加布里尔·拉梅Gabriel Lamé

加布里尔·拉梅是巴黎综合理工学院的学生，1813年入学，1817年毕业。在这些本科岁月里，拉梅就已经在撰写研究论文，他于1816-17年在Gergonne's Journal上发表了他的第一篇论文Mémoire sur les intersections des lignes et des surfacesⓉ（关于线与面交点的回忆录）。从巴黎综合理工学院毕业后，拉梅在巴黎矿业学院学习工程学，1820年从那里毕业。在矿业学院期间，拉梅发表了他的第二部作品，这次是关于他发明的一种计算晶体面之间夹角的方法。

1820年，拉梅与他的同事埃米尔·克拉佩龙一起前往俄罗斯。我们应当为这一事件提供一些背景，因为从表面上看，这对两位年轻数学家来说似乎是一个相当奇怪的职业选择。亚历山大一世从1801年到1825年是俄罗斯的皇帝。法国大革命及其后法国发生的事件向亚历山大展示了科学知识及其在军事技术和工业发展中的应用的重要性。他明白，俄罗斯要强大就必须效仿。他面向欧洲和欧洲科学家，试图推行政策鼓励他们与俄罗斯科学家合作。他鼓励教师前往俄罗斯教授最新的科学理论，并在俄罗斯与欧洲之间建立科学联系。根据这一政策，俄罗斯政府向法国提出请求，法国回应派遣拉梅和埃米尔·克拉佩龙前往圣彼得堡。

拉梅被任命为圣彼得堡交通工程学院教授和工程师。起初，拉梅的处境相当困难，但后来他的访问证明是富有成效的。他讲授分析、物理、力学、化学和工程主题。在那里12年期间，他在俄语和法语期刊上发表了论文，其中一些是与埃米尔·克拉佩龙共同发表的。例如，他们在Journal des voies de communications、Journal du genie civil、Bulletin des sciences mathématiques、Receuil des savants etrangers以及Journal für die reine und angewandte Mathematik（奥古斯都·利奥波德·克雷勒的期刊）于1826年开始出版后在该刊上发表了论文。

在[6]中，叙述了拉梅在圣彼得堡期间发生的一段有趣插曲。它涉及拉梅试图传播奥古斯丁·路易·柯西的严格分析新思想。拉梅任教的研究所的一位教授写了一本书，其中包含布鲁克·泰勒定理的一个证明。拉梅撰写了一份手稿，使用奥古斯丁·路易·柯西的论证批评该证明。拉梅在圣彼得堡工作的另一个方面是他参与帮助制定围绕城市修建桥梁和道路的计划。此时，他更加意识到铁路发展的巨大潜力，这在他返回法国后将成为他极为感兴趣的话题。在此之前，他出席了1830年9月15日英国利物浦-曼彻斯特线开通仪式。

詹姆斯·布拉德利 [4] 对 拉梅 在俄罗斯的时期给出了更多细节。她在论文中总结道：-

……波旁复辟时期法国压抑的氛围使得出国工作对于研究和应用新思想显得更具吸引力。Lame 和 埃米尔·克拉佩龙 抓住了已经扎根俄罗斯的成功的法国工程师们向他们提供的机会，这些工程师带去了巴黎综合理工学院早年的精神。像 Betancourt 和 Bazaine 这样的重要工程师帮助他们在充满科学机遇的土地上追求自己的事业，在那里他们的意识形态信念通过与同胞的接触和讨论得到加强。

1832年，拉梅 回到巴黎，起初他加入了与 埃米尔·克拉佩龙 及另外两人共同组建的一家工程公司。仅仅几个月后，仍在1832年，拉梅 接受了巴黎综合理工学院的物理学讲席。然而，他并未将兴趣局限于教学和研究，因为他仍然是一名随时准备在该领域从事咨询工作的工程师。1836年，他被任命为矿业总工程师，还参与了从巴黎到凡尔赛的铁路以及从巴黎到圣 索菲·热尔曼 的铁路的建设，后者于1837年开通。

1843年，当Louis Puissant去世在几何学部门留下空缺时，拉梅当选为Académie des Sciences院士。次年，他离开了巴黎综合理工学院的物理学讲席，接受了索邦大学数学物理和probability的职位。1851年，他被任命为索邦大学数学物理和概率讲席。

他研究过各种各样的不同主题。他所承担的工程任务中的问题常常引导他去研究数学问题。例如，他对拱顶稳定性和悬索桥设计的研究使他致力于弹性理论。事实上，这并非一时的兴趣，因为拉梅对这一主题做出了重大贡献。另一个例子是他对热传导的研究，这引导他建立了一般曲线坐标理论。

曲线坐标在 拉梅 手中被证明是非常强大的工具。他用它们将 拉普拉斯方程 变换为椭球坐标，从而分离变量并求解所得方程。拉梅 职业生涯的标志是以相当合乎逻辑的方式从一个主题转向另一个主题，但他最终常常研究的是与最初问题相去甚远的题目。曲线坐标的情况就是如此，因为他被引导去研究方程

$(x/a)^{n} + (y/b)^{n} + (z/c)^{n} = 0$

他将其写成非齐次形式为

$(x/a)^{n} + (y/b)^{n} = 1$

其中，由$a = b$得$x^{n} + y^{n} = a^{n}$，因此他被引向费马大定理。尽管他基本上是一位应用数学家，拉梅通过解决$n = 7$的情形对该问题做出了实质性贡献。事实上，他曾一度相信自己已经解决了整个问题，但他忽略了复数某些子环中唯一因子分解的缺失。

他还在微分几何方面做了重要工作，并且在数论的另一项贡献中，他证明了欧几里得算法中的除法次数从不超过较小数位数的五倍。

正如我们上面所指出的，他研究工程数学和弹性理论，其中有两个弹性常数以他的名字命名。他研究了晶体材料中的扩散。

拉梅被许多人认为是当时法国领先的数学家，特别是卡尔·弗里德里希·高斯，他从不轻易给予赞扬，也持这一看法。相当奇怪的是，他在法国以外比在法国国内更受推崇，因为法国人似乎觉得他对数学家来说太实用，而对工程师来说又太理论化。他自己的看法是曲线坐标是他最重要的贡献，但数学史上有奇怪的转折，在拉梅引入它们之后不久，曲线坐标就因夏尔·埃尔米特、菲利克斯·克莱因和马克希莫·博谢引入的推广而过时了。

Gabriel Lamé was a student at the École Polytechnique, entering in 1813 and graduating in 1817. Already during these undergraduate years Lamé was writing research papers, and he published his first paper Mémoire sur les intersections des lignes et des surfaces Ⓣ in Gergonne's Journal in 1816-17. After graduating from the École Polytechnique, Lamé studied engineering at the École des Mines in Paris, graduating from there in 1820. While at the École des Mines Lamé published his second work, this time on a method he had invented to calculate the angles between faces of crystals.

In 1820 Lamé, together with his colleague Émile Clapeyron, went to Russia. We should give some background to this event which, on the face of it, looks rather a strange career move for the two young mathematicians. Alexander I was emperor of Russia from 1801 to 1825. The French Revolution and events in France which followed it, had shown Alexander the importance of scientific knowledge and its applications to military techniques and industrial development. He understood that for Russia to be powerful it must follow suit. He looked towards Europe and European scientists and tried to introduce policies to encourage them to cooperate with Russian scientists. He encouraged teachers to go to Russia to teach the latest scientific theories and to create scientific contacts between Russia and Europe. In line with this policy, the Russian government made a request to France who responded by sending Lamé and Clapeyron to St Petersburg.

Lamé was appointed professor and engineer at the Institut et Corps du Genie des Voies de Communication in St Petersburg. At first things were rather difficult for Lamé but later his visit proved highly productive. He lectured on analysis, physics, mechanics, chemistry, and engineering topics. He published papers in both Russian and French journals during his 12 years there, some jointly with Clapeyron. They published in, for example, the Journal des voies de communications, the Journal du genie civil, the Bulletin des sciences mathématiques, the Receuil des savants etrangers, and Journal für die reine und angewandte Mathematik (Crelle's Journal) after it began publication in 1826.

In [6] an interesting episode which occurred during Lamé's time in St Petersburg is related. It concerns Lamé's attempt to spread Cauchy's new ideas of rigorous analysis. A professor at the Institute where Lamé taught had written a book which contained a proof of Taylor's theorem. Lamé produced a manuscript criticising the proof using Cauchy's arguments. Another side to Lamé's work in St Petersburg was his involvement in helping with plans that were being drawn up for building bridges and roads around the city. At this time he became more aware of the vast potential of railway development, and this would be a topic of great interest to him after his return to France. Before that, he was present when the Liverpool-Manchester line opened in England on 15 September 1830.

Bradley [4] gives a lot more detail regarding Lamé's time in Russia. She concludes in her paper that:-

... the repressive atmosphere in France during the period of the Bourbon restoration had made work abroad seem more attractive for research and the application of new ideas. Lame and Clapeyron seized an opportunity offered to them by successful French engineers already established in Russia who had taken with them the spirit of the early years of the École Polytechnique. Important engineers like Betancourt and Bazaine helped them to pursue their careers in a land of scientific opportunity where their ideological convictions were strengthened through contact and discussion with their compatriots.

In 1832 Lamé returned to Paris and at first he formed part of an engineering firm set up jointly with Clapeyron and two others. After only a few months, and still in 1832, Lamé accepted the chair of physics at the École Polytechnique. He did not restrict his interests to teaching and research, however, for in remained an engineer ready for consulting work in that area. In 1836 he was appointed chief engineer of mines and he was also involved in the building of the railway from Paris to Versailles and of the railway from Paris to St Germain, which was opened in 1837.

Lamé was elected to the Académie des Sciences in 1843 when Louis Puissant died leaving a vacancy in the geometry section. In the following year he left his chair of physics at the École Polytechnique and accepted a post at the Sorbonne in mathematical physics and probability. He was appointed to the chair of mathematical physics and probability at the Sorbonne in 1851.

He worked on a wide variety of different topics. Often problems in the engineering tasks he undertook led him to study mathematical questions. For example his work on the stability of vaults and on the design of suspension bridges led him to work on elasticity theory. In fact this was not a passing interest, for Lamé made substantial contributions to this topic. Another example is his work on the conduction of heat which led him to his general theory of curvilinear coordinates.

Curvilinear coordinates proved a very powerful tool in Lamé's hands. He used them to transform Laplace's equation into ellipsoidal coordinates and so separate the variables and solve the resulting equation. The trademark of Lamé's career was moving from one topic to another in a quite logical way but he often ended up studying problems very far removed from the original. This happened with curvilinear coordinates for he was led to study the equation

$(x/a)^{n} + (y/b)^{n} + (z/c)^{n} = 0$

which, in non-homogeneous form he wrote as

$(x/a)^{n} + (y/b)^{n} = 1$

which, with $a = b$ is $x^{n} + y^{n} = a^{n}$ so he was led to Fermat's last theorem. Although he was basically an applied mathematician, Lamé made a substantial contribution to the problem by solving the case $n = 7$. In fact he believed that he had solved the whole problem at one stage but he had overlooked the lack of unique factorisation in certain subrings of the complex numbers.

He also did important work on differential geometry and, in another contribution to number theory, he showed that the number of divisions in the Euclidean algorithm never exceeds five times the number of digits in the smaller number.

As we noted above, he worked on engineering mathematics and elasticity where two elastic constants are named after him. He studied diffusion in crystalline material.

Lamé was considered the leading French mathematician of his time by many, in particular Gauss who was never one to give praise easily held this opinion. Rather strangely he was more highly thought of outside France than inside, for the French seemed to feel that he was too practical for a mathematician and yet too theoretical for an engineer. His own opinion was that curvilinear coordinates were his most important contribution, but there are strange twists and turns in the history of mathematics and very soon after Lamé introduced them curvilinear coordinates became obsolete through the generalisations introduced by Hermite, Klein, and Bôcher.