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

that some sort of recursive process has been used in a proof, and this probably leads to a wild overestimate.
    In orthodox mathematics, the role played by our ‘plex’ is usually taken over by the exponential function exp x = e x , and 2plexplexplex will look more like exp exp exp 2. However, 10 is replaced by e here, so this statement is a complete lie. However, it’s not hard to complicate it so that it’s true, bearing in mind that e = 10 0.43 or thereabouts. Theorems about repeated exponentials are often rephrased in terms of repeated logarithms, like log log log x (see page 189 for logarithms). For example, it is known that every positive integer, with finitely many exceptions, is a sum of at most
    n log n + n log log n
perfect nth powers - well, ignoring a possible error that is smaller than n. More spectacularly, Carl Pomerance has proved that the number of pairs of amicable numbers (page 110) up to size x is at most
for some constant c.
    Several systems for representing big numbers have been worked out, with names like Steinhaus-Moser, Knuth’s up-arrow and Conway’s chained arrow. The topic is much bigger than you
might expect, which is only appropriate, and you can find much more about it at
    en.wikipedia.org/wiki/Skewes’_number
en.wikipedia.org/wiki/Large_numbers

The Drowning Mathematician
    Which (perhaps unfortunately) reminds me:
    Q: What sound does a drowning mathematician make?
     
    A: ‘log log log log log log log ...’

Mathematical Pirates
    Piracy is probably not the first thing that comes to mind in connection with mathematics. Of course, the peak period for piracy, or its state-sanctioned version, privateering, was also the golden age of the mathematics of navigation. Navigators drew geometric diagrams on charts using compasses and protractors; and they ‘shot the Sun’ with sextants and used mathematical tables to calculate the ship’s latitude. But that’s not the connection I’m after here, which is a curious set of historical links between mathematicians and pirates, centred on one of the all-time greats: Leonhard Euler, a Swiss-born mathematician who worked in Germany and Russia. He lived between 1707 and 1783 and produced more new mathematics than anybody who has ever lived. The connections were discovered by Ed Sandifer, and posted on his wonderful ‘How Euler Did It’ website: www.maa.org/news/howeulerdidit.html
    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