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]]>The teachers of these new students taught two subjects. One was the classical geometry of the Greeks, less useful than it might appear, but universally held up as a model of reasoning in which rigorous argument lead from clearly stated first principles to final conclusions. The other was modern mathematics, in particular calculus, which lacked any such clear structure.

Certainly modern mathematics dealt with numbers but numbers seemed to be drawn from a rag bag of different objects.

Certainly modern mathematics dealt with numbers but numbers seemed to be drawn from a rag bag of different objects. There were numbers like 1 and 2 which everybody understood, and fractions like 1/3 or 3/9 which were the same number and 2/3 and 1/4 which were different numbers. Then there was 0 which was a number like every number except that you were not allowed to divide by it. To these were adjoined the negative numbers which were every bit like positive numbers, provided that you remembered that the product of two negative numbers was a positive number. Negative numbers had no square root unless you allowed mysterious entities called complex numbers which were also exactly like other numbers except when they were not (for example complex numbers usually had three complex cube roots). Finally there were objects like π which were not fractions, or like Euler’s ϒ which may or may not be a fraction, but which everybody agreed were bona fide numbers to be treated exactly the same as other numbers.

It is doubtful if this worried many students then who, like most students now, wished just to pass their exams, get a good job and enjoy themselves. However it did worry some of their professors and during the course of the 19th century, with many fits and starts, they completed the difficult task of rigourising calculus and the linked task of providing a coherent account of the numbers used in calculus.

In 1930, Landau published a little book *Foundations of Analysis* setting out this account at undergraduate level. Landau’s much loved text is still in print but, as Landau says, is written `in merciless telegraph style . . . as befits such easy material’.

There is, I think, room for a more relaxed account which gives some idea of where the ideas come from and why they are used in the way they are used. My book is an attempt at such an account.

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Napoleon used to ask, when considering oﬃcers for promotion, ‘Is he lucky?’ Fourier was certainly lucky in his choice of subject, but he also, as readers of his great book can see, had the clarity of mind, physical intuition and mathematical power required to exploit it.

His Collected Works (**Oeuvres de Fourier**) contain some very interesting anticipations of later work.

One paper which now appears particularly prescient is a paper on the temperature of the terrestrial globe. Why is the world the temperature that it is? It is possible to extract the following argument from his paper. The earth is warmed by the sun’s radiation. The sun is very hot, so why is the earth not very hot?

Answer, because the earth reradiates heat. But if the earth radiates heat, why is it not much colder (as indeed the moon is)? Answer because the atmosphere slows down the process of re-radiation.

However, in his paper, this argument is mixed up with other arguments which lead to diﬀerent conclusions.

As the century proceeded, the question of the earth’s temperature (or at least the temperature of the surface on which we live) became more central. In the sixteenth century, it was believed that the earth had existed for a short time during which its surface temperature was constant. Now evidence accumulated that not only had the earth existed for a long time, but that its temperature had not always been the same. There were indications both of periods of spectacular cold (the ice ages) and, less romantically, of spectacular heat.

By the time **Charles Darwin** wrote **The Origin of Species**, he could include a beautiful passage on the eﬀect of past episodes of global warming on Alpine ﬂora.

Just as Fourier was the ﬁrst to give an interesting answer to why the earth is the temperature it is, so **John Tyndall** (1820-1893) was the ﬁrst to give an interesting answer to why the sky is blue.

His answer has undergone substantial modiﬁcations by **Lord Rayleigh** and **Albert Einstein**, but his general idea of atmospheric scattering has proved correct.

A keen Alpine climber, he was fascinated by glaciers and worked on their ﬂow. Glaciers led him to ice ages and thence to the problem of the earth’s temperature.

By experiment, he was able to identify those gases, primarily water and carbon dioxide, whose presence interferes with the passage of heat radiation. Ice ages could, perhaps, be accounted for by relatively small changes in the chemical composition of the atmosphere.

His results are given in **Heat Considered as a Mode of Motion** and **Contributions to Molecular Physics in the Domain of Radiant Heat**. The Cambridge Library Collection contains these works and several others which reﬂect his desire to popularise science.

It was not until the development of quantum theory that Tyndall’s discoveries could be understood from a theoretical perspective, and not until the middle of the twentieth century that the complexities of the misnamed Greenhouse Eﬀect were understood (what happens in greenhouses is rather diﬀerent).

Mankind may now be in the position of a lobster in a very slowly warming pot but, thanks to people like Fourier, Tyndall and their successors, we do, at least, know what is happening to us.

Go back and read part one: Remembering Joseph Fourier, and part two: Joseph Fourier: Breaking New Ground.

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]]>When asked about his role in the French Revolution, Emmanuel Joseph Sieyès replied ‘I survived it’.

Joseph Fourier’s story would now be forgotten if he had not written one of the most inﬂuential mathematical works of all time. In 1807, Fourier submitted a memoir to the Paris Institute on the subject of heat conduction. Up to now, mathematical physics had dealt with phenomena already considered by Newton.

This was a new topic, and Fourier introduced new methods to deal with it. A committee of several of the greatest mathematicians of the time (including **Lagrange**, whose Mécanique Analytique is in the Cambridge Library Collection, and **Laplace**, whose Exposition du Système du Monde and Essai sur les Probabilités are also reissued) acknowledged the originality but was unconvinced by the methods. In 1811, he resubmitted his work to obtain a prize oﬀered by the Institute, but, although he won the prize, the report on his work was unenthusiastic.

Finally, in 1822, the prize essay was published and recognised as the ground-breaking work it was. (The Cambridge Library Collection includes both the French **Théorie Analytique de la Chaleur** and its English translation.)

One major novelty of his work was the systematic use of a decomposition of a general ‘signal’ (think of the sound of a violin) into the sum of many simpler ‘signals’ (think of the sound of many tuning forks). One of the British physicists who took up Fourier’s ideas and ran with them was **William Thomson** (later Lord Kelvin) of Thomson and Tait’s **Treatise on Natural Philosophy**.

Thomson used Fourier’s ideas to understand why the ﬁrst Atlantic telegraph cable failed and to ensure that the second cables succeeded. When Thomson called Fourier’s work ‘A Mathematical Poem’, he meant it.

Generalisations of Fourier’s ideas underlie the mathematics of Quantum Mechanics. The theory of Wavelets, born within my working life, is yet another generalisation of what came to be called Fourier Series.

Fourier’s claims for the generality of his method lacked precision. Did his method apply to all signals or only ‘well behaved’ signals?

What should ‘well behaved’ mean?

This problem underlay an important part of the work of two of the greatest mathematicians of the nineteenth century, **Gustav Lejeune Dirichlet** and **Bernhard Riemann**. Dirichlet also extended the idea of Fourier Series to give a new method in the theory of numbers – that part of mathematics to which Fermat’s Last Theorem belongs.

It became clear that the question of which signals are well behaved was strongly related to the question of which objects have a well deﬁned volume. (Or to make things a bit simpler, which plane ﬁgures have a well deﬁned area.) When, at the beginning of the twentieth century, Lebesgue discovered a new and powerful approach to these problem he gave his solution to the area problem in Leçons sur l’intégration and its application to Fourier Series in Leçons sur les séries trigonométriques.

Both books are charmingly written (to the eye of a mathematician) and available in the Cambridge Library Collection.

The ﬁnal word on the problem (or, at least the word which Fourier and his contemporaries would have considered ﬁnal) was given by Carleson in a fairly short but very diﬃcult paper written in 1964.

Carleson showed that, as was already known, Fourier’s widest claim was false but, that, even then, something quite remarkable is true.

Part Three coming soon…

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]]>The obscure Fourier has not been completely forgotten. Cambridge University Press still sells his *Theory of the Four Movements*. However, the remaining 236 books brought up by a search of the Cambridge catalogue concern the Fourier at the Academy of Science.

**Joseph Fourier** was not such a dull ﬁgure as Hugo would have us think. He was the orphaned son of a tailor whose obvious gifts won him a good education at the end of which he found further advancement blocked by his lack of good birth. Unsurprisingly, he became a youthful ﬁrebrand at the start of a French Revolution whose twists and turns almost brought him to the guillotine.

When things quietened down a bit, his fellow citizens sent him to the new **École Normale Supérieure** to further his education and, perhaps, to keep him out of trouble. Paris was the centre of the mathematical universe. Fourier thrived and became a Professor.

His quiet life (this is a relative term in the circumstances of the Revolution) was interrupted by a call to join Napoleon’s expedition to Egypt as one of the intellectuals intended to give the expedition a cultural tone. He did, indeed, contribute to the intellectual side, but, more importantly for his future, he showed himself a gifted administrator, helping to organise the munitions workshops which supplied a stranded French army.

On his return to France he took up his old post, but Napoleon soon made him Prefect of the Isère (a *département* centred on Grenoble). Here he drained marshes and built roads, but also helped write the **Description of Egypt**, a twelve-volume report which founded modern Egyptology. He showed a precocious child named Champollion a copy of the **Rosetta Stone**, and later protected the young man from conscription for the Russian campaign.

Cambridge publishes the result in the *Précis du système hiéroglyphique des anciens Egyptiens* and other works on the decipherment of Egyptian hieroglyphics.

After the Restoration, the royalist government maintained Fourier in his position, but, unfortunately, Grenoble was the ﬁrst large town on the route of Napoleon’s triumphant return from Elba. Fourier left by the back gate as Napoleon entered by the front. The end of the Hundred Days saw him without a post and without a pension.

Read Part Two of this three part blog series: Joseph Fourier: Breaking New Ground

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