Joseph Fourier: The Man and the Mathematician
Orphaned when still a child, Joseph Fourier went through the French Revolution, the Napoleonic era, and the following Restoration as a high-ranking public servant.
KeywordsFrench Revolution Retirement Pension Administrative Duty Commanding Officer Neighboring Department
1.1 An adventurous life
Orphaned when still a child, Joseph Fourier went through the French Revolution, the Napoleonic era, and the following Restoration as a high-ranking public servant. He learned on his own “how hard the way up and down another man’s stairs is”1 by experiencing directly how risky it is to please changing lords such as Robespierre, Napoleon, and King Louis XVIII. He started as a convinced Jacobin, maybe impulsive at the time, and ended up as a cautious liberal.
He was a distinguished administrator and diplomat, but more than that he was a scientist whose achievements were truly revolutionary in character. The most relevant aspect of his life, the one for which he is still remembered, is the contribution he made to the advancement of mathematics and physics. It is somewhat miraculous that his most important and creative work took place when he was prefect of Isère, an onerous administrative position he held with first class results. It suffices to mention the construction of the French part of a spectacular road through the Alps from Grenoble to Turin and the draining of twenty million acres of marshes around the village of Bourgoin midway between Lyon and Grenoble, a century-old proposal that nobody previously had been able even to start.
He ranks among the most important scientists of the nineteenth century for his studies on the propagation of heat, the consequences of which reach down to the present day. The question of terrestrial temperature was principally in his mind in establishing the mathematical theory of heat, and the paternity of the expression “greenhouse effect ”— effet de serre—is attributed to Fourier. He was the first on record to hint at it, writing back in 1827: “The problem of global temperatures, one of the most important and difficult of the whole of natural philosophy, is composed by rather different elements that ought to be considered from a unique general point of view.” The novelty of his method, at least initially, perplexed outstanding mathematicians of his time, from Lagrange to Laplace and Poisson. The publication of Fourier’s work was consequently delayed as many as 15 years during which he tenaciously defended, explained, and extended it.
1.2 The beginnings
Jean Baptiste Joseph Fourier was born on March 21, 1768 in the ancient and beautiful town of Auxerre, 150 kilometers south of Paris in a position dominating the river Yonne, with beautiful churches, above all the ancient fifth century abbey St. Germain and the Gothic cathedral of St. Étienne (Fig. 1.1). Joseph was the name of his father, a master taylor of Auxerre, who had been born to shopkeepers in a small town in Lorraine. Joseph might have been attracted to Auxerre by the rich and powerful ecclesiastical establishment of the town, which had had its own bishop since Gallo-Roman times. He probably expected some special consideration in memory of his paternal great uncle Pierre Fourier, a leading figure of the Counter Reformation in Lorraine in the sixteenth and early seventeenth centuries. Joseph Fourier married twice. With his first wife he had three children and with the second one 12. Jean Joseph was the ninth of these. His mother Edmie died in 1777 and his father the following year.
Not yet ten years old, Jean Joseph was left an orphan. By all means the young Fourier could have then been considered “lost.” It did not go that way thanks to a certain Madame Moitton who recommended him to the bishop of Auxerre, thus sparing him a life of apprenticeship and servitude. At 12 he entered the local École Royale Militaire, run by the Benedictines.
The various military schools of the country, 11 in all, were placing special importance on the teaching of science and mathematics, due to the requirements of those pupils who were going to enter the specialist corps of artillery and engineers. They were periodically visited by a panel of inspectors, among them the académicien and mathematician Adrien Marie Legendre (1752–1833). Fourier, an exceptionally gifted student of happy nature and quick mind, soon showed a great passion for mathematics. He was only 13 when he got into the habit of collecting candle ends to use at night during the long hours he spent studying mathematics in some sort of store room. In so doing he succeeded in unintentionally scaring the deputy principal who, one night while making his rounds of the school, saw a light through the keyhole and rushed in, fearing a fire. Fourier’s overall excellence is confirmed by the prizes he won during 1782 and 1783: in rhetoric, mathematics, mechanics, and even singing. Then a year-long period of illness followed, perhaps due to the excessive intensity of his studying.
His main ambition at the time was a military career. At the age of seventeen, having completed his studies, he wished to enter the artillery or the engineers. In spite of the support of the inspectors of the school, the minister of war rejected his application. The fact that he was not a noble appears to have played a role.
As a second choice he decided to enter the Benedictines. To St. Benoît-sur-Loire, seat of an ancient and splendid basilica—hosting the body and relics of St. Benedict, which were transported there from Monte Cassino in the seventh century—Fourier arrived in 1787. He stayed two years preparing for his vows while teaching other novices. During this period of spreading riots and chaos, preparatory to the revolution, Fourier remained indifferent to political news and obtained results in the theory of equations. In 1789 he sent a paper to the Académie des Sciences in Paris. That was the year of the French Revolution and Fourier’s paper was one of the casualties.
The monastic orders, combining great wealth and a steadily shrinking number of inmates, had long been a strong temptation to a government continually on the verge of bankruptcy. During October 1789, by decree of the Constituent Assembly, it was forbidden to take any further religious vows. A few months later all religious orders were suppressed and subsequently all belongings confiscated.
Fourier himself, who in the meantime had manifested some uncertainty about the choice he had made, did not take his vows and returned to Auxerre to teach mathematics, rhetoric, history, and philosophy in his old school, now having as a second title that of collège national. A commissioner of the local directory who visited the college in October 1792 reported favorably on the health of the inmates and the liberal atmosphere. He deplored only the tendency to drive out Latin to make way for mathematics, so much in demand by the parents of the pupils. Fourier carried out his duties there with dedication and remarkable success that earned him a fine reputation.
1.3 The revolutionary
The town of Auxerre fortunately saw little or no bloodletting during the Revolution, but the local Société Populaire, associated with the revolutionary Jacobin party, was one of the most militant provincial clubs in the country. Fourier’s involvement in politics did not occur officially until February 1793. The occasion seems to have been a speech before the local assembly following a decree of the Convention for the draft of 300,000 men. While the quotas of the various departments—the administrative regions into which France was divided in 1790—were fixed, it was left to each department through the vote of the citizens to decide how to meet its quota: by lot, by volunteering, or by other means. This hot issue was debated in Auxerre by a general assembly. Fourier intervened and proposed a plan that was then adopted. In a letter written later, in June 1795 from prison, he makes clear that he fully shared the ideals of the Revolution (Hérivel ): “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 enamored of this cause, in my opinion the greatest and most beautiful which any nation has ever undertaken.”
In March 1793 he was invited to join the local Comité de Surveillance. It is not known whether Fourier wished the invitation or, on the contrary, would have preferred to decline it, after receiving it. Certainly, a refusal would not have been without risks, since by and large the town of Auxerre shared the Republican ideals and a refusal could have identified Fourier as an opponent of the patriot party. From what is known of his later involvement in politics, it is more likely that he eagerly embraced the chance to take part in the defense of the Republic, which was threatened by military reverses in Belgium and by the rebellion in Vendée.
Fourier was not a fanatic; he firmly believed in the ideals of the Revolution and to that he devoted his intelligence, eloquence, and zeal but always retained an independent judgment. This together with his juvenile impulsiveness led him into a dangerous situation. He was sent to the neighboring department of Loiret with the mission of collecting horses for the war effort. He completed that “with every possible success.”
This move produced an opposite effect to what I had intended. In a reply sent to me I was reminded of a law which forbade any official to abandon his post and my resignation was rejected. At the same time other persons openly accused me of abandoning my colleagues at a moment when my help was about to become the most useful to them. I was reproached with the feebleness of my conduct and some even doubted the purity of my intentions.
Unfortunately on his way through Orléans, in the course of his mission, he became involved in a local dispute. To Orléans, plagued by near starvation and declared in a “state of rebellion,” a representative of the people was sent by the Convention. Immediately upon arrival he “purged” the administrative corps of the city, made numerous arrests, and threatened to bring in a movable guillotine like the one in Paris. Thereafter he turned against members of the local Société Populaire, members of his own party. It was at this point that Fourier, “behaving in conformity with the principles of Revolution” took the defense “perhaps imprudent but at least disinterested” of the heads of three local families. Fourier was immediately denounced to the Convention, his commission revoked, and himself declared “incapable of receiving such commissions [in the future].” This took place on October 29, thirteen days after the execution of the queen.
Fourier, in fear, returned to Auxerre where he would have faced the greatest possible danger if the Société Populaire and the Comité de Surveillance had not successfully intervened on behalf of their “young and learned compatriot.” He remained a member of the local revolutionary party and kept on teaching in his old school, which was going to be closed down in August 1794. April of that year saw the execution of Danton and of his associates amid the mounting Terror.
Nothing is known of Fourier’s feeling during this period except for what he wrote afterward, when he claimed he had spoken out in Auxerre against the excesses of the Revolution. By June 1794 he had become president of the Revolutionary Committee in Auxerre and, therefore, the foremost representative of the Terror in town. This high position is known from an entry in the local archives reporting his arrest: Fourier, always feeling injustice at the decree of the Convention that declared him unfit for “similar commissions” in the future, had gone in person to Robespierre in Paris to plead his case. Perhaps he made a bad impression on him. Certainly upon his return to Auxerre on July 4 he was arrested. Because of the high reputation he enjoyed in town, many interceded in his favor and he was released, only to be rearrested a few days later. Then an official delegation was sent to Saint Just in Paris to demand his release, but salvation came from a totally different route: on July 27 (9 Thermidor) the Convention ordered the arrest of Robespierre and Saint Just. They were to be promptly guillotined the next day without trial. The Terror was over. Fourier’s life was saved and he regained his freedom.
1.4 At the École Normale
Fourier, back in Auxerre, was soon going to return to Paris for an event that would have a great impact on his life: the opening of the École Normale.
Fourier was likely to have few regrets at the prospect of leaving Auxerre where, as former president of the Revolutionary Committee, he was a marked man. Compared to the violent whirlpools of the Revolution, at the École Normale must have felt like paradise. The professors were chosen from among the foremost men in the country: Joseph Louis Lagrange (1736–1813), Pierre Simon Laplace (1749–1827) and Gaspard Monge (1746–1816) taught mathematics and for chemistry there was Claude Louis Berthollet (1748–1822). All of them were going to play a role in Fourier’s life in the years to come.
When the pupils [of the École Normale] gathered together, France had only just emerged from beneath the axe of Robespierre. The agents of this tyranny were everywhere regarded with abhorrence: but the fear which they had inspired, joined to a fear of their return to power, retained for them some vestige of credit. They profited from this, by seizing the opportunity of quitting the scene of their vexatious act. Several had themselves named pupils of the École Normale. They carried there with the ignorance proper to them the hate, distrust and contempt which followed them everywhere. Beside them were men full of wisdom, talent and enlightenment, men whose names were celebrated in all Europe.
Fourier, an experienced teacher himself, gives an account of the lecturing habits and idiosyncrasies of the professors. Lagrange, whom Fourier followed with eyes full of admiration, as deserved by “the first among European men of science,” had a “rather poor reception” from the students. Incapable of preserving order, he showed his Italian origin by “a very pronounced accent” (he was born in Turin and spent his youth there). He could make some rather comic sentences, such as “There are still on this matter many important things to say, but I shall not say them.” Nevertheless to Fourier “the hesitation and simplicity of a child,” that sometimes could be seen in Lagrange, made only more apparent “the extraordinary man he is.”
The École Normale holds its sessions at the Jardin des Plantes, in a middle-sized place of circular shape; the pupils who are very numerous are seated in rows on the tiers of a very high amphitheater; there is no room for everyone and every day there are a fair number who find the door closed; if one is obliged to leave during the sessions, one cannot enter again. At the back of the room, within an enclosure separated by a railing, are seated several Parisian scientists and the professors. In front, on a slightly higher platform are three armchairs for the professors who are to speak and their assistants. Behind them, and on a second platform, are the representatives of the people in the uniform of deputies on detached service. The session opens at 11 o’clock when one of the deputies arrives; there is much applause at this moment and when the professor takes his place. The lessons are almost always interrupted and terminated by applause. The pupils keep their hats on, the professor who is speaking is uncovered; three quarters of an hour or an hour later, a second professor takes his place, then a third, and the usher announces that the session is ended.
Of Laplace, “among the first rank men of science,” Fourier writes that “he speaks with precision, but not without a certain difficulty” and that “the mathematical teaching he gives has nothing extraordinary about it and is very rapid.” Berthollet, acknowledged as “the greatest chemist we have either in France or abroad,” Fourier continues, “only speaks with extreme difficulty, hesitates and repeats himself ten times in one sentence” to conclude that “his course is only understood by those who study much or understand already.” Monge instead “speaks with a loud voice” and “the science about which he lectures [descriptive geometry] is presented with infinite care and he expounds it with all possible clarity.”
He reports on others including a certain Sicard, “well known as a teacher of deaf-mutes, full of enthusiasm and patience” but “mad.” “His theory of grammar, which is brilliant in certain respects, is one of the craziest I know of” (Hérivel ).
On May 1795, a few months after inauguration, the school was closed due to objective difficulties. In particular the seminars, which had been intended as the backbone of the system, were a failure because so few students had learned enough to be able to contribute. For the majority of the pupils the École Normale had been a waste of time, but for Fourier it was a turning point of his career: At those seminars he made his mark. Meanwhile disturbing rumors were coming from Auxerre.
1.5 From imprisonment to the École Polytechnique
The fall of Robespierre marked the beginning of the settling of scores with those associated with him. Many had used their power to commit all sorts of injustices and atrocities: It had been the “Terror.” In Auxerre, Fourier’s opponents wasted no time. On March 20, 1795 in an address to the National Convention, arguing that the pupils at the École Normale were chosen under the reign of Robespierre and his protégés and that Fourier “had long professed the atrocious principles and infernal maxims of the tyrants,” they protested his preparations to become a teacher of their children. They wanted to prevent him from securing a teaching position and the indemnities, due to him as a pupil at the École Normale, suspended. On May 12, an order was issued in Auxerre regarding the disarmament of a number of terrorists including Fourier, followed on May 30 by a second-order commanding the detention of all those terrorists who had failed to comply with the original order.
Fourier, in an attempt to disarm his enemies, wrote to the municipality asking to which duly constituted authority he should present himself to be disarmed; he even resigned a new position given to him at the École Centrale des Travaux Publiques (Central School of Public Works). All in vain. On the night of June 7 he was awakened by armed guards and taken to prison without notification and having scarcely been given the time to dress himself. The pretext was failure to present himself to be disarmed; the true charge, that of having inspired terrorism in the years 1793–1794.
As an example of this he pointed to his early imprisonment under Robespierre and added, “I have experienced terror more than I inspired it” and “I owed to 9 Thermidor both life and liberty.” Finally, he wrote with anguish, “To exclude me from a school of mathematics is to take away from me an entirely legitimate possession which I have acquired by my work and which I retain by cultivating it daily.”
There remain therefore those citizens who being nobles or priests or relations of émigrés found themselves included under the law of September 17 and who experienced a temporary detention when they showed themselves declared enemies of the Revolution. They accuse me of not having opposed their arrest and never pardon me for having signed the warrants for their arrests. They pretend to believe I could have released them and wanted me to make use of the trust which had been placed in me. Being unable to accuse us of abusing powers they reproach us with excessive rigor, but far from having merited this insult I believe that I have accorded to humanity, friendship, generosity even, all that was allowed by the letter of the law and the rigor of the times.
By August Fourier was released, the reasons unknown. Perhaps it was an intervention of Lagrange or Monge, or more likely the change in the Convention of the political climate with the end of repression against the Jacobins in face of a mounting royalist threat. Fourier resumed his teaching position at the École Centrale, soon to be renamed École Polytechnique by a decree of September 1, 1795. The École Centrale des Travaux Publiques had been instituted by a decree of the Convention of March 11, 1794. It did not have the general aims of the École Normale since it was a military academy: Its graduates, with 3 years training in science, engineering, and applied arts, were intended to provide the military elite. It took only about 400 students annually. Opened in November 1794 with a trial run of fifty students, it was converted into a full organization when the École Normale closed.
The director was Monge and with his support Fourier was appointed to an assistant teaching post, from which he had resigned in vain to avoid arrest. Now back, he helped run the course in descriptive geometry—by Monge, mathematician and military engineer — that dealt largely with the use of science and mathematics in military contexts, the art of attack and defense, and the organization of simulated battle situations. Also he taught courses in Lagrange’s curricula in analysis and was involved with the selection of the entrants to the École. Fourier believed they should have “outstanding talents regardless of how much they have actually been taught,” an opinion similar to the one expressed by Monge at the time of the setting up of the school. Among those entrants was Siméon-Denis Poisson (1781–1840) who was to be Fourier’s pupil at the École, his deputy as professor of analysis during his absence in Egypt, and finally Fourier’s opponent over some mathematical questions concerning the analytical theory of heat. During this period at the École, Fourier once again built up a fine reputation for his teaching and in 1797 succeeded Lagrange in the chair of analysis and mechanics. While continuing his investigation into the theory of equations he also began work on problems in applied mathematics and in 1798 published his first paper in the Journal de l’École Polytechnique.
1.6 Fourier, Napoleon, and the Egyptian campaign
The Minister of Interior to Citizen Fourier Professor at the École Polytechnique:
The minister of the interior was obeying the orders of the Directory (a five-man executive committee that gave the name to the new regime begun in October 1795) instructing him to “put at the disposition of General Bonaparte engineers, artists and other subordinates of your ministry together with the different things he will demand of you for the purpose of the expedition to which he has been assigned.”
Citizen, the Executive Directory having in the present circumstances a particular need of your talent and of your zeal, has just disposed of you for the sake of public service. You should prepare yourself and be ready to depart at the first order. If you are actually charged with any employment or if you occupy any place at the expense of the Republic you will conserve it during your mission and the salary attached to it will be paid to your family.
In February 1799 Bonaparte launched an ill-fated campaign in Syria, in July he stopped a Turkish invasion at the battle of Abukir, and in August he left for Paris upon learning of a troubled situation there. He brought with him the trusted Monge and Berthollet. Fourier was kept out of the plan until the very last moments. When he finally learned of it, he became so agitated at the thought of being left behind that he followed them into the street and could hardly be persuaded to let them go.
Before departing, Bonaparte found time to plan a mixed scientific and literary expedition to Upper Egypt under the joint leadership of Fourier and the assistant secretary of the Institute. The discoveries of the expedition, which spent about two months in the fall of 1799 investigating the monuments and inscriptions in Upper Egypt, would years later form the basis of the Description of Egypt (Fig. 1.3).
Bonaparte left behind a difficult situation, challenged by the English forces and civilian unrest. General Jean Baptiste Kléber, the new commander in chief, was assassinated in Cairo in June 1800 and was succeeded by General Jacques François Menou. Under both commanders Fourier was given many important administrative and judicial positions. By Menou he was put virtually in control of all nonmilitary affairs and entrusted with a most relevant assignment, the conduct of the French side of the negotiations with the Egyptian beys. He was successful in persuading them to sign an alliance with the French at a time when they were none too strong militarily. At the negotiations the chief bey Mourâd was represented by his wife, the beautiful and intelligent Sitty-Nefiçah with whom Fourier negotiated also the freedom of some of her slaves who were of interest to the French generals.
1.7 The prefect of Isère and the Analytical Theory of Heat
The Department of Isère, crossed by the river of the same name and bordering Italian territory, had its center in Grenoble. It was one of the 83 regions into which France had been divided by a decree of the Constituent Assembly of February 3, 1790. In 1799 the departmental directories were abolished and replaced by a single person, the prefect, sole representative of the executive power. Fourier’s feelings about the position, which Bonaparte presented as “an earnest of my confidence in citizen Fourier,” are not known. He accepted it and departed for the remote town of Grenoble. His tasks as prefect were varied and demanding, even though he could count on administrative officers and in particular on three private secretaries. He had to oversee the enforcement of the laws and the implementation of the directives constantly issued from Paris, mainly pertaining to taxes and recruitment, first for the consular and later the imperial armies. At the same time he had to put forward to the central government the needs and requests of the area, backward in development but independent in spirit. Also he was supposed to keep the government in Paris informed about the preservation of order. Soon Bonaparte’s regime began to develop into a form of police state and Fourier had to execute unpleasant orders concerning the opening of letters, the suppression of antigovernmental pamphlets, and the supervision of the local official news sheet Annals of the department of Isère, administrative, political and literary journal to keep away scandals and revolution.
Much more pleasant it must have been for him to get to know the prominent local people, even though he had to compile notebooks on them for the Ministry of the Interior. Indeed the policy of Napoleon was to unite the greatest number of Frenchmen, regardless of their original sympathies and status, in support of his policy and person. From the beginning Fourier was on excellent terms with the nobility who praised his old world manners, the charm of his conversation, and appreciated the many services he did them like helping émigrés, returning from exile, in gaining back their original properties and allowing immunity from the recruitment of the guards of honor in exchange for money, to be used in raising a body of paid volunteers. He was also popular among the wealthy middle class, the so-called bourgeoisie, for his excellent administration and was on good terms with the local clergy. At a ceremony in the cathedral of Grenoble in which Fourier received the oaths of the curés nominated in the diocese, the bishop celebrated the mass while incense was given to the prefect. On the other hand Fourier’s relations with the members of the local Jacobin party are unknown.
The year 1804 was eventful. In February Bonaparte visited Grenoble and in the following May crowned himself emperor. Fourier wrote to Napoleon to tell him that 82,084 of the electorate supported the plebiscite on hereditary descent, with twelve dissenters, and he spent more than three months in the capital for the celebrations. Then he became involved with the visit in November of Pope Pius VII to Paris, since the route passed through Isère.
Besides political affairs Fourier attended to the development and welfare of the department, which was much in need. He reopened schools and colleges and revitalized the mining and craft industries. His major achievement was the draining of the marshes around the village of Bourgoin midway between Lyon and Grenoble. The huge area, covering some twenty million acres, was rather useless and at that time responsible for annual epidemics of fever. There had been projects planned and attempted many times over more than a century but agreement among the 37 communes bordering the swamps was never achieved. Fourier succeeded where his predecessors had failed, thanks to his persistence, patience, and diplomatic ability. The negotiations stretched over a period of some four years during which he visited all the communes involved, one by one, and met individually most of the inhabitants to convince them to temporarily give up their rights of pasture for the sake of a better future. In 1807 a treaty was finally signed and in 1812 work was completed. The value of the land greatly increased as well as the health of the inhabitants. During the drainage archeological remains were found and Fourier instructed Jean Champollion-Figeac , then librarian of Grenoble, to preserve them in the library. Fourier had Champollion as a cultural friend in Grenoble, introduced him to Egyptology, and used the power of his position to spare him conscription. In 1822 Champollion would do his celebrated work on the decipherment of Egyptian hieroglyphics.
Another large scale project initiated by Fourier was the opening on the French side of an important and spectacular road through the Alps, the Grenoble–Turin, via Briançon and Pinerolo. Fourier overcame the opposition of the minister of the interior, himself a native of the countryside in question, by appealing directly to Napoleon. Knowing the man, Fourier had him presented with a map and a one page memoir setting forth the advantages, not least the military ones. Two days later the request was granted. In Fourier’s times, nevertheless, the road’s construction did not go beyond Briançon, partly due to the fall of Napoleon in 1815, but when completed it was the quickest and shortest route from Lyon to Turin. Later Fourier recorded that of all his prefectorial assignments this one gave him the greatest satisfaction.
Another time-consuming accomplishment was his contribution to the Description of Egypt. Fourier had been chosen to unite and publish the collection of works—which had to be done by another former member of the Institute, due to his many prefectorial duties—and to write the general introduction. This was essentially a survey of the history of the ancient civilization to the time of the French expedition and renaissance under French patronage. Fourier took great care and for the final polishing even isolated himself in a country residence, having to deny rumors of illegal holidays to the minister of the interior. He knew how Napoleon would scrutinize it. Finally in the autumn of 1809 the preliminary discourse was submitted for Napoleon’s approval. The emperor kept it for a long time and made corrections in his own hand, to conform the description of the Egyptian campaign to his personal view. The work was published in 1810. Fourier received the title of baron and an annual pension.
Fourier must have enjoyed his achievements and the demonstrations of loyalty and affection he received, but he always hoped to be allowed to resign from the job and to return to an intellectual life. Monge and Berthollet in vain tried to persuade Napoleon, and Fourier even thought of going into exile. In the end he stayed at his post and tried to make life as interesting as possible for himself as well as for others.
During those very same years in which he was intensively solicited by various engagements, imposed on him by external factors, Fourier found the time to take up a problem totally new with respect to his preceding research and was able to pursue it in the silence of the mind to complete solution. This process peaked in the memoir titled Analytical Theory of Heat, his masterpiece dated 1807. The reason for this new scientific interest is not known, since Fourier never described how or when he came to be motivated to address this problem. It is known nevertheless that he always regretted the Egyptian heat and never quite got used to the chilling climate of the Alps. Soon he suffered bad attacks of rheumatism. Thus he never went out, even in the hottest weather, without his overcoat and was often accompanied by a servant with another coat in reserve. In Grenoble, where the winters are more severe than in Paris, his concern for adequate heating must have been even greater. From this to the question of loss and propagation of heat in solids and radiation in space, to the problem of conserving heat, the step can be short.
He started working on it already in the spring of 1802, when he just had arrived in Grenoble, and had difficulties until 1804. Seemingly only in the second half of 1804 did he find a new approach. During the next 3 years Fourier obtained, in the brief intervals of research time available, the main body of his contribution to mathematical physics. Presumably he used some of the time of his retreat to work on the introductory paper on the Egypt volumes to write a memoir Theory of the Propagation of Heat in Solid Bodies (Fig. 1.5). He presented it to the Institut de France, reading an abstract in front of the First Class. (The full text is reproduced in Grattan-Guinness .)
The permanent secretary for the mathematical and physical sciences of the Institute was the astronomer Jean Baptiste Joseph Delambre (1749–1822). He asked Lagrange , Laplace , Silvestre François Lacroix (1756–1843), and Monge to examine the paper. The members of the commission, if anything, had already formed a high opinion of Fourier, but the memoir was not well received. Lagrange rejected the very foundation of its mathematical side, namely the use of the trigonometric series (now called Fourier series). He thought it was not rigorous and never changed his mind. Also some criticism was directed against Fourier’s derivation of the equation of the motion of heat in a continuous solid. Laplace, who had carried out experiments on the subject, seems to have taken that position. But the chief opponent here was Jean Baptiste Biot (1774–1862), a protégé of Laplace, who had published in 1804 a short paper containing not only experimental results but some efforts at a mathematical formulation of heat diffusion. Fourier made a reference to it but confined himself to its experimental results. While Biot did not derive the right equation and did not attempt to solve the one he obtained, it appears that Fourier got some hint out of Biot’s mathematical model. That must have been all the more evident to Biot who, on top of everything, was aware that Fourier himself had written an equation in an earlier draft of his memoir, possibly sent to him or to Poisson. Thus Biot became transformed into an opponent.
Nothing has survived of the commission’s criticisms, except for Fourier’s replies preserved in two letters to Laplace and Lagrange concerning the convergence of trigonometric series, the most critical issue. Fourier also sent to the Institute additional notes and wrote to Delambre asking for the date of publication of his manuscript. Nothing happened for the simple reason that the commission failed to report. There only appeared in the Bulletin of the Philomathic Society an unenthusiastic summary and review by Poisson who was interested in heat diffusion himself. A few months after Fourier had presented his manuscript, in a report to Napoleon on the development of science since the Revolution, Delambre wrote that in mathematics the chances of progress were very slight, with insurmountable difficulties preventing major breakthroughs and leaving only minor points of detail to be cleared up.
A new turn took place at the beginning of 1810 when the Institute announced the subject of its grand prize in mathematics for the year 1811: The mathematical theory of heat and its experimental verification. There appears to be no trace of the decision process that led to setting the subject of the prize. Certainly the members of the First Class of the Institute were divided on the ground that no report had yet been made on the 1807 memoir, but in the end that was the decision. Fourier submitted the old memoir, revised and extended by new sections on the cooling of infinite solids and on terrestrial and radiant heat; in all it was 215 pages. The commission, made up of Lagrange , Laplace , Lacroix , Malus , and René-Just Haüy (1743–1822), had to examine another candidate also. The prize was awarded to Fourier, but the motivation must have left him disappointed. There were no doubts about the importance of his work but ambivalent feelings remained, which were clearly stated in the report: “This theory contains the true differential equations of the transmission of heat both in the interior of bodies and at their surface. The novelty of the subject combined with its importance has made the class determined to crown this work, while observing however that the manner in which the author arrives at these equations is not exempt from difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigor.” (Hérivel ).
Out of Fourier’s disappointment a letter to the Institute likely grew, for although the letter disappeared, Delambre’s reply remained: The commissioners had full power in such a matter. Fourier understood that the Institute was in no hurry to publish his memoir (indeed only in 1815 on his return to Paris could he get Delambre started on it). Thus he began to prepare a third version of his research in the form of a book and at the same time he obtained new results on other aspects of the problem. In spite of his administrative duties he worked on this in the following three years and was about to complete it when the political situation turned dramatic once more.
1.8 Fourier and the “Hundred Days”
After the disastrous Russian campaign of 1812 and the debacle of the battle of Leipzig of 1813, in January 1814 France was being attacked at its frontiers for the first time since 1795. Fourier, among other imperial officials, must have wondered about Napoleon’s fate and his own. The extraordinary successes Napoleon achieved with his army of young conscripts during the first 3 months of 1814 proved insufficient to reverse the trend. The French people, tired of his interminable wars, were not behind him any longer and on March 31 Paris surrendered to allied troops while Napoleon had moved east to attack their rearguard. In the meantime Grenoble was besieged by Austrian forces.
When the news of the surrender of Paris and the abdication of Napoleon were learned, the Austrian forces occupied Grenoble on behalf of Louis XVIII, brother of the executed Louis XVI, who had since become king of France. The return of the king was unexpected and Fourier must have been concerned about his position and even personal condition, but he was allowed to continue as prefect, certainly due to the integrity of his conduct and the support he had among members of the old nobility. The situation therefore became the more embarrassing when it was learned that, on his way to exile in Elba, Napoleon was going to pass through Grenoble. Fourier let him know, through the prefect of Lyon, that it would be dangerous to pass through Grenoble owing to the excited condition of the population; at the same time, however, he gave instructions to prepare for Napoleon’s stay in the prefecture. Then in a state of extreme upset he waited. Suddenly, the day Napoleon was supposed to come, a messenger arrived announcing that Napoleon would not pass through Grenoble after all. Fourier retired for the rest of the day to recover.
Some time later he was confirmed in his position of prefect and even received a visit by the king’s brother, Count d’Artois, later to become King Charles X. His position secured, he just had to deal with some unrest on the issue of returning the land, sold during the revolution, to their rightful owners, nobles, and clergy. Then on March 1, 1815 a letter from the prefect of the neighboring department of Var reached Fourier. It read: “Bonaparte at the head of 1,700 men disembarked yesterday at the Gulf Juan, reached Grasse this morning and according to those soldiers who have been questioned is heading for Lyon by Saint-Vallier, Digne, and Grenoble.” The letter ended with assurances that the news, no matter how extraordinary it might seem, was entirely true. This time there was no room for diplomacy. Fourier had to take a clear stance. It was 4 p.m. By 7 p.m. he had worked out various contingency plans for the defense of the department in collaboration with the major of Grenoble and the commanding officer of the garrison. He also had started a letter to the minister of the interior, which was completed by 7 a.m. the next day after a troubled night. He assured the minister that the inhabitants of Grenoble were firmly behind the king and that no motive of fear would have turned him from his duty toward king and country. He asked for instructions and added, “I know personally the audacious enemy who threatens us and I do not doubt that before very long he will send us emissaries.” That did not happen.
By the afternoon Fourier had a proclamation put up in town in which he reminded the citizens of their duty to the king and those who might be inclined to forget the severe punishments they would incur, in conformity with the law.
In spite of the sincere efforts of Fourier and other prefects, Napoleon’s advance in the south of France proved to be irresistible. On March 7 the gates were forced open and Napoleon entered Grenoble amidst the enthusiasm of the populace. Meanwhile Fourier was leaving the town from another gate, with him the commanding officer of the garrison. Before leaving he had a room prepared for Napoleon in the prefect’s residence. Also he left a letter in which he expressed his feelings of obligation toward the king and his wishes not to offend his old master.
It was the night of March 7 and Fourier was headed toward Lyon to join the Count d’Artois. Napoleon, irritated by Fourier’s failure to greet him in Grenoble, suspended him from office and required him to evacuate the territory of the 7th military division within five days. Failure to do so meant arrest as an enemy of the nation. On the evening of March 8 Napoleon was already moving toward Lyon with an ever-increasing army. The Count d’Artois quickly retreated. Maybe it became clear to Fourier that the king’s cause was lost in southern France or maybe he was told that the worst was over after Napoleon had been presented with his letter, for on March 9 he turned back to Grenoble and on the morning of the 10th met with Napoleon in Bourgoin. Helpless, he could only accept Napoleon’s wishes. The meeting did not go badly: The next day he learned he had been made prefect of the Rhône and was installed at Lyon on March 12. He carried out all reasonable administrative duties requested by his position, such as recruitment for the imperial army and surveillance of political suspects; at the same time he tried to reduce as much as possible the injustice and suffering associated with the change of regime. But his new post did not last long. Before the end of May he resigned, in disagreement with the harsh orders of the new minister of the interior, the mathematician and French Revolutionary figure Lazare Carnot. Apparently they had to do with a purge of administrators, some in his own prefecture, suspected of royalist sympathies. Whatever the reason, his credits were not entirely destroyed. On June 10 Napoleon granted him a retirement pension of 6,000 francs.
1.9 Return to Paris
Fourier went to Paris where he could enjoy freedom from administrative duties and opportunities to meet again with his old acquaintances such as Laplace, Monge , and Berthollet. But on 18 June Napoleon was defeated at Waterloo. The Napleonic era was over, the Bourbon monarchy back.
Soon Fourier found himself in a desperate financial position. The retirement pension granted to him by Napoleon was totally lost, the first installment being due on July 1. Also the 4,000 francs that went with his Napoleonic barony of 1809 was annulled. He had little money when he came to Paris, due to his generosity and the habit of living up to the top of his income. It seems that he thought of emigrating to England, to be far from political persecution, when a friend came to help. It was the prefect of the Seine, count of Chabrol, who had entered the École Polytechnique in 1794 when Fourier taught there and also joined the Egyptian campaign. In disregard of reactions from the extreme right, he offered Fourier the directorship of the Statistical Bureau. Fourier accepted and held the position for the rest of his life. The salary was modest, but Fourier must have considered himself fortunate. It went quite differently for Monge, faithful and prominent Bonapartist: In spite of his merits for having resurrected the French system of higher education, he was forbidden to enter his beloved École Polytechnique, was forced out of the Académie des Sciences, and died in poverty.
Now what Fourier cared about most was to firmly establish himself in the scientific community and also to obtain a pension for his many services to the state in teaching, in the Egyptian campaign, and as prefect of Isère. He applied to the minister of the interior on November 20, 1815, recalling all the above, hoping that his Jacobin activities of 1793–94 had been forgotten, justifying his action during the Hundred Days as having preserved Lyon “from greatest disasters” and attributing his dismissal to “unjust and arbitrary” measures required of him. Fourier’s application was acknowledged and after considerable time refused. He did not give up and wrote again suggesting that “no political motive should efface the memory of so many services from which the State and many generations will receive real and lasting advantages.” The reply addressed to “Baron Fourier” stated that “the King had recently adjourned his decision on this matter.” Later in response to his insistence, the technicality that he had retired before reaching the age of sixty backed up a final refusal. Only years later was he granted a pension, curiously from the minister of police, for “important services of information” that we may suppose to be of statistical nature.
In the meantime on May 27, 1816 Fourier was elected “free académicien”—a position different from the ordinary one that required the vacancy of a “chair”—to the scientific class of the Institut de France. In vain, since King Louis XVIII did not confirm the election. Shortly afterward the old academies were restored in place of the classes of the Institut de France—with both Monge and Carnot excluded—and already by May 1817 a vacancy had arisen for physics in the Académie des Sciences. Fourier was elected by an overwhelming majority of 47 votes out of 50. This time he was better prepared and obtained the king’s approval, being supported by an old friend of his from Grenoble, the viscount Dubouchage, now minister of marine. In a letter of recommendation Dubouchage had stated, referring to Fourier’s activities as prefect in Isère, that “his conduct has won him the especial gratitude of the families most devoted to the royal cause who found themselves most exposed to oppressive measures.” Fourier was now firmly established, had another source of income, and could throw himself into work. In the next 5 years he submitted to the Académie eight memoirs, in mathematics and statistics, and sat on a large number of commissions.
In August 1822 Delambre, the permanent secretary of the Académie for the mathematical sciences, died and Fourier was elected in his place with 38 votes while his opponent Biot had only 10. In this new position he was responsible for all the official correspondence on the mathematical side, for composing éloges as he did for Delambre and Laplace, and for producing the final reports on the state of mathematics. He also found time to publish a number of papers, but although he extended his studies to thermoelectric effects he did not show any interest in a pamphlet published by Sadi Carnot (1796–1832). Lazare Carnot’s son was putting forth the idea that change of heat could be a source of motive power: His theory of cyclic processes was a great contribution to science indeed.
Life in Paris was not without scientific controversies, partly due to the fact that Fourier’s work had not been published. Only in 1822 did his book Analytical Theory of Heat appear, soon followed by his 1811 prize paper in the Mémoires de l’ Académie Royale des Sciences. By that time much more had been achieved on the subject by Fourier and by others, so on the opening page he added a footnote pointing out that it was an unaltered printing of his manuscript. It had happened that Poisson in a paper of 1815, mentioning Fourier’s prize essay, echoed the criticisms of the commission’s report and attempted to give an alternative treatment of the propagation of heat in solids. Also Biot in his Traité de Physique of 1816 claimed to have been the first, back in 1804, to enunciate and apply the equation for the steady state distribution of the temperature in an iron bar heated at one end. Claims like these always cause great resentment because of the strong emotional bonding that ties every researcher to his own scientific contributions. Fourier reaffirmed the validity and priority of his solution. Here is Fourier’s punch line: “One does not extend the bounds of science by presenting in a form said to be different results which one has not found oneself and, above all, by forestalling the true author in publication,” and with a double-sided compliment to Poisson “M. Poisson has too much talent to exercise it on the work of others. Science waits and will obtain from him discoveries of great superior order.”
With the great Lagrange dead and in 1827 Laplace, there was still much talent left on the Parisian mathematical scene. Besides Poisson, their natural heir, and besides the outsider Fourier there was another outsider of value Augustin Louis Cauchy (1789–1857). His initial work appeared in the early 1810s and continued ceaselessly for almost half a century. In Cauchy extraordinary scientific talent lived alongside smallness of character: He was always ready to put down the work of others to prove the superiority of his own, which in the great majority of cases was clear anyhow. In 1814 Cauchy wrote an important essay on using complex variables in the evaluation of integrals that laid the foundation of his theory of complex variables. The paper met with opposition from Legendre , remained with the secretariat of the Institut de France, and received a poor summary from Poisson, who was interested in the problem too. Then, very much like Fourier, Cauchy won a prize with a paper on the motion of water waves that also remained with the secretariat. In this paper he began to develop integral solutions to partial differential equations and two years later, in 1817, in another paper he found what is now called “Fourier’s integral theorem,” which was in Fourier’s unpublished prize manuscript. Cauchy did have access to Fourier’s memoir, having been nominated the preceding year to one of the vacancies of the Académie arising from the expulsion of Monge and Lazare Carnot. Fourier did not wait to acquaint him with his priority and Cauchy acknowledged it.
Two young and talented mathematicians then briefly appeared on the mathematical scene: Galois and Abel. Évariste Galois (1809–1832), known for that part of algebra named “Galois theory” after him, in 1829 submitted to the Académie a paper solving what was a long-standing open problem on the solutions of algebraic equations. Fourier was then permanent secretary. The paper was sent to Cauchy to examine but got lost. Twice again Galois submitted it to receive three years later, and only six months before his death in a duel, the assessment “unintelligible” signed by Poisson. Things went only slightly better for the Norwegian Niels Henrik Abel (1802–1829). In 1826 he submitted to the Académie his masterpiece on transcendental functions. The paper, passed to Legendre and Cauchy, who never looked at it, got published after various misadventures 35 years later, in 1841. Abel could not rejoice, having died some ten years earlier.
In Paris Fourier added new acquaintances to the old ones. Devoted friends of his were the engineer and mathematician Claude Louis Marie Henri Navier (1785–1836) to whom Fourier’s papers passed on his death, Peter Gustav Lejeune Dirichlet (1805–1859) who made important contributions to some of Fourier’s ideas in pure mathematics, Joseph Liouville (1809–1882) who was interested in heat diffusion—his first paper dealt with the case of the inhomogeneous bars—and Jacques Charles François Sturm (1803–1855), later to work with Liouville on partial differential equations. Fourier who never married, but was said to be fond of the company of intelligent women, particularly cared about the friendship of Sophie Germain (1776–1831), a self-taught mathematician of considerable talent who corresponded with Gauss and Lagrange . Their friendship lasted from at least 1820 until Fourier’s death and a number of his letters to her have survived. Maybe Sophie Germain, who died of cancer, was the suffering woman that Fourier recommended to a certain Doctor Herminier in an undated letter. He wrote “She is worthy of all your interest by reason of the rarest and most beautiful qualities. For myself, who love her tenderly, I would be most grateful for anything you could do for her.”
Fourier at last received many honors. In 1823 he was made a foreign member of the Royal Society, in 1827 was elected to the Académie Française and the Académie de Médecine, and on the death of Laplace, was made president of the Conseil de Perfectionnement of the École Polytechnique. But in his last five years he suffered increasingly, mainly from chronic rheumatism, which pushed him to wear heavy woolen clothes and keep the stove in his apartment lit at all times. In addition he developed breathing problems. To be able to work, in his last months, he had to be placed in a boxlike chair from which only his head and arms protruded. He worked relentlessly, producing almost illegible manuscripts: Now he finally had the opportunity to devote himself to mathematics and he wanted to secure his fame by working as long as he could. Things looked different from some forty years earlier when, young and unknown, he had written in a letter to Bonard: “Yesterday was my 21st birthday, at that age Newton and Pascal had [already] acquired many claims to immortality.” But time was running out. On May 4, 1830 he was stricken by some attack while descending stairs and died on the 16th in his apartment opposite the Jardin du Luxembourg, now 73 Boulevard St. Michel.
It seems nice and truthful what Grattan-Guinness  writes of Fourier: “He preserved his honor in difficult times and when he died he left behind him a memory of gratitude among those who had been under his care as well as important problems for his scientific colleagues.”
“come è duro calle lo scendere e ’l salir per l’altrui scale” The Divine Comedy, Paradise, XVII, 59–60, by Dante Alighieri (1265–1321).
The present day École Normale descends from the homonymous school established 16 years later during the Napoleonic era, dedicated to training professors of secondary and higher education.