The Story of Maths (2008) s01e02 Episode Script

The Genius of the East

From measuring time to understanding our position in the universe, from mapping the Earth to navigating the seas, from man's earliest inventions to today's advanced technologies, mathematics has been the pivot on which human life depends.
The first steps of man's mathematical journey were taken by the ancient cultures of Egypt, Mesopotamia and Greece - cultures which created the basic language of number and calculation.
But when ancient Greece fell into decline, mathematical progress juddered to a halt.
But that was in the West.
In the East, mathematics would reach dynamic new heights.
But in the West, much of this mathematical heritage has been conveniently forgotten or shaded from view.
Due credit has not been given to the great mathematical breakthroughs that ultimately changed the world we live in.
This is the untold story of the mathematics of the East that would transform the West and give birth to the modern world.
The Great Wall of China stretches for thousands of miles.
Nearly 2,000 years in the making, this vast, defensive wall was begun in 220BC to protect China's growing empire.
The Great Wall of China is an amazing feat of engineering built over rough and high countryside.
As soon as they started building, the ancient Chinese realised they had to make calculations about distances, angles of elevation and amounts of material.
So it isn't surprising that this inspired some very clever mathematics to help build Imperial China.
At the heart of ancient Chinese mathematics was an incredibly simple number system which laid the foundations for the way we count in the West today.
When a mathematician wanted to do a sum, he would use small bamboo rods.
These rods were arranged to represent the numbers one to nine.
They were then placed in columns, each column representing units, tens, hundreds, thousands and so on.
So the number 924 was represented by putting the symbol 4 in the units column, the symbol 2 in the tens column and the symbol 9 in the hundreds column.
This is what we call a decimal place-value system, and it's very similar to the one we use today.
We too use numbers from one to nine, and we use their position to indicate whether it's units, tens, hundreds or thousands.
But the power of these rods is that it makes calculations very quick.
In fact, the way the ancient Chinese did their calculations is very similar to the way we learn today in school.
Not only were the ancient Chinese the first to use a decimal place-value system, but they did so over 1,000 years before we adopted it in the West.
But they only used it when calculating with the rods.
When writing the numbers down, the ancient Chinese didn't use the place-value system.
Instead, they used a far more laborious method, in which special symbols stood for tens, hundreds, thousands and so on.
So the number 924 would be written out as nine hundreds, two tens and four.
Not quite so efficient.
The problem was that the ancient Chinese didn't have a concept of zero.
They didn't have a symbol for zero.
It just didn't exist as a number.
Using the counting rods, they would use a blank space where today we would write a zero.
The problem came with trying to write down this number, which is why they had to create these new symbols for tens, hundreds and thousands.
Without a zero, the written number was extremely limited.
But the absence of zero didn't stop the ancient Chinese from making giant mathematical steps.
In fact, there was a widespread fascination with number in ancient China.
According to legend, the first sovereign of China, the Yellow Emperor, had one of his deities create mathematics in 2800BC, believing that number held cosmic significance.
And to this day, the Chinese still believe in the mystical power of numbers.
Odd numbers are seen as male, even numbers, female.
The number four is to be avoided at all costs.
The number eight brings good fortune.
And the ancient Chinese were drawn to patterns in numbers, developing their own rather early version of sudoku.
It was called the magic square.
Legend has it that thousands of years ago, Emperor Yu was visited by a sacred turtle that came out of the depths of the Yellow River.
On its back were numbers arranged into a magic square, a little like this.
In this square, which was regarded as having great religious significance, all the numbers in each line - horizontal, vertical and diagonal - all add up to the same number - 15.
Now, the magic square may be no more than a fun puzzle, but it shows the ancient Chinese fascination with mathematical patterns, and it wasn't too long before they were creating even bigger magic squares with even greater magical and mathematical powers.
But mathematics also played a vital role in the running of the emperor's court.
The calendar and the movement of the planets were of the utmost importance to the emperor, influencing all his decisions, even down to the way his day was planned, so astronomers became prized members of the imperial court, and astronomers were always mathematicians.
Everything in the emperor's life was governed by the calendar, and he ran his affairs with mathematical precision.
The emperor even got his mathematical advisors to come up with a system to help him sleep his way through the vast number of women he had in his harem.
Never one to miss a trick, the mathematical advisors decided to base the harem on a mathematical idea called a geometric progression.
Maths has never had such a fun purpose! Legend has it that in the space of 15 nights, the emperor had to sleep with 121 women .
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the empress, three senior consorts, nine wives, 27 concubines and 81 slaves.
The mathematicians would soon have realised that this was a geometric progression - a series of numbers in which you get from one number to the next by multiplying the same number each time - in this case, three.
Each group of women is three times as large as the previous group, so the mathematicians could quickly draw up a rota to ensure that, in the space of 15 nights, the emperor slept with every woman in the harem.
The first night was reserved for the empress.
The next was for the three senior consorts.
The nine wives came next, and then the 27 concubines were chosen in rotation, nine each night.
And then finally, over a period of nine nights, the 81 slaves were dealt with in groups of nine.
Being the emperor certainly required stamina, a bit like being a mathematician, but the object is clear - to procure the best possible imperial succession.
The rota ensured that the emperor slept with the ladies of highest rank closest to the full moon, when their yin, their female force, would be at its highest and be able to match his yang, or male force.
The emperor's court wasn't alone in its dependence on mathematics.
It was central to the running of the state.
Ancient China was a vast and growing empire with a strict legal code, widespread taxation and a standardised system of weights, measures and money.
The empire needed a highly trained civil service, competent in mathematics.
And to educate these civil servants was a mathematical textbook, probably written in around 200BC - the Nine Chapters.
The book is a compilation of 246 problems in practical areas such as trade, payment of wages and taxes.
And at the heart of these problems lies one of the central themes of mathematics, how to solve equations.
Equations are a little bit like cryptic crosswords.
You're given a certain amount of information about some unknown numbers, and from that information you've got to deduce what the unknown numbers are.
For example, with my weights and scales, I've found out that one plum .
.
together with three peaches weighs a total of 15 grams.
But .
.
two plums together with one peach weighs a total of 10g.
From this information, I can deduce what a single plum weighs and a single peach weighs, and this is how I do it.
If I take the first set of scales, one plum and three peaches weighing 15g, and double it, I get two plums and six peaches weighing 30g.
If I take this and subtract from it the second set of scales - that's two plums and a peach weighing 10g - I'm left with an interesting result - no plums.
Having eliminated the plums, I've discovered that five peaches weighs 20g, so a single peach weighs 4g, and from this I can deduce that the plum weighs 3g.
The ancient Chinese went on to apply similar methods to larger and larger numbers of unknowns, using it to solve increasingly complicated equations.
What's extraordinary is that this particular system of solving equations didn't appear in the West until the beginning of the 19th century.
In 1809, while analysing a rock called Pallas in the asteroid belt, Carl Friedrich Gauss, who would become known as the prince of mathematics, rediscovered this method which had been formulated in ancient China centuries earlier.
Once again, ancient China streets ahead of Europe.
But the Chinese were to go on to solve even more complicated equations involving far larger numbers.
In what's become known as the Chinese remainder theorem, the Chinese came up with a new kind of problem.
In this, we know the number that's left when the equation's unknown number is divided by a given number - say, three, five or seven.
Of course, this is a fairly abstract mathematical problem, but the ancient Chinese still couched it in practical terms.
So a woman in the market has a tray of eggs, but she doesn't know how many eggs she's got.
What she does know is that if she arranges them in threes, she has one egg left over.
If she arranges them in fives, she gets two eggs left over.
But if she arranged them in rows of seven, she found she had three eggs left over.
The ancient Chinese found a systematic way to calculate that the smallest number of eggs she could have had in the tray is 52.
But the more amazing thing is that you can capture such a large number, like 52, by using these small numbers like three, five and seven.
This way of looking at numbers would become a dominant theme over the last two centuries.
By the 6th century AD, the Chinese remainder theorem was being used in ancient Chinese astronomy to measure planetary movement.
But today it still has practical uses.
Internet cryptography encodes numbers using mathematics that has its origins in the Chinese remainder theorem.
By the 13th century, mathematics was long established on the curriculum, with over 30 mathematics schools scattered across the country.
The golden age of Chinese maths had arrived.
And its most important mathematician was called Qin Jiushao.
Legend has it that Qin Jiushao was something of a scoundrel.
He was a fantastically corrupt imperial administrator who crisscrossed China, lurching from one post to another.
Repeatedly sacked for embezzling government money, he poisoned anyone who got in his way.
Qin Jiushao was reputedly described as as violent as a tiger or a wolf and as poisonous as a scorpion or a viper so, not surprisingly, he made a fierce warrior.
For ten years, he fought against the invading Mongols, but for much of that time he was complaining that his military life took him away from his true passion.
No, not corruption, but mathematics.
Qin started trying to solve equations that grew out of trying to measure the world around us.
Quadratic equations involve numbers that are squared, or to the power of two - say, five times five.
The ancient Mesopotamians had already realised that these equations were perfect for measuring flat, two-dimensional shapes, like Tiananmen Square.
But Qin was interested in more complicated equations - cubic equations.
These involve numbers which are cubed, or to the power of three - say, five times five times five, and they were perfect for capturing three-dimensional shapes, like Chairman Mao's mausoleum.
Qin found a way of solving cubic equations, and this is how it worked.
Say Qin wants to know the exact dimensions of Chairman Mao's mausoleum.
He knows the volume of the building and the relationships between the dimensions.
In order to get his answer, Qin uses what he knows to produce a cubic equation.
He then makes an educated guess at the dimensions.
Although he's captured a good proportion of the mausoleum, there are still bits left over.
Qin takes these bits and creates a new cubic equation.
He can now refine his first guess by trying to find a solution to this new cubic equation, and so on.
Each time he does this, the pieces he's left with get smaller and smaller and his guesses get better and better.
What's striking is that Qin's method for solving equations wasn't discovered in the West until the 17th century, when Isaac Newton came up with a very similar approximation method.
The power of this technique is that it can be applied to even more complicated equations.
Qin even used his techniques to solve an equation involving numbers up to the power of ten.
This was extraordinary stuff - highly complex mathematics.
Qin may have been years ahead of his time, but there was a problem with his technique.
It only gave him an approximate solution.
That might be good enough for an engineer - not for a mathematician.
Mathematics is an exact science.
We like things to be precise, and Qin just couldn't come up with a formula to give him an exact solution to these complicated equations.
China had made great mathematical leaps, but the next great mathematical breakthroughs were to happen in a country lying to the southwest of China - a country that had a rich mathematical tradition that would change the face of maths for ever.
India's first great mathematical gift lay in the world of number.
Like the Chinese, the Indians had discovered the mathematical benefits of the decimal place-value system and were using it by the middle of the 3rd century AD.
It's been suggested that the Indians learned the system from Chinese merchants travelling in India with their counting rods, or they may well just have stumbled across it themselves.
It's all such a long time ago that it's shrouded in mystery.
We may never know how the Indians came up with their number system, but we do know that they refined and perfected it, creating the ancestors for the nine numerals used across the world now.
Many rank the Indian system of counting as one of the greatest intellectual innovations of all time, developing into the closest thing we could call a universal language.
But there was one number missing, and it was the Indians who would introduce it to the world.
The earliest known recording of this number dates from the 9th century, though it was probably in practical use for centuries before.
This strange new numeral is engraved on the wall of small temple in the fort of Gwalior in central India.
So here we are in one of the holy sites of the mathematical world, and what I'm looking for is in this inscription on the wall.
Up here are some numbers, and here's the new number.
It's zero.
It's astonishing to think that before the Indians invented it, there was no number zero.
To the ancient Greeks, it simply hadn't existed.
To the Egyptians, the Mesopotamians and, as we've seen, the Chinese, zero had been in use but as a placeholder, an empty space to show a zero inside a number.
The Indians transformed zero from a mere placeholder into a number that made sense in its own right - a number for calculation, for investigation.
This brilliant conceptual leap would revolutionise mathematics.
Now, with just ten digits - zero to nine - it was suddenly possible to capture astronomically large numbers in an incredibly efficient way.
But why did the Indians make this imaginative leap? Well, we'll never know for sure, but it's possible that the idea and symbol that the Indians use for zero came from calculations they did with stones in the sand.
When stones were removed from the calculation, a small, round hole was left in its place, representing the movement from something to nothing.
But perhaps there is also a cultural reason for the invention of zero.
HORNS BLOW AND DRUMS BANG METALLIC BEATING For the ancient Indians, the concepts of nothingness and eternity lay at the very heart of their belief system.
BELL CLANGS AND SILENCE FALLS In the religions of India, the universe was born from nothingness, and nothingness is the ultimate goal of humanity.
So it's perhaps not surprising that a culture that so enthusiastically embraced the void should be happy with the notion of zero.
The Indians even used the word for the philosophical idea of the void, shunya, to represent the new mathematical term "zero".
In the 7th century, the brilliant Indian mathematician Brahmagupta proved some of the essential properties of zero.
Brahmagupta's rules about calculating with zero are taught in schools all over the world to this day.
One plus zero equals one.
One minus zero equals one.
One times zero is equal to zero.
But Brahmagupta came a cropper when he tried to do one divided by zero.
After all, what number times zero equals one? It would require a new mathematical concept, that of infinity, to make sense of dividing by zero, and the breakthrough was made by a 12th-century Indian mathematician called Bhaskara II, and it works like this.
If I take a fruit and I divide it into halves, I get two pieces, so one divided by a half is two.
If I divide it into thirds, I get three pieces.
So when I divide it into smaller and smaller fractions, I get more and more pieces, so ultimately, when I divide by a piece which is of zero size, I'll have infinitely many pieces.
So for Bhaskara, one divided by zero is infinity.
But the Indians would go further in their calculations with zero.
For example, if you take three from three and get zero, what happens when you take four from three? It looks like you have nothing, but the Indians recognised that this was a new sort of nothing - negative numbers.
The Indians called them "debts", because they solved equations like, "If I have three batches of material and take four away, "how many have I left?" This may seem odd and impractical, but that was the beauty of Indian mathematics.
Their ability to come up with negative numbers and zero was because they thought of numbers as abstract entities.
They weren't just for counting and measuring pieces of cloth.
They had a life of their own, floating free of the real world.
This led to an explosion of mathematical ideas.
The Indians' abstract approach to mathematics soon revealed a new side to the problem of how to solve quadratic equations.
That is equations including numbers to the power of two.
Brahmagupta's understanding of negative numbers allowed him to see that quadratic equations always have two solutions, one of which could be negative.
Brahmagupta went even further, solving quadratic equations with two unknowns, a question which wouldn't be considered in the West until 1657, when French mathematician Fermat challenged his colleagues with the same problem.
Little did he know that they'd been beaten to a solution by Brahmagupta 1,000 years earlier.
Brahmagupta was beginning to find abstract ways of solving equations, but astonishingly, he was also developing a new mathematical language to express that abstraction.
Brahmagupta was experimenting with ways of writing his equations down, using the initials of the names of different colours to represent unknowns in his equations.
A new mathematical language was coming to life, which would ultimately lead to the x's and y's which fill today's mathematical journals.
But it wasn't just new notation that was being developed.
Indian mathematicians were responsible for making fundamental new discoveries in the theory of trigonometry.
The power of trigonometry is that it acts like a dictionary, translating geometry into numbers and back.
Although first developed by the ancient Greeks, it was in the hands of the Indian mathematicians that the subject truly flourished.
At its heart lies the study of right-angled triangles.
In trigonometry, you can use this angle here to find the ratios of the opposite side to the longest side.
There's a function called the sine function which, when you input the angle, outputs the ratio.
So for example in this triangle, the angle is about 30 degrees, so the output of the sine function is a ratio of one to two, telling me that this side is half the length of the longest side.
The sine function enables you to calculate distances when you're not able to make an accurate measurement.
To this day, it's used in architecture and engineering.
The Indians used it to survey the land around them, navigate the seas and, ultimately, chart the depths of space itself.
It was central to the work of observatories, like this one in Delhi, where astronomers would study the stars.
The Indian astronomers could use trigonometry to work out the relative distance between Earth and the moon and Earth and the sun.
You can only make the calculation when the moon is half full, because that's when it's directly opposite the sun, so that the sun, moon and Earth create a right-angled triangle.
Now, the Indians could measure that the angle between the sun and the observatory was one-seventh of a degree.
The sine function of one-seventh of a degree gives me the ratio of 400:1.
This means the sun is 400 times further from Earth than the moon is.
So using trigonometry, the Indian mathematicians could explore the solar system without ever having to leave the surface of the Earth.
The ancient Greeks had been the first to explore the sine function, listing precise values for some angles, but they couldn't calculate the sines of every angle.
The Indians were to go much further, setting themselves a mammoth task.
The search was on to find a way to calculate the sine function of any angle you might be given.
The breakthrough in the search for the sine function of every angle would be made here in Kerala in south India.
In the 15th century, this part of the country became home to one of the most brilliant schools of mathematicians to have ever worked.
Their leader was called Madhava, and he was to make some extraordinary mathematical discoveries.
The key to Madhava's success was the concept of the infinite.
Madhava discovered that you could add up infinitely many things with dramatic effects.
Previous cultures had been nervous of these infinite sums, but Madhava was happy to play with them.
For example, here's how one can be made up by adding infinitely many fractions.
I'm heading from zero to one on my boat, but I can split my journey up into infinitely many fractions.
So I can get to a half, then I can sail on a quarter, then an eighth, then a sixteenth, and so on.
The smaller the fractions I move, the nearer to one I get, but I'll only get there once I've added up infinitely many fractions.
Physically and philosophically, it seems rather a challenge to add up infinitely many things, but the power of mathematics is to make sense of the impossible.
By producing a language to articulate and manipulate the infinite, you can prove that after infinitely many steps you'll reach your destination.
Such infinite sums are called infinite series, and Madhava was doing a lot of research into the connections between these series and trigonometry.
First, he realised that he could use the same principle of adding up infinitely many fractions to capture one of the most important numbers in mathematics - pi.
Pi is the ratio of the circle's circumference to its diameter.
It's a number that appears in all sorts of mathematics, but is especially useful for engineers, because any measurements involving curves soon require pi.
So for centuries, mathematicians searched for a precise value for pi.
It was in 6th-century India that the mathematician Aryabhata gave a very accurate approximation for pi - namely 3.
1416.
He went on to use this to make a measurement of the circumference of the Earth, and he got it as 24,835 miles, which amazingly is only 70 miles away from its true value.
But it was in Kerala in the 15th century that Madhava realised he could use infinity to get an exact formula for pi.
By successively adding and subtracting different fractions, Madhava could hone in on an exact formula for pi.
First, he moved four steps up the number line.
That took him way past pi.
So next he took four-thirds of a step, or one-and-one-third steps, back.
Now he'd come too far the other way.
So he headed forward four-fifths of a step.
Each time, he alternated between four divided by the next odd number.
He zigzagged up and down the number line, getting closer and closer to pi.
He discovered that if you went through all the odd numbers, infinitely many of them, you would hit pi exactly.
I was taught at university that this formula for pi was discovered by the 17th-century German mathematician Leibniz, but amazingly, it was actually discovered here in Kerala two centuries earlier by Madhava.
He went on to use the same sort of mathematics to get infinite-series expressions for the sine formula in trigonometry.
And the wonderful thing is that you can use these formulas now to calculate the sine of any angle to any degree of accuracy.
It seems incredible that the Indians made these discoveries centuries before Western mathematicians.
And it says a lot about our attitude in the West to non-Western cultures that we nearly always claim their discoveries as our own.
What is clear is the West has been very slow to give due credit to the major breakthroughs made in non-Western mathematics.
Madhava wasn't the only mathematician to suffer this way.
As the West came into contact more and more with the East during the 18th and 19th centuries, there was a widespread dismissal and denigration of the cultures they were colonising.
The natives, it was assumed, couldn't have anything of intellectual worth to offer the West.
It's only now, at the beginning of the 21st century, that history is being rewritten.
But Eastern mathematics was to have a major impact in Europe, thanks to the development of one of the major powers of the medieval world.
In the 7th century, a new empire began to spread across the Middle East.
The teachings of the Prophet Mohammed inspired a vast and powerful Islamic empire which soon stretched from India in the east to here in Morocco in the west.
And at the heart of this empire lay a vibrant intellectual culture.
A great library and centre of learning was established in Baghdad.
Called the House of Wisdom, its teaching spread throughout the Islamic empire, reaching schools like this one here in Fez.
Subjects studied included astronomy, medicine, chemistry, zoology and mathematics.
The Muslim scholars collected and translated many ancient texts, effectively saving them for posterity.
In fact, without their intervention, we may never have known about the ancient cultures of Egypt, Babylon, Greece and India.
But the scholars at the House of Wisdom weren't content simply with translating other people's mathematics.
They wanted to create a mathematics of their own, to push the subject forward.
Such intellectual curiosity was actively encouraged in the early centuries of the Islamic empire.
The Koran asserted the importance of knowledge.
Learning was nothing less than a requirement of God.
In fact, the needs of Islam demanded mathematical skill.
The devout needed to calculate the time of prayer and the direction of Mecca to pray towards, and the prohibition of depicting the human form meant that they had to use much more geometric patterns to cover their buildings.
The Muslim artists discovered all the different sorts of symmetry that you can depict on a two-dimensional wall.
The director of the House of Wisdom in Baghdad was a Persian scholar called Muhammad Al-Khwarizmi.
Al-Khwarizmi was an exceptional mathematician who was responsible for introducing two key mathematical concepts to the West.
Al-Khwarizmi recognised the incredible potential that the Hindu numerals had to revolutionise mathematics and science.
His work explaining the power of these numbers to speed up calculations and do things effectively was so influential that it wasn't long before they were adopted as the numbers of choice amongst the mathematicians of the Islamic world.
In fact, these numbers have now become known as the Hindu-Arabic numerals.
These numbers - one to nine and zero - are the ones we use today all over the world.
But Al-Khwarizmi was to create a whole new mathematical language.
It was called algebra and was named after the title of his book Al-jabr W'al-muqabala, or Calculation By Restoration Or Reduction.
Algebra is the grammar that underlies the way that numbers work.
It's a language that explains the patterns that lie behind the behaviour of numbers.
It's a bit like a code for running a computer program.
The code will work whatever the numbers you feed in to the program.
For example, mathematicians might have discovered that if you take a number and square it, that's always one more than if you'd taken the numbers either side and multiplied those together.
For example, five times five is 25, which is one more than four times six - 24.
Six times six is always one more than five times seven and so on.
But how can you be sure that this is going to work whatever numbers you take? To explain the pattern underlying these calculations, let's use the dyeing holes in this tannery.
If we take a square of 25 holes, running five by five, and take one row of five away and add it to the bottom, we get six by four with one left over.
But however many holes there are on the side of the square, we can always move one row of holes down in a similar way to be left with a rectangle of holes with one left over.
Algebra was a huge breakthrough.
Here was a new language to be able to analyse the way that numbers worked.
Previously, the Indians and the Chinese had considered very specific problems, but Al-Khwarizmi went from the specific to the general.
He developed systematic ways to be able to analyse problems so that the solutions would work whatever the numbers that you took.
This language is used across the mathematical world today.
Al-Khwarizmi's great breakthrough came when he applied algebra to quadratic equations - that is equations including numbers to the power of two.
The ancient Mesopotamians had devised a cunning method to solve particular quadratic equations, but it was Al-Khwarizmi's abstract language of algebra that could finally express why this method always worked.
This was a great conceptual leap and would ultimately lead to a formula that could be used to solve any quadratic equation, whatever the numbers involved.
The next mathematical Holy Grail was to find a general method that could solve all cubic equations - equations including numbers to the power of three.
It was an 11th-century Persian mathematician who took up the challenge of cracking the problem of the cubic.
His name was Omar Khayyam, and he travelled widely across the Middle East, calculating as he went.
But he was famous for another, very different, reason.
Khayyam was a celebrated poet, author of the great epic poem the Rubaiyat.
It may seem a bit odd that a poet was also a master mathematician.
After all, the combination doesn't immediately spring to mind.
But there's quite a lot of similarity between the disciplines.
Poetry, with its rhyming structure and rhythmic patterns, resonates strongly with constructing a logical mathematical proof.
Khayyam's major mathematical work was devoted to finding the general method to solve all cubic equations.
Rather than looking at particular examples, Khayyam carried out a systematic analysis of the problem, true to the algebraic spirit of Al-Khwarizmi.
Khayyam's analysis revealed for the first time that there were several different sorts of cubic equation.
But he was still very influenced by the geometric heritage of the Greeks.
He couldn't separate the algebra from the geometry.
In fact, he wouldn't even consider equations in higher degrees, because they described objects in more than three dimensions, something he saw as impossible.
Although the geometry allowed him to analyse these cubic equations to some extent, he still couldn't come up with a purely algebraic solution.
It would be another 500 years before mathematicians could make the leap and find a general solution to the cubic equation.
And that leap would finally be made in the West - in Italy.
During the centuries in which China, India and the Islamic empire had been in the ascendant, Europe had fallen under the shadow of the Dark Ages.
All intellectual life, including the study of mathematics, had stagnated.
But by the 13th century, things were beginning to change.
Led by Italy, Europe was starting to explore and trade with the East.
With that contact came the spread of Eastern knowledge to the West.
It was the son of a customs official that would become Europe's first great medieval mathematician.
As a child, he travelled around North Africa with his father, where he learnt about the developments of Arabic mathematics and especially the benefits of the Hindu-Arabic numerals.
When he got home to Italy he wrote a book that would be hugely influential in the development of Western mathematics.
That mathematician was Leonardo of Pisa, better known as Fibonacci, and in his Book Of Calculating, Fibonacci promoted the new number system, demonstrating how simple it was compared to the Roman numerals that were in use across Europe.
Calculations were far easier, a fact that had huge consequences for anyone dealing with numbers - pretty much everyone, from mathematicians to merchants.
But there was widespread suspicion of these new numbers.
Old habits die hard, and the authorities just didn't trust them.
Some believed that they would be more open to fraud - that you could tamper with them.
Others believed that they'd be so easy to use for calculations that it would empower the masses, taking authority away from the intelligentsia who knew how to use the old sort of numbers.
The city of Florence even banned them in 1299, but over time, common sense prevailed, the new system spread throughout Europe, and the old Roman system slowly became defunct.
At last, the Hindu-Arabic numerals, zero to nine, had triumphed.
Today Fibonacci is best known for the discovery of some numbers, now called the Fibonacci sequence, that arose when he was trying to solve a riddle about the mating habits of rabbits.
Suppose a farmer has a pair of rabbits.
Rabbits take two months to reach maturity, and after that they give birth to another pair of rabbits each month.
So the problem was how to determine how many pairs of rabbits there will be in any given month.
Well, during the first month you have one pair of rabbits, and since they haven't matured, they can't reproduce.
During the second month, there is still only one pair.
But at the beginning of the third month, the first pair reproduces for the first time, so there are two pairs of rabbits.
At the beginning of the fourth month, the first pair reproduces again, but the second pair is not mature enough, so there are three pairs.
In the fifth month, the first pair reproduces and the second pair reproduces for the first time, but the third pair is still too young, so there are five pairs.
The mating ritual continues, but what you soon realise is the number of pairs of rabbits you have in any given month is the sum of the pairs of rabbits that you have had in each of the two previous months, so the sequence goes 1123 5813 213455and so on.
The Fibonacci numbers are nature's favourite numbers.
It's not just rabbits that use them.
The number of petals on a flower is invariably a Fibonacci number.
They run up and down pineapples if you count the segments.
Even snails use them to grow their shells.
Wherever you find growth in nature, you find the Fibonacci numbers.
But the next major breakthrough in European mathematics wouldn't happen until the early 16th century.
It would involve finding the general method that would solve all cubic equations, and it would happen here in the Italian city of Bologna.
The University of Bologna was the crucible of European mathematical thought at the beginning of the 16th century.
Pupils from all over Europe flocked here and developed a new form of spectator sport - the mathematical competition.
Large audiences would gather to watch mathematicians challenge each other with numbers, a kind of intellectual fencing match.
But even in this questioning atmosphere it was believed that some problems were just unsolvable.
It was generally assumed that finding a general method to solve all cubic equations was impossible.
But one scholar was to prove everyone wrong.
His name was Tartaglia, but he certainly didn't look the heroic architect of a new mathematics.
At the age of 12, he'd been slashed across the face with a sabre by a rampaging French army.
The result was a terrible facial scar and a devastating speech impediment.
In fact, Tartaglia was the nickname he'd been given as a child and means "the stammerer".
Shunned by his schoolmates, Tartaglia lost himself in mathematics, and it wasn't long before he'd found the formula to solve one type of cubic equation.
But Tartaglia soon discovered that he wasn't the only one to believe he'd cracked the cubic.
A young Italian called Fior was boasting that he too held the secret formula for solving cubic equations.
When news broke about the discoveries made by the two mathematicians, a competition was arranged to pit them against each other.
The intellectual fencing match of the century was about to begin.
The trouble was that Tartaglia only knew how to solve one sort of cubic equation, and Fior was ready to challenge him with questions about a different sort.
But just a few days before the contest, Tartaglia worked out how to solve this different sort, and with this new weapon in his arsenal he thrashed his opponent, solving all the questions in under two hours.
Tartaglia went on to find the formula to solve all types of cubic equations.
News soon spread, and a mathematician in Milan called Cardano became so desperate to find the solution that he persuaded a reluctant Tartaglia to reveal the secret, but on one condition - that Cardano keep the secret and never publish.
But Cardano couldn't resist discussing Tartaglia's solution with his brilliant student, Ferrari.
As Ferrari got to grips with Tartaglia's work, he realised that he could use it to solve the more complicated quartic equation, an amazing achievement.
Cardano couldn't deny his student his just rewards, and he broke his vow of secrecy, publishing Tartaglia's work together with Ferrari's brilliant solution of the quartic.
Poor Tartaglia never recovered and died penniless, and to this day, the formula that solves the cubic equation is known as Cardano's formula.
Tartaglia may not have won glory in his lifetime, but his mathematics managed to solve a problem that had bewildered the great mathematicians of China, India and the Arab world.
It was the first great mathematical breakthrough to happen in modern Europe.
The Europeans now had in their hands the new language of algebra, the powerful techniques of the Hindu-Arabic numerals and the beginnings of the mastery of the infinite.
It was time for the Western world to start writing its own mathematical stories in the language of the East.
The mathematical revolution was about to begin.
You can learn more about The Story Of Maths with the Open University at open2.
net.