Why Beauty is Truth by Ian Stewart

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.”

5
THE CUNNING FOX
    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 ofthe 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