00:00Our world is made up of patterns and sequences.

00:04They're all around us.

00:09Animals travel across the earth in ever-changing formations.

00:14Landscapes are constantly altering.

00:18One of the reasons mathematics began was because we needed to find a way

00:21of making sense of these natural patterns.

00:27The most basic concepts of maths - space and quantity -

00:32are hard-wired into our brains.

00:36Even animals have a sense of distance and number,

00:39assessing when their pack is outnumbered, and whether to fight or fly,

00:45calculating whether their prey is within striking distance.

00:50Understanding maths is the difference between life and death.

00:56But it was man who took these basic concepts

00:59and started to build upon these foundations.

01:01At some point, humans started to spot patterns,

01:05to make connections, to count and to order the world around them.

01:08With this, a whole new mathematical universe began to emerge.

01:20This is the River Nile.

01:22It's been the lifeline of Egypt for millennia.

01:26I've come here because it's where some of the first signs

01:29of mathematics as we know it today emerged.

01:34People abandoned nomadic life and began settling here as early as 6000BC.

01:39The conditions were perfect for farming.

01:47The most important event for Egyptian agriculture each year was the flooding of the Nile.

01:53So this was used as a marker to start each new year.

01:58Egyptians did record what was going on over periods of time,

02:03so in order to establish a calendar like this,

02:05you need to count how many days, for example,

02:09happened in-between lunar phases,

02:11or how many days happened in-between two floodings of the Nile.

02:19Recording the patterns for the seasons was essential,

02:23not only to their management of the land, but also their religion.

02:27The ancient Egyptians who settled on the Nile banks

02:30believed it was the river god, Hapy, who flooded the river each year.

02:34And in return for the life-giving water,

02:37the citizens offered a portion of the yield as a thanksgiving.

02:43As settlements grew larger, it became necessary to find ways to administer them.

02:48Areas of land needed to be calculated, crop yields predicted,

02:52taxes charged and collated.

02:54In short, people needed to count and measure.

02:59The Egyptians used their bodies to measure the world,

03:02and it's how their units of measurements evolved.

03:06A palm was the width of a hand,

03:08a cubit an arm length from elbow to fingertips.

03:13Land cubits, strips of land measuring a cubit by 100,

03:16were used by the pharaoh's surveyors to calculate areas.

03:23There's a very strong link between bureaucracy

03:26and the development of mathematics in ancient Egypt.

03:29And we can see this link right from the beginning,

03:32from the invention of the number system,

03:34throughout Egyptian history, really.

03:37For the Old Kingdom, the only evidence we have

03:40are metrological systems, that is measurements for areas, for length.

03:44This points to a bureaucratic need to develop such things.

03:50It was vital to know the area of a farmer's land so he could be taxed accordingly.

03:55Or if the Nile robbed him of part of his land, so he could request a rebate.

04:00It meant that the pharaoh's surveyors were often calculating

04:03the area of irregular parcels of land.

04:07It was the need to solve such practical problems

04:10that made them the earliest mathematical innovators.

04:18The Egyptians needed some way to record the results of their calculations.

04:25Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo,

04:29I was on the hunt for those that recorded some of the first numbers in history.

04:34They were difficult to track down.

04:39But I did find them in the end.

04:45The Egyptians were using a decimal system, motivated by the 10 fingers on our hands.

04:51The sign for one was a stroke,

04:5310, a heel bone, 100, a coil of rope, and 1,000, a Lotus plant.

04:59How much is this T-shirt?

05:0325!Yes!So that would be 2 knee bones and 5 strokes.

05:09So you're not gonna charge me anything up here?Here, one million!

05:17One million, yeah, that's pretty big!

05:20The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed.

05:27They had no concept of a place value,

05:31so one stroke could only represent one unit,

05:35Although you can write a million with just one character,

05:38rather than the seven that we use, if you want to write a million minus one,

05:42then the poor old Egyptian scribe has got to write nine strokes,

05:46nine heel bones, nine coils of rope, and so on,

05:49a total of 54 characters.

05:54Despite the drawback of this number system, the Egyptians were brilliant problem solvers.

06:01We know this because of the few records that have survived.

06:05The Egyptian scribes used sheets of papyrus

06:08to record their mathematical discoveries.

06:11This delicate material made from reeds decayed over time

06:15and many secrets perished with it.

06:18But there's one revealing document that has survived.

06:22The Rhind Mathematical Papyrus is the most important document

06:26we have today for Egyptian mathematics.

06:29We get a good overview of what types of problems

06:33the Egyptians would have dealt with in their mathematics.

06:37We also get explicitly stated how multiplications and divisions were carried out.

06:44The papyri show how to multiply two large numbers together.

06:49But to illustrate the method, let's take two smaller numbers.

06:53Let's do three times six.

06:56The scribe would take the first number, three, and put it in one column.

07:02In the second column, he would place the number one.

07:05Then he would double the numbers in each column, so three becomes six...

07:13..and six would become 12.

07:20And then in the second column, one would become two,

07:23and two becomes four.

07:28Now, here's the really clever bit.

07:30The scribe wants to multiply three by six.

07:33So he takes the powers of two in the second column,

07:37which add up to six. That's two plus four.

07:40Then he moves back to the first column, and just takes

07:43those rows corresponding to the two and the four.

07:46So that's six and the 12.

07:48He adds those together to get the answer of 18.

07:53But for me, the most striking thing about this method

07:56is that the scribe has effectively written that second number in binary.

08:00Six is one lot of four, one lot of two, and no units.

08:08The Egyptians have understood the power of binary over 3,000 years

08:12before the mathematician and philosopher Leibniz would reveal their potential.

08:16Today, the whole technological world depends on the same principles

08:21that were used in ancient Egypt.

08:25The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC.

08:31Its problems are concerned with finding solutions to everyday situations.

08:36Several of the problems mention bread and beer,

08:39which isn't surprising as Egyptian workers were paid in food and drink.

08:43One is concerned with how to divide nine loaves

08:46equally between 10 people, without a fight breaking out.

08:51I've got nine loaves of bread here.

08:53I'm gonna take five of them and cut them into halves.

08:58Of course, nine people could shave a 10th off their loaf

09:00and give the pile of crumbs to the 10th person.

09:04But the Egyptians developed a far more elegant solution -

09:07take the next four and divide those into thirds.

09:13But two of the thirds I am now going to cut into fifths,

09:16so each piece will be one fifteenth.

09:21Each person then gets one half

09:32It is through such seemingly practical problems

09:35that we start to see a more abstract mathematics developing.

09:38Suddenly, new numbers are on the scene - fractions -

09:41and it isn't too long before the Egyptians are exploring the mathematics of these numbers.

09:48Fractions are clearly of practical importance to anyone dividing quantities for trade in the market.

09:54To log these transactions, the Egyptians developed notation which recorded these new numbers.

10:02One of the earliest representations of these fractions

10:05came from a hieroglyph which had great mystical significance.

10:09It's called the Eye of Horus.

10:13Horus was an Old Kingdom god, depicted as half man, half falcon.

10:20According to legend, Horus' father was killed by his other son, Seth.

10:24Horus was determined to avenge the murder.

10:28During one particularly fierce battle,

10:30Seth ripped out Horus' eye, tore it up and scattered it over Egypt.

10:35But the gods were looking favourably on Horus.

10:38They gathered up the scattered pieces and reassembled the eye.

10:45Each part of the eye represented a different fraction.

10:49Each one, half the fraction before.

10:52Although the original eye represented a whole unit,

10:56the reassembled eye is 1/64 short.

10:59Although the Egyptians stopped at 1/64,

11:03implicit in this picture

11:05is the possibility of adding more fractions,

11:08halving them each time, the sum getting closer and closer to one,

11:13but never quite reaching it.

11:16This is the first hint of something called a geometric series,

11:20and it appears at a number of points in the Rhind Papyrus.

11:23But the concept of infinite series would remain hidden

11:26until the mathematicians of Asia discovered it centuries later.

11:34Having worked out a system of numbers, including these new fractions,

11:38it was time for the Egyptians to apply their knowledge

11:41to understanding shapes that they encountered day to day.

11:44These shapes were rarely regular squares or rectangles,

11:48and in the Rhind Papyrus, we find the area of a more organic form, the circle.

11:53What is astounding in the calculation

11:57of the area of the circle is its exactness, really.

12:00How they would have found their method is open to speculation,

12:04because the texts we have

12:06do not show us the methods how they were found.

12:10This calculation is particularly striking because it depends

12:14on seeing how the shape of the circle

12:16can be approximated by shapes that the Egyptians already understood.

12:21The Rhind Papyrus states that a circular field

12:24with a diameter of nine units

12:26is close in area to a square with sides of eight.

12:30But how would this relationship have been discovered?

12:34My favourite theory sees the answer in the ancient game of mancala.

12:39Mancala boards were found carved on the roofs of temples.

12:43Each player starts with an equal number of stones,

12:47and the object of the game is to move them round the board,

12:50capturing your opponent's counters on the way.

12:54As the players sat around waiting to make their next move,

12:58perhaps one of them realised that sometimes the balls fill the circular holes

13:01of the mancala board in a rather nice way.

13:03He might have gone on to experiment with trying to make larger circles.

13:09Perhaps he noticed that 64 stones, the square of 8,

13:13can be used to make a circle with diameter nine stones.

13:17By rearranging the stones, the circle has been approximated by a square.

13:22And because the area of a circle ispi times the radius squared,

13:26the Egyptian calculation gives us the first accurate value for pi.

13:31The area of the circle is 64. Divide this by the radius squared,

13:36in this case 4.5 squared, and you get a value for pi.

13:40So 64 divided by 4.5 squared is 3.16,

13:44just a little under two hundredths away from its true value.

13:48But the really brilliant thing is, the Egyptians

13:51are using these smaller shapes to capture the larger shape.

13:59But there's one imposing andmajestic symbol of Egyptian

14:02mathematics we haven't attempted to unravel yet -

14:07I've seen so many pictures that I couldn't believe I'd be impressed by them.

14:12But meeting them face to face, you understand why they're called

14:15one of the Seven Wonders of the Ancient World.

14:18They're simply breathtaking.

14:20And how much more impressive they must have been in their day,

14:23when the sides were as smooth as glass, reflecting the desert sun.

14:28To me it looks like there might be mirror pyramidshiding underneath the desert,

14:34which would complete the shapes to make perfectlysymmetrical octahedrons.

14:38Sometimes, in the shimmer of the desert heat, you can almost see these shapes.

14:45It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician.

14:52The pyramids are just a little short to create these perfect shapes,

14:57but some have suggested another important mathematical concept

15:00might be hidden inside the proportions of the Great Pyramid - the golden ratio.

15:06Two lengths are in the golden ratio, if the relationship of the longest

15:11to the shortest is the same as the sum of the two to the longest side.

15:16Such a ratio has been associated with the perfect proportions one finds

15:21all over the natural world, as well as in the work of artists,

15:25architects and designers for millennia.

15:31Whether the architects of the pyramids were conscious of this important mathematical idea,

15:36or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know.

15:41For me, the most impressive thing about the pyramidsis the mathematical brilliance

15:46that went into making them, including the first inkling

15:49of one of the great theorems of the ancient world, Pythagoras' theorem.

15:55In order to get perfect right-angled corners on their buildings

15:58and pyramids, the Egyptians would haveused a rope with knots tied in it.

16:03At some point, the Egyptians realised that if they took a triangle with sides

16:07marked with three knots, four knots and five knots, it guaranteed them aperfect right-angle.

16:14This is because three squared, plus four squared, is equal to five squared.

16:19So we've got a perfect Pythagorean triangle.

16:24In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle.

16:30But I'm pretty sure that the Egyptians hadn't got

16:32this sweeping generalisation of their 3, 4, 5 triangle.

16:37We would not expect to find the general proof

16:41because this is not the style of Egyptian mathematics.

16:44Every problem was solved using concrete numbers and then

16:48if a verification would be carried out at the end, it would use the result

16:52and these concrete, given numbers,

16:54there's no general proof within the Egyptian mathematical texts.

17:00It would be some 2,000 years before the Greeks and Pythagoras

17:03would prove that all right-angled triangles shared certain properties.

17:08This wasn't the only mathematical idea that theEgyptians would anticipate.

17:12In a 4,000-year-old document called the Moscow papyrus, we find a formula for the volume

17:19of a pyramid with its peak sliced off,which shows the first hint ofcalculus at work.

17:25For a culture like Egypt that is famous for its pyramids, you would expect problems like this

17:32to have been a regular feature within the mathematical texts.

17:35The calculation of the volume of a truncated pyramid is one of the most

17:40advanced bits, according to our modern standards of mathematics,

17:45that we have from ancient Egypt.

17:48The architects and engineers would certainly have wanted such a formula

17:52to calculate the amount of materials required to build it.

17:55But it's a mark of the sophistication

17:58of Egyptian mathematics that they were ableto produce such a beautiful method.

18:08To understand how they derived their formula, start with a pyramid

18:12built such that the highest point sits directly over one corner.

18:17Three of these can be put together to make a rectangular box,

18:22so the volume of the skewed pyramid is a third the volume of the box.

18:27That is, the height, times the length, times the width, divided by three.

18:33Now comes an argument which shows the very first hints of the calculus at work,

18:38thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.

18:44Suppose you could cut the pyramid into slices, you could then slide

18:48the layers across to make the more symmetrical pyramid you see in Giza.

18:54However, the volume of the pyramid has not changed, despite the rearrangement of the layers.

18:58So the same formula works.

19:04The Egyptians were amazing innovators,

19:08and their ability to generate new mathematics was staggering.

19:11For me, they revealed the power of geometry and numbers, and made the first moves

19:16towards some of the exciting mathematical discoveries to come.

19:20But there was another civilisation that had mathematics to rival that of Egypt.

19:25And we know much more about their achievements.

19:33This is Damascus, over 5,000 years old,

19:37and still vibrant and bustling today.

19:40It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.

19:46The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.

19:52In order to expand and run their empire, they became masters of managing and manipulating numbers.

20:00We have law codes for instance that tell us

20:03about the way society is ordered.

20:05The people we know most about are the scribes, the professionally literate

20:09and numerate people who kept the records for the wealthy families and for the temples and palaces.

20:14Scribe schools existed from around 2500BC.

20:19Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.

20:26Scribe records were kept on clay tablets,

20:29which allowed the Babylonians to manage and advance their empire.

20:33However, many of the tablets we have today aren't official documents, but children's exercises.

20:40It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.

20:46So, this is a geometrical textbook from about the 18th century BC.

20:51I hope you can see that there are lots of pictures on it.

20:54And underneath each picture is a text that sets a problem about the picture.

20:58So for instance this one here says, I drew a square, 60 units long,

21:04and inside it, I drew four circles - what are their areas?

21:10This little tablet here was written 1,000 years at least later than the tablet here,

21:16but has a very interesting relationship.

21:19It also has four circles on,

21:21in a square, roughly drawn, but this isn't a textbook, it's a school exercise.

21:26The adult scribe who's teaching the student is being given this

21:30as an example of completed homework or something like that.

21:35Like the Egyptians, the Babylonians appeared interested

21:38in solving practical problems to do with measuring and weighing.

21:42The Babylonian solutions to these problems are written like mathematical recipes.

21:46A scribe would simply follow and record a set of instructions to get a result.

21:52Here's an example of the kind of problem they'd solve.

21:56I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them.

22:00Instead, I'm gonna take four times their weight and add them to the scales.

22:07Now I'm gonna add 20 gin. Gin was the ancient Babylonian measure of weight.

22:13I'm gonna take half of everything here and then add it again...

22:17That's two bundles, and ten gin.

22:19Everything on this side is equal to one mana. One mana was 60 gin.

22:25And here, we have one of the first mathematical equations in history,

22:29everything on this side is equal to one mana.

22:32But how much does the bundle of cinnamon sticks weigh?

22:35Without any algebraic language, they were able to manipulate

22:38the quantities to be able to prove that the cinnamon sticks weighed five gin.

22:44In my mind, it's this kind of problem which gives mathematics a bit of a bad name.

22:49You can blame those ancient Babylonians for all those tortuous problems you had at school.

22:54But the ancient Babylonian scribes excelled at this kind of problem.

22:59Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60.

23:09The Babylonians invented their number system, like the Egyptians, by using their fingers.

23:14But instead of counting through the 10 fingers on their hand,

23:17Babylonians found a moreintriguing way to count body parts.

23:20They used the 12 knuckles on one hand,

23:23and the five fingers on the other to be able to count

23:2512 times 5, ie 60 different numbers.

23:29So for example, this number would have been 2 lots of 12, 24,

23:34and then, 1, 2, 3, 4, 5, to make 29.

23:41The number 60 had another powerful property.

23:45It can be perfectly divided in a multitude of ways.

23:50I can arrange them in 2 rows of 30.

24:08The divisibility of 60 makes it a perfect base in which to do arithmetic.

24:14The base 60 system was so successful, we still use elements of it today.

24:20Every time we want to tell the time, we recognise units of 60 -

24:2460 seconds in a minute, 60 minutes in an hour.

24:28But the most important feature of the Babylonians' number system was that it recognised place value.

24:33Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording,

24:39the position of each Babylonian number records the power of 60.

24:50Instead of inventing new symbols for bigger and bigger numbers,

24:53they would write 1-1-1, so this number would be 3,661.

25:03The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky.

25:16The Babylonians' calendar was based on the cycles of the moon.

25:20They needed a way of recording astronomically large numbers.

25:24Month by month, year by year, these cycles were recorded.

25:28From about 800BC, there were complete lists of lunar eclipses.

25:34The Babylonian system of measurement was quite sophisticated at that time.

25:40They had a system of angular measurement,

25:42360 degrees in a full circle, each degree was divided

25:46into 60 minutes, a minute was further divided into 60 seconds.

25:51So they had a regular system for measurement, and it was in perfect harmony with their number system,

25:57so it's well suited not only for observation but also for calculation.

26:01But in order to calculate and cope with these large numbers,

26:05the Babylonians needed to invent a new symbol.

26:09And in so doing, they prepared the ground for one of the great

26:13breakthroughs in the history of mathematics - zero.

26:16In the early days, the Babylonians, in order to mark an empty place in

26:20the middle of a number, would simply leave a blank space.

26:23So they needed a way of representing nothing in the middle of a number.

26:29So they used a sign, as a sort of breathing marker, a punctuation mark,

26:34and it comes to mean zero in the middle of a number.

26:38This was the first time zero, in any form, had appeared in the mathematical universe.

26:44But it would be over a 1,000 years before this little place holder would become a number in its own right.

26:59Having established such a sophisticated system of numbers,

27:03they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.

27:11Babylonian engineers and surveyors found ingenious ways of

27:15accessing water, and channelling it to the crop fields.

27:19Yet again, they used mathematics to come up with solutions.

27:24The Orontes valley in Syria is still an agricultural hub,

27:28and the old methods of irrigation are being exploited today, just as they were thousands of years ago.

27:35Many of the problems in Babylonian mathematics

27:38are concerned with measuring land, and it's here we see for the first time

27:43the use of quadratic equations, one of the greatest legacies of Babylonian mathematics.

27:49Quadratic equations involve things where the unknown quantity

27:52you're trying to identify is multiplied by itself.

27:56We call this squaring because it gives the area of a square,

27:59and it's in the context of calculating the area of land

28:02that thesequadratic equations naturally arise.

28:10Here's a typical problem.

28:12If a field has an area of 55 units

28:15and one side is six units longer than the other,

28:19how long is the shorter side?

28:23The Babylonian solution was to reconfigure the field as a square.

28:27Cut three units off the end

28:31and move this round.

28:33Now, there's a three-by-three piece missing, so let's add this in.

28:39The area of the field has increased by nine units.

28:43This makes the new area 64.

28:47So the sides of the square are eight units.

28:51The problem-solver knows that they've added three to this side.

28:54So, the original length must be five.

28:59It may not look like it, but this is one of the first quadratic equations in history.

29:06In modern mathematics, I would use the symbolic language of algebra to solve this problem.

29:11The amazing feat of the Babylonians is that they were using these geometric games to find the value,

29:16without any recourse to symbols or formulas.

29:19The Babylonians were enjoying problem-solving for its own sake.

29:23They were falling in love with mathematics.

29:38The Babylonians' fascination with numbers soon found a place in their leisure time, too.

29:43They were avid game-players.

29:45The Babylonians and their descendants have been playing

29:47a version of backgammon for over 5,000 years.

29:52The Babylonians played board games,

29:55from very posh board games in royal tombs to little bits of board games found in schools,

30:01to board games scratched on the entrances of palaces,

30:05so that the guardsman must have played when they were bored,

30:09and they used dice to move their counters round.

30:13People who played games were using numbers in their leisure time to try and outwit their opponent,

30:18doing mental arithmetic very fast,

30:21and so they were calculating in their leisure time,

30:26without even thinking about it as being mathematical hard work.

30:33'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.'

30:39It's up to you.Six... I need to move something.

30:42'But it wasn't as easy as I thought.'

30:45Ah! What the hell was that?

30:47Yeah.This is one, this is two.

30:51Now you're in trouble.

30:53So I can't move anything. You cannot move these.

31:04'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.'

31:12Put it there. Good game.

31:16The Babylonians are recognised as one of the first cultures

31:19to use symmetrical mathematical shapes to make dice,

31:23but there is more heated debates about whether they might also

31:26have been the first to discover the secrets of another important shape.

31:30The right-angled triangle.

31:36We've already seen how the Egyptians use a 3-4-5 right-angled triangle.

31:41But what the Babylonians knew about this shape and others like it is much more sophisticated.

31:46This is the most famous and controversial ancient tablet we have.

31:51It's called Plimpton 322.

31:54Many mathematicians are convinced it shows the Babylonians

31:58could well have known the principle regarding right-angled triangles,

32:02that the square on the diagonal is the sum of the squares on the sides,

32:06and known it centuries before the Greeks claimed it.

32:11This is a copy of arguably the most famous Babylonian tablet,

32:15which is Plimpton 322,

32:17and these numbers here reflect the width or height of a triangle,

32:21this being the diagonal, the other side would be over here,

32:26and the square of this column plus the square of the number in this column

32:32equals the square of the diagonal.

32:35They are arranged in an order of steadily decreasing angle,

32:40on a very uniform basis, showing that somebody

32:43had a lot of understanding of how the numbers fit together.

32:53Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.

32:59It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem,

33:05and it's a conclusion that generations of historians have been seduced by.

33:10But there could be a much simpler explanation

33:13for the sets of three numbers which fulfil Pythagoras' theorem.

33:16It's not a systematic explanation of Pythagorean triples, it's simply

33:21a mathematics teacher doing some quite complicated calculations,

33:26but in order to produce some very simple numbers,

33:30in order to set his students problems about right-angled triangles,

33:35and in that sense it's about Pythagorean triples only incidentally.

33:42The most valuable clues to what they understood could lie elsewhere.

33:48This small school exercise tablet is nearly 4,000 years old

33:52and reveals just what the Babylonians did know about right-angled triangles.

33:57It uses a principle of Pythagoras' theorem to find the value of an astounding new number.

34:07Drawn along the diagonal is a really very good approximation to the square root of two,

34:14and so that shows us that it was known and used in school environments.

34:20Why's this important?

34:22Because the square root of two is what we now call an irrational number,

34:27that is, if we write it out in decimals, or even in sexigesimal places,

34:33it doesn't end, the numbers go on forever after the decimal point.

34:38The implications of this calculation are far-reaching.

34:42Firstly, it means the Babylonians knew something of Pythagoras' theorem

34:471,000 years before Pythagoras.

34:48Secondly, the fact that they can calculate this number to an accuracy of four decimal places

34:54shows an amazing arithmetic facility, as well as a passion for mathematical detail.

35:01The Babylonians' mathematical dexterity was astounding,

35:05and for nearly 2,000 years they spearheaded intellectual progress in the ancient world.

35:12But when their imperial power began to wane, so did their intellectual vigour.

35:25By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.

35:34This is Palmyra in central Syria, a once-great city built by the Greeks.

35:42The mathematical expertise needed to build structures with such geometric perfection is impressive.

35:51Just like the Babylonians before them, the Greeks were passionate about mathematics.

35:59The Greeks were clever colonists.

36:02They took the best from the civilisations they invaded

36:05to advance their own power and influence,

36:07but they were soon making contributions themselves.

36:11In my opinion, their greatest innovation was to do with a shift in the mind.

36:16What they initiated would influence humanity for centuries.

36:20They gave us the power of proof.

36:23Somehow they decided that they had to have a deductive system for their mathematics

36:28and the typical deductive system was to begin with certain axioms, which you assume are true.

36:34It's as if you assume a certain theorem is true without proving it.

36:38And then, using logical methods and very careful steps,

36:43from these axioms you prove theorems

36:46and from those theorems you prove more theorems, and it just snowballs.

36:52Proof is what gives mathematics its strength.

36:56It's the power or proof which means that the discoveries of the Greeks

37:00are as true today as they were 2,000 years ago.

37:04I needed to head west into the heart of the old Greek empire to learn more.

37:17For me, Greek mathematics has always been heroic and romantic.

37:24I'm on my way to Samos, less than a mile from the Turkish coast.

37:29This place has become synonymous with the birth of Greek mathematics,

37:34and it's down to the legend of one man.

37:40His name is Pythagoras.

37:42The legends that surround his life and work have contributed

37:45to the celebrity status he has gained over the last 2,000 years.

37:49He's credited, rightly or wrongly, with beginning the transformation

37:54from mathematics as a tool for accounting to the analytic subject we recognise today.

38:03Pythagoras is a controversial figure.

38:06Because he left no mathematical writings, many have questioned

38:09whether he indeed solved any of the theorems attributed to him.

38:14He founded a school in Samos in the sixth century BC,

38:17but his teachings were considered suspect and the Pythagoreans a bizarre sect.

38:24There is good evidence that there were schools of Pythagoreans,

38:28and they may have looked more like sects than what we associate with philosophical schools,

38:35because they didn't just share knowledge, they also shared a way of life.

38:40There may have been communal living and they all seemed to have been

38:45involved in the politics of their cities.

38:49One feature that makes them unusual in the ancient world is that they included women.

38:55But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians -

39:01the properties of right-angled triangles.

39:05What's known as Pythagoras' theorem

39:07states that if you take any right-angled triangle,

39:10build squares on all the sides, then the area of the largest square

39:14is equal to the sum of the squares on the two smaller sides.

39:22It's at this point for me that mathematics is born

39:25and a gulf opens up between the other sciences,

39:29and the proof is as simple as it is devastating in its implications.

39:33Place four copies of the right-angled triangle

39:37on top of this surface.

39:39The square that you now see

39:40has sides equal to the hypotenuse of the triangle.

39:44By sliding these triangles around,

39:46we see how we can break the area of the large square up

39:49into the sum of two smaller squares,

39:52whose sides are given by the two short sides of the triangle.

39:56In other words, the square on the hypotenuse is equal to the sum

40:01of the squares on the other sides. Pythagoras' theorem.

40:07It illustrates one of the characteristic themes of Greek mathematics -

40:11the appeal to beautiful arguments in geometry rather than a reliance on number.

40:20Pythagoras may have fallen out of favour and many of the discoveries accredited to him

40:25have been contested recently, but there's one mathematical theory that I'm loath to take away from him.

40:31It's to do with music and the discoveryof the harmonic series.

40:36The story goes that, walking past a blacksmith's one day,

40:40Pythagoras heard anvils being struck,

40:42and noticed how the notes being produced sounded in perfect harmony.

40:47He believed that there must be some rational explanation

40:51to make sense of why the notes sounded so appealing.

40:55The answer was mathematics.

41:02Experimenting with a stringed instrument, Pythagoras discovered that the intervals between

41:07harmonious musical notes were always represented as whole-number ratios.

41:14And here's how he might have constructed his theory.

41:19First, play a note on the open string.

41:24Next, take half the length.

41:28The note almost sounds the same as the first note.

41:31In fact it's an octave higher, but the relationship is so strong, we give these notes the same name.

41:36Now take a third the length.

41:40We get another note which sounds harmonious next to the first two,

41:44but take a length of string which is not in a whole-number ratio and all we get is dissonance.

41:55According to legend, Pythagoras was so excited by this discovery

42:00that he concluded the whole universe was built from numbers.

42:03But he and his followers were in for a rather unsettling challenge to their world view

42:09and it came about as a result of the theorem which bears Pythagoras' name.

42:16Legend has it, one of his followers, a mathematician called Hippasus,

42:21set out to find the length of the diagonal

42:24for a right-angled triangle with two sides measuring one unit.

42:28Pythagoras' theorem implied that the length of the diagonal was a number whose square was two.

42:34The Pythagoreans assumed that the answer would be a fraction,

42:38but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it.

42:45Eventually he realised his mistake.

42:47It was the assumption that the value was a fraction at all which was wrong.

42:52The value of the square root of two was the number that the Babylonians etched into the Yale tablet.

42:58However, they didn't recognise the special character of this number.

43:04It was an irrational number.

43:10The discovery of this new number, and others like it, is akin to an explorer

43:13discovering a new continent, or a naturalist finding a new species.

43:18But these irrational numbers didn't fit the Pythagorean world view.

43:22Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy,

43:28but Hippasus let slip the discovery

43:31and was promptly drowned for his attempts to broadcast their research.

43:36But these mathematical discoveries could not be easily suppressed.

43:41Schools of philosophy and science started to flourish all over Greece, building on these foundations.

43:47The most famous of these was the Academy.

43:51Plato founded this school in Athens in 387 BC.

43:56Although we think of him today as a philosopher, he was one of mathematics' most important patrons.

44:03Plato was enraptured by the Pythagorean world view

44:06and considered mathematics the bedrock of knowledge.

44:11Some people would say that Plato is the most influential figure

44:16for our perception of Greek mathematics.

44:19He argued that mathematics is an important form of knowledge

44:24and does have a connection with reality.

44:26So by knowing mathematics, we know more about reality.

44:32In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking

44:38the secrets of the universe, a view still held by scientists today.

44:42Indeed, the importance Plato attached to geometry is encapsulated

44:46in the sign that was mounted above the Academy, "Let no-one ignorant of geometry enter here."

44:56Plato proposed that the universe could be crystallised into five regular symmetrical shapes.

45:02These shapes, which we now call the Platonic solids,

45:05were composed of regular polygons, assembled to create

45:08three-dimensional symmetrical objects.

45:12The tetrahedron represented fire.

45:14The icosahedron, made from 20 triangles, represented water.

45:19The stable cube was Earth.

45:21The eight-faced octahedron was air.

45:25And the fifth Platonic solid, the dodecahedron,

45:28made out of 12 pentagons, was reserved for the shape

45:31that captured Plato's view of the universe.

45:38Plato's theory would have a seismic influence and continued to inspire

45:42mathematicians and astronomers for over 1,500 years.

45:47In addition to the breakthroughs made in the Academy,

45:50mathematical triumphs were also emerging from the edge of the Greek empire,

45:55and owed as much to the mathematical heritage of the Egyptians as the Greeks.

46:00Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC,

46:07and its famous library soon gained a reputation to rival Plato's academy.

46:13The kings of Alexandria were prepared to invest in the arts and culture,

46:20in technology, mathematics, grammar,

46:24because patronage for cultural pursuits

46:28was one way of showing that you were a more prestigious ruler,

46:36and had a better entitlement to greatness.

46:41The old library and its precious contents were destroyed

46:44when the Muslims conquered Egypt in the 7th Century.

46:48But its spirit is alive in a new building.

46:52Today, the library remains a place of discovery and scholarship.

47:01Mathematicians and philosophers flocked to Alexandria,

47:04driven by their thirst for knowledge and the pursuit of excellence.

47:07The patrons of the library were the first professional scientists,

47:11individuals who were paid for their devotion to research.

47:15But of all those early pioneers,

47:17my hero is the enigmatic Greek mathematician Euclid.

47:25We know very little about Euclid's life,

47:27but his greatest achievements were as a chronicler of mathematics.

47:31Around 300 BC, he wrote the most important text book of all time -

47:37The Elements. In The Elements,

47:39we find the culmination of the mathematical revolution

47:43which had taken place in Greece.

47:47It's built on a series of mathematical assumptions, called axioms.

47:51For example, a line can be drawn between any two points.

47:56From these axioms, logical deductions are made and mathematical theorems established.

48:04The Elements contains formulas for calculating the volumes of cones

48:08and cylinders, proofs about geometric series,

48:12perfect numbers and primes.

48:14The climax of The Elements is a proof that there are only five Platonic solids.

48:22For me, this last theorem captures the power of mathematics.

48:26It's one thing to build five symmetrical solids,

48:29quite another to come up with a watertight, logical argument for why there can't be a sixth.

48:35The Elements unfolds like a wonderful, logical mystery novel.

48:39But this is a story which transcends time.

48:42Scientific theories get knocked down, from one generation to the next,

48:46but the theorems in The Elements are as true today as they were 2,000 years ago.

48:52When you stop and think about it, it's really amazing.

48:56It's the same theorems that we teach.

48:57We may teach them in a slightly different way, we may organise them differently,

49:02but it's Euclidean geometry that is still valid,

49:06and even in higher mathematics, when you go to higher dimensional spaces,

49:10you're still using Euclidean geometry.

49:14Alexandria must have been an inspiring place for the ancient scholars,

49:18and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port.

49:24One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.

49:32He would become a mathematical visionary.

49:35The best Greek mathematicians, they were always pushing the limits,

49:40pushing the envelope.

49:44did what he could with polygons,

49:50He then moved on to centres of gravity.

49:52He then moved on to the spiral.

49:57This instinct to try and mathematise everything

50:03is something that I see as a legacy.

50:08One of Archimedes' specialities was weapons of mass destruction.

50:12They were used against the Romans when they invaded his home of Syracuse in 212 BC.

50:18He also designed mirrors, which harnessed the power of the sun,

50:22to set the Roman ships on fire.

50:25But to Archimedes, these endeavours were mere amusements in geometry.

50:30He had loftier ambitions.

50:35Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake

50:42and not for the ignoble trade of engineering or the sordid quest for profit.

50:46One of his finest investigations into pure mathematics

50:50was to produce formulas to calculate the areas of regular shapes.

50:56Archimedes' method was to capture new shapes by using shapes he already understood.

51:02So, for example, to calculate the area of a circle,

51:05he would enclose it inside a triangle, and then by doubling the number of sides on the triangle,

51:10the enclosing shape would get closer and closer to the circle.

51:14Indeed, we sometimes call a circle

51:16a polygon with an infinite number of sides.

51:19But by estimating the area of the circle, Archimedes is, in fact,

51:23getting a value for pi, the most important number in mathematics.

51:29However, it was calculating the volumes of solid objects where Archimedes excelled.

51:35He found a way to calculate the volume of a sphere

51:38by slicing it up and approximating each slice as a cylinder.

51:42He then added up the volumes of the slices

51:45to get an approximate value for the sphere.

51:49But his act of genius was to see what happens

51:52if you make the slices thinner and thinner.

51:54In the limit, the approximation becomes an exact calculation.

52:03But it was Archimedes' commitment to mathematics that would be his undoing.

52:10Archimedes was contemplating a problem about circles traced in the sand.

52:15When a Roman soldier accosted him,

52:18Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.

52:24But the Roman soldier was not interested in Archimedes' problem and killed him on the spot.

52:29Even in death, Archimedes' devotion to mathematics was unwavering.

52:55By the middle of the 1st Century BC,

52:59the Romans had tightened their grip on the old Greek empire.

53:03They were less smitten with the beauty of mathematics

53:05and were more concerned with its practical applications.

53:09This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.

53:15But one mathematician was determined to keep the legacy of the Greeks alive.

53:19Hypatia was exceptional, a female mathematician,

53:24and a pagan in the piously Christian Roman empire.

53:29Hypatia was very prestigious and very influential in her time.

53:34She was a teacher with a lot of students, a lot of followers.

53:40She was politically influential in Alexandria.

53:44So it's this combination of...

53:47high knowledge and high prestige that may have made her

53:53a figure of hatred for...

54:04One morning during Lent, Hypatia was dragged off her chariot

54:08by a zealous Christian mob and taken to a church.

54:12There, she was tortured and brutally murdered.

54:18The dramatic circumstances of her life and death

54:21fascinated later generations.

54:24Sadly, her cult status eclipsed her mathematical achievements.

54:30She was, in fact, a brilliant teacher and theorist,

54:33and her death dealt a final blow to the Greek mathematical heritage of Alexandria.

54:46My travels have taken me on a fascinating journey to uncover

54:50the passion and innovation of the world's earliest mathematicians.

54:55It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece

55:00that are the foundations on which my subject is built today.

55:04But this is just the beginning of my mathematical odyssey.

55:08The next leg of my journey lies east, in the depths of Asia,

55:12where mathematicians scaled even greater heights

55:15in pursuit of knowledge.

55:17With this new era came a new language of algebra and numbers,

55:21better suited to telling the next chapter in the story of maths.

55:26You can learn more about the story of maths

55:29with the Open University at...

55:42This is what we call a decimal
place-value system,

55:45and it's very similar
to the one we use today.

55:48We too use numbers from one to nine,
and we use their position

55:51to indicate whether it's units,
tens, hundreds or thousands.

55:55But the power of these rods is that
it makes calculations very quick.

55:59In fact, the way the ancient
Chinese did their calculations

56:03is very similar to the way
we learn today in school.

56:11Not only were the ancient Chinese

56:12the first to use a decimal
place-value system,

56:16but they did so over 1,000 years
before we adopted it in the West.

56:20But they only used it
when calculating with the rods.

56:24When writing the numbers down,

56:27the ancient Chinese
didn't use the place-value system.

56:32Instead, they used a far
more laborious method,

56:36in which special symbols stood for
tens, hundreds, thousands and so on.

56:42So the number 924
would be written out

56:45as nine hundreds,
two tens and four.

56:50Not quite so efficient.

56:55that the ancient Chinese didn't
have a concept of zero.

56:58They didn't have a symbol for zero.
It just didn't exist as a number.

57:01Using the counting rods,

57:03they would use a blank space
where today we would write a zero.

57:07The problem came with trying to
write down this number, which is why

57:11they had to create these new symbols
for tens, hundreds and thousands.

57:14Without a zero, the written
number was extremely limited.

57:21But the absence of zero
didn't stop

57:24the ancient Chinese from
making giant mathematical steps.

57:28In fact,
there was a widespread fascination

57:31with number in ancient China.

57:33According to legend,
the first sovereign of China,

57:37the Yellow Emperor,
had one of his deities

57:40create mathematics in 2800BC,

57:43believing that number held cosmic
significance. And to this day,

57:48the Chinese still believe in
the mystical power of numbers.

57:55Odd numbers are seen as male,
even numbers, female.

58:00The number four
is to be avoided at all costs.

58:03The number eight
brings good fortune.

58:07And the ancient Chinese were
drawn to patterns in numbers,

58:10developing their own
rather early version of sudoku.

58:16It was called the magic square.

58:23Legend has it that thousands of
years ago, Emperor Yu was visited

58:27by a sacred turtle that came out
of the depths of the Yellow River.

58:31On its back were numbers

58:33arranged into a magic square,
a little like this.

58:46which was regarded as having
great religious significance,

58:50all the numbers in each line -
horizontal, vertical and diagonal -

58:54all add up to the same number - 15.

59:01Now, the magic square may be
no more than a fun puzzle,

59:04but it shows
the ancient Chinese fascination

59:06with mathematical patterns,
and it wasn't too long

59:09before they were creating
even bigger magic squares

59:12with even greater magical
and mathematical powers.

59:23But mathematics also played

59:26a vital role in the running
of the emperor's court.

59:30The calendar and the movement
of the planets

59:34were of the utmost
importance to the emperor,

59:37influencing all his decisions, even
down to the way his day was planned,

59:42so astronomers became prized
members of the imperial court,

59:46and astronomers were
always mathematicians.

59:53Everything in the emperor's life
was governed by the calendar,

59:57and he ran his affairs
with mathematical precision.

01:00:01The emperor even got
his mathematical advisors

01:00:04to come up with a system
to help him sleep his way

01:00:07through the vast number of women
he had in his harem.

01:00:10Never one to miss a trick,
the mathematical advisors decided

01:00:13to base the harem on a mathematical
idea called a geometric progression.

01:00:18Maths has never had
such a fun purpose!

01:00:21Legend has it that
in the space of 15 nights,

01:00:25the emperor had to sleep
with 121 women...

01:00:37three senior consorts,

01:00:48The mathematicians
would soon have realised

01:00:51that this was a geometric
progression - a series of numbers

01:00:54in which you get
from one number to the next

01:00:57by multiplying the same number
each time - in this case, three.

01:01:03Each group of women is three times
as large as the previous group,

01:01:07so the mathematicians could quickly
draw up a rota to ensure that,

01:01:11in the space of 15 nights,

01:01:13the emperor slept
with every woman in the harem.

01:01:18The first night
was reserved for the empress.

01:01:22The next was for the three
senior consorts.

01:01:25The nine wives came next,

01:01:27and then the 27 concubines were
chosen in rotation, nine each night.

01:01:34And then finally,
over a period of nine nights,

01:01:37the 81 slaves were dealt with
in groups of nine.

01:01:46Being the emperor certainly
required stamina,

01:01:49a bit like being a mathematician,

01:01:51but the object is clear -

01:01:53to procure the best
possible imperial succession.

01:01:57The rota ensured that the emperor

01:01:59slept with the ladies of highest
rank closest to the full moon,

01:02:03when their yin, their female force,

01:02:05would be at its highest and be able
to match his yang, or male force.

01:02:15The emperor's court wasn't alone
in its dependence on mathematics.

01:02:19It was central to the running
of the state.

01:02:22Ancient China was a vast and growing
empire with a strict legal code,

01:02:27widespread taxation

01:02:28and a standardised system
of weights, measures and money.

01:02:34The empire needed

01:02:36a highly trained civil service,
competent in mathematics.

01:02:42And to educate these civil servants
was a mathematical textbook,

01:02:46probably written in around 200BC -
the Nine Chapters.

01:02:53The book is a compilation
of 246 problems

01:02:57in practical areas such as trade,
payment of wages and taxes.

01:03:04And at the heart
of these problems lies

01:03:07one of the central themes of
mathematics, how to solve equations.

01:03:15Equations are a little bit
like cryptic crosswords.

01:03:18You're given a certain amount
of information

01:03:20about some unknown numbers,
and from that information

01:03:23you've got to deduce what
the unknown numbers are.

01:03:26For example,
with my weights and scales,

01:03:28I've found out that one plum...

01:03:31..together with three peaches

01:03:34weighs a total of 15 grams.

01:03:44together with one peach

01:03:47weighs a total of 10g.

01:03:49From this information, I can
deduce what a single plum weighs

01:03:53and a single peach weighs,
and this is how I do it.

01:03:59If I take the first set of scales,

01:04:01one plum and three peaches
weighing 15g,

01:04:04and double it, I get two plums
and six peaches weighing 30g.

01:04:13If I take this and subtract from it
the second set of scales -

01:04:17that's two plums
and a peach weighing 10g -

01:04:19I'm left with
an interesting result -

01:04:24Having eliminated the plums,

01:04:27I've discovered that
five peaches weighs 20g,

01:04:30so a single peach weighs 4g,

01:04:33and from this I can deduce
that the plum weighs 3g.

01:04:37The ancient Chinese went on
to apply similar methods

01:04:41to larger and larger numbers
of unknowns,

01:04:44using it to solve increasingly
complicated equations.

01:04:49What's extraordinary is

01:04:51that this particular
system of solving equations

01:04:54didn't appear in the West until
the beginning of the 19th century.

01:04:58In 1809, while analysing a rock
called Pallas in the asteroid belt,

01:05:02Carl Friedrich Gauss,

01:05:04who would become known
as the prince of mathematics,

01:05:07rediscovered this method

01:05:08which had been formulated
in ancient China centuries earlier.

01:05:12Once again, ancient China
streets ahead of Europe.

01:05:20But the Chinese
were to go on to solve

01:05:22even more complicated equations
involving far larger numbers.

01:05:26In what's become known as
the Chinese remainder theorem,

01:05:29the Chinese came up
with a new kind of problem.

01:05:34In this, we know the number
that's left

01:05:37when the equation's unknown number
is divided by a given number -

01:05:41say, three, five or seven.

01:05:45Of course, this is a fairly
abstract mathematical problem,

01:05:49but the ancient Chinese still
couched it in practical terms.

01:05:55So a woman in the market has
a tray of eggs,

01:05:58but she doesn't know
how many eggs she's got.

01:06:01What she does know is that
if she arranges them in threes,

01:06:04she has one egg left over.

01:06:07If she arranges them in fives,
she gets two eggs left over.

01:06:11But if she arranged them
in rows of seven,

01:06:14she found she had
three eggs left over.

01:06:17The ancient Chinese found a
systematic way to calculate

01:06:21that the smallest number of eggs she
could have had in the tray is 52.

01:06:25But the more amazing thing is
that you can capture

01:06:28such a large number, like 52,

01:06:30by using these small numbers
like three, five and seven.

01:06:33This way of looking at numbers

01:06:35would become a dominant theme
over the last two centuries.

01:06:48By the 6th century AD, the Chinese
remainder theorem was being used

01:06:52in ancient Chinese astronomy
to measure planetary movement.

01:06:56But today it still
has practical uses.

01:06:59Internet cryptography
encodes numbers using mathematics

01:07:04that has its origins
in the Chinese remainder theorem.

01:07:16By the 13th century,

01:07:18mathematics was long established
on the curriculum,

01:07:21with over 30 mathematics schools
scattered across the country.

01:07:25The golden age of
Chinese maths had arrived.

01:07:31And its most important mathematician
was called Qin Jiushao.

01:07:37Legend has it that Qin Jiushao
was something of a scoundrel.

01:07:42He was a fantastically
corrupt imperial administrator

01:07:46who crisscrossed China,
lurching from one post to another.

01:07:49Repeatedly sacked for embezzling
government money,

01:07:53he poisoned anyone
who got in his way.

01:07:58Qin Jiushao
was reputedly described as

01:08:01as violent as a tiger or a wolf

01:08:03and as poisonous
as a scorpion or a viper

01:08:06so, not surprisingly,
he made a fierce warrior.

01:08:09For ten years, he fought
against the invading Mongols,

01:08:12but for much of that time he was
complaining that his military life

01:08:16took him away
from his true passion.

01:08:18No, not corruption, but mathematics.

01:08:33Qin started trying
to solve equations

01:08:35that grew out of trying
to measure the world around us.

01:08:38Quadratic equations involve numbers

01:08:40that are squared, or to the power
of two - say, five times five.

01:08:46The ancient Mesopotamians

01:08:48had already realised
that these equations

01:08:50were perfect for measuring flat,
two-dimensional shapes,

01:08:54like Tiananmen Square.

01:08:59But Qin was interested

01:09:01in more complicated equations -
cubic equations.

01:09:06These involve numbers
which are cubed,

01:09:09or to the power of three -
say, five times five times five,

01:09:14and they were perfect for capturing
three-dimensional shapes,

01:09:18like Chairman Mao's mausoleum.

01:09:22Qin found a way
of solving cubic equations,

01:09:24and this is how it worked.

01:09:30Say Qin wants to know

01:09:33the exact dimensions
of Chairman Mao's mausoleum.

01:09:38He knows the volume of the building

01:09:41and the relationships
between the dimensions.

01:09:45In order to get his answer,

01:09:48Qin uses what he knows
to produce a cubic equation.

01:09:52He then makes
an educated guess at the dimensions.

01:09:56Although he's captured a good
proportion of the mausoleum,

01:10:00there are still bits left over.

01:10:04Qin takes these bits
and creates a new cubic equation.

01:10:08He can now refine his first guess

01:10:10by trying to find a solution to
this new cubic equation, and so on.

01:10:17Each time he does this,
the pieces he's left with

01:10:20get smaller and smaller and his
guesses get better and better.

01:10:27What's striking is that Qin's
method for solving equations

01:10:30wasn't discovered in the West
until the 17th century,

01:10:33when Isaac Newton came up with a
very similar approximation method.

01:10:38The power of this technique is

01:10:40that it can be applied
to even more complicated equations.

01:10:44Qin even used his techniques
to solve an equation

01:10:48involving numbers
up to the power of ten.

01:10:50This was extraordinary stuff -
highly complex mathematics.

01:10:57Qin may have been years
ahead of his time,

01:10:59but there was a problem
with his technique.

01:11:02It only gave him
an approximate solution.

01:11:04That might be good enough for an
engineer - not for a mathematician.

01:11:08Mathematics is an exact science.
We like things to be precise,

01:11:12and Qin just couldn't
come up with a formula

01:11:15to give him an exact solution
to these complicated equations.

01:11:26China had made
great mathematical leaps,

01:11:29but the next great mathematical
breakthroughs were to happen

01:11:33in a country lying
to the southwest of China -

01:11:35a country that had a rich
mathematical tradition

01:11:39that would change
the face of maths for ever.

01:12:12India's first great mathematical
gift lay in the world of number.

01:12:17Like the Chinese, the Indians had
discovered the mathematical benefits

01:12:21of the decimal place-value system

01:12:23and were using it by the middle
of the 3rd century AD.

01:12:29It's been suggested that
the Indians learned the system

01:12:33from Chinese merchants travelling
in India with their counting rods,

01:12:37or they may well just have
stumbled across it themselves.

01:12:41It's all such a long time ago
that it's shrouded in mystery.

01:12:47We may never know how the Indians
came up with their number system,

01:12:50but we do know that they refined
and perfected it,

01:12:53creating the ancestors for the nine
numerals used across the world now.

01:12:57Many rank the Indian
system of counting

01:13:00as one of the greatest intellectual
innovations of all time,

01:13:03developing into the closest thing
we could call a universal language.

01:13:26But there was one number missing,

01:13:28and it was the Indians who
would introduce it to the world.

01:13:38The earliest known recording of this
number dates from the 9th century,

01:13:43though it was probably in
practical use for centuries before.

01:13:48This strange new numeral
is engraved on the wall

01:13:52of small temple in the fort
of Gwalior in central India.

01:14:00So here we are in one of the holy
sites of the mathematical world,

01:14:04and what I'm looking for
is in this inscription on the wall.

01:14:08Up here are some numbers, and...

01:14:11here's the new number.

01:14:20It's astonishing to think
that before the Indians invented it,

01:14:24there was no number zero.

01:14:26To the ancient Greeks,
it simply hadn't existed.

01:14:30To the Egyptians, the Mesopotamians
and, as we've seen, the Chinese,

01:14:34zero had been in use but as
a placeholder, an empty space

01:14:38to show a zero inside a number.

01:14:44The Indians transformed zero
from a mere placeholder

01:14:47into a number
that made sense in its own right -

01:14:49a number for calculation,
for investigation.

01:14:53This brilliant conceptual leap
would revolutionise mathematics.

01:15:01Now, with just ten digits - zero
to nine - it was suddenly possible

01:15:05to capture astronomically large
numbers

01:15:08in an incredibly efficient way.

01:15:13But why did the Indians
make this imaginative leap?

01:15:17Well, we'll never know for sure,

01:15:19but it's possible that the idea and
symbol that the Indians use for zero

01:15:23came from calculations they did
with stones in the sand.

01:15:26When stones were removed
from the calculation,

01:15:29a small, round hole was left
in its place,

01:15:32representing the movement
from something to nothing.

01:15:38But perhaps there is also a cultural
reason for the invention of zero.

01:15:43HORNS BLOW AND DRUMS BANG

01:15:51For the ancient Indians, the
concepts of nothingness and eternity

01:15:56lay at the very heart
of their belief system.

01:16:03BELL CLANGS AND SILENCE FALLS

01:16:08In the religions of India, the
universe was born from nothingness,

01:16:12and nothingness is
the ultimate goal of humanity.

01:16:15So it's perhaps not surprising

01:16:17that a culture that so
enthusiastically embraced the void

01:16:21should be happy
with the notion of zero.

01:16:24The Indians even used the word for
the philosophical idea of the void,

01:16:29shunya, to represent
the new mathematical term "zero".

01:16:46In the 7th century, the brilliant
Indian mathematician Brahmagupta

01:16:51proved some of the essential
properties of zero.

01:17:00Brahmagupta's rules
about calculating with zero

01:17:03are taught in schools
all over the world to this day.

01:17:08One plus zero equals one.

01:17:12One minus zero equals one.

01:17:15One times zero is equal to zero.

01:17:23But Brahmagupta came a cropper when
he tried to do one divided by zero.

01:17:27After all, what number
times zero equals one?

01:17:30It would require a new mathematical
concept, that of infinity,

01:17:34to make sense of dividing by zero,

01:17:36and the breakthrough was made by a
12th-century Indian mathematician

01:17:40called Bhaskara II,
and it works like this.

01:17:43If I take a fruit and I divide
it into halves, I get two pieces,

01:17:50so one divided by a half is two.

01:17:53If I divide it into thirds,
I get three pieces.

01:17:56So when I divide it into smaller
and smaller fractions,

01:17:59I get more and more pieces,
so ultimately,

01:18:03when I divide by a piece

01:18:05which is of zero size,
I'll have infinitely many pieces.

01:18:09So for Bhaskara,
one divided by zero is infinity.

01:18:21But the Indians would go further
in their calculations with zero.

01:18:26For example, if you take three
from three and get zero,

01:18:31what happens when you take
four from three?

01:18:34It looks like you have nothing,

01:18:36but the Indians recognised
that this

01:18:39was a new sort of nothing -
negative numbers.

01:18:42The Indians called them "debts",
because they solved equations like,

01:18:46"If I have three batches
of material and take four away,

01:18:49"how many have I left?"

01:18:55This may seem odd and impractical,

01:18:57but that was the beauty
of Indian mathematics.

01:19:00Their ability to come up
with negative numbers and zero

01:19:03was because they thought of
numbers as abstract entities.

01:19:07They weren't just for counting
and measuring pieces of cloth.

01:19:10They had a life of their own,
floating free of the real world.

01:19:13This led to an explosion
of mathematical ideas.

01:19:29The Indians' abstract approach
to mathematics soon revealed

01:19:33a new side to the problem of
how to solve quadratic equations.

01:19:37That is equations including
numbers to the power of two.

01:19:42Brahmagupta's understanding of
negative numbers allowed him to see

01:19:46that quadratic equations
always have two solutions,

01:19:49one of which could be negative.

01:19:54Brahmagupta went even further,

01:19:55solving quadratic equations
with two unknowns,

01:19:58a question which wouldn't be
considered in the West until 1657,

01:20:02when French mathematician Fermat

01:20:04challenged his colleagues
with the same problem.

01:20:07Little did he know that they'd
been beaten to a solution

01:20:10by Brahmagupta
1,000 years earlier.

01:20:18Brahmagupta was beginning to find
abstract ways of solving equations,

01:20:23but astonishingly,
he was also developing

01:20:26a new mathematical language
to express that abstraction.

01:20:31Brahmagupta was experimenting with
ways of writing his equations down,

01:20:35using the initials
of the names of different colours

01:20:39to represent unknowns
in his equations.

01:20:43A new mathematical language
was coming to life,

01:20:46which would ultimately lead
to the x's and y's

01:20:48which fill today's
mathematical journals.

01:21:05But it wasn't just new
notation that was being developed.

01:21:12Indian mathematicians
were responsible for making

01:21:14fundamental new discoveries
in the theory of trigonometry.

01:21:21The power of trigonometry
is that it acts like a dictionary,

01:21:25translating geometry into numbers
and back.

01:21:28Although first developed by the
ancient Greeks,

01:21:32it was in the hands
of the Indian mathematicians

01:21:34that the subject truly flourished.

01:21:36At its heart lies the study
of right-angled triangles.

01:21:42In trigonometry,
you can use this angle here

01:21:46to find the ratios of the opposite
side to the longest side.

01:21:51There's a function
called the sine function

01:21:53which, when you input the angle,
outputs the ratio.

01:21:56So for example in this triangle,
the angle is about 30 degrees,

01:22:00so the output of the sine function
is a ratio of one to two,

01:22:04telling me that this side is half
the length of the longest side.

01:22:11The sine function
enables you to calculate distances

01:22:15when you're not able to make
an accurate measurement.

01:22:20To this day, it's used
in architecture and engineering.

01:22:24The Indians used it
to survey the land around them,

01:22:26navigate the seas and, ultimately,
chart the depths of space itself.

01:22:33It was central to the work
of observatories,

01:22:36like this one in Delhi,

01:22:38where astronomers
would study the stars.

01:22:41The Indian astronomers
could use trigonometry

01:22:43to work out the relative distance
between Earth and the moon

01:22:47and Earth and the sun.

01:22:48You can only make the calculation
when the moon is half full,

01:22:52because that's when it's
directly opposite the sun,

01:22:55so that the sun, moon and Earth
create a right-angled triangle.

01:23:01Now, the Indians could measure

01:23:03that the angle between the sun
and the observatory

01:23:06was one-seventh of a degree.

01:23:09The sine function of
one-seventh of a degree

01:23:13gives me the ratio of 400:1.

01:23:16This means the sun is 400 times
further from Earth than the moon is.

01:23:22So using trigonometry,

01:23:24the Indian mathematicians
could explore the solar system

01:23:27without ever having
to leave the surface of the Earth.

01:23:37The ancient Greeks had been the
first to explore the sine function,

01:23:41listing precise values
for some angles,

01:23:45but they couldn't calculate
the sines of every angle.

01:23:49The Indians were to go much further,
setting themselves a mammoth task.

01:23:54The search was on to find a way

01:23:56to calculate the sine function
of any angle you might be given.

01:24:16The breakthrough in the search for
the sine function of every angle

01:24:20would be made here in
Kerala in south India.

01:24:23In the 15th century,
this part of the country

01:24:26became home to one of the most
brilliant schools of mathematicians

01:24:30to have ever worked.

01:24:33Their leader was called Madhava,
and he was to make

01:24:37some extraordinary
mathematical discoveries.

01:24:44The key to Madhava's success
was the concept of the infinite.

01:24:48Madhava discovered that you could
add up infinitely many things

01:24:51with dramatic effects.

01:24:53Previous cultures had been nervous
of these infinite sums,

01:24:56but Madhava
was happy to play with them.

01:24:59For example,
here's how one can be made up

01:25:01by adding
infinitely many fractions.

01:25:05I'm heading from zero
to one on my boat,

01:25:10but I can split my journey up
into infinitely many fractions.

01:25:14So I can get to a half,

01:25:17then I can sail on a quarter,

01:25:20then an eighth, then a sixteenth,
and so on.

01:25:23The smaller the fractions I move,
the nearer to one I get,

01:25:28but I'll only get there once I've
added up infinitely many fractions.

01:25:34Physically and philosophically,

01:25:37it seems rather a challenge
to add up infinitely many things,

01:25:40but the power of mathematics is
to make sense of the impossible.

01:25:44By producing a language

01:25:46to articulate and manipulate
the infinite,

01:25:48you can prove
that after infinitely many steps

01:25:51you'll reach your destination.

01:25:56Such infinite sums are called
infinite series, and Madhava

01:26:00was doing a lot of research
into the connections

01:26:03between these series
and trigonometry.

01:26:07First, he realised that
he could use the same principle

01:26:11of adding up infinitely many
fractions to capture

01:26:13one of the most important
numbers in mathematics - pi.

01:26:19Pi is the ratio of the circle's
circumference to its diameter.

01:26:24It's a number that appears
in all sorts of mathematics,

01:26:28but is especially useful
for engineers,

01:26:31because any measurements
involving curves soon require pi.

01:26:37So for centuries, mathematicians
searched for a precise value for pi.

01:26:47It was in 6th-century India
that the mathematician Aryabhata

01:26:51gave a very accurate approximation
for pi - namely 3.1416.

01:26:56He went on to use this

01:26:57to make a measurement
of the circumference of the Earth,

01:27:00and he got it as 24,835 miles,

01:27:04which amazingly is only 70 miles
away from its true value.

01:27:08But it was in Kerala
in the 15th century

01:27:11that Madhava realised
he could use infinity

01:27:14to get an exact formula for pi.

01:27:19By successively adding
and subtracting different fractions,

01:27:23Madhava could hone in
on an exact formula for pi.

01:27:28First, he moved four steps
up the number line.

01:27:33That took him way past pi.

01:27:36So next he took
four-thirds of a step,

01:27:40or one-and-one-third
steps, back.

01:27:43Now he'd come too far
the other way.

01:27:46So he headed forward
four-fifths of a step.

01:27:50Each time, he alternated between
four divided by the next odd number.

01:28:01He zigzagged up and down
the number line,

01:28:05getting closer and closer to pi.

01:28:07He discovered that if you went
through all the odd numbers,

01:28:10infinitely many of them,
you would hit pi exactly.

01:28:18I was taught at university
that this formula for pi

01:28:21was discovered by the 17th-century
German mathematician Leibniz,

01:28:25but amazingly, it was actually
discovered here in Kerala

01:28:28two centuries earlier by Madhava.

01:28:30He went on to use
the same sort of mathematics

01:28:33to get infinite-series expressions

01:28:35for the sine formula
in trigonometry.

01:28:37And the wonderful thing is that
you can use these formulas now

01:28:41to calculate the sine of any angle
to any degree of accuracy.

01:28:55It seems incredible that
the Indians made these discoveries

01:28:59centuries before
Western mathematicians.

01:29:05And it says a lot about our attitude
in the West to non-Western cultures

01:29:09that we nearly always
claim their discoveries as our own.

01:29:13What is clear is the West has
been very slow to give due credit

01:29:17to the major breakthroughs
made in non-Western mathematics.

01:29:21Madhava wasn't the only
mathematician to suffer this way.

01:29:24As the West came into contact
more and more with the East

01:29:27during the 18th and 19th centuries,

01:29:29there was a widespread dismissal
and denigration

01:29:32of the cultures
they were colonising.

01:29:34The natives, it was assumed,
couldn't have anything

01:29:36of intellectual worth
to offer the West.

01:29:39It's only now, at the beginning
of the 21st century,

01:29:42that history is being rewritten.

01:29:43But Eastern mathematics was to have
a major impact in Europe,

01:29:48thanks to the development
of one of the major powers

01:29:51of the medieval world.

01:30:16In the 7th century,
a new empire began to spread

01:30:19across the Middle East.

01:30:21The teachings
of the Prophet Mohammed

01:30:24inspired a vast
and powerful Islamic empire

01:30:27which soon stretched
from India in the east

01:30:29to here in Morocco
in the west.

01:30:40And at the heart of this empire
lay a vibrant intellectual culture.

01:30:50A great library and centre of
learning was established in Baghdad.

01:30:55Called the House of Wisdom,
its teaching spread

01:30:58throughout the Islamic empire,

01:31:00reaching schools
like this one here in Fez.

01:31:04Subjects studied included astronomy,
medicine,

01:31:07chemistry, zoology

01:31:12The Muslim scholars collected
and translated many ancient texts,

01:31:17effectively saving
them for posterity.

01:31:19In fact, without their intervention,
we may never have known

01:31:22about the ancient cultures of
Egypt, Babylon, Greece and India.

01:31:26But the scholars at the
House of Wisdom weren't content

01:31:29simply with translating
other people's mathematics.

01:31:32They wanted to create
a mathematics of their own,

01:31:35to push the subject forward.

01:31:41Such intellectual curiosity
was actively encouraged

01:31:45in the early centuries
of the Islamic empire.

01:31:50The Koran asserted
the importance of knowledge.

01:31:53Learning was nothing less
than a requirement of God.

01:32:00In fact, the needs of Islam
demanded mathematical skill.

01:32:04The devout needed to calculate
the time of prayer

01:32:06and the direction of Mecca
to pray towards,

01:32:09and the prohibition
of depicting the human form

01:32:12meant that they had to use

01:32:14much more geometric patterns
to cover their buildings.

01:32:17The Muslim artists discovered all
the different sorts of symmetry

01:32:21that you can depict
on a two-dimensional wall.

01:32:32The director of the House of Wisdom
in Baghdad

01:32:35was a Persian scholar
called Muhammad Al-Khwarizmi.

01:32:42Al-Khwarizmi was an exceptional
mathematician who was responsible

01:32:47for introducing two key
mathematical concepts to the West.

01:32:51Al-Khwarizmi recognised
the incredible potential

01:32:54that the Hindu numerals had

01:32:56to revolutionise
mathematics and science.

01:32:59His work explaining
the power of these numbers

01:33:01to speed up calculations
and do things effectively

01:33:04was so influential that it wasn't
long before they were adopted

01:33:08as the numbers of choice amongst the
mathematicians of the Islamic world.

01:33:12In fact, these numbers
have now become known

01:33:14as the Hindu-Arabic numerals.

01:33:17These numbers -
one to nine and zero -

01:33:20are the ones we use today
all over the world.

01:33:28But Al-Khwarizmi was to create
a whole new mathematical language.

01:33:35It was called algebra

01:33:37and was named after the title of
his book Al-jabr W'al-muqabala,

01:33:41or Calculation By Restoration
Or Reduction.

01:33:49Algebra is the grammar that
underlies the way that numbers work.

01:33:55It's a language
that explains the patterns

01:33:57that lie behind
the behaviour of numbers.

01:34:00It's a bit like a code
for running a computer program.

01:34:04The code will work whatever the
numbers you feed in to the program.

01:34:09For example, mathematicians
might have discovered

01:34:13that if you take a number
and square it,

01:34:15that's always one more
than if you'd taken

01:34:18the numbers either side
and multiplied those together.

01:34:21For example, five times five is 25,

01:34:24which is one more
than four times six - 24.

01:34:28Six times six is always one more
than five times seven and so on.

01:34:32But how can you be sure

01:34:33that this is going to work
whatever numbers you take?

01:34:37To explain the pattern underlying
these calculations,

01:34:39let's use the dyeing holes
in this tannery.

01:34:50If we take a square of
25 holes, running five by five,

01:34:55and take one row of five away
and add it to the bottom,

01:34:59we get six by four
with one left over.

01:35:04But however many holes there
are on the side of the square,

01:35:07we can always move one row of holes
down in a similar way

01:35:11to be left with a rectangle
of holes with one left over.

01:35:17Algebra was a huge breakthrough.

01:35:19Here was a new language

01:35:21to be able to analyse
the way that numbers worked.

01:35:24Previously, the Indians
and the Chinese

01:35:26had considered
very specific problems,

01:35:29but Al-Khwarizmi went
from the specific to the general.

01:35:32He developed systematic ways
to be able to analyse problems

01:35:36so that the solutions would work
whatever the numbers that you took.

01:35:39This language is used
across the mathematical world today.

01:35:45Al-Khwarizmi's great breakthrough
came when he applied algebra

01:35:49to quadratic equations -

01:35:51that is equations including
numbers to the power of two.

01:35:54The ancient Mesopotamians
had devised

01:35:57a cunning method to solve
particular quadratic equations,

01:36:01but it was Al-Khwarizmi's
abstract language of algebra

01:36:05that could finally express
why this method always worked.

01:36:10This was a great conceptual leap

01:36:13and would ultimately lead to a
formula that could be used to solve

01:36:16any quadratic equation,
whatever the numbers involved.

01:36:29The next mathematical Holy Grail

01:36:31was to find a general method that
could solve all cubic equations -

01:36:35equations including numbers
to the power of three.

01:36:56It was an 11th-century
Persian mathematician

01:36:59who took up the challenge of
cracking the problem of the cubic.

01:37:07His name was Omar Khayyam,
and he travelled widely

01:37:10across the Middle East,
calculating as he went.

01:37:16But he was famous for another,
very different, reason.

01:37:20Khayyam was a celebrated poet,

01:37:23author of the great
epic poem the Rubaiyat.

01:37:29It may seem a bit odd that a poet
was also a master mathematician.

01:37:34After all, the combination
doesn't immediately spring to mind.

01:37:37But there's quite a lot of
similarity between the disciplines.

01:37:41Poetry, with its rhyming structure
and rhythmic patterns,

01:37:44resonates strongly with constructing
a logical mathematical proof.

01:37:51Khayyam's major mathematical work

01:37:54was devoted to finding the general
method to solve all cubic equations.

01:37:59Rather than looking
at particular examples,

01:38:02Khayyam carried out a systematic
analysis of the problem,

01:38:07true to the algebraic spirit
of Al-Khwarizmi.

01:38:12Khayyam's analysis revealed
for the first time

01:38:15that there were several
different sorts of cubic equation.

01:38:18But he was still very influenced

01:38:20by the geometric heritage
of the Greeks.

01:38:23He couldn't separate the algebra
from the geometry.

01:38:26In fact, he wouldn't even consider
equations in higher degrees,

01:38:29because they described objects
in more than three dimensions,

01:38:32something he saw as impossible.

01:38:34Although the geometry allowed him

01:38:36to analyse these cubic equations
to some extent,

01:38:39he still couldn't come up
with a purely algebraic solution.

01:38:44It would be another 500 years before
mathematicians could make the leap

01:38:50and find a general solution
to the cubic equation.

01:38:55And that leap would finally be made
in the West - in Italy.

01:39:14During the centuries in which China,
India and the Islamic empire

01:39:17had been in the ascendant,

01:39:19Europe had fallen under
the shadow of the Dark Ages.

01:39:25All intellectual life, including the
study of mathematics, had stagnated.

01:39:34But by the 13th century,
things were beginning to change.

01:39:40Led by Italy, Europe was starting
to explore and trade with the East.

01:39:45With that contact came the spread
of Eastern knowledge to the West.

01:39:50It was the son of a customs official

01:39:52that would become Europe's first
great medieval mathematician.

01:39:55As a child, he travelled around
North Africa with his father,

01:39:59where he learnt about the
developments of Arabic mathematics

01:40:02and especially the benefits
of the Hindu-Arabic numerals.

01:40:05When he got home to Italy
he wrote a book

01:40:07that would be hugely influential

01:40:09in the development
of Western mathematics.

01:40:28That mathematician was
Leonardo of Pisa,

01:40:30better known as Fibonacci,

01:40:33and in his Book Of Calculating,

01:40:36Fibonacci promoted
the new number system,

01:40:39demonstrating how simple it was
compared to the Roman numerals

01:40:43that were in use across Europe.

01:40:45Calculations were far easier,
a fact that had huge consequences

01:40:50for anyone dealing with numbers -

01:40:54pretty much everyone,
from mathematicians to merchants.

01:40:57But there was widespread
suspicion of these new numbers.

01:41:01Old habits die hard, and the
authorities just didn't trust them.

01:41:05Some believed that they would
be more open to fraud -

01:41:08that you could tamper with them.

01:41:09Others believed that they'd be
so easy to use for calculations

01:41:13that it would empower the masses,
taking authority away

01:41:16from the intelligentsia who knew
how to use the old sort of numbers.

01:41:26The city of Florence
even banned them in 1299,

01:41:30but over time,
common sense prevailed,

01:41:33the new system spread
throughout Europe,

01:41:36and the old Roman system
slowly became defunct.

01:41:39At last, the Hindu-Arabic numerals,
zero to nine, had triumphed.

01:41:47Today Fibonacci is best known for
the discovery of some numbers,

01:41:50now called the Fibonacci sequence,
that arose when he was trying

01:41:54to solve a riddle
about the mating habits of rabbits.

01:41:57Suppose a farmer
has a pair of rabbits.

01:41:59Rabbits take two months
to reach maturity,

01:42:02and after that they give birth to
another pair of rabbits each month.

01:42:06So the problem was how to determine

01:42:08how many pairs of rabbits there
will be in any given month.

01:42:13Well, during the first month
you have one pair of rabbits,

01:42:18and since they haven't matured,
they can't reproduce.

01:42:23During the second month,
there is still only one pair.

01:42:27But at the beginning of the
third month, the first pair

01:42:30reproduces for the first time,
so there are two pairs of rabbits.

01:42:35At the beginning
of the fourth month,

01:42:37the first pair reproduces again,

01:42:39but the second pair is not mature
enough, so there are three pairs.

01:42:45In the fifth month,
the first pair reproduces

01:42:48and the second pair
reproduces for the first time,

01:42:52but the third pair is still too
young, so there are five pairs.

01:42:57The mating ritual continues,

01:42:59but what you soon realise is

01:43:01the number of pairs of rabbits
you have in any given month

01:43:04is the sum of the pairs of
rabbits that you have had

01:43:08in each of the two previous months,
so the sequence goes...

01:43:2021...34...55...and so on.

01:43:24The Fibonacci numbers are
nature's favourite numbers.

01:43:28It's not just rabbits that use them.

01:43:30The number of petals on a flower
is invariably a Fibonacci number.

01:43:34They run up and down pineapples
if you count the segments.

01:43:38Even snails use them
to grow their shells.

01:43:41Wherever you find growth in nature,
you find the Fibonacci numbers.

01:43:50But the next major breakthrough
in European mathematics

01:43:53wouldn't happen
until the early 16th century.

01:43:58finding the general method that
would solve all cubic equations,

01:44:03and it would happen here
in the Italian city of Bologna.

01:44:09The University of Bologna
was the crucible

01:44:12of European mathematical thought at
the beginning of the 16th century.

01:44:19Pupils from all over Europe
flocked here and developed

01:44:23a new form of spectator sport -
the mathematical competition.

01:44:30Large audiences would gather to
watch mathematicians

01:44:33challenge each other with numbers, a
kind of intellectual fencing match.

01:44:38But even in this
questioning atmosphere

01:44:41it was believed that some problems
were just unsolvable.

01:44:45It was generally assumed
that finding a general method

01:44:50to solve all cubic equations
was impossible.

01:44:53But one scholar
was to prove everyone wrong.

01:44:59His name was Tartaglia,

01:45:01but he certainly didn't look

01:45:03the heroic architect
of a new mathematics.

01:45:06At the age of 12, he'd been
slashed across the face

01:45:10with a sabre
by a rampaging French army.

01:45:12The result
was a terrible facial scar

01:45:15and a devastating speech
impediment.

01:45:17In fact, Tartaglia was the nickname
he'd been given as a child

01:45:21and means "the stammerer".

01:45:28Shunned by his schoolmates,

01:45:30Tartaglia lost himself
in mathematics, and it wasn't long

01:45:36before he'd found the formula
to solve one type of cubic equation.

01:45:42But Tartaglia soon discovered

01:45:43that he wasn't the only one
to believe he'd cracked the cubic.

01:45:47A young Italian called Fior
was boasting

01:45:50that he too held the secret
formula for solving cubic equations.

01:45:55When news broke
about the discoveries

01:45:58made by the two mathematicians,

01:46:01a competition was arranged to
pit them against each other.

01:46:05The intellectual fencing match
of the century was about to begin.

01:46:16The trouble was that Tartaglia

01:46:18only knew how to solve one sort
of cubic equation,

01:46:21and Fior was ready to challenge him

01:46:23with questions
about a different sort.

01:46:26But just a few days
before the contest,

01:46:28Tartaglia worked out how to
solve this different sort,

01:46:31and with this new weapon in his
arsenal he thrashed his opponent,

01:46:34solving all the questions
in under two hours.

01:46:40Tartaglia went on

01:46:42to find the formula to solve
all types of cubic equations.

01:46:47News soon spread,
and a mathematician in Milan

01:46:49called Cardano became so
desperate to find the solution

01:46:53that he persuaded a reluctant
Tartaglia to reveal the secret,

01:46:58but on one condition -

01:47:00that Cardano keep the secret
and never publish.

01:47:06But Cardano couldn't resist

01:47:08discussing Tartaglia's solution
with his brilliant student, Ferrari.

01:47:12As Ferrari got to grips
with Tartaglia's work,

01:47:15he realised that he could use it
to solve

01:47:18the more complicated quartic
equation, an amazing achievement.

01:47:21Cardano couldn't deny his
student his just rewards,

01:47:24and he broke his vow of secrecy,
publishing Tartaglia's work

01:47:28together with Ferrari's
brilliant solution of the quartic.

01:47:34Poor Tartaglia never recovered
and died penniless,

01:47:38and to this day, the formula
that solves the cubic equation

01:47:41is known as Cardano's formula.

01:47:53Tartaglia may not have won glory
in his lifetime,

01:47:56but his mathematics managed to
solve a problem that had bewildered

01:48:00the great mathematicians
of China, India and the Arab world.

01:48:06It was the first
great mathematical breakthrough

01:48:10to happen in modern Europe.

01:48:15The Europeans now had in their
hands the new language of algebra,

01:48:19the powerful techniques
of the Hindu-Arabic numerals

01:48:23and the beginnings
of the mastery of the infinite.

01:48:26It was time for the Western world

01:48:27to start writing
its own mathematical stories

01:48:30in the language of the East.

01:48:31The mathematical revolution
was about to begin.