In 1859 you could buy £1000 of stock in Field's company for £30. The submarine cable to India guaranteed by the British Government failed, costing the taxpayer £800000. Worldwide eleven thousand miles of undersea cable had been laid and only three thousand miles were operating. And in 1861 civil war broke out in the United States.

Yet Field still continued to try and raise interest in his project travelling back and forth across the Atlantic over 30 times (and being sea sick each time). By 1864 things were moving again. Because of the Civil War most of the capital was raised in Britain, half being put up by the firm which was to make the cable. The cable was redesigned in accordance with Thomson's theories. Even more important, strict quality control was exercised: the copper used was so pure that for the next 50 years ‘telegraphist's copper’ was the purest available.

Once again the British Government supported the project-the importance of quick communications in controlling an empire was evident to everybody. (Before it failed the old cable had carried an order countermanding troop movements which had saved a seventh of the cost of the cable.) Public interest was intense: when Thomson was delayed at his laboratory, the Glasgow London train would be held until he arrived. (Only Thoreau in America and Arnold in England felt that rapid communication was not needed until people had something worthwhile to communicate.)

The new cable was mechanically much stronger but also heavier. Only one ship was large enough to handle it and that was Brunei's Great Eastern.

In ancient times the extent of a city or an armed camp was often given in terms of its perimeter (so that a town would be described as requiring so many thousand paces to walk round). In the same way, according to Proclus, certain socialistic communities used to divide land so that each family received a plot of equal perimeter and it may have been in this context that it was first realised that a square contains a much greater area than a long thin rectangle of the same perimeter.

Once it was understood that figures with the same perimeter may contain different areas it was natural to ask whether there exists a figure of maximum area. It is not hard to guess that the answer is a circle but a guess is not a proof. The isoperimetric problem thus asks for a proof that among all figures of equal perimeter the circle has greatest area.

This question formed the subject of one of the last substantial investigations of the golden age of Greek geometry. In it Zenodorus proved that the circle has greater area than any polygon of the same perimeter.

We might expect that a purely geometrical approach could not go much further in the absence of precise notions of area and length. However, in 1841 Steiner showed how simple geometric considerations could be used to prove the following theorem.

In Chapters 39 and 40 we shall see how Sturm and Liouville extended the ideas of Fourier discussed in the last chapter. Liouville worked in many other mathematical fields as well and his results here were often so simple and basic that they have been completely absorbed into the general body of mathematics. It thus seems appropriate to recall some of his achievements.

At the age of 27 he founded the Journal des mathématiques pures et appliquées, which became, under his editorship, one of the major journals of the nineteenth century. He edited and published Galois' manuscripts in his journal and, just as importantly, gave a series of lectures organising and interpreting Galois theory for the general mathematical public.

In complex variable theory he used simple general arguments to bring order to the subject of multiply periodic functions. The result, called by his name which states that a bounded analytic function is constant, was known to Cauchy, but Liouville was the first to demonstrate its power. Some of the flavour of his work in this field is conveyed in the following theorem.

The technique of using a windmill to grind corn is a complex one. Not only must the windmill sails be always set to catch the wind but their speed of rotation must be regulated so that the millstones grind at their optimum speed. Originally all these tasks were done by the miller but slowly simple semiautomatic and automatic devices were introduced to take over part of the work.

This change was most marked among British millers of the eighteenth century and it was by them that the most difficult problem, that of keeping the sails revolving at the proper speed, was solved. A centrifugal pendulum was employed to measure the angular velocity of the millstone and, working through additional mechanisms, to adjust the sail setting so as to reduce the sail speed if the angular velocity was too large or increase it if the angular velocity was too small.

This invention was taken up by James Watt and formed one of the chain of ingenious improvements to Newcomen's steam engine to which we referred in Chapter 42. Initially enclosed, to hide the secret from the eyes of Watt's competitors, the rotating centrifugal pendulum was soon proudly displayed as the visible symbol of man's control over steam.

Naturally, attempts were made to produce new or improved versions of ‘Watt's governor’, but not all were successful.