约瑟夫·傅里叶Joseph Fourier

约瑟夫·傅里叶的父亲是欧塞尔的一名裁缝。他的第一任妻子与他育有三个孩子，她去世后，他又再婚，傅里叶是第二次婚姻所生十二个孩子中的第九个。傅里叶九岁时母亲去世，第二年父亲也去世了。

他最初在帕莱学校接受教育，这所学校由大教堂的音乐教师开办。在那里，傅里叶学习拉丁语和法语，并表现出极大的潜力。1780年，他进入欧塞尔的皇家军事学校，起初展现出文学天赋，但很快，到十三岁时，数学成了他真正的兴趣所在。到14岁时，他已经完成了对艾蒂安·贝祖的Cours de mathématiquesⓉ（《数学教程》）六卷本的学习。1783年，他因学习夏尔·博叙的De la mécanique en généralⓉ（《论一般力学》）而获得一等奖。

1787年，傅里叶决定接受神职培训，进入圣伯努瓦河畔圣伯努瓦的本笃会修道院。然而他对数学的兴趣仍在继续，并与欧塞尔的数学教授C L Bonard通信。傅里叶不确定自己接受神职培训的决定是否正确。他向巴黎的J.É. 蒙蒂克拉提交了一篇关于代数的论文，而他写给Bonard的信表明，他真正想要在数学上产生重大影响。在一封信中傅里叶写道

昨天是我21岁生日，在那个年纪，艾萨克·牛顿和布莱兹·帕斯卡已经获得了许多足以不朽的成就。

傅里叶没有立下宗教誓愿。1789年离开圣伯努瓦后，他造访巴黎，并在Académie Royale des Sciences宣读了一篇关于代数方程的论文。1790年，他成为他曾就读的本笃会学院——欧塞尔皇家军事学校的教师。直到此时，傅里叶内心一直存在冲突：他应当过宗教生活还是从事数学研究。然而在1793年，当他涉足政治并加入当地的革命委员会时，这一冲突中又加入了第三个因素。正如他所写的：——

随着平等这一自然观念的发展，人们有可能构想出一种崇高的希望：在我们中间建立一个摆脱国王和祭司的自由政府，并把欧洲长期被篡夺的土地从这双重枷锁下解放出来。我欣然爱上了这一事业，在我看来，这是任何民族所从事过的最伟大、最美好的事业。

傅里叶对法国大革命引发的恐怖统治感到不满，他试图辞去委员会职务。然而这被证明是不可能的，傅里叶此时已与大革命紧密纠缠在一起，无法抽身。大革命是一件复杂的事情，有许多派别，目标大体相似，却彼此激烈对立。傅里叶在奥尔良时为某一派别的成员辩护。一封描述事件的信件写道：-

傅里叶，一个充满才智、口才和热情的年轻人，被派往卢瓦雷。……看来傅里叶……登上了某些受欢迎的讲坛。他很能言善辩，如果他提出的是欧塞尔协会的观点，那他并没有做什么该受责备的事……

这一事件产生了严重后果，但此后傅里叶回到欧塞尔，继续在革命委员会工作，并继续在学院任教。1794年7月他被逮捕，指控与奥尔良事件有关，随后被监禁。傅里叶担心他会走上断头台，但在罗伯斯庇尔本人被送上断头台之后，政治变化使傅里叶获释。

1794年晚些时候，傅里叶被提名到巴黎的师范学校学习。这所学校是为培训教师而设立的，并打算作为其他师范学校的典范。学校于1795年1月开学，傅里叶无疑是学生中最有能力的，而这些学生的能力参差不齐。他的老师是约瑟夫·拉格朗日，傅里叶将其描述为

欧洲科学界第一人，

以及皮埃尔·西蒙·拉普拉斯，傅里叶对他的评价较低，还有加斯帕尔·蒙日，傅里叶描述为

声音洪亮，活跃、机智且学识渊博。

傅里叶开始在法兰西学院任教，并与约瑟夫·拉格朗日、皮埃尔·西蒙·拉普拉斯和加斯帕尔·蒙日关系良好，开始了进一步的数学研究。他被任命到中央公共工程学院的一个职位，该学院由拉扎尔·卡诺和加斯帕尔·蒙日领导，不久后更名为巴黎综合理工学院。然而，他早先被捕的余波仍在，他再次被捕入狱。他的获释被归因于多种不同原因，他的学生的请求，约瑟夫·拉格朗日、皮埃尔·西蒙·拉普拉斯或加斯帕尔·蒙日的请求，或者政治气候的变化。事实上，这三者可能都起了作用。

到1795年9月1日，傅里叶回到巴黎综合理工学院任教。1797年，他接替约瑟夫·拉格朗日被任命为分析与力学讲席。他作为杰出的讲师而闻名，但在此期间他似乎没有进行原创性研究。

1798年，傅里叶作为科学顾问加入拿破仑入侵埃及的军队。加斯帕尔·蒙日和艾蒂安-路易·马吕斯也是远征军的一部分。远征起初非常成功。1798年6月10日占领马耳他，7月1日攻占亚历山大，尼罗河三角洲迅速被占领。然而，1798年8月1日，法国舰队在尼罗河战役中被纳尔逊的舰队彻底摧毁，因此拿破仑发现自己被困在他所占领的土地上。傅里叶担任行政官，建立了法国式的政治制度和行政机构。他特别帮助在埃及建立教育设施，并进行考古探索。

在开罗期间，傅里叶帮助建立了开罗研究所，并且是数学部的十二名成员之一，其他人包括加斯帕尔·蒙日、艾蒂安-路易·马吕斯和拿破仑本人。傅里叶被选为研究所秘书，在法国占领埃及的整个期间一直担任这一职位。傅里叶还负责整理在埃及期间取得的科学和文学发现。

拿破仑于1799年放弃军队返回巴黎，很快在法国掌握了绝对权力。傅里叶于1801年带着远征军的残部返回法国，并恢复了他在巴黎综合理工学院的分析学教授职位。然而拿破仑对于傅里叶如何为他服务有其他想法，并写道：-

……伊泽尔省省长最近去世，我想通过任命公民傅里叶担任此职来表达我对他的信任。

傅里叶对离开学术界和巴黎的前景并不高兴，但无法拒绝拿破仑的要求。他去了格勒诺布尔，在那里他作为省长的职责繁多且多样。他在这个行政职位上的两项最大成就是监督排干布尔关沼泽的工程，以及监督从格勒诺布尔到都灵的新公路的建设。他还花了很多时间撰写Description of Egypt，该书直到1810年才完成，当时拿破仑在出版前对其进行了修改，在某些地方改写了历史。到第二版出现时，所有对拿破仑的提及都已被删除。

正是在格勒诺布尔期间，傅里叶完成了关于热理论的重要数学工作。他关于这一主题的工作大约始于1804年，到1807年，他已经完成了重要论文On the Propagation of Heat in Solid Bodies。这篇论文于1807年12月21日在巴黎研究院宣读，并成立了一个由约瑟夫·拉格朗日、皮埃尔·西蒙·拉普拉斯、加斯帕尔·蒙日和西尔维斯特·佛朗索瓦·拉克鲁瓦组成的委员会来报告这项工作。如今，这篇论文受到高度评价，但在当时却引起了争议。

委员会对这项工作感到不满有两个原因。第一个反对意见由约瑟夫·拉格朗日和皮埃尔·西蒙·拉普拉斯在1808年提出，针对的是傅里叶将函数展开为三角级数，即我们现在所称的傅里叶级数。傅里叶的进一步澄清仍未能说服他们。正如[4]所指出的：-

所有这些论述都写得极为清晰——从逻辑而非书法的角度来看——以至于它们未能说服皮埃尔·西蒙·拉普拉斯和约瑟夫·拉格朗日……这很好地表明了傅里叶观点的独创性。

第二个反对意见由让-巴蒂斯特·毕奥针对傅里叶对热传递方程的推导提出。傅里叶没有引用让-巴蒂斯特·毕奥1804年关于此主题的论文，但让-巴蒂斯特·毕奥的论文肯定是不正确的。皮埃尔·西蒙·拉普拉斯以及后来的西莫恩·德尼·泊松也有类似的反对意见。

研究院将固体中的热传播定为1811年数学奖的竞赛题目。傅里叶提交了他1807年的论文，以及关于无限固体冷却和地面与辐射热的附加工作。只收到另一份参赛作品，负责决定奖项的委员会——约瑟夫·拉格朗日、皮埃尔·西蒙·拉普拉斯、艾蒂安-路易·马吕斯、Haüy和阿德里安-马里·勒让德——将奖项授予了傅里叶。然而报告并非完全赞同，其中写道：-

……作者得出这些方程的方式并非没有困难，而且他积分这些方程的分析在普遍性甚至严格性方面仍有不足之处。

有了这份相当褒贬不一的报告，巴黎方面没有采取行动出版傅里叶的著作。

当拿破仑战败并前往厄尔巴岛流放时，他的路线本应经过格勒诺布尔。傅里叶设法避免了这一棘手的对抗，他传话说这对拿破仑来说会很危险。当傅里叶得知拿破仑从厄尔巴岛逃脱并正率军向格勒诺布尔进军时，他极为担忧。他试图说服格勒诺布尔的民众反对拿破仑并效忠国王。然而，当拿破仑从一座城门进城时，傅里叶匆忙从另一座城门离开了。

拿破仑对傅里叶感到愤怒，他本希望傅里叶会欢迎他的回归。傅里叶设法凭借口才赢得了双方的欢心，拿破仑任命他为罗讷省省长。然而，傅里叶在接到命令后很快辞职，命令可能来自拉扎尔·卡诺，要求他撤除所有同情保皇派的行政官员。不过，他不可能与拿破仑和拉扎尔·卡诺完全闹翻，因为1815年6月10日，拿破仑授予他一笔6000法郎的养老金，从7月1日起支付。然而，拿破仑于7月1日战败，傅里叶没有收到任何钱。他回到了巴黎。

傅里叶于1817年当选为Académie des Sciences会士。1822年，让·巴蒂斯特·约瑟夫·德朗布尔——时任Académie des Sciences数学部秘书——去世，傅里叶与让-巴蒂斯特·毕奥和弗朗索瓦·阿拉戈一同申请该职位。在弗朗索瓦·阿拉戈退出后，选举使傅里叶轻松获胜。傅里叶成为秘书后不久，Académie于1822年出版了他的获奖论文Théorie analytique de la chaleur Ⓣ（热的解析理论）。然而，这并非傅里叶的政治操作，因为让·巴蒂斯特·约瑟夫·德朗布尔在去世前已安排付印。

在巴黎的最后八年里，傅里叶恢复了数学研究，并发表了多篇论文，其中一些属于纯数学，另一些则涉及应用数学主题。然而，他的生活并非一帆风顺，因为他的热理论仍然引发争议。让-巴蒂斯特·毕奥声称优先于傅里叶，而傅里叶毫不费力地证明这一主张是错误的。然而，西莫恩·德尼·泊松既攻击了傅里叶的数学技巧，又声称拥有另一种理论。傅里叶写了Historical PrécisⓉ（历史概述）作为对这些主张的回应，但尽管这部作品曾向多位数学家展示，却从未出版。

傅里叶关于让-巴蒂斯特·毕奥和西莫恩·德尼·泊松所提出主张的看法见下文，参见[4]：-

在提出异议之后，[让-巴蒂斯特·毕奥和西莫恩·德尼·泊松]现在承认这些结果是精确的，但他们抗议说，他们发明了另一种阐述这些结果的方法，而且这种方法非常出色，是真正的方法。如果他们曾以重要而普遍的见解照亮物理学的这一分支，并大大完善了偏微分方程的分析，如果他们曾通过精良的实验确立了热学理论的一个主要要素……他们就有权评判我的工作并加以纠正。我会非常乐意服从……但是，人们并不能通过以所谓不同的形式呈现并非自己发现的结果，尤其是通过在发表上抢先于真正的作者，来拓展科学的边界。

傅里叶的工作为后来关于三角级数和实变函数论的研究提供了推动力。

Joseph Fourier's father was a tailor in Auxerre. After the death of his first wife, with whom he had three children, he remarried and Joseph was the ninth of the twelve children of this second marriage. Joseph's mother died went he was nine years old and his father died the following year.

His first schooling was at Pallais's school, run by the music master from the cathedral. There Joseph studied Latin and French and showed great promise. He proceeded in 1780 to the École Royale Militaire of Auxerre where at first he showed talents for literature but very soon, by the age of thirteen, mathematics became his real interest. By the age of 14 he had completed a study of the six volumes of Bézout's Cours de mathématiques Ⓣ. In 1783 he received the first prize for his study of Bossut's De la mécanique en général Ⓣ.

In 1787 Fourier decided to train for the priesthood and entered the Benedictine abbey of St Benoit-sur-Loire. His interest in mathematics continued, however, and he corresponded with C L Bonard, the professor of mathematics at Auxerre. Fourier was unsure if he was making the right decision in training for the priesthood. He submitted a paper on algebra to Montucla in Paris and his letters to Bonard suggest that he really wanted to make a major impact in mathematics. In one letter Fourier wrote

Yesterday was my 21st birthday, at that age Newton and Pascal had already acquired many claims to immortality.

Fourier did not take his religious vows. Having left St Benoit in 1789, he visited Paris and read a paper on algebraic equations at the Académie Royale des Sciences. In 1790 he became a teacher at the Benedictine college, École Royale Militaire of Auxerre, where he had studied. Up until this time there had been a conflict inside Fourier about whether he should follow a religious life or one of mathematical research. However in 1793 a third element was added to this conflict when he became involved in politics and joined the local Revolutionary Committee. As he wrote:-

As the natural ideas of equality developed it was possible to conceive the sublime hope of establishing among us a free government exempt from kings and priests, and to free from this double yoke the long-usurped soil of Europe. I readily became enamoured of this cause, in my opinion the greatest and most beautiful which any nation has ever undertaken.

Certainly Fourier was unhappy about the Terror which resulted from the French Revolution and he attempted to resign from the committee. However this proved impossible and Fourier was now firmly entangled with the Revolution and unable to withdraw. The revolution was a complicated affair with many factions, with broadly similar aims, violently opposed to each other. Fourier defended members of one faction while in Orléans. A letter describing events relates:-

Citizen Fourier, a young man full of intelligence, eloquence and zeal, was sent to Loiret. ... It seems that Fourier ... got up on certain popular platforms. He can talk very well and if he put forward the views of the Society of Auxerre he has done nothing blameworthy...

This incident was to have serious consequences but after it Fourier returned to Auxerre and continued to work on the revolutionary committee and continued to teach at the College. In July 1794 he was arrested, the charges relating to the Orléans incident, and he was imprisoned. Fourier feared the he would go to the guillotine but, after Robespierre himself went to the guillotine, political changes resulted in Fourier being freed.

Later in 1794 Fourier was nominated to study at the École Normale in Paris. This institution had been set up for training teachers and it was intended to serve as a model for other teacher-training schools. The school opened in January 1795 and Fourier was certainly the most able of the pupils whose abilities ranged widely. He was taught by Lagrange, whom Fourier described as

the first among European men of science,

and also by Laplace, who Fourier rated less highly, and by Monge who Fourier described as

having a loud voice and is active, ingenious and very learned.

Fourier began teaching at the Collège de France and, having excellent relations with Lagrange, Laplace and Monge, began further mathematical research. He was appointed to a position at the École Centrale des Travaux Publics, the school being under the direction of Lazare Carnot and Gaspard Monge, which was soon to be renamed École Polytechnique. However, repercussions of his earlier arrest remained and he was arrested again and imprisoned. His release has been put down to a variety of different causes, pleas by his pupils, pleas by Lagrange, Laplace or Monge or a change in the political climate. In fact all three may have played a part.

By 1 September 1795 Fourier was back teaching at the École Polytechnique. In 1797 he succeeded Lagrange in being appointed to the chair of analysis and mechanics. He was renowned as an outstanding lecturer but he does not appear to have undertaken original research during this time.

In 1798 Fourier joined Napoleon's army in its invasion of Egypt as scientific adviser. Monge and Malus were also part of the expeditionary force. The expedition was at first a great success. Malta was occupied on 10 June 1798, Alexandria taken by storm on 1 July, and the delta of the Nile quickly taken. However, on 1 August 1798 the French fleet was completely destroyed by Nelson's fleet in the Battle of the Nile, so that Napoleon found himself confined to the land that he was occupying. Fourier acted as an administrator as French type political institutions and administration was set up. In particular he helped establish educational facilities in Egypt and carried out archaeological explorations.

While in Cairo Fourier helped found the Cairo Institute and was one of the twelve members of the mathematics division, the others included Monge, Malus and Napoleon himself. Fourier was elected secretary to the Institute, a position he continued to hold during the entire French occupation of Egypt. Fourier was also put in charge of collating the scientific and literary discoveries made during the time in Egypt.

Napoleon abandoned his army and returned to Paris in 1799, he soon held absolute power in France. Fourier returned to France in 1801 with the remains of the expeditionary force and resumed his post as Professor of Analysis at the École Polytechnique. However Napoleon had other ideas about how Fourier might serve him and wrote:-

... the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place.

Fourier was not happy at the prospect of leaving the academic world and Paris but could not refuse Napoleon's request. He went to Grenoble where his duties as Prefect were many and varied. His two greatest achievements in this administrative position were overseeing the operation to drain the swamps of Bourgoin and supervising the construction of a new highway from Grenoble to Turin. He also spent much time working on the Description of Egypt which was not completed until 1810 when Napoleon made changes, rewriting history in places, to it before publication. By the time a second edition appeared every reference to Napoleon would have been removed.

It was during his time in Grenoble that Fourier did his important mathematical work on the theory of heat. His work on the topic began around 1804 and by 1807 he had completed his important memoir On the Propagation of Heat in Solid Bodies. The memoir was read to the Paris Institute on 21 December 1807 and a committee consisting of Lagrange, Laplace, Monge and Lacroix was set up to report on the work. Now this memoir is very highly regarded but at the time it caused controversy.

There were two reasons for the committee to feel unhappy with the work. The first objection, made by Lagrange and Laplace in 1808, was to Fourier's expansions of functions as trigonometrical series, what we now call Fourier series. Further clarification by Fourier still failed to convince them. As is pointed out in [4]:-

All these are written with such exemplary clarity - from a logical as opposed to calligraphic point of view - that their inability to persuade Laplace and Lagrange ... provides a good index of the originality of Fourier's views.

The second objection was made by Biot against Fourier's derivation of the equations of transfer of heat. Fourier had not made reference to Biot's 1804 paper on this topic but Biot's paper is certainly incorrect. Laplace, and later Poisson, had similar objections.

The Institute set as a prize competition subject the propagation of heat in solid bodies for the 1811 mathematics prize. Fourier submitted his 1807 memoir together with additional work on the cooling of infinite solids and terrestrial and radiant heat. Only one other entry was received and the committee set up to decide on the award of the prize, Lagrange, Laplace, Malus, Haüy and Legendre, awarded Fourier the prize. The report was not however completely favourable and states:-

... the manner in which the author arrives at these equations is not exempt of difficulties and that his analysis to integrate them still leaves something to be desired on the score of generality and even rigour.

With this rather mixed report there was no move in Paris to publish Fourier's work.

When Napoleon was defeated and on his way to exile in Elba, his route should have been through Grenoble. Fourier managed to avoid this difficult confrontation by sending word that it would be dangerous for Napoleon. When he learnt of Napoleon's escape from Elba and that he was marching towards Grenoble with an army, Fourier was extremely worried. He tried to persuade the people of Grenoble to oppose Napoleon and give their allegiance to the King. However as Napoleon marched into the town by one gate Fourier left in haste by another.

Napoleon was angry with Fourier whom he had hoped would welcome his return. Fourier was able to talk his way into favour with both sides and Napoleon made him Prefect of the Rhône. However Fourier soon resigned on receiving orders, possibly from Carnot, that the was to remove all administrators with royalist sympathies. He could not have completely fallen out with Napoleon and Carnot, however, for on 10 June 1815, Napoleon awarded him a pension of 6000 francs, payable from 1 July. However Napoleon was defeated on 1 July and Fourier did not receive any money. He returned to Paris.

Fourier was elected to the Académie des Sciences in 1817. In 1822 Delambre, who was the Secretary to the mathematical section of the Académie des Sciences, died and Fourier together with Biot and Arago applied for the post. After Arago withdrew the election gave Fourier an easy win. Shortly after Fourier became Secretary, the Académie published his prize winning essay Théorie analytique de la chaleur Ⓣ in 1822. This was not a piece of political manoeuvring by Fourier however since Delambre had arranged for the printing before he died.

During Fourier's eight last years in Paris he resumed his mathematical researches and published a number of papers, some in pure mathematics while some were on applied mathematical topics. His life was not without problems however since his theory of heat still provoked controversy. Biot claimed priority over Fourier, a claim which Fourier had little difficulty showing to be false. Poisson, however, attacked both Fourier's mathematical techniques and also claimed to have an alternative theory. Fourier wrote Historical Précis Ⓣ as a reply to these claims but, although the work was shown to various mathematicians, it was never published.

Fourier's views on the claims of Biot and Poisson are given in the following, see [4]:-

Having contested the various results [Biot and Poisson] now recognise that they are exact but they protest that they have invented another method of expounding them and that this method is excellent and the true one. If they had illuminated this branch of physics by important and general views and had greatly perfected the analysis of partial differential equations, if they had established a principal element of the theory of heat by fine experiments ... they would have the right to judge my work and to correct it. I would submit with much pleasure .. But 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.

Fourier's work provided the impetus for later work on trigonometric series and the theory of functions of a real variable.