Professor Stewart's Hoard of Mathematical Treasures (10 page)

Euler made major advances in mechanics, including extensive applications of the principle of least action, credited to Pierre-Louis Moreau de Maupertuis, an influential French mathematician, writer and philosopher. Maupertuis associated a quantity called ‘action’ with the motion of any mechanical
system, and observed that the actual motion of the system minimises the action, compared with all alternative motions. When a stone bounces down a hill, for instance, the total action is less than it would have been if the stone had started by bouncing uphill for a time, or if it had wandered off sideways, or whatever. Maupertuis was President of the Berlin Academy of Sciences during the period when Euler was in Berlin, and knew Euler well. His father, René Moreau, made the family fortune in the 1690s by attacking British ships, on a privateering licence from the King of France, and married into the aristocracy.

Maupertuis wearing Lapp gear on his 1736 expedition to Lapland, which proved that the Earth is slightly flattened at the poles.

Euler wrote widely about ships,
¹⁰
and in particular analysed their stability, a beautiful application of hydrostatics. His work was not merely theoretical: it had a significant influence on Russian naval shipbuilding. In 1773, he published the
Théorie Complette de la Construction et de la Manoeuvre des Vaissaux Mise à la Portée de Ceux qui s’Appliquent à la Navigation.
In 1776, Henry Watson translated the book into English as
A Complete Theory of the Construction and Properties of Vessels, with Practical Conclusions for the 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.

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

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 intoxicated for increased gullibility.) The con-artist places three cups (or glasses) upright on the bar:

He inverts the centre cup
and explains that he will now turn all three of them to the upside-down position in exactly three moves, where each move inverts exactly two cups. They need not be adjacent: any two will do. (Of course, this can be done in one move - invert the two end cups - but the requirement to use three moves is part of the misdirection.)

The three moves are:

Now the con-artist begins to work on the mug. He casually turns the middle cup upright to get
and invites the mug to repeat the trick, with a small bet on the side to make things more interesting.