The Frontiers of Space The Story of Maths

I'm walking in the mountains of the moon.

I'm on the trail of the Renaissance artist, Piero della Francesca,

so I've come to the town in northern Italy which Piero made his own.

There it is, Urbino.

I've come here to see some of Piero's finest works,

masterpieces of art, but also masterpieces of mathematics.

The artists and architects of the early Renaissance brought back

the use of perspective, a technique that had been lost for 1,000 years,

but using it properly turned out to be a lot

more difficult than they'd imagined.

Piero was the first major painter to fully understand perspective.

That's because he was a mathematician as well as an artist.

I came here to see his masterpiece,

The Flagellation of Christ, but there was a problem.

I've just been to see The Flagellation, and it's an

absolutely stunning picture, but unfortunately, for various

kind of Italian reasons, we're not allowed to go and film in there.

But this is a maths programme, after all, and not an arts programme,

so I've used a bit of mathematics to bring this picture alive.

We can't go to the picture, but we can make the picture come to us.

The problem of perspective is how

to represent the three-dimensional world on a two-dimensional canvas.

To give a sense of depth, a sense of the third dimension,

Piero used mathematics.

How big is he going to paint Christ,

if this group of men here were a certain distance away

from these men in the foreground?

Get it wrong and the illusion of perspective is shattered.

It's far from obvious how a three-dimensional world

can be accurately represented on a two-dimensional surface.

Look at how the parallel lines in the three-dimensional world

are no longer parallel in the two-dimensional canvas, but meet

at a vanishing point.

And this is what the tiles in the picture really look like.

What is emerging here is a new

mathematical language which allows us to map one thing into another.

The power of perspective unleashed a new way to see the world,

a perspective that would cause a mathematical revolution.

Piero's work was the beginning of a new way to understand geometry,

but it would take another 200 years

before other mathematicians would continue where he left off.

Our journey has come north.

By the 17th century, Europe had taken over

from the Middle East as the world's powerhouse of mathematical ideas.

Great strides had been made in the geometry

of objects fixed in time and space.

In France, Germany, Holland and Britain,

the race was now on to understand the mathematics of objects in motion

and the pursuit of this new mathematics started here in this

village in the centre of France.

Only the French would name a village after a mathematician.

Imagine in England a town called

Newton or Ball or Cayley. I don't think so!

But in France, they really value their mathematicians.

This is the village of Descartes in the Loire Valley.

It was renamed after the famous philosopher

and mathematician 200 years ago.

Descartes himself was born here in 1596, a sickly child who lost

his mother when very young, so he was allowed to stay in bed every

morning until 11.00am, a practice he tried to continue all his life.

To do mathematics, sometimes you just need to remove

all distractions, to float off into a world of shapes and patterns.

Descartes thought that the bed was the best place to achieve

this meditative state.

I think I know what he means.

The house where Descartes undertook his bedtime meditations

is now a museum dedicated to all things Cartesian.

Come with me.

Its exhibition pieces arranged, by curator Sylvie Garnier, show how

his philosophical, scientific and mathematical ideas all fit together.

It also features less familiar aspects

of Descartes' life and career.

So he decided to be a soldier...in the army,

in the Protestant Army

and too in the Catholic Army, not a problem for him

because no patriotism.

Sylvie is putting it very nicely,

but Descartes was in fact a mercenary.

He fought for the German Protestants, the French Catholics

and anyone else who would pay him.

Very early one autumn morning in 1628, he was in the Bavarian Army

camped out on a cold river bank.

Inspiration very often strikes in very strange places.

The story is told how Descartes couldn't sleep one night,

maybe because he was getting up so late

or perhaps he was celebrating St Martin's Eve

and had just drunk too much.

Problems were tumbling around in his mind.

He was thinking about his favourite subject, philosophy.

He was finding it very frustrating.

How can you actually know anything at all?!

Then he slips into a dream...

and in the dream he understood that the key was to build philosophy

on the indisputable facts of mathematics.

Numbers, he realised, could brush away the cobwebs of uncertainty.

He wanted to publish all his radical ideas, but he was worried how they'd

be received in Catholic France, so he packed his bags and left.

Descartes found a home here in Holland.

He'd been one of the champions of the new scientific revolution

which rejected the dominant view that the sun went around the earth,

an opinion that got scientists like Galileo

into deep trouble with the Vatican.

Descartes reckoned that here amongst the Protestant Dutch

he would be safe, especially

at the old university town of Leiden

where they valued maths and science.

I've come to Leiden too.

Unfortunately, I'm late!

Hello. Yeah, I'm sorry.

I got a puncture. It took me a bit of time, yeah, yeah.

Henk Bos is one of Europe's most eminent Cartesian scholars.

He's not surprised the French scholar ended up in Leiden.

He came to talk with people and some people were open to his ideas.

This was not only mathematic. It was also a mechanics specially.

He merged algebra and geometry.

-Right.

-So you could have formulas and figures and go back and forth.

-So a sort of dictionary between the two?

-Yeah, yeah.

This dictionary, which was finally published here in Holland in 1637,

included mainly controversial

philosophical ideas, but the most radical thoughts

were in the appendix, a proposal to link algebra and geometry.

Each point in two dimensions can be described by two numbers,

one giving the horizontal location, the second number giving the point's

vertical location.

As the point moves around a circle, these coordinates change,

but we can write down an equation that identifies the changing value

of these numbers at any point in the figure.

Suddenly, geometry has turned into algebra.

Using this transformation

from geometry into numbers, you could tell, for example,

if the curve on this bridge was part of a circle or not.

You didn't need to use your eyes.

Instead, the equations of the curve would reveal its secrets,

but it wouldn't stop there.

Descartes had unlocked the possibility of navigating geometries

of higher dimensions, worlds our eyes will never see but are central

to modern technology and physics.

There's no doubt that Descartes was one of the giants of mathematics.

Unfortunately, though, he wasn't the nicest of men.

I think he was not an easy person, so...

And he could be... he was very much concerned about

his image. He was entirely

self-convinced that he was right, also when he was wrong and his first

reaction would be that the other one was stupid that hadn't understood it.

Descartes may not have been the most congenial person,

but there's no doubt that his insight into the connection

between algebra and geometry transformed mathematics forever.

For his mathematical revolution to work, though, he needed one other

vital ingredient.

To find that, I had to say goodbye to Henk and Leiden and go to church.

CHORAL SINGING

I'm not a believer myself, but there's little doubt

that many mathematicians from the time of Descartes

had strong religious convictions.

Maybe it's just a coincidence,

but perhaps it's because mathematics and religion are both building ideas

upon an undisputed set of axioms - one plus one equals two. God exists.

I think I know which set of axioms I've got my faith in.

In the 17th century,

there was a Parisian monk who went to the same school as Descartes.

He loved mathematics as much as he loved God.

Indeed, he saw maths and science as evidence of the existence of God,

Marin Mersenne was a first-class mathematician.

One of his discoveries in prime numbers is still named after him.

But he's also celebrated for his correspondence.

From his monastery in Paris, Mersenne acted like some kind of

17th century internet hub, receiving ideas and then sending them on.

It's not so different now.

We sit like mathematical monks thinking about our ideas, then

sending a message to a colleague and hoping for some reply.

There was a spirit of mathematical communication in 17th century Europe

which had not been seen since the Greeks.

Mersenne urged people to read Descartes' new work on geometry.

He also did something just as important.

He publicised some new findings on the properties of numbers

by an unknown amateur who would end up rivalling Descartes as the

greatest mathematician of his time, Pierre de Fermat.

Here in Beaumont-de-Lomagne

near Toulouse, residents and visitors have come

out to celebrate the life and work of the village's most famous son.

But I'm not too sure what these gladiators are doing here!

And the appearance of this camel came as a bit of a surprise too.

The man himself would have hardly approved of

the ideas of using fun and games to advance an interest in mathematics.

Unlike the aristocratic Descartes, Fermat wouldn't have considered it

worthless or common to create a festival of mathematics.

Maths in action, that one.

It's beautiful, really nice, yeah.

Fermat's greatest contribution to mathematics was to virtually invent

modern number theory.

He devised a wide range of conjectures

and theorems about numbers including his famous Last Theorem,

the proof of which would puzzle mathematicians for over 350 years,

but it's little help to me now.

Getting it apart is the easy bit.

It's putting it together, isn't it, that's the difficult bit.

How many bits have I got? I've got six bits.

I think what I need to do is put some symmetry into this.

I'm afraid he's going to tell me how to do it and I don't want to see.

I hate being told how to do a problem. I don't want to look.

And he's laughing at me now because I can't do it.

That's very unfair!

Here we go.

Can I put them together?

I got it!

Now that's the buzz of doing mathematics when

the thing clicks together and suddenly you see the right answer.

Remarkably, Fermat only tackled mathematics in his spare time.

By day he was a magistrate.

Battling with mathematical problems was his hobby and his passion.

The wonderful thing about mathematics is

you can do it anywhere.

You don't have to have a laboratory.

You don't even really need a library.

Fermat used to do much of his work while sitting at the kitchen table

or praying in his local church or up here on his roof.

He may have looked like an amateur,

but he took his mathematics very seriously indeed.

Fermat managed to find several new patterns in numbers

that had defeated mathematicians for centuries.

One of my favourite theorems of Fermat

is all to do with prime numbers.

If you've got a prime number which when you divide it by four

leaves remainder one, then Fermat showed you could

always rewrite this number as two square numbers added together.

For example, I've got 13 cloves of garlic here,

a prime number which has remainder one when I divide it by four.

Fermat proved you can rewrite this number as two square numbers added

together, so 13 can be rewritten

as three squared plus two squared, or four plus nine.

The amazing thing is that Fermat proved this will work however big

the prime number is. Provided it has remainder one on division by four,

you can always rewrite that number

as two square numbers added together.

Ah, my God!

What I love about this sort of day is the playfulness of mathematics

and Fermat certainly enjoyed playing around with numbers. He loved

looking for patterns in numbers and then the puzzle side of mathematics,

he wanted to prove that these patterns would be there forever.

But as well as being the basis for fun and games in the years to come,

Fermat's mathematics would have some very serious applications.

One of his theorems, his Little Theorem, is

the basis of the codes that protect our credit cards on the internet.

Technology we now rely on today all comes from the scribblings

of a 17th-century mathematician.

But the usefulness of Fermat's mathematics is nothing compared to

that of our next great mathematician and he comes not from France at all,

but from its great rival.

In the 17th century, Britain was emerging as a world power.

Its expansion and ambitions required new methods of measurement

and computation and that gave a great boost to mathematics.

The university towns of Oxford and Cambridge

were churning out mathematicians who were in great demand

and the greatest of them was Isaac Newton.

I'm here in Grantham, where Isaac Newton grew up,

and they're very proud of him here.

They have a wonderful statue to him.

They've even got

the Isaac Newton Shopping Centre, with a nice apple logo up there.

There's a school that he went to with a nice blue plaque

and there's a museum over here in the Town Hall, although, actually,

one of the other famous residents here, Margaret Thatcher,

has got as big a display as Isaac Newton.

In fact, the Thatcher cups have

sold out and there's loads of Newton ones still left,

so I thought I would support mathematics by buying a Newton cup.

And Newton's maths does need support.

-Newton's very famous here. Do you know what he's famous for?

-No.

-No, I don't.

-Discovering gravity.

-Gravity?

-Gravity, yes.

-Gravity?

-Apple tree and all that, gravity.

-'That pretty much summed it up.

'If people know about Newton's work at all, it is his physics,

'his laws of gravity in motion, not his mathematics.'

-I'm in a rush!

-You're in a rush. OK.

Acceleration, you see? One of Newton's laws!

Eight miles south of Grantham,

in the village of Woolsthorpe, where Newton was born,

I met up with someone who does share my passion for his mathematics.

This is the house.

Wow, beautiful. 'Jackie Stedall is a Newton fan and more than willing

'to show me around the house where Newton was brought up.'

So here is the...

you might call it the dining room. I'm sure they didn't call it that,

but the room where they ate, next to the kitchen.

Of course, there would have been a huge fire in there.

Yes! Gosh, I wish it was there now!

His father was an illiterate farmer,

but he died shortly before Newton was born.

Otherwise, the young Isaac's fate might have been very different.

And here's his room.

Oh, lovely, wow.

-They present it really nicely.

-Yes.

-It's got a real feel of going back in time.

-It does, yes.

I can see he's as scruffy as I am. Look at the state of that bed.

That's how, I think, I left my bed this morning.

Newton hated his stepfather, but it was this man who ensured

he became a mathematician rather than a sheep farmer.

I don't think he was particularly remarkable as a child.

-OK.

-So there's hope for all those kids out there.

-Yes, yes.

I think he had a sort of average school report.

He had very few close friends. I don't feel he's someone

I particularly would have wanted to meet,

but I do love his mathematics. It's wonderful.

Newton came back to Lincolnshire from Cambridge

during the Great Plague of 1665 when he was just 22 years old.

In two miraculous years here, he developed a new theory of light,

discovered gravitation

and scribbled out a revolutionary approach to maths, the calculus.

It works like this.

I'm going to accelerate this car from 0 to 60 as quickly as I can.

The speedometer is showing me that the speed's changing all the time,

but this is only an average speed.

How can I tell precisely what my speed is

at any particular instant? Well, here's how.

As the car races along the road, we can draw a graph above the road

where the height above each point in the road records how long it took

the car to get to that point.

I can calculate the average speed between

two points, A and B, on my journey by recording the distance travelled

and dividing by the time it took to get between these two points,

but what about the precise speed at the first point, A?

If I move point B closer and closer to the first point, I take a smaller

and smaller window of time and the speed gets closer

and closer to the true value, but eventually, it looks like

I have to calculate 0 divided by 0.

The calculus allows us to make sense of this calculation.

It enables us to work out the exact speed and also the precise distance

travelled at any moment in time.

I mean, it does make sense, the things we take for granted so much,

things like... if I drop this apple...

Its distance is changing and its

speed is changing and calculus can deal with all of that.

Which is quite in contrast to the Greeks.

It was a very static geometry.

-Yes, it is.

-And here we see...

so the calculus is used by

every engineer, physicist, because it can describe the moving world.

Yes, and it's the only way really you can deal with the mathematics of

motion or with change.

There's a lot of mathematics in this apple!

Newton's calculus enables us to really understand

the changing world, the orbits of planets, the motions of fluids.

Through the power of the calculus, we have a way of describing, with

mathematical precision, the complex, ever-changing natural world.

But it would take 200 years to realise its full potential.

Newton himself decided not to publish, but just to circulate

his thoughts among friends.

His reputation, though, gradually spread.

He became a professor, an MP, and then Warden of the Royal Mint

here in the City of London.

On his regular trips to the Royal Society from the Royal Mint,

he preferred to think about theology and alchemy rather than mathematics.

Developing the calculus just got crowded out

by all his other interests until he heard about a rival...

a rival who was also a member of the Royal Society and who came up

with exactly the same idea as him,

Gottfried Leibniz.

Every word Leibniz wrote has been preserved and catalogued

in his hometown of Hanover in northern Germany.

His actual manuscripts are kept under lock and key,

particularly the manuscript which shows how Leibniz

also discovered the miracle of calculus, shortly after Newton.

What age was he when he wrote...

He was 29 years old and that's the time, within two months, he developed

-differential calculus and integral calculus.

-In two months?

-Yeah.

-Fast and furious, when it comes, er...

Yeah.

There is a little scrap of paper over here. What's that one?

-A letter or...

-That's a small manuscript of Leibniz's notes.

"Sometimes it happens that in the morning lying in the bed,

"I have so many ideas that it takes the whole morning and sometimes

"even longer to note all these ideas and bring them to paper."

I suppose, that's beautiful.

I suppose that he liked to lie in the bed in the morning.

-A true mathematician.

-Yeah.

He spends his time thinking in bed.

I see you've got some paintings down here.

A painting.

This is what he looked like. Right.

Even though he didn't become quite the 17th century celebrity

that Newton did, it wasn't such a bad life.

Leibniz worked for the Royal Family

of Hanover and travelled around Europe representing their interests.

This gave him plenty of time to indulge in

his favourite intellectual pastimes, which were wide, even for the time.

He devised a plan for reunifying the Protestant and Roman Catholic

churches, a proposal for France to conquer Egypt and contributions to

philosophy and logic which are still highly rated today.

-He wrote all these letters?

-Yeah.

-That's absolutely extraordinary.

He must have cloned himself. I can't believe there was just one Leibniz!

'But Leibniz was not just man of words.

'He was also one of the first people

'to invent practical calculating machines

'that worked on the binary system, true forerunners of the computer.

'300 years later, the engineering department at Leibniz University

'in Hanover have put them together following Leibniz's blueprint.'

I love all the ball bearings, so these are going to be all

of our zeros and ones. So a ball bearing is a one.

Only zero and one. Now we represent a number 127.

-In binary, it means that we have the first seven digits in one.

-Yeah.

-And now I give the number one.

-OK.

Now we add 127 plus one - is 128, which is two, power eight.

-Oh, OK. So there's going to be lots of action.

-Would you show this here?

This is the money shot.

So we're going to add one. Oops. Here we go. They're all carrying.

So this 128 is two power eight.

Excellent, so 127 in binary is 1, 1, 1, 1, 1, 1, 1, which is

all the ball bearings here.

To add one it all gets

carried, this goes to 0, 0, 0, 0, and we have a power of two here.

So this mechanism gets rid of all the ball bearings that you

-don't need. It's like pinball, mathematical pinball.

-Exactly.

I love this machine!

After a hard day's work, Leibniz often came here,

the famous gardens of Herrenhausen,

now in the middle of Hanover, but then on the outskirts of the city.

There's something about mathematics and walking.

I don't know, you've been working at your desk all day, all morning

on some problem and your head's all

fuzzy, and you just need to come and have a walk.

You let your subconscious mind kind of take over and sometimes

you get your breakthrough just looking at the trees or whatever.

I've had some of my best ideas whilst walking in my local park,

so I'm hoping to get a little bit of inspiration here on Leibniz's

local stomping ground.

I didn't get the chance to purge my mind of mathematical challenges

because in the years since Leibniz lived here,

someone has built a maze.

Well, there is a mathematical formula for getting out of a maze,

which is if you put your left hand on the side of the maze and just

keep it there, keep on winding round, you eventually get out.

That's the theory, at least. Let's see whether it works!

Leibniz had no such distractions.

Within five years, he'd worked out the details of the calculus,

seemingly independent from Newton,

although he knew about Newton's work,

but unlike Newton, Leibniz was quite happy to make his work known

and so mathematicians across Europe heard about the calculus first

from him and not from Newton, and that's when all the trouble started.

Throughout mathematical history, there have been lots of priority

disputes and arguments.

It may seem a little bit petty and schoolboyish.

We really want our name to be on that theorem.

This is our one chance for a little bit of immortality because that

theorem's going to last forever and that's why we dedicate so much time

to trying to crack these things.

Somehow we can't believe that somebody else

has got it at the same time as us.

These are our theorems, our babies, our children and we

don't want to share the credit.

Back in London, Newton certainly didn't want

to share credit with Leibniz, who he thought of as a Hanoverian upstart.

After years of acrimony and accusation, the Royal Society

in London was asked to adjudicate between the rival claims.

The Royal Society gave Newton credit

for the first discovery of the calculus

and Leibniz credit for the first publication,

but in their final judgment, they accused Leibniz of plagiarism.

However, that might have had something to do with the fact that

the report was written by their President, one Sir Isaac Newton.

Leibniz was incredibly hurt.

He admired Newton and never really recovered.

He died in 1716.

Newton lived on another 11 years and was buried in the grandeur of

Westminster Abbey.

Leibniz's memorial, by contrast,

is here in this small church in Hanover.

The irony is that it's Leibniz's mathematics which

eventually triumphs, not Newton's.

I'm a big Leibniz fan.

Quite often revolutions in mathematics are about producing the

right language to capture a new vision and that's what

Leibniz was so good at.

Leibniz's notation, his way of writing the calculus,

captured its true spirit.

It's still the one we use in maths today.

Newton's notation was, for many mathematicians, clumsy and difficult

to use and so while British mathematics loses its way a little,

the story of maths switches to the very heart of Europe, Basel.

In its heyday in the 18th century, the free city of Basel in

Switzerland was the commercial hub of the entire Western world.

Around this maelstrom of trade, there developed a tradition of

learning, particularly learning which connected with commerce

and one family summed all this up.

It's kind of curious - artists often have children who are artists.

Musicians, their children are often musicians, but us mathematicians,

our children don't tend to be mathematicians.

I'm not sure why it is.

At least that's my view, although others dispute it.

What no-one disagrees with

is there is one great dynasty of mathematicians, the Bernoullis.

In the 18th and 19th centuries they produced half a dozen

outstanding mathematicians, any of which we would have been

proud to have had in Britain, and they all came from Basel.

You might have great minds like Newton and Leibniz who make

these fundamental breakthroughs, but you also need the disciples

who take that message, clarify it, realise its implications,

then spread it wide. The family were originally merchants,

and this is one of their houses.

It's now part of the University of Basel

and it's been completely refurbished, apart from one room,

which has been kept very much as the family would have used it.

Dr Fritz Nagel, keeper of the Bernoulli Archive,

has promised to show it to me.

-If we can find it.

-No, we're on the wrong floor.

Wrong floor, OK. Right!

Oh, look.

Can we take an apple?

'No, wrong mathematician.

'Eventually, we got there.'

This is where the Bernoullis would have done

some of their mathematics.

'I was really just being polite.

'The only thing of interest was an old stove.'

Now, of the Bernoullis, which is your favourite?

My favourite Bernoulli is Johann I.

He is the most smart mathematician.

Perhaps his brother Jakob was the mathematician

with the deeper insight into problems,

but Johann found elegant solutions.

The brothers didn't like each other much, but both worshipped Leibniz.

They corresponded with him, stood up for him

against Newton's allies, and spread his calculus throughout Europe.

Leibnitz was very happy to have found two gifted mathematicians

outside of his personal circle of friends who mastered his calculus

and could distribute it in the scientific community.

-That was very important for Leibniz.

-And important for maths, too.

Without the Bernoullis, it would have taken much longer for calculus

to become what it is today, a cornerstone of mathematics.

At least, that is Dr Nagel's contention.

And he is a great Bernoulli fan.

He has arranged for me to meet Professor Daniel Bernoulli,

the latest member of the family,

whose famous name ensures he gets some odd e-mails.

Another one of which I got was,

"Professor Bernoulli, can you give me a hand with calculus?"

To find a Bernoulli, you expect them to be able to do calculus.

'But this Daniel Bernoulli is a professor of geology.

'The maths gene seems to have truly died out.

'And during our very hearty dinner,

'I found myself wandering back to maths.'

It is a bit unfair on the Bernoullis to describe them simply

as disciples of Leibniz.

One of their many great contributions to mathematics

was to develop the calculus to solve a classic problem of the day.

Imagine a ball rolling down a ramp.

The task is to design a ramp that will get the ball

from the top to the bottom in the fastest time possible.

You might think that a straight ramp would be quickest.

Or possibly a curved one like this

that gives the ball plenty of downward momentum.

In fact, it's neither of these.

Calculus shows that it is what we call a cycloid,

the path traced by a point on the rim of a moving bicycle wheel.

This application of the calculus by the Bernoullis, which became known

as the calculus of variation,

has become one of the most powerful aspects of the mathematics

of Leibniz and Newton. Investors use it to maximise profits.

Engineers exploit it to minimise energy use.

Designers apply it to optimise construction.

It has now become one of the linchpins

of our modern technological world.

Meanwhile, things were getting more interesting in the restaurant.

Here is my second surprise.

Let me introduce Mr Leonhard Euler.

Daniel Bernoulli.

'Leonhard Euler, one of the most famous names in mathematics.

'This Leonhard is a descendant

'of the original Leonhard Euler, star pupil of Johann Bernoulli.'

I am the ninth generation,

the fourth Leonhard in our family

after Leonard Euler I, the mathematician.

OK. And yourself, are you a mathematician?

Actually, I am a business analyst.

I can't study mathematics with my name.

Everyone will expect you to prove that the Riemann hypothesis!

Perhaps it's just as well that Leonhard decided

not to follow in the footsteps of his illustrious ancestor.

He'd have had a lot to live up to.

I am going in a boat across the Rhine,

and I'm feeling a little bit worse for wear.

Last night's dinner with Mr Euler and Professor Bernoulli

degenerated into toasts to all the theorems the Bernoullis and Eulers

have proved, and by God, they have proved quite a lot of them!

Never again.

I was getting disapproving glances from my fellow passengers as well.

Luckily, it was only a short trip.

Not like the trip that Euler took in 1728 to start a new life.

Euler may have been the prodigy of Johann Bernoulli,

but there was no room for him in the city.

If your name wasn't Bernoulli,

there was little chance of getting a job in mathematics here in Basel.

But Daniel, the son of Johann Bernoulli,

was a great friend of Euler

and managed to get him a job at his university.

But to get there would take seven weeks,

because Daniel's university was in Russia.

It wasn't an intellectual powerhouse like Berlin or Paris,

but St Petersburg was by no means unsophisticated in the 18th century.

Peter the Great had created a city very much in the European style.

And every fashionable city at the time had a scientific academy.

Peter's Academy is now a museum.

It includes several rooms full of the kind of grotesque curiosities

that are usually kept out of the public display in the West.

But in the 1730s,

this building was a centre for ground-breaking research.

It is where Euler found his intellectual home.

# I am sure that there could never be a more contented man than me... #

Many of the ideas that were bubbling away at the time -

calculus of variation,

Fermat's theory of numbers - crystallised in Euler's hands.

But he was also creating incredibly modern mathematics,

topology and analysis.

Much of the notation that I use today as a mathematician

was created by Euler, numbers like e and i.

Euler also popularised the use of the symbol pi.

He even combined these numbers together

in one of the most beautiful formulas of mathematics,

e to the power of i times pi is equal to -1.

An amazing feat of mathematical alchemy.

His life, in fact, is full of mathematical magic.

Euler applied his skills to an immense range of topics,

from prime numbers to optics to astronomy.

He devised a new system of weights and measures, wrote a textbook

on mechanics, and even found time to develop a new theory of music.

I think of him as the Mozart of maths.

And that view is shared by the mathematician Nikolai Vavilov,

who met me at the house that was given to Euler

by Catherine the Great.

Euler lived here from '66 to '83, which means from the year

he came back to St Petersburg to the year he died.

And he was a member of the Russian Academy of Sciences,

and their greatest mathematician.

That is exactly what it says.

-What is it now?

-It is a school.

Shall we go in and see?

OK.

'I'm not sure Nikolai entirely approved. But nothing ventured...'

Perhaps we should talk to the head teacher.

The head didn't mind at all.

I rather got the impression that she was used

to people dropping in to talk about Euler.

She even had a couple of very able pupils suspiciously close to hand.

These two young ladies are ready to tell a few words about the scientist

and about this very building.

They certainly knew their stuff.

They had undertaken an entire classroom project on Euler,

his long life, happy marriage and 13 children.

And then his tragedies - only five of his children

survived to adulthood.

His first wife, who he adored, died young.

He started losing most of his eyesight.

So for the last years of his life, he still continued to work, actually.

He continued his mathematical research.

I read a quote that said now with his blindness,

he hasn't got any distractions,

he can finally get on with his mathematics. A positive attitude.

It was a totally unexpected and charming visit.

Although I couldn't resist sneaking back and correcting

one of the equations on the board when everyone else had left.

To demonstrate one of my favourite Euler theorems, I needed a drink.

It concerns calculating infinite sums,

the discovery that shot Euler to the top of the mathematical pops

when it was announced in 1735.

Take one shot glass full of vodka and add it to this tall glass here.

Next, take a glass which is a quarter full, or a half squared,

and add it to the first glass.

Next, take a glass which is a ninth full, or a third squared,

and add that one.

Now, if I keep on adding infinitely many glasses where each one

is a fraction squared, how much will be in this tall glass here?

It was called the Basel problem

after the Bernoullis tried and failed to solve it.

Daniel Bernoulli knew that you would not get an infinite amount of vodka.

He estimated that the total would come to about one and three fifths.

But then Euler came along.

Daniel was close, but mathematics is about precision.

Euler calculated that the total height of the vodka

would be exactly pi squared divided by six.

It was a complete surprise.

What on earth did adding squares of fractions

have to do with the special number pi?

But Euler's analysis showed that they were two sides

of the same equation.

One plus a quarter plus a ninth plus a sixteenth

and so on to infinity is equal to pi squared over six.

That's still quite a lot of vodka, but here goes.

Euler would certainly be a hard act to follow.

Mathematicians from two countries would try.

Both France and Germany were caught up in the age of revolution

that was sweeping Europe in the late 18th century.

Both desperately needed mathematicians.

But they went about supporting mathematics rather differently.

Here in France,

the Revolution emphasised the usefulness of mathematics.

Napoleon recognised that if you were going to have

the best military machine, the best weaponry,

then you needed the best mathematicians.

Napoleon's reforms gave mathematics a big boost.

But this was a mathematics that was going to serve society.

Here in the German states, the great educationalist Wilhelm von Humboldt

was also committed to mathematics, but a mathematics that was detached

from the demands of the State and the military.

Von Humboldt's educational reforms valued mathematics for its own sake.

In France, they got wonderful mathematicians, like Joseph Fourier,

whose work on sound waves we still benefit from today.

MP3 technology is based on Fourier analysis.

But in Germany, they got, at least in my opinion,

the greatest mathematician ever.

Quaint and quiet,

the university town of Gottingen may seem like a bit of a backwater.

But this little town has been home to some of the giants of maths,

including the man who's often described

as the Prince of Mathematics, Carl Friedrich Gauss.

Few non-mathematicians, however, seem to know anything about him.

Not in Paris.

Qui s'appelle Carl Friedrich Gauss?

-Non.

-Non?

'Not in Oxford.'

-I've heard the name but I couldn't tell you.

-No idea.

-No idea?

-No.

'And I'm afraid to say, not even in modern Germany.'

-Nein.

-Nein? OK.

-I don't know.

-You don't know?

But in Gottingen, everyone knows who Gauss is.

He's the local hero.

His father was a stonemason

and it's likely that Gauss would have become one, too.

But his singular talent was recognised by his mother,

and she helped ensure

that he received the best possible education.

Every few years in the news, you hear about a new prodigy

who's passed their A-levels at ten, gone to university at 12,

but nobody compares to Gauss.

Already at the age of 12, he was criticising Euclid's geometry.

At 15, he discovered a new pattern in prime numbers

which had eluded mathematicians for 2,000 years.

And at 19, he discovered the construction of a 17-sided figure

which nobody had known before this time.

His early successes encouraged Gauss to keep a diary.

Here at the University of Gottingen,

you can still read it if you can understand Latin.

Fortunately, I had help.

The first entry is in 1796.

-Is it possible to lift it up?

-Yes, but be careful.

It's really one of the most valuable things that this library possesses.

-Yes, I can believe that.

-He writes beautifully.

It is aesthetically very pleasing,

even if people don't understand what it is.

I'm going to put this down. It's very delicate.

The diary proves that some of Gauss' ideas

were 100 years ahead of their time.

Here are some sines and integrals. Very different sort of mathematics.

Yes, this was the first intimations of the theory

of elliptic functions, which was one of his other great developments.

And here you see something that is basically

the Riemann zeta function appearing.

Wow, gosh! That's very impressive.

The zeta function has become a vital element in our present understanding

of the distribution of the building blocks of all numbers, the primes.

There is somewhere in the diary here where he says,

"I have made this wonderful discovery

"and incidentally, a son was born today."

We see his priorities!

Yes, indeed!

I think I know a few mathematicians like that, too.

My priorities, though, for the rest of the afternoon were clear.

I needed another walk.

Fortunately, Gottingen is surrounded by good woodland trails.

It was a perfect setting for me

to think more about Gauss' discoveries.

Gauss' mathematics has touched many parts of the mathematical world,

but I'm going to just choose one of them, a fun one - imaginary numbers.

In the 16th and 17th century, European mathematicians

imagined the square root of minus one and gave it the symbol i.

They didn't like it much, but it solved equations

that couldn't be solved any other way.

Imaginary numbers have helped us to understand radio waves,

to build bridges and aeroplanes.

They're even the key to quantum physics,

the science of the sub-atomic world.

They've provided a map to see how things really are.

But back in the early 19th century, they had no map, no picture

of how imaginary numbers connected with real numbers.

Where is this new number?

There's no room on the number line for the square root of minus one.

I've got the positive numbers running out here,

the negative numbers here.

The great step is to create a new direction of numbers,

perpendicular to the number line,

and that's where the square root of minus one is.

Gauss was not the first to come up with this two-dimensional picture

of numbers, but he was the first person to explain it all clearly.

He gave people a picture to understand

how imaginary numbers worked.

And once they'd developed this picture,

their immense potential could really be unleashed.

Guten Morgen. Ein Kaffee, bitte.

His maths led to a claim and financial security for Gauss.

He could have gone anywhere, but he was happy enough

to settle down and spend the rest of his life in sleepy Gottingen.

Unfortunately, as his fame developed,

so his character deteriorated.

A naturally conservative, shy man,

he became increasingly distrustful and grumpy.

Many young mathematicians across Europe regarded Gauss as a god

and they would send in their theorems,

their conjectures, even some proofs.

But most of the time, he wouldn't respond, and even when he did,

it was generally to say either that they'd got it wrong

or he'd proved it already.

His dismissal or lack of interest in the work of lesser mortals

sometimes discouraged some very talented mathematicians

from pursuing their ideas.

But occasionally, Gauss also failed

to follow up on his own insights, including one very important insight

that might have transformed the mathematics of his time.

15 kilometres outside Gottingen stands what is known today

as the Gauss Tower.

Wow, that is stunning.

It is really a fantastic view here, yes.

Gauss took on many projects for the Hanoverian government,

including the first proper survey of all the lands of Hanover.

Was this Gauss' choice to do this surveying?

For a mathematician, it sounds like the last thing I'd want to do.

He wanted to do it.

The major point in doing this was to discover the shape of the earth.

But he also started speculating

about something even more revolutionary - the shape of space.

So he's thinking there may not be anything flat in the universe?

Yes. And if we were living in a curved universe,

there wouldn't be anything flat.

This led Gauss to question one of the central tenets of mathematics -

Euclid's geometry.

He realised that this geometry, far from universal,

depended on the idea of space as flat.

It just didn't apply to a universe that was curved.

But in the early 19th century, Euclid's geometry

was seen as God-given and Gauss didn't want any trouble.

So he never published anything.

Another mathematician, though, had no such fears.

In mathematics, it's often helpful to be part of a community

where you can talk to and bounce ideas off others.

But inside such a mathematical community,

it can sometimes be difficult to come up with that one idea

that completely challenges the status quo,

and then the breakthrough often comes from somewhere else.

Mathematics can be done in some pretty weird places.

I'm in Transylvania,

which is fairly appropriate, cos I'm in search of a lone wolf.

Janos Bolyai spent much of his life

hundreds of miles away from the mathematical centres of excellence.

This is the only portrait of him that I was able to find.

Unfortunately, it isn't actually him.

It's one that the Communist Party in Romania started circulating

when people got interested in his theories in the 1960s.

They couldn't find a picture of Janos.

So they substituted a picture of somebody else instead.

Born in 1802, Janos was the son of Farkas Bolyai,

who was a maths teacher.

He realised his son was a mathematical prodigy,

so he wrote to his old friend Carl Friedrich Gauss,

asking him to tutor the boy.

Sadly, Gauss declined.

So instead of becoming a professional mathematician,

Janos joined the Army.

But mathematics remained his first love.

Maybe there's something about the air here because Bolyai carried on

doing his mathematics in his spare time.

He started to explore what he called imaginary geometries,

where the angles in triangles add up to less than 180.

The amazing thing is that these imaginary geometries

make perfect mathematical sense.

Bolyai's new geometry has become known as hyperbolic geometry.

The best way to imagine it is a kind of mirror image of a sphere

where lines curve back on each other.

It's difficult to represent it since we are so used

to living in space which appears to be straight and flat.

In his hometown of Targu Mures,

I went looking for more about Bolyai's revolutionary mathematics.

His memory is certainly revered here.

The museum contains a collection of Bolyai-related artefacts,

some of which might be considered distinctly Transylvanian.

It's still got some hair on it.

It's kind of a little bit gruesome.

But the object I like most here

is a beautiful model of Bolyai's geometry.

You got the shortest distance between here and here

if you stick on this surface. It's not a straight line,

but this curved line which of bends into the triangle.

Here is a surface where the shortest distances which define the triangle

add up to less than 180.

Bolyai published his work in 1831.

His father sent his old friend Gauss a copy.

Gauss wrote back straightaway giving his approval,

but Gauss refused to praise the young Bolyai,

because he said the person he should be praising was himself.

He had worked it all out a decade or so before.

Actually, there is a letter from Gauss

to another friend of his where he says,

"I regard this young geometer boy

"as a genius of the first order."

But Gauss never thought to tell Bolyai that.

And young Janos was completely disheartened.

Another body blow soon followed.

Somebody else had developed exactly the same idea,

but had published two years before him -

the Russian mathematician Nicholas Lobachevsky.

It was all downhill for Bolyai after that.

With no recognition or career, he didn't publish anything else.

Eventually, he went a little crazy.

In 1860, Janos Bolyai died in obscurity.

Gauss, by contrast, was lionised after his death.

A university, the units used to measure magnetic induction,

even a crater on the moon would be named after him.

During his lifetime, Gauss lent his support

to very few mathematicians.

But one exception was another of Gottingen's mathematical giants -

Bernhard Riemann.

His father was a minister

and he would remain a sincere Christian all his life.

But Riemann grew up a shy boy who suffered from consumption.

His family was large and poor and the only thing

the young boy had going for him was an excellence at maths.

That was his salvation.

Many mathematicians like Riemann had very difficult childhoods,

were quite unsociable. Their lives seemed to be falling apart.

It was mathematics that gave them a sense of security.

Riemann spent much of his early life in the town of Luneburg

in northern Germany.

This was his local school, built as a direct result

of Humboldt's educational reforms in the early 19th century.

Riemann was one of its first pupils.

The head teacher saw a way of bringing out the shy boy.

He was given the freedom of the school's library.

It opened up a whole new world to him.

One of the books he found in there

was a book by the French mathematician Legendre,

all about number theory.

His teacher asked him how he was getting on with it.

He replied, "I have understood all 859 pages of this wonderful book."

It was a strategy that obviously suited Riemann

because he became a brilliant mathematician.

One of his most famous contributions to mathematics was a lecture in 1852

on the foundations of geometry. In the lecture,

Riemann first described what geometry actually was

and its relationship with the world.

He then sketched out what geometry could be -

a mathematics of many different kinds of space,

only one of which would be the flat Euclidian space

in which we appear to live.

He was just 26 years old.

Was it received well? Did people recognise the revolution?

There was no way that people could actually

make these ideas concrete.

That only occurred 50, 60 years after this, with Einstein.

So this is the beginning, really, of the revolution

-which ends with Einstein's relativity.

-Exactly.

Riemann's mathematics changed how we see the world.

Suddenly, higher dimensional geometry appeared.

The potential was there from Descartes,

but it was Riemann's imagination that made it happen.

He began without putting any restriction

on the dimensions whatsoever. This was something quite new,

his way of thinking about things.

Someone like Bolyai was really thinking about new geometries,

but new two-dimensional geometries.

New two-dimensional geometries. Riemann then broke away

from all the limitations of two or three dimensions

and began to think in in higher dimensions.

And this was quite new.

Multi-dimensional space is at the heart

of so much mathematics done today.

In geometry, number theory, and several other branches of maths,

Riemann's ideas still perplex and amaze.

He died, though, in 1866.

He was only 39 years old.

Today, the results of Riemann's mathematics are everywhere.

Hyperspace is no longer science fiction, but science fact.

In Paris, they have even tried to visualise what shapes

in higher dimensions might look like.

Just as the Renaissance artist Piero would have drawn a square

inside a square to represent a cube on the two-dimensional canvas,

the architect here at La Defense has built a cube inside a cube

to represent a shadow of the four-dimensional hypercube.

It is with Riemann's work that we finally have

the mathematical glasses to be able to explore

such worlds of the mind.

It's taken a while to make these glasses fit,

but without this golden age of mathematics,

from Descartes to Riemann, there would be no calculus,

no quantum physics, no relativity, none of the technology we use today.

But even more important than that,

their mathematics blew away the cobwebs

and allowed us to see the world as it really is -

a world much stranger than we ever thought.

You can learn more about the story of maths

at the Open University at:

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