle is 64. Divide this by the radius squared, in this case 4.5 squared, and you get a value for pi. So 64 divided by 4.5 squared is 3.16, just a little under two hundredths away from its true value. But the really brilliant thing is the Egyptians are using these smaller shapes to capture the larger shape. But there's one imposing and majestic symbol of Egyptian mathematics. we haven't attempted to run it the pyramid i've seen so many pictures that i couldn't believe i'd be impressed by them but meeting them face to face you understand why they're called one of the seven wonders of the ancient world they're simply breathtaking and how much more impressive they must have been in their day when the sides were as smooth as glass reflecting the desert sun to me it looks like there might be mirror pyramids hiding underneath the desert, which will complete these shapes to make perfectly symmetrical octahedrons. Sometimes in the shimmer of the desert heat, you can almost see these shapes. It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician. The pyramids are just a little short to create these perfect shapes, but some have suggested that another important mathematical concept might be hidden inside the proportions of the Great Pyramid. The Golden Ratio. Two lengths are in the Golden Ratio if the relationship of the longest to the shortest is the same as the sum of the two to the longest side. Such a ratio has been associated with the perfect proportions one finds all over the natural world, as well as in the work of artists, architects and designers for millennia. Whether the architects of the pyramids were conscious of this important mathematical idea, or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know. For me, the most impressive thing about the pyramids is the mathematical brilliance that went into making them, including the first inkling of one of the great theorems of the ancient world, Pythagoras'theorem. In order to get perfect right-angled corners on their buildings and pyramids, the Egyptians would have used a rope. with knots tied in it. At some point the Egyptians realised that if they took a triangle with sides marked with 3 knots, 4 knots and 5 knots, it guaranteed them a perfect right angle. This is because 3 squared plus 4 squared is equal to 5 squared. So we've got a perfect Pythagorean triangle. In fact any triangle whose sides satisfy this relationship will give me a 90 degree angle. But I'm pretty sure that the Egyptians hadn't got this sweeping generalization of their 3-4-5 triangle. We would not expect to find a general proof because this is not the style of Egyptian mathematics. Every problem was solved using concrete numbers, and then if a verification would be carried out at the end, it would use the result and these concrete given numbers. There's no general proof within the Egyptian mathematical texts. It would be some 2,000 years before the Greeks and Pythagoras would prove that all right-angled triangles shared certain properties. This wasn't the only mathematical idea that the Egyptians would anticipate. In a 4,000-year-old document called the Moscow Papyrus, we find a formula for the volume of a pyramid with its peak sliced off, which shows the first hint of calculus at work. For a culture like Egypt that is famous for its pyramids, you would expect problems like this to have been a regular feature within the mathematical texts. The calculation of the volume of a truncated pyramid is one of the most advanced bits according to our modern standards of mathematics that we have from ancient Egypt. The architects and engineers would certainly have wanted such a formula to calculate the amount of materials required to build it. But it's a mark of the sophistication of Egyptian mathematics that they were able to produce such a beautiful method. To understand how they derived their form, start with a pyramid built such that the highest point sits directly over one corner. Three of these can be put together to make a rectangular box. So the volume of the skewed pyramid is a third the volume of the box. That is, the height times the length times the width divided by three. Now comes an argument which shows the very first hints of the calculus at work thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory. Suppose you could cut the pyramid into slices. You could then slide the layers across to make the more symmetrical pyramid you see in Giza. However, the volume of the pyramid has not changed, despite the rearrangement of the layers. So the same formula works. The Egyptians were amazing innovators, and their ability to generate new mathematics was staggering. For me, they revealed the power of geometry and numbers and made the first moves towards some of the exciting mathematical discoveries to come. But there was another civilisation that had mathematics to rival that of Egypt, and we know much more about their achievements. This is Damascus. Over 5,000 years old and still vibrant and bustling today. It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt. The Babylonians controlled much of modern-day Iraq, Iran and Syria from 1800 BC. In order to expand and run their empire, they became masters of managing and manipulating numbers. We have law codes, for instance, that tell us about the way society is ordered. Now, the people we know most about are the scribes, the professionally literate, innumerate people who kept the records for the wealthy families and for the temples and palaces. Scribe schools existed from around 2500 BC. Aspiring scribes were sent there as children and learned how to read, write and work with numbers. Scribe records were kept on clay tablets, which allowed the Babylonians to manage and advance their empire. However, many of the tablets we have today aren't official documents, but children's exercises. It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics. So this is a geometrical textbook from about the 18th century BC, and I hope you can see that there are lots of pictures on it. And underneath each picture is a text that sets a problem about the picture. So, for instance, this one here says... I drew a square 60 units long and inside it I drew four circles. What are their areas? This little tablet here was written a thousand years at least later than the tablet here. has a very interesting relationship to it. It also has four circles on in a square, roughly drawn, but this isn't a textbook, it's a school exercise so that the adult scribe who's teaching the student is being given this as an example of completed homework or something like that. Like the Egyptians, the Babylonians appeared interested in solving practical problems to do with measuring and weighing. The Babylonian solutions to these problems are written like mathematical recipes. A scribe would simply follow and record a set of instructions. to get a result. Here's an example of the kind of problem they'd solve. Now I've got a bundle of cinnamon sticks here, but I'm not going to weigh them. Instead, I'm going to take four times their weight and add them to the scales. Now I'm going to add 20 gin. Gin was the ancient Babylonian measure of weight. I'm going to take half of everything here and add it again. So that's two bundles and 10 gin. Now everything on this side is equal to one manna. One manna was six. And here we have one of the first mathematical equations in history. Everything on this side is equal to one manna. But how much does the bundle of cinnamon sticks weigh? Without any algebraic language, they were able to manipulate the quantities to be able to prove... that the cinnamon sticks weighed 5 jinn. In my mind, it's this kind of problem which gives mathematics a bit of a bad name. You can blame those ancient Babylonians for all those torturous problems you had at school. But the ancient Babylonian scribes excelled at this kind of problem. Intriguingly, they weren't using powers of 10 like the Egyptians. They were using powers of 60. The Babylonians invented their number system, like the Egyptians, by using their fingers. But instead of counting through the ten fingers on their hand, Babylonians found a much more intriguing way to count body parts. They used the twelve knuckles on one hand and the five fingers on the other to be able to count twelve times five, i.e. sixty different numbers. So, for example, this number would have been two lots of twelve, twenty-four, and then one, two, three, four, five, to make twenty-nine. But the number 60 had another powerful property. It can be perfectly divided in a multitude of ways. Here are 60 beans. I can arrange them in two rows of 30, three rows of 20, four rows of 15, five rows of 12, or six rows of 10. The divisibility of 60 makes it a perfect base in which to do arithmetic. The base 60 system was so successful, we still use elements of it today. Every time we want to tell the time, we recognize units of 60. 60 seconds in a minute, 60 minutes in an hour. The most important feature of the Babylonians number system was that it recognized place value. Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording, the position of each Babylonian number records the power of 60 you're counting. Instead of inventing new symbols for bigger and bigger numbers, they would write 111. So this number would be 3661. The catalyst for this discovery was the Babylonians'desire to chart the course of the night sky. The Babylonians'calendar was based on the cycles of the moon. And so they needed a way of recording astronomically large numbers. Month by month, year by year, these cycles were recorded. From about 800 BC, there were complete lists of lunar eclipses. The Babylonian system of measurement was quite sophisticated at that time. They had a system of angular measurement, 360 degrees in a full circle. Each degree was divided into 60 minutes. A minute was further divided into 60 seconds. So they had a regular system for measurement, and it was in perfect harmony with their number system. So it was well suited not only for observation but also for calculation. But in order to calculate and cope with these large numbers, the Babylonians needed to invent a new symbol. And in so doing, they prepared the ground for one of the great breakthroughs in the history of mathematics, zero. In the early days, the Babylonians, in order to mark an empty place in the middle of a number, would simply leave a blank space. So they needed a way of representing nothing in the middle of a number. So they used a sign as a sort of breathing mark or a punctuation mark, and it comes to mean zero in the middle of a number. This was the first time zero in any form had appeared in the mathematical universe. But it would be over a thousand years before this little placeholder would become a number in its own right. Having established such a sophisticated system of numbers, they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia. Babylonian engineers and surveyors found ingenious ways of accessing water and channeling it to the crop fields. Yet again they used mathematics to come up with solutions. The Orontes Valley in Syria is still an agricultural hub, and the old methods of irrigation are being exploited today just as they were thousands of years ago. Many of the problems in Babylonian mathematics are concerned with measuring land, and it's here we see for the first time the use of quadratic equations, one of the greatest legacies of Babylonian mathematics. Quadratic equations... involve things where the unknown quantity you're trying to identify is multiplied by itself. We call this squaring because it gives the area of a square. And it's in the context of calculating the area of land that these quadratic equations naturally arise. Here's a typical problem. If a field has an area of 55 units, and one side is six units longer than the other, how long is the shorter side? The Babylonian solution was to reconfigure the field as a square. Cut three units off the end, and move this round. Now, there's a 3x3 piece missing, so let's add this in. The area of the field has increased by 9 units. This makes the new area 64. So the size of the square are 8 units. The problem solver knows that they've added 3 to this side. So, the original length must be 5. It may not look like it, but this is one of the first quadratic equations in history. In modern mathematics, I would use the symbolic language of algebra to solve this problem. But the amazing feat of the Babylonians is that they were using these geometric games to find the value, without any recourse to symbols or formulas. The Babylonians were enjoying problem solving for its own sake. They were falling in love with mathematics. The Babylonians'fascination with numbers soon found a place in their leisure time too. They were avid game players. The Babylonians and their descendants have been playing a version of backgammon for over 5,000 years. They played board games, from very posh board games in royal tombs, to little bits of board games found in schools, to board games scratched on the entrances of palaces, so that the guardsmen must have played at times when they were bored. And they used dice to move their counters round. People who played games were using numbers in their leisure time to try and outwit their opponent, doing mental arithmetic very fast. And so they were calculating in their leisure time without even thinking about it as being mathematical. hardware. I hadn't played backgammon for ages, but I reckon my maths would be good enough to give me a fighting chance. But it wasn't as easy as it seemed. as I thought. Oh! What the hell is that? Yeah, yeah, this is... This is one, this is two. Now you are in trouble. So I can't move anything, can I? I can't move these anyway. You cannot move these. You're gonna play... I just gotta... Oh, gosh. Exactly. There you go. You seem to be... All right. Three and four, so... Just like the ancient Babylonians, my opponents were masters of tactical mathematics. Yeah. So, I'm gonna go ahead and... I'm gonna go ahead and... Put it there. All right, good game. The Babylonians are recognised as one of the first cultures to use symmetrical mathematical shapes to make dice. But there is more heated debate about whether they might also have been the first to discover the secrets of another important shape, the right-angled triangle. We've already seen how the Egyptians used a 3-4-5 right-angled triangle. But what the Babylonians knew about this shape, and others like it, is much more sophisticated. This is the most famous and controversial ancient tablet we have. It's called Plimpton 322. Many mathematicians are convinced it shows the Babylonians could well have known the principle regarding right-angled triangles, that the square on the diagonal is the sum of the squares on the sides, and known it centuries before the Greeks claimed it. This is a copy of the arguably most famous Babylonian tablet, which is Plimpton 322, and these numbers here reflect the width or height of a triangle, this being the diagonal. The other side would be over here. The square of this column plus the square of the number in this column equals the square of the diagonal. They are arranged in an order of steadily decreasing angle on a very uniform basis, showing that somebody had a lot of understanding of how the numbers fit together. Here were 15 perfect Pythagorean triangles, all of whose sides had whole number lengths. It's tempting to think that the Babylonians were the first custodians of Pythagoras'theorem, and it's a conclusion that generations of historians have been seduced by. But there could be a much simpler explanation for the sets of three numbers which fulfil Pythagoras'theorem. It's not a systematic exploration of Pythagorean triples. It's simply a mathematics teacher doing some quite complicated calculations, but in order to produce some very simple numbers in order to set his students'problems about right-angled triangles. And in that sense, it's about Pythagorean triples, only incidentally. The most valuable clues to what they understood. could lie elsewhere. This small school exercise tablet is nearly 4,000 years old and reveals just what the Babylonians did know about right-angled triangles. It uses the principle of Pythagoras'theorem to find the value of an astounding... Drawn along the diagonal is a really very good approximation to the square root of 2. And so that shows us that it was known and used in school environments. Now why is this important? It's because the square root of 2 is what we now call an irrational number. That is, if we write it out in decimals or even in sexagesimal places, it doesn't end. The numbers go on forever after the decimal point. The implications of this calculation are far-reaching. Firstly, it means the Babylonians knew something of Pythagoras'theorem a thousand years before Pythagoras. Secondly, the fact that they can calculate this number to an accuracy of four decimal places shows an amazing arithmetic facility as well as a passion for mathematical detail. The Babylonians'mathematical dexterity was astounding and for nearly 2,000 years they spearheaded intellectual progress in the ancient world. But when their imperial power began to wane, so did their intellectual vigor. By 330 BC, the Greeks had advanced their imperial reach into old Mesopotamia. This is Palmyra in central Syria, a once great city built by the Greeks. The mathematical expertise needed to build structures with such geometric perfection is impressive. Just like the Babylonians before them, the Greeks were passionate about mathematics. The Greeks were clever colonists. They took the best from the civilizations they invaded to advance their own power and influence. But they were soon making contributions themselves. In my opinion, their greatest innovation was to do with a shift in the mind. What they initiated would influence humanity for centuries. They gave us the power of proof. Somehow they decided that they had to have a deductive system for their mathematics. And the typical deductive system was to begin with certain axioms, which you assume are true. It's as if you assume a certain theorem is true without proving it. And then using logical methods... And very careful steps from these actions you prove theorems. And then from those theorems you prove more theorems. And it just snowballs that way. Proof is what gives mathematics its strength. It's the power of proof which means that the discoveries of the Greeks are as true today as they were 2,000 years ago. I needed to head west into the heart of the old Greek empire to learn more. For me, Greek mathematics has always been heroic and romantic. I'm on my way to Samos, less than a mile from the Turkish coast. This place has become synonymous with the birth of Greek mathematics. And it's down to the legend of one man. His name is Pythagoras. The legends that surround his life and work have contributed to the celebrity status he has gained over the last 2000 years. He's credited, rightly or wrongly, with beginning the transformation from mathematics as a tool for accounting to the analytic subject we recognise today. Pythagoras is a controversial figure. Because he left no mathematical writings, many have questioned whether he indeed solved any of the theorems attributed to him. He founded a school in Samos in the 6th century BC, but his teachings were considered suspect, and the Pythagoreans a bizarre sect. There is good evidence that there were schools of Pythagoreans. They may have looked more like sects than what we associate with philosophical schools, because they didn't just share knowledge, they also shared... a way of life. They may have been communal living and they all seem to have been involved in the politics of their cities. One feature that makes them unusual in the ancient world is that they included women. But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians, the properties of right-angled triangles. What's known as Pythagoras'theorem states that if you take any right-angled triangle, build squares on all the sides, then the area of the largest square is equal to the sum of the squares on the two smaller sides. It's at this point for me that mathematics is born, and a gulf opens up between the other sciences. And the proof is as simple as it is devastating in its implications. Place four copies of the right-angled triangle on top of this surface. The square that you now see has sides equal to the hypotenuse of the triangle. By sliding these triangles around, we see how we can break the area of the large square up into the sum of two smaller squares. whose sides are given by the two short sides of the triangle. In other words, the square on the hypotenuse is equal to the sum of the squares on the other sides. Pythagoras'theorem. It illustrates one of the characteristic themes of Greek mathematics, the appeal to beautiful arguments in geometry rather than a reliance on number. Pythagoras may have fallen out of favor and many of the discoveries accredited to him have been contested recently but there's one mathematical theory that I'm loathe to take away from him. It's to do with music and the discovery of the harmonic series. The story goes that walking past a blacksmith's one day Pythagoras heard anvils being struck and notice how the notes being produced sounded in perfect harmony. He believed that there must be some rational explanation to make sense of why the notes sounded so appealing. The answer was mathematics. Experimenting with a stringed instrument, Pythagoras discovered that the intervals between harmonious musical notes were always represented as whole number ratios. And here's how he might have constructed his theory. First, play a note on the open string. Next, take half the length. The note almost sounds the same as the first note. In fact, it's an octave higher, but the relationship is so strong we give these notes the same name. Now, take a third the length. We get another note which sounds harmonious next to the first two. But take a length of string which is not in a whole number ratio, and all we get is dissonance. According to legend, Pythagoras was so excited by this discovery that he concluded the whole universe was built from numbers. But he and his followers were in for a rather unsettling challenge to their world view. And it came about as a result of a theorem which bears Pythagoras'name. Legend has it one of his followers, a mathematician called Hyppasus, set out to find the length of the diagonal for a right-angled triangle with two sides measuring one unit. Pythagoras's theorem implied that the length of the diagonal was a number whose square was two. The Pythagoreans assumed that the answer would be a fraction, but when Hyppasus tried to express it in this way, no matter how he tried, he couldn't capture it. Eventually he realized his mistake. It was the assumption that the value was a fraction at all which was wrong. The value of the square root of 2 was the number that the Babylonians etched into the Yale tablet. However, they didn't recognize the special character of this number. But, Hyppasus did. It was an irrational number. The discovery of this new number, and others like it, is akin to an explorer discovering a new continent, or a naturalist finding a new species. But these irrational numbers didn't fit the Pythagorean world view. Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy. But Hippasus let slip the discovery, and was promptly drowned for his attempts to broadcast their research. But these mathematical discoveries could not be easily suppressed. Schools of philosophy and science started to flourish all over Greece, building on these foundations. The most famous of these was the Academy. Plato founded this school in Athens in 387 BC. Although we think of him today as a philosopher, he was one of mathematics'most important patrons. Plato was enraptured by the Pythagorean worldview and considered mathematics the bedrock of knowledge. Some people would say that Plato is possibly the most influential figure for our perception of Greek mathematics. He argued that mathematics is an important form of knowledge and does have a connection with reality. So by knowing mathematics we'll know more about reality. In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking the secrets of the universe, a view still held by scientists today. Indeed, the importance Plato attached to geometry is encapsulated in the sign that was mounted above the academy. Let no one ignorant of geometry enter here. Plato proposed that the universe could be crystallized into five regular symmetrical shapes. These shapes, which we now call the platonic solids, were composed of regular polygons assembled to create three-dimensional symmetrical objects. The tetrahedron represented fire. The icosahedron, made from 20 triangles, represented water. The stable cube was Earth. The eight-faced octahedron was air. And the fifth platonic solid, the dodecahedron, made out of 12 pentagons, was reserved for the shape which captured Plato's view of the universe. Plato's theory would have a seismic influence and continue to inspire mathematicians and astronomers for over 1500 years. In addition to the breakthroughs made in the Academy, mathematical triumphs were also emerging from the edge of the Greek Empire and owed as much to the mathematical heritage of the Egyptians as the Greeks. Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC and its famous library soon gained a reputation to rival Plato's Academy. The kings of Alexandria were... prepared to invest in the arts, in culture, in technology, mathematics, grammar, because patronage for cultural pursuits was one way of showing that you were a more prestigious ruler and had a better entitlement to greatness. were destroyed when the Muslims conquered Egypt in the 7th century. But its spirit is alive in a new building. Today, the library remains a place of discovery and scholarship. Mathematicians and philosophers flocked to Alexandria, driven by their thirst for knowledge and the pursuit of excellence. The patrons of the library were the first professional scientists. individuals who have paid for their devotion to research. But of all those early pioneers, my hero is the enigmatic Greek mathematician Euclid. We know very little about Euclid's life, but his greatest achievements were as a chronicler of mathematics. Around 300 BC, he wrote the most important textbook of all time, The Elements. In The Elements... we find the culmination of the mathematical revolution which had taken place in Greece. It's built on a series of mathematical assumptions called axioms. For example, a line can be drawn between any two points. From these axioms, logical deductions are made and mathematical theorems established. The elements contains formulas for calculating the volumes of cones and cylinders. Proofs about geometric series, perfect numbers and primes. The climax of the elements is a proof that there are only five platonic solids. For me, this last theorem captures the power of mathematics. It's one thing to build five symmetrical solids, quite another to come up with a watertight, logical argument for why there can't be a sixth. The elements unfold like a wonderful, logical mystery novel. This is a story which transcends time. Scientific theories get knocked down from one generation to the next. But the theorems and the elements are as true today as they were 2,000 years ago. When you stop and think about it, it's really amazing that it's the same theorems that we teach. We may teach them in a slightly different way. We may organize them differently. But it's Euclidean geometry that is still valid, and even in higher mathematics. When you go to higher dimensional spaces, you're still using Euclidean geometry. Alexandria must have been an inspiring place for the ancient scholars, and Euclid's fame would have attracted even more eager young intellectuals to the Egyptian port. One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes. He would become a mathematical visionary. The best Greek mathematicians, they were always pushing the limits, pushing the envelope. So, Archimedes did what he could with polygons, with solids. He then moved on to centres of gravity, or he then moved on to the spiral. This instinct to... try and mathematize everything is something that I see as a legacy. One of Archimedes'specialities was weapons of mass destruction. They were used against the Romans when they invaded his home of Syracuse in 212 BC. He also designed mirrors which harnessed the power of the sun to set the Roman ships on fire. But to Archimedes, these endeavors were mere amusements in geometry. He had loftier ambitions. Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake and not for the ignoble trade of engineering or the sordid quest for profit. One of his finest investigations into pure mathematics was to produce formulas to calculate the areas of regular shapes. Archimedes'method was to capture new shapes... by using shapes he already understood. So for example, to calculate the area of a circle, he would enclose it inside a triangle. And then by doubling the number of sides on the triangle, the enclosing shape would get closer and closer to the circle. Indeed, we sometimes call a circle a polygon with an infinite number of sides. But by estimating the area of the circle, Archimedes is in fact getting a value for pi, the most important number in mathematics. However, it was calculating the volumes of solid objects where Archimedes excelled. He found a way to calculate the volume of a sphere by slicing it up and approximating each slice as a cylinder. He then added up the volumes of the slices to get an approximate value for the sphere. But his act of genius was to see what happens if you make the slices thinner and thinner. In the limit, the approximation becomes an exact... calculation but it was our comedies commitment to mathematics it will be his undoing Archimedes was contemplating a problem about circles traced in the sand when a Roman soldier accosted him. Archimedes was so engrossed in his problem that he insisted he be allowed to finish his theorem. But the Roman soldier was not interested in Archimedes'problem and killed him on the spot. Even in death, Archimedes'devotion to mathematics was unwavering. By the middle of the first century BC, the Romans had tightened their grip on the old Greek Empire. They were less smitten with the beauty of mathematics, and were more concerned with its practical applications. This pragmatic attitude signalled the beginning of the end for the great library of Alexandria. But one mathematician was determined to keep the legacy of the Greeks alive. Hypatia was exceptional, a female mathematician and a pagan in the piously Christian Roman Empire. Aitresha was very prestigious and very influential in her time. She was a teacher with a lot of students, a lot of followers. She was politically influential in Alexandria. So it's this combination of high knowledge and high prestige that may have made her a figure of hatred for the Christian mob. One morning during Lent, Hypatia was dragged off her chariot by a zealous Christian mob and taken to a church. There, she was tortured and brutally murdered. The dramatic circumstances of her life and death fascinated later generations. Sadly, her cult status eclipsed her mathematical achievements. She was in fact a brilliant teacher and theorist, and her death dealt a final blow to the Greek mathematical heritage of Alexandria. My travels have taken me on a fascinating journey to uncover the passion and innovation of the world's earliest mathematicians. It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece that are the foundations on which my subject is built today. But this is just the beginning of my mathematical odyssey. The next leg of my journey lies east, in the depths of Asia, where mathematicians scaled even greater heights in pursuit of knowledge. With this new era came a new language of algebra and numbers, better suited to telling the next chapter in the story of maths. You can learn more about the story of maths with the Open University at open2.net. And the story of maths continues here on BBC4 next Monday at the same time. Next tonight, British novelists become agents of provocation as they describe a time when nothing was sacred in their own words.

I see myself as a pattern searcher, hunting down the hidden structures that lie behind the apparent chaos and complexity of the world around us. In my search for pattern and order, I draw upon the work of the great mathematicians who've gone before me. People belonging to cultures across the globe, whose innovations created the language the universe is written in. I want to take you on a journey through time and space, and track the growth of mathematics, from its awakening to the sophisticated subject we know today.

Using computer generated imagery, we'll explore the trailblazing discoveries that allowed the earliest civilizations to understand the world mathematically. This is the story of maths. Our world is made up of patterns and sequences. They're all around us. Day becomes night.

Animals travel across the Earth in ever-changing formations. Landscapes are constantly altering. One of the reasons mathematics began was because we needed to find a way of making sense of these natural patterns.

The most basic concepts of maths, space and quantity, are hardwired into our brains. Even animals have a sense of distance and number, assessing when their pack is outnumbered and whether to fight or fly. Calculating whether their prey is within striking distance. Understanding maths is the difference between life and death. But it was man who took these basic concepts and started to build upon these foundations.

At some point, humans started to spot patterns, to make connections, to count and to order the world around them. And with this, a whole new mathematical universe began to emerge. This is the River Nile.

It's been the lifeline of Egypt for millennia. I've come here because it's where some of the first signs of mathematics as we know it today emerged. People abandoned nomadic life and began settling here as early as 6000 BC. The conditions were perfect for farming. The most important event for Egyptian agriculture each year was the flooding of the Nile.

So this was used as a marker to start each new year. Egyptians did record what was going on over periods of time. So in order to establish a calendar like this, you need to count how many days, for example, happened in between lunar phases, or how many days happened in between two floodings of the Nile. Recording the patterns for seasons was essential, not only to their management of the land, but also their religion. The ancient Egyptians who settled on the Nile banks believed it was the river god Happy who flooded the river each year.

And in return for the life-giving water, the citizens offered a portion of the yield as a thanksgiving. As settlements grew larger, it became necessary to find ways to administer them. Areas of land needed to be calculated, crop yields predicted, taxes charged and collated. In short, people needed to count and measure.

The Egyptians used their bodies to measure the world, and it's how their units of measurement evolved. A palm was the width of a hand, a cubit an arm. from elbow to fingertips. Land cubits, strips of land measuring a cubit by a hundred, were used by the Faroes'surveyors to calculate areas. There's a very strong link between bureaucracy and the development of mathematics in ancient Egypt and we can see this link right from the beginning from the invention of the number system throughout Egyptian history really.

For the Old Kingdom the only evidence we have are metrological systems, that is measurements for areas for length. This points to a bureaucratic need to develop such things. It was vital to know the area of a farmer's land so he could be taxed accordingly. Or the Nile robbed him of part of his land so he could request a rebate. It meant that the pharaoh's surveyors were often calculating the area of irregular parcels of land.

And it was the need to solve such practical problems that made them the earliest mathematical innovators. The Egyptians needed some way to record the results of their calculations. Amongst all the hieroglyphs that cover the tourist souvenirs scattered around the city of Cairo, I was on the hunt for those that recorded some of the first numbers in history. They were difficult to track down.

But I did find them in the end. The Egyptians were using a decimal system, motivated by the ten fingers on our hands. The sign for one was a stroke, ten a heel bump. 100 a coil of rope and a thousand and Lotus plans yeah how much is this t-shirt 25 so that'd be two neighbors and five strokes so you So you're not going to charge me anything up here? Here, one million.

One million? Oh, my God. No, no, no. Is this one million? One million.

Yeah, that's pretty big. The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed. They had no concept of a place value.

So one stroke could only represent one unit, not a hundred or a thousand. Although you can write a million with just one character, rather than the seven that we use, if you want to write a million minus one, then the poor old Egyptian scribe has got to write nine strokes, nine heel bones, nine coils of rope, and so on, a total of 54 characters. Despite the drawback of this number system, the Egyptians were brilliant problem solvers. We know this because of the few records that have survived.

The Egyptian scribes used sheets of papyrus to record their mathematical discoveries. This delicate material made from reeds decayed over time, and many secrets perished with it. But there's one revealing document that has survived. The Rhind Mathematical Papyrus is the most important document we have today for Egyptian mathematics.

We get a good overview of what types of problems the Egyptians would have dealt with in their mathematics. We also get explicitly stated how multiplications and divisions were carried out. The papyri show how to multiply two large numbers together. But to illustrate the method, let's take two smaller numbers. Let's do 3 times 6. The scribe would take the first number 3 and put it in one column.

In the second column, he would place the number 1. Then he would double the numbers in each column. So 3 becomes 6. And 6 would become 12. And then in the second column, 1 would become 2, and 2 becomes 4. Now here's the really clever bit. The scribe wants to multiply 3 by 6. So he takes the powers of 2 in the second column, which add up to 6. So that's 2 plus 4. Then he moves back to the first column and just takes those rows corresponding to the 2 and the 4. So that's 6 and the 12. He adds those together to get the answer of 18. But for me, the most striking thing about this method is that the scribe has effectively written that second number in binary.

6 is 1 lot of 4, 1 lot of 2, and no units, which is 1, 1, 0. The Egyptians have understood the power of binary over 3,000 years before the mathematician and philosopher Leibniz would reveal their potential. Today, the whole technological world depends on the same principles that were used in ancient Egypt. The Rhine Papyrus was recorded by a scribe called Achmes around 1650 BC and his problems are concerned with finding solutions to everyday situations. Several of the problems mention bread and beer, which isn't surprising as Egyptian workers were paid in food and drink.

One is concerned with how to divide nine loaves equally between ten people without a fight breaking out. I've got nine loaves of bread here. I'm going to take five of them. and cut them into halves. Of course, nine people could shave a tenth off their loaf and give the pile of crumbs to the tenth person, but the Egyptians developed a far more elegant solution.

Take the next four and divide those into thirds. For two of the thirds, I'm now going to cut into fifths, so each piece will be one-fifteenth. Each person then gets one half, and one third, and one fifteenth. It's through such seemingly practical problems that we start to see a more abstract mathematics developing. Suddenly new numbers are on the scene, fractions.

And it isn't too long before the Egyptians are exploring the mathematics of these numbers. Fractions are clearly of practical importance to anyone dividing quantities for trade in the market. To log these transactions, the Egyptians developed notation which recorded these new numbers.

One of the earliest representations of these fractions came from a hieroglyph which had great mystical significance. It's called the Eye of Horus. Horus was an old kingdom god, depicted as half man, half falcon. According to legend, Horus's father was killed by his other son, Seth.

Horus was determined to avenge the murder. During one particularly fierce battle, Seth ripped out Horus's eye, tore it up and scattered it over Egypt. But the gods were looking favourably on Horus. They gathered up the scattered pieces and reassembled the eye. Each part of the eye represented a different fraction, each one half the fraction before.

Although the original eye represented a whole unit, the reassembled eye is one sixty-fourth short. Although the Egyptians stopped at one over sixty-four, implicit in this picture is the possibility of adding more fractions, halving them each time, the sum getting closer and closer to one, but never quite reaching it. This is the first hint of something called a geometric series, and it appears at a number of points in the Rhine papyrus.

But the concept of infinite series would remain hidden until the mathematicians of Asia discovered it centuries later. Having worked out a system of numbers, including these new fractions, it was time for the Egyptians to apply their knowledge to understanding shapes that they encountered day to day. These shapes were rarely regular squares or rectangles. And in the Rhine papyrus we find the area of a more organic form, the circle. What is astounding in the calculation of the area of a circle is its exactness, really.

How they would have found their method is open to speculation because the texts we have do not show us the methods how they were found. This calculation is particularly striking because it depends on seeing how the shape of the circle can be approximated by shapes the Egyptians already understood. The Rhine Papyrus states that a circular field with a diameter of 9 units is close in area to a square with sides of 8. But how would this relationship have been discovered? My favorite theory sees the answer in the ancient game of Mancala. Mancala boards were found carved on the roofs of temples.

Each player starts with an equal number of stones, and the object of the game is to move them around the board, capturing your opponent's counters on the way. As the players sat around waiting to make their next move, perhaps one of them realized that sometimes the balls fill the circular holes of the Mancala board in a rather nice way. He might have gone on to experiment with trying to make larger circles. Perhaps he noticed that 64 stones, the square of 8, can be used to make a circle with diameter 9 stones.

By rearranging the stones, the circle has been approximated by a square. And because the area of a circle is pi times the radius squared, the Egyptian calculation gives us the first accurate value for pi. The area of the circle is 64. Divide this by the radius squared, in this case 4.5 squared, and you get a value for pi.

So 64 divided by 4.5 squared is 3.16, just a little under two hundredths away from its true value. But the really brilliant thing is the Egyptians are using these smaller shapes to capture the larger shape. But there's one imposing and majestic symbol of Egyptian mathematics.

we haven't attempted to run it the pyramid i've seen so many pictures that i couldn't believe i'd be impressed by them but meeting them face to face you understand why they're called one of the seven wonders of the ancient world they're simply breathtaking and how much more impressive they must have been in their day when the sides were as smooth as glass reflecting the desert sun to me it looks like there might be mirror pyramids hiding underneath the desert, which will complete these shapes to make perfectly symmetrical octahedrons. Sometimes in the shimmer of the desert heat, you can almost see these shapes. It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician.

The pyramids are just a little short to create these perfect shapes, but some have suggested that another important mathematical concept might be hidden inside the proportions of the Great Pyramid. The Golden Ratio. Two lengths are in the Golden Ratio if the relationship of the longest to the shortest is the same as the sum of the two to the longest side.

Such a ratio has been associated with the perfect proportions one finds all over the natural world, as well as in the work of artists, architects and designers for millennia. Whether the architects of the pyramids were conscious of this important mathematical idea, or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know. For me, the most impressive thing about the pyramids is the mathematical brilliance that went into making them, including the first inkling of one of the great theorems of the ancient world, Pythagoras'theorem.

In order to get perfect right-angled corners on their buildings and pyramids, the Egyptians would have used a rope. with knots tied in it. At some point the Egyptians realised that if they took a triangle with sides marked with 3 knots, 4 knots and 5 knots, it guaranteed them a perfect right angle.

This is because 3 squared plus 4 squared is equal to 5 squared. So we've got a perfect Pythagorean triangle. In fact any triangle whose sides satisfy this relationship will give me a 90 degree angle. But I'm pretty sure that the Egyptians hadn't got this sweeping generalization of their 3-4-5 triangle. We would not expect to find a general proof because this is not the style of Egyptian mathematics.

Every problem was solved using concrete numbers, and then if a verification would be carried out at the end, it would use the result and these concrete given numbers. There's no general proof within the Egyptian mathematical texts. It would be some 2,000 years before the Greeks and Pythagoras would prove that all right-angled triangles shared certain properties. This wasn't the only mathematical idea that the Egyptians would anticipate.

In a 4,000-year-old document called the Moscow Papyrus, we find a formula for the volume of a pyramid with its peak sliced off, which shows the first hint of calculus at work. For a culture like Egypt that is famous for its pyramids, you would expect problems like this to have been a regular feature within the mathematical texts. The calculation of the volume of a truncated pyramid is one of the most advanced bits according to our modern standards of mathematics that we have from ancient Egypt. The architects and engineers would certainly have wanted such a formula to calculate the amount of materials required to build it. But it's a mark of the sophistication of Egyptian mathematics that they were able to produce such a beautiful method.

To understand how they derived their form, start with a pyramid built such that the highest point sits directly over one corner. Three of these can be put together to make a rectangular box. So the volume of the skewed pyramid is a third the volume of the box. That is, the height times the length times the width divided by three. Now comes an argument which shows the very first hints of the calculus at work thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.

Suppose you could cut the pyramid into slices. You could then slide the layers across to make the more symmetrical pyramid you see in Giza. However, the volume of the pyramid has not changed, despite the rearrangement of the layers.

So the same formula works. The Egyptians were amazing innovators, and their ability to generate new mathematics was staggering. For me, they revealed the power of geometry and numbers and made the first moves towards some of the exciting mathematical discoveries to come. But there was another civilisation that had mathematics to rival that of Egypt, and we know much more about their achievements.

This is Damascus. Over 5,000 years old and still vibrant and bustling today. It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt. The Babylonians controlled much of modern-day Iraq, Iran and Syria from 1800 BC. In order to expand and run their empire, they became masters of managing and manipulating numbers.

We have law codes, for instance, that tell us about the way society is ordered. Now, the people we know most about are the scribes, the professionally literate, innumerate people who kept the records for the wealthy families and for the temples and palaces. Scribe schools existed from around 2500 BC. Aspiring scribes were sent there as children and learned how to read, write and work with numbers.

Scribe records were kept on clay tablets, which allowed the Babylonians to manage and advance their empire. However, many of the tablets we have today aren't official documents, but children's exercises. It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics. So this is a geometrical textbook from about the 18th century BC, and I hope you can see that there are lots of pictures on it.

And underneath each picture is a text that sets a problem about the picture. So, for instance, this one here says... I drew a square 60 units long and inside it I drew four circles. What are their areas?

This little tablet here was written a thousand years at least later than the tablet here. has a very interesting relationship to it. It also has four circles on in a square, roughly drawn, but this isn't a textbook, it's a school exercise so that the adult scribe who's teaching the student is being given this as an example of completed homework or something like that.

Like the Egyptians, the Babylonians appeared interested in solving practical problems to do with measuring and weighing. The Babylonian solutions to these problems are written like mathematical recipes. A scribe would simply follow and record a set of instructions. to get a result.

Here's an example of the kind of problem they'd solve. Now I've got a bundle of cinnamon sticks here, but I'm not going to weigh them. Instead, I'm going to take four times their weight and add them to the scales. Now I'm going to add 20 gin.

Gin was the ancient Babylonian measure of weight. I'm going to take half of everything here and add it again. So that's two bundles and 10 gin.

Now everything on this side is equal to one manna. One manna was six. And here we have one of the first mathematical equations in history.

Everything on this side is equal to one manna. But how much does the bundle of cinnamon sticks weigh? Without any algebraic language, they were able to manipulate the quantities to be able to prove... that the cinnamon sticks weighed 5 jinn. In my mind, it's this kind of problem which gives mathematics a bit of a bad name.

You can blame those ancient Babylonians for all those torturous problems you had at school. But the ancient Babylonian scribes excelled at this kind of problem. Intriguingly, they weren't using powers of 10 like the Egyptians.

They were using powers of 60. The Babylonians invented their number system, like the Egyptians, by using their fingers. But instead of counting through the ten fingers on their hand, Babylonians found a much more intriguing way to count body parts. They used the twelve knuckles on one hand and the five fingers on the other to be able to count twelve times five, i.e. sixty different numbers.

So, for example, this number would have been two lots of twelve, twenty-four, and then one, two, three, four, five, to make twenty-nine. But the number 60 had another powerful property. It can be perfectly divided in a multitude of ways. Here are 60 beans.

I can arrange them in two rows of 30, three rows of 20, four rows of 15, five rows of 12, or six rows of 10. The divisibility of 60 makes it a perfect base in which to do arithmetic. The base 60 system was so successful, we still use elements of it today. Every time we want to tell the time, we recognize units of 60. 60 seconds in a minute, 60 minutes in an hour. The most important feature of the Babylonians number system was that it recognized place value.

Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording, the position of each Babylonian number records the power of 60 you're counting. Instead of inventing new symbols for bigger and bigger numbers, they would write 111. So this number would be 3661. The catalyst for this discovery was the Babylonians'desire to chart the course of the night sky. The Babylonians'calendar was based on the cycles of the moon.

And so they needed a way of recording astronomically large numbers. Month by month, year by year, these cycles were recorded. From about 800 BC, there were complete lists of lunar eclipses. The Babylonian system of measurement was quite sophisticated at that time.

They had a system of angular measurement, 360 degrees in a full circle. Each degree was divided into 60 minutes. A minute was further divided into 60 seconds. So they had a regular system for measurement, and it was in perfect harmony with their number system.

So it was well suited not only for observation but also for calculation. But in order to calculate and cope with these large numbers, the Babylonians needed to invent a new symbol. And in so doing, they prepared the ground for one of the great breakthroughs in the history of mathematics, zero.

In the early days, the Babylonians, in order to mark an empty place in the middle of a number, would simply leave a blank space. So they needed a way of representing nothing in the middle of a number. So they used a sign as a sort of breathing mark or a punctuation mark, and it comes to mean zero in the middle of a number. This was the first time zero in any form had appeared in the mathematical universe. But it would be over a thousand years before this little placeholder would become a number in its own right.

Having established such a sophisticated system of numbers, they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia. Babylonian engineers and surveyors found ingenious ways of accessing water and channeling it to the crop fields. Yet again they used mathematics to come up with solutions.

The Orontes Valley in Syria is still an agricultural hub, and the old methods of irrigation are being exploited today just as they were thousands of years ago. Many of the problems in Babylonian mathematics are concerned with measuring land, and it's here we see for the first time the use of quadratic equations, one of the greatest legacies of Babylonian mathematics. Quadratic equations...

involve things where the unknown quantity you're trying to identify is multiplied by itself. We call this squaring because it gives the area of a square. And it's in the context of calculating the area of land that these quadratic equations naturally arise. Here's a typical problem.

If a field has an area of 55 units, and one side is six units longer than the other, how long is the shorter side? The Babylonian solution was to reconfigure the field as a square. Cut three units off the end, and move this round.

Now, there's a 3x3 piece missing, so let's add this in. The area of the field has increased by 9 units. This makes the new area 64. So the size of the square are 8 units.

The problem solver knows that they've added 3 to this side. So, the original length must be 5. It may not look like it, but this is one of the first quadratic equations in history. In modern mathematics, I would use the symbolic language of algebra to solve this problem. But the amazing feat of the Babylonians is that they were using these geometric games to find the value, without any recourse to symbols or formulas.

The Babylonians were enjoying problem solving for its own sake. They were falling in love with mathematics. The Babylonians'fascination with numbers soon found a place in their leisure time too.

They were avid game players. The Babylonians and their descendants have been playing a version of backgammon for over 5,000 years. They played board games, from very posh board games in royal tombs, to little bits of board games found in schools, to board games scratched on the entrances of palaces, so that the guardsmen must have played at times when they were bored.

And they used dice to move their counters round. People who played games were using numbers in their leisure time to try and outwit their opponent, doing mental arithmetic very fast. And so they were calculating in their leisure time without even thinking about it as being mathematical. hardware.

I hadn't played backgammon for ages, but I reckon my maths would be good enough to give me a fighting chance. But it wasn't as easy as it seemed. as I thought.

Oh! What the hell is that? Yeah, yeah, this is...

This is one, this is two. Now you are in trouble. So I can't move anything, can I?

I can't move these anyway. You cannot move these. You're gonna play...

I just gotta... Oh, gosh. Exactly. There you go. You seem to be...

All right. Three and four, so... Just like the ancient Babylonians, my opponents were masters of tactical mathematics. Yeah.

So, I'm gonna go ahead and... I'm gonna go ahead and... Put it there.

All right, good game. The Babylonians are recognised as one of the first cultures to use symmetrical mathematical shapes to make dice. But there is more heated debate about whether they might also have been the first to discover the secrets of another important shape, the right-angled triangle. We've already seen how the Egyptians used a 3-4-5 right-angled triangle.

But what the Babylonians knew about this shape, and others like it, is much more sophisticated. This is the most famous and controversial ancient tablet we have. It's called Plimpton 322. Many mathematicians are convinced it shows the Babylonians could well have known the principle regarding right-angled triangles, that the square on the diagonal is the sum of the squares on the sides, and known it centuries before the Greeks claimed it. This is a copy of the arguably most famous Babylonian tablet, which is Plimpton 322, and these numbers here reflect the width or height of a triangle, this being the diagonal. The other side would be over here.

The square of this column plus the square of the number in this column equals the square of the diagonal. They are arranged in an order of steadily decreasing angle on a very uniform basis, showing that somebody had a lot of understanding of how the numbers fit together. Here were 15 perfect Pythagorean triangles, all of whose sides had whole number lengths.

It's tempting to think that the Babylonians were the first custodians of Pythagoras'theorem, and it's a conclusion that generations of historians have been seduced by. But there could be a much simpler explanation for the sets of three numbers which fulfil Pythagoras'theorem. It's not a systematic exploration of Pythagorean triples.

It's simply a mathematics teacher doing some quite complicated calculations, but in order to produce some very simple numbers in order to set his students'problems about right-angled triangles. And in that sense, it's about Pythagorean triples, only incidentally. The most valuable clues to what they understood. could lie elsewhere. This small school exercise tablet is nearly 4,000 years old and reveals just what the Babylonians did know about right-angled triangles.

It uses the principle of Pythagoras'theorem to find the value of an astounding... Drawn along the diagonal is a really very good approximation to the square root of 2. And so that shows us that it was known and used in school environments. Now why is this important? It's because the square root of 2 is what we now call an irrational number.

That is, if we write it out in decimals or even in sexagesimal places, it doesn't end. The numbers go on forever after the decimal point. The implications of this calculation are far-reaching. Firstly, it means the Babylonians knew something of Pythagoras'theorem a thousand years before Pythagoras. Secondly, the fact that they can calculate this number to an accuracy of four decimal places shows an amazing arithmetic facility as well as a passion for mathematical detail.

The Babylonians'mathematical dexterity was astounding and for nearly 2,000 years they spearheaded intellectual progress in the ancient world. But when their imperial power began to wane, so did their intellectual vigor. By 330 BC, the Greeks had advanced their imperial reach into old Mesopotamia. This is Palmyra in central Syria, a once great city built by the Greeks. The mathematical expertise needed to build structures with such geometric perfection is impressive.

Just like the Babylonians before them, the Greeks were passionate about mathematics. The Greeks were clever colonists. They took the best from the civilizations they invaded to advance their own power and influence. But they were soon making contributions themselves.

In my opinion, their greatest innovation was to do with a shift in the mind. What they initiated would influence humanity for centuries. They gave us the power of proof.

Somehow they decided that they had to have a deductive system for their mathematics. And the typical deductive system was to begin with certain axioms, which you assume are true. It's as if you assume a certain theorem is true without proving it.

And then using logical methods... And very careful steps from these actions you prove theorems. And then from those theorems you prove more theorems. And it just snowballs that way. Proof is what gives mathematics its strength.

It's the power of proof which means that the discoveries of the Greeks are as true today as they were 2,000 years ago. I needed to head west into the heart of the old Greek empire to learn more. For me, Greek mathematics has always been heroic and romantic. I'm on my way to Samos, less than a mile from the Turkish coast.

This place has become synonymous with the birth of Greek mathematics. And it's down to the legend of one man. His name is Pythagoras.

The legends that surround his life and work have contributed to the celebrity status he has gained over the last 2000 years. He's credited, rightly or wrongly, with beginning the transformation from mathematics as a tool for accounting to the analytic subject we recognise today. Pythagoras is a controversial figure.

Because he left no mathematical writings, many have questioned whether he indeed solved any of the theorems attributed to him. He founded a school in Samos in the 6th century BC, but his teachings were considered suspect, and the Pythagoreans a bizarre sect. There is good evidence that there were schools of Pythagoreans.

They may have looked more like sects than what we associate with philosophical schools, because they didn't just share knowledge, they also shared... a way of life. They may have been communal living and they all seem to have been involved in the politics of their cities. One feature that makes them unusual in the ancient world is that they included women.

But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians, the properties of right-angled triangles. What's known as Pythagoras'theorem states that if you take any right-angled triangle, build squares on all the sides, then the area of the largest square is equal to the sum of the squares on the two smaller sides. It's at this point for me that mathematics is born, and a gulf opens up between the other sciences. And the proof is as simple as it is devastating in its implications.

Place four copies of the right-angled triangle on top of this surface. The square that you now see has sides equal to the hypotenuse of the triangle. By sliding these triangles around, we see how we can break the area of the large square up into the sum of two smaller squares. whose sides are given by the two short sides of the triangle. In other words, the square on the hypotenuse is equal to the sum of the squares on the other sides.

Pythagoras'theorem. It illustrates one of the characteristic themes of Greek mathematics, the appeal to beautiful arguments in geometry rather than a reliance on number. Pythagoras may have fallen out of favor and many of the discoveries accredited to him have been contested recently but there's one mathematical theory that I'm loathe to take away from him.

It's to do with music and the discovery of the harmonic series. The story goes that walking past a blacksmith's one day Pythagoras heard anvils being struck and notice how the notes being produced sounded in perfect harmony. He believed that there must be some rational explanation to make sense of why the notes sounded so appealing. The answer was mathematics.

Experimenting with a stringed instrument, Pythagoras discovered that the intervals between harmonious musical notes were always represented as whole number ratios. And here's how he might have constructed his theory. First, play a note on the open string. Next, take half the length. The note almost sounds the same as the first note.

In fact, it's an octave higher, but the relationship is so strong we give these notes the same name. Now, take a third the length. We get another note which sounds harmonious next to the first two. But take a length of string which is not in a whole number ratio, and all we get is dissonance. According to legend, Pythagoras was so excited by this discovery that he concluded the whole universe was built from numbers.

But he and his followers were in for a rather unsettling challenge to their world view. And it came about as a result of a theorem which bears Pythagoras'name. Legend has it one of his followers, a mathematician called Hyppasus, set out to find the length of the diagonal for a right-angled triangle with two sides measuring one unit.

Pythagoras's theorem implied that the length of the diagonal was a number whose square was two. The Pythagoreans assumed that the answer would be a fraction, but when Hyppasus tried to express it in this way, no matter how he tried, he couldn't capture it. Eventually he realized his mistake. It was the assumption that the value was a fraction at all which was wrong. The value of the square root of 2 was the number that the Babylonians etched into the Yale tablet.

However, they didn't recognize the special character of this number. But, Hyppasus did. It was an irrational number.

The discovery of this new number, and others like it, is akin to an explorer discovering a new continent, or a naturalist finding a new species. But these irrational numbers didn't fit the Pythagorean world view. Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy.

But Hippasus let slip the discovery, and was promptly drowned for his attempts to broadcast their research. But these mathematical discoveries could not be easily suppressed. Schools of philosophy and science started to flourish all over Greece, building on these foundations.

The most famous of these was the Academy. Plato founded this school in Athens in 387 BC. Although we think of him today as a philosopher, he was one of mathematics'most important patrons.

Plato was enraptured by the Pythagorean worldview and considered mathematics the bedrock of knowledge. Some people would say that Plato is possibly the most influential figure for our perception of Greek mathematics. He argued that mathematics is an important form of knowledge and does have a connection with reality. So by knowing mathematics we'll know more about reality. In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking the secrets of the universe, a view still held by scientists today.

Indeed, the importance Plato attached to geometry is encapsulated in the sign that was mounted above the academy. Let no one ignorant of geometry enter here. Plato proposed that the universe could be crystallized into five regular symmetrical shapes.

These shapes, which we now call the platonic solids, were composed of regular polygons assembled to create three-dimensional symmetrical objects. The tetrahedron represented fire. The icosahedron, made from 20 triangles, represented water. The stable cube was Earth.

The eight-faced octahedron was air. And the fifth platonic solid, the dodecahedron, made out of 12 pentagons, was reserved for the shape which captured Plato's view of the universe. Plato's theory would have a seismic influence and continue to inspire mathematicians and astronomers for over 1500 years. In addition to the breakthroughs made in the Academy, mathematical triumphs were also emerging from the edge of the Greek Empire and owed as much to the mathematical heritage of the Egyptians as the Greeks. Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC and its famous library soon gained a reputation to rival Plato's Academy.

The kings of Alexandria were... prepared to invest in the arts, in culture, in technology, mathematics, grammar, because patronage for cultural pursuits was one way of showing that you were a more prestigious ruler and had a better entitlement to greatness. were destroyed when the Muslims conquered Egypt in the 7th century. But its spirit is alive in a new building.

Today, the library remains a place of discovery and scholarship. Mathematicians and philosophers flocked to Alexandria, driven by their thirst for knowledge and the pursuit of excellence. The patrons of the library were the first professional scientists.

individuals who have paid for their devotion to research. But of all those early pioneers, my hero is the enigmatic Greek mathematician Euclid. We know very little about Euclid's life, but his greatest achievements were as a chronicler of mathematics. Around 300 BC, he wrote the most important textbook of all time, The Elements. In The Elements...

we find the culmination of the mathematical revolution which had taken place in Greece. It's built on a series of mathematical assumptions called axioms. For example, a line can be drawn between any two points.

From these axioms, logical deductions are made and mathematical theorems established. The elements contains formulas for calculating the volumes of cones and cylinders. Proofs about geometric series, perfect numbers and primes.

The climax of the elements is a proof that there are only five platonic solids. For me, this last theorem captures the power of mathematics. It's one thing to build five symmetrical solids, quite another to come up with a watertight, logical argument for why there can't be a sixth.

The elements unfold like a wonderful, logical mystery novel. This is a story which transcends time. Scientific theories get knocked down from one generation to the next.

But the theorems and the elements are as true today as they were 2,000 years ago. When you stop and think about it, it's really amazing that it's the same theorems that we teach. We may teach them in a slightly different way.

We may organize them differently. But it's Euclidean geometry that is still valid, and even in higher mathematics. When you go to higher dimensional spaces, you're still using Euclidean geometry. Alexandria must have been an inspiring place for the ancient scholars, and Euclid's fame would have attracted even more eager young intellectuals to the Egyptian port. One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.

He would become a mathematical visionary. The best Greek mathematicians, they were always pushing the limits, pushing the envelope. So, Archimedes did what he could with polygons, with solids. He then moved on to centres of gravity, or he then moved on to the spiral.

This instinct to... try and mathematize everything is something that I see as a legacy. One of Archimedes'specialities was weapons of mass destruction. They were used against the Romans when they invaded his home of Syracuse in 212 BC. He also designed mirrors which harnessed the power of the sun to set the Roman ships on fire.

But to Archimedes, these endeavors were mere amusements in geometry. He had loftier ambitions. Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake and not for the ignoble trade of engineering or the sordid quest for profit.

One of his finest investigations into pure mathematics was to produce formulas to calculate the areas of regular shapes. Archimedes'method was to capture new shapes... by using shapes he already understood.

So for example, to calculate the area of a circle, he would enclose it inside a triangle. And then by doubling the number of sides on the triangle, the enclosing shape would get closer and closer to the circle. Indeed, we sometimes call a circle a polygon with an infinite number of sides. But by estimating the area of the circle, Archimedes is in fact getting a value for pi, the most important number in mathematics.

However, it was calculating the volumes of solid objects where Archimedes excelled. He found a way to calculate the volume of a sphere by slicing it up and approximating each slice as a cylinder. He then added up the volumes of the slices to get an approximate value for the sphere. But his act of genius was to see what happens if you make the slices thinner and thinner.

In the limit, the approximation becomes an exact... calculation but it was our comedies commitment to mathematics it will be his undoing Archimedes was contemplating a problem about circles traced in the sand when a Roman soldier accosted him. Archimedes was so engrossed in his problem that he insisted he be allowed to finish his theorem. But the Roman soldier was not interested in Archimedes'problem and killed him on the spot.

Even in death, Archimedes'devotion to mathematics was unwavering. By the middle of the first century BC, the Romans had tightened their grip on the old Greek Empire. They were less smitten with the beauty of mathematics, and were more concerned with its practical applications.

This pragmatic attitude signalled the beginning of the end for the great library of Alexandria. But one mathematician was determined to keep the legacy of the Greeks alive. Hypatia was exceptional, a female mathematician and a pagan in the piously Christian Roman Empire.

Aitresha was very prestigious and very influential in her time. She was a teacher with a lot of students, a lot of followers. She was politically influential in Alexandria. So it's this combination of high knowledge and high prestige that may have made her a figure of hatred for the Christian mob.

One morning during Lent, Hypatia was dragged off her chariot by a zealous Christian mob and taken to a church. There, she was tortured and brutally murdered. The dramatic circumstances of her life and death fascinated later generations.

Sadly, her cult status eclipsed her mathematical achievements. She was in fact a brilliant teacher and theorist, and her death dealt a final blow to the Greek mathematical heritage of Alexandria. My travels have taken me on a fascinating journey to uncover the passion and innovation of the world's earliest mathematicians.

It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece that are the foundations on which my subject is built today. But this is just the beginning of my mathematical odyssey. The next leg of my journey lies east, in the depths of Asia, where mathematicians scaled even greater heights in pursuit of knowledge. With this new era came a new language of algebra and numbers, better suited to telling the next chapter in the story of maths. You can learn more about the story of maths with the Open University at open2.net.

And the story of maths continues here on BBC4 next Monday at the same time. Next tonight, British novelists become agents of provocation as they describe a time when nothing was sacred in their own words.

Transcript for:The Evolution of Mathematics through History

Transcript for:
The Evolution of Mathematics through History