110 A WORD FROM OUR FOUNDER (The following passage is translated from the intro-

duction to Fourier’s Théorie Analytique de la Chaleur.)

The equations of heat conduction like those of sound or small oscillations for liquids belong to one of the most recently discovered branches of science which it is most important to extend. Having established these differential equations we must obtain their solution which involves passing from a general to a particular solution obeying all the boundary conditions. This difficult investigation requires a special calculus based on new theorems . . .. The method which results leaves nothing vague or undetermined in the solutions; it leads us to the point where we can obtain numerical values, a necessary property of any such method, without which we obtain nothing but useless formal transformations.

These same theorems which enable us to solve the heat equation have immediate applications to questions in general analysis and dynamics whose solutions have long been sought.

The profound study of nature is the most fruitful source of mathematical discovery. Not only does this study, by proposing a fixed goal for our research, have the advantage of excluding vague questions and calculations without result. It is also a sure way of shaping Analysis itself, and discovering which parts it is most important to know and must always remain part of the subject. These fundamental principles are those which are found in all natural phenomena.

We see, for example, that the same expression which the geometers have studied abstractly and which, from this point of view belongs to pure mathematics also describes the motion of light through the atmosphere, determines the laws governing heat conduction in solids and is involved in all the main questions of probability theory.

The analytic equations, unknown to the classical geometers, which Descartes was the first to introduce for the study of curves and surfaces do not only appear in the study of geometrical figures and rational mechanics, they cover all general systems. There can not be a language more universal or more simple, more free from error and obscurity, that is to say more worthy of expressing the relationship between natural phenomena.

From this point of view, mathematical analysis is as extensive as nature herself. It defines all observable relationships, measures time, space, force, temperature. This difficult science grows slowly but once ground has been gained it is never relinquished. It grows, without ceasing, in size and strength, amid the constant error and confusion of the human spirit.

Its main property is clarity; analysis has no means of expressing confused notions. Analysis brings together the most disparate phenomena and discovers the hidden analogies which unify them. If the material escapes us like air or light because of its extreme fineness, if the bodies are placed far from us in the immensity of space, if man desires to know the aspect of the heavens for times separated by many centuries, if the effects of gravity and temperature occur in the interior of the solid earth at depths which will remain for ever inaccessible, yet mathematical analysis can still grasp the laws governing the phenomena. Analysis makes them actual and measurable and seems to be a faculty of human reason meant to compensate for the brevity of life and the imperfection of our senses. And which is still more remarkable it follows the same path when applied to all phenomena and interprets them all in the same language as if to attest the unity and simplicity of the design of the universe and to make still more evident that unchanging order which presides over all natural laws.

54 LORD KELVIN

Fourier believed that the main aim of mathematics should be the understanding of nature and that the purpose of understanding nature should be the benefit of mankind. Yet, interesting though Fourier’s own work was, he could point to no direct practical benefit from it. The credit for showing how powerful were the tools that Fourier had forged belongs, above all, to William Thomson, Lord Kelvin.

The life of William Thomson was linked with the University of Glasgow from 1832, when his father became professor there. Thomson was not a late developer. At the age of ten he and his 12-year-old brother enrolled in the University of Glasgow. Prizes in Greek, logic (what we would now call philosophy), mathematics, astronomy and physics marked his progress. ’A boy’, he said later, ’should have learnt by the age of twelve to write his own language with accuracy and some elegance; he should have a reading knowledge of French, should be able to translate Latin and easy Greek authors and should have some acquaintance with German.’ Towards the end of his time as a student in Glasgow he fell under the influence of an inspiring physics teacher called Nichol.

… after I had attended in 1839 Nichol’s… Class, I had become filled with the utmost admiration for the splendour and poetry of Fourier. Nichol was not a mathematician and did not profess to have really read Fourier, but he was capable of perceiving his greatness and of understanding what he was driving at, and of making us appreciate it. I asked Nichol if he thought I could read Fourier. He replied ’perhaps’. He thought the book a work of most transcendent merit. So on the Ist of May the very day when the prizes were given, I took Fourier out of the University Library, and in a fortnight I had mastered it—gone right through it. (This and the next quotation are from Silvanus P. Thompson’s Life of Lord Kelvin.)

At the age of seventeen Thomson entered Cambridge University. (It was not uncommon for good students to first study at a Scottish University and then transfer to Cambridge. A cynic might say that people like Maxwell and Thomson were educated in Scotland and examined in Cambridge.) The examinations in Cambridge were fiercely competitive exercises in problem solving against the clock. The best candidates were trained as for an athletics contest. Thomson’s coach might declare that his candidate had mathematical abilities which would outshine any man in England but the coach of Thomson’s rival boasted, more pertinently, that he had a candidate who could beat any man in Europe in an examination. Thomson (like Maxwell later) came second. During his time in Cambridge he had however won the Colquhoun sculls as a rower, published sixteen papers and, most important of all, obtained from his coach, a day before he left, two copies of Green’s Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism.

Thomson’s father had by now conceived the ambition of seeing his son elected to a chair in Glasgow. To improve his chances he was sent to France to learn experimental work in the laboratory of Regnault, to meet as many influential people as possible and, since the electors would place great emphasis on teaching ability, to learn the art of giving popular lectures. Thomson’s abilities won him a ready welcome and his copies of Green’s Essay created a great stir.

Eager steps were heard without and with a hasty rap upon the door a panting visitor rushed in. It was Sturm, in a state of high excitement. ’Vouz avez le Memoire de Green’, he exclaimed, ’M. Liouville me l’a dit!’. The Essay was produced, and Sturm eagerly scanned its contents, turning over page after page. ’Ah voilà mon affaire’, he cried, jumping from his seat as he caught sight of the formula in which Green had anticipated his theorem of the equivalent distribution.

The methods of Green and Fourier formed the foundation of Thomson’s mathematical methods. The foundations for his deepest physical insights were laid in Paris. Liouville encouraged him to investigate the relation between Faraday’s ’lines of force’ view of electrical phenomena and the classical continental ’action at a distance’, and, by reading Clapeyron, he became aware of Sadi Carnot’s view of heat.

Meanwhile his father’s campaign had borne fruit and at the age of twenty two, swept in on a wave of testimonials from, among others, De Morgan, Cayley, Hamilton, Boole, Sylvester, Stokes, Regnault and Liouville, Thomson became Professor of Natural Sciences at the University of Glasgow. Nor were his sponsors mistaken. For the next 20 years Thomson was to dominate physics.

There is a curious anonymity about physical ideas which makes it difficult to appreciate fully the originality of their discoverers. A novel always retains the signature of its author, a proof in pure mathematics may still delight us by its ingenuity or power after a hundred (consider Liouville’s proof given as Theorem 38.2 of the existence of transcendental numbers) or a thousand years (consider Euclid’s proof that there exist an infinity of primes), but the physics breakthrough of today is the common sense of tomorrow.

In electricity Thomson provided the link between Faraday and Maxwell. He was able to mathematise Faraday’s laws and to show the formal analogy between problems in heat flow and electricity. Thus the work of Fourier on heat immediately gave rise to theorems on electricity and the work of Green on potential theory immediately gave rise to theorems on heat flow. Similarly methods used to deal with linear and rotational displacements in elastic solids could be applied to give results on electricity and magnetism.

Maxwell, who from the end of his undergraduate days conducted a constant correspondence with Thomson first as pupil to master and then as equal to equal, felt sure that Thomson must have a complete theory of electrodynamics ’lying in loose papers and neglected only till you have worked out Heat or got a little spare time’. (The correspondence was published as Origins of Clerk Maxwell’s Electric Ideas, Larmor, Cambridge University Press, 1937. It includes a useful note from Maxwell on the habits of peacocks.) However, Thomson did not have such a theory and could never fully accept Maxwell’s ideas.

Thomson’s other major contribution to fundamental physics was his combination of the almost forgotten work of Carnot with the work of Joule on the conservation of energy to lay the foundations of Thermodynamics. Thomson himself was particularly proud of an argument developed with his brother which predicted the fall in the freezing point of water under pressure.

Other discoveries and inventions of Thomson are dealt with elsewhere in this book. We note in passing the discovery of what is now called Stokes’ theorem and the first mathematical discription of the oscillation of an electric circuit. After his work on the Atlantic cables (to be described in Chapters 65 and 66) he turned increasingly towards the practical applications of physics. He acted as consultant engineer to undersea telegraph and overland electric power transmission schemes (for example he worked out the optimum diameter of a transmission line). He was also unsurpassed as a designer of the rugged but accurate instruments required by the new electrical industries and took an active part in running the firm of Kelvin and Wright which manufactured them.

As a teacher he was inspiring rather than methodical. By a tradition which he supported, his audience included non specialists such as future doctors and theologians. Since he refused to talk down to his audience, some of them may not have understood very much but they all appreciated that they were being taught by a great man. Particularly enjoyed were lectures on acoustics (illustrated by a performance on the French horn), impulse (in which a large rifle was fired at a ballistic pendulum) and an experiment in which a large rubber sheet was slowly filled with water until it burst.

His teaching laboratory was the first in Britain. (The proposal to introduce experimental work into the Cambridge undergraduate courses aroused strong opposition. ’If [the student] does not believe the statements of his tutor probably a clergyman of mature knowledge, recognised ability and blameless character his suspicion is irrational and manifests a want of the power of appreciating evidence, a want fatal to his success in that branch of science which he is supposed to be cultivating.’ (Todhunter, First essay in ’The Conflict of Studies’, Macmillan 1873; since Todhunter is no fool the essay is worth reading.))

In conjunction with his friend Tait he wrote the famous Thomson and Tait (or T and T’ as it was known by everybody from Maxwell downwards) Treatise on Natural Philosophy. Originally intended as three moderately sized volumes which would cover all of mathematical and experimental physics, it became, as these things tend to become, two large volumes on mechanics.

As Maxwell said in his review of the second edition, the great merit of this work lay in rescuing a large chunk of physics from the mathematicians. Thanks to Thomson and Tait dynamical theorems have been dragged out of the sanctuary of profound mathematics in which they lay so long enshrined and have been set to do all kinds of work easy as well as difficult, throughout the whole range of physical science, …

The two northern wizards were the first who, without compunction or dread, uttered in their mother tongue the true and proper names of those dynamical concepts which the magicians of old were wont to invoke only by the aid of muttered symbols and inarticulate equations. And now the feeblest among us can repeat the words of power and take part in dynamical discussions which but a few years ago we should have left for our betters.

(Maxwell’s Collected Works, p. 782, Vol. 1.) As in mechanics so elsewhere Thomson’s career marked the break away of mathematical physics and mathematical engineering from mathematics into separate disciplines with their own methods and ideas. Thomson’s contemporaries considered him almost a second Newton, but he would, I think, have been happy with the modern estimate of him as Fourier’s successor and (with Faraday) Maxwell’s predecessor.

65 THE TRANSATLANTIC CABLE I At the beginning of the nineteenth century a new

industrial Europe was taking shape. Its capitalists knew that time is money and its politicians that time is power. During the Napoleonic wars the French and then the English set up systems of semaphores for transmitting military information but with this exception messages could travel no faster than a horse or ship. The invention of the electric telegraph in the 1830s changed all this and within ten years a network of telegraph wires covered England, western Europc and the more settled part of the USA. (Telegraphs were particularly important for the efficient and safe control of the new railways.)

It was thus inevitable that attempts would be made to provide underwater links between the various separate systems. The first cable between Britain and France was laid in 1850. The operators found the greatest difficulty in transmitting even a few words. After 12 hours the enterprise was brought to an abrupt conclusion when a trawler accidentally caught and cut the cable. Undeterred the railway engineer Crampton put up money for a new attempt with a heavier armoured cable of his own design.

Crampton’s cable was a complete success and a spurt of submarine cable laying began. On the whole, the short lines worked, but their operators found that signals could not be transmitted along submarine cables as fast as along land lines without becoming confused. (It was the attempt to transmit at normal speed which had led to the jumbled messages along the first Anglo French cable.) The practical men just noted this fact as an annoyance while the scientific men observed that Faraday had predicted some retardation on account of the increased capacitance of undersea cables.

The record of the longer lines was, at best, mixed. Inspite of this an American called Cyrus J. Fields proposed a telegraph line linking Europe and America. Oceanographic surveys showed that the bottom of the Atlantic was suitable for cable laying and by connecting circuits of the existing land telegraph lines people had produced a telegraph line of the length of the proposed cable through which signals had been passed extremely rapidly. The British government offered a subsidy and money was rapidly raised. Contracts were placed and the business of spinning 2500 miles of cable put in hand.

The mechanical problems of laying such a cable were considerable. Airy, the Astronomer Royal stated that ’it was a mathematical impossibility to submerge the cable successfully at so great a depth’. (Airy had a bad track record he failed to look for Neptune when it was predicted by Adams, he disbelieved Hamilton’s prediction of double refraction, he predicted that the Crystal Palace would be destroyed by hail and he advised the designer of the Tay Bridge about wind speeds without taking into account gusting.) However, the laying was in the hands of an engineering genius Charles Bright and it was the electrical side which was to cause the worst problems.

Faraday had indeed predicted signal retardation but he and others like Morse had in mind a model of a submarine cable as a hosepipe which took longer to fill with water (signal) as it got longer. The remedy was thus to use a thin wire (so that less electricity was needed to charge it) and high voltages to push the signal through. Faraday’s opinion was shared by Morse and by the electrical adviser Dr Whitehouse.

Thomson’s researches had given him a clearer mathematical picture of the problem. The current flow in a telegraph wire in air is governed (approximately) by the wave equation

c^{2} \frac{∂^{2}ϕ}{∂x^{2}} = \frac{∂^{2}ϕ}{∂t^{2}} .

As we saw in Chapter 64 a pulse ϕ(x, t) = f(t − c^{−1}x) travels at a well defined speed c with no change of shape or magnitude with time. Thus signals can be sent as close together as our transmitter can make them and our receiver distinguish them. (Trained operators could manage up to 40 words a minute in Morse code.)

However in undersea cables (of the type proposed) capacitative effects dominate and current flow is governed (approximately) by the heat equation

K \frac{∂^{2}ϕ}{∂x^{2}} = \frac{∂ϕ}{∂t} .

We discussed this problem in Chapter 62 ending up with a series of remarks on the nature of the solution for a heat pulse. In particular, looking at statement (5) at the end of Chapter 62, we are led to the following prediction. (5)’ For a very long submarine cable of length x the main effect of any sent electric pulse will last for a time T seconds proportional to x2 (and inversely proportional to K).

If we try to transmit a further signal during this time 7 the two signals will be jumbled up when received. We are thus roughly limited to sending one signal every T seconds. The maximum rate of signalling is thus inversely proportional to the square of the length of the cable (and directly proportional to K). Thus in leaping from submarine cables of 30 miles length to cables of length 1500 miles retardation effects become not 50 times but 2500 times worse. Moreover, contrary to received opinion, increasing the voltage (i.e. increasing the pulse height) would, if anything, make things worse. Finally, since K turns out to be directly proportional to the conductivity, the wire should have as large a diameter as possible.

Whitehouse, whose professional reputation was now involved, denied these conclusions. The shareholders of Field’s company were largely British and contained many Glaswegians who voted Thomson onto the board of directors. But even as a director Thomson had no authority over the technical advisers. Moreover the production of the cable was now underway on principles contrary to Thomson’s own. Testing the completed cable, he was astonished to find some sections conducted only half as well as others. (This was not due to dishonesty or incompetence, the manufacturers were supplying copper to the then highest commercial standards of purity. The need for really high conductivity had not arisen before the Atlantic cable.) Orders were given for greater care to be taken in future.

Realising that the success of the enterprise would depend on a fast, sensitive detector, Thomson set about to invent one. The problem with an ordinary galvanometer is the high inertia of the needle (which thus requires a large kick to make it swing and a long time to stop swinging.) Thomson came up with the mirror galvanometer in which the pointer is replaced by a beam of light. This marvellous idea so impressed Maxwell as to inspire one of his verses (see Appendix D). The other directors who believed, or hoped, that Whitehouse was right, were polite but not enthusiastic about Thomson’s activity on their behalf.

In a first attempt in 1857 the cable snapped after 335 miles had been laid. In 1858 new attempts were foiled by storm and breakages but the company went doggedly on and on 5 August 1858, Europe and America were linked by cable. On 16 August it carried a 99-word message of greeting from Queen Victoria to President Buchanan. But that 99-word message took 16 \frac{1}{2} hours to get through. In vain Whitehouse tried to get his receiver to work only Thomson’s galvanometer was sensitive enough to interpret the minute and blurred messages coming through. In vain Whitehouse used his two thousand volt induction coils to try to push messages through faster after four weeks of this treatment the cable gave up the ghost; 2500 tons of cable and L350 000 of capital lay useless on the ocean floor.

66 THE TRANSATLANTIC CABLE II In 1859 you could buy L1000 of stock in Field’s com-

pany for L30. The submarine cable to India guaranteed by the British Government failed, costing the taxpayer L800 000. 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 Brunel’s Great Eastern. Five times larger than any other existing ship she had been built to steam to Australia without refuelling but bad luck and the opening of the Suez Canal had prevented her profitable use and ruined a succession of owners. A separate company was set up to buy her at a knockdown price and lease her to the cable company in return for a shareholding.

But this time there was a competitor. The Western Union Company had decided to build a cable along the overland route across America, Alaska, the Bering Straits, Siberia and Russia to reach Europe the long way round. The Western Union’s views on competition were simple. ’The public has been slow to learn how disastrous opposition to the Western Union Telegraph Company has been. Competitive lines cannot make a go of it and they might as well quit trying’ (quoted in A.F. Harlow Old Wires and New Wawes, Appleton Century 1936).

The commercial success of the cable would thus depend on the rate at which messages could be transmitted. Thomson had predicted the problems of the first cable by mathematics. On the basis of the same mathematics he now promised the company a rate of eight or even 12 words a minute. Half a million pounds was being staked on the correctness of the solution of a partial differential equation.

On 23 July, 1865 the Great Eastern set sail. All went well for nine days and then, when 1250 miles had been laid, the cable parted. For a fortnight the expedition fished for the cable in two miles of water. Three times they caught and lifted it but each time the lifting gear failed. Leaving a buoy to mark the spot the expedition sailed for home.

Thomson and the other engineers were now confident that the cable would do all that was required, since communication with the home base had been perfect until the final break. They also felt sure that they could recover the old cable. The company decided to build and lay a new cable and then go back and complete the old one. By now Field had tried three times to lay his cable and failed each time, yet the public subscribed half of the L600 000 required within two weeks.

This time all went well. Cable laying started on 12 July 1866 and the cable was landed on the morning of the 27th. On the 28th the cable was open for business and earned L1000. A director’s meeting of the Western Union ordered all work on their projected line to be stopped. Their crews had laid 300 miles of cable in Alaska, 350 miles in Siberia,400 miles in the Canadian waste but all this was abandoned at a loss of $3 000 000.

On I September after three weeks of effort the old cable was recovered and on 8 September a second perfect cable linked America and Europe. A wave of knighthoods swept over the engineers and directors. Investors who had seen their shares drop to almost nothing now received a return of 16% (in an age when government stock paid 22%). Even more importantly the cable manufacturers had established a commanding technical lead in the next 80 years no less than 90% of all undersea cables were supplied by that one company.

Thomson’s knowledge and instruments had contributed greatly to the success of the new industry. In turn the patents which he held made him a wealthy man. He could now indulge his love of the sea from his own yacht. (Though even here he and his friend Helmholtz studied the theory of water waves, “’which”, [said] Helmholtz, “he loved to treat as a kind of race between us”. When Thomson had to go ashore at Inverary for some hours, as he left hc said: “Now, mind, Helmholtz, you’re not to work at waves while I’m away” (quoted in S.P. Thompson, p. 614).’)

Thomson enjoyed the wealth and fame that his work brought him. (His elevation to the Peerage in 1892 was to leave future generations with the vague impression that there had been three great Victorian scientists called, Thomson, Sir William Thomson and Lord Kelvin.) But he delighted in technological progress itself and in that fact that ’… science has tended very much to accelerate the results, and to give the world the benefit of those results earlier than it could have had them, if left to struggle for them and try for them by repeated efforts and repeated failures unguided by such principles as can be evolved from the abstract investigations of science (speech, on being made a freeman of Glasgow in S.P. Thompson, p. 405).’

56 THE AGE OF THE EARTH I It is a very natural assumption that the earth and

the species on it, including man, were all created at the same time. The age of the earth would then be approximately the length of man’s recorded history, a few thousand years, and its geography and landscape essentially unchanged since the beginning.

However in the first decades of the nineteenth century geological evidence for great changes in the past began to build up. Large areas of land had once been under water, mountain ranges had been thrown up from lowlands and the evidence of fossils showed the past existence of species with no living counterparts.

At first it seemed that such vast changes must have been the outcome of tremendous and violent forces. According to this view the world that we live in has been shaped by a series of cataclysms (the last of which was often identified with Noah’s flood) and the age of the earth must be reckoned in tens of thousands of years.

However Lyell in his Principles of Geology presented detailed and convincing arguments for an opposing view. He sought to ’explain the former changes by causes now in operation’. According to his theory processes such as slow erosion by wind and water, gradual deposition of sediment by rivers and the cumulative effect of earthquakes and volcanic action combined over very long periods of time to produce the vast changes recorded in the earth’s surface.

Lyell’s theory was called ’uniformitarian’ because it rejected the idea that geological processes had been more violent in the past than they are today. But the rejection of vast forces was only made possible by the acceptance of vast times. The rate of action of very slow forces is, obviously, very hard to measure but uniformitarian theories demanded that the age of the earth be measured certainly in terms of millions of years, probably in hundreds of millions, perhaps in thousands of millions of years or more.

Lyell was unable to produce a uniformitarian biological theory to complement his uniformitarian geological theory. He was able to account for the disapperance of species in the geological record but not for the emergence of new species. A solution to this problem was provided by Darwin with his theory of evolution by natural selection.

Darwin’s theory too, required a great age for the earth. ’He who can read Sir Charles Lyell’s grand work on the Principles of Geology,… yet does not admit how incomprehensible vast has been the past period of time, may at once close this volume’ (Darwin, On the Origin of Species, First Edition, p.282). To allow time for natural selection to operate, the age of the earth must be measured in many hundreds of millions of years.

But such demands for endless time run counter to the laws of thermodynamics. Every day the sun radiates immense amounts of energy. By the law of conservation of energy there must be some source of this energy and when the source is exhausted the process must cease. Kelvin, as one of the founders of thermodynamics, was fascinated by this problem. Simple calculation shows that chemical processes (such as in the burning of coal) are totally insufficient as a source and he was forced to conclude that the only available mechanism was the conversion of gravitational potential energy into heat as the sun contracted.

On this assumption his calculations showed that ’[it is] on the whole most probable that the sun has not illuminated the earth for 100 000 000 years and almost certain that [it] has not done so for 500 000 000 years’. Moreover, Kelvin believed that the sun was cooling so that the climate would become more extreme (and the associated geological processes more violent) as we go back in history.

However, Kelvin’s most compelling argument, though still based on the principle of conservation of energy, concerned the earth rather than the sun. It is well known that the temperature of the earth increases with depth and this implies a continual loss of heat from the interior, by conduction outwards through or into the upper crust. Hence, since the upper crust does not become hotter from year to year there must be a… loss of heat from the whole earth. It is possible that no cooling may result from this loss of heat but only an exhaustion of potential energy which in this case could scarcely be other than chemical.

But there is no reasonable mechanism to keep a chemical reaction going at a steady pace for millions of years so, Kelvin concludes, ’the… view… that the earth is merely a warm chemically inert body cooling, is clearly to be preferred in the present state of science’. (The quotations are from Appendices D and E of Thomson [Kelvin] and Tait ’Treatise on Natural Philosophy’.)

However high the initial temperature of the earth, geological history could only begin when the outer crust had solidified. Did the earth solidify first from the outside leaving a liquid core or did convection currents ensure that the earth solidified at a uniform temperature throughout? Many early Victorian geologists pictured the earth as a molten mass on which floated a thin skin of solidified rock and this plausible picture survives in the popular mind today. However, Kelvin and others had already shown that such a picture was dynamically inconsistent with known facts. (Indeed, Kelvin’s computations revealed a body with the rigidity of a steel ball of similar size.) The rejection of this model together with the fact that molten rock contracts on solidifying led Kelvin to believe that the earth was a solid body and that it had solidified at a more or less uniform temperature. In the next chapter we shall see how, starting from this hypothesis, he was able to estimate the age of the earth.

58 THE AGE OF THE EARTH III

The reception of Kelvin’s calculations is described by J.D. Burchfield in his very interesting book Lord Kelvin and The Age of the Earth (Macmillan, 1975) on which this chapter is based. Before Kelvin few geologists had thought deeply about the problem of geological time partly from lack of data but mainly perhaps because the human mind finds it hard to contrast a vast time of ten million years with a vast time of a thousand million years.

As the force of Kelvin’s argument sank in, geologists began to examine their uniformitarian theories more critically. Certainly if the rate of geological change had been the same throughout geological history as that observed today, 100 000 000 years would not suffice to shape the present state of the earth’s surface. But as Kelvin himself pointed out, if the earth is losing energy then in earlier times geological forces like earthquakes and volcanoes were probably much more violent. Thus by modifying uniformitarian theory so as to allow past geological forces, though still the same in kind, to be different in degree from those observed today it was possible to speed up the rate of geological change in the past and compress the earth’s geological history into Kelvin’s 100 000 000 years.

The problems posed to Darwin’s theory of evolution by natural selection were more serious. ‘I am greatly troubled at the short duration of the world according to [Kelvin], for I require for my theoretical views a very long period before the Cambrian formation (Darwin quoted in Burchfield, p.75).’ In the fifth edition of The Origin of Species, Darwin attempted to adjust to the new time scale by allowing greater scope for evolution by processes other than natural selection. He also pointed out that more extreme geological conditions might increase the pressures of natural selection and so increase the rate of evolution.

But in the end he was forced to ask for a suspension of judgement. In the final chapter of The Origin of Species in which Darwin reviewed the argument of his book he now added the following words.

With respect to the lapse of time not having been sufficient since our planet was consolidated for the assumed amount of organic change, and this objection, as argued by [Kelvin], is probably one of the gravest yet advanced, I can only say, firstly that we do not know at what rate species change as measured by years, and secondly, that many philosophers are not as yet willing to admit that we know enough of the constitution of the universe and of the interior of our globe to speculate with safety on its past duration. (Darwin The Origin of Species, Sixth Edition, p.409.)

Kelvin reinforced his case with a third argument drawn from the tidal retardation of the earth’s rotation. But here he exposed the chief weakness of his arguments. Huxley was able to quote Laplace, Adams and earlier work of Kelvin himself as disagreeing both about the cause and the extent of this supposed phenomenon.

I do not presume (Huxley wrote) to throw the slightest doubt upon the accuracy of any of these calculations made by such distinguished mathematicians as those who have made the suggestions I have cited. On the contrary, it is necessary to my argument to assume that they are all correct. But I desire to point out that this seems to be one of the many cases in which the admitted accuracy of mathematical processes is allowed to throw a wholly inadmissible appearance of authority over the results obtained by them. Mathematics may be compared to a mill of exquisite workmanship, which grinds you stuff of any degree of fineness; but nevertheless, what you get out depends on what you put in; and as the grandest mill in the world will not extract wheat-flour from peascods, so pages of formulae will not get a definite result out of loose data. (Quarterly Journal of the Geological Society of London, Vol. 25, 1869)

However Kelvin’s estimates were the best available and for the next thirty years geology took its time from physics, and biology its time from geology. Tait only echoed the general opinion when he wrote: ’Let us then hear no more nonsense about the interference of mathematicians in matters with which they have no concern; rather let them be lauded for condescending from their proud preeminence to help out of the rut the too ponderous waggon of some scientific brother (quoted in Burchfield, p.93.).’

But even as the geologists readjusted their time scales to fit Kelvin’s first estimate for the age of the earth, he and his followers began to adjust it down until at the end of the nineteenth century the best physical estimates of the age of the earth and sun were about 20 million years whilst the minimum the geologists could allow was closer to Kelvin’s original 100 million years.

Then in 1904 Rutherford announced that the radioactive decay of radium was accompanied by the release of immense amounts of energy and speculated that this could replace the heat lost from the surface of the earth. Kelvin’s argument would then only give a minimum for the earth’s age. ’The discovery of the radio-active elements … thus increases the possible limit of the duration of life on this planet, and allows the time claimed by the geologist and biologist for the process of evolution. (Rutherford quoted in Burchfield, p. 164.)’

Rutherford was fond of recounting the story of his lecture to the Royal Institution, I came into the room which was half dark, and presently spotted Lord Kelvin in the audience and realised I was in for trouble at the last part of the speech dealing with the age of the earth, where my views conflicted with his. To my relief, Kelvin fell fast asleep, but as I came to the important point, I saw the old bird sit up, open an eye and take a baleful glance at me! Then a sudden inspiration came, and I said Lord Kelvin had limited the age of the earth, provided no new source of heat was discovered. That prophetic utterance refers to what we are now considering tonight, radium! Behold! the old boy beamed upon me. (Eve Rutherford, Macmillan 1939, p.107.)

A problem for the geologists was now replaced by a problem for the physicists. How could the principle of conservation of energy be reconciled with the immense release of energy which occurs in radioactive decay? The answer was provided by a theory which was just beginning to be gossiped about. ’One day.. . Rutherford began twitting Wien about relativity. Wien explained that Newton was wrong in the matter of relative motion, which was not the joint velocities u + v, but that expression, according to Einstein must be divided by 1+uv/c^{2}. Wien added, “But no Anglo Saxon can understand relativity!” “No!” laughed Rutherford, “they have too much good sense” (Eve, p.193).’ Einstein’s theory of relativity extended the principle of conservation of energy by taking matter as a form of energy. It is the conversion of matter to heat which maintains the earth’s internal temperature and supplies the energy radiated by the sun.

Radio-active dating now leads geologists to give the earth an age of about two or three thousand million years. Indeed they now have so much time that they believe that the present day must be a period of particularly intense geological activity. And in this way Darwin’s theories have received a remarkable confirmation.

91 THE FUTURE OF MATHEMATICS VIEWED FROM 1800

The discovery of the calculus was followed by a century of brilliant exploitation led by men like Euler and Lagrange. But towards the end of that century it became clear that several problems of physics and pure mathematics were resisting all attacks. In addition the lack of any clear and agreed foundation for the calculus led to paradoxes and obscurities which might be ignored but could not be resolved. In a much quoted letter to d’Alembert in 1781, Lagrange wrote

It seems to me that the mine is already almost too deep, and unless we discover new seams we shall sooner or later have to abandon it. Today Physics and Chemistry offer more brilliant and more easily exploited riches; and it seems that the taste of the century has turned entirely in that direction. It is not impossible that the mathematical positions in the Academies will one day become what the University chairs in Arabic are now. (p. 386, Vol. 13 of Lagrange’s Collected Works).

These sentiments were echoed by Delambre in his summary for the Academy of Sciences of the state of the mathematical sciences (’Rapport Historique sur les Progrès des Sciences Mathématiques Depuis 1789’). Speaking of the calculus he described the general view of such discoveries. ’Prepared by the work of many centuries, when they are sufficiently ripe to be plucked by genius, their first effect is to excite admiration and their second to impose on following generations immense labours whose glory will never equal their difficulty.’

It would be difficult (he continues) and perhaps rash to try and analyse the chances which the future offers for the progress of mathematics; in almost all its branches we are halted by insurmountable difficulties and only improvements of detail seem to remain for us. All movements which cannot be reduced to small oscillations about a stable point subject to simple laws, all solutions which we are not given initially to a first approximation, all these things seem beyond our grasp…

After remarking that the fundamental principles of hydrodynamics still eluded mathematicians, he goes on:- ’AII these difficulties seem to show that the power of our methods is almost exhausted, just as the power of ordinary algebra was in the time of Leibnitz and Newton and that we need some new method to enable us to study transcendental functions and the equations which contain them.’ Delambre was mathematical secretary to the Academy of Sciences and his report appeared in 1810. Three years previously the prefect of Isère had submitted to the Academy a memoir on the conduction of heat which was received without enthusiasm. Looking back, we now see this memoir as heralding the surge of new mathematical methods and results which were to mark the new century.

92 WHO WAS FOURIER? I

(The contents of the next two chapters are based on the excellent biography by Herivel Joseph Fourier The Man and the Physicist, Oxford 1975.)

Joseph Fourier was born in 1768 in Auxerre, the ninth child of a master tailor. Although the death of his father left him an orphan at the age of ten, his intelligence gained him a free place at the local Benedictine school. At the end of a brilliant school career he applied to enter the artillery only to be informed that such a profession was only open to those of noble blood and was closed to him ’even if he were a second Newton’.

Fourier began to prepare to enter the Benedictine teaching order but, whatever his plans may have been, the course of his life was violently altered by the outbreak of the French Revolution. ’As the natural ideas of equality developed, it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests and to free from this double yoke the long usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and the most beautiful which any nation has undertaken.’

However, the Revolution was soon threatened by problems of its own making. The collapse of royal authority and the effects of revolutionary zeal and political misjudgement created administrative chaos whilst at the same time driving France into war against most of Europe. A succession of military defeats aggravated by treachery, economic problems and internal revolution put the Republic in danger and brought Robespierre’s party to power. In the next two years France was reorganised for war and central power reasserted. Both at a national and a local level, educated men were needed to fill the administrative vacuum caused by the revolution Fourier, who was now teaching at his old school, became first an active member of the revolutionary committee of Auxerre and then its president.

The situation of the new Republic called for ruthless measures which the government, conscious of its own revolutionary virtues, was well prepared to take. Treachery was fought by a political terror in which opponents both to the left and right were executed and, as the definition of treachery was extended, it became clear that no one was safe. Fourier himself was arrested, released and then rearrested. A deputation from Auxerre which, with considerable courage, went to Paris to plead his case, was told—’Yes, he speaks well, but we no longer have any need of musical patriots.’ Only the fall of Robespierre saved Fourier’s head.

However Fourier’s release did not mark the end of his troubles. As coup d’état followed coup d’état, and the revolution swung erratically to the right he would remain a marked man. No one had been executed in Auxerre but Fourier had been an agent of the terror there. His arrest was on a charge of Hébertism and the Hébertists were to the left of Robespierre. The word ’terrorist’ then, like ’Trotskyist’ now, denoted a defeated yet feared opponent.

Luckily an opportunity to leave Auxerre now presented itself A new college (the Ecole Normale) was being set up in Paris to help train teachers and Fourier was nominated as a pupil by a neighbouring district. Fourier could now study under men like Lagrange, Monge and Laplace and escape his terrorist past. Fourier’s talents were soon noted, but the college was not successful and its closure was followed by further problems for Fourier.

’We shudder when we think that the pupils of the Ecole Normale were chosen under the reign of Robespierre and his protégés. It is only too true that Balme and Fourier, pupils of the department of Yonne have long professed the atrocious principles and infernal maxims of the tyrants. Nevertheless they prepare to become teachers of our children. Is it not to vomit their poison in the bosom of innocence (From an address to the National Convention, quoted by Herivel).’ Fourier was again arrested, released, rearrested and finally, following yet another political swing, released to become a teacher at the new Ecole Polytechnique.

Here Fourier remained for three years. That his talent was recognised is shown by the fact that he succeeded Lagrange in the Chair of Analysis and Mechanics. This quiet interlude was ended by a government order to join the invasion of Egypt. Ostensibly intended to liberate Egypt from the Turks and to threaten the British position in India, the expedition may have been seen by the government as a way of keeping a troublesome general as far away as possible and by the general (Napoleon) as the first step toward becoming Emperor of the East. Fourier was one of a group of scientists and intellectuals intended to form part of the immense cultural benefits that France was to bestow on Egypt.

Unfortunately the effects of stunning French victories on land were negated by Nelson’s destruction of the French invasion fleet. Learning of serious military and domestic problems in France, Napoleon deserted his army and rushed home to ’save France’. A rapid and successful coup d’état enabled him to read the dispatches in which General Kléber (his successor in Egypt) explained exactly what he thought of this behaviour.

Both before and after Napoleon’s departure, Fourier occupied several important administrative and political posts in Egypt. When the French expedition finally surrendered in 1801 and Fourier was repatriated, Napoleon offered him the post of Prefect of the Department of the Isère centred round Grenoble. (France had been divided into 83 Departments and each Prefect governed his Department on behalf of the central government.) Although he could have continued as Professor at the Polytechnique, Fourier accepted the offer. Herivel suggests that Egypt had given him a taste for administration and that he hoped to rise higher. Herivel also suggests that Fourier’s close association with Kléber after Napoleon’s departure accounts for the fact that these hopes were not fulfilled.

Fourier seems to have been a popular and efficient Prefect. His greatest achievement during his 14 years of office was by reconciling the conflicting interests of some forty communities to enable the swamps of Bourgoin to be drained. The draining of twenty thousand acres of swamps resulted in major economic and health benefits and was achieved during a period more noted for grandiose paper plans than for concrete achievements. Fourier’s other administrative memorial was a new road across the Alps (now Route N91).

Apart from his prefectorial duties Fourier helped organise the Description of Egypt. This work written by the intellectuals attached to the Egyptian expedition did much to inspire European interest in Egypt and was thus one of the two permanent results of that expedition. (The other was the discovery of the Rosetta Stone, a trilingual inscription which was to provide the key to the deciphering of hieroglyphics.) Fourier’s main contribution was the general introduction—a survey of Egyptian history up to modern times. (An Egyptologist with whom I discussed this described the introduction as a masterpiece and a turning point in the subject. He was surprised to hear that Fourier also had a reputation as a mathematician.) On a personal level he encouraged Champollion, a linguistic infant prodigy, to take up Egyptology and used his position as prefect to preserve his protegé from conscription. It was Champollion who eventually deciphered the Rosetta Stone.

Napoleon’s domestic policy carried to its natural conclusion a political compromise which had been developing since the fall of Robespierre. The old powers of the church and the nobility were to be reconciled with the new power of the bourgeoisie and an administration loyal to the state alone. Henceforward men were to be equal before the law but the state renounced any attempt to achieve economic equality, or indeed any form of economic control. Fourier who had risen through his own talent but who ’could give lessons in theology to bishops and politeness to the pre-1790 parliamentarians’ both symbolised and in his position as Prefect aided the new compromise.

93 WHO WAS FOURIER? II

’Yesterday was my 21st birthday, at that age Newton and Pascal had [already] acquired many claims to immortality.’ Fourteen years after he wrote the postscript above, Fourier was prefect of Isère but had still no claim to the immortality he had craved as a young man. The work on the zeros of algebraic polynomials which he had pursued since his time at Auxerre still gains him a footnote in algebra textbooks—but a footnote is not immortality. His lecturing while at the Ecole Polytechnique had been much praised—but a good lecture is the most ephemeral of triumphs.

However, in 1804 he took up the subject of the propagation of heat. The field of Newtonian mechanics had already been worked over by several masters but the physics of heat, light, electricity and magnetism had still to be brought under the rule of mathematics. The choice of one of these fields thus requires no explanation. The particular choice of heat may have been connected with an obsessional need for warmth which Fourier acquired in Egypt.

In three remarkable years Fourier found the fundamental equations for heat conduction, developed new methods to solve them, applied these methods in a wide variety of cases and produced experimental evidence to support his solutions. At the end of 1807 he had submitted a memoir containing this work to the Academy. A commission consisting of Lagrange, Laplace, Monge and Lacroix was set up to examine it.

Instead of the enthusiastic acceptance that Fourier may have hoped for, he ran into two major criticisms. On the mathematical side neither Laplace nor Lagrange could accept the validity of his use of Fourier series. For example Laplace could not believe that cos x could possibly be expressed using a sine series. On the physical side Laplace and his pupils had already attacked the problem of heat conduction from a different angle. Unable to accept that Fourier’s approach was superior, Laplace, Biot and Poisson attacked his derivation of the equations of conduction as lacking in rigour.

In 1811 the Academy gave Fourier a second chance to promote his theory by offering its grand prize in mathematics for work on the theory of heat conduction Fourier resubmitted his earlier essay together with some further results. (The most important addition was the introduction of the Fourier transform and its use, in the manner of Chapter 55, in studying the cooling of infinite solids.) However, although Fourier gained the prize, the accompanying report made it clear that Lagrange and Laplace had not withdrawn their objections.

The attitude of Lagrange and Laplace has been severely criticised by writers of modern texts on mathematical methods as (to quote one of them) ’typical in its demonstration of the difference between pure and applied mathematicians’. Since Laplace was the greatest mathematical physicist of the age and the author of the Traité de Mécanique Céleste, whilst Lagrange (who admittedly often worked in pure mathematics) was the author of the equally influential Mécanique Analytique this seems a curious judgement.

In fact Laplace and, particularly, Lagrange, did not simply doubt the rigour but also the truth of Fourier’s assertions concerning expansions in trigonometric series. They may also have doubted the utility of such expansions. Why should Fourier’s expansion in trigonometric series be superior to an expansion in continued fractions, or infinite products, or formal power series or any of the hundred and one pretty tricks of the eighteenth century mathematicians? With hindsight we can see (or at least persuade ourselves that we can see) why Fourier series are much better adapted than Taylor series to a whole class of physical problems. Laplace and Lagrange could not see into the future and their doubts are surely more a tribute to the originality of Fourier’s methods than a reproach to mathematicians who Fourier greatly respected (and, in Lagrange’s case, admired).

Soon new political worries were added to Fourier’s academic ones (the Academy had published neither his first nor his second memoir leaving his results open to plagiarism or rediscovery). Napoleon had continued the aggressive foreign policy of the revolution with results depicted in vast battle pieces by a generation of French painters and, on a smaller scale, by Goya. So long as French forces were victorious the cost of the policy would be borne by the defeated powers and not by France but from 1812 onwards the tide of war turned. In 1814 enemy troops entered Paris, Napoleon was forced to abdicate and Louis XVIII installed as a constitutional monarch.

The new king understood, even if some of his followers did not, that the Napoleonic domestic compromise would have to form the basis of his rule. Nor could all of France’s experienced administrators be replaced by the few men who had stayed loyal to him in exile. Fourier remained in Grenoble as prefect of the Isère.

However the last act of the Napoleonic adventure was still to be played. The new government was short of money and made the mistake of trying to economise on the pensions of Napoleon’s old armies, and, worse, the pension of Napoleon himself. Tired of his role as a ruler of the little kingdom of Elba, Napoleon returned to France and began a march towards Paris. Troops sent out to arrest him went over to him without firing a shot. Grenoble was the first large town on Napoleon’s line of march. Fourier’s attempts to organise resistance proved fruitless and he and the commanding general left Grenoble by one gate as Napoleon entered by another.

Whilst at Grenoble, Fourier had acted as a loyal and indeed zealous servant of the King. But the continuing collapse of the Royalist position seems to have persuaded him to throw in his lot with Napoleon and five days later we find him as Napoleon’s prefect of the Rhône.

Twenty years before when Fourier was president of the revolutionary committee of Auxerre, the republic had lacked everything from experienced soldiers to gunpowder; but hope and enthusiasm had saved France. Now there was material and organisation but only Napoleon’s promise of a new regime that was to be at once revolutionary, liberal and imperial. Passive and active resistance increased and Fourier himself was dismissed for failure to comply with orders. Napoleon was defeated at Waterloo and the King returned.

Fourier was now without a job and in deep disfavour with the new regime. The pension to which his years of public service would normally have entitled him was refused on technical grounds. An internal minute concerning his case sums up the government’s attitude. ’The prefect of Grenoble at the time of the arrival of Bonaparte should not be surprised not to have a pension.’

The low point of Fourier’s fortunes did not last long. The prefect of Paris was an old pupil and friend and had gained sufficient credit with the new government to be able to appoint him as Director of the Statistical Bureau of the Seine. (There are demographers who only know of Fourier as an important figure in the development of French governmental statistics.) In 1816 he was elected to the Academy of Sciences but the King refused to confirm his election. In 1817 the Academy tried again and this time his election was allowed. The physical and mathematical results of his theory of heat began to gain general acceptance. In 1822 his prize essay was at last published and he was elected permanent mathematical secretary of the Academy.

Fourier was now able to enter upon the role of grand old man encouraging younger talent (Liouville, Sturm, Dirichlet, Navier) and attaining that curious summit of French distinction, election to the Académie Française. He did not completely stop mathematical work and some of his later work concerns problems in what is now called linear programming. (Darboux, the nineteenth-century editor of his collected work dearly wondered why such a great mind should be interested in such out of the way puzzles.)

’There was at the Academy of Sciences’, wrote Victor Hugo, seeking a contrast with the growing fame of the socialist Charles Fourier, ’a celebrated Fourier whom posterity has forgotten’. But posterity has not forgotten Joseph Fourier the Egyptologist, mathematician, physicist and public servant.

[Thomas William Körner: Fourier analysis, Cambridge University Press, Cambridge 1988]