Fourier’s 250th Anniversary

The 250th anniversary of Joseph Fourier’s birth has been added to the French national commemorations of 2018 by the High committee of the French Academy.

Portrait of Joseph Fourier, mathematician

March 21 marks the 250th birthday of one of the most influential mathematicians in history. He accompanied Napoleon on his expedition to Egypt, revolutionized science’s understanding of heat transfer, developed the mathematical tools used today to create CT and MRI scan images, and discovered the greenhouse effect.

He wrote of mathematics: “There cannot be a language more universal and more simple, more free from errors and obscurities … Mathematical analysis is as extensive as nature itself, and it defines all perceptible relations.”

Jean-Baptiste-Joseph Fourier is the most illustrious citizen of Auxerre, the principal city of western Burgundy, where he was born on March 21, 1768. Both his father Joseph, who was a master tailor originally from Lorraine, and his mother Edmie died before he was ten years old. Fortunately certain local citizens took an interest in the boy’s education and secured him a place in the progressive École Royale Militaire, one of a number run by the Benedictine and other monastic orders. Science and mathematics were taught there, among other subjects, and, while the boy displayed all-round ability, he had a special gift for mathematics. He went on from there to complete his studies in Paris at the College Montagu. His aim was to join other the artillery or the engineers, the branches of the army supposedly open to all classes of society, but when he applied he was turned down, despite a strong recommendation from Legendre, who was an inspector of the Écoles Militaires. Although he could have been rejected on medical grounds the reason given by the minister was that only candidates of noble birth were acceptable.

After this setback Fourier embarked on a career in the church. He became a novice at the famous Benedictine Abbey of St. Benoît-sur-Loire, where he was called on to teach elementary mathematics to the other novices. After taking monastic vows he became known as Abbé Fourier (Father or Reverend), but instead of pursuing a career in the church he returned to Auxerre to teach at the École Militaire. By this time he was twenty-one and had already read a research paper at a meeting of the Paris Academy.

During the first years of the Revolution, Fourier was prominent in local affairs. His courageous defence of victims of the Terror led to his arrest by order of the Committee for Public Safety in 1794. A personal appeal to Robespierre was unsuccessful, but he was released after Robespierre himself was guillotined. Fourier then went as a student to the short-lived École Normale. The innovative teaching methods used there made a strong impression on him and it gave him the opportunity to meet some of the foremost mathematicians of the day, including Lagrange, Laplace and Monge. Fourier was amused when it emerged that, due to administrative error, the proud Laplace had been enrolled as a student rather than a professor. The next year, when the École Polytechnique opened its doors, under its original name of the École Centrale des Travaux Publiques, Fourier was appointed assistant lecturer to support the teaching of Lagrange and Monge. However, before long he fell victim to the reaction against the previous regime and was arrested again. He had an anxious time in prison but his colleagues at the École successfully sought his release.

In 1798 he was selected to join an expedition to an undisclosed destination. This proved to be Napoleon’s Egyptian adventure, Campagne d’Égypte. Once the newly formed Institut d’Egypte was established in Cairo, with Monge as its president and Fourier as permanent secretary, the cultural arm of the expedition set to work studying the antiquities, some of which were appropriated. On top of this activity Fourier was also entrusted with some negotiations of a diplomatic nature, and he even found time to think about mathematics. He proposed that a report be published on the work of the Institut d’Égypte, and on his return to France was consulted as to its organisation and deputed to write a historical preface describing the rediscovery of the wonders of the ancient civilisation. When the Description de l’Égypte (a twelve-volume report which founded modern Egyptology) was published, Fourier’s elegant preface, somewhat edited by Napoleon, appeared at the front of it.

Meanwhile Fourier had resumed his work at the École Polytechnique. Before long, however, Napoleon, who had been impressed by his capacity for administration, decided to appoint him prefect of the Departement of Isère, based at Grenoble and extending to what was then the Italian border. The office of prefect is a demanding one but it was during this period that Fourier wrote his classic monograph on heat diffusion entitled On the propagation of heat in solid bodies and presented it to the Paris Academy in 1807. It was examined by Lagrange, Laplace, Lacroix and Monge. Lagrange was adamant in his rejection of several of its features (essentially the central concept of trigonometric or, as we say, of Fourier series) and so its publication in full was blocked; only an inadequate five-page summary appeared, written by Poisson. Outclassed as rivals in the theory of heat diffusion, Poisson and Biot tried for years to belittle Fourier’s achievements. Later he received a prize from the academy for the work, but it was not until 1822 that Fourier’s theory of heat diffusion was published.

Pierre-Simon de Laplace (L) and Joseph Louis Lagrange (R) were not initially convinced by Fourier’s work.

To quote from the preface to the Théorie Analytique de la Chaleur, this ‘great mathematical poem’ as Clerk Maxwell described it:

First causes are not known to us, but they are subjected to simple and constant laws that can be studied by observation and whose study is the goal of Natural Philosophy… Heat penetrates, as does gravity, all the substances of the universe; its rays occupy all regions of space. The aim of our work is to expose the mathematical laws that this element follows… But whatever the extent of the mechanical theories, they do not apply at all to the effects of heat. They constitute a special order of phenomena that cannot be explained by principles of movement and of equilibrium… The differential equations for the propagation of heat express the most general conditions and reduce physical questions to problems in pure Analysis that is properly the object of the theory.

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 cable succeeded.

As prefect, Fourier’s administrative achievements included securing the agreement of thirty-seven different communities to the drainage of a huge area of marshland to make valuable agricultural land, and the planning of a spectacular highway between Grenoble and Turin, of which only the French section was built. Napoleon conferred on him the title of baron, in recognition of his excellent work as prefect.

Fourier was still at Grenoble in 1814 when Napoleon fell from power. The city happened to be directly on the route of the party escorting the Emperor from Paris to the south and thence to Elba; to avoid and embarrassing meeting with his former chief, Fourier negotiated a detour in the route. But no such detour was possible when Napoleon returned on his march to Paris in 1815, and so Fourier compromised, fulfilling his duties as prefect by ordering the preparation of the defences – which he knew to be futile – and then leaving the town by one gate as Napoleon entered by another. His handling of this awkward situation did not affect their relationship. In fact the Emperor promptly gave him the title of count and appointed him prefect of the neighbouring Département of the Rhône, based at Lyon. However before the end of the Hundred Days Fourier had resigned his new title and appointment in protest against the severities of the regime and returned to Paris to concentrate on scientific work.

This was the low point in Fourier’s life. For a short while he was without employment, subsisting on a small pension, and out of favour politically. However a former student at the École Polytechnique and companion in Egypt was now prefect of the Département of the Seine. He appointed Fourier director of the Statistical Bureau of the Seine, a post without arduous duties but with a salary sufficient for his needs.

Fourier’s last burst of creative activity came in 1817/18 when he achieved an effective insight into the relation between integral-transform solutions to differential equations and the operational calculus. There was at that time a three-cornered race in progress between Fourier, Poisson and Cauchy to develop such techniques. In a crushing response to a criticism by Poisson, Fourier exhibited integral-transform solutions of several equations which had long defied analysis, and paved the way for Cauchy to develop a systematic theory, en route to the calculus of residues.

In 1816 Fourier was elected to the reconstituted Académie des Sciences, but Louis XVIII could not forgive his acceptance of the Rhône prefecture from Napoleon and at first refused to approve the election. Diplomatic negotiation eventually resolved the situation and his renomination the next year was approved. He also had some trouble with the second edition of the Description de l’Égypte (for now his references to Napoleon needed revision) but in general his reputation was recovering rapidly. He was left in a position of strength after the decline of the Société d’Arcueil, and gained the support of Laplace against the enmity of Poisson. In 1822 he was elected to the powerful position of permanent secretary of the Académie des Sciences. In 1827, like d’Alembert and Laplace before him, he was elected to the literary Académie Française. Outside France he was elected to the Royal Society of London.

Fourier’s health was never robust, and towards the end of his life he began to display peculiar symptoms which are thought to have been due to a disease of the thyroid gland called myxoedema, possibly contracted in Egypt. As well as certain physical symptoms, the disorder can lead to a dulling of the memory, apparent in the rambling papers he wrote towards the end of his life. Perhaps it was also partly responsible for the unfortunate incident which occurred in February 1830 when he apparently mislaid the second paper on the solution of equations sent to the Academy by Galois for the competition for the Grand Prix de Mathématiques. The prize was awarded jointly to Niels Abel (posthumously) and Carl Jacobi for their work on elliptic functions.

Fourier was terminally ill by that time. Early in May 1830 he suffered a collapse and his condition deteriorated until he died on May 16, at the age of sixty-two. The funeral service took place at the church of St Jacques de Haut Pas and he was buried in the cemetery of Père Lachaise, close to the grave of Monge.

Today, Fourier’s name is inscribed on the Eiffel Tower. But more importantly, it is immortalized in Fourier’s law and the Fourier transform, enduring emblems of his belief that mathematics holds the key to the universe.

Visualisation of an approximation of a square wave by taking the first 1, 2, 3 and 4 terms of its Fourier series

Thiis interactive animation will keep little bears occupied for hours 🙂

Fourier’s law states that heat transfers through a material at a rate proportional to both the difference in temperature between different areas and to the area across which the transfer takes place. For example, people who are overheated can cool off quickly by getting to a cool place and exposing as much of their body to it as possible.

Fourier’s work enables scientists to predict the future distribution of heat. Heat is transferred through different materials at different rates. For example, brass has a high thermal conductivity. Air is poorly conductive, which is why it’s frequently used in insulation.

Remarkably, Fourier’s equation applies widely to matter, whether in the form of solid, liquid or gas. It powerfully shaped scientists’ understanding of both electricity and the process of diffusion. It also transformed scientists’ understanding of flow in nature generally – from water’s passage through porous rocks to the movement of blood through capillaries.

Fourier applications

Modern medical imaging machines rely on another mathematical discovery of Fourier’s, the “Fourier transform”.

In CT scans, doctors send X-ray beams through a patient from multiple different directions. Some X-rays emerge from the other side, where they can be measured, while others are blocked by structures within the body.

With many such measurements taken at many different angles, it becomes possible to determine the degree to which each tiny block of tissue blocked the beam. For example, bone blocks most of the X-rays, while the lungs block very little. Through a complex series of computations, it’s possible to reconstruct the measurements into two-dimensional images of a patient’s internal anatomy.

Thanks to Fourier and today’s powerful computers, doctors can create almost instantaneous images of the brain, the pulmonary arteries, the appendix and other parts of the body. This in turn makes it possible to confirm or rule out the presence of issues such as blood clots in the pulmonary arteries or inflammation of the appendix.

Fourier is also regarded as the first scientist to notice what we today call the greenhouse effect.

His interest was piqued when he observed that a planet as far away from the sun as Earth should be considerably cooler. He hypothesized that something about the Earth – in particular, its atmosphere – must enable it to trap solar radiation that would otherwise simply radiate back out into space.

Fourier created a model of the Earth involving a box with a glass cover. Over time, the temperature in the box rose above that of the surrounding air, suggesting that the glass continually trapped heat. Because his model resembled a greenhouse in some respects, this phenomenon came to be called the “greenhouse effect”.

Simply put: The earth is warmed by the sun’s radiation. The sun is very hot, so why is the earth not very hot? Because the earth reradiates heat. But if the earth radiates heat, why is it not much colder (as the moon is)? Because the atmosphere slows down the process of re-radiation.

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.

John Tyndall (1820-1893).
Smithsonian Institution’s digital collection of portraits

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.

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 20th 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.

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