The Frontiers of Space The Story of Maths


The Frontiers of Space

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I'm walking in the mountains of the moon.

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

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

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There it is, Urbino.

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

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masterpieces of art, but also masterpieces of mathematics.

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The artists and architects of the early Renaissance brought back

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

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but using it properly turned out to be a lot

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more difficult than they'd imagined.

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Piero was the first major painter to fully understand perspective.

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

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I came here to see his masterpiece,

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The Flagellation of Christ, but there was a problem.

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

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absolutely stunning picture, but unfortunately, for various

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

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

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

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

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The problem of perspective is how

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

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

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Piero used mathematics.

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How big is he going to paint Christ,

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if this group of men here were a certain distance away

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from these men in the foreground?

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Get it wrong and the illusion of perspective is shattered.

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

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can be accurately represented on a two-dimensional surface.

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

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

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at a vanishing point.

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

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What is emerging here is a new

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

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

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a perspective that would cause a mathematical revolution.

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

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but it would take another 200 years

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before other mathematicians would continue where he left off.

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Our journey has come north.

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By the 17th century, Europe had taken over

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

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Great strides had been made in the geometry

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of objects fixed in time and space.

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In France, Germany, Holland and Britain,

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

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and the pursuit of this new mathematics started here in this

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village in the centre of France.

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Only the French would name a village after a mathematician.

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Imagine in England a town called

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

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But in France, they really value their mathematicians.

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This is the village of Descartes in the Loire Valley.

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It was renamed after the famous philosopher

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and mathematician 200 years ago.

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

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

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

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To do mathematics, sometimes you just need to remove

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

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Descartes thought that the bed was the best place to achieve

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this meditative state.

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I think I know what he means.

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The house where Descartes undertook his bedtime meditations

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is now a museum dedicated to all things Cartesian.

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Come with me.

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

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his philosophical, scientific and mathematical ideas all fit together.

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It also features less familiar aspects

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of Descartes' life and career.

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

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in the Protestant Army

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

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because no patriotism.

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Sylvie is putting it very nicely,

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but Descartes was in fact a mercenary.

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He fought for the German Protestants, the French Catholics

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and anyone else who would pay him.

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

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camped out on a cold river bank.

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Inspiration very often strikes in very strange places.

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

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maybe because he was getting up so late

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or perhaps he was celebrating St Martin's Eve

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and had just drunk too much.

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Problems were tumbling around in his mind.

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He was thinking about his favourite subject, philosophy.

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He was finding it very frustrating.

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How can you actually know anything at all?!

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Then he slips into a dream...

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

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on the indisputable facts of mathematics.

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

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

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

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Descartes found a home here in Holland.

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

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

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an opinion that got scientists like Galileo

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into deep trouble with the Vatican.

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Descartes reckoned that here amongst the Protestant Dutch

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he would be safe, especially

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at the old university town of Leiden

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where they valued maths and science.

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I've come to Leiden too.

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Unfortunately, I'm late!

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Hello. Yeah, I'm sorry.

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

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

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

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

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

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He merged algebra and geometry.

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-Right.

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

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-So a sort of dictionary between the two?

-Yeah, yeah.

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

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included mainly controversial

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philosophical ideas, but the most radical thoughts

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

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

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

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vertical location.

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

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

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of these numbers at any point in the figure.

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Suddenly, geometry has turned into algebra.

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Using this transformation

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

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

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You didn't need to use your eyes.

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

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but it wouldn't stop there.

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Descartes had unlocked the possibility of navigating geometries

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

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to modern technology and physics.

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

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

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

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

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his image. He was entirely

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

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

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Descartes may not have been the most congenial person,

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

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between algebra and geometry transformed mathematics forever.

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

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vital ingredient.

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

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CHORAL SINGING

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

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that many mathematicians from the time of Descartes

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had strong religious convictions.

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Maybe it's just a coincidence,

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

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

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

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In the 17th century,

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

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He loved mathematics as much as he loved God.

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

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Marin Mersenne was a first-class mathematician.

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

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But he's also celebrated for his correspondence.

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

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

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It's not so different now.

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We sit like mathematical monks thinking about our ideas, then

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

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

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which had not been seen since the Greeks.

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

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He also did something just as important.

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He publicised some new findings on the properties of numbers

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

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greatest mathematician of his time, Pierre de Fermat.

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Here in Beaumont-de-Lomagne

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near Toulouse, residents and visitors have come

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

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

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

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The man himself would have hardly approved of

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

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

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worthless or common to create a festival of mathematics.

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Maths in action, that one.

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It's beautiful, really nice, yeah.

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Fermat's greatest contribution to mathematics was to virtually invent

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modern number theory.

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He devised a wide range of conjectures

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and theorems about numbers including his famous Last Theorem,

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

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but it's little help to me now.

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Getting it apart is the easy bit.

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

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

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

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

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

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

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That's very unfair!

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Here we go.

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Can I put them together?

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I got it!

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Now that's the buzz of doing mathematics when

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

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Remarkably, Fermat only tackled mathematics in his spare time.

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By day he was a magistrate.

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Battling with mathematical problems was his hobby and his passion.

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The wonderful thing about mathematics is

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you can do it anywhere.

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You don't have to have a laboratory.

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You don't even really need a library.

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

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

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He may have looked like an amateur,

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but he took his mathematics very seriously indeed.

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Fermat managed to find several new patterns in numbers

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that had defeated mathematicians for centuries.

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One of my favourite theorems of Fermat

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is all to do with prime numbers.

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

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leaves remainder one, then Fermat showed you could

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always rewrite this number as two square numbers added together.

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

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

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

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together, so 13 can be rewritten

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

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

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

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you can always rewrite that number

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as two square numbers added together.

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Ah, my God!

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

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and Fermat certainly enjoyed playing around with numbers. He loved

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

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

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

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Fermat's mathematics would have some very serious applications.

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One of his theorems, his Little Theorem, is

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

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Technology we now rely on today all comes from the scribblings

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of a 17th-century mathematician.

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

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

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but from its great rival.

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

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Its expansion and ambitions required new methods of measurement

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and computation and that gave a great boost to mathematics.

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The university towns of Oxford and Cambridge

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were churning out mathematicians who were in great demand

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and the greatest of them was Isaac Newton.

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

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and they're very proud of him here.

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They have a wonderful statue to him.

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They've even got

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

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

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

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one of the other famous residents here, Margaret Thatcher,

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has got as big a display as Isaac Newton.

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In fact, the Thatcher cups have

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

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

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And Newton's maths does need support.

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

-No.

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-No, I don't.

-Discovering gravity.

-Gravity?

-Gravity, yes.

-Gravity?

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-Apple tree and all that, gravity.

-'That pretty much summed it up.

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

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

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-I'm in a rush!

-You're in a rush. OK.

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

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Eight miles south of Grantham,

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in the village of Woolsthorpe, where Newton was born,

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

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This is the house.

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

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

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So here is the...

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

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

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

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Yes! Gosh, I wish it was there now!

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His father was an illiterate farmer,

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but he died shortly before Newton was born.

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

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And here's his room.

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Oh, lovely, wow.

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-They present it really nicely.

-Yes.

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

-It does, yes.

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

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

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

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he became a mathematician rather than a sheep farmer.

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

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-OK.

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

-Yes, yes.

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I think he had a sort of average school report.

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

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I particularly would have wanted to meet,

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

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Newton came back to Lincolnshire from Cambridge

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

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

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discovered gravitation

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

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It works like this.

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

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

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but this is only an average speed.

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How can I tell precisely what my speed is

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

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

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

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the car to get to that point.

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I can calculate the average speed between

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

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

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

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

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and smaller window of time and the speed gets closer

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

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I have to calculate 0 divided by 0.

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The calculus allows us to make sense of this calculation.

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

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travelled at any moment in time.

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

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

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Its distance is changing and its

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

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Which is quite in contrast to the Greeks.

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It was a very static geometry.

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-Yes, it is.

-And here we see...

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so the calculus is used by

0:20:270:20:29

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

0:20:290:20:33

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

0:20:330:20:36

motion or with change.

0:20:360:20:38

There's a lot of mathematics in this apple!

0:20:380:20:40

Newton's calculus enables us to really understand

0:20:420:20:46

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

0:20:460:20:50

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

0:20:500:20:54

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

0:20:540:20:58

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

0:21:040:21:09

Newton himself decided not to publish, but just to circulate

0:21:090:21:12

his thoughts among friends.

0:21:120:21:14

His reputation, though, gradually spread.

0:21:140:21:17

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

0:21:170:21:21

here in the City of London.

0:21:210:21:23

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

0:21:250:21:28

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

0:21:280:21:33

Developing the calculus just got crowded out

0:21:330:21:35

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

0:21:350:21:39

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

0:21:410:21:46

with exactly the same idea as him,

0:21:460:21:48

Gottfried Leibniz.

0:21:480:21:50

Every word Leibniz wrote has been preserved and catalogued

0:21:500:21:54

in his hometown of Hanover in northern Germany.

0:21:540:21:57

His actual manuscripts are kept under lock and key,

0:21:570:22:01

particularly the manuscript which shows how Leibniz

0:22:010:22:04

also discovered the miracle of calculus, shortly after Newton.

0:22:040:22:09

What age was he when he wrote...

0:22:090:22:11

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

0:22:110:22:16

-differential calculus and integral calculus.

-In two months?

0:22:160:22:19

-Yeah.

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

0:22:190:22:21

Yeah.

0:22:210:22:23

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

0:22:230:22:26

-A letter or...

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

0:22:260:22:29

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

0:22:320:22:37

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

0:22:370:22:40

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

0:22:400:22:45

I suppose, that's beautiful.

0:22:450:22:47

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

0:22:470:22:51

-A true mathematician.

-Yeah.

0:22:510:22:53

He spends his time thinking in bed.

0:22:530:22:55

I see you've got some paintings down here.

0:22:550:22:58

A painting.

0:22:580:23:00

This is what he looked like. Right.

0:23:000:23:02

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

0:23:030:23:07

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

0:23:070:23:10

Leibniz worked for the Royal Family

0:23:100:23:12

of Hanover and travelled around Europe representing their interests.

0:23:120:23:16

This gave him plenty of time to indulge in

0:23:160:23:19

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

0:23:190:23:23

He devised a plan for reunifying the Protestant and Roman Catholic

0:23:230:23:26

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

0:23:260:23:32

philosophy and logic which are still highly rated today.

0:23:320:23:36

-He wrote all these letters?

-Yeah.

-That's absolutely extraordinary.

0:23:360:23:39

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

0:23:390:23:43

'But Leibniz was not just man of words.

0:23:430:23:46

'He was also one of the first people

0:23:460:23:47

'to invent practical calculating machines

0:23:470:23:49

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

0:23:490:23:54

'300 years later, the engineering department at Leibniz University

0:23:540:23:58

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

0:23:580:24:02

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

0:24:020:24:04

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

0:24:040:24:06

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

0:24:060:24:10

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

-Yeah.

0:24:100:24:15

-And now I give the number one.

-OK.

0:24:150:24:18

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

0:24:180:24:24

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

-Would you show this here?

0:24:240:24:28

This is the money shot.

0:24:280:24:30

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

0:24:300:24:33

So this 128 is two power eight.

0:24:330:24:36

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

0:24:360:24:42

all the ball bearings here.

0:24:420:24:44

To add one it all gets

0:24:440:24:46

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

0:24:460:24:50

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

0:24:500:24:53

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

-Exactly.

0:24:530:24:56

I love this machine!

0:24:560:24:58

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

0:25:030:25:08

the famous gardens of Herrenhausen,

0:25:080:25:10

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

0:25:100:25:14

There's something about mathematics and walking.

0:25:140:25:17

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

0:25:170:25:21

on some problem and your head's all

0:25:210:25:22

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

0:25:220:25:25

You let your subconscious mind kind of take over and sometimes

0:25:250:25:27

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

0:25:270:25:31

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

0:25:310:25:35

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

0:25:350:25:39

local stomping ground.

0:25:390:25:40

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

0:25:440:25:47

because in the years since Leibniz lived here,

0:25:470:25:49

someone has built a maze.

0:25:490:25:50

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

0:25:500:25:53

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

0:25:530:25:57

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

0:25:570:26:00

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

0:26:000:26:03

Leibniz had no such distractions.

0:26:110:26:13

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

0:26:130:26:17

seemingly independent from Newton,

0:26:170:26:19

although he knew about Newton's work,

0:26:190:26:21

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

0:26:210:26:26

and so mathematicians across Europe heard about the calculus first

0:26:260:26:29

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

0:26:290:26:35

Throughout mathematical history, there have been lots of priority

0:26:350:26:39

disputes and arguments.

0:26:390:26:40

It may seem a little bit petty and schoolboyish.

0:26:400:26:43

We really want our name to be on that theorem.

0:26:430:26:46

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

0:26:460:26:49

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

0:26:490:26:54

to trying to crack these things.

0:26:540:26:55

Somehow we can't believe that somebody else

0:26:550:26:57

has got it at the same time as us.

0:26:570:27:00

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

0:27:000:27:03

don't want to share the credit.

0:27:030:27:06

Back in London, Newton certainly didn't want

0:27:060:27:08

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

0:27:080:27:13

After years of acrimony and accusation, the Royal Society

0:27:130:27:16

in London was asked to adjudicate between the rival claims.

0:27:160:27:21

The Royal Society gave Newton credit

0:27:210:27:23

for the first discovery of the calculus

0:27:230:27:25

and Leibniz credit for the first publication,

0:27:250:27:28

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

0:27:280:27:33

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

0:27:330:27:36

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

0:27:360:27:41

Leibniz was incredibly hurt.

0:27:440:27:46

He admired Newton and never really recovered.

0:27:460:27:50

He died in 1716.

0:27:500:27:52

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

0:27:520:27:56

Westminster Abbey.

0:27:560:27:58

Leibniz's memorial, by contrast,

0:27:580:28:00

is here in this small church in Hanover.

0:28:000:28:02

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

0:28:020:28:06

eventually triumphs, not Newton's.

0:28:060:28:08

I'm a big Leibniz fan.

0:28:110:28:13

Quite often revolutions in mathematics are about producing the

0:28:130:28:16

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

0:28:160:28:19

Leibniz was so good at.

0:28:190:28:21

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

0:28:210:28:25

captured its true spirit.

0:28:250:28:27

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

0:28:270:28:29

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

0:28:290:28:34

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

0:28:340:28:38

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

0:28:380:28:43

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

0:28:480:28:52

Switzerland was the commercial hub of the entire Western world.

0:28:520:28:56

Around this maelstrom of trade, there developed a tradition of

0:28:560:28:59

learning, particularly learning which connected with commerce

0:28:590:29:03

and one family summed all this up.

0:29:030:29:06

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

0:29:060:29:11

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

0:29:110:29:15

our children don't tend to be mathematicians.

0:29:150:29:17

I'm not sure why it is.

0:29:170:29:19

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

0:29:190:29:23

What no-one disagrees with

0:29:230:29:25

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

0:29:250:29:30

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

0:29:300:29:33

outstanding mathematicians, any of which we would have been

0:29:330:29:37

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

0:29:370:29:41

You might have great minds like Newton and Leibniz who make

0:29:410:29:44

these fundamental breakthroughs, but you also need the disciples

0:29:440:29:48

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

0:29:480:29:51

then spread it wide. The family were originally merchants,

0:29:510:29:55

and this is one of their houses.

0:29:550:29:57

It's now part of the University of Basel

0:29:570:30:00

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

0:30:000:30:03

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

0:30:030:30:07

Dr Fritz Nagel, keeper of the Bernoulli Archive,

0:30:070:30:09

has promised to show it to me.

0:30:090:30:12

-If we can find it.

-No, we're on the wrong floor.

0:30:120:30:15

Wrong floor, OK. Right!

0:30:150:30:17

Oh, look.

0:30:170:30:19

Can we take an apple?

0:30:190:30:21

'No, wrong mathematician.

0:30:210:30:24

'Eventually, we got there.'

0:30:240:30:26

This is where the Bernoullis would have done

0:30:260:30:28

some of their mathematics.

0:30:280:30:30

'I was really just being polite.

0:30:300:30:33

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

0:30:330:30:36

Now, of the Bernoullis, which is your favourite?

0:30:360:30:40

My favourite Bernoulli is Johann I.

0:30:400:30:44

He is the most smart mathematician.

0:30:440:30:49

Perhaps his brother Jakob was the mathematician

0:30:490:30:54

with the deeper insight into problems,

0:30:540:30:57

but Johann found elegant solutions.

0:30:570:30:59

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

0:30:590:31:03

They corresponded with him, stood up for him

0:31:030:31:06

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

0:31:060:31:10

Leibnitz was very happy to have found two gifted mathematicians

0:31:100:31:15

outside of his personal circle of friends who mastered his calculus

0:31:150:31:20

and could distribute it in the scientific community.

0:31:200:31:23

-That was very important for Leibniz.

-And important for maths, too.

0:31:230:31:28

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

0:31:280:31:32

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

0:31:320:31:36

At least, that is Dr Nagel's contention.

0:31:360:31:38

And he is a great Bernoulli fan.

0:31:380:31:41

He has arranged for me to meet Professor Daniel Bernoulli,

0:31:410:31:44

the latest member of the family,

0:31:440:31:46

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

0:31:460:31:49

Another one of which I got was,

0:31:490:31:51

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

0:31:510:31:54

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

0:31:540:31:58

'But this Daniel Bernoulli is a professor of geology.

0:31:580:32:02

'The maths gene seems to have truly died out.

0:32:020:32:05

'And during our very hearty dinner,

0:32:050:32:07

'I found myself wandering back to maths.'

0:32:070:32:11

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

0:32:110:32:14

as disciples of Leibniz.

0:32:140:32:16

One of their many great contributions to mathematics

0:32:160:32:18

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

0:32:180:32:23

Imagine a ball rolling down a ramp.

0:32:230:32:26

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

0:32:260:32:29

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

0:32:290:32:32

You might think that a straight ramp would be quickest.

0:32:320:32:36

Or possibly a curved one like this

0:32:360:32:37

that gives the ball plenty of downward momentum.

0:32:370:32:40

In fact, it's neither of these.

0:32:400:32:42

Calculus shows that it is what we call a cycloid,

0:32:420:32:45

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

0:32:450:32:49

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

0:32:490:32:53

as the calculus of variation,

0:32:530:32:55

has become one of the most powerful aspects of the mathematics

0:32:550:32:58

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

0:32:580:33:01

Engineers exploit it to minimise energy use.

0:33:010:33:05

Designers apply it to optimise construction.

0:33:050:33:08

It has now become one of the linchpins

0:33:080:33:10

of our modern technological world.

0:33:100:33:12

Meanwhile, things were getting more interesting in the restaurant.

0:33:120:33:17

Here is my second surprise.

0:33:170:33:18

Let me introduce Mr Leonhard Euler.

0:33:180:33:22

Daniel Bernoulli.

0:33:220:33:23

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

0:33:230:33:27

'This Leonhard is a descendant

0:33:270:33:29

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

0:33:290:33:34

I am the ninth generation,

0:33:340:33:36

the fourth Leonhard in our family

0:33:360:33:39

after Leonard Euler I, the mathematician.

0:33:390:33:42

OK. And yourself, are you a mathematician?

0:33:420:33:44

Actually, I am a business analyst.

0:33:440:33:47

I can't study mathematics with my name.

0:33:470:33:51

Everyone will expect you to prove that the Riemann hypothesis!

0:33:510:33:55

Perhaps it's just as well that Leonhard decided

0:33:550:33:58

not to follow in the footsteps of his illustrious ancestor.

0:33:580:34:02

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

0:34:020:34:04

I am going in a boat across the Rhine,

0:34:130:34:15

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

0:34:150:34:17

Last night's dinner with Mr Euler and Professor Bernoulli

0:34:170:34:21

degenerated into toasts to all the theorems the Bernoullis and Eulers

0:34:210:34:25

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

0:34:250:34:28

Never again.

0:34:280:34:30

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

0:34:300:34:34

Luckily, it was only a short trip.

0:34:340:34:37

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

0:34:370:34:41

Euler may have been the prodigy of Johann Bernoulli,

0:34:410:34:45

but there was no room for him in the city.

0:34:450:34:47

If your name wasn't Bernoulli,

0:34:470:34:49

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

0:34:490:34:53

But Daniel, the son of Johann Bernoulli,

0:34:530:34:55

was a great friend of Euler

0:34:550:34:57

and managed to get him a job at his university.

0:34:570:35:00

But to get there would take seven weeks,

0:35:000:35:03

because Daniel's university was in Russia.

0:35:030:35:05

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

0:35:080:35:11

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

0:35:110:35:17

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

0:35:170:35:21

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

0:35:210:35:26

Peter's Academy is now a museum.

0:35:270:35:30

It includes several rooms full of the kind of grotesque curiosities

0:35:300:35:34

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

0:35:340:35:38

But in the 1730s,

0:35:380:35:39

this building was a centre for ground-breaking research.

0:35:390:35:44

It is where Euler found his intellectual home.

0:35:440:35:46

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

0:35:500:35:57

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

0:35:580:36:00

calculus of variation,

0:36:000:36:02

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

0:36:020:36:06

But he was also creating incredibly modern mathematics,

0:36:060:36:09

topology and analysis.

0:36:090:36:12

Much of the notation that I use today as a mathematician

0:36:120:36:15

was created by Euler, numbers like e and i.

0:36:150:36:19

Euler also popularised the use of the symbol pi.

0:36:190:36:23

He even combined these numbers together

0:36:230:36:25

in one of the most beautiful formulas of mathematics,

0:36:250:36:28

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

0:36:280:36:32

An amazing feat of mathematical alchemy.

0:36:320:36:36

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

0:36:360:36:39

Euler applied his skills to an immense range of topics,

0:36:390:36:43

from prime numbers to optics to astronomy.

0:36:430:36:46

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

0:36:460:36:49

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

0:36:490:36:54

I think of him as the Mozart of maths.

0:36:590:37:01

And that view is shared by the mathematician Nikolai Vavilov,

0:37:010:37:04

who met me at the house that was given to Euler

0:37:040:37:07

by Catherine the Great.

0:37:070:37:10

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

0:37:100:37:14

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

0:37:140:37:17

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

0:37:170:37:22

and their greatest mathematician.

0:37:220:37:24

That is exactly what it says.

0:37:240:37:27

-What is it now?

-It is a school.

0:37:270:37:29

Shall we go in and see?

0:37:290:37:30

OK.

0:37:300:37:33

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

0:37:330:37:38

Perhaps we should talk to the head teacher.

0:37:380:37:41

The head didn't mind at all.

0:37:460:37:48

I rather got the impression that she was used

0:37:480:37:50

to people dropping in to talk about Euler.

0:37:500:37:53

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

0:37:530:37:57

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

0:37:570:38:02

and about this very building.

0:38:020:38:04

They certainly knew their stuff.

0:38:040:38:06

They had undertaken an entire classroom project on Euler,

0:38:060:38:09

his long life, happy marriage and 13 children.

0:38:090:38:13

And then his tragedies - only five of his children

0:38:130:38:16

survived to adulthood.

0:38:160:38:17

His first wife, who he adored, died young.

0:38:170:38:21

He started losing most of his eyesight.

0:38:210:38:23

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

0:38:260:38:31

He continued his mathematical research.

0:38:310:38:34

I read a quote that said now with his blindness,

0:38:340:38:36

he hasn't got any distractions,

0:38:360:38:38

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

0:38:380:38:42

It was a totally unexpected and charming visit.

0:38:420:38:46

Although I couldn't resist sneaking back and correcting

0:38:460:38:49

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

0:38:490:38:53

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

0:38:540:38:59

It concerns calculating infinite sums,

0:38:590:39:02

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

0:39:020:39:06

when it was announced in 1735.

0:39:060:39:08

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

0:39:110:39:15

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

0:39:170:39:22

and add it to the first glass.

0:39:220:39:24

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

0:39:250:39:30

and add that one.

0:39:300:39:31

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

0:39:310:39:36

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

0:39:360:39:43

It was called the Basel problem

0:39:430:39:45

after the Bernoullis tried and failed to solve it.

0:39:450:39:47

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

0:39:470:39:52

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

0:39:520:39:57

But then Euler came along.

0:39:570:39:59

Daniel was close, but mathematics is about precision.

0:39:590:40:03

Euler calculated that the total height of the vodka

0:40:030:40:06

would be exactly pi squared divided by six.

0:40:060:40:10

It was a complete surprise.

0:40:130:40:15

What on earth did adding squares of fractions

0:40:150:40:17

have to do with the special number pi?

0:40:170:40:20

But Euler's analysis showed that they were two sides

0:40:200:40:23

of the same equation.

0:40:230:40:25

One plus a quarter plus a ninth plus a sixteenth

0:40:250:40:29

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

0:40:290:40:34

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

0:40:340:40:38

Euler would certainly be a hard act to follow.

0:40:430:40:46

Mathematicians from two countries would try.

0:40:460:40:49

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

0:40:490:40:53

that was sweeping Europe in the late 18th century.

0:40:530:40:56

Both desperately needed mathematicians.

0:40:560:40:59

But they went about supporting mathematics rather differently.

0:40:590:41:04

Here in France,

0:41:040:41:05

the Revolution emphasised the usefulness of mathematics.

0:41:050:41:09

Napoleon recognised that if you were going to have

0:41:090:41:12

the best military machine, the best weaponry,

0:41:120:41:14

then you needed the best mathematicians.

0:41:140:41:17

Napoleon's reforms gave mathematics a big boost.

0:41:170:41:21

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

0:41:210:41:24

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

0:41:250:41:30

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

0:41:300:41:33

from the demands of the State and the military.

0:41:330:41:36

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

0:41:360:41:42

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

0:41:420:41:46

whose work on sound waves we still benefit from today.

0:41:460:41:49

MP3 technology is based on Fourier analysis.

0:41:490:41:53

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

0:41:530:41:56

the greatest mathematician ever.

0:41:560:41:58

Quaint and quiet,

0:42:010:42:03

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

0:42:030:42:08

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

0:42:080:42:12

including the man who's often described

0:42:120:42:14

as the Prince of Mathematics, Carl Friedrich Gauss.

0:42:140:42:19

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

0:42:190:42:23

Not in Paris.

0:42:230:42:25

Qui s'appelle Carl Friedrich Gauss?

0:42:250:42:27

-Non.

-Non?

0:42:270:42:28

'Not in Oxford.'

0:42:280:42:30

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

-No idea.

-No idea?

-No.

0:42:300:42:34

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

0:42:340:42:37

-Nein.

-Nein? OK.

0:42:370:42:39

-I don't know.

-You don't know?

0:42:390:42:41

But in Gottingen, everyone knows who Gauss is.

0:42:410:42:44

He's the local hero.

0:42:440:42:47

His father was a stonemason

0:42:470:42:49

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

0:42:490:42:52

But his singular talent was recognised by his mother,

0:42:520:42:55

and she helped ensure

0:42:550:42:57

that he received the best possible education.

0:42:570:43:01

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

0:43:010:43:05

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

0:43:050:43:08

but nobody compares to Gauss.

0:43:080:43:10

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

0:43:100:43:13

At 15, he discovered a new pattern in prime numbers

0:43:130:43:16

which had eluded mathematicians for 2,000 years.

0:43:160:43:20

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

0:43:200:43:24

which nobody had known before this time.

0:43:240:43:26

His early successes encouraged Gauss to keep a diary.

0:43:300:43:34

Here at the University of Gottingen,

0:43:340:43:36

you can still read it if you can understand Latin.

0:43:360:43:40

Fortunately, I had help.

0:43:400:43:42

The first entry is in 1796.

0:43:440:43:46

-Is it possible to lift it up?

-Yes, but be careful.

0:43:460:43:49

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

0:43:490:43:54

-Yes, I can believe that.

-He writes beautifully.

0:43:540:43:56

It is aesthetically very pleasing,

0:43:560:43:59

even if people don't understand what it is.

0:43:590:44:02

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

0:44:020:44:05

The diary proves that some of Gauss' ideas

0:44:050:44:08

were 100 years ahead of their time.

0:44:080:44:10

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

0:44:100:44:15

Yes, this was the first intimations of the theory

0:44:150:44:20

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

0:44:200:44:25

And here you see something that is basically

0:44:250:44:28

the Riemann zeta function appearing.

0:44:280:44:30

Wow, gosh! That's very impressive.

0:44:300:44:34

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

0:44:340:44:38

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

0:44:380:44:43

There is somewhere in the diary here where he says,

0:44:430:44:47

"I have made this wonderful discovery

0:44:470:44:49

"and incidentally, a son was born today."

0:44:490:44:52

We see his priorities!

0:44:520:44:53

Yes, indeed!

0:44:530:44:55

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

0:44:550:44:58

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

0:45:000:45:03

I needed another walk.

0:45:030:45:05

Fortunately, Gottingen is surrounded by good woodland trails.

0:45:050:45:08

It was a perfect setting for me

0:45:080:45:10

to think more about Gauss' discoveries.

0:45:100:45:13

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

0:45:220:45:26

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

0:45:260:45:31

In the 16th and 17th century, European mathematicians

0:45:310:45:34

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

0:45:340:45:40

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

0:45:400:45:42

that couldn't be solved any other way.

0:45:420:45:45

Imaginary numbers have helped us to understand radio waves,

0:45:460:45:49

to build bridges and aeroplanes.

0:45:490:45:52

They're even the key to quantum physics,

0:45:520:45:54

the science of the sub-atomic world.

0:45:540:45:56

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

0:45:560:46:01

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

0:46:010:46:05

of how imaginary numbers connected with real numbers.

0:46:050:46:08

Where is this new number?

0:46:080:46:10

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

0:46:100:46:14

I've got the positive numbers running out here,

0:46:140:46:16

the negative numbers here.

0:46:160:46:17

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

0:46:170:46:21

perpendicular to the number line,

0:46:210:46:23

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

0:46:230:46:26

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

0:46:280:46:32

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

0:46:320:46:36

He gave people a picture to understand

0:46:360:46:38

how imaginary numbers worked.

0:46:380:46:40

And once they'd developed this picture,

0:46:400:46:43

their immense potential could really be unleashed.

0:46:430:46:46

Guten Morgen. Ein Kaffee, bitte.

0:46:460:46:49

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

0:46:490:46:53

He could have gone anywhere, but he was happy enough

0:46:530:46:56

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

0:46:560:47:01

Unfortunately, as his fame developed,

0:47:010:47:03

so his character deteriorated.

0:47:030:47:06

A naturally conservative, shy man,

0:47:060:47:08

he became increasingly distrustful and grumpy.

0:47:080:47:12

Many young mathematicians across Europe regarded Gauss as a god

0:47:120:47:16

and they would send in their theorems,

0:47:160:47:18

their conjectures, even some proofs.

0:47:180:47:20

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

0:47:200:47:23

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

0:47:230:47:26

or he'd proved it already.

0:47:260:47:28

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

0:47:280:47:32

sometimes discouraged some very talented mathematicians

0:47:320:47:35

from pursuing their ideas.

0:47:350:47:38

But occasionally, Gauss also failed

0:47:380:47:40

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

0:47:400:47:45

that might have transformed the mathematics of his time.

0:47:450:47:48

15 kilometres outside Gottingen stands what is known today

0:47:500:47:53

as the Gauss Tower.

0:47:530:47:55

Wow, that is stunning.

0:47:550:47:57

It is really a fantastic view here, yes.

0:47:570:48:01

Gauss took on many projects for the Hanoverian government,

0:48:010:48:05

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

0:48:050:48:09

Was this Gauss' choice to do this surveying?

0:48:090:48:12

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

0:48:120:48:16

He wanted to do it.

0:48:160:48:17

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

0:48:170:48:23

But he also started speculating

0:48:230:48:25

about something even more revolutionary - the shape of space.

0:48:250:48:29

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

0:48:290:48:34

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

0:48:340:48:37

there wouldn't be anything flat.

0:48:370:48:40

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

0:48:400:48:44

Euclid's geometry.

0:48:440:48:47

He realised that this geometry, far from universal,

0:48:470:48:50

depended on the idea of space as flat.

0:48:500:48:52

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

0:48:520:48:56

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

0:48:560:48:59

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

0:48:590:49:03

So he never published anything.

0:49:030:49:05

Another mathematician, though, had no such fears.

0:49:050:49:09

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

0:49:110:49:16

where you can talk to and bounce ideas off others.

0:49:160:49:19

But inside such a mathematical community,

0:49:190:49:22

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

0:49:220:49:25

that completely challenges the status quo,

0:49:250:49:28

and then the breakthrough often comes from somewhere else.

0:49:280:49:33

Mathematics can be done in some pretty weird places.

0:49:330:49:36

I'm in Transylvania,

0:49:360:49:38

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

0:49:380:49:42

Janos Bolyai spent much of his life

0:49:420:49:45

hundreds of miles away from the mathematical centres of excellence.

0:49:450:49:49

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

0:49:490:49:53

Unfortunately, it isn't actually him.

0:49:530:49:56

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

0:49:560:50:00

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

0:50:000:50:04

They couldn't find a picture of Janos.

0:50:040:50:06

So they substituted a picture of somebody else instead.

0:50:060:50:09

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

0:50:110:50:15

who was a maths teacher.

0:50:150:50:17

He realised his son was a mathematical prodigy,

0:50:170:50:20

so he wrote to his old friend Carl Friedrich Gauss,

0:50:200:50:23

asking him to tutor the boy.

0:50:230:50:25

Sadly, Gauss declined.

0:50:250:50:28

So instead of becoming a professional mathematician,

0:50:280:50:31

Janos joined the Army.

0:50:310:50:33

But mathematics remained his first love.

0:50:330:50:37

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

0:50:400:50:44

doing his mathematics in his spare time.

0:50:440:50:46

He started to explore what he called imaginary geometries,

0:50:460:50:50

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

0:50:500:50:55

The amazing thing is that these imaginary geometries

0:50:550:50:58

make perfect mathematical sense.

0:50:580:51:00

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

0:51:040:51:09

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

0:51:090:51:12

where lines curve back on each other.

0:51:120:51:15

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

0:51:150:51:18

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

0:51:180:51:21

In his hometown of Targu Mures,

0:51:230:51:25

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

0:51:250:51:29

His memory is certainly revered here.

0:51:290:51:33

The museum contains a collection of Bolyai-related artefacts,

0:51:330:51:36

some of which might be considered distinctly Transylvanian.

0:51:360:51:40

It's still got some hair on it.

0:51:400:51:42

It's kind of a little bit gruesome.

0:51:420:51:45

But the object I like most here

0:51:450:51:46

is a beautiful model of Bolyai's geometry.

0:51:460:51:50

You got the shortest distance between here and here

0:51:500:51:54

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

0:51:540:51:56

but this curved line which of bends into the triangle.

0:51:560:51:59

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

0:51:590:52:03

add up to less than 180.

0:52:030:52:06

Bolyai published his work in 1831.

0:52:060:52:09

His father sent his old friend Gauss a copy.

0:52:090:52:12

Gauss wrote back straightaway giving his approval,

0:52:120:52:16

but Gauss refused to praise the young Bolyai,

0:52:160:52:19

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

0:52:190:52:22

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

0:52:220:52:26

Actually, there is a letter from Gauss

0:52:260:52:29

to another friend of his where he says,

0:52:290:52:32

"I regard this young geometer boy

0:52:320:52:34

"as a genius of the first order."

0:52:340:52:37

But Gauss never thought to tell Bolyai that.

0:52:370:52:41

And young Janos was completely disheartened.

0:52:410:52:44

Another body blow soon followed.

0:52:440:52:47

Somebody else had developed exactly the same idea,

0:52:470:52:49

but had published two years before him -

0:52:490:52:52

the Russian mathematician Nicholas Lobachevsky.

0:52:520:52:55

It was all downhill for Bolyai after that.

0:52:570:53:00

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

0:53:000:53:04

Eventually, he went a little crazy.

0:53:040:53:06

In 1860, Janos Bolyai died in obscurity.

0:53:080:53:13

Gauss, by contrast, was lionised after his death.

0:53:150:53:19

A university, the units used to measure magnetic induction,

0:53:190:53:22

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

0:53:220:53:25

During his lifetime, Gauss lent his support

0:53:280:53:31

to very few mathematicians.

0:53:310:53:33

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

0:53:330:53:38

Bernhard Riemann.

0:53:380:53:41

His father was a minister

0:53:480:53:49

and he would remain a sincere Christian all his life.

0:53:490:53:54

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

0:53:540:53:58

His family was large and poor and the only thing

0:53:580:54:00

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

0:54:000:54:04

That was his salvation.

0:54:040:54:07

Many mathematicians like Riemann had very difficult childhoods,

0:54:070:54:11

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

0:54:110:54:14

It was mathematics that gave them a sense of security.

0:54:140:54:18

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

0:54:210:54:24

in northern Germany.

0:54:240:54:26

This was his local school, built as a direct result

0:54:260:54:30

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

0:54:300:54:34

Riemann was one of its first pupils.

0:54:340:54:37

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

0:54:370:54:41

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

0:54:410:54:44

It opened up a whole new world to him.

0:54:440:54:46

One of the books he found in there

0:54:460:54:48

was a book by the French mathematician Legendre,

0:54:480:54:51

all about number theory.

0:54:510:54:53

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

0:54:530:54:55

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

0:54:550:55:01

It was a strategy that obviously suited Riemann

0:55:010:55:04

because he became a brilliant mathematician.

0:55:040:55:07

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

0:55:070:55:12

on the foundations of geometry. In the lecture,

0:55:120:55:16

Riemann first described what geometry actually was

0:55:160:55:20

and its relationship with the world.

0:55:200:55:22

He then sketched out what geometry could be -

0:55:220:55:25

a mathematics of many different kinds of space,

0:55:250:55:28

only one of which would be the flat Euclidian space

0:55:280:55:31

in which we appear to live.

0:55:310:55:32

He was just 26 years old.

0:55:320:55:36

Was it received well? Did people recognise the revolution?

0:55:360:55:40

There was no way that people could actually

0:55:400:55:42

make these ideas concrete.

0:55:420:55:45

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

0:55:450:55:50

So this is the beginning, really, of the revolution

0:55:500:55:53

-which ends with Einstein's relativity.

-Exactly.

0:55:530:55:56

Riemann's mathematics changed how we see the world.

0:55:560:56:01

Suddenly, higher dimensional geometry appeared.

0:56:010:56:04

The potential was there from Descartes,

0:56:040:56:06

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

0:56:060:56:11

He began without putting any restriction

0:56:110:56:15

on the dimensions whatsoever. This was something quite new,

0:56:150:56:18

his way of thinking about things.

0:56:180:56:21

Someone like Bolyai was really thinking about new geometries,

0:56:210:56:24

but new two-dimensional geometries.

0:56:240:56:26

New two-dimensional geometries. Riemann then broke away

0:56:260:56:30

from all the limitations of two or three dimensions

0:56:300:56:35

and began to think in in higher dimensions.

0:56:350:56:37

And this was quite new.

0:56:370:56:39

Multi-dimensional space is at the heart

0:56:390:56:41

of so much mathematics done today.

0:56:410:56:44

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

0:56:440:56:48

Riemann's ideas still perplex and amaze.

0:56:480:56:51

He died, though, in 1866.

0:56:520:56:55

He was only 39 years old.

0:56:550:56:59

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

0:56:590:57:02

Hyperspace is no longer science fiction, but science fact.

0:57:020:57:07

In Paris, they have even tried to visualise what shapes

0:57:070:57:11

in higher dimensions might look like.

0:57:110:57:13

Just as the Renaissance artist Piero would have drawn a square

0:57:150:57:18

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

0:57:180:57:22

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

0:57:220:57:27

to represent a shadow of the four-dimensional hypercube.

0:57:270:57:31

It is with Riemann's work that we finally have

0:57:310:57:34

the mathematical glasses to be able to explore

0:57:340:57:37

such worlds of the mind.

0:57:370:57:39

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

0:57:420:57:44

but without this golden age of mathematics,

0:57:440:57:47

from Descartes to Riemann, there would be no calculus,

0:57:470:57:50

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

0:57:500:57:55

But even more important than that,

0:57:550:57:57

their mathematics blew away the cobwebs

0:57:570:58:00

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

0:58:000:58:04

a world much stranger than we ever thought.

0:58:040:58:07

You can learn more about the story of maths

0:58:110:58:13

at the Open University at:

0:58:130:58:16

Subtitles by Red Bee Media Ltd

0:58:260:58:29

Email [email protected]

0:58:290:58:33

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