The Story of Maths - 3. The Frontiers of Space - Subtitles

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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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00:14:55,520 –> 00:14:58,840
he wanted to prove that these patterns would be there forever.

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00:15:00,800 –> 00:15:04,640
But as well as being the basis for fun and games in the years to come,

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00:15:04,640 –> 00:15:09,200
Fermat’s mathematics would have some very serious applications.

242
00:15:09,200 –> 00:15:11,120
One of his theorems, his Little Theorem, is

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00:15:11,120 –> 00:15:16,040
the basis of the codes that protect our credit cards on the internet.

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00:15:16,040 –> 00:15:20,080
Technology we now rely on today all comes from the scribblings

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00:15:20,080 –> 00:15:22,520
of a 17th-century mathematician.

246
00:15:24,520 –> 00:15:28,160
But the usefulness of Fermat’s mathematics is nothing compared to

247
00:15:28,160 –> 00:15:33,200
that of our next great mathematician and he comes not from France at all,

248
00:15:33,200 –> 00:15:34,880
but from its great rival.

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00:15:38,280 –> 00:15:43,040
In the 17th century, Britain was emerging as a world power.

250
00:15:43,040 –> 00:15:46,600
Its expansion and ambitions required new methods of measurement

251
00:15:46,600 –> 00:15:51,960
and computation and that gave a great boost to mathematics.

252
00:15:51,960 –> 00:15:53,840
The university towns of Oxford and Cambridge

253
00:15:53,840 –> 00:15:58,720
were churning out mathematicians who were in great demand

254
00:15:58,720 –> 00:16:02,680
and the greatest of them was Isaac Newton.

255
00:16:06,760 –> 00:16:09,360
I’m here in Grantham, where Isaac Newton grew up,

256
00:16:09,360 –> 00:16:11,240
and they’re very proud of him here.

257
00:16:11,240 –> 00:16:13,120
They have a wonderful statue to him.

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They’ve even got

259
00:16:14,680 –> 00:16:18,920
the Isaac Newton Shopping Centre, with a nice apple logo up there.

260
00:16:18,920 –> 00:16:21,920
There’s a school that he went to with a nice blue plaque

261
00:16:21,920 –> 00:16:25,480
and there’s a museum over here in the Town Hall, although, actually,

262
00:16:25,480 –> 00:16:28,480
one of the other famous residents here, Margaret Thatcher,

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00:16:28,480 –> 00:16:30,920
has got as big a display as Isaac Newton.

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00:16:30,920 –> 00:16:32,600
In fact, the Thatcher cups have

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00:16:32,600 –> 00:16:36,640
sold out and there’s loads of Newton ones still left,

266
00:16:36,640 –> 00:16:41,320
so I thought I would support mathematics by buying a Newton cup.

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00:16:41,320 –> 00:16:43,920
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.

269
00:16:49,360 –> 00:16:53,920

• No, I don’t.
• Discovering gravity.
• Gravity?
• Gravity, yes.
• Gravity?

270
00:16:53,920 –> 00:16:58,360

• Apple tree and all that, gravity.
• ‘That pretty much summed it up.

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00:16:58,360 –> 00:17:01,560
‘If people know about Newton’s work at all, it is his physics,

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00:17:01,560 –> 00:17:05,160
‘his laws of gravity in motion, not his mathematics.’

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00:17:05,160 –> 00:17:07,360

• I’m in a rush!
• You’re in a rush. OK.

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00:17:07,360 –> 00:17:10,680
Acceleration, you see? One of Newton’s laws!

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00:17:18,200 –> 00:17:20,080
Eight miles south of Grantham,

276
00:17:20,080 –> 00:17:22,960
in the village of Woolsthorpe, where Newton was born,

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00:17:22,960 –> 00:17:26,040
I met up with someone who does share my passion for his mathematics.

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00:17:26,040 –> 00:17:28,080
This is the house.

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00:17:28,080 –> 00:17:32,120
Wow, beautiful. ‘Jackie Stedall is a Newton fan and more than willing

280
00:17:32,120 –> 00:17:35,360
‘to show me around the house where Newton was brought up.’

281
00:17:35,360 –> 00:17:37,360
So here is the…

282
00:17:37,360 –> 00:17:40,840
you might call it the dining room. I’m sure they didn’t call it that,

283
00:17:40,840 –> 00:17:43,560
but the room where they ate, next to the kitchen.

284
00:17:43,560 –> 00:17:45,520
Of course, there would have been a huge fire in there.

285
00:17:45,520 –> 00:17:48,120
Yes! Gosh, I wish it was there now!

286
00:17:48,120 –> 00:17:50,640
His father was an illiterate farmer,

287
00:17:50,640 –> 00:17:53,120
but he died shortly before Newton was born.

288
00:17:53,120 –> 00:17:57,080
Otherwise, the young Isaac’s fate might have been very different.

289
00:17:57,080 –> 00:17:59,080
And here’s his room.

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00:17:59,080 –> 00:18:01,480
Oh, lovely, wow.

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00:18:01,480 –> 00:18:03,760

• They present it really nicely.
• Yes.

292
00:18:03,760 –> 00:18:07,440

• It’s got a real feel of going back in time.
• It does, yes.

293
00:18:07,440 –> 00:18:10,440
I can see he’s as scruffy as I am. Look at the state of that bed.

294
00:18:10,440 –> 00:18:13,480
That’s how, I think, I left my bed this morning.

295
00:18:13,480 –> 00:18:18,160
Newton hated his stepfather, but it was this man who ensured

296
00:18:18,160 –> 00:18:21,240
he became a mathematician rather than a sheep farmer.

297
00:18:21,240 –> 00:18:23,480
I don’t think he was particularly remarkable as a child.

298
00:18:23,480 –> 00:18:26,800

• OK.
• So there’s hope for all those kids out there.
• Yes, yes.

299
00:18:26,800 –> 00:18:28,400
I think he had a sort of average school report.

300
00:18:28,400 –> 00:18:32,280
He had very few close friends. I don’t feel he’s someone

301
00:18:32,280 –> 00:18:33,880
I particularly would have wanted to meet,

302
00:18:33,880 –> 00:18:37,760
but I do love his mathematics. It’s wonderful.

303
00:18:37,760 –> 00:18:40,320
Newton came back to Lincolnshire from Cambridge

304
00:18:40,320 –> 00:18:46,600
during the Great Plague of 1665 when he was just 22 years old.

305
00:18:46,600 –> 00:18:50,880
In two miraculous years here, he developed a new theory of light,

306
00:18:50,880 –> 00:18:52,400
discovered gravitation

307
00:18:52,400 –> 00:18:57,960
and scribbled out a revolutionary approach to maths, the calculus.

308
00:18:57,960 –> 00:18:59,880
It works like this.

309
00:18:59,880 –> 00:19:03,920
I’m going to accelerate this car from 0 to 60 as quickly as I can.

310
00:19:03,920 –> 00:19:07,520
The speedometer is showing me that the speed’s changing all the time,

311
00:19:07,520 –> 00:19:09,360
but this is only an average speed.

312
00:19:09,360 –> 00:19:11,480
How can I tell precisely what my speed is

313
00:19:11,480 –> 00:19:15,400
at any particular instant? Well, here’s how.

314
00:19:15,400 –> 00:19:20,320
As the car races along the road, we can draw a graph above the road

315
00:19:20,320 –> 00:19:23,560
where the height above each point in the road records how long it took

316
00:19:23,560 –> 00:19:26,400
the car to get to that point.

317
00:19:26,400 –> 00:19:28,840
I can calculate the average speed between

318
00:19:28,840 –> 00:19:33,240
two points, A and B, on my journey by recording the distance travelled

319
00:19:33,240 –> 00:19:37,760
and dividing by the time it took to get between these two points,

320
00:19:37,760 –> 00:19:42,000
but what about the precise speed at the first point, A?

321
00:19:43,520 –> 00:19:48,200
If I move point B closer and closer to the first point, I take a smaller

322
00:19:48,200 –> 00:19:51,440
and smaller window of time and the speed gets closer

323
00:19:51,440 –> 00:19:55,240
and closer to the true value, but eventually, it looks like

324
00:19:55,240 –> 00:19:59,320
I have to calculate 0 divided by 0.

325
00:19:59,320 –> 00:20:03,920
The calculus allows us to make sense of this calculation.

326
00:20:03,920 –> 00:20:08,320
It enables us to work out the exact speed and also the precise distance

327
00:20:08,320 –> 00:20:11,280
travelled at any moment in time.

328
00:20:11,280 –> 00:20:15,080
I mean, it does make sense, the things we take for granted so much,

329
00:20:15,080 –> 00:20:16,720
things like… if I drop this apple…

330
00:20:16,720 –> 00:20:18,280
Its distance is changing and its

331
00:20:18,280 –> 00:20:20,920
speed is changing and calculus can deal with all of that.

332
00:20:20,920 –> 00:20:22,480
Which is quite in contrast to the Greeks.

333
00:20:22,480 –> 00:20:25,120
It was a very static geometry.

334
00:20:25,120 –> 00:20:27,000

• Yes, it is.
• And here we see…

335
00:20:27,000 –> 00:20:29,880
so the calculus is used by

336
00:20:29,880 –> 00:20:33,200
every engineer, physicist, because it can describe the moving world.

337
00:20:33,200 –> 00:20:36,720
Yes, and it’s the only way really you can deal with the mathematics of

338
00:20:36,720 –> 00:20:38,480
motion or with change.

339
00:20:38,480 –> 00:20:40,080
There’s a lot of mathematics in this apple!

340
00:20:42,360 –> 00:20:46,040
Newton’s calculus enables us to really understand

341
00:20:46,040 –> 00:20:50,600
the changing world, the orbits of planets, the motions of fluids.

342
00:20:50,600 –> 00:20:54,200
Through the power of the calculus, we have a way of describing, with

343
00:20:54,200 –> 00:20:58,840
mathematical precision, the complex, ever-changing natural world.

344
00:21:04,800 –> 00:21:09,080
But it would take 200 years to realise its full potential.

345
00:21:09,080 –> 00:21:12,640
Newton himself decided not to publish, but just to circulate

346
00:21:12,640 –> 00:21:14,960
his thoughts among friends.

347
00:21:14,960 –> 00:21:17,240
His reputation, though, gradually spread.

348
00:21:17,240 –> 00:21:21,480
He became a professor, an MP, and then Warden of the Royal Mint

349
00:21:21,480 –> 00:21:23,640
here in the City of London.

350
00:21:25,600 –> 00:21:28,760
On his regular trips to the Royal Society from the Royal Mint,

351
00:21:28,760 –> 00:21:33,120
he preferred to think about theology and alchemy rather than mathematics.

352
00:21:33,120 –> 00:21:35,440
Developing the calculus just got crowded out

353
00:21:35,440 –> 00:21:39,720
by all his other interests until he heard about a rival…

354
00:21:41,800 –> 00:21:46,080
a rival who was also a member of the Royal Society and who came up

355
00:21:46,080 –> 00:21:48,800
with exactly the same idea as him,

356
00:21:48,800 –> 00:21:50,960
Gottfried Leibniz.

357
00:21:50,960 –> 00:21:54,240
Every word Leibniz wrote has been preserved and catalogued

358
00:21:54,240 –> 00:21:57,800
in his hometown of Hanover in northern Germany.

359
00:21:57,800 –> 00:22:01,040
His actual manuscripts are kept under lock and key,

360
00:22:01,040 –> 00:22:04,360
particularly the manuscript which shows how Leibniz

361
00:22:04,360 –> 00:22:09,720
also discovered the miracle of calculus, shortly after Newton.

362
00:22:09,720 –> 00:22:11,520
What age was he when he wrote…

363
00:22:11,520 –> 00:22:16,720
He was 29 years old and that’s the time, within two months, he developed

364
00:22:16,720 –> 00:22:19,640

• differential calculus and integral calculus.
• In two months?

365
00:22:19,640 –> 00:22:21,600

• Yeah.
• Fast and furious, when it comes, er…

366
00:22:21,600 –> 00:22:23,240
Yeah.

367
00:22:23,240 –> 00:22:26,440
There is a little scrap of paper over here. What’s that one?

368
00:22:26,440 –> 00:22:29,840

• A letter or…
• That’s a small manuscript of Leibniz’s notes.

369
00:22:32,560 –> 00:22:37,280
“Sometimes it happens that in the morning lying in the bed,

370
00:22:37,280 –> 00:22:40,960
“I have so many ideas that it takes the whole morning and sometimes

371
00:22:40,960 –> 00:22:45,760
“even longer to note all these ideas and bring them to paper.”

372
00:22:45,760 –> 00:22:47,280
I suppose, that’s beautiful.

373
00:22:47,280 –> 00:22:51,480
I suppose that he liked to lie in the bed in the morning.

374
00:22:51,480 –> 00:22:53,400

• A true mathematician.
• Yeah.

375
00:22:53,400 –> 00:22:55,680
He spends his time thinking in bed.

376
00:22:55,680 –> 00:22:58,640
I see you’ve got some paintings down here.

377
00:22:58,640 –> 00:23:00,280
A painting.

378
00:23:00,280 –> 00:23:02,360
This is what he looked like. Right.

379
00:23:03,880 –> 00:23:07,280
Even though he didn’t become quite the 17th century celebrity

380
00:23:07,280 –> 00:23:10,560
that Newton did, it wasn’t such a bad life.

381
00:23:10,560 –> 00:23:12,520
Leibniz worked for the Royal Family

382
00:23:12,520 –> 00:23:16,600
of Hanover and travelled around Europe representing their interests.

383
00:23:16,600 –> 00:23:19,040
This gave him plenty of time to indulge in

384
00:23:19,040 –> 00:23:23,400
his favourite intellectual pastimes, which were wide, even for the time.

385
00:23:23,400 –> 00:23:26,960
He devised a plan for reunifying the Protestant and Roman Catholic

386
00:23:26,960 –> 00:23:32,000
churches, a proposal for France to conquer Egypt and contributions to

387
00:23:32,000 –> 00:23:36,280
philosophy and logic which are still highly rated today.

388
00:23:36,280 –> 00:23:39,880

• He wrote all these letters?
• Yeah.
• That’s absolutely extraordinary.

389
00:23:39,880 –> 00:23:43,080
He must have cloned himself. I can’t believe there was just one Leibniz!

390
00:23:43,080 –> 00:23:46,040
‘But Leibniz was not just man of words.

391
00:23:46,040 –> 00:23:47,640
‘He was also one of the first people

392
00:23:47,640 –> 00:23:49,480
‘to invent practical calculating machines

393
00:23:49,480 –> 00:23:54,520
‘that worked on the binary system, true forerunners of the computer.

394
00:23:54,520 –> 00:23:58,680
‘300 years later, the engineering department at Leibniz University

395
00:23:58,680 –> 00:24:02,880
‘in Hanover have put them together following Leibniz’s blueprint.’

396
00:24:02,880 –> 00:24:04,760
I love all the ball bearings, so these are going to be all

397
00:24:04,760 –> 00:24:06,680
of our zeros and ones. So a ball bearing is a one.

398
00:24:06,680 –> 00:24:10,720
Only zero and one. Now we represent a number 127.

399
00:24:10,720 –> 00:24:15,960

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

400
00:24:15,960 –> 00:24:18,880

• And now I give the number one.
• OK.

401
00:24:18,880 –> 00:24:24,360
Now we add 127 plus one - is 128, which is two, power eight.

402
00:24:24,360 –> 00:24:28,000

• Oh, OK. So there’s going to be lots of action.
• Would you show this here?

403
00:24:28,000 –> 00:24:30,480
This is the money shot.

404
00:24:30,480 –> 00:24:33,560
So we’re going to add one. Oops. Here we go. They’re all carrying.

405
00:24:33,560 –> 00:24:36,520
So this 128 is two power eight.

406
00:24:36,520 –> 00:24:42,360
Excellent, so 127 in binary is 1, 1, 1, 1, 1, 1, 1, which is

407
00:24:42,360 –> 00:24:44,320
all the ball bearings here.

408
00:24:44,320 –> 00:24:46,320
To add one it all gets

409
00:24:46,320 –> 00:24:50,920
carried, this goes to 0, 0, 0, 0, and we have a power of two here.

410
00:24:50,920 –> 00:24:53,080
So this mechanism gets rid of all the ball bearings that you

411
00:24:53,080 –> 00:24:56,680

• don’t need. It’s like pinball, mathematical pinball.
• Exactly.

412
00:24:56,680 –> 00:24:58,200
I love this machine!

413
00:25:03,680 –> 00:25:08,120
After a hard day’s work, Leibniz often came here,

414
00:25:08,120 –> 00:25:10,080
the famous gardens of Herrenhausen,

415
00:25:10,080 –> 00:25:14,800
now in the middle of Hanover, but then on the outskirts of the city.

416
00:25:14,800 –> 00:25:17,400
There’s something about mathematics and walking.

417
00:25:17,400 –> 00:25:21,040
I don’t know, you’ve been working at your desk all day, all morning

418
00:25:21,040 –> 00:25:22,640
on some problem and your head’s all

419
00:25:22,640 –> 00:25:25,040
fuzzy, and you just need to come and have a walk.

420
00:25:25,040 –> 00:25:27,760
You let your subconscious mind kind of take over and sometimes

421
00:25:27,760 –> 00:25:31,880
you get your breakthrough just looking at the trees or whatever.

422
00:25:31,880 –> 00:25:35,160
I’ve had some of my best ideas whilst walking in my local park,

423
00:25:35,160 –> 00:25:39,120
so I’m hoping to get a little bit of inspiration here on Leibniz’s

424
00:25:39,120 –> 00:25:40,760
local stomping ground.

425
00:25:44,240 –> 00:25:47,120
I didn’t get the chance to purge my mind of mathematical challenges

426
00:25:47,120 –> 00:25:49,240
because in the years since Leibniz lived here,

427
00:25:49,240 –> 00:25:50,440
someone has built a maze.

428
00:25:50,440 –> 00:25:53,520
Well, there is a mathematical formula for getting out of a maze,

429
00:25:53,520 –> 00:25:57,200
which is if you put your left hand on the side of the maze and just

430
00:25:57,200 –> 00:26:00,760
keep it there, keep on winding round, you eventually get out.

431
00:26:00,760 –> 00:26:03,760
That’s the theory, at least. Let’s see whether it works!

432
00:26:11,080 –> 00:26:13,600
Leibniz had no such distractions.

433
00:26:13,600 –> 00:26:17,320
Within five years, he’d worked out the details of the calculus,

434
00:26:17,320 –> 00:26:19,160
seemingly independent from Newton,

435
00:26:19,160 –> 00:26:21,680
although he knew about Newton’s work,

436
00:26:21,680 –> 00:26:26,200
but unlike Newton, Leibniz was quite happy to make his work known

437
00:26:26,200 –> 00:26:29,440
and so mathematicians across Europe heard about the calculus first

438
00:26:29,440 –> 00:26:35,680
from him and not from Newton, and that’s when all the trouble started.

439
00:26:35,680 –> 00:26:39,200
Throughout mathematical history, there have been lots of priority

440
00:26:39,200 –> 00:26:40,800
disputes and arguments.

441
00:26:40,800 –> 00:26:43,800
It may seem a little bit petty and schoolboyish.

442
00:26:43,800 –> 00:26:46,600
We really want our name to be on that theorem.

443
00:26:46,600 –> 00:26:49,800
This is our one chance for a little bit of immortality because that

444
00:26:49,800 –> 00:26:54,120
theorem’s going to last forever and that’s why we dedicate so much time

445
00:26:54,120 –> 00:26:55,920
to trying to crack these things.

446
00:26:55,920 –> 00:26:57,800
Somehow we can’t believe that somebody else

447
00:26:57,800 –> 00:27:00,000
has got it at the same time as us.

448
00:27:00,000 –> 00:27:03,040
These are our theorems, our babies, our children and we

449
00:27:03,040 –> 00:27:06,000
don’t want to share the credit.

450
00:27:06,000 –> 00:27:08,440
Back in London, Newton certainly didn’t want

451
00:27:08,440 –> 00:27:13,040
to share credit with Leibniz, who he thought of as a Hanoverian upstart.

452
00:27:13,040 –> 00:27:16,160
After years of acrimony and accusation, the Royal Society

453
00:27:16,160 –> 00:27:21,120
in London was asked to adjudicate between the rival claims.

454
00:27:21,120 –> 00:27:23,080
The Royal Society gave Newton credit

455
00:27:23,080 –> 00:27:25,240
for the first discovery of the calculus

456
00:27:25,240 –> 00:27:28,880
and Leibniz credit for the first publication,

457
00:27:28,880 –> 00:27:33,400
but in their final judgment, they accused Leibniz of plagiarism.

458
00:27:33,400 –> 00:27:36,640
However, that might have had something to do with the fact that

459
00:27:36,640 –> 00:27:41,920
the report was written by their President, one Sir Isaac Newton.

460
00:27:44,040 –> 00:27:46,440
Leibniz was incredibly hurt.

461
00:27:46,440 –> 00:27:50,400
He admired Newton and never really recovered.

462
00:27:50,400 –> 00:27:52,440
He died in 1716.

463
00:27:52,440 –> 00:27:56,200
Newton lived on another 11 years and was buried in the grandeur of

464
00:27:56,200 –> 00:27:58,240
Westminster Abbey.

465
00:27:58,240 –> 00:28:00,360
Leibniz’s memorial, by contrast,

466
00:28:00,360 –> 00:28:02,520
is here in this small church in Hanover.

467
00:28:02,520 –> 00:28:06,040
The irony is that it’s Leibniz’s mathematics which

468
00:28:06,040 –> 00:28:08,800
eventually triumphs, not Newton’s.

469
00:28:11,040 –> 00:28:13,720
I’m a big Leibniz fan.

470
00:28:13,720 –> 00:28:16,920
Quite often revolutions in mathematics are about producing the

471
00:28:16,920 –> 00:28:19,680
right language to capture a new vision and that’s what

472
00:28:19,680 –> 00:28:21,520
Leibniz was so good at.

473
00:28:21,520 –> 00:28:25,280
Leibniz’s notation, his way of writing the calculus,

474
00:28:25,280 –> 00:28:27,360
captured its true spirit.

475
00:28:27,360 –> 00:28:29,960
It’s still the one we use in maths today.

476
00:28:29,960 –> 00:28:34,320
Newton’s notation was, for many mathematicians, clumsy and difficult

477
00:28:34,320 –> 00:28:38,600
to use and so while British mathematics loses its way a little,

478
00:28:38,600 –> 00:28:43,360
the story of maths switches to the very heart of Europe, Basel.

479
00:28:48,560 –> 00:28:52,280
In its heyday in the 18th century, the free city of Basel in

480
00:28:52,280 –> 00:28:56,840
Switzerland was the commercial hub of the entire Western world.

481
00:28:56,840 –> 00:28:59,640
Around this maelstrom of trade, there developed a tradition of

482
00:28:59,640 –> 00:29:03,520
learning, particularly learning which connected with commerce

483
00:29:03,520 –> 00:29:06,400
and one family summed all this up.

484
00:29:06,400 –> 00:29:11,160
It’s kind of curious - artists often have children who are artists.

485
00:29:11,160 –> 00:29:15,480
Musicians, their children are often musicians, but us mathematicians,

486
00:29:15,480 –> 00:29:17,680
our children don’t tend to be mathematicians.

487
00:29:17,680 –> 00:29:19,720
I’m not sure why it is.

488
00:29:19,720 –> 00:29:23,000
At least that’s my view, although others dispute it.

489
00:29:23,000 –> 00:29:25,000
What no-one disagrees with

490
00:29:25,000 –> 00:29:30,080
is there is one great dynasty of mathematicians, the Bernoullis.

491
00:29:30,080 –> 00:29:33,760
In the 18th and 19th centuries they produced half a dozen

492
00:29:33,760 –> 00:29:37,040
outstanding mathematicians, any of which we would have been

493
00:29:37,040 –> 00:29:41,800
proud to have had in Britain, and they all came from Basel.

494
00:29:41,800 –> 00:29:44,960
You might have great minds like Newton and Leibniz who make

495
00:29:44,960 –> 00:29:48,440
these fundamental breakthroughs, but you also need the disciples

496
00:29:48,440 –> 00:29:51,680
who take that message, clarify it, realise its implications,

497
00:29:51,680 –> 00:29:55,480
then spread it wide. The family were originally merchants,

498
00:29:55,480 –> 00:29:57,440
and this is one of their houses.

499
00:29:57,440 –> 00:30:00,360
It’s now part of the University of Basel

500
00:30:00,360 –> 00:30:03,440
and it’s been completely refurbished, apart from one room,

501
00:30:03,440 –> 00:30:07,360
which has been kept very much as the family would have used it.

502
00:30:07,360 –> 00:30:09,720
Dr Fritz Nagel, keeper of the Bernoulli Archive,

503
00:30:09,720 –> 00:30:12,480
has promised to show it to me.

504
00:30:12,480 –> 00:30:15,120

• If we can find it.
• No, we’re on the wrong floor.

505
00:30:15,120 –> 00:30:17,440
Wrong floor, OK. Right!

506
00:30:17,440 –> 00:30:19,560
Oh, look.

507
00:30:19,560 –> 00:30:21,440
Can we take an apple?

508
00:30:21,440 –> 00:30:24,000
‘No, wrong mathematician.

509
00:30:24,000 –> 00:30:26,480
‘Eventually, we got there.’

510
00:30:26,480 –> 00:30:28,840
This is where the Bernoullis would have done

511
00:30:28,840 –> 00:30:30,600
some of their mathematics.

512
00:30:30,600 –> 00:30:33,680
‘I was really just being polite.

513
00:30:33,680 –> 00:30:36,400
‘The only thing of interest was an old stove.’

514
00:30:36,400 –> 00:30:40,200
Now, of the Bernoullis, which is your favourite?

515
00:30:40,200 –> 00:30:44,080
My favourite Bernoulli is Johann I.

516
00:30:44,080 –> 00:30:49,640
He is the most smart mathematician.

517
00:30:49,640 –> 00:30:54,160
Perhaps his brother Jakob was the mathematician

518
00:30:54,160 –> 00:30:57,160
with the deeper insight into problems,

519
00:30:57,160 –> 00:30:59,800
but Johann found elegant solutions.

520
00:30:59,800 –> 00:31:03,920
The brothers didn’t like each other much, but both worshipped Leibniz.

521
00:31:03,920 –> 00:31:06,560
They corresponded with him, stood up for him

522
00:31:06,560 –> 00:31:10,960
against Newton’s allies, and spread his calculus throughout Europe.

523
00:31:10,960 –> 00:31:15,440
Leibnitz was very happy to have found two gifted mathematicians

524
00:31:15,440 –> 00:31:20,640
outside of his personal circle of friends who mastered his calculus

525
00:31:20,640 –> 00:31:23,680
and could distribute it in the scientific community.

526
00:31:23,680 –> 00:31:28,320

• That was very important for Leibniz.
• And important for maths, too.

527
00:31:28,320 –> 00:31:32,440
Without the Bernoullis, it would have taken much longer for calculus

528
00:31:32,440 –> 00:31:36,200
to become what it is today, a cornerstone of mathematics.

529
00:31:36,200 –> 00:31:38,760
At least, that is Dr Nagel’s contention.

530
00:31:38,760 –> 00:31:41,240
And he is a great Bernoulli fan.

531
00:31:41,240 –> 00:31:44,520
He has arranged for me to meet Professor Daniel Bernoulli,

532
00:31:44,520 –> 00:31:46,960
the latest member of the family,

533
00:31:46,960 –> 00:31:49,680
whose famous name ensures he gets some odd e-mails.

534
00:31:49,680 –> 00:31:51,320
Another one of which I got was,

535
00:31:51,320 –> 00:31:54,440
“Professor Bernoulli, can you give me a hand with calculus?”

536
00:31:54,440 –> 00:31:58,560
To find a Bernoulli, you expect them to be able to do calculus.

537
00:31:58,560 –> 00:32:02,640
‘But this Daniel Bernoulli is a professor of geology.

538
00:32:02,640 –> 00:32:05,880
‘The maths gene seems to have truly died out.

539
00:32:05,880 –> 00:32:07,880
‘And during our very hearty dinner,

540
00:32:07,880 –> 00:32:11,200
‘I found myself wandering back to maths.’

541
00:32:11,200 –> 00:32:14,400
It is a bit unfair on the Bernoullis to describe them simply

542
00:32:14,400 –> 00:32:16,040
as disciples of Leibniz.

543
00:32:16,040 –> 00:32:18,960
One of their many great contributions to mathematics

544
00:32:18,960 –> 00:32:23,800
was to develop the calculus to solve a classic problem of the day.

545
00:32:23,800 –> 00:32:26,360
Imagine a ball rolling down a ramp.

546
00:32:26,360 –> 00:32:29,320
The task is to design a ramp that will get the ball

547
00:32:29,320 –> 00:32:32,440
from the top to the bottom in the fastest time possible.

548
00:32:32,440 –> 00:32:36,080
You might think that a straight ramp would be quickest.

549
00:32:36,080 –> 00:32:37,920
Or possibly a curved one like this

550
00:32:37,920 –> 00:32:40,720
that gives the ball plenty of downward momentum.

551
00:32:40,720 –> 00:32:42,880
In fact, it’s neither of these.

552
00:32:42,880 –> 00:32:45,960
Calculus shows that it is what we call a cycloid,

553
00:32:45,960 –> 00:32:49,640
the path traced by a point on the rim of a moving bicycle wheel.

554
00:32:49,640 –> 00:32:53,360
This application of the calculus by the Bernoullis, which became known

555
00:32:53,360 –> 00:32:55,520
as the calculus of variation,

556
00:32:55,520 –> 00:32:58,600
has become one of the most powerful aspects of the mathematics

557
00:32:58,600 –> 00:33:01,560
of Leibniz and Newton. Investors use it to maximise profits.

558
00:33:01,560 –> 00:33:05,240
Engineers exploit it to minimise energy use.

559
00:33:05,240 –> 00:33:08,560
Designers apply it to optimise construction.

560
00:33:08,560 –> 00:33:10,680
It has now become one of the linchpins

561
00:33:10,680 –> 00:33:12,840
of our modern technological world.

562
00:33:12,840 –> 00:33:17,160
Meanwhile, things were getting more interesting in the restaurant.

563
00:33:17,160 –> 00:33:18,760
Here is my second surprise.

564
00:33:18,760 –> 00:33:22,000
Let me introduce Mr Leonhard Euler.

565
00:33:22,000 –> 00:33:23,720
Daniel Bernoulli.

566
00:33:23,720 –> 00:33:27,920
‘Leonhard Euler, one of the most famous names in mathematics.

567
00:33:27,920 –> 00:33:29,600
‘This Leonhard is a descendant

568
00:33:29,600 –> 00:33:34,080
‘of the original Leonhard Euler, star pupil of Johann Bernoulli.’

569
00:33:34,080 –> 00:33:36,640
I am the ninth generation,

570
00:33:36,640 –> 00:33:39,840
the fourth Leonhard in our family

571
00:33:39,840 –> 00:33:42,440
after Leonard Euler I, the mathematician.

572
00:33:42,440 –> 00:33:44,840
OK. And yourself, are you a mathematician?

573
00:33:44,840 –> 00:33:47,840
Actually, I am a business analyst.

574
00:33:47,840 –> 00:33:51,920
I can’t study mathematics with my name.

575
00:33:51,920 –> 00:33:55,320
Everyone will expect you to prove that the Riemann hypothesis!

576
00:33:55,320 –> 00:33:58,600
Perhaps it’s just as well that Leonhard decided

577
00:33:58,600 –> 00:34:02,240
not to follow in the footsteps of his illustrious ancestor.

578
00:34:02,240 –> 00:34:04,600
He’d have had a lot to live up to.

579
00:34:13,000 –> 00:34:15,000
I am going in a boat across the Rhine,

580
00:34:15,000 –> 00:34:17,560
and I’m feeling a little bit worse for wear.

581
00:34:17,560 –> 00:34:21,120
Last night’s dinner with Mr Euler and Professor Bernoulli

582
00:34:21,120 –> 00:34:25,480
degenerated into toasts to all the theorems the Bernoullis and Eulers

583
00:34:25,480 –> 00:34:28,600
have proved, and by God, they have proved quite a lot of them!

584
00:34:28,600 –> 00:34:30,880
Never again.

585
00:34:30,880 –> 00:34:34,800
I was getting disapproving glances from my fellow passengers as well.

586
00:34:34,800 –> 00:34:37,360
Luckily, it was only a short trip.

587
00:34:37,360 –> 00:34:41,960
Not like the trip that Euler took in 1728 to start a new life.

588
00:34:41,960 –> 00:34:45,240
Euler may have been the prodigy of Johann Bernoulli,

589
00:34:45,240 –> 00:34:47,800
but there was no room for him in the city.

590
00:34:47,800 –> 00:34:49,520
If your name wasn’t Bernoulli,

591
00:34:49,520 –> 00:34:53,240
there was little chance of getting a job in mathematics here in Basel.

592
00:34:53,240 –> 00:34:55,600
But Daniel, the son of Johann Bernoulli,

593
00:34:55,600 –> 00:34:57,120
was a great friend of Euler

594
00:34:57,120 –> 00:35:00,360
and managed to get him a job at his university.

595
00:35:00,360 –> 00:35:03,280
But to get there would take seven weeks,

596
00:35:03,280 –> 00:35:05,800
because Daniel’s university was in Russia.

597
00:35:08,280 –> 00:35:11,720
It wasn’t an intellectual powerhouse like Berlin or Paris,

598
00:35:11,720 –> 00:35:17,320
but St Petersburg was by no means unsophisticated in the 18th century.

599
00:35:17,320 –> 00:35:21,440
Peter the Great had created a city very much in the European style.

600
00:35:21,440 –> 00:35:26,080
And every fashionable city at the time had a scientific academy.

601
00:35:27,840 –> 00:35:30,040
Peter’s Academy is now a museum.

602
00:35:30,040 –> 00:35:34,320
It includes several rooms full of the kind of grotesque curiosities

603
00:35:34,320 –> 00:35:38,000
that are usually kept out of the public display in the West.

604
00:35:38,000 –> 00:35:39,960
But in the 1730s,

605
00:35:39,960 –> 00:35:44,400
this building was a centre for ground-breaking research.

606
00:35:44,400 –> 00:35:46,880
It is where Euler found his intellectual home.

607
00:35:50,280 –> 00:35:57,000

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

608
00:35:58,000 –> 00:36:00,840
Many of the ideas that were bubbling away at the time -

609
00:36:00,840 –> 00:36:02,480
calculus of variation,

610
00:36:02,480 –> 00:36:06,560
Fermat’s theory of numbers - crystallised in Euler’s hands.

611
00:36:06,560 –> 00:36:09,560
But he was also creating incredibly modern mathematics,

612
00:36:09,560 –> 00:36:12,040
topology and analysis.

613
00:36:12,040 –> 00:36:15,240
Much of the notation that I use today as a mathematician

614
00:36:15,240 –> 00:36:19,240
was created by Euler, numbers like e and i.

615
00:36:19,240 –> 00:36:23,000
Euler also popularised the use of the symbol pi.

616
00:36:23,000 –> 00:36:25,200
He even combined these numbers together

617
00:36:25,200 –> 00:36:28,120
in one of the most beautiful formulas of mathematics,

618
00:36:28,120 –> 00:36:32,920
e to the power of i times pi is equal to -1.

619
00:36:32,920 –> 00:36:36,600
An amazing feat of mathematical alchemy.

620
00:36:36,600 –> 00:36:39,960
His life, in fact, is full of mathematical magic.

621
00:36:39,960 –> 00:36:43,560
Euler applied his skills to an immense range of topics,

622
00:36:43,560 –> 00:36:46,440
from prime numbers to optics to astronomy.

623
00:36:46,440 –> 00:36:49,840
He devised a new system of weights and measures, wrote a textbook

624
00:36:49,840 –> 00:36:54,520
on mechanics, and even found time to develop a new theory of music.

625
00:36:59,360 –> 00:37:01,440
I think of him as the Mozart of maths.

626
00:37:01,440 –> 00:37:04,800
And that view is shared by the mathematician Nikolai Vavilov,

627
00:37:04,800 –> 00:37:07,360
who met me at the house that was given to Euler

628
00:37:07,360 –> 00:37:10,040
by Catherine the Great.

629
00:37:10,040 –> 00:37:14,360
Euler lived here from ‘66 to ‘83, which means from the year

630
00:37:14,360 –> 00:37:17,640
he came back to St Petersburg to the year he died.

631
00:37:17,640 –> 00:37:22,720
And he was a member of the Russian Academy of Sciences,

632
00:37:22,720 –> 00:37:24,760
and their greatest mathematician.

633
00:37:24,760 –> 00:37:27,360
That is exactly what it says.

634
00:37:27,360 –> 00:37:29,360

• What is it now?
• It is a school.

635
00:37:29,360 –> 00:37:30,920
Shall we go in and see?

636
00:37:30,920 –> 00:37:33,760
OK.

637
00:37:33,760 –> 00:37:38,920
‘I’m not sure Nikolai entirely approved. But nothing ventured…’

638
00:37:38,920 –> 00:37:41,320
Perhaps we should talk to the head teacher.

639
00:37:46,200 –> 00:37:48,320
The head didn’t mind at all.

640
00:37:48,320 –> 00:37:50,680
I rather got the impression that she was used

641
00:37:50,680 –> 00:37:53,200
to people dropping in to talk about Euler.

642
00:37:53,200 –> 00:37:57,040
She even had a couple of very able pupils suspiciously close to hand.

643
00:37:57,040 –> 00:38:02,240
These two young ladies are ready to tell a few words about the scientist

644
00:38:02,240 –> 00:38:04,400
and about this very building.

645
00:38:04,400 –> 00:38:06,200
They certainly knew their stuff.

646
00:38:06,200 –> 00:38:09,880
They had undertaken an entire classroom project on Euler,

647
00:38:09,880 –> 00:38:13,160
his long life, happy marriage and 13 children.

648
00:38:13,160 –> 00:38:16,160
And then his tragedies - only five of his children

649
00:38:16,160 –> 00:38:17,720
survived to adulthood.

650
00:38:17,720 –> 00:38:21,200
His first wife, who he adored, died young.

651
00:38:21,200 –> 00:38:23,640
He started losing most of his eyesight.

652
00:38:26,720 –> 00:38:31,480
So for the last years of his life, he still continued to work, actually.

653
00:38:31,480 –> 00:38:34,560
He continued his mathematical research.

654
00:38:34,560 –> 00:38:36,480
I read a quote that said now with his blindness,

655
00:38:36,480 –> 00:38:38,640
he hasn’t got any distractions,

656
00:38:38,640 –> 00:38:42,480
he can finally get on with his mathematics. A positive attitude.

657
00:38:42,480 –> 00:38:46,200
It was a totally unexpected and charming visit.

658
00:38:46,200 –> 00:38:49,200
Although I couldn’t resist sneaking back and correcting

659
00:38:49,200 –> 00:38:53,640
one of the equations on the board when everyone else had left.

660
00:38:54,960 –> 00:38:59,960
To demonstrate one of my favourite Euler theorems, I needed a drink.

661
00:38:59,960 –> 00:39:02,920
It concerns calculating infinite sums,

662
00:39:02,920 –> 00:39:06,280
the discovery that shot Euler to the top of the mathematical pops

663
00:39:06,280 –> 00:39:08,840
when it was announced in 1735.

664
00:39:11,120 –> 00:39:15,680
Take one shot glass full of vodka and add it to this tall glass here.

665
00:39:17,960 –> 00:39:22,400
Next, take a glass which is a quarter full, or a half squared,

666
00:39:22,400 –> 00:39:24,120
and add it to the first glass.

667
00:39:25,880 –> 00:39:30,240
Next, take a glass which is a ninth full, or a third squared,

668
00:39:30,240 –> 00:39:31,920
and add that one.

669
00:39:31,920 –> 00:39:36,880
Now, if I keep on adding infinitely many glasses where each one

670
00:39:36,880 –> 00:39:43,200
is a fraction squared, how much will be in this tall glass here?

671
00:39:43,200 –> 00:39:45,080
It was called the Basel problem

672
00:39:45,080 –> 00:39:47,760
after the Bernoullis tried and failed to solve it.

673
00:39:47,760 –> 00:39:52,600
Daniel Bernoulli knew that you would not get an infinite amount of vodka.

674
00:39:52,600 –> 00:39:57,280
He estimated that the total would come to about one and three fifths.

675
00:39:57,280 –> 00:39:59,280
But then Euler came along.

676
00:39:59,280 –> 00:40:03,520
Daniel was close, but mathematics is about precision.

677
00:40:03,520 –> 00:40:06,640
Euler calculated that the total height of the vodka

678
00:40:06,640 –> 00:40:10,960
would be exactly pi squared divided by six.

679
00:40:13,040 –> 00:40:15,160
It was a complete surprise.

680
00:40:15,160 –> 00:40:17,800
What on earth did adding squares of fractions

681
00:40:17,800 –> 00:40:20,520
have to do with the special number pi?

682
00:40:20,520 –> 00:40:23,600
But Euler’s analysis showed that they were two sides

683
00:40:23,600 –> 00:40:25,240
of the same equation.

684
00:40:25,240 –> 00:40:29,280
One plus a quarter plus a ninth plus a sixteenth

685
00:40:29,280 –> 00:40:34,560
and so on to infinity is equal to pi squared over six.

686
00:40:34,560 –> 00:40:38,080
That’s still quite a lot of vodka, but here goes.

687
00:40:43,280 –> 00:40:46,440
Euler would certainly be a hard act to follow.

688
00:40:46,440 –> 00:40:49,560
Mathematicians from two countries would try.

689
00:40:49,560 –> 00:40:53,680
Both France and Germany were caught up in the age of revolution

690
00:40:53,680 –> 00:40:56,960
that was sweeping Europe in the late 18th century.

691
00:40:56,960 –> 00:40:59,760
Both desperately needed mathematicians.

692
00:40:59,760 –> 00:41:04,600
But they went about supporting mathematics rather differently.

693
00:41:04,600 –> 00:41:05,960
Here in France,

694
00:41:05,960 –> 00:41:09,560
the Revolution emphasised the usefulness of mathematics.

695
00:41:09,560 –> 00:41:12,280
Napoleon recognised that if you were going to have

696
00:41:12,280 –> 00:41:14,920
the best military machine, the best weaponry,

697
00:41:14,920 –> 00:41:17,720
then you needed the best mathematicians.

698
00:41:17,720 –> 00:41:21,120
Napoleon’s reforms gave mathematics a big boost.

699
00:41:21,120 –> 00:41:24,400
But this was a mathematics that was going to serve society.

700
00:41:25,920 –> 00:41:30,000
Here in the German states, the great educationalist Wilhelm von Humboldt

701
00:41:30,000 –> 00:41:33,840
was also committed to mathematics, but a mathematics that was detached

702
00:41:33,840 –> 00:41:36,360
from the demands of the State and the military.

703
00:41:36,360 –> 00:41:42,200
Von Humboldt’s educational reforms valued mathematics for its own sake.

704
00:41:42,200 –> 00:41:46,080
In France, they got wonderful mathematicians, like Joseph Fourier,

705
00:41:46,080 –> 00:41:49,280
whose work on sound waves we still benefit from today.

706
00:41:49,280 –> 00:41:53,360
MP3 technology is based on Fourier analysis.

707
00:41:53,360 –> 00:41:56,680
But in Germany, they got, at least in my opinion,

708
00:41:56,680 –> 00:41:58,680
the greatest mathematician ever.

709
00:42:01,960 –> 00:42:03,920
Quaint and quiet,

710
00:42:03,920 –> 00:42:08,080
the university town of Gottingen may seem like a bit of a backwater.

711
00:42:08,080 –> 00:42:12,000
But this little town has been home to some of the giants of maths,

712
00:42:12,000 –> 00:42:14,320
including the man who’s often described

713
00:42:14,320 –> 00:42:19,360
as the Prince of Mathematics, Carl Friedrich Gauss.

714
00:42:19,360 –> 00:42:23,240
Few non-mathematicians, however, seem to know anything about him.

715
00:42:23,240 –> 00:42:25,040
Not in Paris.

716
00:42:25,040 –> 00:42:27,000
Qui s’appelle Carl Friedrich Gauss?

717
00:42:27,000 –> 00:42:28,880

• Non.
• Non?

718
00:42:28,880 –> 00:42:30,480
‘Not in Oxford.’

719
00:42:30,480 –> 00:42:34,440

• I’ve heard the name but I couldn’t tell you.
• No idea.
• No idea?
• No.

720
00:42:34,440 –> 00:42:37,480
‘And I’m afraid to say, not even in modern Germany.’

721
00:42:37,480 –> 00:42:39,400

• Nein.
• Nein? OK.

722
00:42:39,400 –> 00:42:41,040

• I don’t know.
• You don’t know?

723
00:42:41,040 –> 00:42:44,600
But in Gottingen, everyone knows who Gauss is.

724
00:42:44,600 –> 00:42:47,040
He’s the local hero.

725
00:42:47,040 –> 00:42:49,440
His father was a stonemason

726
00:42:49,440 –> 00:42:52,560
and it’s likely that Gauss would have become one, too.

727
00:42:52,560 –> 00:42:55,720
But his singular talent was recognised by his mother,

728
00:42:55,720 –> 00:42:57,560
and she helped ensure

729
00:42:57,560 –> 00:43:01,320
that he received the best possible education.

730
00:43:01,320 –> 00:43:05,080
Every few years in the news, you hear about a new prodigy

731
00:43:05,080 –> 00:43:08,240
who’s passed their A-levels at ten, gone to university at 12,

732
00:43:08,240 –> 00:43:10,240
but nobody compares to Gauss.

733
00:43:10,240 –> 00:43:13,680
Already at the age of 12, he was criticising Euclid’s geometry.

734
00:43:13,680 –> 00:43:16,960
At 15, he discovered a new pattern in prime numbers

735
00:43:16,960 –> 00:43:20,240
which had eluded mathematicians for 2,000 years.

736
00:43:20,240 –> 00:43:24,000
And at 19, he discovered the construction of a 17-sided figure

737
00:43:24,000 –> 00:43:26,880
which nobody had known before this time.

738
00:43:30,200 –> 00:43:34,160
His early successes encouraged Gauss to keep a diary.

739
00:43:34,160 –> 00:43:36,120
Here at the University of Gottingen,

740
00:43:36,120 –> 00:43:40,000
you can still read it if you can understand Latin.

741
00:43:40,000 –> 00:43:42,120
Fortunately, I had help.

742
00:43:44,200 –> 00:43:46,960
The first entry is in 1796.

743
00:43:46,960 –> 00:43:49,600

• Is it possible to lift it up?
• Yes, but be careful.

744
00:43:49,600 –> 00:43:54,160
It’s really one of the most valuable things that this library possesses.

745
00:43:54,160 –> 00:43:56,680

• Yes, I can believe that.
• He writes beautifully.

746
00:43:56,680 –> 00:43:59,120
It is aesthetically very pleasing,

747
00:43:59,120 –> 00:44:02,560
even if people don’t understand what it is.

748
00:44:02,560 –> 00:44:05,320
I’m going to put this down. It’s very delicate.

749
00:44:05,320 –> 00:44:08,520
The diary proves that some of Gauss’ ideas

750
00:44:08,520 –> 00:44:10,240
were 100 years ahead of their time.

751
00:44:10,240 –> 00:44:15,520
Here are some sines and integrals. Very different sort of mathematics.

752
00:44:15,520 –> 00:44:20,400
Yes, this was the first intimations of the theory

753
00:44:20,400 –> 00:44:25,040
of elliptic functions, which was one of his other great developments.

754
00:44:25,040 –> 00:44:28,600
And here you see something that is basically

755
00:44:28,600 –> 00:44:30,720
the Riemann zeta function appearing.

756
00:44:30,720 –> 00:44:34,200
Wow, gosh! That’s very impressive.

757
00:44:34,200 –> 00:44:38,880
The zeta function has become a vital element in our present understanding

758
00:44:38,880 –> 00:44:43,600
of the distribution of the building blocks of all numbers, the primes.

759
00:44:43,600 –> 00:44:47,280
There is somewhere in the diary here where he says,

760
00:44:47,280 –> 00:44:49,280
“I have made this wonderful discovery

761
00:44:49,280 –> 00:44:52,000
“and incidentally, a son was born today.”

762
00:44:52,000 –> 00:44:53,640
We see his priorities!

763
00:44:53,640 –> 00:44:55,560
Yes, indeed!

764
00:44:55,560 –> 00:44:58,600
I think I know a few mathematicians like that, too.

765
00:45:00,320 –> 00:45:03,800
My priorities, though, for the rest of the afternoon were clear.

766
00:45:03,800 –> 00:45:05,560
I needed another walk.

767
00:45:05,560 –> 00:45:08,960
Fortunately, Gottingen is surrounded by good woodland trails.

768
00:45:08,960 –> 00:45:10,920
It was a perfect setting for me

769
00:45:10,920 –> 00:45:13,440
to think more about Gauss’ discoveries.

770
00:45:22,400 –> 00:45:26,280
Gauss’ mathematics has touched many parts of the mathematical world,

771
00:45:26,280 –> 00:45:31,320
but I’m going to just choose one of them, a fun one - imaginary numbers.

772
00:45:31,320 –> 00:45:34,920
In the 16th and 17th century, European mathematicians

773
00:45:34,920 –> 00:45:40,120
imagined the square root of minus one and gave it the symbol i.

774
00:45:40,120 –> 00:45:42,760
They didn’t like it much, but it solved equations

775
00:45:42,760 –> 00:45:45,240
that couldn’t be solved any other way.

776
00:45:46,320 –> 00:45:49,760
Imaginary numbers have helped us to understand radio waves,

777
00:45:49,760 –> 00:45:52,000
to build bridges and aeroplanes.

778
00:45:52,000 –> 00:45:54,240
They’re even the key to quantum physics,

779
00:45:54,240 –> 00:45:56,560
the science of the sub-atomic world.

780
00:45:56,560 –> 00:46:01,400
They’ve provided a map to see how things really are.

781
00:46:01,400 –> 00:46:05,560
But back in the early 19th century, they had no map, no picture

782
00:46:05,560 –> 00:46:08,560
of how imaginary numbers connected with real numbers.

783
00:46:08,560 –> 00:46:10,760
Where is this new number?

784
00:46:10,760 –> 00:46:14,240
There’s no room on the number line for the square root of minus one.

785
00:46:14,240 –> 00:46:16,320
I’ve got the positive numbers running out here,

786
00:46:16,320 –> 00:46:17,880
the negative numbers here.

787
00:46:17,880 –> 00:46:21,600
The great step is to create a new direction of numbers,

788
00:46:21,600 –> 00:46:23,560
perpendicular to the number line,

789
00:46:23,560 –> 00:46:26,720
and that’s where the square root of minus one is.

790
00:46:28,880 –> 00:46:32,600
Gauss was not the first to come up with this two-dimensional picture

791
00:46:32,600 –> 00:46:36,720
of numbers, but he was the first person to explain it all clearly.

792
00:46:36,720 –> 00:46:38,760
He gave people a picture to understand

793
00:46:38,760 –> 00:46:40,920
how imaginary numbers worked.

794
00:46:40,920 –> 00:46:43,080
And once they’d developed this picture,

795
00:46:43,080 –> 00:46:46,200
their immense potential could really be unleashed.

796
00:46:46,200 –> 00:46:49,680
Guten Morgen. Ein Kaffee, bitte.

797
00:46:49,680 –> 00:46:53,120
His maths led to a claim and financial security for Gauss.

798
00:46:53,120 –> 00:46:56,360
He could have gone anywhere, but he was happy enough

799
00:46:56,360 –> 00:47:01,680
to settle down and spend the rest of his life in sleepy Gottingen.

800
00:47:01,680 –> 00:47:03,920
Unfortunately, as his fame developed,

801
00:47:03,920 –> 00:47:06,080
so his character deteriorated.

802
00:47:06,080 –> 00:47:08,440
A naturally conservative, shy man,

803
00:47:08,440 –> 00:47:12,760
he became increasingly distrustful and grumpy.

804
00:47:12,760 –> 00:47:16,600
Many young mathematicians across Europe regarded Gauss as a god

805
00:47:16,600 –> 00:47:18,720
and they would send in their theorems,

806
00:47:18,720 –> 00:47:20,720
their conjectures, even some proofs.

807
00:47:20,720 –> 00:47:23,560
But most of the time, he wouldn’t respond, and even when he did,

808
00:47:23,560 –> 00:47:26,480
it was generally to say either that they’d got it wrong

809
00:47:26,480 –> 00:47:28,480
or he’d proved it already.

810
00:47:28,480 –> 00:47:32,600
His dismissal or lack of interest in the work of lesser mortals

811
00:47:32,600 –> 00:47:35,360
sometimes discouraged some very talented mathematicians

812
00:47:35,360 –> 00:47:38,120
from pursuing their ideas.

813
00:47:38,120 –> 00:47:40,240
But occasionally, Gauss also failed

814
00:47:40,240 –> 00:47:45,040
to follow up on his own insights, including one very important insight

815
00:47:45,040 –> 00:47:48,240
that might have transformed the mathematics of his time.

816
00:47:50,400 –> 00:47:53,640
15 kilometres outside Gottingen stands what is known today

817
00:47:53,640 –> 00:47:55,640
as the Gauss Tower.

818
00:47:55,640 –> 00:47:57,960
Wow, that is stunning.

819
00:47:57,960 –> 00:48:01,640
It is really a fantastic view here, yes.

820
00:48:01,640 –> 00:48:05,040
Gauss took on many projects for the Hanoverian government,

821
00:48:05,040 –> 00:48:09,320
including the first proper survey of all the lands of Hanover.

822
00:48:09,320 –> 00:48:12,560
Was this Gauss’ choice to do this surveying?

823
00:48:12,560 –> 00:48:16,120
For a mathematician, it sounds like the last thing I’d want to do.

824
00:48:16,120 –> 00:48:17,320
He wanted to do it.

825
00:48:17,320 –> 00:48:23,280
The major point in doing this was to discover the shape of the earth.

826
00:48:23,280 –> 00:48:25,280
But he also started speculating

827
00:48:25,280 –> 00:48:29,880
about something even more revolutionary - the shape of space.

828
00:48:29,880 –> 00:48:34,720
So he’s thinking there may not be anything flat in the universe?

829
00:48:34,720 –> 00:48:37,280
Yes. And if we were living in a curved universe,

830
00:48:37,280 –> 00:48:40,480
there wouldn’t be anything flat.

831
00:48:40,480 –> 00:48:44,680
This led Gauss to question one of the central tenets of mathematics -

832
00:48:44,680 –> 00:48:47,280
Euclid’s geometry.

833
00:48:47,280 –> 00:48:50,160
He realised that this geometry, far from universal,

834
00:48:50,160 –> 00:48:52,960
depended on the idea of space as flat.

835
00:48:52,960 –> 00:48:56,160
It just didn’t apply to a universe that was curved.

836
00:48:56,160 –> 00:48:59,520
But in the early 19th century, Euclid’s geometry

837
00:48:59,520 –> 00:49:03,320
was seen as God-given and Gauss didn’t want any trouble.

838
00:49:03,320 –> 00:49:05,640
So he never published anything.

839
00:49:05,640 –> 00:49:09,200
Another mathematician, though, had no such fears.

840
00:49:11,960 –> 00:49:16,000
In mathematics, it’s often helpful to be part of a community

841
00:49:16,000 –> 00:49:19,320
where you can talk to and bounce ideas off others.

842
00:49:19,320 –> 00:49:22,160
But inside such a mathematical community,

843
00:49:22,160 –> 00:49:25,400
it can sometimes be difficult to come up with that one idea

844
00:49:25,400 –> 00:49:28,760
that completely challenges the status quo,

845
00:49:28,760 –> 00:49:33,560
and then the breakthrough often comes from somewhere else.

846
00:49:33,560 –> 00:49:36,840
Mathematics can be done in some pretty weird places.

847
00:49:36,840 –> 00:49:38,440
I’m in Transylvania,

848
00:49:38,440 –> 00:49:42,040
which is fairly appropriate, cos I’m in search of a lone wolf.

849
00:49:42,040 –> 00:49:45,200
Janos Bolyai spent much of his life

850
00:49:45,200 –> 00:49:49,520
hundreds of miles away from the mathematical centres of excellence.

851
00:49:49,520 –> 00:49:53,600
This is the only portrait of him that I was able to find.

852
00:49:53,600 –> 00:49:56,600
Unfortunately, it isn’t actually him.

853
00:49:56,600 –> 00:50:00,040
It’s one that the Communist Party in Romania started circulating

854
00:50:00,040 –> 00:50:04,000
when people got interested in his theories in the 1960s.

855
00:50:04,000 –> 00:50:06,480
They couldn’t find a picture of Janos.

856
00:50:06,480 –> 00:50:09,520
So they substituted a picture of somebody else instead.

857
00:50:11,800 –> 00:50:15,520
Born in 1802, Janos was the son of Farkas Bolyai,

858
00:50:15,520 –> 00:50:17,120
who was a maths teacher.

859
00:50:17,120 –> 00:50:20,400
He realised his son was a mathematical prodigy,

860
00:50:20,400 –> 00:50:23,720
so he wrote to his old friend Carl Friedrich Gauss,

861
00:50:23,720 –> 00:50:25,640
asking him to tutor the boy.

862
00:50:25,640 –> 00:50:28,880
Sadly, Gauss declined.

863
00:50:28,880 –> 00:50:31,760
So instead of becoming a professional mathematician,

864
00:50:31,760 –> 00:50:33,920
Janos joined the Army.

865
00:50:33,920 –> 00:50:37,080
But mathematics remained his first love.

866
00:50:40,680 –> 00:50:44,320
Maybe there’s something about the air here because Bolyai carried on

867
00:50:44,320 –> 00:50:46,720
doing his mathematics in his spare time.

868
00:50:46,720 –> 00:50:50,360
He started to explore what he called imaginary geometries,

869
00:50:50,360 –> 00:50:55,040
where the angles in triangles add up to less than 180.

870
00:50:55,040 –> 00:50:58,240
The amazing thing is that these imaginary geometries

871
00:50:58,240 –> 00:51:00,720
make perfect mathematical sense.

872
00:51:04,520 –> 00:51:09,280
Bolyai’s new geometry has become known as hyperbolic geometry.

873
00:51:09,280 –> 00:51:12,800
The best way to imagine it is a kind of mirror image of a sphere

874
00:51:12,800 –> 00:51:15,440
where lines curve back on each other.

875
00:51:15,440 –> 00:51:18,320
It’s difficult to represent it since we are so used

876
00:51:18,320 –> 00:51:21,680
to living in space which appears to be straight and flat.

877
00:51:23,800 –> 00:51:25,480
In his hometown of Targu Mures,

878
00:51:25,480 –> 00:51:29,600
I went looking for more about Bolyai’s revolutionary mathematics.

879
00:51:29,600 –> 00:51:33,040
His memory is certainly revered here.

880
00:51:33,040 –> 00:51:36,760
The museum contains a collection of Bolyai-related artefacts,

881
00:51:36,760 –> 00:51:40,520
some of which might be considered distinctly Transylvanian.

882
00:51:40,520 –> 00:51:42,480
It’s still got some hair on it.

883
00:51:42,480 –> 00:51:45,160
It’s kind of a little bit gruesome.

884
00:51:45,160 –> 00:51:46,760
But the object I like most here

885
00:51:46,760 –> 00:51:50,400
is a beautiful model of Bolyai’s geometry.

886
00:51:50,400 –> 00:51:54,000
You got the shortest distance between here and here

887
00:51:54,000 –> 00:51:56,760
if you stick on this surface. It’s not a straight line,

888
00:51:56,760 –> 00:51:59,160
but this curved line which of bends into the triangle.

889
00:51:59,160 –> 00:52:03,760
Here is a surface where the shortest distances which define the triangle

890
00:52:03,760 –> 00:52:06,040
add up to less than 180.

891
00:52:06,040 –> 00:52:09,440
Bolyai published his work in 1831.

892
00:52:09,440 –> 00:52:12,360
His father sent his old friend Gauss a copy.

893
00:52:12,360 –> 00:52:16,280
Gauss wrote back straightaway giving his approval,

894
00:52:16,280 –> 00:52:19,440
but Gauss refused to praise the young Bolyai,

895
00:52:19,440 –> 00:52:22,560
because he said the person he should be praising was himself.

896
00:52:22,560 –> 00:52:26,200
He had worked it all out a decade or so before.

897
00:52:26,200 –> 00:52:29,760
Actually, there is a letter from Gauss

898
00:52:29,760 –> 00:52:32,200
to another friend of his where he says,

899
00:52:32,200 –> 00:52:34,840
“I regard this young geometer boy

900
00:52:34,840 –> 00:52:37,960
“as a genius of the first order.”

901
00:52:37,960 –> 00:52:41,560
But Gauss never thought to tell Bolyai that.

902
00:52:41,560 –> 00:52:44,520
And young Janos was completely disheartened.

903
00:52:44,520 –> 00:52:47,040
Another body blow soon followed.

904
00:52:47,040 –> 00:52:49,880
Somebody else had developed exactly the same idea,

905
00:52:49,880 –> 00:52:52,000
but had published two years before him -

906
00:52:52,000 –> 00:52:55,080
the Russian mathematician Nicholas Lobachevsky.

907
00:52:57,560 –> 00:53:00,080
It was all downhill for Bolyai after that.

908
00:53:00,080 –> 00:53:04,080
With no recognition or career, he didn’t publish anything else.

909
00:53:04,080 –> 00:53:06,960
Eventually, he went a little crazy.

910
00:53:08,440 –> 00:53:13,160
In 1860, Janos Bolyai died in obscurity.

911
00:53:15,280 –> 00:53:19,040
Gauss, by contrast, was lionised after his death.

912
00:53:19,040 –> 00:53:22,560
A university, the units used to measure magnetic induction,

913
00:53:22,560 –> 00:53:25,520
even a crater on the moon would be named after him.

914
00:53:28,760 –> 00:53:31,600
During his lifetime, Gauss lent his support

915
00:53:31,600 –> 00:53:33,960
to very few mathematicians.

916
00:53:33,960 –> 00:53:38,840
But one exception was another of Gottingen’s mathematical giants -

917
00:53:38,840 –> 00:53:41,840
Bernhard Riemann.

918
00:53:48,280 –> 00:53:49,800
His father was a minister

919
00:53:49,800 –> 00:53:54,080
and he would remain a sincere Christian all his life.

920
00:53:54,080 –> 00:53:58,280
But Riemann grew up a shy boy who suffered from consumption.

921
00:53:58,280 –> 00:54:00,640
His family was large and poor and the only thing

922
00:54:00,640 –> 00:54:04,560
the young boy had going for him was an excellence at maths.

923
00:54:04,560 –> 00:54:07,720
That was his salvation.

924
00:54:07,720 –> 00:54:11,240
Many mathematicians like Riemann had very difficult childhoods,

925
00:54:11,240 –> 00:54:14,960
were quite unsociable. Their lives seemed to be falling apart.

926
00:54:14,960 –> 00:54:18,800
It was mathematics that gave them a sense of security.

927
00:54:21,920 –> 00:54:24,800
Riemann spent much of his early life in the town of Luneburg

928
00:54:24,800 –> 00:54:26,840
in northern Germany.

929
00:54:26,840 –> 00:54:30,440
This was his local school, built as a direct result

930
00:54:30,440 –> 00:54:34,280
of Humboldt’s educational reforms in the early 19th century.

931
00:54:34,280 –> 00:54:37,040
Riemann was one of its first pupils.

932
00:54:37,040 –> 00:54:41,360
The head teacher saw a way of bringing out the shy boy.

933
00:54:41,360 –> 00:54:44,320
He was given the freedom of the school’s library.

934
00:54:44,320 –> 00:54:46,880
It opened up a whole new world to him.

935
00:54:46,880 –> 00:54:48,680
One of the books he found in there

936
00:54:48,680 –> 00:54:51,480
was a book by the French mathematician Legendre,

937
00:54:51,480 –> 00:54:53,000
all about number theory.

938
00:54:53,000 –> 00:54:55,680
His teacher asked him how he was getting on with it.

939
00:54:55,680 –> 00:55:01,360
He replied, “I have understood all 859 pages of this wonderful book.”

940
00:55:01,360 –> 00:55:04,520
It was a strategy that obviously suited Riemann

941
00:55:04,520 –> 00:55:07,080
because he became a brilliant mathematician.

942
00:55:07,080 –> 00:55:12,280
One of his most famous contributions to mathematics was a lecture in 1852

943
00:55:12,280 –> 00:55:16,400
on the foundations of geometry. In the lecture,

944
00:55:16,400 –> 00:55:20,120
Riemann first described what geometry actually was

945
00:55:20,120 –> 00:55:22,160
and its relationship with the world.

946
00:55:22,160 –> 00:55:25,240
He then sketched out what geometry could be -

947
00:55:25,240 –> 00:55:28,240
a mathematics of many different kinds of space,

948
00:55:28,240 –> 00:55:31,240
only one of which would be the flat Euclidian space

949
00:55:31,240 –> 00:55:32,880
in which we appear to live.

950
00:55:32,880 –> 00:55:36,080
He was just 26 years old.

951
00:55:36,080 –> 00:55:40,560
Was it received well? Did people recognise the revolution?

952
00:55:40,560 –> 00:55:42,840
There was no way that people could actually

953
00:55:42,840 –> 00:55:45,040
make these ideas concrete.

954
00:55:45,040 –> 00:55:50,640
That only occurred 50, 60 years after this, with Einstein.

955
00:55:50,640 –> 00:55:53,400
So this is the beginning, really, of the revolution

956
00:55:53,400 –> 00:55:56,960

• which ends with Einstein’s relativity.
• Exactly.

957
00:55:56,960 –> 00:56:01,640
Riemann’s mathematics changed how we see the world.

958
00:56:01,640 –> 00:56:04,400
Suddenly, higher dimensional geometry appeared.

959
00:56:04,400 –> 00:56:06,640
The potential was there from Descartes,

960
00:56:06,640 –> 00:56:11,120
but it was Riemann’s imagination that made it happen.

961
00:56:11,120 –> 00:56:15,160
He began without putting any restriction

962
00:56:15,160 –> 00:56:18,680
on the dimensions whatsoever. This was something quite new,

963
00:56:18,680 –> 00:56:21,320
his way of thinking about things.

964
00:56:21,320 –> 00:56:24,800
Someone like Bolyai was really thinking about new geometries,

965
00:56:24,800 –> 00:56:26,920
but new two-dimensional geometries.

966
00:56:26,920 –> 00:56:30,160
New two-dimensional geometries. Riemann then broke away

967
00:56:30,160 –> 00:56:35,240
from all the limitations of two or three dimensions

968
00:56:35,240 –> 00:56:37,880
and began to think in in higher dimensions.

969
00:56:37,880 –> 00:56:39,400
And this was quite new.

970
00:56:39,400 –> 00:56:41,960
Multi-dimensional space is at the heart

971
00:56:41,960 –> 00:56:44,520
of so much mathematics done today.

972
00:56:44,520 –> 00:56:48,080
In geometry, number theory, and several other branches of maths,

973
00:56:48,080 –> 00:56:51,800
Riemann’s ideas still perplex and amaze.

974
00:56:52,760 –> 00:56:55,920
He died, though, in 1866.

975
00:56:55,920 –> 00:56:59,480
He was only 39 years old.

976
00:56:59,480 –> 00:57:02,960
Today, the results of Riemann’s mathematics are everywhere.

977
00:57:02,960 –> 00:57:07,520
Hyperspace is no longer science fiction, but science fact.

978
00:57:07,520 –> 00:57:11,280
In Paris, they have even tried to visualise what shapes

979
00:57:11,280 –> 00:57:13,880
in higher dimensions might look like.

980
00:57:15,680 –> 00:57:18,640
Just as the Renaissance artist Piero would have drawn a square

981
00:57:18,640 –> 00:57:22,880
inside a square to represent a cube on the two-dimensional canvas,

982
00:57:22,880 –> 00:57:27,360
the architect here at La Defense has built a cube inside a cube

983
00:57:27,360 –> 00:57:31,720
to represent a shadow of the four-dimensional hypercube.

984
00:57:31,720 –> 00:57:34,640
It is with Riemann’s work that we finally have

985
00:57:34,640 –> 00:57:37,120
the mathematical glasses to be able to explore

986
00:57:37,120 –> 00:57:39,360
such worlds of the mind.

987
00:57:42,480 –> 00:57:44,920
It’s taken a while to make these glasses fit,

988
00:57:44,920 –> 00:57:47,320
but without this golden age of mathematics,

989
00:57:47,320 –> 00:57:50,480
from Descartes to Riemann, there would be no calculus,

990
00:57:50,480 –> 00:57:55,240
no quantum physics, no relativity, none of the technology we use today.

991
00:57:55,240 –> 00:57:57,440
But even more important than that,

992
00:57:57,440 –> 00:58:00,800
their mathematics blew away the cobwebs

993
00:58:00,800 –> 00:58:04,520
and allowed us to see the world as it really is -

994
00:58:04,520 –> 00:58:07,680
a world much stranger than we ever thought.

995
00:58:11,080 –> 00:58:13,400
You can learn more about the story of maths

996
00:58:13,400 –> 00:58:16,000
at the Open University at:

997
00:58:26,680 –> 00:58:29,440
Subtitles by Red Bee Media Ltd

998
00:58:29,440 –> 00:58:33,320
Email subtitling@bbc.co.uk

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Subtitles by © Red Bee Media Ltd