BBC. История математики. Язык Вселенной
Throughout history, humankind has struggled to understand the fundamental workings of the material world.
We've endeavoured to discover the rules and patterns that determine the qualities of the objects that surround us, and their complex relationship to us and each other.
Over thousands of years, societies all over the world have found that one discipline above all others yields certain knowledge about the underlying realities of the physical world.
That discipline is mathematics.
I'm Marcus Du Sautoy, and I'm a mathematician.
I see myself as a pattern searcher, hunting down the hidden structures that lie behind the apparent chaos and complexity of the world around us.
In my search for pattern and order, I draw upon the work of the great mathematicians who've gone before me, people belonging to cultures across the globe, whose innovations created the language the universe is written in.
I want to take you on a journey through time and space, and track the growth of mathematics from its awakening to the sophisticated subject we know today.
Using computer generated imagery, we will explore the trailblazing discoveries that allowed the earliest civilisations to understand the world mathematical.
This is the story of maths.
Our world is made up of patterns and sequences.
They're all around us.
Day becomes night.
Animals travel across the earth in ever-changing formations.
Landscapes are constantly altering.
One of the reasons mathematics began was because we needed to find a way of making sense of these natural patterns.
The most basic concepts of maths – space and quantity – are hard-wired into our brains.
Even animals have a sense of distance and number, assessing when their pack is outnumbered, and whether to fight or fly, calculating whether their prey is within striking distance.
Understanding maths is the difference between life and death.
But it was man who took these basic concepts and started to build upon these foundations.
At some point, humans started to spot patterns, to make connections, to count and to order the world around them.
With this, a whole new mathematical universe began to emerge.
This is the River Nile.
It's been the lifeline of Egypt for millennia.
I've come here because it's where some of the first signs of mathematics as we know it today emerged.
People abandoned nomadic life and began settling here as early as 6000BC.
The conditions were perfect for farming.
The most important event for Egyptian agriculture each year was the flooding of the Nile.
So this was used as a marker to start each new year.
Egyptians did record what was going on over periods of time, so in order to establish a calendar like this, you need to count how many days, for example, happened in-between lunar phases, or how many days happened in-between two floodings of the Nile.
Recording the patterns for the seasons was essential, not only to their management of the land, but also their religion.
The ancient Egyptians who settled on the Nile banks believed it was the river god, Hapy, who flooded the river each year.
And in return for the life-giving water, the citizens offered a portion of the yield as a thanksgiving.
As settlements grew larger, it became necessary to find ways to administer them.
Areas of land needed to be calculated, crop yields predicted, taxes charged and collated.
In short, people needed to count and measure.
The Egyptians used their bodies to measure the world, and it's how their units of measurements evolved.
A palm was the width of a hand, a cubit an arm length from elbow to fingertips.
Land cubits, strips of land measuring a cubit by 100, were used by the pharaoh's surveyors to calculate areas.
There's a very strong link between bureaucracy and the development of mathematics in ancient Egypt.
And we can see this link right from the beginning, from the invention of the number system, throughout Egyptian history, really.
For the Old Kingdom, the only evidence we have are metrological systems, that is measurements for areas, for length.
This points to a bureaucratic need to develop such things.
It was vital to know the area of a farmer's land so he could be taxed accordingly.
Or if the Nile robbed him of part of his land, so he could request a rebate.
It meant that the pharaoh's surveyors were often calculating the area of irregular parcels of land.
It was the need to solve such practical problems that made them the earliest mathematical innovators.
The Egyptians needed some way to record the results of their calculations.
Amongst all the hieroglyphs that cover the tourist souvenirs scattered around Cairo, I was on the hunt for those that recorded some of the first numbers in history.
They were difficult to track down.
But I did find them in the end.
The Egyptians were using a decimal system, motivated by the 10 fingers on our hands.
The sign for one was a stroke, 10, a heel bone, 100, a coil of rope, and 1, 000, a Lotus plant.
How much is this T-shirt? Er, 25.
25!Yes!So that would be 2 knee bones and 5 strokes.
So you're not gonna charge me anything up here?Here, one million! One million?My God! This one million.
One million, yeah, that's pretty big! The hieroglyphs are beautiful, but the Egyptian number system was fundamentally flawed.
They had no concept of a place value, so one stroke could only represent one unit, not 100 or 1, 000.
Although you can write a million with just one character, rather than the seven that we use, if you want to write a million minus one, then the poor old Egyptian scribe has got to write nine strokes, nine heel bones, nine coils of rope, and so on, a total of 54 characters.
Despite the drawback of this number system, the Egyptians were brilliant problem solvers.
We know this because of the few records that have survived.
The Egyptian scribes used sheets of papyrus to record their mathematical discoveries.
This delicate material made from reeds decayed over time and many secrets perished with it.
But there's one revealing document that has survived.
The Rhind Mathematical Papyrus is the most important document we have today for Egyptian mathematics.
We get a good overview of what types of problems the Egyptians would have dealt with in their mathematics.
We also get explicitly stated how multiplications and divisions were carried out.
The papyri show how to multiply two large numbers together.
But to illustrate the method, let's take two smaller numbers.
Let's do three times six.
The scribe would take the first number, three, and put it in one column.
In the second column, he would place the number one.
Then he would double the numbers in each column, so three becomes six.
.
.
.
.
and six would become 12.
And then in the second column, one would become two, and two becomes four.
Now, here's the really clever bit.
The scribe wants to multiply three by six.
So he takes the powers of two in the second column, which add up to six.
That's two plus four.
Then he moves back to the first column, and just takes those rows corresponding to the two and the four.
So that's six and the 12.
He adds those together to get the answer of 18.
But for me, the most striking thing about this method is that the scribe has effectively written that second number in binary.
Six is one lot of four, one lot of two, and no units.
Which is 1-1-0.
The Egyptians have understood the power of binary over 3, 000 years before the mathematician and philosopher Leibniz would reveal their potential.
Today, the whole technological world depends on the same principles that were used in ancient Egypt.
The Rhind Papyrus was recorded by a scribe called Ahmes around 1650BC.
Its problems are concerned with finding solutions to everyday situations.
Several of the problems mention bread and beer, which isn't surprising as Egyptian workers were paid in food and drink.
One is concerned with how to divide nine loaves equally between 10 people, without a fight breaking out.
I've got nine loaves of bread here.
I'm gonna take five of them and cut them into halves.
Of course, nine people could shave a 10th off their loaf and give the pile of crumbs to the 10th person.
But the Egyptians developed a far more elegant solution – take the next four and divide those into thirds.
But two of the thirds I am now going to cut into fifths, so each piece will be one fifteenth.
Each person then gets one half and one third and one fifteenth.
It is through such seemingly practical problems that we start to see a more abstract mathematics developing.
Suddenly, new numbers are on the scene – fractions – and it isn't too long before the Egyptians are exploring the mathematics of these numbers.
Fractions are clearly of practical importance to anyone dividing quantities for trade in the market.
To log these transactions, the Egyptians developed notation which recorded these new numbers.
One of the earliest representations of these fractions came from a hieroglyph which had great mystical significance.
It's called the Eye of Horus.
Horus was an Old Kingdom god, depicted as half man, half falcon.
According to legend, Horus' father was killed by his other son, Seth.
Horus was determined to avenge the murder.
During one particularly fierce battle, Seth ripped out Horus' eye, tore it up and scattered it over Egypt.
But the gods were looking favourably on Horus.
They gathered up the scattered pieces and reassembled the eye.
Each part of the eye represented a different fraction.
Each one, half the fraction before.
Although the original eye represented a whole unit, the reassembled eye is 1/64 short.
Although the Egyptians stopped at 1/64, implicit in this picture is the possibility of adding more fractions, halving them each time, the sum getting closer and closer to one, but never quite reaching it.
This is the first hint of something called a geometric series, and it appears at a number of points in the Rhind Papyrus.
But the concept of infinite series would remain hidden until the mathematicians of Asia discovered it centuries later.
Having worked out a system of numbers, including these new fractions, it was time for the Egyptians to apply their knowledge to understanding shapes that they encountered day to day.
These shapes were rarely regular squares or rectangles, and in the Rhind Papyrus, we find the area of a more organic form, the circle.
What is astounding in the calculation of the area of the circle is its exactness, really.
How they would have found their method is open to speculation, because the texts we have do not show us the methods how they were found.
This calculation is particularly striking because it depends on seeing how the shape of the circle can be approximated by shapes that the Egyptians already understood.
The Rhind Papyrus states that a circular field with a diameter of nine units is close in area to a square with sides of eight.
But how would this relationship have been discovered? My favourite theory sees the answer in the ancient game of mancala.
Mancala boards were found carved on the roofs of temples.
Each player starts with an equal number of stones, and the object of the game is to move them round the board, capturing your opponent's counters on the way.
As the players sat around waiting to make their next move, perhaps one of them realised that sometimes the balls fill the circular holes of the mancala board in a rather nice way.
He might have gone on to experiment with trying to make larger circles.
Perhaps he noticed that 64 stones, the square of 8, can be used to make a circle with diameter nine stones.
By rearranging the stones, the circle has been approximated by a square.
And because the area of a circle ispi times the radius squared, the Egyptian calculation gives us the first accurate value for pi.
The area of the circle is 64.
Divide this by the radius squared, in this case 4.
5 squared, and you get a value for pi.
So 64 divided by 4.
5 squared is 3.
16, just a little under two hundredths away from its true value.
But the really brilliant thing is, the Egyptians are using these smaller shapes to capture the larger shape.
But there's one imposing andmajestic symbol of Egyptian mathematics we haven't attempted to unravel yet – the pyramid.
I've seen so many pictures that I couldn't believe I'd be impressed by them.
But meeting them face to face, you understand why they're called one of the Seven Wonders of the Ancient World.
They're simply breathtaking.
And how much more impressive they must have been in their day, when the sides were as smooth as glass, reflecting the desert sun.
To me it looks like there might be mirror pyramidshiding underneath the desert, which would complete the shapes to make perfectlysymmetrical octahedrons.
Sometimes, in the shimmer of the desert heat, you can almost see these shapes.
It's the hint of symmetry hidden inside these shapes that makes them so impressive for a mathematician.
The pyramids are just a little short to create these perfect shapes, but some have suggested another important mathematical concept might be hidden inside the proportions of the Great Pyramid – the golden ratio.
Two lengths are in the golden ratio, if the relationship of the longest to the shortest is the same as the sum of the two to the longest side.
Such a ratio has been associated with the perfect proportions one finds all over the natural world, as well as in the work of artists, architects and designers for millennia.
Whether the architects of the pyramids were conscious of this important mathematical idea, or were instinctively drawn to it because of its satisfying aesthetic properties, we'll never know.
For me, the most impressive thing about the pyramidsis the mathematical brilliance that went into making them, including the first inkling of one of the great theorems of the ancient world, Pythagoras' theorem.
In order to get perfect right-angled corners on their buildings and pyramids, the Egyptians would haveused a rope with knots tied in it.
At some point, the Egyptians realised that if they took a triangle with sides marked with three knots, four knots and five knots, it guaranteed them aperfect right-angle.
This is because three squared, plus four squared, is equal to five squared.
So we've got a perfect Pythagorean triangle.
In fact any triangle whose sides satisfy this relationship will give me an 90-degree angle.
But I'm pretty sure that the Egyptians hadn't got this sweeping generalisation of their 3, 4, 5 triangle.
We would not expect to find the general proof because this is not the style of Egyptian mathematics.
Every problem was solved using concrete numbers and then if a verification would be carried out at the end, it would use the result and these concrete, given numbers, there's no general proof within the Egyptian mathematical texts.
It would be some 2, 000 years before the Greeks and Pythagoras would prove that all right-angled triangles shared certain properties.
This wasn't the only mathematical idea that theEgyptians would anticipate.
In a 4, 000-year-old document called the Moscow papyrus, we find a formula for the volume of a pyramid with its peak sliced off, which shows the first hint ofcalculus at work.
For a culture like Egypt that is famous for its pyramids, you would expect problems like this to have been a regular feature within the mathematical texts.
The calculation of the volume of a truncated pyramid is one of the most advanced bits, according to our modern standards of mathematics, that we have from ancient Egypt.
The architects and engineers would certainly have wanted such a formula to calculate the amount of materials required to build it.
But it's a mark of the sophistication of Egyptian mathematics that they were ableto produce such a beautiful method.
To understand how they derived their formula, start with a pyramid built such that the highest point sits directly over one corner.
Three of these can be put together to make a rectangular box, so the volume of the skewed pyramid is a third the volume of the box.
That is, the height, times the length, times the width, divided by three.
Now comes an argument which shows the very first hints of the calculus at work, thousands of years before Gottfried Leibniz and Isaac Newton would come up with the theory.
Suppose you could cut the pyramid into slices, you could then slide the layers across to make the more symmetrical pyramid you see in Giza.
However, the volume of the pyramid has not changed, despite the rearrangement of the layers.
So the same formula works.
The Egyptians were amazing innovators, and their ability to generate new mathematics was staggering.
For me, they revealed the power of geometry and numbers, and made the first moves towards some of the exciting mathematical discoveries to come.
But there was another civilisation that had mathematics to rival that of Egypt.
And we know much more about their achievements.
This is Damascus, over 5, 000 years old, and still vibrant and bustling today.
It used to be the most important point on the trade routes, linking old Mesopotamia with Egypt.
The Babylonians controlled much of modern-day Iraq, Iran and Syria, from 1800BC.
In order to expand and run their empire, they became masters of managing and manipulating numbers.
We have law codes for instance that tell us about the way society is ordered.
The people we know most about are the scribes, the professionally literate and numerate people who kept the records for the wealthy families and for the temples and palaces.
Scribe schools existed from around 2500BC.
Aspiring scribes were sent there as children, and learned how to read, write and work with numbers.
Scribe records were kept on clay tablets, which allowed the Babylonians to manage and advance their empire.
However, many of the tablets we have today aren't official documents, but children's exercises.
It's these unlikely relics that give us a rare insight into how the Babylonians approached mathematics.
So, this is a geometrical textbook from about the 18th century BC.
I hope you can see that there are lots of pictures on it.
And underneath each picture is a text that sets a problem about the picture.
So for instance this one here says, I drew a square, 60 units long, and inside it, I drew four circles – what are their areas? This little tablet here was written 1, 000 years at least later than the tablet here, but has a very interesting relationship.
It also has four circles on, in a square, roughly drawn, but this isn't a textbook, it's a school exercise.
The adult scribe who's teaching the student is being given this as an example of completed homework or something like that.
Like the Egyptians, the Babylonians appeared interested in solving practical problems to do with measuring and weighing.
The Babylonian solutions to these problems are written like mathematical recipes.
A scribe would simply follow and record a set of instructions to get a result.
Here's an example of the kind of problem they'd solve.
I've got a bundle of cinnamon sticks here, but I'm not gonna weigh them.
Instead, I'm gonna take four times their weight and add them to the scales.
Now I'm gonna add 20 gin.
Gin was the ancient Babylonian measure of weight.
I'm gonna take half of everything here and then add it again.
.
.
That's two bundles, and ten gin.
Everything on this side is equal to one mana.
One mana was 60 gin.
And here, we have one of the first mathematical equations in history, everything on this side is equal to one mana.
But how much does the bundle of cinnamon sticks weigh? Without any algebraic language, they were able to manipulate the quantities to be able to prove that the cinnamon sticks weighed five gin.
In my mind, it's this kind of problem which gives mathematics a bit of a bad name.
You can blame those ancient Babylonians for all those tortuous problems you had at school.
But the ancient Babylonian scribes excelled at this kind of problem.
Intriguingly, they weren't using powers of 10, like the Egyptians, they were using powers of 60.
The Babylonians invented their number system, like the Egyptians, by using their fingers.
But instead of counting through the 10 fingers on their hand, Babylonians found a moreintriguing way to count body parts.
They used the 12 knuckles on one hand, and the five fingers on the other to be able to count 12 times 5, ie 60 different numbers.
So for example, this number would have been 2 lots of 12, 24, and then, 1, 2, 3, 4, 5, to make 29.
The number 60 had another powerful property.
It can be perfectly divided in a multitude of ways.
Here are 60 beans.
I can arrange them in 2 rows of 30.
3 rows of 20.
4 rows of 15.
5 rows of 12.
Or 6 rows of 10.
The divisibility of 60 makes it a perfect base in which to do arithmetic.
The base 60 system was so successful, we still use elements of it today.
Every time we want to tell the time, we recognise units of 60 – 60 seconds in a minute, 60 minutes in an hour.
But the most important feature of the Babylonians' number system was that it recognised place value.
Just as our decimal numbers count how many lots of tens, hundreds and thousands you're recording, the position of each Babylonian number records the power of 60.
Instead of inventing new symbols for bigger and bigger numbers, they would write 1-1-1, so this number would be 3, 661.
The catalyst for this discovery was the Babylonians' desire to chart the course of the night sky.
The Babylonians' calendar was based on the cycles of the moon.
They needed a way of recording astronomically large numbers.
Month by month, year by year, these cycles were recorded.
From about 800BC, there were complete lists of lunar eclipses.
The Babylonian system of measurement was quite sophisticated at that time.
They had a system of angular measurement, 360 degrees in a full circle, each degree was divided into 60 minutes, a minute was further divided into 60 seconds.
So they had a regular system for measurement, and it was in perfect harmony with their number system, so it's well suited not only for observation but also for calculation.
But in order to calculate and cope with these large numbers, the Babylonians needed to invent a new symbol.
And in so doing, they prepared the ground for one of the great breakthroughs in the history of mathematics – zero.
In the early days, the Babylonians, in order to mark an empty place in the middle of a number, would simply leave a blank space.
So they needed a way of representing nothing in the middle of a number.
So they used a sign, as a sort of breathing marker, a punctuation mark, and it comes to mean zero in the middle of a number.
This was the first time zero, in any form, had appeared in the mathematical universe.
But it would be over a 1, 000 years before this little place holder would become a number in its own right.
Having established such a sophisticated system of numbers, they harnessed it to tame the arid and inhospitable land that ran through Mesopotamia.
Babylonian engineers and surveyors found ingenious ways of accessing water, and channelling it to the crop fields.
Yet again, they used mathematics to come up with solutions.
The Orontes valley in Syria is still an agricultural hub, and the old methods of irrigation are being exploited today, just as they were thousands of years ago.
Many of the problems in Babylonian mathematics are concerned with measuring land, and it's here we see for the first time the use of quadratic equations, one of the greatest legacies ofBabylonian mathematics.
Quadratic equations involve things where the unknown quantity you're trying to identify is multiplied by itself.
We call this squaring because it gives the area of a square, and it's in the context of calculating the area of land that thesequadratic equations naturally arise.
Here's a typical problem.
If a field has an area of 55 units and one side is six units longer than the other, how long is the shorter side? The Babylonian solution was to reconfigure the field as a square.
Cut three units off the end and move this round.
Now, there's a three-by-three piece missing, so let's add this in.
The area of the field has increased by nine units.
This makes the new area 64.
So the sides of the square are eight units.
The problem-solver knows that they've added three to this side.
So, the original length must be five.
It may not look like it, but this is one of the first quadratic equations in history.
In modern mathematics, I would use the symbolic language of algebra to solve this problem.
The amazing feat of the Babylonians is that they were using these geometric games to find the value, without any recourse to symbols or formulas.
The Babylonians were enjoying problem-solving for its own sake.
They were falling in love with mathematics.
The Babylonians' fascination with numbers soon found a place in their leisure time, too.
They were avid game-players.
The Babylonians and their descendants have been playing a version of backgammon for over 5, 000 years.
The Babylonians played board games, from very posh board games in royal tombs to little bits of board games found in schools, to board games scratched on the entrances of palaces, so that the guardsman must have played when they were bored, and they used dice to move their counters round.
People who played games were using numbers in their leisure time to try and outwit their opponent, doing mental arithmetic very fast, and so they were calculating in their leisure time, without even thinking about it as being mathematical hard work.
Now's my chance.
'I hadn't played backgammon for ages but I reckoned my maths would give me a fighting chance.
' It's up to you.
Six.
.
.
I need to move something.
'But it wasn't as easy as I thought.
' Ah! What the hell was that? Yeah.
This is one, this is two.
Now you're in trouble.
So I can't move anything.
You cannot move these.
Oh, gosh.
There you go.
Three and four.
'Just like the ancient Babylonians, my opponents were masters of tactical mathematics.
' Yeah.
Put it there.
Good game.
The Babylonians are recognised as one of the first cultures to use symmetrical mathematical shapes to make dice, but there is more heated debates about whether they might also have been the first to discover the secrets of another important shape.
The right-angled triangle.
We've already seen how the Egyptians use a 3-4-5 right-angled triangle.
But what the Babylonians knew about this shape and others like it is much more sophisticated.
This is the most famous and controversial ancient tablet we have.
It's called Plimpton 322.
Many mathematicians are convinced it shows the Babylonians could well have known the principle regarding right-angled triangles, that the square on the diagonal is the sum of the squares on the sides, and known it centuries before the Greeks claimed it.
This is a copy of arguably the most famous Babylonian tablet, which is Plimpton 322, and these numbers here reflect the width or height of a triangle, this being the diagonal, the other side would be over here, and the square of this column plus the square of the number in this column equals the square of the diagonal.
They are arranged in an order of steadily decreasing angle, on a very uniform basis, showing that somebody had a lot of understanding of how the numbers fit together.
Here were 15 perfect Pythagorean triangles, all of whose sides had whole-number lengths.
It's tempting to think that the Babylonians were the first custodians of Pythagoras' theorem, and it's a conclusion that generations of historians have been seduced by.
But there could be a much simpler explanation for the sets of three numbers which fulfil Pythagoras' theorem.
It's not a systematic explanation of Pythagorean triples, it's simply a mathematics teacher doing some quite complicated calculations, but in order to produce some very simple numbers, in order to set his students problems about right-angled triangles, and in that sense it's about Pythagorean triples only incidentally.
The most valuable clues to what they understood could lie elsewhere.
This small school exercise tablet is nearly 4, 000 years old and reveals just what the Babylonians did know about right-angled triangles.
It uses a principle of Pythagoras' theorem to find the value of an astounding new number.
Drawn along the diagonal is a really very good approximation to the square root of two, and so that shows us that it was known and used in school environments.
Why's this important? Because the square root of two is what we now call an irrational number, that is, if we write it out in decimals, or even in sexigesimal places, it doesn't end, the numbers go on forever after the decimal point.
The implications of this calculation are far-reaching.
Firstly, it means the Babylonians knew something of Pythagoras' theorem 1, 000 years before Pythagoras.
Secondly, the fact that they can calculate this number to an accuracy of four decimal places shows an amazing arithmetic facility, as well as a passion for mathematical detail.
The Babylonians' mathematical dexterity was astounding, and for nearly 2, 000 years they spearheaded intellectual progress in the ancient world.
But when their imperial power began to wane, so did their intellectual vigour.
By 330BC, the Greeks had advanced their imperial reach into old Mesopotamia.
This is Palmyra in central Syria, a once-great city built by the Greeks.
The mathematical expertise needed to build structures with such geometric perfection is impressive.
Just like the Babylonians before them, the Greeks were passionate about mathematics.
The Greeks were clever colonists.
They took the best from the civilisations they invaded to advance their own power and influence, but they were soon making contributions themselves.
In my opinion, their greatest innovation was to do with a shift in the mind.
What they initiated would influence humanity for centuries.
They gave us the power of proof.
Somehow they decided that they had to have a deductive system for their mathematics and the typical deductive system was to begin with certain axioms, which you assume are true.
It's as if you assume a certain theorem is true without proving it.
And then, using logical methods and very careful steps, from these axioms you prove theorems and from those theorems you prove more theorems, and it just snowballs.
Proof is what gives mathematics its strength.
It's the power or proof which means that the discoveries of the Greeks are as true today as they were 2, 000 years ago.
I needed to head west into the heart of the old Greek empire to learn more.
For me, Greek mathematics has always been heroic and romantic.
I'm on my way to Samos, less than a mile from the Turkish coast.
This place has become synonymous with the birth of Greek mathematics, and it's down to the legend of one man.
His name is Pythagoras.
The legends that surround his life and work have contributed to the celebrity status he has gained over the last 2, 000 years.
He's credited, rightly or wrongly, with beginning the transformation from mathematics as a tool for accounting to the analytic subject we recognise today.
Pythagoras is a controversial figure.
Because he left no mathematical writings, many have questioned whether he indeed solved any of the theorems attributed to him.
He founded a school in Samos in the sixth century BC, but his teachings were considered suspect and the Pythagoreans a bizarre sect.
There is good evidence that there were schools of Pythagoreans, and they may have looked more like sects than what we associate with philosophical schools, because they didn't just share knowledge, they also shared a way of life.
There may have been communal living and they all seemed to have been involved in the politics of their cities.
One feature that makes them unusual in the ancient world is that they included women.
But Pythagoras is synonymous with understanding something that eluded the Egyptians and the Babylonians – the properties of right-angled triangles.
What's known as Pythagoras' theorem states that if you take any right-angled triangle, build squares on all the sides, then the area of the largest square is equal to the sum of the squares on the two smaller sides.
It's at this point for me that mathematics is born and a gulf opens up between the other sciences, and the proof is as simple as it is devastating in its implications.
Place four copies of the right-angled triangle on top of this surface.
The square that you now see has sides equal to the hypotenuse of the triangle.
By sliding these triangles around, we see how we can break the area of the large square up into the sum of two smaller squares, whose sides are given by the two short sides of the triangle.
In other words, the square on the hypotenuse is equal to the sum of the squares on the other sides.
Pythagoras' theorem.
It illustrates one of the characteristic themes of Greek mathematics – the appeal to beautiful arguments in geometry rather than a reliance on number.
Pythagoras may have fallen out of favour and many of the discoveries accredited to him have been contested recently, but there's one mathematical theory that I'm loath to take away from him.
It's to do with music and the discoveryof the harmonic series.
The story goes that, walking past a blacksmith's one day, Pythagoras heard anvils being struck, and noticed how the notes being produced sounded in perfect harmony.
He believed that there must be some rational explanation to make sense of why the notes sounded so appealing.
The answer was mathematics.
Experimenting with a stringed instrument, Pythagoras discovered that the intervals between harmonious musical notes were always represented as whole-number ratios.
And here's how he might have constructed his theory.
First, play a note on the open string.
MAN PLAYS NOTE Next, take half the length.
The note almost sounds the same as the first note.
In fact it's an octave higher, but the relationship is so strong, we give these notes the same name.
Now take a third the length.
We get another note which sounds harmonious next to the first two, but take a length of string which is not in a whole-number ratio and all we get is dissonance.
According to legend, Pythagoras was so excited by this discovery that he concluded the whole universe was built from numbers.
But he and his followers were in for a rather unsettling challenge to their world view and it came about as a result of the theorem which bears Pythagoras' name.
Legend has it, one of his followers, a mathematician called Hippasus, set out to find the length of the diagonal for a right-angled triangle with two sides measuring one unit.
Pythagoras' theorem implied that the length of the diagonal was a number whose square was two.
The Pythagoreans assumed that the answer would be a fraction, but when Hippasus tried to express it in this way, no matter how he tried, he couldn't capture it.
Eventually he realised his mistake.
It was the assumption that the value was a fraction at all which was wrong.
The value of the square root of two was the number that the Babylonians etched into the Yale tablet.
However, they didn't recognise the special character of this number.
But Hippasus did.
It was an irrational number.
The discovery of this new number, and others like it, is akin to an explorer discovering a new continent, or a naturalist finding a new species.
But these irrational numbers didn't fit the Pythagorean world view.
Later Greek commentators tell the story of how Pythagoras swore his sect to secrecy, but Hippasus let slip the discovery and was promptly drowned for his attempts to broadcast their research.
But these mathematical discoveries could not be easily suppressed.
Schools of philosophy and science started to flourish all over Greece, building on these foundations.
The most famous of these was the Academy.
Plato founded this school in Athens in 387 BC.
Although we think of him today as a philosopher, he was one of mathematics' most important patrons.
Plato was enraptured by the Pythagorean world view and considered mathematics the bedrock of knowledge.
Some people would say that Plato is the most influential figure for our perception of Greek mathematics.
He argued that mathematics is an important form of knowledge and does have a connection with reality.
So by knowing mathematics, we know more about reality.
In his dialogue Timaeus, Plato proposes the thesis that geometry is the key to unlocking the secrets of the universe, a view still held by scientists today.
Indeed, the importance Plato attached to geometry is encapsulated in the sign that was mounted above the Academy, “Let no-one ignorant of geometry enter here.
” Plato proposed that the universe could be crystallised into five regular symmetrical shapes.
These shapes, which we now call the Platonic solids, were composed of regular polygons, assembled to create three-dimensional symmetrical objects.
The tetrahedron represented fire.
The icosahedron, made from 20 triangles, represented water.
The stable cube was Earth.
The eight-faced octahedron was air.
And the fifth Platonic solid, the dodecahedron, made out of 12 pentagons, was reserved for the shape that captured Plato's view of the universe.
Plato's theory would have a seismic influence and continued to inspire mathematicians and astronomers for over 1, 500 years.
In addition to the breakthroughs made in the Academy, mathematical triumphs were also emerging from the edge of the Greek empire, and owed as much to the mathematical heritage of the Egyptians as the Greeks.
Alexandria became a hub of academic excellence under the rule of the Ptolemies in the 3rd century BC, and its famous library soon gained a reputation to rival Plato's academy.
The kings of Alexandria were prepared to invest in the arts and culture, in technology, mathematics, grammar, because patronage for cultural pursuits was one way of showing that you were a more prestigious ruler, and had a better entitlement to greatness.
The old library and its precious contents were destroyed when the Muslims conquered Egypt in the 7th Century.
But its spirit is alive in a new building.
Today, the library remains a place of discovery and scholarship.
Mathematicians and philosophers flocked to Alexandria, driven by their thirst for knowledge and the pursuit of excellence.
The patrons of the library were the first professional scientists, individuals who were paid for their devotion to research.
But of all those early pioneers, my hero is the enigmatic Greek mathematician Euclid.
We know very little about Euclid's life, but his greatest achievements were as a chronicler of mathematics.
Around 300 BC, he wrote the most important text book of all time – The Elements.
In The Elements, we find the culmination of the mathematical revolution which had taken place in Greece.
It's built on a series of mathematical assumptions, called axioms.
For example, a line can be drawn between any two points.
From these axioms, logical deductions are made and mathematical theorems established.
The Elements contains formulas for calculating the volumes of cones and cylinders, proofs about geometric series, perfect numbers and primes.
The climax of The Elements is a proof that there are only five Platonic solids.
For me, this last theorem captures the power of mathematics.
It's one thing to build five symmetrical solids, quite another to come up with a watertight, logical argument for why there can't be a sixth.
The Elements unfolds like a wonderful, logical mystery novel.
But this is a story which transcends time.
Scientific theories get knocked down, from one generation to the next, but the theorems in The Elements are as true today as they were 2, 000 years ago.
When you stop and think about it, it's really amazing.
It's the same theorems that we teach.
We may teach them in a slightly different way, we may organise them differently, but it's Euclidean geometry that is still valid, and even in higher mathematics, when you go to higher dimensional spaces, you're still using Euclidean geometry.
Alexandria must have been an inspiring place for the ancient scholars, and Euclid's fame would have attracted even more eager, young intellectuals to the Egyptian port.
One mathematician who particularly enjoyed the intellectual environment in Alexandria was Archimedes.
He would become a mathematical visionary.
The best Greek mathematicians, they were always pushing the limits, pushing the envelope.
So, Archimedes.
.
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did what he could with polygons, with solids.
He then moved on to centres of gravity.
He then moved on to the spiral.
This instinct to try and mathematise everything is something that I see as a legacy.
One of Archimedes' specialities was weapons of mass destruction.
They were used against the Romans when they invaded his home of Syracuse in 212 BC.
He also designed mirrors, which harnessed the power of the sun, to set the Roman ships on fire.
But to Archimedes, these endeavours were mere amusements in geometry.
He had loftier ambitions.
Archimedes was enraptured by pure mathematics and believed in studying mathematics for its own sake and not for the ignoble trade of engineering or the sordid quest for profit.
One of his finest investigations into pure mathematics was to produce formulas to calculate the areas of regular shapes.
Archimedes' method was to capture new shapes by using shapes he already understood.
So, for example, to calculate the area of a circle, he would enclose it inside a triangle, and then by doubling the number of sides on the triangle, the enclosing shape would get closer and closer to the circle.
Indeed, we sometimes call a circle a polygon with an infinite number of sides.
But by estimating the area of the circle, Archimedes is, in fact, getting a value for pi, the most important number in mathematics.
However, it was calculating the volumes of solid objects where Archimedes excelled.
He found a way to calculate the volume of a sphere by slicing it up and approximating each slice as a cylinder.
He then added up the volumes of the slices to get an approximate value for the sphere.
But his act of genius was to see what happens if you make the slices thinner and thinner.
In the limit, the approximation becomes an exact calculation.
But it was Archimedes' commitment to mathematics that would be his undoing.
Archimedes was contemplating a problem about circles traced in the sand.
When a Roman soldier accosted him, Archimedes was so engrossed in his problem that he insisted that he be allowed to finish his theorem.
But the Roman soldier was not interested in Archimedes' problem and killed him on the spot.
Even in death, Archimedes' devotion to mathematics was unwavering.
By the middle of the 1st Century BC, the Romans had tightened their grip on the old Greek empire.
They were less smitten with the beauty of mathematics and were more concerned with its practical applications.
This pragmatic attitude signalled the beginning of the end for the great library of Alexandria.
But one mathematician was determined to keep the legacy of the Greeks alive.
Hypatia was exceptional, a female mathematician, and a pagan in the piously Christian Roman empire.
Hypatia was very prestigious and very influential in her time.
She was a teacher with a lot of students, a lot of followers.
She was politically influential in Alexandria.
So it's this combination of.
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high knowledge and high prestige that may have made her a figure of hatred for.
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the Christian mob.
One morning during Lent, Hypatia was dragged off her chariot by a zealous Christian mob and taken to a church.
There, she was tortured and brutally murdered.
The dramatic circumstances of her life and death fascinated later generations.
Sadly, her cult status eclipsed her mathematical achievements.
She was, in fact, a brilliant teacher and theorist, and her death dealt a final blow to the Greek mathematical heritage of Alexandria.
My travels have taken me on a fascinating journey to uncover the passion and innovation of the world's earliest mathematicians.
It's the breakthroughs made by those early pioneers of Egypt, Babylon and Greece that are the foundations on which my subject is built today.
But this is just the beginning of my mathematical odyssey.
The next leg of my journey lies east, in the depths of Asia, where mathematicians scaled even greater heights in pursuit of knowledge.
With this new era came a new language of algebra and numbers, better suited to telling the next chapter in the story of maths.
You can learn more about the story of maths with the Open University at.
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