Professor Stewart's Hoard of Mathematical Treasures by Ian Stewart Read Free Book Online Page A

Management of Ships, Made Easy to Navigators. Watson was a prominent and regular contributor to the Ladies’ Diary, which
contained many mathematical games and problems and was widely read by men as well as women. He borrowed enough money to build three ships, based on some of Euler’s work on ship design, and applied to the King of England for a privateer’s licence, to operate near the Philippines. When the King declined, Watson used the ships to carry goods instead. Shortly after, he lost £100,000 (the equivalent of about £15-20 million in today’s money) on a project to modernise the Calcutta docks for the East India Company. The Company let the project go bankrupt and then bought it for peanuts. On his way back to England to sue the Company, Watson caught a fever and died.
    Sir Kenelm Digby was a courtier and diplomat in the reign of King Charles I of England. His link to Euler runs through Fermat, who sent Digby a geometrical problem in 1658. The letter was lost but Digby sent a copy to John Wallis, which has survived. Euler, who made a systematic effort to read everything Fermat wrote, heard of the problem and solved it. Digby had a colourful background. His father, Sir Everard Digby, was executed in 1606 for involvement in the Gunpowder Plot. He dabbled in alchemy, and was a founder of the Royal Society. In 1627-28 he led a privateering expedition to the Mediterranean. Here he seized Spanish, Flemish and Dutch ships, and attacked some French and Venetian ships anchored near the friendly Turkish port of Iskanderun. He returned to England with two ships filled with plunder. However, he also made life difficult for English merchant shipping, by inviting reprisals.

    Fermat’s poser: Draw a rectangle for which AB is √2 times AC, put a semicircle on top, and choose any point P on the semicircle. Construct X and Y as shown. Prove that AY 2 + BX 2 = AB 2 .

    Sandifer also mentions a very tenuous link, through Catherine the Great, who earlier had employed Euler as Court Mathematician, to John Paul Jones, ‘Father of the American Navy’. Jones was charged with piracy by the Dutch because he allegedly attacked shipping under ‘an unknown flag’, but the charge was dropped when the American flag was registered with the appropriate authorities.

The Hairy Ball Theorem
    An important theorem in topology says that you can’t comb a hairy ball smoothly. 11 A proof was given in 1912 by Luitzen Brouwer.

    A failed attempt to comb a hairy ball smoothly. At the north and south poles, the hairs would stick up, which is not allowed.
    Among the consequences of this theorem is the fact that, at any instant, the horizontal wind speed at some point on the Earth must be zero. Bearing in mind that typical winds are non-zero, such a point will almost always be isolated, and it will often be surrounded by a cyclone. So, at any time, there should be at least one cyclone somewhere in the Earth’s atmosphere, for purely topological reasons.
    The theorem also helps to explain why experimental fusion reactors use toroidal magnetic bottles (‘tokamaks’) to contain the superheated plasma. You can comb a hairy torus (or doughnut 12 ) smoothly. There’s more to the physics than that, of course.

    How to comb a hairy doughnut smoothly.
    Years ago, one of my mathematical colleagues explained this theorem to a friend of his, and unwisely pointed out that it applied to the family dog. The dog was called ‘hairy ball’ from that moment on.
    The picture shows a combed sphere with two ‘tufts’ - two places where the hairs don’t lay flat. The theorem says there can’t be no such places, but can there be only one?
     
    Answer on page 287

Cups and Downs
    This puzzle starts with a simple trick involving three cups, which is fun in its own right but also suggests some further questions with surprising answers.
    There is a time-honoured way to make money in a pub, requiring three cups and one mug. (The mug is human, and
should be moderately