by David A. Keston, University of Glasgow, United Kingdom

*"He will have views and prospects to himself
perpetually soliciting his eye, which he can no more help standing still to
look at than he can fly; he will moreover have various Accounts to reconcile:
Anecdotes to pick up: Inscriptions to make out: Stories to weave in:
Traditions to sift: ..."***from "Tristram Shandy" by
Laurence Sterne**

**Index**

- Introduction
- Early life in Auxerre
- The Revolutionary Years
- The Expedition to Egypt
- The Prefecture of Isère
- Fourier's Theory of Heat
- The Memoir on Propagation of Heat of 1807
- Experimental Research
- The Prize (finally awarded in 1812)
- About the Fourier Series
- Conclusion
- Bibliography

The life of Baron Jean Baptiste Joseph Fourier (1768-1830), the mathematical physicist, has to be seen in the context of the French Revolution and its reverberations. One might say his career followed the peaks and troughs of the political wave.

He was in turns: a teacher; a secret policeman; a political prisoner; governor of Egypt; prefect of Isère and Rhône; friend of Napoleon; and secretary of the Académie des Sciences.

His major work, *The Analytic Theory of Heat*,
(*Théorie
analytique de la chaleur*) changed the way scientists think about
functions and successfully stated the equations governing heat transfer in
solids.

His life spanned the eruption and aftermath of the Revolution; Napoleon's rise to power, defeat and brief return (the so-called Hundred Days); and the Restoration of the Bourbon Kings.

Joseph Fourier was born in 1768 in Auxerre in the département of Yonne; a town steeped in history. He was orphaned before he was ten years old and grew up with his aunt and uncle in the same town. On the recommendation of the Bishop of Auxerre he was given a place at the nearby École Royale Militaire. Under teachers of the Benedictine order he showed himself to be a fast and diligent student. He studied mathematics intensly. In the style of the novels of the period, he is reputed to have collected the stubs of candles so as to study late into the night, thus ruining his delicate constitution. It is true to say that he was a sickly child by the standards of the day, suffering from asthma and insomnia.

His teachers saw him as a possible recruit to their order. Yet what he truly
wanted to do was join the army (in either the artillery or the engineers): due
to his lowly background, his father had been a *mere* tailor, he was
prevented from doing so. Confounded, he chose to enter the Church. He went
then to the abbey at St. Benôit-sur-Loire to ready himself to take his
vows, meanwhile acting as mathematics teacher to his fellow novices.

This was in 1787. In the background we must imagine the first ripples of revolutionary discontent beginning to cloud French politics.

Fourier never took his vows. He was ready to, but at about the same time an order from the Constituent Assembly halted further taking of holy orders throughout France. Whether this was a disappointment to him is open to question. Nevertheless the necessary order and practical nature of monastic life were to exert a strong influence on his life's works.

Fourier returned to work as assistant to his old mathematics teacher, Bonard, at the École Royale Militaire, later the Collège Nationale, in Auxerre. The congregation of St. Maur to which the teachers belonged was exempted from the revolutionary clamp-down on the Church as it was a juring order.

Under the Commune, education was swiftly reorganised to allow entry to a broader range of students. This was a continuation, in the revolutionary manner, of the reforms of the deposed Louis XVI. Modern historians see Louis' attempts as one of the root causes of the Revolution. The combination of improved education and a continental recession left a large disgruntled population.

Fourier stayed in Auxerre for the first four years of the Revolution. By this time the call to revolution could not sensibly be ignored. The high ideals of those early years swept him along, but later as France felt the paranoia of the Terror, Fourier was amongst those who tried to resist.

There had been a trade war raging between Britain and France. When the inevitable war was declared, on the 1st of February 1793, it was popular, at least in Paris. The call to revolution became a call to arms under Carnot's levée en masse. Fourier addressed the Popular Society in Auxerre on this subject, urging that the men levied should be true volunteers. Due to his eloquence he was invited to join the society, being deputised to organise the levée in the region. This meant being elected to the committee of surveillance, an organisation later to be used as a secret police force. The municipality trusted him to carry out a number of other missions of importance. One of these missions got him into trouble.

By the summer of that year, France was in turmoil; there were set-backs in the war abroad and in the Vendée a royalist counter-revolution was in full swing (ironically, partly due to resistance to the unpopular levée). Fourier was collecting horses for both war efforts and returning from one mission, passed through Orléans. At the time this was a hot-bed of sans-culottes activity. Through a series of circumstances the leaders of the sans-culottes of Orléans were disliked by certain members of the Convention, including the town's administrator (a Montagnard - supporting the bourgeois and the advancement their money could bring).

In this atmosphere Fourier spoke on the behalf of the leaders of the town's sans-culottes. The local administrator complained to his Parisian connections and eventually Fourier was arrested on a charge of being an Hébertist at a time when Robespierre was increasingly paranoid about other Jacobin clubs. Our hero was imprisoned only to be released after Thermidor 9 (1794) when Robespierre was beheaded.

Due in part to the excesses of the Terror, in which many would-be teachers had
lost their heads, there was a political will for a college to train new
teachers. The École Normale was founded for this purpose. Fourier was
nominated as a pupil for the newly formed school. Most striking to the modern
reader are the guidelines on which the school was run: the lectures were to be
delivered standing; with no prepared notes; the lecturers were to accept
questions from the floor when they arose; and the lecturers themselves were
expected to be actively researching in the field they taught! The first
séance was in January of 1795. Fourier accepted with alacrity perhaps
because it offered him a chance to escape Auxerre for the 'big lights' of
Paris. He made good use of his time there in the first half of 1795 and was
certainly one of the pupils best able to cope with the class of lectures
offered. The political pendulum was on a reactionary swing; in September 1795
Fourier was rearrested, again associated with the events in Orléans
and his subsequent behaviour, this time being charged as a Robespierrist!
This second imprisonment seems to have been the more serious. Fourier writes
later that he truly believed he was to die. Friends and
colleagues
pleaded on his behalf. Again it was to be outside events that saved him:
after Napoleon's *whiff of grape shot* there was another amnesty and
Fourier was released. Using
connections made at the École Normale he went into teaching
mathematics at the École Centrale (later to become the École
Polytechnique)

On the 16th of May 1798, an armada of 180 ships sailed from Toulon. On board were 30,000 soldiers and sailors; Napoleon Bonaparte, his generals and officers; and an entourage of 165 scientific and literary intelligentsia, the so-called Legion of Culture, among them Fourier. Their destination was known only to Napoleon, and a few of his most trusted friends.

Though Nelson, with the British fleet, was hunting the Mediterranean for him Napoleon seemed unperturbed, concentrating on the campaign ahead. For the crews and passengers the story was quite different; the stress was tangible.

It was in this climate that Fourier must first have met Bonaparte. As a
professor at the École Polytechnique, Fourier ranked with Napoleon's
staff and would certainly have been involved in the *institutes*. These
were the soirées for the officers and savants which Napoleon instigated
and took part in. Fourier almost certainly met General Kléber
(Napoleon's 'right hand' in Egypt).

After the capture of Malta the fleet continued towards Egypt. The mission was now obvious, to "liberate" the people of Egypt from their brutal 'uncultured state'. The armada landed at Alexandria, on the 1st of July 1798 (and took it three days later). Thereafter the campaign was plagued with problems. Only a few days after taking Cairo from the Marmelukes, the French fleet was scuttled off Aboukir Bay. The French were effectively exiled; what's more, the people they came to free did not seem to favour their presence!

Even so, Napoleon (with his organising spirit) set to work on the situation in
Cairo. He formed the *divan* - a kind of municipal council - including a
French observer (a post which Fourier would later fill). What was more
important for Fourier, Napoleon created the Institute d'Égypte. The
founding precepts were noble indeed: the nurture of the sciences in Egypt; the
collection of data historical, statistical, etc.; and as a research and
development arm of the army. The latter aim was surely due to Bonaparte
himself.

The campaign in Egypt faltering and dire news arriving from home, Napoleon
returned to France where he was elected Consul. He gave command of Egypt to
Kléber. This was short-lived as Kléber did not endear himself to
the Egyptians and was assassinated. Fourier read the funeral address. The new
commander Menou knew of Fourier's work at the Institute and had him put on the
*Divan* among other jobs of lesser importance, the sum of which however
made Fourier civilian governor of Egypt.

Harassment by the British Expeditionary Force became a threat in March of 1801 when General Abercrombie landed at Aboukir Bay. The French defence was outmanoeuvred at Alexandria. In Cairo the members of the Institute decided to try to return to France. They travelled to Alexandria but as they left port their ship was stopped by the British blockade under Sir Sidney Smith.

In a gesture characteristic of the era Sir Sidney gave the savants safe passage to Toulon (but not their research materials ). Fourier was back in France and it was cold!

Soon after his return Napoleon wrote to him explaining that the first prefect of Isère, Ricard de Séalt, had recently died, and that Fourier would be ideal for the job. Fourier must have been displeased; he had been expecting to return to Paris and the lively scientific discourse of savants of his own calibre. Disappointed though he was, Fourier was cannier than to refuse Bonaparte.

Ricard had not left the affairs of the département of Isère in good order. The administration (with its centre at Grenoble) was a shambles, so that Fourier had to attend to virtually everything himself. He created several posts, which ensured that many of his friends were given jobs in the region. Fourier had many responsibilities: as prefect he was both a public figure and a governor. The demands of the Minister of the Interior (his immediate superior) for statistical information were unending. This information could be anything from the price of sheep at market to the number of known dissenters and members of religious sects. The darker side of the job was the suppression of anti-government literature and censorship of the local newssheet.

He was expected to take care of matters of diplomacy where they concerned Isère; he organised the visits of the king of Spain and Pope Pius VII to the region as well as that of Napoleon.

By 1809 the Parisian government were pressing for completion of the research
work on the Egyptian Campaign, which Fourier was to compile. At one stage
Fourier was researching and carrying out his prefectural responsibilities and
readying the definitive version of the Cairo Institute's report for
publication, the *"Déscription de l'Egypte"* all at the same time.
He got round this by requesting leave in order to complete the Egyptian work.
He certainly used some of this time to continue his experiments.

He was decorated twice: once as a chevalier in Napoleon's Légion d'Honneur, in 1804, and on completion of the Déscription de l`Egypte, in 1809, as a baron (a title which commanded a healthy pension).

His major public works were the draining of a large area of marshland at Bourgoin and the partial construction of a road to Turin. The former may sound simple enough, but since there were many clashing interests between the farmers around the land, the peasants living there and the local nobility, Fourier was forced to visit virtually all these people to bargain with them for their cooperation.

A direct road to Turin was an obvious step now that Italy had been annexed. Fourier finally received permission from the Minister of the Interior to build this road; the minister had been sceptical initially. It was not completed under Fourier because of the abdication of Napoleon in April 1814.

This event placed Fourier in a predicament, for Napoleon's road to exile on Elba ran through Grenoble. Fourier wrote to beg that the route be changed, giving the excuse that there was civil unrest in Grenoble and Bonaparte's passing would stir up trouble. Of course he was more intent on avoiding the embarrassment of seeing his friend defeated. He need not have worried; Napoleon's entourage did not go through Grenoble in the end.

Circumstances were different a year later, when Napoleon escaped from Elba. This time he headed straight for Lyons on a route that must pass Grenoble. Fourier was warned and therefore prepared a token defence of the town at one gate and left by the other, heading towards Lyons (ostensibly to warn the Bourbons of the danger). He then returned, to be intercepted by Napoleon's men. Napoleon was displeased at Fourier's "desertion" but perhaps remembering his own desertion of his friend in Egypt he gave Fourier the prefecture of Rhône.

When Fourier realised the vengeful nature of the government of the Hundred Days, he resigned this post. Napoleon was defeated at Waterloo and sent into exile on St. Helena. The following upheaval in France was mirrored in Fourier's life; he lost his barony, his pension and his reputation. Out of favour and out of work, he had to sell many of his personal belongings to survive. The Bourbons now re-established pettily excluded anyone associated with Napoleon.

In the end Fourier was offered a job in the Bureau of Statistics of the département of the Seine (including Paris) by the comte de Chabrol de Volvic: the prefect of that department and a friend of Fourier's through the École Polytechnique and Egypt. This paid enough to live on while giving him time for research.

When the Académie des Sciences had open elections in 1816, Fourier won the vote; the King however would not approve his membership. The next opportunity to join was after the death of a member, in 1817. He won the election with ease. This time there were no objections. He remained a member for life.

**Mathematical Investigation of Heat Transfer**

Fourier had been interested in the phenomenon of heat transfer from as early as 1802. It would be anecdotal to think that his return to France's cold shores from Egypt was the cause, though from accounts of his years in Isère, it is conceivable that he had contracted myxedema in Egypt; a disease that would make life in the cold of the Jura mountains all but unbearable.

There are two distinct problems in any description of heat propagation: the steady state (where a steady heat is supplied to the body concerned which eventually reaches an equilibrial distribution); and the time-dependent cooling of a body in air of a fixed temperature.

His first attempts were in the nature of an abstract model: no surface heat
transfer; discrete bodies; and heat transferred by some shuttle mechanism from
hot to cold. Starting with two bodies of equal mass (and of the same material)
and adding more bodies thereafter, he hoped to generalise still further to
*n* bodies arranged in a straight line, eventually setting *n* to
infinity. The resulting model was not satisfactory. With hindsight we can say
that he had not included any terms that described why heat was conducted at
all: he had only an *ad hoc* conductivity coefficient.

He was stuck until 1804, when Biot visited him in Grenoble. Biot was himself working on heat propagation in solids and in his work he separated the treatment of the interior heat transfer and the surface effects. What is more important, his work dealt with continuous bodies. With these hints Fourier was well placed to begin research. The simplest problem of this kind was the thin bar (prism), heated at one end and cooled to a steady temperature at the other. Here the model was one-dimensional and surface cooling was always normal to the bar.

Fourier would later complain that Biot did not give references to the works of Amontons or Lambert that both discuss the temperature curve along a bar heated at one end. The former assumed a linearly decreasing curve; the latter argued for a logarithmic effect. Fourier's draft paper of that time made it clear that the linear decrease would imply no net loss of heat to the surrounding air, an obvious flaw. He defined:

where ** K** is coefficient of conductivity and (dy/dx) is the
temperature gradient. Then he described three thin slices of the bar at

and this is identified with the net rate of heat loss to the air = ** 8lhy
dx** (where

where ** y``** is the second (ordinary) derivative

Next in complexity to the thin bar came the thin ring (annulus). This again was effectively one-dimensional. Perhaps feeling more adventurous he considered the cooling problem for the ring too.

In December of 1807 Fourier read a long memoir on "the propagation of heat in solids" before the Class of the Institut de France. It concentrated on heat diffusion between discrete masses and certain special cases of continuous bodies (bar, ring, sphere, cylinder, rectangular prism, and cube).

The diffusion equation used can be stated in three dimensions. The paper was never published since one of the examiners, Lagrange, denounced his use of the Fourier Series to express the initial temperature distribution.

In the case of the thin bar Fourier used the same method as he had in his
earlier draft. In it he still used the "slices" method of deriving the flux.
Later (in his Essay of 1811 and in his book *Analytical Theory of Heat*)
the slices became mathematical sections, thus resolving difficulties with the
description of the heat flux. The problem had been that by using slices -
albeit infinitesimally thin ones - he had assumed a temperature jump (the heat
in any one slice had to come from its immediate neighbours). This was not
physically realisable (not even in theory), thus became a major flaw. The
introduction of mathematical sections in place of slices avoided this trouble.
Unfortunately detractors (chiefly Biot,
Laplace and
Poisson) did not seem to be aware of the significance of this change from
"temperature difference" to "temperature gradient". This was one of the
criticisms was levelled at Fourier's research after 1807 (the other criticism
often made was the difficulty with periodicity).

The examiners pointed out that his works did not give Euler and d'Alembert their due. This Fourier conceded, though he still claimed that his work on trigonometric series was independent since he had had no access to the relevant mathematical works in Grenoble. Certainly his treatment of these series was original.

One omission which did cause him trouble related to Biot's 1804 paper. It
seems that Fourier had sent Biot an early copy of his memoir: one with no
reference to Biot's part in the development. Since they were both working in
the same field, there was a certain amount of rivalry. Feeling slighted, Biot
wrote scathingly about Fourier's memoir in the *"Mercure de France"*, a
public journal. Fourier was outraged. They remained enemies throughout their
careers and took great pleasure deprecating the other's achievements.

In 1809, Poisson, a friend of Biot's and another scientist researching heat propagation, wrote in the Nouveau Bulletin des Sciences (of which he was editor) about the state of knowledge in the field. He cited Biot's work generously; while his remarks about Fourier's investigations deliberately disregarded the very considerable discoveries Fourier had made in the various special cases described in his memoir. The whole article seemed calculated to insult.

The last section of the 1807 memoir was a description of the various experiments which Fourier had undertaken. They follow what seems to be the chronological progression of his research.

Having treated the problem of the thin bar in 1804, this last section began by describing the heated annulus: in the steady state and then as it cooled.

His equipment was basic but effective: a polished iron ring of

For the time-dependent case the ring was placed halfway into a furnace and then removed to an insulating bath of sand. The initial distribution was of uniformly hot around one half and cold around the other. The curve was then seen to iron out as heat flowed from hot to cold. Fourier's theory suggested that very soon this would resolve even further to give a simple sinusoidal pattern that would gradually dampen until the ring was at a uniform temperature. The experiments bore him out: measurements were more troublesome but conclusive. Those small discrepancies that were there, were probably due to the unreliable nature of the thermometers being used (thermometers, of the type Fourier used, tended to be affected by atmospheric pressure both in their manufacture and in their application).

Fourier treated the initial distribution of temperatures around the ring as a superposition of many simple sinusoids that varied from peak to trough to peak an integer number of times along the circumference of the ring. He reasoned that the higher frequency sinusoids would damp out rapidly. A sinusoid with twice the frequency would imply that the distance between hot peak and cold trough was halved and on top of that the temperature gradient would be doubled. As a result, a sinusoidal distribution with twice the frequency would dampen at four times the rate.

Parallel to his theoretical development, Fourier then tackled the experimental rate of cooling of a uniformly heated sphere. He approached this by heating a small polished iron sphere, then allowing it to cool. The sphere was specially drilled so that thermometer readings could be taken at its centre. He found that varying the method of heating had little effect, while blackening the surface would approximately double the cooling rate. To finish, he compared the rates of cooling of a cube and a sphere (where the cube had sides the same length as the diameter of the sphere). This he did by repeating the above experiments with both solids under the same conditions.

The results of both these were less than impressive: the substantial experimental errors were put down to deficiencies of the equipment used especially the thermometers.

In 1810, the Institut de France announced that the Grand Prize in Mathematics for the following year was to be on "the propagation of heat in solid bodies". The ideal title for Fourier. The choice was certainly influenced by the political chicanery of Laplace and Monge, supporters of Fourier's cause, and Lagrange, one of his detractors.

The Commission of judges was named as Lagrange, Laplace, Malus, Haüy, and Lacroix.

The essay repeated the derivations from his earlier works, while correcting
many of the errata. The only major change was in his treatment of the flux, as
mentioned previously. In 1812, he was awarded the prize and the sizeable
honorarium that came with it. He won the prize, but not the outright acclaim
of his
referees. They accepted that Fourier had the right equations, but felt
that his methods were *not without their difficulties*.

His prize winning essay would in the normal course of things now be published.
Due to his heavy prefectural duties, he was unable to see this happen. It was
only when he had returned to Paris for good that he could work on getting what
was now *The Analytical Theory of Heat* printed. Even then the Theory was
only published in 1822 (due to the intransigence of Delambre then the
permanent secretary to the Institute - perhaps at the bidding of Lagrange).
This book contained the analytic side of Fourier's essay only. A companion
volume that would cover his experimental work, problems of terrestial heat,
and practical matters (such as the efficient heating of houses) was never
completed. The actual text of his original essay appeared in two parts in the
Memoirs of the Académie des Sciences in 1824 and 1826. This was only
after Fourier had been elected permanent secretary.

The germ of the idea of Fourier Series is to be found, with hindsight, in the works of Leonard Euler. Euler was interested in interpolation. In a problem with origins in planetary perturbations, he approached and solved problems of the form

the solutions were the well-known trigonometric series.

Euler, too, produced the correct Fourier sine expansion, though from erroneous sources. Lagrange delivered this same result in the context of sound wave propagation.

In 1754, d'Alembert, working on an astronomical problem, obtained a trigonometric cosine expansion for the reciprocal of the distance between two planets in terms of the angle between the vectors from the origin to the planets: this saw the introduction of the integral expression for the Fourier coefficients.

There was great debate on the true meaning of the concept of the function. With many infinite trigonometric series appearing in a variety of situations, the stage was set for Fourier to break new ground with his consistent use of the equations named after him.

These same names were associated with another problem of the day - the
vibrating string. As early as 1753, Bernoulli advocated a
trigonometric series solution, on physical grounds. While Euler and d'Alembert
favoured a functional solution, despite problems at t = 0 (*i.e.,* a
corner in a supposedly differentiable equation). Euler could not accept
Bernoulli's solution because he could not reconcile the periodicity of such a
series with the obvious non-periodicity of the physical problem. He was
certain that such trigonometric series could not represent non-periodic
functions at all. His mistake was his uncertainty about the difference between
algebraic generality over the whole real line and geometric generality over
the fixed length of the string. Fourier resolved this problem of periodicity
in Bernoulli's favour in his memoir of 1807.

Fourier's Analytic Theory of Heat derived and justified the basic equations of heat propagation. Part of its impact was that it did not fit into the scheme of "rational and celestial mechanics", yet it retained a mathematical simplicity.

Predominantly mathematical in flavour, the outstanding nature of the work stemmed from its use of analysis. Partial differential equations were used to represent physical phenomena and then initial and boundary conditions were applied as had been done in many special cases by others. Fourier went further, he distinguished between actions in the "interior" and those at a "surface" boundary (expressing each as a separate equation). He chose coordinate systems freely to take advantage of symmetry arguments.

He added explicit statements of initial conditions so that explicit calculation could be compared with experimental test.

The techniques he used could not have arisen without his contribution: none of his contemporaries was as secure in their use of separation of variables nor as confident in infinite series solutions. The beauty of these techniques was - and remains - their wide applicability.

When Victor Hugo wrote in *Les Misèrables:*

There was a celebrated Fourier at the Academy of Science, whom posterity has forgotten; and in some garret an obscure Fourier, whom the future will recall.

he did not have the benefit of two centuries' hindsight - his Fourier in a garret is Charles Fourier, a political philosopher. Much of the mathematics of the nineteenth century (and indeed the twentieth) is heavily indebted to the Analytic Theory: definite integrals defined as sums; the theory of infinite determinants; and uniform convergence. From a work which was, at least in spirit, physical this was and still is breathtaking. Joseph Fourier's legacy will certainly not be forgotten.

I acknowledge that in creating this page I have heavily leant on a number of very readable sources.

The Bibliography

Other resources:

- Fourier in French
- For short biographies of Mathematicians including Fourier you cannot improve on The History of Mathematics site at my first University, the University of St. Andrews.
- A detailed account of Modern Fourier Theory