Why Beauty is Truth (16 page)

The Great Art
does not include a solution of the quintic equation, in which the unknown appears to the fifth power. But as the degree of the equation increases, the method for solving it gets more and more complicated, so few doubted that with enough ingenuity, the quintic too could be solved—you probably had to use fifth roots, and any formula would be really messy.

Cardano did not spend time seeking such a solution. After 1539 he returned to his numerous other activities, especially medicine. And now his family life fell apart in the most horrific manner: “My [youngest] son, between the day of his marriage and the day of his doom, had been accused of attempting to poison his wife while she was still in the weakness attendant upon childbirth. On the 17th day of February he was apprehended, and fifty-three days after, on April 13th, he was beheaded in prison.” While Cardano was still trying to come to terms with that tragedy, the horror got worse. “One house—mine—witnessed within the space of a few days, three funerals, that of my son, of my little granddaughter, Diaregina, and of the baby's nurse; nor was the infant grandson far from dying.”

For all that, Cardano was incurably optimistic about the human condition: “Nevertheless, I still have so many blessings, that if they were another's he would count himself lucky.”

W
hich road to take? Which subject to study? He loved them both, but he must choose between them. It was a terrible dilemma. The year was 1796, and a brilliant 19-year-old youth was faced with a decision that would affect the rest of his life. He had to decide on a career. Although he came from an ordinary family, Carl Friedrich Gauss knew that he could rise to greatness. Everyone recognized his ability, including the duke of Brunswick, in whose domain Gauss had been born and where his family lived. His problem was that he had too much ability, and he was forced to choose between his two great loves: mathematics and linguistics.

On 30 March, however, the decision was taken out of his hands by a curious, remarkable, and totally unprecedented discovery. On that day, Gauss found a Euclidean construction for a regular polygon with seventeen sides.

This may sound esoteric, but there was not even a hint of it in Euclid. You could find methods for constructing regular polygons with three sides, or four, or five, or six. You could combine the constructions for three and five sides to get fifteen, and repeated bisections would double the number of sides, leading to eight, ten, twelve, sixteen, twenty, . . .

But seventeen was crazy. It was also true, and Gauss knew full well
why
it was true. It all boiled down to two simple properties of the number 17. It is a prime number—its only exact divisors are itself and 1. And it is one greater than a power of two: 17 = 16 + 1 = 2
⁴
+ 1.

If you were a genius like Gauss, you could see why those two unassuming statements implied the existence of a construction, using straightedge and compass, of the regular seventeen-sided polygon. If you were any of
the other great mathematicians who had lived between 500 BCE and 1796, you would not even have got a sniff of any connection. We know this because they didn't.

If Gauss had needed confirmation of his mathematical talent, he certainly had it now. He resolved to become a mathematician.

The Gauss family had moved to Brunswick in 1740 when Carl's grandfather took a job there as a gardener. One of his three sons, Gebhard Dietrich Gauss, also became a gardener, occasionally working at other laboring jobs such as laying bricks and tending canals; at other times he was a “master of waterworks,” a merchant's assistant, and the treasurer of a small insurance fund. The more profitable trades were all controlled by guilds, and newcomers—even second-generation newcomers—were denied access to them. Gebhard married his second wife, Dorothea Benze, a stonemason's daughter working as a maid, in 1776. Their son Johann Friederich Carl (who later always called himself Carl Friedrich) was born in 1777.

Gebhard was honest but obstinate, ill-mannered, and not very bright. Dorothea was intelligent and self-assertive, traits that worked to Carl's advantage. By the time the boy was two, his mother knew she had a prodigy on her hands, and she set her heart on ensuring that he received an education that would allow his talents to flourish. Gebhard would have been happier if Carl had become a bricklayer. Thanks to his mother, Carl rose to fulfill the prediction that his friend, the geometer Wolfgang Bolyai, made to Dorothea when her son was 19, saying that Carl would become “the greatest mathematician in Europe.” She was so overjoyed that she burst into tears.

The boy responded to his mother's devotion, and for the last two decades of her life she lived with him, her eyesight failing until she became totally blind. The eminent mathematician insisted on looking after her himself, and he nursed her until 1839, when she died.

Gauss showed his talents early. At the age of three, he was watching his father, at that point a foreman in charge of a gang of laborers, handing out the weekly wages. Noticing a mistake in the arithmetic, the boy pointed it out to the amazed Gebhard. No one had taught the child numbers. He had taught himself.

A few years later, a schoolmaster named J. G. Büttner set Gauss's class a task that was intended to occupy them for a good few hours, giving
the teacher a well-earned rest. We don't know the exact question, but it was something very similar to this: add up all of the numbers from 1 to 100. Most likely, the numbers were not as nice as that, but there was a hidden pattern to them: they formed an arithmetic progression, meaning that the difference between any two consecutive numbers was always the same. There is a simple but not particularly obvious trick for adding the numbers in an arithmetic progression, but the class had not been taught it, so they had to laboriously add the numbers one at a time.

At least, that's what Büttner expected. He instructed his pupils that as soon as they had finished the assignment, they should place their slate, with the answer, on his desk. While his fellow students sat scribbling things like

1 + 2 = 3

3 + 3 = 6

6 + 4 = 10

with the inevitable mistake

10 + 5 = 14

and running out of space to write in, Gauss thought for a moment, chalked one number on his slate, walked up to the teacher, and slapped the slate face down on the desk.

“There it lies,” he said, went back to his desk, and sat down.

At the end of the lesson, when the teacher collected all the slates, precisely one had the correct answer: Gauss's.

Again, we don't know exactly how Gauss's mind worked, but we can come up with a plausible reconstruction. In all likelihood, Gauss had already thought about sums of that kind and spotted a useful trick. (If not, he was entirely capable of inventing one on the spot.) An easy way to find the answer is to group the numbers in pairs: 1 and 100, 2 and 99, 3 and 98, and so on, all the way to 50 and 51. Every number from 1 to 100 occurs exactly once in some pair, so the sum of all those numbers is the sum of all the pairs. But each pair adds up to 101. And there are 50 pairs. So the total is 50 × 101 = 5050. This (or some equivalent) is what he chalked on his slate.

The point of this tale is not that Gauss was unusually good at arithmetic, though he was; in his later astronomical work he routinely carried out enormous calculations to many decimal places, working with the speed of an idiot savant. But lighting calculation was not his sole talent. What he possessed in abundance was a gift for spotting cryptic patterns in mathematical problems, and using them to find solutions.

Büttner was so astonished that Gauss had seen through his clever ploy that, to his credit, he gave the boy the best arithmetic textbook that money could buy. Within a week, Gauss had gone beyond anything his teacher could handle.

It so happened that Büttner had a 17-year-old assistant, Johann Bartels, whose official duties were to cut quills for writing and to help the boys use them. Unofficially, Bartels had a fascination for mathematics. He was drawn to the brilliant ten-year old, and the two became lifelong friends. They worked on mathematics together, each encouraging the other.

Bartels was on familiar terms with some of the leading lights of Brunswick, and they soon learned that there was an unsung genius in their midst, whose family lived on the brink of poverty. One of these gentlemen, councilor and professor E. A. W. Zimmerman, introduced Gauss to the duke of Brunswick, Carl Wilhelm Ferdinand, in 1791. The duke, charmed and impressed, took it upon himself to pay for Gauss's education, as he occasionally did for the talented sons of the poor.

Mathematics was not the boy's sole talent. By the age of 15 he had become proficient in classical languages, so the duke financed studies in classics at the local gymnasium. (In the old German educational system, a gymnasium was a type of school that prepared its pupils for university entrance. It translates roughly as “high school,” but only paying students were admitted.) Many of Gauss's best works were later written in Latin. In 1792, he entered the Collegium Carolinium in Brunswick, again at the duke's expense.

By the age of 17 he had already discovered an astonishing theorem known as the “law of quadratic reciprocity” in the theory of numbers. It is a basic but rather esoteric regularity in divisibility properties of perfect squares. The pattern had already been noticed by Leonhard Euler, but Gauss was unaware of this and made the discovery entirely on his own. Very few people would even have thought of asking the question. And the boy was thinking very deeply about the theory of equations. In fact, that
was what led him to his construction of the regular 17-gon and thus set him on the road to mathematical immortality.

Between 1795 and 1798, Gauss studied for a degree at the University of Göttingen, once more paid for by Ferdinand. He made few friends, but the friendships he did strike up were deep and long-lasting. It was at Göttingen that Gauss met Bolyai, an accomplished geometer in the Euclidean tradition.

Mathematical ideas came so thick and fast to Gauss that sometimes they seemed to overwhelm him. He would suddenly cease whatever he was doing and stare blankly into the middle distance as a new thought struck him. At one point he worked out some of the theorems that would hold “if Euclidean geometry were not the true one.” At the forefront of his thoughts was a major work that he was composing, the
Disquisitiones Arithmeticae
, and by 1798 it was pretty much finished. But Gauss wanted to make certain that he had given due credit to his predecessors, so he visited the University of Helmstedt, which had a high-quality mathematics library overseen by Johann Pfaff, the best-known mathematician in Germany.

In 1801, after a frustrating delay at the printer's, the
Disquisitiones Arithmeticae
was published, with an effusive and no doubt heartfelt dedication to Duke Ferdinand. The duke's generosity did not end when Carl left the university. Ferdinand paid for his doctoral thesis, which he presented at the University of Helmstedt, to be printed as the regulations required. And when Carl started to worry about how to support himself when he left university, the duke gave him an allowance so that he could continue his researches without having to be bothered about money.

A noteworthy feature of the
Disquisitiones Arithmeticae
is its uncompromising style. The proofs are careful and logical, but the writing makes no concessions to the reader and gives no clue about the intuition behind the theorems. Later, he justified this attitude, which continued throughout his career, on the grounds that “When one has constructed a fine building, the scaffolding should no longer be visible.” Which is all very well if all you want people to do is admire the building. It's not so helpful if you want to teach them how to build their own. Carl Gustav Jacob Jacobi, whose work in complex analysis built on Gauss's ideas, said of his
illustrious predecessor, “He is like the fox, who erases his tracks in the sand with his tail.”