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Throughout history, humankind has struggled 

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to understand the fundamental workings of the material world. 

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We've endeavoured to discover the rules and patterns that determine the qualities 

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of the objects that surround us, and their complex relationship to us and each other. 

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Over thousands of years, societies all over the world have found that one discipline 

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above all others yields certain knowledge 

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about the underlying realities of the physical world. 

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That discipline is mathematics. 

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I'm Marcus Du Sautoy, and I'm a mathematician. 

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I see myself as a pattern searcher, hunting down the hidden structures 

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that lie behind the apparent chaos and complexity of the world around us. 

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In my search for pattern and order, I draw upon the work of the great mathematicians 

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who've gone before me, people belonging to cultures across the globe, 

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whose innovations created the language the universe is written in. 

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I want to take you on a journey through time and space, and track the growth of mathematics 

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from its awakening to the sophisticated subject we know today. 

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Using computer generated imagery, we will explore 

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the trailblazing discoveries that allowed the earliest civilisations 

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to understand the world mathematical. 

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This is the story of maths. 

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Our world is made up of patterns and sequences. 

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They're all around us. 

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Day becomes night. 

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Animals travel across the earth in ever-changing formations. 

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Landscapes are constantly altering. 

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One of the reasons mathematics began was because we needed to find a way 

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of making sense of these natural patterns. 

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The most basic concepts of maths - space and quantity - 

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are hard-wired into our brains. 

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Even animals have a sense of distance and number, 

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assessing when their pack is outnumbered, and whether to fight or fly, 

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calculating whether their prey is within striking distance. 

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Understanding maths is the difference between life and death. 

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But it was man who took these basic concepts 

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and started to build upon these foundations. 

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At some point, humans started to spot patterns, 

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to make connections, to count and to order the world around them. 

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With this, a whole new mathematical universe began to emerge. 

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This is the River Nile. 

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It's been the lifeline of Egypt for millennia. 

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I've come here because it's where some of the first signs 

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of mathematics as we know it today emerged. 

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People abandoned nomadic life and began settling here as early as 6000BC. 

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The conditions were perfect for farming. 

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The most important event for Egyptian agriculture each year was the flooding of the Nile. 

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So this was used as a marker to start each new year. 

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Egyptians did record what was going on over periods of time, 

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so in order to establish a calendar like this, 

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you need to count how many days, for example, 

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happened in-between lunar phases, 

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or how many days happened in-between two floodings of the Nile. 

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Recording the patterns for the seasons was essential, 

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not only to their management of the land, but also their religion. 

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The ancient Egyptians who settled on the Nile banks 

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believed it was the river god, Hapy, who flooded the river each year. 

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And in return for the life-giving water, 

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the citizens offered a portion of the yield as a thanksgiving. 

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As settlements grew larger, it became necessary to find ways to administer them. 

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Areas of land needed to be calculated, crop yields predicted, 

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taxes charged and collated. 

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In short, people needed to count and measure. 

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The Egyptians used their bodies to measure the world, 

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and it's how their units of measurements evolved. 

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A palm was the width of a hand, 

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a cubit an arm length from elbow to fingertips. 

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Land cubits, strips of land measuring a cubit by 100, 

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were used by the pharaoh's surveyors to calculate areas. 

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There's a very strong link between bureaucracy 

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and the development of mathematics in ancient Egypt. 

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And we can see this link right from the beginning, 

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from the invention of the number system, 

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throughout Egyptian history, really. 

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For the Old Kingdom, the only evidence we have 

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are metrological systems, that is measurements for areas, for length. 

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This points to a bureaucratic need to develop such things. 

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It was vital to know the area of a farmer's land so he could be taxed accordingly. 

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Or if the Nile robbed him of part of his land, so he could request a rebate. 

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It meant that the pharaoh's surveyors were often calculating 

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the area of irregular parcels of land. 

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It was the need to solve such practical problems 

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that made them the earliest mathematical innovators. 

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The Egyptians needed some way to record the results of their calculations. 

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Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo, 

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I was on the hunt for those that recorded some of the first numbers in history. 

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They were difficult to track down. 

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But I did find them in the end. 

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The Egyptians were using a decimal system, motivated by the 10 fingers on our hands. 

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The sign for one was a stroke, 

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10, a heel bone, 100, a coil of rope, and 1,000, a Lotus plant. 

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How much is this T-shirt? 

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Er, 25. 

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- 25!
- Yes!
- So that would be 2 knee bones and 5 strokes. 

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- So you're not gonna charge me anything up here?
- Here, one million! 

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- One million?
- My God! 

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This one million. 

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One million, yeah, that's pretty big! 

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The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed. 

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They had no concept of a place value, 

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so one stroke could only represent one unit, 

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not 100 or 1,000. 

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Although you can write a million with just one character, 

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rather than the seven that we use, if you want to write a million minus one, 

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then the poor old Egyptian scribe has got to write nine strokes, 

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nine heel bones, nine coils of rope, and so on, 

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a total of 54 characters. 

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Despite the drawback of this number system, the Egyptians were brilliant problem solvers. 

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We know this because of the few records that have survived. 

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The Egyptian scribes used sheets of papyrus 

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to record their mathematical discoveries. 

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This delicate material made from reeds decayed over time 

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and many secrets perished with it. 

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But there's one revealing document that has survived. 

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The Rhind Mathematical Papyrus is the most important document 

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we have today for Egyptian mathematics. 

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We get a good overview of what types of problems 

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the Egyptians would have dealt with in their mathematics. 

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We also get explicitly stated how multiplications and divisions were carried out. 

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The papyri show how to multiply two large numbers together. 

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But to illustrate the method, let's take two smaller numbers. 

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Let's do three times six. 

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The scribe would take the first number, three, and put it in one column. 

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In the second column, he would place the number one. 

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Then he would double the numbers in each column, so three becomes six... 

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..and six would become 12. 

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And then in the second column, one would become two, 

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and two becomes four. 

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Now, here's the really clever bit. 

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The scribe wants to multiply three by six. 

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So he takes the powers of two in the second column, 

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which add up to six. That's two plus four. 

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Then he moves back to the first column, and just takes 

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those rows corresponding to the two and the four. 

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So that's six and the 12. 

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He adds those together to get the answer of 18. 

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But for me, the most striking thing about this method 

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is that the scribe has effectively written that second number in binary. 

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Six is one lot of four, one lot of two, and no units. 

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Which is 1-1-0. 

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The Egyptians have understood the power of binary over 3,000 years 

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before the mathematician and philosopher Leibniz would reveal their potential. 

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Today, the whole technological world depends on the same principles 

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that were used in ancient Egypt. 

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The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC. 

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Its problems are concerned with finding solutions to everyday situations. 

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Several of the problems mention bread and beer, 

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which isn't surprising as Egyptian workers were paid in food and drink. 

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One is concerned with how to divide nine loaves 

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equally between 10 people, without a fight breaking out. 

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I've got nine loaves of bread here. 

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I'm gonna take five of them and cut them into halves. 

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Of course, nine people could shave a 10th off their loaf 

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and give the pile of crumbs to the 10th person. 

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But the Egyptians developed a far more elegant solution - 

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take the next four and divide those into thirds. 

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But two of the thirds I am now going to cut into fifths, 

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so each piece will be one fifteenth. 

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Each person then gets one half 

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and one third 

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and one fifteenth. 

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It is through such seemingly practical problems 

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that we start to see a more abstract mathematics developing. 

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Suddenly, new numbers are on the scene - fractions - 

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and it isn't too long before the Egyptians are exploring the mathematics of these numbers. 

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Fractions are clearly of practical importance to anyone dividing quantities for trade in the market. 

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To log these transactions, the Egyptians developed notation which recorded these new numbers. 

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One of the earliest representations of these fractions 

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came from a hieroglyph which had great mystical significance. 

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It's called the Eye of Horus. 

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Horus was an Old Kingdom god, depicted as half man, half falcon. 

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According to legend, Horus' father was killed by his other son, Seth. 

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Horus was determined to avenge the murder. 

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During one particularly fierce battle, 

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Seth ripped out Horus' eye, tore it up and scattered it over Egypt. 

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But the gods were looking favourably on Horus. 

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They gathered up the scattered pieces and reassembled the eye. 

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Each part of the eye represented a different fraction. 

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Each one, half the fraction before. 

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Although the original eye represented a whole unit, 

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the reassembled eye is 1/64 short. 

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Although the Egyptians stopped at 1/64, 

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implicit in this picture 

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is the possibility of adding more fractions, 

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halving them each time, the sum getting closer and closer to one, 

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but never quite reaching it. 

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This is the first hint of something called a geometric series, 

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and it appears at a number of points in the Rhind Papyrus. 

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But the concept of infinite series would remain hidden 

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until the mathematicians of Asia discovered it centuries later. 

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Having worked out a system of numbers, including these new fractions, 

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it was time for the Egyptians to apply their knowledge 

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to understanding shapes that they encountered day to day. 

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These shapes were rarely regular squares or rectangles, 

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and in the Rhind Papyrus, we find the area of a more organic form, the circle. 

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What is astounding in the calculation 

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of the area of the circle is its exactness, really. 

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How they would have found their method is open to speculation, 

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because the texts we have 

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do not show us the methods how they were found. 

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This calculation is particularly striking because it depends 

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on seeing how the shape of the circle 

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can be approximated by shapes that the Egyptians already understood. 

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The Rhind Papyrus states that a circular field 

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with a diameter of nine units 

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is close in area to a square with sides of eight. 

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But how would this relationship have been discovered? 

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My favourite theory sees the answer in the ancient game of mancala. 

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Mancala boards were found carved on the roofs of temples. 

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Each player starts with an equal number of stones, 

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and the object of the game is to move them round the board, 

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capturing your opponent's counters on the way. 

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As the players sat around waiting to make their next move, 

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perhaps one of them realised that sometimes the balls fill the circular holes 

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of the mancala board in a rather nice way. 

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He might have gone on to experiment with trying to make larger circles. 

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Perhaps he noticed that 64 stones, the square of 8, 

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can be used to make a circle with diameter nine stones. 

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By rearranging the stones, the circle has been approximated by a square. 

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And because the area of a circle is pi times the radius squared, 

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the Egyptian calculation gives us the first accurate value for pi. 

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The area of the circle is 64. Divide this by the radius squared, 

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in this case 4.5 squared, and you get a value for pi. 

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So 64 divided by 4.5 squared is 3.16, 

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just a little under two hundredths away from its true value. 

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But the really brilliant thing is, the Egyptians 

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are using these smaller shapes to capture the larger shape. 

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But there's one imposing and majestic symbol of Egyptian 

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mathematics we haven't attempted to unravel yet - 

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the pyramid. 

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I've seen so many pictures that I couldn't believe I'd be impressed by them. 

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But meeting them face to face, you understand why they're called 

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one of the Seven Wonders of the Ancient World. 

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They're simply breathtaking. 

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And how much more impressive they must have been in their day, 

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when the sides were as smooth as glass, reflecting the desert sun. 

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To me it looks like there might be mirror pyramids hiding underneath the desert, 

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which would complete the shapes to make perfectly symmetrical octahedrons. 

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Sometimes, in the shimmer of the desert heat, you can almost see these shapes. 

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It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician. 

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The pyramids are just a little short to create these perfect shapes, 

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but some have suggested another important mathematical concept 

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might be hidden inside the proportions of the Great Pyramid - the golden ratio. 

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Two lengths are in the golden ratio, if the relationship of the longest 

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to the shortest is the same as the sum of the two to the longest side. 

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Such a ratio has been associated with the perfect proportions one finds 

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00:17:11,840 --> 00:17:15,840
all over the natural world, as well as in the work of artists, 

246
00:17:15,840 --> 00:17:18,720
architects and designers for millennia. 

247
00:17:22,560 --> 00:17:27,000
Whether the architects of the pyramids were conscious of this important mathematical idea, 

248
00:17:27,000 --> 00:17:32,680
or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know. 

249
00:17:32,680 --> 00:17:37,040
For me, the most impressive thing about the pyramids is the mathematical brilliance 

250
00:17:37,040 --> 00:17:40,600
that went into making them, including the first inkling 

251
00:17:40,600 --> 00:17:44,640
of one of the great theorems of the ancient world, Pythagoras' theorem. 

252
00:17:46,160 --> 00:17:49,160
In order to get perfect right-angled corners on their buildings 

253
00:17:49,160 --> 00:17:54,320
and pyramids, the Egyptians would have used a rope with knots tied in it. 

254
00:17:54,320 --> 00:17:58,200
At some point, the Egyptians realised that if they took a triangle with sides 

255
00:17:58,200 --> 00:18:05,640
marked with three knots, four knots and five knots, it guaranteed them a perfect right-angle. 

256
00:18:05,640 --> 00:18:10,120
This is because three squared, plus four squared, is equal to five squared. 

257
00:18:10,120 --> 00:18:12,840
So we've got a perfect Pythagorean triangle. 

258
00:18:15,160 --> 00:18:20,960
In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle. 

259
00:18:20,960 --> 00:18:23,600
But I'm pretty sure that the Egyptians hadn't got 

260
00:18:23,600 --> 00:18:28,480
this sweeping generalisation of their 3, 4, 5 triangle. 

261
00:18:28,480 --> 00:18:32,240
We would not expect to find the general proof 

262
00:18:32,240 --> 00:18:35,720
because this is not the style of Egyptian mathematics. 

263
00:18:35,720 --> 00:18:39,320
Every problem was solved using concrete numbers and then 

264
00:18:39,320 --> 00:18:43,760
if a verification would be carried out at the end, it would use the result 

265
00:18:43,760 --> 00:18:45,720
and these concrete, given numbers, 

266
00:18:45,720 --> 00:18:49,440
there's no general proof within the Egyptian mathematical texts. 

267
00:18:50,960 --> 00:18:54,080
It would be some 2,000 years before the Greeks and Pythagoras 

268
00:18:54,080 --> 00:18:59,280
would prove that all right-angled triangles shared certain properties. 

269
00:18:59,280 --> 00:19:03,640
This wasn't the only mathematical idea that the Egyptians would anticipate. 

270
00:19:03,640 --> 00:19:10,160
In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume 

271
00:19:10,160 --> 00:19:16,120
of a pyramid with its peak sliced off, which shows the first hint of calculus at work. 

272
00:19:16,120 --> 00:19:22,920
For a culture like Egypt that is famous for its pyramids, you would expect problems like this 

273
00:19:22,920 --> 00:19:26,560
to have been a regular feature within the mathematical texts. 

274
00:19:26,560 --> 00:19:31,280
The calculation of the volume of a truncated pyramid is one of the most 

275
00:19:31,280 --> 00:19:36,480
advanced bits, according to our modern standards of mathematics, 

276
00:19:36,480 --> 00:19:39,080
that we have from ancient Egypt. 

277
00:19:39,080 --> 00:19:43,120
The architects and engineers would certainly have wanted such a formula 

278
00:19:43,120 --> 00:19:46,760
to calculate the amount of materials required to build it. 

279
00:19:46,760 --> 00:19:49,000
But it's a mark of the sophistication 

280
00:19:49,000 --> 00:19:53,760
of Egyptian mathematics that they were able to produce such a beautiful method. 

281
00:19:59,760 --> 00:20:03,760
To understand how they derived their formula, start with a pyramid 

282
00:20:03,760 --> 00:20:08,480
built such that the highest point sits directly over one corner. 

283
00:20:08,480 --> 00:20:13,080
Three of these can be put together to make a rectangular box, 

284
00:20:13,080 --> 00:20:18,240
so the volume of the skewed pyramid is a third the volume of the box. 

285
00:20:18,240 --> 00:20:24,280
That is, the height, times the length, times the width, divided by three. 

286
00:20:24,280 --> 00:20:29,320
Now comes an argument which shows the very first hints of the calculus at work, 

287
00:20:29,320 --> 00:20:35,320
thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory. 

288
00:20:35,320 --> 00:20:39,640
Suppose you could cut the pyramid into slices, you could then slide 

289
00:20:39,640 --> 00:20:44,960
the layers across to make the more symmetrical pyramid you see in Giza. 

290
00:20:44,960 --> 00:20:49,720
However, the volume of the pyramid has not changed, despite the rearrangement of the layers. 

291
00:20:49,720 --> 00:20:52,120
So the same formula works. 

292
00:20:55,360 --> 00:20:58,880
The Egyptians were amazing innovators, 

293
00:20:58,880 --> 00:21:02,080
and their ability to generate new mathematics was staggering. 

294
00:21:02,080 --> 00:21:07,320
For me, they revealed the power of geometry and numbers, and made the first moves 

295
00:21:07,320 --> 00:21:11,760
towards some of the exciting mathematical discoveries to come. 

296
00:21:11,760 --> 00:21:15,960
But there was another civilisation that had mathematics to rival that of Egypt. 

297
00:21:15,960 --> 00:21:20,040
And we know much more about their achievements. 

298
00:21:24,280 --> 00:21:27,880
This is Damascus, over 5,000 years old, 

299
00:21:27,880 --> 00:21:31,280
and still vibrant and bustling today. 

300
00:21:31,280 --> 00:21:36,840
It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt. 

301
00:21:36,840 --> 00:21:43,720
The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC. 

302
00:21:43,720 --> 00:21:51,120
In order to expand and run their empire, they became masters of managing and manipulating numbers. 

303
00:21:51,120 --> 00:21:53,920
We have law codes for instance that tell us 

304
00:21:53,920 --> 00:21:56,200
about the way society is ordered. 

305
00:21:56,200 --> 00:22:00,120
The people we know most about are the scribes, the professionally literate 

306
00:22:00,120 --> 00:22:05,280
and numerate people who kept the records for the wealthy families and for the temples and palaces. 

307
00:22:05,280 --> 00:22:10,320
Scribe schools existed from around 2500BC. 

308
00:22:10,320 --> 00:22:17,240
Aspiring scribes were sent there as children, and learned how to read, write and work with numbers. 

309
00:22:17,240 --> 00:22:20,120
Scribe records were kept on clay tablets, 

310
00:22:20,120 --> 00:22:24,200
which allowed the Babylonians to manage and advance their empire. 

311
00:22:24,200 --> 00:22:31,000
However, many of the tablets we have today aren't official documents, but children's exercises. 

312
00:22:31,000 --> 00:22:37,640
It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics. 

313
00:22:37,640 --> 00:22:42,440
So, this is a geometrical textbook from about the 18th century BC. 

314
00:22:42,440 --> 00:22:44,920
I hope you can see that there are lots of pictures on it. 

315
00:22:44,920 --> 00:22:49,160
And underneath each picture is a text that sets a problem about the picture. 

316
00:22:49,160 --> 00:22:55,800
So for instance this one here says, I drew a square, 60 units long, 

317
00:22:55,800 --> 00:23:01,200
and inside it, I drew four circles - what are their areas? 

318
00:23:01,200 --> 00:23:07,240
This little tablet here was written 1,000 years at least later than the tablet here, 

319
00:23:07,240 --> 00:23:10,120
but has a very interesting relationship. 

320
00:23:10,120 --> 00:23:12,520
It also has four circles on, 

321
00:23:12,520 --> 00:23:17,280
in a square, roughly drawn, but this isn't a textbook, it's a school exercise. 

322
00:23:17,280 --> 00:23:21,400
The adult scribe who's teaching the student is being given this 

323
00:23:21,400 --> 00:23:25,320
as an example of completed homework or something like that. 

324
00:23:26,440 --> 00:23:29,560
Like the Egyptians, the Babylonians appeared interested 

325
00:23:29,560 --> 00:23:32,920
in solving practical problems to do with measuring and weighing. 

326
00:23:32,920 --> 00:23:37,400
The Babylonian solutions to these problems are written like mathematical recipes. 

327
00:23:37,400 --> 00:23:43,000
A scribe would simply follow and record a set of instructions to get a result. 

328
00:23:43,000 --> 00:23:47,760
Here's an example of the kind of problem they'd solve. 

329
00:23:47,760 --> 00:23:51,760
I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them. 

330
00:23:51,760 --> 00:23:56,440
Instead, I'm gonna take four times their weight and add them to the scales. 

331
00:23:58,040 --> 00:24:04,640
Now I'm gonna add 20 gin. Gin was the ancient Babylonian measure of weight. 

332
00:24:04,640 --> 00:24:07,960
I'm gonna take half of everything here and then add it again... 

333
00:24:07,960 --> 00:24:10,280
That's two bundles, and ten gin. 

334
00:24:10,280 --> 00:24:16,320
Everything on this side is equal to one mana. One mana was 60 gin. 

335
00:24:16,320 --> 00:24:20,280
And here, we have one of the first mathematical equations in history, 

336
00:24:20,280 --> 00:24:23,160
everything on this side is equal to one mana. 

337
00:24:23,160 --> 00:24:26,200
But how much does the bundle of cinnamon sticks weigh? 

338
00:24:26,200 --> 00:24:29,480
Without any algebraic language, they were able to manipulate 

339
00:24:29,480 --> 00:24:35,200
the quantities to be able to prove that the cinnamon sticks weighed five gin. 

340
00:24:35,200 --> 00:24:40,560
In my mind, it's this kind of problem which gives mathematics a bit of a bad name. 

341
00:24:40,560 --> 00:24:45,040
You can blame those ancient Babylonians for all those tortuous problems you had at school. 

342
00:24:45,040 --> 00:24:50,200
But the ancient Babylonian scribes excelled at this kind of problem. 

343
00:24:50,200 --> 00:24:57,440
Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60. 

344
00:25:00,120 --> 00:25:05,320
The Babylonians invented their number system, like the Egyptians, by using their fingers. 

345
00:25:05,320 --> 00:25:08,520
But instead of counting through the 10 fingers on their hand, 

346
00:25:08,520 --> 00:25:11,480
Babylonians found a more intriguing way to count body parts. 

347
00:25:11,480 --> 00:25:14,000
They used the 12 knuckles on one hand, 

348
00:25:14,000 --> 00:25:16,400
and the five fingers on the other to be able to count 

349
00:25:16,400 --> 00:25:20,520
12 times 5, ie 60 different numbers. 

350
00:25:20,520 --> 00:25:25,000
So for example, this number would have been 2 lots of 12, 24, 

351
00:25:25,000 --> 00:25:29,120
and then, 1, 2, 3, 4, 5, to make 29. 

352
00:25:32,200 --> 00:25:35,920
The number 60 had another powerful property. 

353
00:25:35,920 --> 00:25:39,360
It can be perfectly divided in a multitude of ways. 

354
00:25:39,360 --> 00:25:41,360
Here are 60 beans. 

355
00:25:41,360 --> 00:25:44,800
I can arrange them in 2 rows of 30. 

356
00:25:48,760 --> 00:25:51,520
3 rows of 20. 

357
00:25:51,520 --> 00:25:53,920
4 rows of 15. 

358
00:25:53,920 --> 00:25:56,160
5 rows of 12. 

359
00:25:56,160 --> 00:25:59,320
Or 6 rows of 10. 

360
00:25:59,320 --> 00:26:04,560
The divisibility of 60 makes it a perfect base in which to do arithmetic. 

361
00:26:04,560 --> 00:26:11,000
The base 60 system was so successful, we still use elements of it today. 

362
00:26:11,000 --> 00:26:15,080
Every time we want to tell the time, we recognise units of 60 - 

363
00:26:15,080 --> 00:26:19,040
60 seconds in a minute, 60 minutes in an hour. 

364
00:26:19,040 --> 00:26:24,800
But the most important feature of the Babylonians' number system was that it recognised place value. 

365
00:26:24,800 --> 00:26:30,200
Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording, 

366
00:26:30,200 --> 00:26:34,320
the position of each Babylonian number records the power of 60. 

367
00:26:41,360 --> 00:26:44,440
Instead of inventing new symbols for bigger and bigger numbers, 

368
00:26:44,440 --> 00:26:50,440
they would write 1-1-1, so this number would be 3,661. 

369
00:26:54,000 --> 00:26:59,680
The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky. 

370
00:27:07,400 --> 00:27:10,840
The Babylonians' calendar was based on the cycles of the moon. 

371
00:27:10,840 --> 00:27:15,200
They needed a way of recording astronomically large numbers. 

372
00:27:15,200 --> 00:27:19,560
Month by month, year by year, these cycles were recorded. 

373
00:27:19,560 --> 00:27:25,720
From about 800BC, there were complete lists of lunar eclipses. 

374
00:27:25,720 --> 00:27:30,480
The Babylonian system of measurement was quite sophisticated at that time. 

375
00:27:30,480 --> 00:27:32,840
They had a system of angular measurement, 

376
00:27:32,840 --> 00:27:36,960
360 degrees in a full circle, each degree was divided 

377
00:27:36,960 --> 00:27:41,920
into 60 minutes, a minute was further divided into 60 seconds. 

378
00:27:41,920 --> 00:27:48,560
So they had a regular system for measurement, and it was in perfect harmony with their number system, 

379
00:27:48,560 --> 00:27:52,200
so it's well suited not only for observation but also for calculation. 

380
00:27:52,200 --> 00:27:56,360
But in order to calculate and cope with these large numbers, 

381
00:27:56,360 --> 00:28:00,720
the Babylonians needed to invent a new symbol. 

382
00:28:00,720 --> 00:28:03,760
And in so doing, they prepared the ground for one of the great 

383
00:28:03,760 --> 00:28:06,880
breakthroughs in the history of mathematics - zero. 

384
00:28:06,880 --> 00:28:11,240
In the early days, the Babylonians, in order to mark an empty place in 

385
00:28:11,240 --> 00:28:14,640
the middle of a number, would simply leave a blank space. 

386
00:28:14,640 --> 00:28:19,960
So they needed a way of representing nothing in the middle of a number. 

387
00:28:19,960 --> 00:28:25,360
So they used a sign, as a sort of breathing marker, a punctuation mark, 

388
00:28:25,360 --> 00:28:28,480
and it comes to mean zero in the middle of a number. 

389
00:28:28,480 --> 00:28:31,680
This was the first time zero, in any form, 

390
00:28:31,680 --> 00:28:35,440
had appeared in the mathematical universe. 

391
00:28:35,440 --> 00:28:42,000
But it would be over a 1,000 years before this little place holder would become a number in its own right. 

392
00:28:50,600 --> 00:28:53,920
Having established such a sophisticated system of numbers, 

393
00:28:53,920 --> 00:28:59,720
they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia. 

394
00:29:02,080 --> 00:29:06,400
Babylonian engineers and surveyors found ingenious ways of 

395
00:29:06,400 --> 00:29:10,400
accessing water, and channelling it to the crop fields. 

396
00:29:10,400 --> 00:29:15,760
Yet again, they used mathematics to come up with solutions. 

397
00:29:15,760 --> 00:29:19,200
The Orontes valley in Syria is still an agricultural hub, 

398
00:29:19,200 --> 00:29:26,320
and the old methods of irrigation are being exploited today, just as they were thousands of years ago. 

399
00:29:26,320 --> 00:29:29,160
Many of the problems in Babylonian mathematics 

400
00:29:29,160 --> 00:29:34,360
are concerned with measuring land, and it's here we see for the first time 

401
00:29:34,360 --> 00:29:39,920
the use of quadratic equations, one of the greatest legacies of Babylonian mathematics. 

402
00:29:39,920 --> 00:29:43,560
Quadratic equations involve things where the unknown quantity 

403
00:29:43,560 --> 00:29:46,920
you're trying to identify is multiplied by itself. 

404
00:29:46,920 --> 00:29:49,880
We call this squaring because it gives the area of a square, 

405
00:29:49,880 --> 00:29:53,040
and it's in the context of calculating the area of land 

406
00:29:53,040 --> 00:29:55,960
that these quadratic equations naturally arise. 

407
00:30:01,320 --> 00:30:03,280
Here's a typical problem. 

408
00:30:03,280 --> 00:30:06,160
If a field has an area of 55 units 

409
00:30:06,160 --> 00:30:10,640
and one side is six units longer than the other, 

410
00:30:10,640 --> 00:30:12,560
how long is the shorter side? 

411
00:30:14,200 --> 00:30:18,640
The Babylonian solution was to reconfigure the field as a square. 

412
00:30:18,640 --> 00:30:21,920
Cut three units off the end 

413
00:30:21,920 --> 00:30:24,760
and move this round. 

414
00:30:24,760 --> 00:30:29,920
Now, there's a three-by-three piece missing, so let's add this in. 

415
00:30:29,920 --> 00:30:34,640
The area of the field has increased by nine units. 

416
00:30:34,640 --> 00:30:38,040
This makes the new area 64. 

417
00:30:38,040 --> 00:30:41,880
So the sides of the square are eight units. 

418
00:30:41,880 --> 00:30:45,320
The problem-solver knows that they've added three to this side. 

419
00:30:45,320 --> 00:30:49,520
So, the original length must be five. 

420
00:30:50,520 --> 00:30:55,600
It may not look like it, but this is one of the first quadratic equations in history. 

421
00:30:57,400 --> 00:31:02,400
In modern mathematics, I would use the symbolic language of algebra to solve this problem. 

422
00:31:02,400 --> 00:31:07,400
The amazing feat of the Babylonians is that they were using these geometric games to find the value, 

423
00:31:07,400 --> 00:31:10,200
without any recourse to symbols or formulas. 

424
00:31:10,200 --> 00:31:13,920
The Babylonians were enjoying problem-solving for its own sake. 

425
00:31:13,920 --> 00:31:17,960
They were falling in love with mathematics. 

426
00:31:29,080 --> 00:31:34,080
The Babylonians' fascination with numbers soon found a place in their leisure time, too. 

427
00:31:34,080 --> 00:31:35,960
They were avid game-players. 

428
00:31:35,960 --> 00:31:38,760
The Babylonians and their descendants have been playing 

429
00:31:38,760 --> 00:31:43,160
a version of backgammon for over 5,000 years. 

430
00:31:43,160 --> 00:31:45,840
The Babylonians played board games, 

431
00:31:45,840 --> 00:31:52,200
from very posh board games in royal tombs to little bits of board games found in schools, 

432
00:31:52,200 --> 00:31:56,280
to board games scratched on the entrances of palaces, 

433
00:31:56,280 --> 00:32:00,520
so that the guardsmen must have played when they were bored, 

434
00:32:00,520 --> 00:32:03,760
and they used dice to move their counters round. 

435
00:32:04,880 --> 00:32:09,800
People who played games were using numbers in their leisure time to try and outwit their opponent, 

436
00:32:09,800 --> 00:32:12,680
doing mental arithmetic very fast, 

437
00:32:12,680 --> 00:32:17,280
and so they were calculating in their leisure time, 

438
00:32:17,280 --> 00:32:21,000
without even thinking about it as being mathematical hard work. 

439
00:32:23,320 --> 00:32:24,600
Now's my chance. 

440
00:32:24,600 --> 00:32:30,000
'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.' 

441
00:32:30,000 --> 00:32:33,560
- It's up to you.
- Six... I need to move something. 

442
00:32:33,560 --> 00:32:36,560
'But it wasn't as easy as I thought.' 

443
00:32:36,560 --> 00:32:38,680
Ah! What the hell was that? 

444
00:32:38,680 --> 00:32:42,440
- Yeah.
- This is one, this is two. 

445
00:32:42,440 --> 00:32:44,200
Now you're in trouble. 

446
00:32:44,200 --> 00:32:47,800
- So I can't move anything. 
- You cannot move these. 

447
00:32:47,800 --> 00:32:49,200
Oh, gosh. 

448
00:32:50,520 --> 00:32:52,320
There you go. 

449
00:32:53,320 --> 00:32:54,960
Three and four. 

450
00:32:54,960 --> 00:33:00,720
'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.' 

451
00:33:00,720 --> 00:33:02,120
Yeah. 

452
00:33:03,120 --> 00:33:05,840
Put it there. Good game. 

453
00:33:07,120 --> 00:33:10,080
The Babylonians are recognised as one of the first cultures 

454
00:33:10,080 --> 00:33:13,840
to use symmetrical mathematical shapes to make dice, 

455
00:33:13,840 --> 00:33:17,440
but there is more heated debate about whether they might also 

456
00:33:17,440 --> 00:33:20,920
have been the first to discover the secrets of another important shape. 

457
00:33:20,920 --> 00:33:24,040
The right-angled triangle. 

458
00:33:27,000 --> 00:33:32,360
We've already seen how the Egyptians use a 3-4-5 right-angled triangle. 

459
00:33:32,360 --> 00:33:37,600
But what the Babylonians knew about this shape and others like it is much more sophisticated. 

460
00:33:37,600 --> 00:33:42,120
This is the most famous and controversial ancient tablet we have. 

461
00:33:42,120 --> 00:33:44,480
It's called Plimpton 322. 

462
00:33:45,480 --> 00:33:49,080
Many mathematicians are convinced it shows the Babylonians 

463
00:33:49,080 --> 00:33:53,360
could well have known the principle regarding right-angled triangles, 

464
00:33:53,360 --> 00:33:57,400
that the square on the diagonal is the sum of the squares on the sides, 

465
00:33:57,400 --> 00:34:00,280
and known it centuries before the Greeks claimed it. 

466
00:34:01,880 --> 00:34:06,320
This is a copy of arguably the most famous Babylonian tablet, 

467
00:34:06,320 --> 00:34:08,040
which is Plimpton 322, 

468
00:34:08,040 --> 00:34:12,680
and these numbers here reflect the width or height of a triangle, 

469
00:34:12,680 --> 00:34:17,520
this being the diagonal, the other side would be over here, 

470
00:34:17,520 --> 00:34:19,880
and the square of this column 

471
00:34:19,880 --> 00:34:23,280
plus the square of the number in this column 

472
00:34:23,280 --> 00:34:26,360
equals the square of the diagonal. 

473
00:34:26,360 --> 00:34:31,120
They are arranged in an order of steadily decreasing angle, 

474
00:34:31,120 --> 00:34:34,000
on a very uniform basis, showing that somebody 

475
00:34:34,000 --> 00:34:38,600
had a lot of understanding of how the numbers fit together. 

476
00:34:44,680 --> 00:34:50,800
Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths. 

477
00:34:50,800 --> 00:34:56,160
It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem, 

478
00:34:56,160 --> 00:35:01,200
and it's a conclusion that generations of historians have been seduced by. 

479
00:35:01,200 --> 00:35:03,960
But there could be a much simpler explanation 

480
00:35:03,960 --> 00:35:07,760
for the sets of three numbers which fulfil Pythagoras' theorem. 

481
00:35:07,760 --> 00:35:12,800
It's not a systematic explanation of Pythagorean triples, it's simply 

482
00:35:12,800 --> 00:35:17,640
a mathematics teacher doing some quite complicated calculations, 

483
00:35:17,640 --> 00:35:21,160
but in order to produce some very simple numbers, 

484
00:35:21,160 --> 00:35:26,120
in order to set his students problems about right-angled triangles, 

485
00:35:26,120 --> 00:35:31,000
and in that sense it's about Pythagorean triples only incidentally. 

486
00:35:33,480 --> 00:35:39,040
The most valuable clues to what they understood could lie elsewhere. 

487
00:35:39,040 --> 00:35:43,360
This small school exercise tablet is nearly 4,000 years old 

488
00:35:43,360 --> 00:35:48,800
and reveals just what the Babylonians did know about right-angled triangles. 

489
00:35:48,800 --> 00:35:54,360
It uses a principle of Pythagoras' theorem to find the value of an astounding new number. 

490
00:35:57,920 --> 00:36:05,000
Drawn along the diagonal is a really very good approximation to the square root of two, 

491
00:36:05,000 --> 00:36:10,880
and so that shows us that it was known and used in school environments. 

492
00:36:10,880 --> 00:36:12,880
Why's this important? 

493
00:36:12,880 --> 00:36:18,440
Because the square root of two is what we now call an irrational number, 

494
00:36:18,440 --> 00:36:23,960
that is, if we write it out in decimals, or even in sexigesimal places, 

495
00:36:23,960 --> 00:36:28,360
it doesn't end, the numbers go on forever after the decimal point. 

496
00:36:29,640 --> 00:36:33,640
The implications of this calculation are far-reaching. 

497
00:36:33,640 --> 00:36:37,920
Firstly, it means the Babylonians knew something of Pythagoras' theorem 

498
00:36:37,920 --> 00:36:39,800
1,000 years before Pythagoras. 

499
00:36:39,800 --> 00:36:45,560
Secondly, the fact that they can calculate this number to an accuracy of four decimal places 

500
00:36:45,560 --> 00:36:50,600
shows an amazing arithmetic facility, as well as a passion for mathematical detail. 

501
00:36:52,200 --> 00:36:56,440
The Babylonians' mathematical dexterity was astounding, 

502
00:36:56,440 --> 00:37:03,080
and for nearly 2,000 years they spearheaded intellectual progress in the ancient world. 

503
00:37:03,080 --> 00:37:08,280
But when their imperial power began to wane, so did their intellectual vigour. 

504
00:37:16,400 --> 00:37:23,280
By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia. 

505
00:37:25,200 --> 00:37:31,000
This is Palmyra in central Syria, a once-great city built by the Greeks. 

506
00:37:33,800 --> 00:37:41,000
The mathematical expertise needed to build structures with such geometric perfection is impressive. 

507
00:37:42,120 --> 00:37:48,320
Just like the Babylonians before them, the Greeks were passionate about mathematics. 

508
00:37:50,520 --> 00:37:53,080
The Greeks were clever colonists. 

509
00:37:53,080 --> 00:37:56,280
They took the best from the civilisations they invaded 

510
00:37:56,280 --> 00:37:58,720
to advance their own power and influence, 

511
00:37:58,720 --> 00:38:01,880
but they were soon making contributions themselves. 

512
00:38:01,880 --> 00:38:07,080
In my opinion, their greatest innovation was to do with a shift in the mind. 

513
00:38:07,080 --> 00:38:11,560
What they initiated would influence humanity for centuries. 

514
00:38:11,560 --> 00:38:14,520
They gave us the power of proof. 

515
00:38:14,520 --> 00:38:18,200
Somehow they decided that they had to have a deductive system 

516
00:38:18,200 --> 00:38:19,640
for their mathematics 

517
00:38:19,640 --> 00:38:21,800
and the typical deductive system 

518
00:38:21,800 --> 00:38:25,720
was to begin with certain axioms, which you assume are true. 

519
00:38:25,720 --> 00:38:29,080
It's as if you assume a certain theorem is true without proving it. 

520
00:38:29,080 --> 00:38:34,600
And then, using logical methods and very careful steps, 

521
00:38:34,600 --> 00:38:37,480
from these axioms you prove theorems 

522
00:38:37,480 --> 00:38:42,400
and from those theorems you prove more theorems, and it just snowballs. 

523
00:38:43,520 --> 00:38:47,000
Proof is what gives mathematics its strength. 

524
00:38:47,000 --> 00:38:51,360
It's the power of proof which means that the discoveries of the Greeks 

525
00:38:51,360 --> 00:38:55,480
are as true today as they were 2,000 years ago. 

526
00:38:55,480 --> 00:39:01,120
I needed to head west into the heart of the old Greek empire to learn more. 

527
00:39:08,720 --> 00:39:14,000
For me, Greek mathematics has always been heroic and romantic. 

528
00:39:15,280 --> 00:39:20,240
I'm on my way to Samos, less than a mile from the Turkish coast. 

529
00:39:20,240 --> 00:39:25,000
This place has become synonymous with the birth of Greek mathematics, 

530
00:39:25,000 --> 00:39:27,920
and it's down to the legend of one man. 

531
00:39:31,000 --> 00:39:33,120
His name is Pythagoras. 

532
00:39:33,120 --> 00:39:36,520
The legends that surround his life and work have contributed 

533
00:39:36,520 --> 00:39:40,320
to the celebrity status he has gained over the last 2,000 years. 

534
00:39:40,320 --> 00:39:44,960
He's credited, rightly or wrongly, with beginning the transformation 

535
00:39:44,960 --> 00:39:50,240
from mathematics as a tool for accounting to the analytic subject we recognise today. 

536
00:39:54,160 --> 00:39:57,160
Pythagoras is a controversial figure. 

537
00:39:57,160 --> 00:40:00,360
Because he left no mathematical writings, many have questioned 

538
00:40:00,360 --> 00:40:04,920
whether he indeed solved any of the theorems attributed to him. 

539
00:40:04,920 --> 00:40:07,960
He founded a school in Samos in the sixth century BC, 

540
00:40:07,960 --> 00:40:13,440
but his teachings were considered suspect and the Pythagoreans a bizarre sect. 

541
00:40:14,960 --> 00:40:19,720
There is good evidence that there were schools of Pythagoreans, 

542
00:40:19,720 --> 00:40:22,360
and they may have looked more like sects 

543
00:40:22,360 --> 00:40:25,920
than what we associate with philosophical schools, 

544
00:40:25,920 --> 00:40:30,920
because they didn't just share knowledge, they also shared a way of life. 

545
00:40:30,920 --> 00:40:36,080
There may have been communal living and they all seemed to have been 

546
00:40:36,080 --> 00:40:40,000
involved in the politics of their cities. 

547
00:40:40,000 --> 00:40:45,440
One feature that makes them unusual in the ancient world is that they included women. 

548
00:40:46,560 --> 00:40:52,280
But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians - 

549
00:40:52,280 --> 00:40:56,040
the properties of right-angled triangles. 

550
00:40:56,040 --> 00:40:58,400
What's known as Pythagoras' theorem 

551
00:40:58,400 --> 00:41:01,360
states that if you take any right-angled triangle, 

552
00:41:01,360 --> 00:41:05,320
build squares on all the sides, then the area of the largest square 

553
00:41:05,320 --> 00:41:09,320
is equal to the sum of the squares on the two smaller sides. 

554
00:41:13,240 --> 00:41:16,680
It's at this point for me that mathematics is born 

555
00:41:16,680 --> 00:41:19,880
and a gulf opens up between the other sciences, 

556
00:41:19,880 --> 00:41:24,600
and the proof is as simple as it is devastating in its implications. 

557
00:41:24,600 --> 00:41:28,080
Place four copies of the right-angled triangle 

558
00:41:28,080 --> 00:41:29,840
on top of this surface. 

559
00:41:29,840 --> 00:41:31,720
The square that you now see 

560
00:41:31,720 --> 00:41:35,440
has sides equal to the hypotenuse of the triangle. 

561
00:41:35,440 --> 00:41:37,600
By sliding these triangles around, 

562
00:41:37,600 --> 00:41:40,720
we see how we can break the area of the large square up 

563
00:41:40,720 --> 00:41:43,160
into the sum of two smaller squares, 

564
00:41:43,160 --> 00:41:47,280
whose sides are given by the two short sides of the triangle. 

565
00:41:47,280 --> 00:41:52,040
In other words, the square on the hypotenuse is equal to the sum 

566
00:41:52,040 --> 00:41:55,840
of the squares on the other sides. Pythagoras' theorem. 

567
00:41:58,040 --> 00:42:02,400
It illustrates one of the characteristic themes of Greek mathematics - 

568
00:42:02,400 --> 00:42:07,600
the appeal to beautiful arguments in geometry rather than a reliance on number. 

569
00:42:11,400 --> 00:42:16,000
Pythagoras may have fallen out of favour and many of the discoveries accredited to him 

570
00:42:16,000 --> 00:42:21,840
have been contested recently, but there's one mathematical theory that I'm loath to take away from him. 

571
00:42:21,840 --> 00:42:25,840
It's to do with music and the discovery of the harmonic series. 

572
00:42:27,680 --> 00:42:31,480
The story goes that, walking past a blacksmith's one day, 

573
00:42:31,480 --> 00:42:33,800
Pythagoras heard anvils being struck, 

574
00:42:33,800 --> 00:42:38,800
and noticed how the notes being produced sounded in perfect harmony. 

575
00:42:38,800 --> 00:42:42,240
He believed that there must be some rational explanation 

576
00:42:42,240 --> 00:42:46,080
to make sense of why the notes sounded so appealing. 

577
00:42:46,080 --> 00:42:48,560
The answer was mathematics. 

578
00:42:53,480 --> 00:42:58,120
Experimenting with a stringed instrument, Pythagoras discovered that the intervals between 

579
00:42:58,120 --> 00:43:02,400
harmonious musical notes were always represented as whole-number ratios. 

580
00:43:05,200 --> 00:43:08,160
And here's how he might have constructed his theory. 

581
00:43:10,720 --> 00:43:13,600
First, play a note on the open string. 

582
00:43:13,600 --> 00:43:15,120
MAN PLAYS NOTE 

583
00:43:15,120 --> 00:43:17,040
Next, take half the length. 

584
00:43:18,960 --> 00:43:22,160
The note almost sounds the same as the first note. 

585
00:43:22,160 --> 00:43:27,120
In fact it's an octave higher, but the relationship is so strong, we give these notes the same name. 

586
00:43:27,120 --> 00:43:28,960
Now take a third the length. 

587
00:43:31,600 --> 00:43:35,640
We get another note which sounds harmonious next to the first two, 

588
00:43:35,640 --> 00:43:41,240
but take a length of string which is not in a whole-number ratio and all we get is dissonance. 

589
00:43:46,600 --> 00:43:51,000
According to legend, Pythagoras was so excited by this discovery 

590
00:43:51,000 --> 00:43:54,440
that he concluded the whole universe was built from numbers. 

591
00:43:54,440 --> 00:44:00,040
But he and his followers were in for a rather unsettling challenge to their world view 

592
00:44:00,040 --> 00:44:05,120
and it came about as a result of the theorem which bears Pythagoras' name. 

593
00:44:07,120 --> 00:44:12,400
Legend has it, one of his followers, a mathematician called Hippasus, 

594
00:44:12,400 --> 00:44:15,480
set out to find the length of the diagonal 

595
00:44:15,480 --> 00:44:19,760
for a right-angled triangle with two sides measuring one unit. 

596
00:44:19,760 --> 00:44:25,520
Pythagoras' theorem implied that the length of the diagonal was a number whose square was two. 

597
00:44:25,520 --> 00:44:29,560
The Pythagoreans assumed that the answer would be a fraction, 

598
00:44:29,560 --> 00:44:36,000
but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it. 

599
00:44:36,000 --> 00:44:38,600
Eventually he realised his mistake. 

600
00:44:38,600 --> 00:44:43,320
It was the assumption that the value was a fraction at all which was wrong. 

601
00:44:43,320 --> 00:44:49,440
The value of the square root of two was the number that the Babylonians etched into the Yale tablet. 

602
00:44:49,440 --> 00:44:53,320
However, they didn't recognise the special character of this number. 

603
00:44:53,320 --> 00:44:55,040
But Hippasus did. 

604
00:44:55,040 --> 00:44:57,560
It was an irrational number. 

605
00:45:00,880 --> 00:45:04,800
The discovery of this new number, and others like it, is akin to an explorer 

606
00:45:04,800 --> 00:45:09,240
discovering a new continent, or a naturalist finding a new species. 

607
00:45:09,240 --> 00:45:13,520
But these irrational numbers didn't fit the Pythagorean world view. 

608
00:45:13,520 --> 00:45:19,120
Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy, 

609
00:45:19,120 --> 00:45:21,840
but Hippasus let slip the discovery 

610
00:45:21,840 --> 00:45:25,600
and was promptly drowned for his attempts to broadcast their research. 

611
00:45:27,080 --> 00:45:32,440
But these mathematical discoveries could not be easily suppressed. 

612
00:45:32,440 --> 00:45:37,920
Schools of philosophy and science started to flourish all over Greece, building on these foundations. 

613
00:45:37,920 --> 00:45:42,360
The most famous of these was the Academy. 

614
00:45:42,360 --> 00:45:47,560
Plato founded this school in Athens in 387 BC. 

615
00:45:47,560 --> 00:45:54,040
Although we think of him today as a philosopher, he was one of mathematics' most important patrons. 

616
00:45:54,040 --> 00:45:57,720
Plato was enraptured by the Pythagorean world view 

617
00:45:57,720 --> 00:46:02,040
and considered mathematics the bedrock of knowledge. 

618
00:46:02,040 --> 00:46:07,200
Some people would say that Plato is the most influential figure 

619
00:46:07,200 --> 00:46:10,080
for our perception of Greek mathematics. 

620
00:46:10,080 --> 00:46:15,120
He argued that mathematics is an important form of knowledge 

621
00:46:15,120 --> 00:46:17,600
and does have a connection with reality. 

622
00:46:17,600 --> 00:46:23,480
So by knowing mathematics, we know more about reality. 

623
00:46:23,480 --> 00:46:29,240
In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking 

624
00:46:29,240 --> 00:46:33,480
the secrets of the universe, a view still held by scientists today. 

625
00:46:33,480 --> 00:46:37,480
Indeed, the importance Plato attached to geometry is encapsulated 

626
00:46:37,480 --> 00:46:43,960
in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here." 

627
00:46:47,520 --> 00:46:53,720
Plato proposed that the universe could be crystallised into five regular symmetrical shapes. 

628
00:46:53,720 --> 00:46:56,640
These shapes, which we now call the Platonic solids, 

629
00:46:56,640 --> 00:46:59,600
were composed of regular polygons, assembled to create 

630
00:46:59,600 --> 00:47:03,080
three-dimensional symmetrical objects. 

631
00:47:03,080 --> 00:47:05,720
The tetrahedron represented fire. 

632
00:47:05,720 --> 00:47:09,960
The icosahedron, made from 20 triangles, represented water. 

633
00:47:09,960 --> 00:47:12,160
The stable cube was Earth. 

634
00:47:12,160 --> 00:47:15,880
The eight-faced octahedron was air. 

635
00:47:15,880 --> 00:47:19,440
And the fifth Platonic solid, the dodecahedron, 

636
00:47:19,440 --> 00:47:22,280
made out of 12 pentagons, was reserved for the shape 

637
00:47:22,280 --> 00:47:26,000
that captured Plato's view of the universe. 

638
00:47:29,600 --> 00:47:33,640
Plato's theory would have a seismic influence and continued to inspire 

639
00:47:33,640 --> 00:47:37,400
mathematicians and astronomers for over 1,500 years. 

640
00:47:38,360 --> 00:47:41,120
In addition to the breakthroughs made in the Academy, 

641
00:47:41,120 --> 00:47:45,040
mathematical triumphs were also emerging from the edge of the Greek empire, 

642
00:47:45,040 --> 00:47:51,520
and owed as much to the mathematical heritage of the Egyptians as the Greeks. 

643
00:47:51,520 --> 00:47:58,000
Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC, 

644
00:47:58,000 --> 00:48:04,320
and its famous library soon gained a reputation to rival Plato's Academy. 

645
00:48:04,320 --> 00:48:11,760
The kings of Alexandria were prepared to invest in the arts and culture, 

646
00:48:11,760 --> 00:48:14,960
in technology, mathematics, grammar, 

647
00:48:14,960 --> 00:48:19,680
because patronage for cultural pursuits 

648
00:48:19,680 --> 00:48:27,000
was one way of showing that you were a more prestigious ruler, 

649
00:48:27,000 --> 00:48:30,320
and had a better entitlement to greatness. 

650
00:48:32,040 --> 00:48:35,360
The old library and its precious contents were destroyed 

651
00:48:35,360 --> 00:48:38,240
But its spirit is alive in a new building. 

652
00:48:40,240 --> 00:48:44,120
Today, the library remains a place of discovery and scholarship. 

653
00:48:48,600 --> 00:48:51,920
Mathematicians and philosophers flocked to Alexandria, 

654
00:48:51,920 --> 00:48:55,080
driven by their thirst for knowledge and the pursuit of excellence. 

655
00:48:55,080 --> 00:48:59,040
The patrons of the library were the first professional scientists, 

656
00:48:59,040 --> 00:49:02,600
individuals who were paid for their devotion to research. 

657
00:49:02,600 --> 00:49:04,720
But of all those early pioneers, 

658
00:49:04,720 --> 00:49:08,880
my hero is the enigmatic Greek mathematician Euclid. 

659
00:49:12,560 --> 00:49:15,120
We know very little about Euclid's life, 

660
00:49:15,120 --> 00:49:19,360
but his greatest achievements were as a chronicler of mathematics. 

661
00:49:19,360 --> 00:49:24,600
Around 300 BC, he wrote the most important text book of all time - 

662
00:49:24,600 --> 00:49:27,080
The Elements. In The Elements, 

663
00:49:27,080 --> 00:49:31,120
we find the culmination of the mathematical revolution 

664
00:49:31,120 --> 00:49:32,960
which had taken place in Greece. 

665
00:49:34,880 --> 00:49:39,240
It's built on a series of mathematical assumptions, called axioms. 

666
00:49:39,240 --> 00:49:44,000
For example, a line can be drawn between any two points. 

667
00:49:44,000 --> 00:49:48,760
From these axioms, logical deductions are made and mathematical theorems established. 

668
00:49:51,880 --> 00:49:56,360
The Elements contains formulas for calculating the volumes of cones 

669
00:49:56,360 --> 00:49:59,400
and cylinders, proofs about geometric series, 

670
00:49:59,400 --> 00:50:02,160
perfect numbers and primes. 

671
00:50:02,160 --> 00:50:06,760
The climax of The Elements is a proof that there are only five Platonic solids. 

672
00:50:09,560 --> 00:50:14,280
For me, this last theorem captures the power of mathematics. 

673
00:50:14,280 --> 00:50:17,080
It's one thing to build five symmetrical solids, 

674
00:50:17,080 --> 00:50:22,600
quite another to come up with a watertight, logical argument for why there can't be a sixth. 

675
00:50:22,600 --> 00:50:26,600
The Elements unfolds like a wonderful, logical mystery novel. 

676
00:50:26,600 --> 00:50:29,720
But this is a story which transcends time. 

677
00:50:29,720 --> 00:50:33,560
Scientific theories get knocked down, from one generation to the next, 

678
00:50:33,560 --> 00:50:39,920
but the theorems in The Elements are as true today as they were 2,000 years ago. 

679
00:50:39,920 --> 00:50:43,480
When you stop and think about it, it's really amazing. 

680
00:50:43,480 --> 00:50:45,160
It's the same theorems that we teach. 

681
00:50:45,160 --> 00:50:49,960
We may teach them in a slightly different way, we may organise them differently, 

682
00:50:49,960 --> 00:50:54,200
but it's Euclidean geometry that is still valid, 

683
00:50:54,200 --> 00:50:58,320
and even in higher mathematics, when you go to higher dimensional spaces, 

684
00:50:58,320 --> 00:51:00,560
you're still using Euclidean geometry. 

685
00:51:02,080 --> 00:51:06,080
Alexandria must have been an inspiring place for the ancient scholars, 

686
00:51:06,080 --> 00:51:12,360
and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port. 

687
00:51:12,360 --> 00:51:18,680
One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes. 

688
00:51:19,640 --> 00:51:23,200
He would become a mathematical visionary. 

689
00:51:23,200 --> 00:51:28,080
The best Greek mathematicians, they were always pushing the limits, 

690
00:51:28,080 --> 00:51:29,560
pushing the envelope. 

691
00:51:29,560 --> 00:51:32,200
So, Archimedes... 

692
00:51:32,200 --> 00:51:35,200
did what he could with polygons, 

693
00:51:35,200 --> 00:51:37,520
with solids. 

694
00:51:37,520 --> 00:51:40,360
He then moved on to centres of gravity. 

695
00:51:40,360 --> 00:51:44,680
He then moved on to the spiral. 

696
00:51:44,680 --> 00:51:50,800
This instinct to try and mathematise everything 

697
00:51:50,800 --> 00:51:54,440
is something that I see as a legacy. 

698
00:51:55,520 --> 00:52:00,280
One of Archimedes' specialities was weapons of mass destruction. 

699
00:52:00,280 --> 00:52:06,360
They were used against the Romans when they invaded his home of Syracuse in 212 BC. 

700
00:52:06,360 --> 00:52:10,200
He also designed mirrors, which harnessed the power of the sun, 

701
00:52:10,200 --> 00:52:12,760
to set the Roman ships on fire. 

702
00:52:12,760 --> 00:52:17,520
But to Archimedes, these endeavours were mere amusements in geometry. 

703
00:52:17,520 --> 00:52:20,280
He had loftier ambitions. 

704
00:52:23,040 --> 00:52:29,560
Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake 

705
00:52:29,560 --> 00:52:33,800
and not for the ignoble trade of engineering or the sordid quest for profit. 

706
00:52:33,800 --> 00:52:37,840
One of his finest investigations into pure mathematics 

707
00:52:37,840 --> 00:52:41,840
was to produce formulas to calculate the areas of regular shapes. 

708
00:52:43,760 --> 00:52:49,480
Archimedes' method was to capture new shapes by using shapes he already understood. 

709
00:52:49,480 --> 00:52:52,720
So, for example, to calculate the area of a circle, 

710
00:52:52,720 --> 00:52:57,920
he would enclose it inside a triangle, and then by doubling the number of sides on the triangle, 

711
00:52:57,920 --> 00:53:02,320
the enclosing shape would get closer and closer to the circle. 

712
00:53:02,320 --> 00:53:04,360
Indeed, we sometimes call a circle 

713
00:53:04,360 --> 00:53:07,360
a polygon with an infinite number of sides. 

714
00:53:07,360 --> 00:53:11,200
But by estimating the area of the circle, Archimedes is, in fact, 

715
00:53:11,200 --> 00:53:15,480
getting a value for pi, the most important number in mathematics. 

716
00:53:16,480 --> 00:53:22,760
However, it was calculating the volumes of solid objects where Archimedes excelled. 

717
00:53:22,760 --> 00:53:25,800
He found a way to calculate the volume of a sphere 

718
00:53:25,800 --> 00:53:30,280
by slicing it up and approximating each slice as a cylinder. 

719
00:53:30,280 --> 00:53:33,120
He then added up the volumes of the slices 

720
00:53:33,120 --> 00:53:36,480
to get an approximate value for the sphere. 

721
00:53:36,480 --> 00:53:39,440
But his act of genius was to see what happens 

722
00:53:39,440 --> 00:53:42,280
if you make the slices thinner and thinner. 

723
00:53:42,280 --> 00:53:47,040
In the limit, the approximation becomes an exact calculation. 

724
00:53:51,080 --> 00:53:56,040
But it was Archimedes' commitment to mathematics that would be his undoing. 

725
00:53:58,120 --> 00:54:02,960
Archimedes was contemplating a problem about circles traced in the sand. 

726
00:54:02,960 --> 00:54:05,600
When a Roman soldier accosted him, 

727
00:54:05,600 --> 00:54:11,640
Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem. 

728
00:54:11,640 --> 00:54:16,920
But the Roman soldier was not interested in Archimedes' problem and killed him on the spot. 

729
00:54:16,920 --> 00:54:21,800
Even in death, Archimedes' devotion to mathematics was unwavering. 

730
00:54:43,360 --> 00:54:46,480
By the middle of the 1st century BC, 

731
00:54:46,480 --> 00:54:50,520
the Romans had tightened their grip on the old Greek empire. 

732
00:54:50,520 --> 00:54:53,320
They were less smitten with the beauty of mathematics 

733
00:54:53,320 --> 00:54:56,640
and were more concerned with its practical applications. 

734
00:54:56,640 --> 00:55:02,520
This pragmatic attitude signalled the beginning of the end for the great library of Alexandria. 

735
00:55:02,520 --> 00:55:06,760
But one mathematician was determined to keep the legacy of the Greeks alive. 

736
00:55:06,760 --> 00:55:11,640
Hypatia was exceptional, a female mathematician, 

737
00:55:11,640 --> 00:55:14,800
and a pagan in the piously Christian Roman empire. 

738
00:55:16,680 --> 00:55:21,560
Hypatia was very prestigious and very influential in her time. 

739
00:55:21,560 --> 00:55:27,440
She was a teacher with a lot of students, a lot of followers. 

740
00:55:27,440 --> 00:55:31,680
She was politically influential in Alexandria. 

741
00:55:31,680 --> 00:55:34,560
So it's this combination of... 

742
00:55:34,560 --> 00:55:40,840
high knowledge and high prestige that may have made her 

743
00:55:40,840 --> 00:55:44,400
a figure of hatred for... 

744
00:55:44,400 --> 00:55:46,080
the Christian mob. 

745
00:55:51,760 --> 00:55:55,800
One morning during Lent, Hypatia was dragged off her chariot 

746
00:55:55,800 --> 00:55:59,840
by a zealous Christian mob and taken to a church. 

747
00:55:59,840 --> 00:56:03,560
There, she was tortured and brutally murdered. 

748
00:56:06,280 --> 00:56:09,880
The dramatic circumstances of her life and death 

749
00:56:09,880 --> 00:56:12,000
fascinated later generations. 

750
00:56:12,000 --> 00:56:17,680
Sadly, her cult status eclipsed her mathematical achievements. 

751
00:56:17,680 --> 00:56:20,720
She was, in fact, a brilliant teacher and theorist, 

752
00:56:20,720 --> 00:56:26,440
and her death dealt a final blow to the Greek mathematical heritage of Alexandria. 

753
00:56:33,800 --> 00:56:37,680
My travels have taken me on a fascinating journey to uncover 

754
00:56:37,680 --> 00:56:42,880
the passion and innovation of the world's earliest mathematicians. 

755
00:56:42,880 --> 00:56:47,920
It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece 

756
00:56:47,920 --> 00:56:52,320
that are the foundations on which my subject is built today. 

757
00:56:52,320 --> 00:56:55,760
But this is just the beginning of my mathematical odyssey. 

758
00:56:55,760 --> 00:56:59,400
The next leg of my journey lies east, in the depths of Asia, 

759
00:56:59,400 --> 00:57:02,560
where mathematicians scaled even greater heights 

760
00:57:02,560 --> 00:57:04,800
in pursuit of knowledge. 

761
00:57:04,800 --> 00:57:08,720
With this new era came a new language of algebra and numbers, 

762
00:57:08,720 --> 00:57:12,920
better suited to telling the next chapter in the story of maths. 

763
00:57:12,920 --> 00:57:16,600
You can learn more about the story of maths 

764
00:57:16,600 --> 00:57:19,840
with the Open University at... 

765
00:57:36,040 --> 00:57:39,080
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