In Our Time Read online

  MELVYN BRAGG: Do you think it would help the world if we all got buried?

  NICK LANE: Yes. Except we’d end up with lots of oxygen in the atmosphere and everything would burn.

  GAUSS

  Carl Friedrich Gauss, by those who know about these 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 baffled 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’s geometry and, on the way, for inventing the first telegraph. Later, his importance to the mathematical foundations of the theory of relativity was overwhelming, as Einstein acknowledged.

  With Melvyn to discuss Gauss were: Marcus du Sautoy, professor of mathematics and Simonyi professor for the public understanding of science at the University of Oxford; Colva Roney-Dougal, reader in pure mathematics at the University of St Andrews; and Nick Evans, professor of theoretical physics at the University of Southampton.

  Occasionally on In Our Time, Melvyn asks what is known of the famous subject’s childhood and the answer comes back, ‘Nothing.’ In the case of Gauss, there were rather more fragments than usual. Colva Roney-Dougal told how his parents referred to his date of birth as eight days before Ascension, from which he later worked out the actual day (30 April 1777). His father could read and write and do arithmetic, though, and one of the first stories told about Gauss’s early brilliance came from when he was with his father. Some of these stories have more weight than others.

  COLVA RONEY-DOUGAL: 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 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.’

  MELVYN BRAGG: Do you believe that?

  COLVA RONEY-DOUGAL: Maybe …

  MELVYN BRAGG: Well, let’s move on then …

  Gauss had taught himself to read and write by that time, we heard, so this story was plausible. Then, according to a story that Gauss loved to tell, when he was at school, around the age of six, his teacher asked the children to add up the numbers from 1 to 100 on their slates. Gauss immediately walked up with the right answer on his slate: 5,050. He worked it out by spotting that 1 plus 100, the first number and the last number, is 101. Carrying on in that way, pairing up each number at the beginning with the corresponding number from the end, he noticed the sum is always 101. Gauss reasoned that there were fifty pairs and so the sum of all of the numbers from 1 up to 100 is 50 times 101, which is 5,050.

  The consequence of that, as Melvyn put it to Colva Roney-Dougal, was that the teacher thought this was a very, very clever boy and, when Gauss was about eleven, he took him to the Duke of Brunswick, the local duke, and said, ‘Will you please look after his education because nobody else will?’ Gauss, according to another story, explained to the duke how to tell that the square root of 2 is not a fraction. The duke, she said, was very impressed and agreed to sponsor his education, putting him through university and his doctorate. He studied intensely.

  MARCUS DU SAUTOY: On his fourteenth 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 Gauss.

  Log tables, as older listeners would have remembered, were used before calculators to change complicated multiplication into addition. Gauss noticed the table of prime numbers at the end, the numbers that are only divisible by themselves and 1, ‘the atoms of mathematics’, as Marcus du Sautoy put it.

  MARCUS DU SAUTOY: 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.

  Until then, we heard, people had been trying to find a formula for the primes, to generate them, but Gauss could not find one so asked instead how many primes there were. He was interested in finding out if there was a way to predict the probability that a number would be prime as the numbers climbed higher and higher. He spotted that the logarithms would tell him the chance that a number would be prime and help him to count how many primes there were as he climbed higher and higher. This was his conjecture, it was right, although not yet published, and it was not proved until the end of the nineteenth century.

  MARCUS DU SAUTOY: This is staggering, this he discovered when he was fifteen. A fifteen-year-old boy had completely changed our perspective on the prime numbers and it’s this perspective we use today. This was a completely new insight about the most fundamental numbers on the mathematical books. And Gauss wrote, ‘You have no idea how much poetry is in a table of logarithms.’

  He enjoyed data, we heard, and he was quite an experimental mathematician. He liked just messing around with numbers and trying to see the patterns there.

  For Nick Evans, a key part of Gauss’s genius was just that he was obsessed, and could not really think about anything else. When he was a young child, he used a turnip to make a candle so he could study maths at night. There is the 10,000-hours hypothesis, which is that anyone can become really good at something if they spend five years of their life dedicated to it. Gauss had done that by the age of ten, and he had a potent mixture of intelligence and dedication.

  NICK EVANS: You can actually go and ask Gauss himself, because people did, and he said, ‘No, I don’t think I’m that much smarter than everybody else.’ What he said was, ‘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.

  He was recognised as a genius from before his teens. At university, he was such a prodigious person that he would suck the mathematical knowledge out of his teachers quite quickly. He left, living on his own for a while to dedicate everything to maths. He was engrossed.

  NICK EVANS: 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 [she] said, ‘Your wife’s dying, you need to come.’ And his response was, ‘Can’t she wait?’

  While still in his teens, Gauss took on problems that had baffled the ancient Greeks and everyone since. One, Colva Roney-Dougal told us, was about regular polygons, and the Greeks had been fascinated with the question of which of these shapes could be constructed using a straight edge and a pair of compasses only. The Greeks knew how to make an equilateral triangle, a square, a pentagon, how to make a regular fifteen-sided shape and how to double the sides of any one they could already make, but they couldn’t make one with seven sides or nine sides or eleven sides or thirteen sides, and people had been trying since Euclid.

  COLVA RONEY-DOUGAL: Young Gauss comes along – this is while he’s at university, he’s eighteen at the time – and he finds a construction of a seventeen-sided shape. Now this is 2,500 years after people started thinking about the question and then this sprightly young eighteen-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 schoolteachers saying, ‘By the way, he’s only eighteen.’

  While the log table and primes idea were still in his private papers at that point, this discovery about polygons was published and started to make his reputation. Then, in 1801, Gauss showed something even stronger. He worked out exactly which shapes we can and cannot make and gave a complete justification for the ones we cannot make. Among the primes, we can do only 3, 5, 17 (which was the new one), 257 is the ne
xt one and then 65,537 is the one after that. ‘And, as far as we know,’ Colva Roney-Dougal concluded, ‘at the moment, that’s it.’

  1801 was the year in which Gauss had a work published, Disquisitiones Arithmeticae, which collected together all his great thoughts about number theory. Marcus du Sautoy said that this was a subject that he studies now as a research mathematician, so he credits Gauss with forming his discipline. He invented something that, we were told, could be seen as a sort of clock calculator, where if we say it is 9 o’clock now and we are going to meet someone in four hours’ time, we actually say we are going to meet somebody at 1 o’clock, we don’t keep on adding the number. Gauss realised that, with this kind of calculator, you could use a clock with seven hours on, for example, rather than twelve, which would be a very powerful way to explore properties of solving equations and is now called modular arithmetic.

  MARCUS DU SAUTOY: Gauss 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 was 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 you couldn’t understand where rabbits were being pulled out of hats. And Gauss always used to say, ‘Well, yes, but an architect doesn’t leave up the scaffolding when he has built a building.’

  Gauss felt he was justified in this, but it caused him problems. He hated criticism and only published when he was sure he was right, and that meant he did not get the recognition he wanted. One of the last entries in his diaries is Marcus du Sautoy’s favourite because Gauss showed he understood how to solve an equation called an elliptic curve, the first example of something called the Riemann hypothesis for finite fields, which was discovered and proved at the beginning of the twentieth century. ‘For me, it is absolutely staggering that he’s beginning a journey that culminates in one of the great theorems that is proved in the twentieth century.’ That, too, went unpublished in his lifetime.

  MARCUS DU SAUTOY: I think his conservatism actually held back mathematics by fifty years probably, that if he’d been more open, there would have been a massive explosion. The things he did were amazing but what he knew was even greater.

  COLVA RONEY-DOUGAL: He’s got some things other people would be rushing to publish because they’re fascinating in their own right, but he feels [that if] 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.’

  He did, though, get recognition for his astronomy. As Nick Evans explained, telescopes had really improved since around 1800 and people were starting to find bits of rock in the solar system, smaller than planets, of which one was Ceres, around 1,000km across and in the asteroid belt. Astronomers observed it just before it disappeared behind the sun, so did not have a chance to plot its orbit and thereby predict where it would emerge. They thought it was lost. Gauss, though, matched the little data available to a range of ellipses, developing something called the least squares fitting method, which is now used across science to explain how to draw the best line through a bunch of data points. He then predicted where it would reappear and, a few months later, he was proved right. That brought him new fame.

  NICK EVANS: That [problem] made him think about not just the perfect case, but actually [about] experimentalists and the fact that they have errors. 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 telling you the probability 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, to this day, our understanding of errors and how they occur in experiments across all of science.

  Gauss had, meanwhile, struck up correspondence with a French mathematician called Monsieur Le Blanc and this led to what Colva Roney-Dougal said was one of her favourite stories about Gauss. There was a young French mathematician called Sophie Germain, a woman living at a time when the establishment assumed it was not possible to be a woman in science, and she had become fascinated by mathematics while shut up for her protection during the French Revolution. She had read of the death of Archimedes at the hand of a Roman soldier and been impressed that maths could be so fascinating to Archimedes that he did not realise he was in danger and was killed while studying geometries.

  COLVA RONEY-DOUGAL: Now Napoleon invaded Brunswick, and Sophie Germain was very worried that the fate of Archimedes might befall Gauss. She got in touch with a general in the invading army and told him to send someone to make sure that Gauss was safe. The person duly 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 turned out that his correspondent M. Le Blanc was Sophie Germain. Gauss sent her a letter, talking about her superior genius and amazing abilities at maths and saying that the customs and prejudices of the day made it so much harder for a woman to do maths than a man, and that he was overwhelmed by her abilities. When, about thirty years after that, Sophie Germain was dead, Gauss was asked to nominate someone for an honorary degree, he said he was incredibly sad that he could not nominate Sophie Germain because she would have been the best person imaginable to receive a degree. That was more than thirty-five years before the first woman was awarded a PhD in mathematics in Europe.

  While Gauss survived Napoleon’s invasion, his sponsor was not so lucky. He died in battle while leading Prussian forces at Auerstedt, aged seventy-one. That meant Gauss had to find a new way of making a living. As Marcus du Sautoy put it, Gauss thought there was no way he was going to get a teaching job at university because all he would do was teach boring low-level maths to undergraduates. He took up the role of head of the Göttingen Observatory.

  MARCUS DU SAUTOY: I blame Ceres a bit for dragging this wonderful mathematician away to doing things like physics and stuff like this, because he’d spent 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.’ Later on, he got asked to do a survey of Hanover (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 to make yet another extraordinary discovery about geometry.

  It was out on this survey of Hanover that Gauss began to think the universe might be curved in some way. Gauss had been thinking about the geometry of Euclid and the parallel postulate, which says that, if we take any line and a point off that line, then there’s only one unique line through that point, which is parallel to the first line. Gauss started to ponder whether there were other geometries with no parallel lines, or perhaps many, and came up with an idea of new geometries, what we now call non-Euclidian geometries.

  MARCUS DU SAUTOY: In Euclid’s geometry, if I draw a triangle, the angles add up, as we all 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 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 Gauss had discovered new geometries with different properties from Euclid and he wondered whether maybe the universe is like that.

  Einstein also believed there was a curved geometry in the universe and it was many years later, when we measured light from stars, that we understood that the geometry Gauss discovered was the one that models the way the universe works. Marcus du Sautoy added that this was another idea he did not publish (‘silly fellow’) and, when Nikolai Lobachevsky and János Bolyai announced these new geometries,
Gauss said of Bolyai that to praise him would be to praise himself as he had already discovered them twenty years before, something that the diaries showed. Also from his surveys of Hanover came ideas about curvature, and an easy practical consequence of those, Colva Roney-Dougal told us, was that we cannot make flat maps of the whole globe.

  Around 1830, as Nick Evans put it, ‘This great theoretical physicist managed to pull himself away from pure mathematics.’ Magnetism had been interesting since about 1820 when Hans Christian Ørsted had shown that a current-carrying wire generates a magnetic field. Gauss and his friend Wilhelm Eduard Weber built a famous magnetism lab that stripped out all the iron, which could be magnetised, and replaced it with copper, an idea that was copied in labs across Europe. He looked at the theory of electricity, which is very like the theory of magnetism, and he went back over the mathematical structure and devised something that is now called Gauss’s Law of Electricity.

  NICK EVANS: Think of an electric charge as being like a hedgehog, with lots of spikes coming out of it. 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.

  The reason this is useful, Nick Evans explained, is that we can now think about taking lots of hedgehogs and sticking them to a wall, as charges on a wall, and now those spines can only escape in one direction, like the bristles on the end of a brush, straight out, parallel to each other, and so the density stays the same, which means that the electric forces do not fall away in that example. It became a way to think about electric problems and, almost immediately, to see the answers, at least in some cases.