0:00:23 > 0:00:27I'm walking in the mountains of the moon.
0:00:29 > 0:00:33I'm on the trail of the Renaissance artist, Piero della Francesca,
0:00:33 > 0:00:38so I've come to the town in northern Italy which Piero made his own.
0:00:38 > 0:00:41There it is, Urbino.
0:00:41 > 0:00:45I've come here to see some of Piero's finest works,
0:00:45 > 0:00:50masterpieces of art, but also masterpieces of mathematics.
0:00:52 > 0:00:55The artists and architects of the early Renaissance brought back
0:00:55 > 0:01:00the use of perspective, a technique that had been lost for 1,000 years,
0:01:00 > 0:01:03but using it properly turned out to be a lot
0:01:03 > 0:01:05more difficult than they'd imagined.
0:01:05 > 0:01:10Piero was the first major painter to fully understand perspective.
0:01:10 > 0:01:15That's because he was a mathematician as well as an artist.
0:01:15 > 0:01:17I came here to see his masterpiece,
0:01:17 > 0:01:21The Flagellation of Christ, but there was a problem.
0:01:21 > 0:01:24I've just been to see The Flagellation, and it's an
0:01:24 > 0:01:27absolutely stunning picture, but unfortunately, for various
0:01:27 > 0:01:30kind of Italian reasons, we're not allowed to go and film in there.
0:01:30 > 0:01:34But this is a maths programme, after all, and not an arts programme,
0:01:34 > 0:01:38so I've used a bit of mathematics to bring this picture alive.
0:01:38 > 0:01:43We can't go to the picture, but we can make the picture come to us.
0:01:43 > 0:01:46The problem of perspective is how
0:01:46 > 0:01:51to represent the three-dimensional world on a two-dimensional canvas.
0:01:51 > 0:01:54To give a sense of depth, a sense of the third dimension,
0:01:54 > 0:01:57Piero used mathematics.
0:01:57 > 0:02:00How big is he going to paint Christ,
0:02:00 > 0:02:03if this group of men here were a certain distance away
0:02:03 > 0:02:05from these men in the foreground?
0:02:07 > 0:02:11Get it wrong and the illusion of perspective is shattered.
0:02:11 > 0:02:14It's far from obvious how a three-dimensional world
0:02:14 > 0:02:19can be accurately represented on a two-dimensional surface.
0:02:19 > 0:02:23Look at how the parallel lines in the three-dimensional world
0:02:23 > 0:02:26are no longer parallel in the two-dimensional canvas, but meet
0:02:26 > 0:02:28at a vanishing point.
0:02:30 > 0:02:33And this is what the tiles in the picture really look like.
0:02:39 > 0:02:41What is emerging here is a new
0:02:41 > 0:02:45mathematical language which allows us to map one thing into another.
0:02:45 > 0:02:49The power of perspective unleashed a new way to see the world,
0:02:49 > 0:02:53a perspective that would cause a mathematical revolution.
0:02:55 > 0:02:59Piero's work was the beginning of a new way to understand geometry,
0:02:59 > 0:03:02but it would take another 200 years
0:03:02 > 0:03:05before other mathematicians would continue where he left off.
0:03:13 > 0:03:16Our journey has come north.
0:03:16 > 0:03:19By the 17th century, Europe had taken over
0:03:19 > 0:03:23from the Middle East as the world's powerhouse of mathematical ideas.
0:03:23 > 0:03:26Great strides had been made in the geometry
0:03:26 > 0:03:27of objects fixed in time and space.
0:03:27 > 0:03:30In France, Germany, Holland and Britain,
0:03:30 > 0:03:35the race was now on to understand the mathematics of objects in motion
0:03:35 > 0:03:38and the pursuit of this new mathematics started here in this
0:03:38 > 0:03:42village in the centre of France.
0:03:42 > 0:03:45Only the French would name a village after a mathematician.
0:03:45 > 0:03:47Imagine in England a town called
0:03:47 > 0:03:50Newton or Ball or Cayley. I don't think so!
0:03:50 > 0:03:54But in France, they really value their mathematicians.
0:03:54 > 0:03:57This is the village of Descartes in the Loire Valley.
0:03:57 > 0:04:00It was renamed after the famous philosopher
0:04:00 > 0:04:02and mathematician 200 years ago.
0:04:02 > 0:04:07Descartes himself was born here in 1596, a sickly child who lost
0:04:07 > 0:04:11his mother when very young, so he was allowed to stay in bed every
0:04:11 > 0:04:18morning until 11.00am, a practice he tried to continue all his life.
0:04:18 > 0:04:20To do mathematics, sometimes you just need to remove
0:04:20 > 0:04:25all distractions, to float off into a world of shapes and patterns.
0:04:25 > 0:04:28Descartes thought that the bed was the best place to achieve
0:04:28 > 0:04:30this meditative state.
0:04:30 > 0:04:32I think I know what he means.
0:04:36 > 0:04:39The house where Descartes undertook his bedtime meditations
0:04:39 > 0:04:43is now a museum dedicated to all things Cartesian.
0:04:43 > 0:04:46Come with me.
0:04:46 > 0:04:50Its exhibition pieces arranged, by curator Sylvie Garnier, show how
0:04:50 > 0:04:57his philosophical, scientific and mathematical ideas all fit together.
0:04:57 > 0:04:58It also features less familiar aspects
0:04:58 > 0:05:00of Descartes' life and career.
0:05:00 > 0:05:03So he decided to be a soldier...in the army,
0:05:05 > 0:05:08in the Protestant Army
0:05:08 > 0:05:14and too in the Catholic Army, not a problem for him
0:05:14 > 0:05:17because no patriotism.
0:05:17 > 0:05:20Sylvie is putting it very nicely,
0:05:20 > 0:05:23but Descartes was in fact a mercenary.
0:05:23 > 0:05:26He fought for the German Protestants, the French Catholics
0:05:26 > 0:05:29and anyone else who would pay him.
0:05:29 > 0:05:34Very early one autumn morning in 1628, he was in the Bavarian Army
0:05:34 > 0:05:37camped out on a cold river bank.
0:05:37 > 0:05:40Inspiration very often strikes in very strange places.
0:05:40 > 0:05:44The story is told how Descartes couldn't sleep one night,
0:05:44 > 0:05:46maybe because he was getting up so late
0:05:46 > 0:05:48or perhaps he was celebrating St Martin's Eve
0:05:48 > 0:05:50and had just drunk too much.
0:05:50 > 0:05:52Problems were tumbling around in his mind.
0:05:52 > 0:05:55He was thinking about his favourite subject, philosophy.
0:05:55 > 0:05:57He was finding it very frustrating.
0:05:57 > 0:06:01How can you actually know anything at all?!
0:06:01 > 0:06:04Then he slips into a dream...
0:06:06 > 0:06:10and in the dream he understood that the key was to build philosophy
0:06:10 > 0:06:14on the indisputable facts of mathematics.
0:06:14 > 0:06:19Numbers, he realised, could brush away the cobwebs of uncertainty.
0:06:19 > 0:06:23He wanted to publish all his radical ideas, but he was worried how they'd
0:06:23 > 0:06:28be received in Catholic France, so he packed his bags and left.
0:06:31 > 0:06:34Descartes found a home here in Holland.
0:06:34 > 0:06:37He'd been one of the champions of the new scientific revolution
0:06:37 > 0:06:41which rejected the dominant view that the sun went around the earth,
0:06:41 > 0:06:44an opinion that got scientists like Galileo
0:06:44 > 0:06:47into deep trouble with the Vatican.
0:06:47 > 0:06:50Descartes reckoned that here amongst the Protestant Dutch
0:06:50 > 0:06:53he would be safe, especially
0:06:53 > 0:06:55at the old university town of Leiden
0:06:55 > 0:06:58where they valued maths and science.
0:06:58 > 0:07:00I've come to Leiden too.
0:07:00 > 0:07:02Unfortunately, I'm late!
0:07:02 > 0:07:05Hello. Yeah, I'm sorry.
0:07:05 > 0:07:09I got a puncture. It took me a bit of time, yeah, yeah.
0:07:09 > 0:07:13Henk Bos is one of Europe's most eminent Cartesian scholars.
0:07:13 > 0:07:16He's not surprised the French scholar ended up in Leiden.
0:07:16 > 0:07:20He came to talk with people and some people were open to his ideas.
0:07:20 > 0:07:24This was not only mathematic. It was also a mechanics specially.
0:07:24 > 0:07:26He merged algebra and geometry.
0:07:26 > 0:07:31- Right.- So you could have formulas and figures and go back and forth.
0:07:31 > 0:07:36- So a sort of dictionary between the two?- Yeah, yeah.
0:07:39 > 0:07:41This dictionary, which was finally published here in Holland in 1637,
0:07:41 > 0:07:42included mainly controversial
0:07:42 > 0:07:46philosophical ideas, but the most radical thoughts
0:07:46 > 0:07:51were in the appendix, a proposal to link algebra and geometry.
0:07:53 > 0:07:58Each point in two dimensions can be described by two numbers,
0:07:58 > 0:08:02one giving the horizontal location, the second number giving the point's
0:08:02 > 0:08:04vertical location.
0:08:04 > 0:08:08As the point moves around a circle, these coordinates change,
0:08:08 > 0:08:12but we can write down an equation that identifies the changing value
0:08:12 > 0:08:15of these numbers at any point in the figure.
0:08:15 > 0:08:18Suddenly, geometry has turned into algebra.
0:08:18 > 0:08:20Using this transformation
0:08:20 > 0:08:24from geometry into numbers, you could tell, for example,
0:08:24 > 0:08:27if the curve on this bridge was part of a circle or not.
0:08:27 > 0:08:29You didn't need to use your eyes.
0:08:29 > 0:08:33Instead, the equations of the curve would reveal its secrets,
0:08:33 > 0:08:35but it wouldn't stop there.
0:08:35 > 0:08:39Descartes had unlocked the possibility of navigating geometries
0:08:39 > 0:08:43of higher dimensions, worlds our eyes will never see but are central
0:08:43 > 0:08:46to modern technology and physics.
0:08:46 > 0:08:50There's no doubt that Descartes was one of the giants of mathematics.
0:08:50 > 0:08:55Unfortunately, though, he wasn't the nicest of men.
0:08:55 > 0:08:59I think he was not an easy person, so...
0:08:59 > 0:09:04And he could be... he was very much concerned about
0:09:04 > 0:09:07his image. He was entirely
0:09:07 > 0:09:12self-convinced that he was right, also when he was wrong and his first
0:09:12 > 0:09:16reaction would be that the other one was stupid that hadn't understood it.
0:09:16 > 0:09:19Descartes may not have been the most congenial person,
0:09:19 > 0:09:22but there's no doubt that his insight into the connection
0:09:22 > 0:09:27between algebra and geometry transformed mathematics forever.
0:09:27 > 0:09:31For his mathematical revolution to work, though, he needed one other
0:09:31 > 0:09:32vital ingredient.
0:09:32 > 0:09:38To find that, I had to say goodbye to Henk and Leiden and go to church.
0:09:38 > 0:09:39CHORAL SINGING
0:09:44 > 0:09:46I'm not a believer myself, but there's little doubt
0:09:46 > 0:09:49that many mathematicians from the time of Descartes
0:09:49 > 0:09:52had strong religious convictions.
0:09:56 > 0:09:58Maybe it's just a coincidence,
0:09:58 > 0:10:02but perhaps it's because mathematics and religion are both building ideas
0:10:02 > 0:10:08upon an undisputed set of axioms - one plus one equals two. God exists.
0:10:08 > 0:10:11I think I know which set of axioms I've got my faith in.
0:10:14 > 0:10:16In the 17th century,
0:10:16 > 0:10:19there was a Parisian monk who went to the same school as Descartes.
0:10:19 > 0:10:22He loved mathematics as much as he loved God.
0:10:22 > 0:10:27Indeed, he saw maths and science as evidence of the existence of God,
0:10:27 > 0:10:31Marin Mersenne was a first-class mathematician.
0:10:31 > 0:10:34One of his discoveries in prime numbers is still named after him.
0:10:36 > 0:10:41But he's also celebrated for his correspondence.
0:10:41 > 0:10:44From his monastery in Paris, Mersenne acted like some kind of
0:10:44 > 0:10:4917th century internet hub, receiving ideas and then sending them on.
0:10:49 > 0:10:51It's not so different now.
0:10:51 > 0:10:55We sit like mathematical monks thinking about our ideas, then
0:10:55 > 0:10:59sending a message to a colleague and hoping for some reply.
0:11:00 > 0:11:05There was a spirit of mathematical communication in 17th century Europe
0:11:05 > 0:11:08which had not been seen since the Greeks.
0:11:08 > 0:11:13Mersenne urged people to read Descartes' new work on geometry.
0:11:13 > 0:11:15He also did something just as important.
0:11:15 > 0:11:20He publicised some new findings on the properties of numbers
0:11:20 > 0:11:23by an unknown amateur who would end up rivalling Descartes as the
0:11:23 > 0:11:26greatest mathematician of his time, Pierre de Fermat.
0:11:32 > 0:11:35Here in Beaumont-de-Lomagne
0:11:35 > 0:11:37near Toulouse, residents and visitors have come
0:11:37 > 0:11:42out to celebrate the life and work of the village's most famous son.
0:11:42 > 0:11:46But I'm not too sure what these gladiators are doing here!
0:11:46 > 0:11:50And the appearance of this camel came as a bit of a surprise too.
0:11:50 > 0:11:53The man himself would have hardly approved of
0:11:53 > 0:11:57the ideas of using fun and games to advance an interest in mathematics.
0:11:57 > 0:12:01Unlike the aristocratic Descartes, Fermat wouldn't have considered it
0:12:01 > 0:12:05worthless or common to create a festival of mathematics.
0:12:05 > 0:12:07Maths in action, that one.
0:12:07 > 0:12:10It's beautiful, really nice, yeah.
0:12:10 > 0:12:14Fermat's greatest contribution to mathematics was to virtually invent
0:12:14 > 0:12:16modern number theory.
0:12:16 > 0:12:18He devised a wide range of conjectures
0:12:18 > 0:12:21and theorems about numbers including his famous Last Theorem,
0:12:21 > 0:12:27the proof of which would puzzle mathematicians for over 350 years,
0:12:27 > 0:12:29but it's little help to me now.
0:12:29 > 0:12:31Getting it apart is the easy bit.
0:12:31 > 0:12:33It's putting it together, isn't it, that's the difficult bit.
0:12:33 > 0:12:36How many bits have I got? I've got six bits.
0:12:38 > 0:12:42I think what I need to do is put some symmetry into this.
0:12:42 > 0:12:45I'm afraid he's going to tell me how to do it and I don't want to see.
0:12:45 > 0:12:48I hate being told how to do a problem. I don't want to look.
0:12:48 > 0:12:52And he's laughing at me now because I can't do it.
0:12:52 > 0:12:54That's very unfair!
0:12:54 > 0:12:55Here we go.
0:12:56 > 0:12:59Can I put them together?
0:12:59 > 0:13:01I got it!
0:13:01 > 0:13:03Now that's the buzz of doing mathematics when
0:13:03 > 0:13:08the thing clicks together and suddenly you see the right answer.
0:13:08 > 0:13:12Remarkably, Fermat only tackled mathematics in his spare time.
0:13:12 > 0:13:15By day he was a magistrate.
0:13:15 > 0:13:19Battling with mathematical problems was his hobby and his passion.
0:13:21 > 0:13:23The wonderful thing about mathematics is
0:13:23 > 0:13:24you can do it anywhere.
0:13:24 > 0:13:26You don't have to have a laboratory.
0:13:26 > 0:13:28You don't even really need a library.
0:13:28 > 0:13:31Fermat used to do much of his work while sitting at the kitchen table
0:13:31 > 0:13:35or praying in his local church or up here on his roof.
0:13:35 > 0:13:38He may have looked like an amateur,
0:13:38 > 0:13:41but he took his mathematics very seriously indeed.
0:13:41 > 0:13:44Fermat managed to find several new patterns in numbers
0:13:44 > 0:13:46that had defeated mathematicians for centuries.
0:13:46 > 0:13:50One of my favourite theorems of Fermat
0:13:50 > 0:13:52is all to do with prime numbers.
0:13:52 > 0:13:55If you've got a prime number which when you divide it by four
0:13:55 > 0:13:58leaves remainder one, then Fermat showed you could
0:13:58 > 0:14:02always rewrite this number as two square numbers added together.
0:14:02 > 0:14:05For example, I've got 13 cloves of garlic here,
0:14:05 > 0:14:09a prime number which has remainder one when I divide it by four.
0:14:09 > 0:14:13Fermat proved you can rewrite this number as two square numbers added
0:14:13 > 0:14:17together, so 13 can be rewritten
0:14:17 > 0:14:22as three squared plus two squared, or four plus nine.
0:14:22 > 0:14:26The amazing thing is that Fermat proved this will work however big
0:14:26 > 0:14:31the prime number is. Provided it has remainder one on division by four,
0:14:31 > 0:14:33you can always rewrite that number
0:14:33 > 0:14:36as two square numbers added together.
0:14:39 > 0:14:42Ah, my God!
0:14:44 > 0:14:47What I love about this sort of day is the playfulness of mathematics
0:14:47 > 0:14:51and Fermat certainly enjoyed playing around with numbers. He loved
0:14:51 > 0:14:55looking for patterns in numbers and then the puzzle side of mathematics,
0:14:55 > 0:14:58he wanted to prove that these patterns would be there forever.
0:15:00 > 0:15:04But as well as being the basis for fun and games in the years to come,
0:15:04 > 0:15:09Fermat's mathematics would have some very serious applications.
0:15:09 > 0:15:11One of his theorems, his Little Theorem, is
0:15:11 > 0:15:16the basis of the codes that protect our credit cards on the internet.
0:15:16 > 0:15:20Technology we now rely on today all comes from the scribblings
0:15:20 > 0:15:22of a 17th-century mathematician.
0:15:24 > 0:15:28But the usefulness of Fermat's mathematics is nothing compared to
0:15:28 > 0:15:33that of our next great mathematician and he comes not from France at all,
0:15:33 > 0:15:34but from its great rival.
0:15:38 > 0:15:43In the 17th century, Britain was emerging as a world power.
0:15:43 > 0:15:46Its expansion and ambitions required new methods of measurement
0:15:46 > 0:15:51and computation and that gave a great boost to mathematics.
0:15:51 > 0:15:53The university towns of Oxford and Cambridge
0:15:53 > 0:15:58were churning out mathematicians who were in great demand
0:15:58 > 0:16:02and the greatest of them was Isaac Newton.
0:16:06 > 0:16:09I'm here in Grantham, where Isaac Newton grew up,
0:16:09 > 0:16:11and they're very proud of him here.
0:16:11 > 0:16:13They have a wonderful statue to him.
0:16:13 > 0:16:14They've even got
0:16:14 > 0:16:18the Isaac Newton Shopping Centre, with a nice apple logo up there.
0:16:18 > 0:16:21There's a school that he went to with a nice blue plaque
0:16:21 > 0:16:25and there's a museum over here in the Town Hall, although, actually,
0:16:25 > 0:16:28one of the other famous residents here, Margaret Thatcher,
0:16:28 > 0:16:30has got as big a display as Isaac Newton.
0:16:30 > 0:16:32In fact, the Thatcher cups have
0:16:32 > 0:16:36sold out and there's loads of Newton ones still left,
0:16:36 > 0:16:41so I thought I would support mathematics by buying a Newton cup.
0:16:41 > 0:16:43And Newton's maths does need support.
0:16:43 > 0:16:49- Newton's very famous here. Do you know what he's famous for?- No.
0:16:49 > 0:16:53- No, I don't.- Discovering gravity. - Gravity?- Gravity, yes.- Gravity?
0:16:53 > 0:16:58- Apple tree and all that, gravity. - 'That pretty much summed it up.
0:16:58 > 0:17:01'If people know about Newton's work at all, it is his physics,
0:17:01 > 0:17:05'his laws of gravity in motion, not his mathematics.'
0:17:05 > 0:17:07- I'm in a rush!- You're in a rush. OK.
0:17:07 > 0:17:10Acceleration, you see? One of Newton's laws!
0:17:18 > 0:17:20Eight miles south of Grantham,
0:17:20 > 0:17:22in the village of Woolsthorpe, where Newton was born,
0:17:22 > 0:17:26I met up with someone who does share my passion for his mathematics.
0:17:26 > 0:17:28This is the house.
0:17:28 > 0:17:32Wow, beautiful. 'Jackie Stedall is a Newton fan and more than willing
0:17:32 > 0:17:35'to show me around the house where Newton was brought up.'
0:17:35 > 0:17:37So here is the...
0:17:37 > 0:17:40you might call it the dining room. I'm sure they didn't call it that,
0:17:40 > 0:17:43but the room where they ate, next to the kitchen.
0:17:43 > 0:17:45Of course, there would have been a huge fire in there.
0:17:45 > 0:17:48Yes! Gosh, I wish it was there now!
0:17:48 > 0:17:50His father was an illiterate farmer,
0:17:50 > 0:17:53but he died shortly before Newton was born.
0:17:53 > 0:17:57Otherwise, the young Isaac's fate might have been very different.
0:17:57 > 0:17:59And here's his room.
0:17:59 > 0:18:01Oh, lovely, wow.
0:18:01 > 0:18:03- They present it really nicely.- Yes.
0:18:03 > 0:18:07- It's got a real feel of going back in time.- It does, yes.
0:18:07 > 0:18:10I can see he's as scruffy as I am. Look at the state of that bed.
0:18:10 > 0:18:13That's how, I think, I left my bed this morning.
0:18:13 > 0:18:18Newton hated his stepfather, but it was this man who ensured
0:18:18 > 0:18:21he became a mathematician rather than a sheep farmer.
0:18:21 > 0:18:23I don't think he was particularly remarkable as a child.
0:18:23 > 0:18:26- OK.- So there's hope for all those kids out there.- Yes, yes.
0:18:26 > 0:18:28I think he had a sort of average school report.
0:18:28 > 0:18:32He had very few close friends. I don't feel he's someone
0:18:32 > 0:18:33I particularly would have wanted to meet,
0:18:33 > 0:18:37but I do love his mathematics. It's wonderful.
0:18:37 > 0:18:40Newton came back to Lincolnshire from Cambridge
0:18:40 > 0:18:46during the Great Plague of 1665 when he was just 22 years old.
0:18:46 > 0:18:50In two miraculous years here, he developed a new theory of light,
0:18:50 > 0:18:52discovered gravitation
0:18:52 > 0:18:57and scribbled out a revolutionary approach to maths, the calculus.
0:18:57 > 0:18:59It works like this.
0:18:59 > 0:19:03I'm going to accelerate this car from 0 to 60 as quickly as I can.
0:19:03 > 0:19:07The speedometer is showing me that the speed's changing all the time,
0:19:07 > 0:19:09but this is only an average speed.
0:19:09 > 0:19:11How can I tell precisely what my speed is
0:19:11 > 0:19:15at any particular instant? Well, here's how.
0:19:15 > 0:19:20As the car races along the road, we can draw a graph above the road
0:19:20 > 0:19:23where the height above each point in the road records how long it took
0:19:23 > 0:19:26the car to get to that point.
0:19:26 > 0:19:28I can calculate the average speed between
0:19:28 > 0:19:33two points, A and B, on my journey by recording the distance travelled
0:19:33 > 0:19:37and dividing by the time it took to get between these two points,
0:19:37 > 0:19:42but what about the precise speed at the first point, A?
0:19:43 > 0:19:48If I move point B closer and closer to the first point, I take a smaller
0:19:48 > 0:19:51and smaller window of time and the speed gets closer
0:19:51 > 0:19:55and closer to the true value, but eventually, it looks like
0:19:55 > 0:19:59I have to calculate 0 divided by 0.
0:19:59 > 0:20:03The calculus allows us to make sense of this calculation.
0:20:03 > 0:20:08It enables us to work out the exact speed and also the precise distance
0:20:08 > 0:20:11travelled at any moment in time.
0:20:11 > 0:20:15I mean, it does make sense, the things we take for granted so much,
0:20:15 > 0:20:16things like... if I drop this apple...
0:20:16 > 0:20:18Its distance is changing and its
0:20:18 > 0:20:20speed is changing and calculus can deal with all of that.
0:20:20 > 0:20:22Which is quite in contrast to the Greeks.
0:20:22 > 0:20:25It was a very static geometry.
0:20:25 > 0:20:27- Yes, it is.- And here we see...
0:20:27 > 0:20:29so the calculus is used by
0:20:29 > 0:20:33every engineer, physicist, because it can describe the moving world.
0:20:33 > 0:20:36Yes, and it's the only way really you can deal with the mathematics of
0:20:36 > 0:20:38motion or with change.
0:20:38 > 0:20:40There's a lot of mathematics in this apple!
0:20:42 > 0:20:46Newton's calculus enables us to really understand
0:20:46 > 0:20:50the changing world, the orbits of planets, the motions of fluids.
0:20:50 > 0:20:54Through the power of the calculus, we have a way of describing, with
0:20:54 > 0:20:58mathematical precision, the complex, ever-changing natural world.
0:21:04 > 0:21:09But it would take 200 years to realise its full potential.
0:21:09 > 0:21:12Newton himself decided not to publish, but just to circulate
0:21:12 > 0:21:14his thoughts among friends.
0:21:14 > 0:21:17His reputation, though, gradually spread.
0:21:17 > 0:21:21He became a professor, an MP, and then Warden of the Royal Mint
0:21:21 > 0:21:23here in the City of London.
0:21:25 > 0:21:28On his regular trips to the Royal Society from the Royal Mint,
0:21:28 > 0:21:33he preferred to think about theology and alchemy rather than mathematics.
0:21:33 > 0:21:35Developing the calculus just got crowded out
0:21:35 > 0:21:39by all his other interests until he heard about a rival...
0:21:41 > 0:21:46a rival who was also a member of the Royal Society and who came up
0:21:46 > 0:21:48with exactly the same idea as him,
0:21:48 > 0:21:50Gottfried Leibniz.
0:21:50 > 0:21:54Every word Leibniz wrote has been preserved and catalogued
0:21:54 > 0:21:57in his hometown of Hanover in northern Germany.
0:21:57 > 0:22:01His actual manuscripts are kept under lock and key,
0:22:01 > 0:22:04particularly the manuscript which shows how Leibniz
0:22:04 > 0:22:09also discovered the miracle of calculus, shortly after Newton.
0:22:09 > 0:22:11What age was he when he wrote...
0:22:11 > 0:22:16He was 29 years old and that's the time, within two months, he developed
0:22:16 > 0:22:19- differential calculus and integral calculus.- In two months?
0:22:19 > 0:22:21- Yeah.- Fast and furious, when it comes, er...
0:22:21 > 0:22:23Yeah.
0:22:23 > 0:22:26There is a little scrap of paper over here. What's that one?
0:22:26 > 0:22:29- A letter or...- That's a small manuscript of Leibniz's notes.
0:22:32 > 0:22:37"Sometimes it happens that in the morning lying in the bed,
0:22:37 > 0:22:40"I have so many ideas that it takes the whole morning and sometimes
0:22:40 > 0:22:45"even longer to note all these ideas and bring them to paper."
0:22:45 > 0:22:47I suppose, that's beautiful.
0:22:47 > 0:22:51I suppose that he liked to lie in the bed in the morning.
0:22:51 > 0:22:53- A true mathematician.- Yeah.
0:22:53 > 0:22:55He spends his time thinking in bed.
0:22:55 > 0:22:58I see you've got some paintings down here.
0:22:58 > 0:23:00A painting.
0:23:00 > 0:23:02This is what he looked like. Right.
0:23:03 > 0:23:07Even though he didn't become quite the 17th century celebrity
0:23:07 > 0:23:10that Newton did, it wasn't such a bad life.
0:23:10 > 0:23:12Leibniz worked for the Royal Family
0:23:12 > 0:23:16of Hanover and travelled around Europe representing their interests.
0:23:16 > 0:23:19This gave him plenty of time to indulge in
0:23:19 > 0:23:23his favourite intellectual pastimes, which were wide, even for the time.
0:23:23 > 0:23:26He devised a plan for reunifying the Protestant and Roman Catholic
0:23:26 > 0:23:32churches, a proposal for France to conquer Egypt and contributions to
0:23:32 > 0:23:36philosophy and logic which are still highly rated today.
0:23:36 > 0:23:39- He wrote all these letters?- Yeah. - That's absolutely extraordinary.
0:23:39 > 0:23:43He must have cloned himself. I can't believe there was just one Leibniz!
0:23:43 > 0:23:46'But Leibniz was not just man of words.
0:23:46 > 0:23:47'He was also one of the first people
0:23:47 > 0:23:49'to invent practical calculating machines
0:23:49 > 0:23:54'that worked on the binary system, true forerunners of the computer.
0:23:54 > 0:23:58'300 years later, the engineering department at Leibniz University
0:23:58 > 0:24:02'in Hanover have put them together following Leibniz's blueprint.'
0:24:02 > 0:24:04I love all the ball bearings, so these are going to be all
0:24:04 > 0:24:06of our zeros and ones. So a ball bearing is a one.
0:24:06 > 0:24:10Only zero and one. Now we represent a number 127.
0:24:10 > 0:24:15- In binary, it means that we have the first seven digits in one.- Yeah.
0:24:15 > 0:24:18- And now I give the number one.- OK.
0:24:18 > 0:24:24Now we add 127 plus one - is 128, which is two, power eight.
0:24:24 > 0:24:28- Oh, OK. So there's going to be lots of action.- Would you show this here?
0:24:28 > 0:24:30This is the money shot.
0:24:30 > 0:24:33So we're going to add one. Oops. Here we go. They're all carrying.
0:24:33 > 0:24:36So this 128 is two power eight.
0:24:36 > 0:24:42Excellent, so 127 in binary is 1, 1, 1, 1, 1, 1, 1, which is
0:24:42 > 0:24:44all the ball bearings here.
0:24:44 > 0:24:46To add one it all gets
0:24:46 > 0:24:50carried, this goes to 0, 0, 0, 0, and we have a power of two here.
0:24:50 > 0:24:53So this mechanism gets rid of all the ball bearings that you
0:24:53 > 0:24:56- don't need. It's like pinball, mathematical pinball.- Exactly.
0:24:56 > 0:24:58I love this machine!
0:25:03 > 0:25:08After a hard day's work, Leibniz often came here,
0:25:08 > 0:25:10the famous gardens of Herrenhausen,
0:25:10 > 0:25:14now in the middle of Hanover, but then on the outskirts of the city.
0:25:14 > 0:25:17There's something about mathematics and walking.
0:25:17 > 0:25:21I don't know, you've been working at your desk all day, all morning
0:25:21 > 0:25:22on some problem and your head's all
0:25:22 > 0:25:25fuzzy, and you just need to come and have a walk.
0:25:25 > 0:25:27You let your subconscious mind kind of take over and sometimes
0:25:27 > 0:25:31you get your breakthrough just looking at the trees or whatever.
0:25:31 > 0:25:35I've had some of my best ideas whilst walking in my local park,
0:25:35 > 0:25:39so I'm hoping to get a little bit of inspiration here on Leibniz's
0:25:39 > 0:25:40local stomping ground.
0:25:44 > 0:25:47I didn't get the chance to purge my mind of mathematical challenges
0:25:47 > 0:25:49because in the years since Leibniz lived here,
0:25:49 > 0:25:50someone has built a maze.
0:25:50 > 0:25:53Well, there is a mathematical formula for getting out of a maze,
0:25:53 > 0:25:57which is if you put your left hand on the side of the maze and just
0:25:57 > 0:26:00keep it there, keep on winding round, you eventually get out.
0:26:00 > 0:26:03That's the theory, at least. Let's see whether it works!
0:26:11 > 0:26:13Leibniz had no such distractions.
0:26:13 > 0:26:17Within five years, he'd worked out the details of the calculus,
0:26:17 > 0:26:19seemingly independent from Newton,
0:26:19 > 0:26:21although he knew about Newton's work,
0:26:21 > 0:26:26but unlike Newton, Leibniz was quite happy to make his work known
0:26:26 > 0:26:29and so mathematicians across Europe heard about the calculus first
0:26:29 > 0:26:35from him and not from Newton, and that's when all the trouble started.
0:26:35 > 0:26:39Throughout mathematical history, there have been lots of priority
0:26:39 > 0:26:40disputes and arguments.
0:26:40 > 0:26:43It may seem a little bit petty and schoolboyish.
0:26:43 > 0:26:46We really want our name to be on that theorem.
0:26:46 > 0:26:49This is our one chance for a little bit of immortality because that
0:26:49 > 0:26:54theorem's going to last forever and that's why we dedicate so much time
0:26:54 > 0:26:55to trying to crack these things.
0:26:55 > 0:26:57Somehow we can't believe that somebody else
0:26:57 > 0:27:00has got it at the same time as us.
0:27:00 > 0:27:03These are our theorems, our babies, our children and we
0:27:03 > 0:27:06don't want to share the credit.
0:27:06 > 0:27:08Back in London, Newton certainly didn't want
0:27:08 > 0:27:13to share credit with Leibniz, who he thought of as a Hanoverian upstart.
0:27:13 > 0:27:16After years of acrimony and accusation, the Royal Society
0:27:16 > 0:27:21in London was asked to adjudicate between the rival claims.
0:27:21 > 0:27:23The Royal Society gave Newton credit
0:27:23 > 0:27:25for the first discovery of the calculus
0:27:25 > 0:27:28and Leibniz credit for the first publication,
0:27:28 > 0:27:33but in their final judgment, they accused Leibniz of plagiarism.
0:27:33 > 0:27:36However, that might have had something to do with the fact that
0:27:36 > 0:27:41the report was written by their President, one Sir Isaac Newton.
0:27:44 > 0:27:46Leibniz was incredibly hurt.
0:27:46 > 0:27:50He admired Newton and never really recovered.
0:27:50 > 0:27:52He died in 1716.
0:27:52 > 0:27:56Newton lived on another 11 years and was buried in the grandeur of
0:27:56 > 0:27:58Westminster Abbey.
0:27:58 > 0:28:00Leibniz's memorial, by contrast,
0:28:00 > 0:28:02is here in this small church in Hanover.
0:28:02 > 0:28:06The irony is that it's Leibniz's mathematics which
0:28:06 > 0:28:08eventually triumphs, not Newton's.
0:28:11 > 0:28:13I'm a big Leibniz fan.
0:28:13 > 0:28:16Quite often revolutions in mathematics are about producing the
0:28:16 > 0:28:19right language to capture a new vision and that's what
0:28:19 > 0:28:21Leibniz was so good at.
0:28:21 > 0:28:25Leibniz's notation, his way of writing the calculus,
0:28:25 > 0:28:27captured its true spirit.
0:28:27 > 0:28:29It's still the one we use in maths today.
0:28:29 > 0:28:34Newton's notation was, for many mathematicians, clumsy and difficult
0:28:34 > 0:28:38to use and so while British mathematics loses its way a little,
0:28:38 > 0:28:43the story of maths switches to the very heart of Europe, Basel.
0:28:48 > 0:28:52In its heyday in the 18th century, the free city of Basel in
0:28:52 > 0:28:56Switzerland was the commercial hub of the entire Western world.
0:28:56 > 0:28:59Around this maelstrom of trade, there developed a tradition of
0:28:59 > 0:29:03learning, particularly learning which connected with commerce
0:29:03 > 0:29:06and one family summed all this up.
0:29:06 > 0:29:11It's kind of curious - artists often have children who are artists.
0:29:11 > 0:29:15Musicians, their children are often musicians, but us mathematicians,
0:29:15 > 0:29:17our children don't tend to be mathematicians.
0:29:17 > 0:29:19I'm not sure why it is.
0:29:19 > 0:29:23At least that's my view, although others dispute it.
0:29:23 > 0:29:25What no-one disagrees with
0:29:25 > 0:29:30is there is one great dynasty of mathematicians, the Bernoullis.
0:29:30 > 0:29:33In the 18th and 19th centuries they produced half a dozen
0:29:33 > 0:29:37outstanding mathematicians, any of which we would have been
0:29:37 > 0:29:41proud to have had in Britain, and they all came from Basel.
0:29:41 > 0:29:44You might have great minds like Newton and Leibniz who make
0:29:44 > 0:29:48these fundamental breakthroughs, but you also need the disciples
0:29:48 > 0:29:51who take that message, clarify it, realise its implications,
0:29:51 > 0:29:55then spread it wide. The family were originally merchants,
0:29:55 > 0:29:57and this is one of their houses.
0:29:57 > 0:30:00It's now part of the University of Basel
0:30:00 > 0:30:03and it's been completely refurbished, apart from one room,
0:30:03 > 0:30:07which has been kept very much as the family would have used it.
0:30:07 > 0:30:09Dr Fritz Nagel, keeper of the Bernoulli Archive,
0:30:09 > 0:30:12has promised to show it to me.
0:30:12 > 0:30:15- If we can find it. - No, we're on the wrong floor.
0:30:15 > 0:30:17Wrong floor, OK. Right!
0:30:17 > 0:30:19Oh, look.
0:30:19 > 0:30:21Can we take an apple?
0:30:21 > 0:30:24'No, wrong mathematician.
0:30:24 > 0:30:26'Eventually, we got there.'
0:30:26 > 0:30:28This is where the Bernoullis would have done
0:30:28 > 0:30:30some of their mathematics.
0:30:30 > 0:30:33'I was really just being polite.
0:30:33 > 0:30:36'The only thing of interest was an old stove.'
0:30:36 > 0:30:40Now, of the Bernoullis, which is your favourite?
0:30:40 > 0:30:44My favourite Bernoulli is Johann I.
0:30:44 > 0:30:49He is the most smart mathematician.
0:30:49 > 0:30:54Perhaps his brother Jakob was the mathematician
0:30:54 > 0:30:57with the deeper insight into problems,
0:30:57 > 0:30:59but Johann found elegant solutions.
0:30:59 > 0:31:03The brothers didn't like each other much, but both worshipped Leibniz.
0:31:03 > 0:31:06They corresponded with him, stood up for him
0:31:06 > 0:31:10against Newton's allies, and spread his calculus throughout Europe.
0:31:10 > 0:31:15Leibnitz was very happy to have found two gifted mathematicians
0:31:15 > 0:31:20outside of his personal circle of friends who mastered his calculus
0:31:20 > 0:31:23and could distribute it in the scientific community.
0:31:23 > 0:31:28- That was very important for Leibniz. - And important for maths, too.
0:31:28 > 0:31:32Without the Bernoullis, it would have taken much longer for calculus
0:31:32 > 0:31:36to become what it is today, a cornerstone of mathematics.
0:31:36 > 0:31:38At least, that is Dr Nagel's contention.
0:31:38 > 0:31:41And he is a great Bernoulli fan.
0:31:41 > 0:31:44He has arranged for me to meet Professor Daniel Bernoulli,
0:31:44 > 0:31:46the latest member of the family,
0:31:46 > 0:31:49whose famous name ensures he gets some odd e-mails.
0:31:49 > 0:31:51Another one of which I got was,
0:31:51 > 0:31:54"Professor Bernoulli, can you give me a hand with calculus?"
0:31:54 > 0:31:58To find a Bernoulli, you expect them to be able to do calculus.
0:31:58 > 0:32:02'But this Daniel Bernoulli is a professor of geology.
0:32:02 > 0:32:05'The maths gene seems to have truly died out.
0:32:05 > 0:32:07'And during our very hearty dinner,
0:32:07 > 0:32:11'I found myself wandering back to maths.'
0:32:11 > 0:32:14It is a bit unfair on the Bernoullis to describe them simply
0:32:14 > 0:32:16as disciples of Leibniz.
0:32:16 > 0:32:18One of their many great contributions to mathematics
0:32:18 > 0:32:23was to develop the calculus to solve a classic problem of the day.
0:32:23 > 0:32:26Imagine a ball rolling down a ramp.
0:32:26 > 0:32:29The task is to design a ramp that will get the ball
0:32:29 > 0:32:32from the top to the bottom in the fastest time possible.
0:32:32 > 0:32:36You might think that a straight ramp would be quickest.
0:32:36 > 0:32:37Or possibly a curved one like this
0:32:37 > 0:32:40that gives the ball plenty of downward momentum.
0:32:40 > 0:32:42In fact, it's neither of these.
0:32:42 > 0:32:45Calculus shows that it is what we call a cycloid,
0:32:45 > 0:32:49the path traced by a point on the rim of a moving bicycle wheel.
0:32:49 > 0:32:53This application of the calculus by the Bernoullis, which became known
0:32:53 > 0:32:55as the calculus of variation,
0:32:55 > 0:32:58has become one of the most powerful aspects of the mathematics
0:32:58 > 0:33:01of Leibniz and Newton. Investors use it to maximise profits.
0:33:01 > 0:33:05Engineers exploit it to minimise energy use.
0:33:05 > 0:33:08Designers apply it to optimise construction.
0:33:08 > 0:33:10It has now become one of the linchpins
0:33:10 > 0:33:12of our modern technological world.
0:33:12 > 0:33:17Meanwhile, things were getting more interesting in the restaurant.
0:33:17 > 0:33:18Here is my second surprise.
0:33:18 > 0:33:22Let me introduce Mr Leonhard Euler.
0:33:22 > 0:33:23Daniel Bernoulli.
0:33:23 > 0:33:27'Leonhard Euler, one of the most famous names in mathematics.
0:33:27 > 0:33:29'This Leonhard is a descendant
0:33:29 > 0:33:34'of the original Leonhard Euler, star pupil of Johann Bernoulli.'
0:33:34 > 0:33:36I am the ninth generation,
0:33:36 > 0:33:39the fourth Leonhard in our family
0:33:39 > 0:33:42after Leonard Euler I, the mathematician.
0:33:42 > 0:33:44OK. And yourself, are you a mathematician?
0:33:44 > 0:33:47Actually, I am a business analyst.
0:33:47 > 0:33:51I can't study mathematics with my name.
0:33:51 > 0:33:55Everyone will expect you to prove that the Riemann hypothesis!
0:33:55 > 0:33:58Perhaps it's just as well that Leonhard decided
0:33:58 > 0:34:02not to follow in the footsteps of his illustrious ancestor.
0:34:02 > 0:34:04He'd have had a lot to live up to.
0:34:13 > 0:34:15I am going in a boat across the Rhine,
0:34:15 > 0:34:17and I'm feeling a little bit worse for wear.
0:34:17 > 0:34:21Last night's dinner with Mr Euler and Professor Bernoulli
0:34:21 > 0:34:25degenerated into toasts to all the theorems the Bernoullis and Eulers
0:34:25 > 0:34:28have proved, and by God, they have proved quite a lot of them!
0:34:28 > 0:34:30Never again.
0:34:30 > 0:34:34I was getting disapproving glances from my fellow passengers as well.
0:34:34 > 0:34:37Luckily, it was only a short trip.
0:34:37 > 0:34:41Not like the trip that Euler took in 1728 to start a new life.
0:34:41 > 0:34:45Euler may have been the prodigy of Johann Bernoulli,
0:34:45 > 0:34:47but there was no room for him in the city.
0:34:47 > 0:34:49If your name wasn't Bernoulli,
0:34:49 > 0:34:53there was little chance of getting a job in mathematics here in Basel.
0:34:53 > 0:34:55But Daniel, the son of Johann Bernoulli,
0:34:55 > 0:34:57was a great friend of Euler
0:34:57 > 0:35:00and managed to get him a job at his university.
0:35:00 > 0:35:03But to get there would take seven weeks,
0:35:03 > 0:35:05because Daniel's university was in Russia.
0:35:08 > 0:35:11It wasn't an intellectual powerhouse like Berlin or Paris,
0:35:11 > 0:35:17but St Petersburg was by no means unsophisticated in the 18th century.
0:35:17 > 0:35:21Peter the Great had created a city very much in the European style.
0:35:21 > 0:35:26And every fashionable city at the time had a scientific academy.
0:35:27 > 0:35:30Peter's Academy is now a museum.
0:35:30 > 0:35:34It includes several rooms full of the kind of grotesque curiosities
0:35:34 > 0:35:38that are usually kept out of the public display in the West.
0:35:38 > 0:35:39But in the 1730s,
0:35:39 > 0:35:44this building was a centre for ground-breaking research.
0:35:44 > 0:35:46It is where Euler found his intellectual home.
0:35:50 > 0:35:57# I am sure that there could never be a more contented man than me... #
0:35:58 > 0:36:00Many of the ideas that were bubbling away at the time -
0:36:00 > 0:36:02calculus of variation,
0:36:02 > 0:36:06Fermat's theory of numbers - crystallised in Euler's hands.
0:36:06 > 0:36:09But he was also creating incredibly modern mathematics,
0:36:09 > 0:36:12topology and analysis.
0:36:12 > 0:36:15Much of the notation that I use today as a mathematician
0:36:15 > 0:36:19was created by Euler, numbers like e and i.
0:36:19 > 0:36:23Euler also popularised the use of the symbol pi.
0:36:23 > 0:36:25He even combined these numbers together
0:36:25 > 0:36:28in one of the most beautiful formulas of mathematics,
0:36:28 > 0:36:32e to the power of i times pi is equal to -1.
0:36:32 > 0:36:36An amazing feat of mathematical alchemy.
0:36:36 > 0:36:39His life, in fact, is full of mathematical magic.
0:36:39 > 0:36:43Euler applied his skills to an immense range of topics,
0:36:43 > 0:36:46from prime numbers to optics to astronomy.
0:36:46 > 0:36:49He devised a new system of weights and measures, wrote a textbook
0:36:49 > 0:36:54on mechanics, and even found time to develop a new theory of music.
0:36:59 > 0:37:01I think of him as the Mozart of maths.
0:37:01 > 0:37:04And that view is shared by the mathematician Nikolai Vavilov,
0:37:04 > 0:37:07who met me at the house that was given to Euler
0:37:07 > 0:37:10by Catherine the Great.
0:37:10 > 0:37:14Euler lived here from '66 to '83, which means from the year
0:37:14 > 0:37:17he came back to St Petersburg to the year he died.
0:37:17 > 0:37:22And he was a member of the Russian Academy of Sciences,
0:37:22 > 0:37:24and their greatest mathematician.
0:37:24 > 0:37:27That is exactly what it says.
0:37:27 > 0:37:29- What is it now?- It is a school.
0:37:29 > 0:37:30Shall we go in and see?
0:37:30 > 0:37:33OK.
0:37:33 > 0:37:38'I'm not sure Nikolai entirely approved. But nothing ventured...'
0:37:38 > 0:37:41Perhaps we should talk to the head teacher.
0:37:46 > 0:37:48The head didn't mind at all.
0:37:48 > 0:37:50I rather got the impression that she was used
0:37:50 > 0:37:53to people dropping in to talk about Euler.
0:37:53 > 0:37:57She even had a couple of very able pupils suspiciously close to hand.
0:37:57 > 0:38:02These two young ladies are ready to tell a few words about the scientist
0:38:02 > 0:38:04and about this very building.
0:38:04 > 0:38:06They certainly knew their stuff.
0:38:06 > 0:38:09They had undertaken an entire classroom project on Euler,
0:38:09 > 0:38:13his long life, happy marriage and 13 children.
0:38:13 > 0:38:16And then his tragedies - only five of his children
0:38:16 > 0:38:17survived to adulthood.
0:38:17 > 0:38:21His first wife, who he adored, died young.
0:38:21 > 0:38:23He started losing most of his eyesight.
0:38:26 > 0:38:31So for the last years of his life, he still continued to work, actually.
0:38:31 > 0:38:34He continued his mathematical research.
0:38:34 > 0:38:36I read a quote that said now with his blindness,
0:38:36 > 0:38:38he hasn't got any distractions,
0:38:38 > 0:38:42he can finally get on with his mathematics. A positive attitude.
0:38:42 > 0:38:46It was a totally unexpected and charming visit.
0:38:46 > 0:38:49Although I couldn't resist sneaking back and correcting
0:38:49 > 0:38:53one of the equations on the board when everyone else had left.
0:38:54 > 0:38:59To demonstrate one of my favourite Euler theorems, I needed a drink.
0:38:59 > 0:39:02It concerns calculating infinite sums,
0:39:02 > 0:39:06the discovery that shot Euler to the top of the mathematical pops
0:39:06 > 0:39:08when it was announced in 1735.
0:39:11 > 0:39:15Take one shot glass full of vodka and add it to this tall glass here.
0:39:17 > 0:39:22Next, take a glass which is a quarter full, or a half squared,
0:39:22 > 0:39:24and add it to the first glass.
0:39:25 > 0:39:30Next, take a glass which is a ninth full, or a third squared,
0:39:30 > 0:39:31and add that one.
0:39:31 > 0:39:36Now, if I keep on adding infinitely many glasses where each one
0:39:36 > 0:39:43is a fraction squared, how much will be in this tall glass here?
0:39:43 > 0:39:45It was called the Basel problem
0:39:45 > 0:39:47after the Bernoullis tried and failed to solve it.
0:39:47 > 0:39:52Daniel Bernoulli knew that you would not get an infinite amount of vodka.
0:39:52 > 0:39:57He estimated that the total would come to about one and three fifths.
0:39:57 > 0:39:59But then Euler came along.
0:39:59 > 0:40:03Daniel was close, but mathematics is about precision.
0:40:03 > 0:40:06Euler calculated that the total height of the vodka
0:40:06 > 0:40:10would be exactly pi squared divided by six.
0:40:13 > 0:40:15It was a complete surprise.
0:40:15 > 0:40:17What on earth did adding squares of fractions
0:40:17 > 0:40:20have to do with the special number pi?
0:40:20 > 0:40:23But Euler's analysis showed that they were two sides
0:40:23 > 0:40:25of the same equation.
0:40:25 > 0:40:29One plus a quarter plus a ninth plus a sixteenth
0:40:29 > 0:40:34and so on to infinity is equal to pi squared over six.
0:40:34 > 0:40:38That's still quite a lot of vodka, but here goes.
0:40:43 > 0:40:46Euler would certainly be a hard act to follow.
0:40:46 > 0:40:49Mathematicians from two countries would try.
0:40:49 > 0:40:53Both France and Germany were caught up in the age of revolution
0:40:53 > 0:40:56that was sweeping Europe in the late 18th century.
0:40:56 > 0:40:59Both desperately needed mathematicians.
0:40:59 > 0:41:04But they went about supporting mathematics rather differently.
0:41:04 > 0:41:05Here in France,
0:41:05 > 0:41:09the Revolution emphasised the usefulness of mathematics.
0:41:09 > 0:41:12Napoleon recognised that if you were going to have
0:41:12 > 0:41:14the best military machine, the best weaponry,
0:41:14 > 0:41:17then you needed the best mathematicians.
0:41:17 > 0:41:21Napoleon's reforms gave mathematics a big boost.
0:41:21 > 0:41:24But this was a mathematics that was going to serve society.
0:41:25 > 0:41:30Here in the German states, the great educationalist Wilhelm von Humboldt
0:41:30 > 0:41:33was also committed to mathematics, but a mathematics that was detached
0:41:33 > 0:41:36from the demands of the State and the military.
0:41:36 > 0:41:42Von Humboldt's educational reforms valued mathematics for its own sake.
0:41:42 > 0:41:46In France, they got wonderful mathematicians, like Joseph Fourier,
0:41:46 > 0:41:49whose work on sound waves we still benefit from today.
0:41:49 > 0:41:53MP3 technology is based on Fourier analysis.
0:41:53 > 0:41:56But in Germany, they got, at least in my opinion,
0:41:56 > 0:41:58the greatest mathematician ever.
0:42:01 > 0:42:03Quaint and quiet,
0:42:03 > 0:42:08the university town of Gottingen may seem like a bit of a backwater.
0:42:08 > 0:42:12But this little town has been home to some of the giants of maths,
0:42:12 > 0:42:14including the man who's often described
0:42:14 > 0:42:19as the Prince of Mathematics, Carl Friedrich Gauss.
0:42:19 > 0:42:23Few non-mathematicians, however, seem to know anything about him.
0:42:23 > 0:42:25Not in Paris.
0:42:25 > 0:42:27Qui s'appelle Carl Friedrich Gauss?
0:42:27 > 0:42:28- Non.- Non?
0:42:28 > 0:42:30'Not in Oxford.'
0:42:30 > 0:42:34- I've heard the name but I couldn't tell you.- No idea.- No idea?- No.
0:42:34 > 0:42:37'And I'm afraid to say, not even in modern Germany.'
0:42:37 > 0:42:39- Nein.- Nein? OK.
0:42:39 > 0:42:41- I don't know.- You don't know?
0:42:41 > 0:42:44But in Gottingen, everyone knows who Gauss is.
0:42:44 > 0:42:47He's the local hero.
0:42:47 > 0:42:49His father was a stonemason
0:42:49 > 0:42:52and it's likely that Gauss would have become one, too.
0:42:52 > 0:42:55But his singular talent was recognised by his mother,
0:42:55 > 0:42:57and she helped ensure
0:42:57 > 0:43:01that he received the best possible education.
0:43:01 > 0:43:05Every few years in the news, you hear about a new prodigy
0:43:05 > 0:43:08who's passed their A-levels at ten, gone to university at 12,
0:43:08 > 0:43:10but nobody compares to Gauss.
0:43:10 > 0:43:13Already at the age of 12, he was criticising Euclid's geometry.
0:43:13 > 0:43:16At 15, he discovered a new pattern in prime numbers
0:43:16 > 0:43:20which had eluded mathematicians for 2,000 years.
0:43:20 > 0:43:24And at 19, he discovered the construction of a 17-sided figure
0:43:24 > 0:43:26which nobody had known before this time.
0:43:30 > 0:43:34His early successes encouraged Gauss to keep a diary.
0:43:34 > 0:43:36Here at the University of Gottingen,
0:43:36 > 0:43:40you can still read it if you can understand Latin.
0:43:40 > 0:43:42Fortunately, I had help.
0:43:44 > 0:43:46The first entry is in 1796.
0:43:46 > 0:43:49- Is it possible to lift it up? - Yes, but be careful.
0:43:49 > 0:43:54It's really one of the most valuable things that this library possesses.
0:43:54 > 0:43:56- Yes, I can believe that. - He writes beautifully.
0:43:56 > 0:43:59It is aesthetically very pleasing,
0:43:59 > 0:44:02even if people don't understand what it is.
0:44:02 > 0:44:05I'm going to put this down. It's very delicate.
0:44:05 > 0:44:08The diary proves that some of Gauss' ideas
0:44:08 > 0:44:10were 100 years ahead of their time.
0:44:10 > 0:44:15Here are some sines and integrals. Very different sort of mathematics.
0:44:15 > 0:44:20Yes, this was the first intimations of the theory
0:44:20 > 0:44:25of elliptic functions, which was one of his other great developments.
0:44:25 > 0:44:28And here you see something that is basically
0:44:28 > 0:44:30the Riemann zeta function appearing.
0:44:30 > 0:44:34Wow, gosh! That's very impressive.
0:44:34 > 0:44:38The zeta function has become a vital element in our present understanding
0:44:38 > 0:44:43of the distribution of the building blocks of all numbers, the primes.
0:44:43 > 0:44:47There is somewhere in the diary here where he says,
0:44:47 > 0:44:49"I have made this wonderful discovery
0:44:49 > 0:44:52"and incidentally, a son was born today."
0:44:52 > 0:44:53We see his priorities!
0:44:53 > 0:44:55Yes, indeed!
0:44:55 > 0:44:58I think I know a few mathematicians like that, too.
0:45:00 > 0:45:03My priorities, though, for the rest of the afternoon were clear.
0:45:03 > 0:45:05I needed another walk.
0:45:05 > 0:45:08Fortunately, Gottingen is surrounded by good woodland trails.
0:45:08 > 0:45:10It was a perfect setting for me
0:45:10 > 0:45:13to think more about Gauss' discoveries.
0:45:22 > 0:45:26Gauss' mathematics has touched many parts of the mathematical world,
0:45:26 > 0:45:31but I'm going to just choose one of them, a fun one - imaginary numbers.
0:45:31 > 0:45:34In the 16th and 17th century, European mathematicians
0:45:34 > 0:45:40imagined the square root of minus one and gave it the symbol i.
0:45:40 > 0:45:42They didn't like it much, but it solved equations
0:45:42 > 0:45:45that couldn't be solved any other way.
0:45:46 > 0:45:49Imaginary numbers have helped us to understand radio waves,
0:45:49 > 0:45:52to build bridges and aeroplanes.
0:45:52 > 0:45:54They're even the key to quantum physics,
0:45:54 > 0:45:56the science of the sub-atomic world.
0:45:56 > 0:46:01They've provided a map to see how things really are.
0:46:01 > 0:46:05But back in the early 19th century, they had no map, no picture
0:46:05 > 0:46:08of how imaginary numbers connected with real numbers.
0:46:08 > 0:46:10Where is this new number?
0:46:10 > 0:46:14There's no room on the number line for the square root of minus one.
0:46:14 > 0:46:16I've got the positive numbers running out here,
0:46:16 > 0:46:17the negative numbers here.
0:46:17 > 0:46:21The great step is to create a new direction of numbers,
0:46:21 > 0:46:23perpendicular to the number line,
0:46:23 > 0:46:26and that's where the square root of minus one is.
0:46:28 > 0:46:32Gauss was not the first to come up with this two-dimensional picture
0:46:32 > 0:46:36of numbers, but he was the first person to explain it all clearly.
0:46:36 > 0:46:38He gave people a picture to understand
0:46:38 > 0:46:40how imaginary numbers worked.
0:46:40 > 0:46:43And once they'd developed this picture,
0:46:43 > 0:46:46their immense potential could really be unleashed.
0:46:46 > 0:46:49Guten Morgen. Ein Kaffee, bitte.
0:46:49 > 0:46:53His maths led to a claim and financial security for Gauss.
0:46:53 > 0:46:56He could have gone anywhere, but he was happy enough
0:46:56 > 0:47:01to settle down and spend the rest of his life in sleepy Gottingen.
0:47:01 > 0:47:03Unfortunately, as his fame developed,
0:47:03 > 0:47:06so his character deteriorated.
0:47:06 > 0:47:08A naturally conservative, shy man,
0:47:08 > 0:47:12he became increasingly distrustful and grumpy.
0:47:12 > 0:47:16Many young mathematicians across Europe regarded Gauss as a god
0:47:16 > 0:47:18and they would send in their theorems,
0:47:18 > 0:47:20their conjectures, even some proofs.
0:47:20 > 0:47:23But most of the time, he wouldn't respond, and even when he did,
0:47:23 > 0:47:26it was generally to say either that they'd got it wrong
0:47:26 > 0:47:28or he'd proved it already.
0:47:28 > 0:47:32His dismissal or lack of interest in the work of lesser mortals
0:47:32 > 0:47:35sometimes discouraged some very talented mathematicians
0:47:35 > 0:47:38from pursuing their ideas.
0:47:38 > 0:47:40But occasionally, Gauss also failed
0:47:40 > 0:47:45to follow up on his own insights, including one very important insight
0:47:45 > 0:47:48that might have transformed the mathematics of his time.
0:47:50 > 0:47:5315 kilometres outside Gottingen stands what is known today
0:47:53 > 0:47:55as the Gauss Tower.
0:47:55 > 0:47:57Wow, that is stunning.
0:47:57 > 0:48:01It is really a fantastic view here, yes.
0:48:01 > 0:48:05Gauss took on many projects for the Hanoverian government,
0:48:05 > 0:48:09including the first proper survey of all the lands of Hanover.
0:48:09 > 0:48:12Was this Gauss' choice to do this surveying?
0:48:12 > 0:48:16For a mathematician, it sounds like the last thing I'd want to do.
0:48:16 > 0:48:17He wanted to do it.
0:48:17 > 0:48:23The major point in doing this was to discover the shape of the earth.
0:48:23 > 0:48:25But he also started speculating
0:48:25 > 0:48:29about something even more revolutionary - the shape of space.
0:48:29 > 0:48:34So he's thinking there may not be anything flat in the universe?
0:48:34 > 0:48:37Yes. And if we were living in a curved universe,
0:48:37 > 0:48:40there wouldn't be anything flat.
0:48:40 > 0:48:44This led Gauss to question one of the central tenets of mathematics -
0:48:44 > 0:48:47Euclid's geometry.
0:48:47 > 0:48:50He realised that this geometry, far from universal,
0:48:50 > 0:48:52depended on the idea of space as flat.
0:48:52 > 0:48:56It just didn't apply to a universe that was curved.
0:48:56 > 0:48:59But in the early 19th century, Euclid's geometry
0:48:59 > 0:49:03was seen as God-given and Gauss didn't want any trouble.
0:49:03 > 0:49:05So he never published anything.
0:49:05 > 0:49:09Another mathematician, though, had no such fears.
0:49:11 > 0:49:16In mathematics, it's often helpful to be part of a community
0:49:16 > 0:49:19where you can talk to and bounce ideas off others.
0:49:19 > 0:49:22But inside such a mathematical community,
0:49:22 > 0:49:25it can sometimes be difficult to come up with that one idea
0:49:25 > 0:49:28that completely challenges the status quo,
0:49:28 > 0:49:33and then the breakthrough often comes from somewhere else.
0:49:33 > 0:49:36Mathematics can be done in some pretty weird places.
0:49:36 > 0:49:38I'm in Transylvania,
0:49:38 > 0:49:42which is fairly appropriate, cos I'm in search of a lone wolf.
0:49:42 > 0:49:45Janos Bolyai spent much of his life
0:49:45 > 0:49:49hundreds of miles away from the mathematical centres of excellence.
0:49:49 > 0:49:53This is the only portrait of him that I was able to find.
0:49:53 > 0:49:56Unfortunately, it isn't actually him.
0:49:56 > 0:50:00It's one that the Communist Party in Romania started circulating
0:50:00 > 0:50:04when people got interested in his theories in the 1960s.
0:50:04 > 0:50:06They couldn't find a picture of Janos.
0:50:06 > 0:50:09So they substituted a picture of somebody else instead.
0:50:11 > 0:50:15Born in 1802, Janos was the son of Farkas Bolyai,
0:50:15 > 0:50:17who was a maths teacher.
0:50:17 > 0:50:20He realised his son was a mathematical prodigy,
0:50:20 > 0:50:23so he wrote to his old friend Carl Friedrich Gauss,
0:50:23 > 0:50:25asking him to tutor the boy.
0:50:25 > 0:50:28Sadly, Gauss declined.
0:50:28 > 0:50:31So instead of becoming a professional mathematician,
0:50:31 > 0:50:33Janos joined the Army.
0:50:33 > 0:50:37But mathematics remained his first love.
0:50:40 > 0:50:44Maybe there's something about the air here because Bolyai carried on
0:50:44 > 0:50:46doing his mathematics in his spare time.
0:50:46 > 0:50:50He started to explore what he called imaginary geometries,
0:50:50 > 0:50:55where the angles in triangles add up to less than 180.
0:50:55 > 0:50:58The amazing thing is that these imaginary geometries
0:50:58 > 0:51:00make perfect mathematical sense.
0:51:04 > 0:51:09Bolyai's new geometry has become known as hyperbolic geometry.
0:51:09 > 0:51:12The best way to imagine it is a kind of mirror image of a sphere
0:51:12 > 0:51:15where lines curve back on each other.
0:51:15 > 0:51:18It's difficult to represent it since we are so used
0:51:18 > 0:51:21to living in space which appears to be straight and flat.
0:51:23 > 0:51:25In his hometown of Targu Mures,
0:51:25 > 0:51:29I went looking for more about Bolyai's revolutionary mathematics.
0:51:29 > 0:51:33His memory is certainly revered here.
0:51:33 > 0:51:36The museum contains a collection of Bolyai-related artefacts,
0:51:36 > 0:51:40some of which might be considered distinctly Transylvanian.
0:51:40 > 0:51:42It's still got some hair on it.
0:51:42 > 0:51:45It's kind of a little bit gruesome.
0:51:45 > 0:51:46But the object I like most here
0:51:46 > 0:51:50is a beautiful model of Bolyai's geometry.
0:51:50 > 0:51:54You got the shortest distance between here and here
0:51:54 > 0:51:56if you stick on this surface. It's not a straight line,
0:51:56 > 0:51:59but this curved line which of bends into the triangle.
0:51:59 > 0:52:03Here is a surface where the shortest distances which define the triangle
0:52:03 > 0:52:06add up to less than 180.
0:52:06 > 0:52:09Bolyai published his work in 1831.
0:52:09 > 0:52:12His father sent his old friend Gauss a copy.
0:52:12 > 0:52:16Gauss wrote back straightaway giving his approval,
0:52:16 > 0:52:19but Gauss refused to praise the young Bolyai,
0:52:19 > 0:52:22because he said the person he should be praising was himself.
0:52:22 > 0:52:26He had worked it all out a decade or so before.
0:52:26 > 0:52:29Actually, there is a letter from Gauss
0:52:29 > 0:52:32to another friend of his where he says,
0:52:32 > 0:52:34"I regard this young geometer boy
0:52:34 > 0:52:37"as a genius of the first order."
0:52:37 > 0:52:41But Gauss never thought to tell Bolyai that.
0:52:41 > 0:52:44And young Janos was completely disheartened.
0:52:44 > 0:52:47Another body blow soon followed.
0:52:47 > 0:52:49Somebody else had developed exactly the same idea,
0:52:49 > 0:52:52but had published two years before him -
0:52:52 > 0:52:55the Russian mathematician Nicholas Lobachevsky.
0:52:57 > 0:53:00It was all downhill for Bolyai after that.
0:53:00 > 0:53:04With no recognition or career, he didn't publish anything else.
0:53:04 > 0:53:06Eventually, he went a little crazy.
0:53:08 > 0:53:13In 1860, Janos Bolyai died in obscurity.
0:53:15 > 0:53:19Gauss, by contrast, was lionised after his death.
0:53:19 > 0:53:22A university, the units used to measure magnetic induction,
0:53:22 > 0:53:25even a crater on the moon would be named after him.
0:53:28 > 0:53:31During his lifetime, Gauss lent his support
0:53:31 > 0:53:33to very few mathematicians.
0:53:33 > 0:53:38But one exception was another of Gottingen's mathematical giants -
0:53:38 > 0:53:41Bernhard Riemann.
0:53:48 > 0:53:49His father was a minister
0:53:49 > 0:53:54and he would remain a sincere Christian all his life.
0:53:54 > 0:53:58But Riemann grew up a shy boy who suffered from consumption.
0:53:58 > 0:54:00His family was large and poor and the only thing
0:54:00 > 0:54:04the young boy had going for him was an excellence at maths.
0:54:04 > 0:54:07That was his salvation.
0:54:07 > 0:54:11Many mathematicians like Riemann had very difficult childhoods,
0:54:11 > 0:54:14were quite unsociable. Their lives seemed to be falling apart.
0:54:14 > 0:54:18It was mathematics that gave them a sense of security.
0:54:21 > 0:54:24Riemann spent much of his early life in the town of Luneburg
0:54:24 > 0:54:26in northern Germany.
0:54:26 > 0:54:30This was his local school, built as a direct result
0:54:30 > 0:54:34of Humboldt's educational reforms in the early 19th century.
0:54:34 > 0:54:37Riemann was one of its first pupils.
0:54:37 > 0:54:41The head teacher saw a way of bringing out the shy boy.
0:54:41 > 0:54:44He was given the freedom of the school's library.
0:54:44 > 0:54:46It opened up a whole new world to him.
0:54:46 > 0:54:48One of the books he found in there
0:54:48 > 0:54:51was a book by the French mathematician Legendre,
0:54:51 > 0:54:53all about number theory.
0:54:53 > 0:54:55His teacher asked him how he was getting on with it.
0:54:55 > 0:55:01He replied, "I have understood all 859 pages of this wonderful book."
0:55:01 > 0:55:04It was a strategy that obviously suited Riemann
0:55:04 > 0:55:07because he became a brilliant mathematician.
0:55:07 > 0:55:12One of his most famous contributions to mathematics was a lecture in 1852
0:55:12 > 0:55:16on the foundations of geometry. In the lecture,
0:55:16 > 0:55:20Riemann first described what geometry actually was
0:55:20 > 0:55:22and its relationship with the world.
0:55:22 > 0:55:25He then sketched out what geometry could be -
0:55:25 > 0:55:28a mathematics of many different kinds of space,
0:55:28 > 0:55:31only one of which would be the flat Euclidian space
0:55:31 > 0:55:32in which we appear to live.
0:55:32 > 0:55:36He was just 26 years old.
0:55:36 > 0:55:40Was it received well? Did people recognise the revolution?
0:55:40 > 0:55:42There was no way that people could actually
0:55:42 > 0:55:45make these ideas concrete.
0:55:45 > 0:55:50That only occurred 50, 60 years after this, with Einstein.
0:55:50 > 0:55:53So this is the beginning, really, of the revolution
0:55:53 > 0:55:56- which ends with Einstein's relativity.- Exactly.
0:55:56 > 0:56:01Riemann's mathematics changed how we see the world.
0:56:01 > 0:56:04Suddenly, higher dimensional geometry appeared.
0:56:04 > 0:56:06The potential was there from Descartes,
0:56:06 > 0:56:11but it was Riemann's imagination that made it happen.
0:56:11 > 0:56:15He began without putting any restriction
0:56:15 > 0:56:18on the dimensions whatsoever. This was something quite new,
0:56:18 > 0:56:21his way of thinking about things.
0:56:21 > 0:56:24Someone like Bolyai was really thinking about new geometries,
0:56:24 > 0:56:26but new two-dimensional geometries.
0:56:26 > 0:56:30New two-dimensional geometries. Riemann then broke away
0:56:30 > 0:56:35from all the limitations of two or three dimensions
0:56:35 > 0:56:37and began to think in in higher dimensions.
0:56:37 > 0:56:39And this was quite new.
0:56:39 > 0:56:41Multi-dimensional space is at the heart
0:56:41 > 0:56:44of so much mathematics done today.
0:56:44 > 0:56:48In geometry, number theory, and several other branches of maths,
0:56:48 > 0:56:51Riemann's ideas still perplex and amaze.
0:56:52 > 0:56:55He died, though, in 1866.
0:56:55 > 0:56:59He was only 39 years old.
0:56:59 > 0:57:02Today, the results of Riemann's mathematics are everywhere.
0:57:02 > 0:57:07Hyperspace is no longer science fiction, but science fact.
0:57:07 > 0:57:11In Paris, they have even tried to visualise what shapes
0:57:11 > 0:57:13in higher dimensions might look like.
0:57:15 > 0:57:18Just as the Renaissance artist Piero would have drawn a square
0:57:18 > 0:57:22inside a square to represent a cube on the two-dimensional canvas,
0:57:22 > 0:57:27the architect here at La Defense has built a cube inside a cube
0:57:27 > 0:57:31to represent a shadow of the four-dimensional hypercube.
0:57:31 > 0:57:34It is with Riemann's work that we finally have
0:57:34 > 0:57:37the mathematical glasses to be able to explore
0:57:37 > 0:57:39such worlds of the mind.
0:57:42 > 0:57:44It's taken a while to make these glasses fit,
0:57:44 > 0:57:47but without this golden age of mathematics,
0:57:47 > 0:57:50from Descartes to Riemann, there would be no calculus,
0:57:50 > 0:57:55no quantum physics, no relativity, none of the technology we use today.
0:57:55 > 0:57:57But even more important than that,
0:57:57 > 0:58:00their mathematics blew away the cobwebs
0:58:00 > 0:58:04and allowed us to see the world as it really is -
0:58:04 > 0:58:07a world much stranger than we ever thought.
0:58:11 > 0:58:13You can learn more about the story of maths
0:58:13 > 0:58:16at the Open University at:
0:58:26 > 0:58:29Subtitles by Red Bee Media Ltd
0:58:29 > 0:58:33Email subtitling@bbc.co.uk