夏尔·埃尔米特Charles Hermite

夏尔·埃尔米特的父亲是Ferdinand Hermite，母亲是Madeleine Lallemand。Ferdinand Hermite是一名受过训练的工程师，他在Dieuse附近的一个盐矿以这一身份工作。与Madeleine结婚后，他加入了她的家族所从事的布商行业。然而，他是一个有艺术气质的人，一直想以艺术为职业。他让妻子照看布商生意，自己则从事艺术。埃尔米特是父母七个孩子中的第六个，当他大约七岁时，父母离开Dieuse，搬到南锡，因为生意已迁往那里。

教育并不是埃尔米特父母的首要大事，但尽管他们对子女的教育没有太多个人兴趣，他们还是为子女提供了良好的学校教育。埃尔米特让父母有些担忧，因为他的右脚有缺陷，这意味着他行动困难。显然，这会给他的求职带来问题。然而他性格开朗，以愉快的微笑面对自己的残疾。

埃尔米特曾就读于南锡公学，随后前往巴黎，进入亨利公学。1840至1841年间，他在路易大帝公学学习，而大约十五年前埃瓦里斯特·伽罗瓦也曾在此就读。事实上，在那里教他数学的是Louis Richard，此人还曾教过埃瓦里斯特·伽罗瓦。在某些方面，埃尔米特与埃瓦里斯特·伽罗瓦相似，因为他更愿意阅读莱昂哈德·欧拉、卡尔·弗里德里希·高斯和约瑟夫·拉格朗日的论文，而不是为正式考试而学习。

埃尔米特忽视了他本应专注的学业，但他展现出了非凡的研究能力，在路易大帝学院期间发表了两篇论文。也像埃瓦里斯特·伽罗瓦一样，他被求解代数方程的问题所吸引，其中一篇论文试图证明quintic不能用根式求解。尽管在同一所学校，他对埃瓦里斯特·伽罗瓦的贡献并不熟悉，这一点也不奇怪，因为当时的数学界完全不知道这些贡献。然而，他本应合理地知道保罗·鲁菲尼和尼尔斯·阿贝尔对这个问题的贡献，但显然他并不知道。

再次像埃瓦里斯特·伽罗瓦一样，埃尔米特想就读于巴黎综合理工学院，他花了一年时间准备考试。1841-42年，他由乌惹内·查尔斯·卡塔兰辅导，埃尔米特 certainly fared better than 埃瓦里斯特·伽罗瓦 had done for he passed. 然而这并不是一次光荣的通过，因为他在排名名单中只获得了第六十八名。在巴黎综合理工学院学习一年后，埃尔米特因残疾被拒绝继续学业。显然这是一个不公平的决定，一些重要人物准备为他争取，为他争取继续作为巴黎综合理工学院学生的权利。决定被推翻，以便他可以继续学业，但施加了严格的条件。埃尔米特认为这些条件不可接受，并决定不从巴黎综合理工学院毕业。

埃尔米特此时与重要数学家结交，并经常拜访约瑟夫约瑟·伯特兰。就个人而言，这非常重要，因为他将娶约瑟夫约瑟·伯特兰的妹妹。从数学角度来看更重要的是，他开始与卡尔·古斯塔夫·雅各布·雅可比通信，尽管在正规教育中并不出色，但他已经在进行研究，跻身世界级顶尖数学家之列。他与卡尔·古斯塔夫·雅各布·雅可比交换的信件表明，埃尔米特已经发现了一些由theta函数满足的微分方程，并且他正在使用Fourier series来研究它们。他已经找到了用theta函数表示的方程的一般解。埃尔米特可能还是本科生，但很可能他大约1843年的想法帮助约瑟夫·刘维尔得出了1844年的重要结果，其中包括现在被称为约瑟夫·刘维尔定理的结果。

在花了五年时间攻读学位后，他参加并通过了业士学位和学士学位的考试，并于1847年获得这些学位。第二年，他被任命到巴黎综合理工学院，这个机构大约四年前曾试图阻止他继续学业；他被任命为复习教师和入学考试考官。

埃尔米特在数论、代数、正交多项式和椭圆函数方面做出了重要贡献。在被任命到巴黎综合理工学院后的十年里，他发现了他最重要的数学成果。1848年，他证明双周期函数可以表示为周期整函数的商。1849年，埃尔米特向Académie des Sciences提交了一篇论文，将奥古斯丁·路易·柯西的留数技巧应用于双周期函数。雅克·夏尔·弗朗索瓦·施图姆和奥古斯丁·路易·柯西在1851年对这篇论文给出了好评，但与约瑟夫·刘维尔的优先权争议似乎阻止了它的发表。

埃尔米特做出重要贡献的另一个主题是二次型理论。这引导他研究不变量理论，并发现了一个与二元形式有关的互反律。凭借对二次型和不变式论的理解，他在1855年创立了变换理论。他在这一主题上的成果提供了数论、theta函数和abelian functions变换之间的联系。

1856年7月14日，埃尔米特当选为Académie des Sciences院士。然而，尽管取得了这一成就，1856年对埃尔米特来说是糟糕的一年，因为他感染了天花。正是奥古斯丁·路易·柯西以其坚定的宗教信仰帮助埃尔米特度过了危机。这对埃尔米特产生了深远影响，他在奥古斯丁·路易·柯西的影响下皈依了罗马天主教。奥古斯丁·路易·柯西也是一位非常坚定的保皇派，埃尔米特受其影响也成为了保皇派。我们在本文前面曾与埃瓦里斯特·伽罗瓦作过比较，但持有保皇派观点的埃尔米特现在完全反对坚定的共和派埃瓦里斯特·伽罗瓦所持有的观点。

我们必须提到的埃尔米特的下一个数学成果是他理应闻名的一个成果。虽然五次代数方程不能solved in radicals——这一结果由保罗·鲁菲尼和尼尔斯·阿贝尔证明——但埃尔米特在1858年表明，五次代数方程可以用椭圆函数求解。他将这些结果应用于数论，特别是二次型的类数关系。

1862年，埃尔米特被任命为巴黎综合理工学院的专任讲师，这一职位是专为他设立的。次年，他成为那里的考官。1869年，他成为教授，接替让-马里·迪阿梅尔担任巴黎综合理工学院和索邦大学的分析学教授。埃尔米特于1876年辞去巴黎综合理工学院的讲席，但继续担任索邦大学的讲席，直到1897年退休。在1890年代，埃尔米特对下一代数学家发现的新结果变得不那么感兴趣了。

1870年代，埃尔米特回到了他职业生涯早期感兴趣的问题，如关于逼近和插值的问题。1873年，埃尔米特发表了第一个证明$e$是超越的数的证明。这是另一个他理应闻名的成果。使用与埃尔米特类似的方法，费迪南德·冯·林德曼在1882年证明了π也是超越数。许多科学史家遗憾地表示，埃尔米特尽管做了大部分艰苦工作，却未能用它来证明那个本会给他带来数学界之外声誉的结果。埃尔米特现在最为人所知的是若干以他命名的数学对象：埃尔米特多项式、埃尔米特微分方程、埃尔米特插值公式和埃尔米特matrices。

埃尔米特认为数学的某些领域比其他领域有趣得多。雅克·阿达马与他的老师埃尔米特不同，在数学的所有领域都有工作，他谈到埃尔米特对几何学的厌恶：-

[埃尔米特]对几何学有一种积极的憎恨，并且曾奇怪地责备我写了一篇几何学论文。

埃尔米特最大的爱好是分析，毫不奇怪，他对卡尔·魏尔斯特拉斯极为尊重。当约斯塔·米塔格-莱弗勒来到巴黎跟随他学习时，埃尔米特热情地迎接他，但说道：-

先生，您弄错了，您应该跟随卡尔·魏尔斯特拉斯在柏林学习。他是我们所有人的导师。

儒勒·昂利·庞加莱几乎可以肯定是埃尔米特最著名的学生。他曾提出，埃尔米特的思维并非以逻辑方式推进。他写道：-

但把埃尔米特称为逻辑学家！在我看来，没有什么比这更违背事实了。方法似乎总是以某种神秘的方式在他头脑中诞生。

雅克·阿达马像儒勒·昂利·庞加莱一样，对数学被发现的方式非常感兴趣。他还就埃尔米特做出发现的方式说了这样的话：-

埃尔米特过去常常指出，[生物学]即使对数学家来说也可能是一门极其有用的学问，因为在这两类研究中，过程之间可能出现隐藏的、最终富有成果的类比。

雅克·阿达马对作为教师的埃尔米特极为尊敬。他说：-

我认为，那些从未听过他讲课的人无法体会埃尔米特的教学有多么宏大，洋溢着对科学的热情，这种热情似乎在他的声音中获得了生命，而他从不吝于将这种美传达给我们，因为他自己从存在的深处如此深切地感受到了它。

[埃尔米特]给我们留下了深刻的印象，不仅以他的方法和卡尔·魏尔斯特拉斯的方法，还以他对科学的热情与热爱；在我们简短而富有成果的交谈中，埃尔米特喜欢对我说这样的话：“偏离天意所指引的道路者必将毁灭。”这是一个极其虔诚的人说的话，但像我这样的无神论者也能很好地理解，尤其是当他另一些时候又补充道：“在数学中，我们的角色与其说是主人，不如说是仆人。”不言而喻，随着岁月流逝和我的科学工作展开，我逐渐越来越深刻地理解了他这些话的贴切与深意。

Cross在评论[10]时写道，该书收录了埃尔米特致约斯塔·米塔格-莱弗勒的125封信：-

于是，19世纪下半叶巴黎和法国数学界最引人注目、最具影响力的人物之一就此展现出来，甚至可以说，在他发表著作的1842—1901年这一时期，他是核心人物。从文本中散发出来的是[埃尔米特]的谦逊、他的天主教信仰、他对（人数众多的）家庭的关切、他愿意为那些他看出其价值的同事而斗争，以及他对家庭、价值和原则而非单纯影响力的忠诚。

在家庭生活方面，埃尔米特娶了Louise Bertrand，即约瑟·伯特兰的姐妹。他们的两个女儿之一嫁给了埃米尔·皮卡。迪尔克·扬·斯特勒伊克写道：-

埃尔米特与家人一起过着退隐的生活。他的工作时间用于数学研究和教学。他对数学的看法是柏拉图意义上的实在论：数学家像博物学家一样，发现一个外部世界，就他而言，是一个理念的世界。因此，埃尔米特不喜欢格奥尔格·康托尔的世界，那是一个新数学世界被创造出来的世界。

Charles Hermite's father was Ferdinand Hermite and his mother was Madeleine Lallemand. Ferdinand Hermite was a trained engineer and he worked in this capacity in a salt mine near Dieuse. After he married Madeleine he joined in the draper's trade in which her family were involved. However he was an artistic man who always wanted to pursue art as a career. He had his wife look after the draper's business and he took up art. Charles was the sixth of his parents seven children and when he was about seven years old his parents left Dieuse and went to live in Nancy to where the business had moved.

Education was not a high priority for Charles's parents but despite not taking too much personal interest in their children's education, nevertheless they did provide them with good schooling. Charles was something of a worry to his parents for he had a defect in his right foot which meant that he moved around only with difficulty. It was clear that this would present him with problems in finding a career. However he had a happy disposition and bore his disability with a cheerful smile.

Charles attended the Collège de Nancy, then went to Paris where he attended the Collège Henri. In 1840-41 he studied at the Collège Louis-le-Grand where some fifteen years earlier Galois had studied. In fact he was taught mathematics there by Louis Richard who had taught Galois. In some ways Hermite was similar to Galois for he preferred to read papers by Euler, Gauss and Lagrange rather than work for his formal examinations.

If Hermite neglected the studies that he should have concentrated on, he was showing remarkable research ability publishing two papers while at Louis-le-Grand. Also like Galois he was attracted by the problem of solving algebraic equations and one of the two papers attempted to show that the quintic cannot be solved in radicals. That he was unfamiliar with Galois's contributions, despite being at the same school, is not at all surprising since the mathematical community were completely unaware of them at this time. However he might reasonably have known of the contributions of Ruffini and Abel to this question, but apparently he did not.

Again like Galois, Hermite wanted to study at the École Polytechnique and he took a year preparing for the examinations. He was tutored by Catalan in 1841-42 and certainly Hermite fared better than Galois had done for he passed. However it was not a glorious pass for he only attained sixty-eighth place in the ordered list. After one year at the École Polytechnique Hermite was refused the right to continue his studies because of his disability. Clearly this was an unfair decision and some important people were prepared to take up his case and fight for him to have the right to continue as a student at the École Polytechnique. The decision was reversed so that he could continue his studies but strict conditions were imposed. Hermite did not find these conditions acceptable and decided that he would not graduate from the École Polytechnique.

Hermite made friends with important mathematicians at this time and frequently visited Joseph Bertrand. On a personal note this was highly significant for he would marry Joseph Bertrand's sister. More significantly from a mathematical point of view he began corresponding with Jacobi and, despite not shining in his formal education, he was already producing research which was ranking as a leading world-class mathematician. The letters he exchanged with Jacobi show that Hermite had discovered some differential equations satisfied by theta-functions and he was using Fourier series to study them. He had found general solutions to the equations in terms of theta-functions. Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville's theorem.

After spending five years working towards his degree he took and passed the examinations for the baccalauréat and licence which he was awarded in 1847. In the following year he was appointed to the École Polytechnique, the institution which had tried to prevent him continuing his studies some four years earlier; he was appointed répétiteur and admissions examiner.

Hermite made important contributions to number theory and algebra, orthogonal polynomials, and elliptic functions. He discovered his most significant mathematical results over the ten years following his appointment to the École Polytechnique. In 1848 he proved that doubly periodic functions can be represented as quotients of periodic entire functions. In 1849 Hermite submitted a memoir to the Académie des Sciences which applied Cauchy's residue techniques to doubly periodic functions. Sturm and Cauchy gave a good report on this memoir in 1851 but a priority dispute with Liouville seems to have prevented its publication.

Another topic on which Hermite worked and made important contributions was the theory of quadratic forms. This led him to study invariant theory and he found a reciprocity law relating to binary forms. With his understanding of quadratic forms and invariant theory he created a theory of transformations in 1855. His results on this topic provided connections between number theory, theta functions, and the transformations of abelian functions.

On 14 July 1856 Hermite was elected to the Académie des Sciences. However, despite this achievement, 1856 was a bad year for Hermite for he contracted smallpox. It was Cauchy who, with his strong religious conviction, helped Hermite through the crisis. This had a profound effect on Hermite who, under Cauchy's influence, turned to the Roman Catholic religion. Cauchy was also a very staunch royalist and Hermite was influenced by him to also become a royalist. We made comparisons with Galois earlier on in this article, but with royalist views, Hermite was now completely opposed to the views which the staunch republican Galois had held.

The next mathematical result by Hermite which we must mention is one for which he is rightly famous. Although an algebraic equation of the fifth degree cannot be solved in radicals, a result which was proved by Ruffini and Abel, Hermite showed in 1858 that an algebraic equation of the fifth degree could be solved using elliptic functions. He applied these results to number theory, in particular to class number relations of quadratic forms.

In 1862 Hermite was appointed maître de conférence at the École Polytechnique, a position which had been specially created for him. In the following year he became an examiner there. The year 1869 saw him become a professor when he succeeded Duhamel as professor of analysis both at the École Polytechnique and at the Sorbonne. Hermite resigned his chair at the École Polytechnique in 1876 but continued to hold the chair at the Sorbonne until he retired in 1897. In the 1890s Hermite became much less interested in the new results found by the mathematicians of the next generation.

The 1870s saw Hermite return to problems which had interested him earlier in his career such as problems concerning approximation and interpolation. In 1873 Hermite published the first proof that $e$ is a transcendental number. This is another result for which he is rightly famous. Using method's similar to those of Hermite, Lindemann established in 1882 that π was also transcendental. Many historians of science regret that Hermite, despite doing most of the hard work, failed to use it to prove the result on which would have brought him fame outside the world of mathematics. Hermite is now best known for a number of mathematical entities that bear his name: Hermite polynomials, Hermite's differential equation, Hermite's formula of interpolation and Hermitian matrices.

For Hermite certain areas of mathematics were much more interesting than other areas. Hadamard, who unlike his teacher Hermite worked in all areas of mathematics, spoke of Hermite's dislike for geometry:-

[Hermite] had a kind of positive hatred of geometry and once curiously reproached me with having made a geometrical memoir.

Hermite's great love was for analysis and, not surprisingly, he had a great respect for Weierstrass. When Mittag-Leffler arrived in Paris to study with him, Hermite greeted him warmly but said:-

You have made a mistake, sir, you should follow Weierstrass's course in Berlin. He is the master of us all.

Poincaré is almost certainly the best known of Hermite's students. He once suggested that Hermite's mind did not proceed in logical fashion. He wrote:-

But to call Hermite a logician! Nothing can appear to me more contrary to the truth. Methods always seemed to be born in his mind in some mysterious way.

Hadamard like Poincaré was very interested in the way that mathematics was discovered. He also had this to say about the way that Hermite made his discoveries:-

Hermite used to observe [that biology] may be a most useful study even for mathematicians, as hidden and eventually fruitful analogies may appear between processes in both kinds of studies.

Hadamard had great respect for Hermite as a teacher. He said:-

I do not think that those who never listened to him can realise how magnificent Hermite's teaching was, overflowing with enthusiasm for science, which seemed to come to life in his voice and whose beauty he never failed to communicate to us, since he felt it so much himself to the very depth of his being.

[Hermite] was making a deep impression on us, not only with his methods and those of Weierstrass, but also with his enthusiasm and love of science; in our brief but fruitful conversations, Hermite loved to direct to me remarks such as: "He who strays from the paths traced by providence crashes." These were the words of a profoundly religious man, but an atheist like me understood them very well, especially when he added at other times: "In mathematics, our role is more of servant than of master." It goes without saying that gradually, as years and my scientific work unfolded, I came to understand more and more deeply the aptness and scope of his words.

Cross, reviewing [10] where 125 letters from Hermite to Mittag-Leffler are reproduced, writes:-

So there stands revealed one of the most engaging and influential men in Parisian and French mathematics in the second half of the 19th century, one might even say the central character for the period in which he published, 1842-1901. What radiates from the text is [Hermite's] humility, his Catholicism, his concern for his (very extended) family, his willingness to fight for colleagues whose merit he discerns, and his devotion to family, merit, and principle rather than simple influence.

In terms of his family life Hermite had married Louise Bertrand, Joseph Bertrand's sister. One of their two daughters married Émile Picard. Struik writes:-

Hermite lived a retired life, with his family. His working hours were devoted to mathematical research and teaching. His outlook on mathematics was realistic in the Platonic sense: a mathematician, like a naturalist, discovers an outside world, in his case a world of ideas. Hermite, therefore, disliked Cantor's world, in which a new mathematical world was created.