In Our Time - Carl Friedrich Gauss

Episode Date: November 30, 2017

In a programme first broadcast in 2017, Melvyn Bragg and guests discuss Gauss (1777-1855), widely viewed as one of the greatest mathematicians of all time. He was a child prodigy, correcting his father's accounts before he was 3, dumbfounding his teachers with the speed of his mental arithmetic, and gaining a wealthy patron who supported his education. He wrote on number theory when he was 21, with his Disquisitiones Arithmeticae, which has influenced developments since. Among his achievements, he was the first to work out how to make a 17-sided polygon, he predicted the orbit of the minor planet Ceres, rediscovering it, he found a way of sending signals along a wire, using electromagnetism, the first electromagnetic telegraph, and he advanced the understanding of parallel lines on curved surfaces. With Marcus du Sautoy Professor of Mathematics and Simonyi Professor for the Public Understanding of Science at the University of OxfordColva Roney-Dougal Reader in Pure Mathematics at the University of St AndrewsAnd Nick Evans Professor of Theoretical Physics at the University of SouthamptonProducer: Simon Tillotson.

This is the BBC. Thanks for downloading this episode of In Our Time. There's a reading list to go with it on our website, and you can get news about our programmes if you follow us on Twitter at BBC In Our Time. I hope you enjoy the programmes. Hello, Karl Friedrich Gauss, by those who know about this matters, is considered the greatest mathematician of his time and arguably of all time. He was born in 1777 in Brunswick, Germany,

to parents too poor to pay for his education, but his brilliance brought him a royal patron and sponsor and as a teenager, he solved problems that had babbled everyone since the ancient Greeks. By the time he died in 1855, he'd been called the Prince of Mathematicians for advances in number theory, for predicting where to find asteroids, for thinking beyond Euclid, geometry, and on the way inventing the first telegraph. Later, his importance of the mathematical foundations of the theory of relativity was overwhelming, as Einstein acknowledged.

With me to discuss Gass are Marcus Usoitoi, Professor of Mathematics and Simony Professor for the public understanding of science at the University of Oxford, Colber Roni Doogle, reader in pure mathematics at the University of St Andrews, and Nick Evans, Professor of Theoretical Physics at the University of Southampton. Colmer Roni Doegel. I've said Giles' parents were poor. What else do we know about his early childhood? Well, we know lots of fragments about his early childhood.

His parents were insufficiently educated to be able to remember the date of his birth, is a fairly telling example. his mother could remember that it was eight days before ascension, and Gauss was later able to work out his own birthday from that fact. His father could read and write and do arithmetic, though. So one of the first stories told about Gauss's early brilliance is that when he was aged three, he was sat in the corner of a room

while his father was handing out the weekly wages to the employees at the brick factory where he worked. And so the father was having to work out how many hours each person had done and overtime, and Gauss suddenly pipes up, Papa, the calculation is wrong, age three. Do you believe that? Maybe. Maybe.

He'd certainly taught himself to read and write by then, so it's plausible. Yeah. And? And so when Gauss started school at the age of six, a couple of years later in his first arithmetic class, his teacher, Butner, was clearly waiting for some peace and quiet and set all of the students to add up the numbers from one to 100 on their slates. He thought this would get him in a number.

hour of quiet, Gauss immediately walked up to the front with just a single number on his slate to the teacher's amazement, and not only was there only one number on the slate, the number was correct. Now, that number was 550, and let me explain how it could be that the young Gauss had worked that number out. What had spotted... When you say young, this is... This is age nine or ten. Yeah, yeah. We can count it the non-apocryphal. Well, it was a story he loved to tell, and it had consequences, which makes it more likely that it's true. How did he He worked it out by spotting that 1 plus 100, so the first number and the last number is 101. 2 plus 99, so the second number and the penultimate number, is also 101.

3 plus 98 is also 101. So carrying on in that way, pairing up each number at the beginning with the corresponding number from the end, you always get that the sum is 101. So the young Gauss then reasoned that there's 50 pairs, because you go up from 1 up to 50 and down from 100, down to 51, each adding up to 101. And so the sum of all of the numbers from one up to 100 is 50 times 101, which is 50-50. The consequence of that, which makes it not apocryphal, as the teacher thought this is a very, very clever boy,

and he went to the Duke of Brunswick, the local Duke, and said, will you please look after his education because nobody else will? Indeed. And what happened there? Well, the Duke was very impressed by the young gouse. He asked him to do some mental arithmetic and possibly also, to demonstrate a proof to him. And having been struck by his cleverness,

the Duke actually agreed to sponsor his education, and it was only because of the Duke's sponsorship that Gouse was able to continue and go on to higher college. And the Duke continued to sponsor Gouse all the way up to his death in 1806, so he put Gouse through university and PhD and everything. Just as a lot of an example of this prodigious youth, how did he convince the Duke?

Well, there's a story told. Well, let's tell you, it's quite a good story. The square root of two story. Yeah. That he explained to the Duke how to tell that the square root of two, so the number that when you multiply it by itself gives you two, is not a fraction. And the way that you do this is you imagine that the square root of two is a fraction.

So it's some whole number A divided by some whole number B. And what do you do? Well, the square root of two is A over B, so you square it. So two is A squared over B squared. then you multiply across and you get that 2B squared is A squared and you go R so A squared is 2B squared so A must be even So I can think of A as being twice C for some number C I square that and I've now got 2B squared is 4C squared

I divide by 2B squared now is 2C squared so B is also even I listen to the hanging on to your every word column I think I think we're how old was he when he did this He must have been about 11 I think at that point So we've got the prodigy proven with that. With that doubt, please hold on. There's more to come. Marcus, what happened on his 14th birthday? Well, on his 14th birthday, he got given a book of logarithm tables for his birthday present. The kind of thing we mathematicians like to get for our birthdays.

And this book of logarithm tables began to obsess the young girls. I mean, logarithm tables, if you're old enough, you probably remember them. They're the things that you used to use as calculators, because logarithms change much. multiplication, which can be very complicated, into addition. So this is the important point about logarithms. So engineers would use them to facilitate calculation. But it wasn't the logarithms in this book of tables that obsess Gauss, because at the back of the book was a table of prime numbers.

And this has been one of the big open challenges ever since the ancient Greeks to understand prime numbers. And Gauss started... Sorry, can we just... Yes, a prime number is a number, which is only divisible by itself. and one, the indivisible numbers. They are the atoms of mathematics. This table of primes is a bit like the periodic table in chemistry.

1-3-5 and on we go. 2-3-5. But interestingly, you say one because actually I've seen this book of tables in Gerting and Library and it contains one, which today we don't say as a prime, but in Gouse's day it was considered as something you couldn't divide. But we now recognise primes as building blocks and one is not counted. But it's interesting how times change. And so Gauss thought that one was a prime.

But the staggering thing is that Gauss managed to find a connection between the primes at the back of this book and the logarithms at the front of the book. And I think this really illustrates the brilliance of Gauss's mind. I think somebody who is a great thinker, is somebody who changes the question. We've been trying to find a formula for the primes to generate them, a pattern in there.

And Gauss said, well, it doesn't seem to be one of these. Let's change the question and ask how many primes there are. Now, you might say this is stupid. The ancient Greeks have proved there are infinitely many of these. But he was interested. Now, how many primes are there up to 10, up to 100, up to 1,000? Is there a pattern in the way these primes thin out? So, actually, if you count up to 100, there are 25. So 1 in 4 numbers up to 100 is a prime number. So there's a 1 in 4 chance that a number is prime if you choose a number between 1 and 100. Up to 1,000, they start to get rarer. It's now down to 1 in 6. So Gowell's was interested, is there a way to predict the probability that a number will be prime as you climb higher and higher? And he spotted actually the probability that a number is prime is given by the logarithms at the front of the book.

So the logarithms tell you the chance that a number will be prime and help you to count how many primes there are as you climb higher and higher. He couldn't prove this. It became a conjecture that this was a connection and wasn't proved until the end of the 19th. century. But this is staggering. He discovered when he was 15. A 15-year-old boy had completely changed our perspective on the prime numbers, and it's this perspective

we use today. And I think the examples that Colvers told it already are things that we already knew. This was a completely new insight about the most fundamental numbers on the mathematical books. And Gauss actually wrote that

you have no idea how much poetry is in a table of logarithms. And is there, have you any inkling about where this came from, how he got, the word genius is used by you very rarely, thank goodness, but you apply it to him from the age of 15th.

I would do, yes. Did we know any idea where that might have come from? Well, I think there's something very special about the way he was looking at these primes because he was somebody who enjoyed data. He enjoyed experimenting. He was quite an experimental mathematician. And I think that's, even today,

quite unique. So he liked just messing around with numbers and trying to see the patterns there. Nick Evans could you take that out? Yeah so I think we've already heard examples of what really drove this guy which is that he was just fascinated by numbers from the first moment he found them. They were

like toys that he liked to play with. And that for me is a key part of genius. It's just this guy was obsessed. He couldn't really think about anything else. So for example there's the 10,000 hours hypothesis that anyone can become really good at something if they spend basically five years of their life dedicated to it.

Well, this kid had done that by age 10, and he was just miles ahead of everybody else. So, I mean, it's an interesting question about genius, whether it's just constantly thinking about it that's important or it's real intelligence. I mean, it's probably a mixture of the two. But it's interesting, you can actually go and ask Gouse himself because people did. And he said, no, I don't think I'm that much smarter than everybody else. What he said is that those who do great mathematics spend vast and deep and constant energy and attention on it. And that's what he did

all through his life. It's not unlike Newton's remark when he kept me asking about how did he arrive at gravity. But it's by thinking on it continuously. They're very close, aren't they? But simply concentrating hard all the time. Can you give us some more about how this boy was growing up? What

more numbers he was playing with, Nick? What other numbers he was playing with? Well, so I mean he was renowned for basically having memorized multiplications beyond any level that other people would do so that when he would do calculations, he could do them in his head whilst other people were having to go to their log tables and so on, just because he loved the structure and just to know these things.

How did he... So this Duke of Brunswick is supporting him, and he goes to the local university. Do we know anything about his life there, what he did, what teachers were like, how he was being brought on? Yeah, so I think he was such a prodigious person that he would suck the mathematical knowledge out of his teachers quite quickly. And so he spent a lot of his time in the library just reading through the literature, doing his own thing, working his own stuff out. And indeed at one point he sort of came home from the university saying, well, I've got everything that I need from there.

I need to just go and live in a flat on my own. So he didn't go back to his family. This was an example of him. you wanted to dedicate everything to maths. So he went and he lived in a flat on his own and just sat there and thought about maths. So the question I was going to ask is how did he organise his life? He organised his life by doing nothing else at that stage.

Yeah, I think that's right. I mean, there are some good examples of that. When he was a young child, he used a turnip to make a candle so he could study maths at night. This thing when he came back from university. And then there's a story which is a little bit mean from the end of his life when his second wife was dying from tuberculosis. when allegedly the maid came down and he was studying his mathematics and said,

you know, your wife's dying, you need to come. And his response was, well, can't she wait? And I think it's not so much meant to be a mean story as just showing just how engrossed he got in the mathematics. Was it, when did he begin to be recognised as this phenomenon at university, Mark has talked about him being 15, you know, we're talking about as a boy, a boy genius, just getting a hold of him before we move on into the theories. 12 or so was the point when the Archduke, sorry, the Duke Ferdinand was the guy who was starting to fund him. And at that point it was already clear. He was a genius.

And it was clear all round. Thank you very much. Colva. What was important about Gauss's work with polygons? Can you please start with telling us about polygons? Absolutely. So a polygon or a regular polygon, which is the ones Gauss worked on, is a shape where all the sides are the same length and all the angles are the same. the example most familiar to our listeners will be an equilateral triangle. So all the sides of the triangle are the same length and the angles are the same. That's a regular three gone.

The regular four gone is the square. And the regular five gone is the pentagon that we all know. So the Greeks had been fascinated with the question of which of these shapes can you construct using ruler and compass only. So by ruler, they don't mean one with measurements on it. They just mean a straight edge that you can use to draw a straight line. and by compass they mean a pair of compasses, so you can stick a point in and use the other end to draw a circle.

Now, it's not very hard to work out a way to make an equilateral triangle using only a straight line and some compasses. And the Greeks also knew how to make a square, how to make a pentagon, how to make a regular 15-sided shape, and how to double the sides of any one they could already make. So given a triangle, they could make a hexagon, for example. But they couldn't make one with seven sides,

or nine sides or 11 sides or 13 sides. And so ever since the time of Euclid, people had been trying to find ways to make these. Why couldn't they do that? They couldn't find a way to do it. They couldn't prove that it wasn't doable, but they also couldn't find any way to do it either, if that makes sense. Perfect.

And so a lot of people had tried along the way. And young Gaus comes along. This is whilst he's at university, he's 18 at the time, and he finds a construction of a 17-sided shape. Now, this is two and a half thousand years after people started thinking about the question and then this sprightly young 18-year-old just comes up with it. There's a notice in the local

paper explaining that he's worked out how to do it with a side note from one of his school teachers saying, by the way, he's only 18. So this was the first thing that Gous did that actually reached the general public view, if that makes sense. So the earlier log table stuff is just in his private records. And it didn't get published until 1801,

but what Gouse then showed in 1801 was something even stronger. He managed to work out exact which shapes we can and can't make, and at least for the ones we can make, he gave a complete justification for why we can make them. So amongst the primes, we can do only. Three, five, they were known to the Greeks. 17, which was the new one.

257 is the next one. And then 65,537 is the one after that. And as far as we know at the moment, that's it. Marcus, can you tell us, tell us about number theory. Why was he so influential number theory? And again, what is it first?

Number theory is the exploration and properties of numbers and in particular things like prime numbers, the building blocks. And actually this discovery of the construction of a 17-sided figure excited Gowse so much that he started his own personal diary recording all of the discoveries that he was making about mathematics. And this diary starts with that entry, but it's full of just discoveries. that he was making about properties of numbers,

new properties about prime numbers, how to solve equations that the ancient Greeks have been looking at. And there was one thing in particular, a particular way of approaching the problems. Yes, absolutely. One thing about these diaries, which you say are beautiful. Yes, I've seen the diaries and gurdening it. By no means published everything that was in the diaries,

but he had to say the right, right but few. He published it when he was perfectly sure that it was all. Yes. So in the diary are a lot of things that he didn't publish in his diary. Actually, what it culminated in was a publication. And I think this is a publication of something called Disquantione's Arithmetici. It was published in 1801. And it collects together all his great thoughts about number theory.

And actually, before Gauss did this, number theory wasn't really a subject that, in its own right, it was sort of a random collection of strange things that people like Pierre de Fermat had discovered about numbers. And it was bringing together all of these ideas that made it. suddenly a subject kind of in its own right within mathematics. It's a subject that I study as a research mathematician. So I really credit Gauss with forming this thing, which is now what I spend most of my time exploring. But when he published this book, which had some very interesting things about the ways to look at solving equations, so he invented something a bit, which I might

call a sort of clock calculator. So on a clock, if you say, okay, it's nine o'clock now and I'm going to meet you in four hours time, you actually say you're going to meet somebody at one o'clock. So you don't say, you don't keep on adding the numbers. You do look up what the remainder. You might say 13 o'clock if you're working on 24 hour clock. But the idea is that you take, you subtract the multiples of 12 off until you get something which is less than 12. At Grouse realized that this kind of calculator don't have to use a clock with 12 hours on. You could use a clock with seven hours on, that it was a very powerful way to explore properties of solving equations. And this is a tool. It's called modular arithmetic. And it's a tool number theorists use

every day now. And it really goes back to Gauss. Gals pulled all of these discoveries about numbers together in this book. And he was really hoping to impress the French Academy, because France was really where mathematics was being done at this time. And the French Academy were really dismissive of this book. It was quite a cryptic book because Gauss didn't like to explain where he got his ideas from. And so people called this book a book of seven seals because somehow it just, you know, you couldn't understand where rabbits were being pulled out of hats. And Gauss always used to say, well, yeah, but an architect

doesn't leave up the scaffolding when he has built a building. So he felt he was justified in this, the magic that he was creating. But I think it really caused him problems. because he wanted this recognition and it didn't really get it. And I think it caused him to then be very conservative about what he published. That, as you say, few but ripe, he only, he hated criticism and he only published, he was absolutely sure that he was right about something. And I think it goes back to this reception of Dysquentione's Arithmetici.

Nick Evans, he became famous for his work on the Asteroid Series. Yes, that's right. Can you tell us the story behind that? So around 1800 telescopes had got sufficiently good that people were starting to find smaller bits of rock in our solar system. And Sures is one of these. It's part of the asteroid belt. It's lump of rock about 1,000 kilometres across.

Now what normally happened is that people would look at a planet and they'd watch it for a year as it went around the sun and they'd plot the orbit and they'd say, oh look, it's an ellipse like Newton said. But in this case, they found Serez just before it was going to go behind the sun. They only had a few percent of the orbit and then it disappeared behind the sun. And they were like...

How much of the percent? You said a few percent? Yeah, one or two percent of them. And so they... At that point they were like, well, okay, we've lost it. We're never going to find this again. But what Gowse did was she says, no, Newton told us that this thing has to be following an ellipse. And here in my hand, I have all of the ellipses that is possible.

And what I'm going to do is I'm going to try and fit those to the bit of data that we have and see if I can work out which one of those ellipses it was following. So he had to develop a lot of the... of mathematical methodology to do that, something called the least squares fitting method, which is now used across science to explain how you draw the best line through a bunch of data points. Anyway, so he did that, and he predicted where this planet would reappear on the other side of the sun. And then... How much later was he predicting it, in a matter of days? A few months. A few months. A few months. Yeah. And they, well, there were a few cloudy days, I think, but eventually

they looked for it and they found it exactly where he'd predicted it and so that was sort of a big success that sort of flooded round Europe that everybody knew that he'd achieved this What did that, because as I read from yours and others notes, this made it, this brought him great fame. What did great fame consist in

at that time? I think it was just knowledge amongst all of the academics in Europe and beyond the people as well of the work that he had done. So, for example, there's a story that the Duke at this point had to compete with the St. Petersburg Observatory who wanted to steal him away. And so the Duke increased his salary at that point.

And all of the people in Hanover knew about this and were saying, well, why on earths the Duke spending the money on this? So, yeah, he was well known. How hard was it to do what he did about series in plotting it when it was behind the sun for all that time? Well, so he had to really invent new mathematical techniques for how you connect data to a theory that you have underlying it. And that made him have to think about not just the perfect case, but actually experimentalists and the fact that they have errors. So if you get 100 people to measure the length of a race, they will get slightly different times. And he came up with something called the Gaussian distribution, which is basically telling you the problem. that you'll be a little bit out in the time

or the probability that someone will be a long way out in the time. And that is basically to this day our understanding of errors and how they occur in experiments across all of science. He certainly got there first of a lot of things, isn't he? Colver, at that time he had a strong relationship by letter, corresponders, with the French mathematician, Monsieur Leblanc. Can you tell us about Monsieur Leblanc? Yes.

So this is one of my favourite people's stories about Gout. I think it sheds a lovely light in his character. So I'm going to slightly spoil the story by starting with who Monsieur LeBlanc actually was. So there was a young French mathematician called Sophie Germain. Now, you'll notice that's a female name. And at the time, it wasn't really possible to be a woman in science. And so she'd got fascinated by mathematics during the French Revolution. She'd been shut indoors and for her own safety and needed to read.

She'd read of the death of alchemides at the hands of a Roman soldier and had been very impressed that maths could be so fascinating that it could lead to Archimedes being so busy studying geometries on the ground that he got himself killed. So she read Dysquizizciones and started writing with Gauss, and she covered up her identity because she always had to cover up her identity because women couldn't do maths. Now, Napoleon invaded Brunswick,

and Sophie Germain was very worried that the fate of Archimedes might befall Gauss. and so she got in touch with a general in the invading army and told him to send someone to make sure that Gauss was safe. So the person, Julie, arrives at Gauss's house and says that a woman in Paris has come to check that he's okay, and Gauss says, well, I don't know any woman in Paris. And then it turns out that it's indeed Sophie Germain, and he sends her the most beautiful letter

talking about her superior genius and amazing abilities at maths and saying that the customs and prejudices of the day make it so much harder for a woman to do math than a man, that he's overwhelmed by her abilities. About 30 years after that, after Sophie Germain's death, the University of Göttingen is making some honorary degrees to celebrate its centenary.

And at that point, Gouc says he's incredibly sad that he can't nominate Sophie Germain because she would have been the best person imaginable to receive a degree. As a measure of how ahead of its time that was, that was more than 35 years before the first woman got a PhD in mathematics in Europe. So Gowse was really very open-minded when he saw genuine ability.

There was another impact of Napoleon. The Duke of Brunswick got on his horse and the age of 71, went into battle with Napoleon, got killed. So he lost his sponsor. What then? But he had to get a job. Yeah. And he said, well, there's no way I'm going to get a teaching job at university

because all you end up doing is teaching really boring, low-level maths to undergraduates. So he wanted to find some place that would allow him to... just keep on thinking. So he actually took up the head of the observatory in Göttingen. And actually, I blame series a bit for dragging this wonderful mathematician away to doing things like physics and stuff like this. Because, you know, he'd spend a lot of time doing astronomy. And he called these things clods of Earth. I've got to spend my time tracking clods of Earth. And actually, later on, he got asked to do a survey of Hanover to map out. I mean, what a waste of time for this genius. But this was what he had to do in this position.

But he used this actually to make yet another extraordinary discovery, which is about geometry. Something he'd actually been thinking about as a young boy already. If we go back to the geometry of Euclid, there's something called the parallel postulate, which says that if I take any line and a point off that line, Euclid says there's just one unique line through that point, which is parallel to the first line.

And everyone had been trying to prove that this was a, a consequence of the other axioms of geometry. And Gauss felt he tried to do that, couldn't do it. And he thought, well, maybe actually it's not a consequence. And he started to think maybe there are other geometries out there where there aren't any parallel lines, or maybe where there are many parallel lines through these points. And he started to play with this and came up with an idea of new geometries, what we call non-Euclidean geometries. And when it was out on this survey of Hanover, he began to think maybe the universe might be curved in some ways. So an example of a non-Euclidean geometry is the surface of the earth. In Euclid's geometry, if I draw a triangle, the angles add up, as we

will learn at school, to 180. But let me draw a triangle on the surface of the earth. I'm going to take a point at the pole, north pole, two points on the equator. The straight lines here are now lines of longitude, curved lines. And if I measure the angles in here, I've got two 90-degree angles at the equator, they already add up to 180, and then I've got an angle at the top. So Galser discovered new geometries with different properties from Euclid, and he wondered whether maybe the universe is like that. And during the survey, he took lights on mountaintops and tried to see whether the triangles that they created might have angles adding up to less than 100, more than 180, and we now know this is on too small a scale. This is what Einstein also believed about a curve geometry

in the universe and it was many years later when we looked at the eclipse and measured light from stars coming that we understand that the geometry that gals discovered is actually the geometry that models the way the universe works i wonder what it was like inside his head it just goes on we are going on he never stopped did he well one of the striking things about this is he didn't publish it silly fellow and it turned out that two other people so bolyai hungarian lived in Transylvania and a Russian Lovacevsky announced the discovery of these new geometries

Gauss said about Pollyai, what a wonderful young geometry, but to praise him, to praise myself, because I've already discovered this 20 years earlier. And the Dyrus proved who was telling the truth? Well, yeah, exactly, so. Nick, he was it, then we go to magnetism. Okay, what did he do?

Yes, say this great theoretical physicist managed to pull himself away from pure mathematics. So, magnetism. Magnetism had been interesting since about 1820 when Ersted had shown that a current carrying wire generates a magnetic field. But Gauss only got interested in it around 1830 because of his friendship with an experimental physicist called Weber who he actually wrote a very strong reference letter for

in order to get him appointed at Gauss's university. And then they started working on magnetism. They built a rather famed magnetism lab. So the problem is that if you have lots of iron in your lab, then when you try to do magnetic experiments, that iron becomes magnetized and it messes up your experiments. They had to strip all the iron out and replace it with copper, which is first of all expensive, but it became very famed. And throughout Europe, in Paris and so on, people started at Greenwich as well. They started to build copies of this lab.

But anyway, while he was doing that, he went back and he looked at the theory of electricity, which is very like the theory of magnetism, and he went back over the mathematical structure. He reinvented some of the things which had been previously discovered and ordered them. And so now he's famed in physics circles for something called Gauss's Law of Electricity. So to give you a quick idea of that,

think of an electric charge as being like a hedgehog. There are lots of spikes coming out of it. So where the spikes are near the charge at the centre by the hedgehog, they're close together and the electric forces are strong. Then as you go away, the spikes spread out in space and that reflects the fact that the forces

become weaker as you go away from a charge. Now the reason this is useful is that you can now think about taking lots of hedgehogs and sticking them to a wall charges on a wall and now those spines, they have to escape but now there's only one direction they can go so it's like the bristles

on the end of a brush. All of these spikes are going straight out parallel to each other and so they don't dilute. The density stays the same and that means that the electric forces don't fall away in that example. So it's a way by which you can just think about these electric problems and almost immediately see the answers, at least in some cases.

Right, Colbert. How many more secrets were in the diary that were not revealed until after his death and he had pre-discovered stuff? Loads, essentially. I mean, Gauss on some level, had prefigured a huge amount of what went on right through up to 1880. It's a measure of how much he did that his collected works were still being

printed until 1929. So huge amounts of work was done, not just in the diaries but also elsewhere. For example, many of the major developments of what we now think of as complex analysis, which are credited to Arbel and others were already done by Gauss

maybe 25 years before but never published and it's only via looking in the diaries that we have the evidence that all of those happened. To keep with these diaries, Marcus, which you're great admirer. Does it reveal the way he worked? Does he let himself know how he worked? Does he keep him in that a secret? There are a lot of sneakers in there. So you see things like Eureka. And then there's this kind of cryptic little thing, is a number equals, and then he's drawn three triangles. And we now know what he discovered is that every number can be written as the sum of what we call three triangular numbers.

Triangular numbers actually relate back to that very early story of adding up one to a hundred. That's the hundredth triangle. So quite a lot of the thing is written in quite cryptic form. There are some things we don't even know now quite what they were that he discovered. But actually my favourite is the last entry of the diary because he understands how to solve an equation called an elliptic curve, which is y squared equals x cubed minus x. He wants to try and find some numbers which solve this. And he makes a discovery about this. This turns out to be something, the first example of something called,

the Riemann hypothesis for finite fields, which was discovered and proved in the beginning of the 20th century. The fact that it is the last entry in Gow's diary, for me, is absolutely staggering that he's beginning a journey that culminates on one of the great theorems that is proved in the 20th century. But one of the problems of the whole of this is that he was such a perfectionist, and he really hated criticism, this kind of few but ripe. In a way, we've heard how incredible he's advanced so many different bits of mathematics and science. And yet, actually, he'd been a bit more open with his discoveries. For example, he'd discovered a new picture for complex numbers, imaginary numbers,

kind of two-dimensional picture of numbers, which was a great tool for doing this new theory, and it fed a lot of his ideas, but he didn't tell anybody about this. I think actually his conservatism held back mathematics by 50 years, probably, that if he'd been more open, there'd been a massive explosion. The things he did were amazing, but what he knew was even greater. Colva. I think one of the things with Gauss, and you can see it going all the way back to his first book on number theory, Dixiziones,

is that he doesn't want to publish fragments. He only wants to publish when he has a complete theory of everything. And so what we see in the diary and in his notes on geometry and other things, when he's got some things which other people would be rushing to publish because they're fascinating in their own right, but he feels like he can't yet tell the whole story the way it ought to be told, then he won't publish. He's on record as saying the first proof you find is not the proof you should publish. You should wait until you've found the right proof, the best proof,

the one that shows why this is as it is. Can we just nip in before I ask the next question? He married and he was a lovely marriage, they had three children, everything was good about it, and his wife died of child, but he married a good friend of hers and they had more children. So he seems to be a man who was capable of being very happily married and getting on with that part of his life and taking a lot of care about.

His elderly mother came to live with him and so on. So that's going on as well. But he really didn't want his children to become mathematicians. Why? Because he didn't want to spoil. You know, he knew he was a great mathematician and he thought he would let down the name of Gauss if his children tried to be mathematicians,

which I think is pretty horrific. Well, I don't know that's terrific. I mean, looking after them. Xeroxas machines always, you know, make bad copies of the original. Paul has no an emmon. We can develop that. I've got to go to Nick. But the invention of the telegraph, let's go to that, Nick. What did he do about? This was another game that he played with Weber, where he strung a piece of wire between his observatory where he lived and the lab where Weber did his experiments.

And then they developed the ability to basically send pulses of electric current down this wire and then to detect that at the other end. And this was the first example of a telegraph where they could say. send pulses down and, you know, see them a few kilometres. So, primitive Morse code, was it really? Yeah, so indeed. They started off by just sending time signals, but then they developed their own version of the Morse code. It was before Morse code. It was different.

But they could communicate, and they would send messages like, you know, one of the research assistants is coming over to your lab and so on. Of course, you know, once they'd done it, they sort of thought they'd sort of done it, and it was for other people to take it on. So it didn't really progress from there. Apparently, eventually, the Y got hit by lightning, and that was the end of it. But they'd understood everything that was needed to do it.

Colby, I wanted to come in. Yeah, I mean, they did briefly have grand plans that you could send messages around Germany using the train lines, the rails that the trains would run on. And sadly, I think Gauss was not maybe the best at communicating with business people, so that never quite came to light. But they did have a very early vision of sending messages around the whole of Germany. There's something called Gaussian curvature, which you can't wait to hear about.

So, Gaussian curvature. This is a kind of notion of curvature that's even more complicated than the stuff that Marcus was talking about before. So it came out of Gauss's surveys of Hanover. He got interested in curvature where the curvature can be lumpy and bumpy, exactly like the surface of the earth. There can be hills and valleys, and it's not smoothly curved the same way everywhere.

He found a way to associate a number with each point on a surface, where if the number was zero, that meant that at that point on the surface, it was completely flat, just like the table in front of us here. If the number was positive, then it was as if you were sitting on or inside a sphere. So, for example, at the top of a mountain or at the bottom of a lake. If the number was negative, it was like being a saddle point. So imagine going through a mountain pass where you've got mountains going up either side of you and down in front and behind you, so it's curving down one way and up the other.

And what Gouse was able to show, having associated these numbers with every point, is that if you now pick up this surface, so imagine taking a film over the surface of the earth and lifting it up, then no matter how you bend or fold it, provided you don't cut or stretch it, the curvature at each point on that surface won't change, no matter how much you crumple and fold the whole thing.

An easy practical consequence of this, which listeners might already know, is that we can't make flat maps of the whole globe. So when I think of the globe as a whole, the curvature at every point, well it's obviously the curvature of a sphere, because it's a ball, which means it's positive, whereas the curvature of a flat piece of paper, a map, is zero at every point.

So that's a sort of a two-line proof that you can't possibly turn a flat sphere, a round sphere onto a flat map. Marcus, can you develop the notion that he would have shifted mathematics on by 50 years had he published more? One of the things was the centre of mathematics shifted from France to Germany because of him and his influence. local influence and from that small university he emanated great power. But can you just develop that a bit? Yes, I think, you know, the centre of mathematical activity when Gauss started was France.

The French Revolution had really initiated great new discoveries, but it was a very much utilitarian view of mathematics. Mathematics was going to serve the state and actually French mathematicians are rather dismissed if they were just interested in maths for its own sake. But we see a real change and Gauss is part of this and Humboldt who was defining the education system within Germany said no I think we should value sort of pure knowledge for its own sake in our education system and this actually helped transfer kind of the centre of mathematics from France to Germany because it allowed people like Gauss to think of just ideas for their own sake which then would go on to be applied in interesting new ways that no one had ever thought of and so Gowell

In Göttingen, Gertiggen was this tiny little place, not a centre of Europe, but Gowse begins to make this really the hub of mathematics. If I was going to spend anywhere, you know, where would I want to be in the 19th century to do my mathematics? I would choose Gertingen. And it's from Gals right up to the beginning of the Second World War, when Hitler just blew the place to pieces by kicking out all the Jewish mathematicians and it just collapsed as a centre. This was where all the great mathematics was being done,

especially why a lot of Gauss's students, people like Bernhardt-Riedman, dedicated. So I think Gauss can be credited with really making this shift not only from France to Germany, but also from a kind of utilitarian view of maths being useful

to mass being the queen of the sciences. Yes, it serves the sciences, but it is kind of at the top. It's the regal top of science. Is it back to Galileo? I've discovered the secret of the universe. And it's all written in mathematics. Yes, I think that's true.

Can I begin to talk briskly about his legacy? Nick, what do you think is his greatest legacy? Well, he's had a huge impact on all bits of science. His work on electricity and magnetism eventually led to the discovery of Maxwell's equations and radio, which I think we all agree is very important. But all of modern-day electronics and the lighting in your houses goes back to those equations.

His work on error analysis is basically what defines sciences these days. the careful study of data and its application to theories. And then his work on non-Nuclidean geometry, which did feed in through his students, led eventually to general relativity, the Einstein's theory of gravity. And at the moment, we're seeing black holes colliding with gravitational wave detectors,

all of which goes back to that word. Colber? Too many to enumerate. He founded my own discipline, namely group theory with his study of modular arithmetic. He founded modern number theory. He founded the study of complex numbers. He made the shape of 20th century mathematics what it is now. Thank you very much.

Thank you, Colveroni-Dougall. Markes-Sotoid, Nick Evans. Next week, it's our listener week. We'll be discussing one of the many ideas you've suggested this autumn. That will be revealed next Thursday morning. Thank you for listening. And the In Our Time podcast gets some extra time now with a few minutes of bonus material from Melvin. and his guests.

Right, away you go. What did we miss out? Anything, Marcus? I think we did pretty well, actually. Considering how broad his, I think you did very well in covering. I had another practical example of what Gaussian curvature tells us. So I talked about the mat-making thing and that you can't turn a flat, a round sphere onto a flat plane.

Another example is even more practical, and it's why corrugated iron works the way it does. So if you walk down past... I've often thought how that way. If you walk down past building sites, you'll often see corrugated iron making a flat. fence and why did they corrugate it? Why is it not just flat sheets of metal? And the answer comes from Gaussian curvature. So the wiggles that they put on the iron, it started out as a flat

piece of iron. Gousis theorem says that therefore it always remains flat in the sense of gousy and curvature. So once you've got wiggles going one way, if it were to start sacking or bending over, then little bits of it would look like little bits of spheres. And Gausus theorem says it never can look like little bits of spheres because it started out flat. And so it's Gousis theorem that shows you that corrugated iron can't sag as badly as a flat sheet would, and likewise by corrugated cardboard is much, much more rigid than flat cardboard ever would be. They retain the knowledge that they started out flat, and so the wiggles stopped them bending. The thing I really miss that we should have talked about is, and I did contemplate putting this

in, but I think it was just too hardcore, is quadratic reciprocity. You heard Colbertic right, that, you know, it's, quadratic reciprocity, which is about these clock arithmetic. So you're interested, well, what numbers on the clock? So if I've got a seven hour clock, for example, so I've got numbers, one, two, three, four, five, all the way up to, well, seven will put it zero. So which of those are squares? So what numbers actually, you know, if I square a number on this clock would actually hit those times? And Gowse found a wonderful way to understand which of the hours on the clock would be the square of another number on that clock and it's a beautiful so um i think uh five if you took a prime number clock for example um five

will be a square on that prime number clock even only if the prime p is a square on the five hour clock he won is this kind of like this is the reciprocity kind of relation and he called this i think the golden theorem or something i mean it's just a beautiful discovery um about the kind of uh a new way to look at these kind of numbers um but i just thought it was a bit hardcore for... I almost understood, though. Ah, cool. There you go. Right, should have done it. So as an example,

two is not a square if we consider it in the ordinary whole numbers, but if we consider it on the seven clock, as Marcus said, then three squared is nine, and nine becomes the same as two, and suddenly three squares to two, and so suddenly now two is a square. There's another thing which is related to this, I think, which is about, if you take a one-seventh and write that as a decimal number, you know it's going to be some decimal numbers and then sort of repeat. It turns out there are six numbers and then it's a recurring decimal.

This is the maximum it could have been. So for example, if you do one over nine, that's I think 0.111. So that repeats itself just after one. But the maximum, so you do one over a prime number, the maximum number of recurring decimals can be p-minus one. And he was very interested in it's still an open problem actually, which primes actually must. have the maximum number. So I think if you do one over 17 or one over a 19, it requires, you know, P minus one of these recurring decimals. I mean, you might say, well, what's the point of all that? But that's the extraordinary thing. The point of it is that many of these things are fed into things like cryptography. So a very practical application, which Gouse would have never thought about, but, you know, it was what he would have used on his telegraph.

Nick? Yeah, we should also talk about pension schemes. Oh, yeah. Oh, his investments. At the end of his wife, his university's widows scheme was going bust. And he came in and did a probabilistic analysis of how likely people were to die and therefore how you should run the fund, which basically underlies the way people run pension schemes. Oh, we miss it. And also, he was this indigent professor who made a fortune on the stock market. Yeah, he died a wealthy man having...

He worked it out. The students at Gertingen used to call him the newspaper eater because he would come in every morning. morning and work his way to all the newspapers seeing what stocks were up and down and invested accordingly. And made a fortune. That's all just messy, practical stuff. Not what why Gow's interest. But it shows something of the variety of his

thinking about it. Yeah, distraction. I'm glad to say that on the programme because we've been invaded by people from the sitting saying, how did you do it? He's had in his diaries. Yeah, it's funny. So for somebody that insisted on few but ripe, he did actually published volumes. I mean, between the published works and all of the

things that he left for publication after his death, there were books and books and books of stuff. It's lovely to go to, I mean, if anyone gets a chance to go to Gertingen, go to the library, you can get them to bring up this diary. And it's just, I mean, it was very moving to see this thing with

all of these ideas in, including little scribbles, of course, it wasn't just all maths, there's kind of little shopping lists and a little picture that he drew a face and things like this. So it's really just makes the guy human. You know, for us, he is this kind of almost godlike figure in our subject. But it's just, I think for me, seeing these original documents brought him alive as a person. Thank you very much. We're about to be rudely interrupted,

aren't we, by our producer, Simon? Yeah, I'll take your tea and coffee, but I got a question, was he really the greatest mathematician, as we've suggested. Yeah. Yes. Well, of course, that's debatable, but I mean, I think, you know, he concerns. He concerns, certainly be, who else are you going to choose? I'd say yes, but who would like to your coffee? Yeah, no, I think... Are you being asked that on the Twitter's a general matter?

I'm interested who would be comparable. Archimedes? Yeah, so I... No, not comedies. Newton. Newton has some case for that. Great theoretical physicists, so Einstein maybe. So I couldn't have done anything without cows.

And Einstein himself said that, right? Have you got enough for your reply? Coffee would be great. Yeah. Hello, I'm Neil McGregor, and I'd like to invite you to listen to my new 30-part series about faith and society. For the whole of human history, believing and belonging have gone together. And in this series, I'm looking at objects and places to see how those shared beliefs have helped to build communities and also to divide them. It's called Living with the Gods, but it's just as much about how.

we live with each other. You can download the programmes from the Radio 4 website or on the IPlayer radio app. And there's also a free podcast to which you can subscribe. Search online for Living with the Gods.