奥古斯丁·路易·柯西Augustin-Louis Cauchy

当奥古斯丁·路易·柯西还是幼童时，由于法国大革命前后的政治事件，巴黎是一个难以居住的地方。他四岁时，父亲因担心在巴黎有生命危险，将全家搬到了阿尔克伊。那里的生活很艰难，他在一封信中写道[4]：-

我们从来没有超过半磅的面包——有时甚至更少。我们用分配到的少量硬饼干和大米来补充。

他们很快回到了巴黎，柯西的父亲积极参与了年幼的柯西的教育。皮埃尔·西蒙·拉普拉斯和约瑟夫·拉格朗日是柯西家的访客，尤其是约瑟夫·拉格朗日似乎对年幼的柯西的数学教育产生了兴趣。约瑟夫·拉格朗日建议柯西的父亲，他的儿子在开始认真研究数学之前，应该在语言方面打下良好的基础。1802年，柯西进入先贤祠中央学校，在那里学习了两年古典语言。

从1804年起，柯西参加了数学课程，并于1805年参加了巴黎理工学院的入学考试。他由让-巴蒂斯特·毕奥考核，名列第二。在巴黎理工学院，他参加了西尔维斯特·佛朗索瓦·拉克鲁瓦、de Prony和Hachette的课程，而他的分析导师是安德烈-马里·安培。1807年，他从巴黎理工学院毕业，进入了工程学校巴黎路桥学校。他是一名优秀的学生，由于他的实际工作，他被分配到Ourcq运河项目，在Pierre Girard手下工作。

1810年，柯西在瑟堡找到了第一份工作，为拿破仑入侵英国的舰队修建港口设施。他随身带了一本皮埃尔·西蒙·拉普拉斯的Mécanique CélesteⓉ（天体力学）和一本约瑟夫·拉格朗日的Théorie des FonctionsⓉ（函数论）。那段时间柯西很忙，他在写给家里的信中谈到自己的日常职责时说[4]：-

我每天早上四点起床，从那时起就一直忙个不停。……我并不厌倦工作，相反，工作使我精力充沛，我身体非常健康……

柯西是一位虔诚的天主教徒，他对宗教的态度已经给他带来了麻烦。在1810年写给母亲的一封信中，他说：-

所以他们声称，我的虔诚使我变得骄傲、自大和自恋。……现在在宗教方面没有人来打扰我，也没有人再向我提起它了……

除了繁重的工作量之外，柯西还从事数学研究，他在1811年证明了凸多面体的角由其面决定。他随后提交了关于这个主题的第一篇论文，在阿德里安-马里·勒让德和艾蒂安-路易·马吕斯的鼓励下，他又在1812年提交了一篇关于多边形和多面体的论文。柯西觉得，如果要在数学研究上给人留下印象，就必须回到巴黎。1812年9月，他在生病后回到了巴黎。看来这场病不是身体上的，而很可能是心理性质的，导致了严重的抑郁。

回到Paris Cauchy后，他研究了对称函数，并于1812年11月提交了一篇关于这个主题的论文。这篇论文于1815年发表在《巴黎理工学院学报》上。然而，他本应在1813年2月恢复健康后返回瑟堡，这与他的数学抱负不符。他向de Prony提出的在巴黎路桥学校担任副教授的请求被拒绝了，但他被允许继续作为工程师参与Ourcq运河项目，而不是返回瑟堡。Pierre Girard显然对他之前在这个项目上的工作感到满意，并支持这一变动。

学术生涯是柯西想要的，他申请了经度局的一个职位。他没有得到这个职位，阿德里安-马里·勒让德被任命了。他也未能被任命到研究所的几何部门，该职位由路易·普安索获得。柯西获得了进一步的病假，带薪休假九个月，然后政治事件阻止了Ourcq运河的工作，因此柯西能够完全投身于研究几年。

其他职位也空了出来，但1814年的一个职位给了安德烈-马里·安培，而拿破仑·波拿巴辞职后出现的科学院力学职位给了莫拉尔。在这次最后的选举中，柯西没有获得53票中的任何一票。他的数学产出依然强劲，1814年他发表了关于定积分的论文，这篇论文后来成为他复变函数理论的基础。

1815年，柯西在巴黎综合理工学院的力学讲席竞争中输给了雅可·比内，但随后被任命为该学院的分析学助理教授。他负责二年级的课程。1816年，他凭借一篇关于波动的著作赢得了法国科学院的大奖。然而，当他向科学院提交一篇论文，解决了皮埃尔·德·费马就多边形数向马兰·梅森提出的一个主张时，他才真正获得了声誉。此时政治帮助柯西进入了科学院，当时拉扎尔·卡诺和加斯帕尔·蒙日失宠于政治并被解职，柯西填补了其中一个位置。

1817年，当让-巴蒂斯特·毕奥离开巴黎前往苏格兰设得兰群岛探险时，柯西填补了他在法兰西公学院的职位。在那里，他讲授了他早先发现但未发表的积分方法。除了对积分的严格定义外，柯西还是第一个对无穷级数收敛条件进行严格研究的人。他1821年的教材Cours d'analyseⓉ（《分析教程》）是为巴黎综合理工学院的学生编写的，旨在尽可能严格地发展微积分的基本定理。他于1826年在Sur un nouveau genre de calcul analogue au calcul infinitésimalⓉ（《关于一种类似于无穷小计算的新计算》）中开始了对留数计算的研究，而1829年在Leçons sur le Calcul DifférentielⓉ（《微分学讲义》）中，他首次定义了复变量的复函数。

柯西与其他科学家的关系并不特别好。他坚定的天主教观点使他站在耶稣会士一边反对Académie des Sciences。他会把宗教带入自己的科学工作中，例如他在1824年作关于光的理论的报告时就攻击作者，因为该作者认为艾萨克·牛顿不相信人有灵魂。一位记者这样描述他：-

……看到一位似乎履行着向异教徒传教的传教士般可敬职责的院士，确实是一件稀奇的事。

关于柯西如何对待同事的一个例子，由让-维克托·彭赛列给出，其关于射影几何的工作在1820年受到了柯西的批评：-

……我设法在我的过于严厉的评判者的住所接近他……就在他正要离开的时候……在这段非常短暂而迅速的散步中，我很快察觉到，我丝毫没有赢得他作为科学家的好感或尊重……没等我说出任何别的话，他就突然走开了，让我参考他即将出版的《⟦E1⟧》，据他说，在那里‘这个问题将得到非常恰当的探讨’。

在这一时期，他对待埃瓦里斯特·伽罗瓦和尼尔斯·阿贝尔的方式同样是不幸的。尼尔斯·阿贝尔于1826年访问了该研究所，他这样写道：-

柯西疯了，对他无能为力，尽管眼下他是唯一知道数学应该如何做的人。

⟦N2⟧在[4]中说：-

当尼尔斯·阿贝尔于1829年4月6日过早离世时，柯西仍未就1826年的论文提交报告，尽管阿德里安-马里·勒让德多次提出抗议。他最终于1829年6月29日提交的报告仓促、恶劣且肤浅，既不配他自身的才华，也不配他所评判的研究的真正重要性。

到1830年，巴黎的政治事件和多年的辛勤工作已使他身心俱疲，柯西决定休息一段时间。七月革命后，他于1830年9月离开巴黎，在瑞士短暂停留。在那里，他热情地协助筹建瑞士科学院，但随着该计划卷入政治事件，最终失败了。

法国的政治事件意味着柯西现在必须宣誓效忠新政权，当他未能返回巴黎宣誓时，他失去了在那里的一切职位。1831年，柯西前往都灵，在那里待了一段时间后，他接受了皮埃蒙特国王提供的理论物理讲席。他从1832年起在都灵任教。费德里科·路易吉在都灵参加了这些课程，并写道这些课程[4]：-

非常混乱，突然从一个想法跳到另一个想法，从一个公式跳到下一个公式，没有试图在它们之间建立联系。他的讲解是模糊的云团，不时被纯粹天才的闪光所照亮。……在我注册的三十人中，我是唯一一个坚持到底的。

1833年，柯西从都灵前往布拉格，以追随查理十世并担任其孙子的导师。然而，他在教导王子方面并不十分成功，如下描述所示：-

……考试……每周六进行。……当柯西就一个画法几何问题提问时，王子感到困惑且犹豫不决。……还有关于物理和化学的材料。与数学一样，王子对这些科目表现出极少的兴趣。柯西变得恼怒，大声尖叫和喊叫。王后有时会安抚他，微笑着对他说：‘太大声了，别那么大声。’

在布拉格期间，柯西于1834年应伯纳德·波尔查诺之请与伯纳德·波尔查诺会面一次。在[16]和[18]中讨论了柯西的连续性定义在多大程度上归功于伯纳德·波尔查诺，汉斯·弗洛伊登萨在[18]中认为柯西的定义形成于伯纳德·波尔查诺之前的观点似乎更有说服力。

柯西于1838年返回巴黎，并重新获得了他在科学院的职位，但没有恢复他的教学职位，因为他拒绝宣誓效忠。De Prony于1839年去世，他在经度局的职位空缺。柯西得到了让-巴蒂斯特·毕奥和弗朗索瓦·阿拉戈的大力支持，但西莫恩·德尼·泊松强烈反对他。柯西当选，但在拒绝宣誓后，未被任命，不能参加会议或领取薪水。

1843年西尔维斯特·佛朗索瓦·拉克鲁瓦去世，柯西成为法兰西公学院其数学讲席的候选人。约瑟夫·刘维尔和Libri也是候选人。柯西本应凭其数学才能轻松获任，但他的政治和宗教活动，例如支持耶稣会，成为关键因素。Libri被选中，他显然是三人中数学上最弱的，而约瑟夫·刘维尔在次日写道，他：-

作为一个人和一名数学家，因昨天在法兰西公学院发生的事而深感屈辱。

在此期间，柯西的数学产出少于他自我流放之前的时期。他在微分方程及其在数学物理中的应用方面做了重要工作。他还撰写了数学天文学方面的著作，主要是因为他在经度局职位的候选资格。1840年至1847年间出版的4卷本Exercices d'analyse et de physique mathématique Ⓣ（分析与数学物理练习）被证明极其重要。

1848年路易·菲利普被推翻后，柯西重新获得了他的大学职位。然而，他没有改变自己的观点，继续给同事制造问题。Libri，曾以上述政治方式被任命，辞去了他的讲席并逃离法国。部分原因一定是他即将因偷窃珍贵书籍而被起诉。约瑟夫·刘维尔和柯西在1850年再次成为该讲席的候选人，就像他们在1843年一样。经过一场势均力敌的选举，约瑟夫·刘维尔被任命。随后试图推翻这一决定的尝试导致约瑟夫·刘维尔和柯西之间的关系非常糟糕。

另一次相当愚蠢的争论，这次是与让-马里·迪阿梅尔的争论，给柯西生命的最后几年蒙上了阴影。这场争论是关于非弹性碰撞一个结果的优先权主张。让-马里·迪阿梅尔与柯西关于他在1832年首次给出这些结果的主张进行了争论。让-维克托·彭赛列提到了他自己1826年关于该主题的工作，柯西被证明是错误的。然而柯西从不承认自己错了。Valson在[7]中写道：-

……这场争论给他生命的最后日子带来了一种基本的悲伤和苦涩，只有他的朋友们才知晓……

同样在7中，给出了柯西的女儿描述他去世的一封信：-

直到凌晨3点30分，他一直保持完全清醒，心智能力完全在掌控之中，我的父亲突然念出了耶稣、玛利亚和约瑟夫的神圣名字。他第一次似乎意识到了自己病情的严重性。大约四点钟时，他的灵魂归于上帝。他如此平静地面对死亡，让我们为自己的悲伤感到羞愧。

数学中有许多术语以柯西的名字命名：- 复函数论中的柯西积分定理，偏微分方程解的柯西-柯瓦列夫斯卡娅存在定理，柯西-波恩哈德·黎曼方程和柯西序列。他发表了789篇数学论文，这是一项令人难以置信的成就。这一成就在4中总结如下：-

……如此巨大的科学创造力简直令人震惊，因为它呈现了对当时已知数学所有领域的研究……尽管柯西的科学著作规模庞大且具有丰富的多面性，但它们拥有一个明确的统一主题，一种隐秘的完整性。……柯西的创造天才不仅在他关于实分析和复分析基础的工作中得到了广泛体现——他的名字与这些领域密不可分——而且在许多其他领域也是如此。具体而言，在这方面，我们应该提到他对数学物理发展和理论力学的主要贡献……我们提到……他的两个弹性理论和他对光理论的研究，这些研究要求他发展出全新的数学技术，如约瑟夫·傅里叶变换、matrices的对角化以及留数演算。

他的文集，Oeuvres complètes d'Augustin Cauchy Ⓣ（Augustin Cauchy全集）（1882-1970），共出版了27卷。

Paris was a difficult place to live in when Augustin-Louis Cauchy was a young child due to the political events surrounding the French Revolution. When he was four years old his father, fearing for his life in Paris, moved his family to Arcueil. There things were hard and he wrote in a letter [4]:-

We never have more than a half pound of bread - and sometimes not even that. This we supplement with the little supply of hard crackers and rice that we are allotted.

They soon returned to Paris and Cauchy's father was active in the education of young Augustin-Louis. Laplace and Lagrange were visitors at the Cauchy family home and Lagrange in particular seems to have taken an interest in young Cauchy's mathematical education. Lagrange advised Cauchy's father that his son should obtain a good grounding in languages before starting a serious study of mathematics. In 1802 Augustin-Louis entered the École Centrale du Panthéon where he spent two years studying classical languages.

From 1804 Cauchy attended classes in mathematics and he took the entrance examination for the École Polytechnique in 1805. He was examined by Biot and placed second. At the École Polytechnique he attended courses by Lacroix, de Prony and Hachette while his analysis tutor was Ampère. In 1807 he graduated from the École Polytechnique and entered the engineering school École des Ponts et Chaussées. He was an outstanding student and for his practical work he was assigned to the Ourcq Canal project where he worked under Pierre Girard.

In 1810 Cauchy took up his first job in Cherbourg to work on port facilities for Napoleon's English invasion fleet. He took a copy of Laplace's Mécanique Céleste Ⓣ and one of Lagrange's Théorie des Fonctions Ⓣ with him. It was a busy time for Cauchy, writing home about his daily duties he said [4]:-

I get up at four o'clock each morning and I am busy from then on. ... I do not get tired of working, on the contrary, it invigorates me and I am in perfect health...

Cauchy was a devout Catholic and his attitude to his religion was already causing problems for him. In a letter written to his mother in 1810 he says:-

So they are claiming that my devotion is causing me to become proud, arrogant and self-infatuated. ... I am now left alone about religion and nobody mentions it to me anymore...

In addition to his heavy workload Cauchy undertook mathematical researches and he proved in 1811 that the angles of a convex polyhedron are determined by its faces. He submitted his first paper on this topic then, encouraged by Legendre and Malus, he submitted a further paper on polygons and polyhedra in 1812. Cauchy felt that he had to return to Paris if he was to make an impression with mathematical research. In September of 1812 he returned to Paris after becoming ill. It appears that the illness was not a physical one and was probably of a psychological nature resulting in severe depression.

Back in Paris Cauchy investigated symmetric functions and submitted a memoir on this topic in November 1812. This was published in the Journal of the École Polytechnique in 1815. However he was supposed to return to Cherbourg in February 1813 when he had recovered his health and this did not fit with his mathematical ambitions. His request to de Prony for an associate professorship at the École des Ponts et Chaussées was turned down but he was allowed to continue as an engineer on the Ourcq Canal project rather than return to Cherbourg. Pierre Girard was clearly pleased with his previous work on this project and supported the move.

An academic career was what Cauchy wanted and he applied for a post in the Bureau des Longitudes. He failed to obtain this post, Legendre being appointed. He also failed to be appointed to the geometry section of the Institute, the position going to Poinsot. Cauchy obtained further sick leave, having unpaid leave for nine months, then political events prevented work on the Ourcq Canal so Cauchy was able to devote himself entirely to research for a couple of years.

Other posts became vacant but one in 1814 went to Ampère and a mechanics vacancy at the Institute, which had occurred when Napoleon Bonaparte resigned, went to Molard. In this last election Cauchy did not receive a single one of the 53 votes cast. His mathematical output remained strong and in 1814 he published the memoir on definite integrals that later became the basis of his theory of complex functions.

In 1815 Cauchy lost out to Binet for a mechanics chair at the École Polytechnique, but then was appointed assistant professor of analysis there. He was responsible for the second year course. In 1816 he won the Grand Prix of the French Academy of Sciences for a work on waves. He achieved real fame however when he submitted a paper to the Institute solving one of Fermat's claims on polygonal numbers made to Mersenne. Politics now helped Cauchy into the Academy of Sciences when Carnot and Monge fell from political favour and were dismissed and Cauchy filled one of the two places.

In 1817 when Biot left Paris for an expedition to the Shetland Islands in Scotland Cauchy filled his post at the Collège de France. There he lectured on methods of integration which he had discovered, but not published, earlier. Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral. His text Cours d'analyse Ⓣ in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal Ⓣ while in 1829 in Leçons sur le Calcul Différentiel Ⓣ he defined for the first time a complex function of a complex variable.

Cauchy did not have particularly good relations with other scientists. His staunchly Catholic views had him involved on the side of the Jesuits against the Académie des Sciences. He would bring religion into his scientific work as for example he did on giving a report on the theory of light in 1824 when he attacked the author for his view that Newton had not believed that people had souls. He was described by a journalist who said:-

... it is certain a curious thing to see an academician who seemed to fulfil the respectable functions of a missionary preaching to the heathens.

An example of how Cauchy treated colleagues is given by Poncelet whose work on projective geometry had, in 1820, been criticised by Cauchy:-

... I managed to approach my too rigid judge at his residence ... just as he was leaving ... During this very short and very rapid walk, I quickly perceived that I had in no way earned his regards or his respect as a scientist ... without allowing me to say anything else, he abruptly walked off, referring me to the forthcoming publication of his Leçons à 'École Polytechnique where, according to him, 'the question would be very properly explored'.

Again his treatment of Galois and Abel during this period was unfortunate. Abel, who visited the Institute in 1826, wrote of him:-

Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.

Belhoste in [4] says:-

When Abel's untimely death occurred on April 6, 1829, Cauchy still had not given a report on the 1826 paper, in spite of several protests from Legendre. The report he finally did give, on June 29, 1829, was hasty, nasty, and superficial, unworthy of both his own brilliance and the real importance of the study he had judged.

By 1830 the political events in Paris and the years of hard work had taken their toll and Cauchy decided to take a break. He left Paris in September 1830, after the revolution of July, and spent a short time in Switzerland. There he was an enthusiastic helper in setting up the Académie Helvétique but this project collapsed as it became caught up in political events.

Political events in France meant that Cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to Paris to do so he lost all his positions there. In 1831 Cauchy went to Turin and after some time there he accepted an offer from the King of Piedmont of a chair of theoretical physics. He taught in Turin from 1832. Menabrea attended these courses in Turin and wrote that the courses [4]:-

were very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius. ... of the thirty who enrolled with me, I was the only one to see it through.

In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor his grandson. However he was not very successful in teaching the prince as this description shows:-

... exams .. were given each Saturday. ... When questioned by Cauchy on a problem in descriptive geometry, the prince was confused and hesitant. ... There was also material on physics and chemistry. As with mathematics, the prince showed very little interest in these subjects. Cauchy became annoyed and screamed and yelled. The queen sometimes said to him, soothingly, smilingly, 'too loud, not so loud'.

While in Prague Cauchy had one meeting with Bolzano, at Bolzano's request, in 1834. In [16] and [18] there are discussions on how much Cauchy's definition of continuity is due to Bolzano, Freudenthal's view in [18] that Cauchy's definition was formed before Bolzano's seems the more convincing.

Cauchy returned to Paris in 1838 and regained his position at the Academy but not his teaching positions because he had refused to take an oath of allegiance. De Prony died in 1839 and his position at the Bureau des Longitudes became vacant. Cauchy was strongly supported by Biot and Arago but Poisson strongly opposed him. Cauchy was elected but, after refusing to swear the oath, was not appointed and could not attend meetings or receive a salary.

In 1843 Lacroix died and Cauchy became a candidate for his mathematics chair at the Collège de France. Liouville and Libri were also candidates. Cauchy should have easily been appointed on his mathematical abilities but his political and religious activities, such as support for the Jesuits, became crucial factors. Libri was chosen, clearly by far the weakest of the three mathematically, and Liouville wrote the following day that he was:-

deeply humiliated as a man and as a mathematician by what took place yesterday at the Collège de France.

During this period Cauchy's mathematical output was less than in the period before his self-imposed exile. He did important work on differential equations and applications to mathematical physics. He also wrote on mathematical astronomy, mainly because of his candidacy for positions at the Bureau des Longitudes. The 4-volume text Exercices d'analyse et de physique mathématique Ⓣ published between 1840 and 1847 proved extremely important.

When Louis Philippe was overthrown in 1848 Cauchy regained his university positions. However he did not change his views and continued to give his colleagues problems. Libri, who had been appointed in the political way described above, resigned his chair and fled from France. Partly this must have been because he was about to be prosecuted for stealing valuable books. Liouville and Cauchy were candidates for the chair again in 1850 as they had been in 1843. After a close run election Liouville was appointed. Subsequent attempts to reverse this decision led to very bad relations between Liouville and Cauchy.

Another, rather silly, dispute this time with Duhamel clouded the last few years of Cauchy's life. This dispute was over a priority claim regarding a result on inelastic shocks. Duhamel argued with Cauchy's claim to have been the first to give the results in 1832. Poncelet referred to his own work of 1826 on the subject and Cauchy was shown to be wrong. However Cauchy was never one to admit he was wrong. Valson writes in [7]:-

...the dispute gave the final days of his life a basic sadness and bitterness that only his friends were aware of...

Also in [7] a letter by Cauchy's daughter describing his death is given:-

Having remained fully alert, in complete control of his mental powers, until 3.30 a.m.. my father suddenly uttered the blessed names of Jesus, Mary and Joseph. For the first time, he seemed to be aware of the gravity of his condition. At about four o'clock, his soul went to God. He met his death with such calm that made us ashamed of our unhappiness.

Numerous terms in mathematics bear Cauchy's name:- the Cauchy integral theorem, in the theory of complex functions, the Cauchy-Kovalevskaya existence theorem for the solution of partial differential equations, the Cauchy-Riemann equations and Cauchy sequences. He produced 789 mathematics papers, an incredible achievement. This achievement is summed up in [4] as follows:-

... such an enormous scientific creativity is nothing less than staggering, for it presents research on all the then-known areas of mathematics ... in spite of its vastness and rich multifaceted character, Cauchy's scientific works possess a definite unifying theme, a secret wholeness. ... Cauchy's creative genius found broad expression not only in his work on the foundations of real and complex analysis, areas to which his name is inextricably linked, but also in many other fields. Specifically, in this connection, we should mention his major contributions to the development of mathematical physics and to theoretical mechanics... we mention ... his two theories of elasticity and his investigations on the theory of light, research which required that he develop whole new mathematical techniques such as Fourier transforms, diagonalisation of matrices, and the calculus of residues.

His collected works, Oeuvres complètes d'Augustin Cauchy Ⓣ (1882-1970), were published in 27 volumes.