阿尔弗雷德·克莱布什Alfred Clebsch

阿尔弗雷德·克莱布什的父母是Ernst Friedrich Leopold Clebsch（1802-1863）和Pauline Ramberg（卒于1864年）。Ernst Clebsch的父亲，即本传记主人公的祖父，是Johannn Friedrich Leberecht 克莱布什（1759-1847），一位在Colberg州的外科医生，他在那里的军队医院工作。

克莱布什在柯尼斯堡就读于Altstädtisches 文理中学（Gymnasium），在那里他与同学卡尔·诺伊曼成为朋友，后者是恩斯特·弗朗茨·诺伊曼的儿子，当时恩斯特·弗朗茨·诺伊曼是柯尼斯堡大学的物理学教授。克莱布什从Altstädtisches文理中学毕业后，于1850年进入柯尼斯堡大学数学学院。在这个由卡尔·古斯塔夫·雅各布·雅可比创立的学院中，他通过他的老师奥托·黑塞和Friedrich Julius Richelot（1808-1875）受到了卡尔·古斯塔夫·雅各布·雅可比的影响，这两位老师都是卡尔·古斯塔夫·雅各布·雅可比的学生。事实上，尽管他从未见过卡尔·古斯塔夫·雅各布·雅可比——后者在克莱布什进入柯尼斯堡大学一年后去世——但卡尔·古斯塔夫·雅各布·雅可比通过这两位老师以及克莱布什参与Collected Works of Jacobi的编写这一事实，直接对他产生了影响。在柯尼斯堡，克莱布什由恩斯特·弗朗茨·诺伊曼教授数学物理，后者是他朋友的父亲。他在恩斯特·弗朗茨·诺伊曼的指导下进行博士研究，并凭借其学位论文De motu ellipsoidis in fluido incompressibili viribus quibuslibet impulsi Ⓣ（不可压缩流体中任意力引起的椭球运动）于1854年提交给柯尼斯堡大学而获得学位。在这项工作中，他研究了流体动力学中的一个问题。

1854年毕业后，克莱布什前往柏林，在那里多所中学任教。这在当时是进入学术职业的一条相当常见的途径，克莱布什也一心想要谋求一个大学职位。因此，他在担任中学教师的四年间继续进行研究并发表了多篇文章。他在1854-58年间提交给奥古斯都·利奥波德·克雷勒的期刊、并于1856-59年在该期刊上发表的文章有：Über die Bewegung eines Ellipsoids in einer tropfbaren Flüssigkeit Ⓣ（论液体形成液滴中椭球的运动）（1856）；Zusatz zu dem vorhergehenden Aufsatze Ⓣ（前文补遗）（1857）；Anwendung der elliptischen Functionen auf ein Problem der Geometrie des Raumes Ⓣ（椭圆函数在空间几何问题中的应用）（1857）；Über eine allgemeine Transformation der hydrodynamischen Gleichungen Ⓣ（论流体动力学方程的一般变换）（1857）；Über die Reduction der zwei Variation auf ihre einfachste Form Ⓣ（论将两个变分化为最简形式）（1858）；以及Über diejenigen Probleme der Variationsrechnung, welche nur eine unabhängige Variable enthalten Ⓣ（论那些只含一个自变量的变分法问题）（1858）。他还在1857年于期刊Monatsberichte der Berliner Akademie上发表了Über die Criterien des Maximums und des Minimums in der Variationsrechnung Ⓣ（论变分法中极大值与极小值的判据）。

他的第一个学术职位是在1858年，当时他被任命到柏林大学。短暂任职后他便离开，仍在1858年，他接受了卡尔斯鲁厄理工学院的一个职位。然而，一旦他知道自己能够进入学术职业，他便得以结婚。他的妻子是Dorothe Charlote Mathilde Heinel（1838-1866），Marienburg牧师Heinel的女儿。克莱布什和Dorothe 克莱布什有四个儿子：Ernst Friedrich 克莱布什（1859-1945）；Arthur Friedrich 克莱布什（1860-1931）；Eduard Friedrich 克莱布什（1861-1895），后来成为Ems的医生；以及克莱布什 Friedrich Clebsch（1864-），后来在不来梅的克莱布什 and Schünemann烟草商行做商人。

正如我们上面提到的，克莱布什在柯尼斯堡的博士学位论文是关于流体力学的，而他当学校教师时写的大部分论文主要涉及流体力学和弹性力学。他任教所在的卡尔斯鲁厄理工学院成立于1825年，主要培养工程师和建筑师。克莱布什为卡尔斯鲁厄理工学院带来了许多新思想，其中也许最重要的是建立了数学讨论班。他从1858年到1863年在卡尔斯鲁厄工作，但在离开卡尔斯鲁厄之前，他的研究方向已经发生了变化。甚至在他被任命到卡尔斯鲁厄之前，就有迹象表明克莱布什正通过他在变分法方面的工作转向纯粹数学。他在数学物理课题方面工作的结束，也许最清楚地以1862年出版的Theorie der Elastizität fester KörperⓉ（固体弹性理论）为标志，这是一部关于弹性力学的重要著作。在其中他[1]：-

……处理并推广了杆和板的弹性振动问题。

当克莱布什开始研究变分法和偏微分方程时，纯数学成了他的主要研究课题。克莱布什于1863年搬到吉森大学，在那里他与保罗·哥尔丹合作。他们的合作工作在1866年达到顶峰，完成了一部关于abelian function Theorie der Abelschen Funktionen的重要著作。球谐函数中使用的克莱布什-保罗·哥尔丹系数就是他们这次合作的成果。克莱布什证明了自己是一位杰出的教师，并将教学技巧与研究技巧结合起来，在吉森建立了一个代数几何和不变量理论学派，其中包括保罗·哥尔丹、Alexander Brill、马克斯·诺特、费迪南德·冯·林德曼和雅各布·吕罗特。

正是奥托·黑塞建议克莱布什研究阿瑟·凯莱、詹姆斯·约瑟夫·西尔维斯特和乔治·萨蒙的代数几何，而他尤其被Aronhold对这些理论所做的贡献所吸引。克莱布什回到了尼尔斯·阿贝尔的代数几何方法，而不是波恩哈德·黎曼的几何方法，他采用了代数方法。他以这种方式对阿瑟·凯莱、詹姆斯·约瑟夫·西尔维斯特和乔治·萨蒙著作的解读，使克莱布什对波恩哈德·黎曼的函数论有了辉煌的新解读。菲利克斯·克莱因写道[2]：-

作为克莱布什的第一项成就，我们必须记下他将阿瑟·凯莱和詹姆斯·约瑟夫·西尔维斯特先前在英国所做的工作引入德国。但他不仅将他们的不变量理论以及通过该理论对射影几何的解释移植到德国土壤上；他还将这一理论与波恩哈德·黎曼函数论的基本思想联系起来，使之生动而富有成果。

克莱布什关于代数几何的著作Über die Anwendung der Abelschen Functionen in der Geometrie Ⓣ（论阿贝尔函数在几何中的应用）（1864年）发表在奥古斯都·利奥波德·克雷勒的期刊上，伊戈尔·沙法列维奇在[5]中将其描述为：-

……现代代数几何的诞生之声。

[3]的作者描述了这篇论文中的一些思想：-

克莱布什证明了尼尔斯·阿贝尔定理的逆定理，并利用卡尔·古斯塔夫·雅各布·雅可比和波恩哈德·黎曼发展的函数论方法推导出代数曲线的许多基本性质。这些技巧使他能够相对轻松地重新推导出一些结果，而这些结果曾让像雅各布·施泰纳和奥托·黑塞这样的几何学家付出大得多的努力。

我们再次引用菲利克斯·克莱因 [2]中的话，他在其中解释了克莱布什如何以不同的方式看待波恩哈德·黎曼的思想：-

波恩哈德·黎曼1857年那篇著名的论文以一种略显惊人的新颖形式提出了函数论的新思想，这妨碍了它们立即被接受和承认。他将阿贝尔积分及其反演——阿贝尔函数——的理论建立在他如今以其名字命名的曲面这一概念以及相应的存在性基本定理之上。克莱布什以由方程定义的代数曲线为出发点，使这一理论对他那个时代的数学家更为易懂，并通过他从阿贝尔函数理论中推导出的几何定理为其增添了更具体的意义。

1868年，克莱布什被任命到哥廷根大学。 [3]的作者们写道：-

到1868年，当克莱布什接受哥廷根大学先前由波恩哈德·黎曼担任的讲席时，他和他的学生们在代数几何和不变量理论方面产出了如此多的新成果，以至于他们开始计划创办一份新期刊，旨在让他们的工作获得更多关注。

到达哥廷根后不久，他与卡尔·诺伊曼——他在柯尼斯堡的前老师的儿子——共同创办了Mathematische Annalen，一份极为重要的数学期刊。第一卷于1869年付印；这一卷的第一部分包含海因里希·马丁·韦伯、雅各布·吕罗特、保罗·哥尔丹（2篇论文）、Karl Geiser和Ernst Christian Julius Schering的论文。

事实上，正是在1868年，克莱布什 做出了 菲利克斯·克莱因 认为最重要的贡献 [2]：-

现在让我们转向 克莱布什 方法中在我看来最为重要的那一面，而且这一面确实必须被承认具有重大而持久的价值；我指的是 克莱布什 所获得的、将整个阿贝尔积分理论推广到多变量代数函数理论的工作。通过把他为形如 $F(x, y) = 0$ 的函数、或在齐次坐标下为 $f (x_{1}, x_{2}, x_{3}) = 0$ 的函数所发展出的方法应用于具有四个齐次变量的函数 $f (x_{1}, x_{2}, x_{3}, x_{4}) = 0$，他在1868年发现，也存在一个数 p，它在曲面 f = 0 的所有有理变换下保持不变。克莱布什 通过考虑属于该曲面的二重积分得到了这一结果。显然，从 波恩哈德·黎曼 的观点出发不可能发现这一理论。

就在 克莱布什 提出这些新思想之后，菲利克斯·克莱因 来到哥廷根，跟随他从事博士后工作。克莱布施学派中正在发展的思想对 菲利克斯·克莱因 产生了极其重要的影响。克莱布什 的思想正被他学派中其他有才华的成员迅速扩展。在哥廷根工作八个月后，菲利克斯·克莱因 向 克莱布什 谈到要拓宽视野，并去柏林待一个学期。克莱布什 强烈建议他不要去柏林，菲利克斯·克莱因 意识到哥廷根学派与柏林学派之间存在严重的紧张关系。然而，他违背了 克莱布什 的建议，去柏林待了一个学期。

克莱布什辉煌的职业生涯在1872年突然结束，他死于白喉。马克斯·诺特和Alexander Brill——他在吉森学派的成员——继续了他关于曲线的工作。他关于几何的讲义有两卷在他去世后于1876年和1891年出版。其中一卷的一部分的第二版，以克莱布什为合著者，分三部分于1906年、1910年和1932年出版。

W Burau 在 [1] 中写道，对 克莱布什 的工作作出如下评论：-

…… 克莱布什 描述了各种有理曲面的平面表示，特别是一般三次曲面的平面表示。克莱布什 还必须被归功于代数曲面的第一个双有理不变量，即他引入的几何亏格，作为其上存在的第一类二重积分的最大数目。

最后，我们给出 [3] 的作者们对 克莱布什 的评价：-

克莱布什曾是一位模范教师，将数学天赋与激发和鼓励天才学生的能力结合在一起。如果他活得更久，他很可能轻松成为他那一代的主导人物，德国数学的整个框架可能会以不同的方式演变。在他的领导下，已经为创建德国数学学会制定了具体计划，但在他去世后，这一努力逐渐失去了动力。

Alfred Clebsch's parents were Ernst Friedrich Leopold Clebsch (1802-1863) and Pauline Ramberg (died 1864). Ernst Clebsch's father, the paternal grandfather of the subject of this biography, was Johannn Friedrich Leberecht Clebsch (1759-1847), a surgeon in the state of Colberg where he worked at the military hospital.

Clebsch attended the Altstädtisches Gymnasium in Königsberg where he became friends with his fellow pupil Carl Neumann, the son of Franz Neumann who was at this time professor of physics at the University of Königsberg. Clebsch graduated from the Altstädtisches Gymnasium and entered the school of mathematics at the University of Königsberg in 1850. In this school, founded by Carl Jacobi, he was influenced by Jacobi through his teachers Otto Hesse and Friedrich Julius Richelot (1808-1875) who were both students of Jacobi. In fact although he never met Jacobi, who died one year after Clebsch entered the University of Königsberg, Jacobi was to influence him both through these two teachers and also directly through the fact that Clebsch was to collaborate in the production of the Collected Works of Jacobi. At Königsberg, Clebsch was taught mathematical physics by Franz Neumann, the father of his friend. He undertook research for his doctorate advised by Franz Neumann and he was awarded the degree for his thesis De motu ellipsoidis in fluido incompressibili viribus quibuslibet impulsi Ⓣ which he presented to the University of Königsberg in 1854. In this work he studied a problem in hydrodynamics.

After graduating in 1854 Clebsch went to Berlin where he taught at various secondary schools. This was a fairly common route into the academic profession at this time and Clebsch had every intention of seeking a university post. Consequently, he continued to undertake research and published a number of articles during his four years as a secondary school teacher. Articles which he submitted to Crelle's Journal in 1854-58 and which were published in that journal in 1856-59 were: Über die Bewegung eines Ellipsoids in einer tropfbaren Flüssigkeit Ⓣ (1856); Zusatz zu dem vorhergehenden Aufsatze Ⓣ (1857); Anwendung der elliptischen Functionen auf ein Problem der Geometrie des Raumes Ⓣ (1857); Über eine allgemeine Transformation der hydrodynamischen Gleichungen Ⓣ (1857); Über die Reduction der zwei Variation auf ihre einfachste Form Ⓣ (1858); and Über diejenigen Probleme der Variationsrechnung, welche nur eine unabhängige Variable enthalten Ⓣ (1858). He also published Über die Criterien des Maximums und des Minimums in der Variationsrechnung Ⓣ in the journal Monatsberichte der Berliner Akademie in 1857.

His first academic appointment was in 1858 when he was appointed to the University of Berlin. He left after a short spell and, still in 1858, he took up an appointment at the Polytechnischen Schule in Karlsruhe. However, once he knew that he was able to enter the academic profession he was able to marry. His wife was Dorothe Charlote Mathilde Heinel (1838-1866), the daughter of the Priest Heinel in Marienburg. Alfred and Dorothe Clebsch had four sons: Ernst Friedrich Alfred Clebsch (1859-1945); Arthur Friedrich Alfred Clebsch (1860-1931); Eduard Friedrich Alfred Clebsch (1861-1895), who became a medical doctor in Ems; and Alfred Friedrich Clebsch (1864-), who became a trader in Bremen in the firm Clebsch and Schünemann, Tobacco merchants.

Now as we mentioned above, Clebsch's doctoral dissertation at Königsberg was on hydrodynamics and most of the papers he wrote while a school teacher were on topics mainly concerned with hydrodynamics and elasticity. The Polytechnischen Schule in Karlsruhe where he taught had been founded in 1825 and was mainly involved in training engineers and architects. To the Polytechnischen Schule, Clebsch brought many new ideas, perhaps the most significant of which was the establishment of the mathematical colloquia. He worked in Karlsruhe from 1858 to 1863 but before he left Karlsruhe the direction of his research had changed. Even before his appointment at Karlsruhe there had been signs of Clebsch moving towards pure mathematics with his work on the calculus of variations. The end of his work on topics in mathematical physics is perhaps most clearly defined by the publication of Theorie der Elastizität fester Körper Ⓣ in 1862 which was a major work on elasticity. In it he [1]:-

... treated and extended problems of elastic vibrations of rods and plates.

Pure mathematics became Clebsch's main research topic when he began to study the calculus of variations and partial differential equations. Clebsch moved to the University of Giessen in 1863 and there he collaborated with Paul Gordan. Their joint work culminated in a major work on abelian function Theorie der Abelschen Funktionen in 1866. The Clebsch-Gordan coefficients used in spherical harmonics were introduced by them as a result of this cooperation. Clebsch proved himself an outstanding teacher and combined his teaching skills with his research skills in building a school of algebraic geometry and invariant theory at Giessen which included Paul Gordan, Alexander Brill, Max Noether, Ferdinand von Lindemann and Jacob Lüroth.

It was Otto Hesse who had advised Clebsch to investigate the algebraic geometry of Cayley, Sylvester and Salmon and he was particularly attracted to the contributions that Aronhold had made to their theories. Clebsch went back to Abel's approach to algebraic geometry and, rather than the geometric approach of Riemann. he adopted an algebraic approach. His interpretation of the works of Cayley, Sylvester and Salmon in this way led Clebsch to a brilliant new interpretation of Riemann's function theory. Felix Klein writes [2]:-

As the first achievement of Clebsch we must set down the introduction into Germany of the work done previously by Cayley and Sylvester in England. But he not only transplanted to German soil their theory of invariants and the interpretation of projective geometry by means of this theory; he also brought this theory into live and fruitful correlation with the fundamental ideas of Riemann's theory of functions.

Clebsch's work on algebraic geometry Über die Anwendung der Abelschen Functionen in der Geometrie Ⓣ (1864) was published in Crelle's Journal and is described by Igor Shafarevich in [5] as the:-

... birth cry of modern algebraic geometry.

The authors of [3] describe some of the ideas in this paper:-

Clebsch proved the converse of Abel's theorem and derived many fundamental properties of algebraic curves by utilising function-theoretic methods developed by Jacobi and Riemann. These techniques enabled him to rederive with relative ease a number of results that had cost geometers like Jacob Steiner and Otto Hesse a great deal more effort.

Again we quote from Felix Klein [2] where he explains how Clebsch looked at Riemann's ideas in a different way:-

Riemann's celebrated memoir of 1857 presented the new ideas on the theory of functions in a somewhat startling novel form that prevented their immediate acceptance and recognition. He based the theory of the Abelian integrals and their inverses, the Abelian functions, on the idea of the surface now so well known by his name, and on the corresponding fundamental theorems of existence. Clebsch, by taking as his starting-point an algebraic curve defined by its equation, made the theory more accessible to the mathematicians of his time, and added a more concrete interest to it by the geometrical theorems that he deduced from the theory of Abelian functions.

In 1868 Clebsch was appointed to the University of Göttingen. The authors of [3] write:-

By 1868, when Clebsch accepted the chair formerly held by Riemann at Göttingen, he and his entourage of students were turning out so much new material on algebraic geometry and invariant theory that they began making plans for the inauguration of a new journal designed to give their work more visibility.

Shortly after arriving in Göttingen, together with Carl Neumann, the son of his former teacher at Königsberg, he co-founded Mathematische Annalen, a mathematics journal of major importance. The first volume appeared in print in 1869; the first part of this first volume contained papers by Heinrich Weber, Jacob Lüroth, Paul Gordan (2 papers), Karl Geiser, and Ernst Christian Julius Schering.

In fact it was in 1868 that Clebsch made the contribution that Klein considered the most significant [2]:-

Let us now turn to that side of Clebsch's method which appears to me to be the most important, and which certainly must be recognised as being of great and permanent value; I mean the generalisation, obtained by Clebsch, of the whole theory of Abelian integrals to the theory of algebraic functions with several variables. By applying the methods he had developed for functions of the form $F(x, y) = 0$, or in homogeneous coordinates, $f (x_{1}, x_{2}, x_{3}) = 0$, to functions with four homogeneous variables $f (x_{1}, x_{2}, x_{3}, x_{4}) = 0$, he found in 1868, that there also exists a number p that remains invariant under all rational transformations of the surface f = 0. Clebsch arrives at this result by considering double integrals belonging to the surface. It is evident that this theory could not have been found from Riemann's point of view.

It was just after Clebsch had produced these new ideas that Felix Klein arrived in Göttingen to undertake postdoctoral work with him. The ideas being developed in Clebsh's school had a highly significant influence on Klein. Clebsch's ideas were being rapidly extended by the other talented members of his school. After spending eight months working in Göttingen, Klein spoke to Clebsch about broadening his horizons and spending a semester in Berlin. Clebsch strongly advised him not to go to Berlin and Klein realised that there were serious tensions between the Göttingen school and the one in Berlin. However, he went against Clebsch's advice and spent a semester to Berlin.

Sadly Clebsch's brilliant career came to a sudden end in 1872 when he died of diphtheria. Max Noether and Alexander Brill, who were members of his school at Giessen, continued his work on curves. Two volumes of his lectures on geometry were published after his death in 1876 and 1891. A second edition of part of one of these volumes, with Clebsch as joint author, was published in three parts in 1906, 1910 and 1932.

W Burau, writing in [1], makes the following comments about Clebsch's work:-

... Clebsch described the plane representations of various rational surfaces, especially that of the general cubic surface. Clebsch must also be credited with the first birational invariant of an algebraic surface, the geometric genus that he introduced as the maximal number of double integrals of the first kind existing on it.

Finally we give the assessment of Clebsch from the authors of [3]:-

Clebsch had been a model teacher, combining mathematical genius with an ability to inspire and encourage gifted students. Had he lived longer, he might easily have become the dominant figure of his generation, and the whole framework for mathematics in Germany might have evolved differently. Under his leadership, concrete plans had already been made for the creation of a German mathematical society, but after his death, this effort gradually lost momentum.