费迪南德·冯·林德曼Ferdinand von Lindemann

费迪南德·冯·林德曼的父亲也叫冯·林德曼，在他出生时是汉诺威文理中学的现代语言教师。他的母亲是埃米莉·克鲁修斯，是文理中学校长的女儿。当冯·林德曼（本传记的主人公）两岁时，他的父亲被任命为什未林一家煤气厂的厂长。全家搬到了那个城镇，冯·林德曼在那里度过了童年，并在什未林上学。

正如19世纪下半叶德国学生的标准做法，冯·林德曼从一所大学转到另一所大学。他于1870年在哥廷根开始学习，在那里深受阿尔弗雷德·克莱布什的影响。他有幸受教于阿尔弗雷德·克莱布什，因为后者于1868年才被任命到哥廷根，不幸的是他于1872年去世。后来，冯·林德曼在编辑和修订阿尔弗雷德·克莱布什的几何学讲座笔记以供1876年出版时，能够利用他参加这些讲座时所做的笔记。

冯·林德曼还在埃尔朗根和慕尼黑学习。在埃尔朗根，他攻读博士学位，并在菲利克斯·克莱因的指导下，写了一篇关于非欧几里得线几何及其与非欧运动学和静力学联系的学位论文。该学位于1873年因学位论文Über unendlich kleine Bewegungen und über Kraftsysteme bei allgemeiner projektivischer Massbestimmung Ⓣ（论一般射影质量确定中的无穷小运动和力系）而授予。

获得博士学位后，冯·林德曼出发访问英国和法国的重要数学中心。在英国，他访问了牛津、剑桥和伦敦，而在法国，他在巴黎度过了一段时间，受到米歇尔·沙勒、约瑟·伯特兰、卡米耶·若尔当和夏尔·埃尔米特的影响。回到德国后，冯·林德曼为他的教授资格论文（Habilitation）工作。该学位由维尔茨堡大学于1877年授予，同年晚些时候，他被任命为弗莱堡大学的编外教授。他于1879年在弗莱堡晋升为正教授。

冯·林德曼于1883年成为柯尼斯堡大学教授。阿道夫·赫维兹和大卫·希尔伯特都在他在那里时加入了柯尼斯堡的教职员队伍。在柯尼斯堡期间，他与Elizabeth Küssner结婚，她是一名女演员，也是当地学校教师的女儿。1893年，冯·林德曼接受了慕尼黑大学的讲席，他将在那里度过余下的职业生涯。

冯·林德曼的主要工作是在几何和分析领域。他以证明π是超越的而闻名，即π不是任何具有有理的系数的代数方程的根。squaring the circle问题，即仅用ruler and compasses构造一个与给定圆面积相等的正方形，曾是希腊数学的经典问题之一。1873年，即冯·林德曼获得博士学位的同年，夏尔·埃尔米特发表了他的证明，证明$e$是超越数。此后不久，冯·林德曼在巴黎拜访了夏尔·埃尔米特，并讨论了他证明中使用的方法。使用与夏尔·埃尔米特类似的方法，冯·林德曼在1882年确立了π也是超越数。

事实上，他的证明基于$e$是超越数的证明，以及$e^{i\pi} = -1$这一事实。许多科学史家遗憾地指出，夏尔·埃尔米特尽管完成了大部分艰苦工作，却未能迈出最后一步来证明这一结果，而这一结果本会为他带来数学界之外的声誉。这一声誉反而落在了冯·林德曼身上，但许多人觉得他是一位明显不如夏尔·埃尔米特的数学家，只是运气好，偶然发现了一个著名结果。尽管这有一定道理，但许多人确实创造了自己的运气，在冯·林德曼的情况下，人们必须高度赞扬他发现了夏尔·埃尔米特未能看到的诀窍。

兰伯特在1761年证明了π是无理数，但这不足以证明用尺规化圆为方是不可能的，因为某些代数数可以用尺规构造。冯·林德曼关于π是超越数的证明最终确立了用尺规化圆为方是不可解的。他在1882年的论文Über die Ludolphsehe ZahlⓉ（论鲁道夫数（即π））中发表了他的证明。

物理学也是冯·林德曼感兴趣的领域。他研究电子理论，并在这个问题上与阿诺·索末菲发生冲突。Eckert在[4]中利用手稿材料，包括与阿诺·索末菲的通信，考察了冯·林德曼对物理学的贡献。

冯·林德曼的另一个研究兴趣是数学史。他还与妻子合作进行翻译工作。特别是，他们翻译并修订了儒勒·昂利·庞加莱的一些著作。

冯·林德曼于1894年当选为巴伐利亚科学院准成员，次年成为正式成员。1912年，圣安德鲁斯大学授予他荣誉学位。

Wussing在[1]中写道：-

冯·林德曼是现代德国教育体系的奠基人之一。他强调研讨班的发展，并在讲座中传达最新研究成果。他还指导了六十多名博士生，其中包括大卫·希尔伯特。

大卫·希尔伯特是冯·林德曼在柯尼斯堡的博士生。他的另一名博士生是奥斯卡·佩龙，后者在慕尼黑跟随他学习。

Ferdinand von Lindemann's father, also named Ferdinand Lindemann, was a modern language teacher at the Gymnasium in Hannover at the time of his birth. His mother was Emilie Crusius, the daughter of the headmaster of the Gymnasium. When Ferdinand (the subject of this biography) was two years old his father was appointed as director of a gasworks in Schwerin. The family moved to that town where Ferdinand spent his childhood years and he attended school in Schwerin.

As was the standard practice of students in Germany in the second half of the 19th century, Lindemann moved from one university to another. He began his studies in Göttingen in 1870 and there he was much influenced by Clebsch. He was fortunate to be taught by Clebsch for he had only been appointed to Göttingen in 1868 and sadly he died in 1872. Later Lindemann was able to make use of the lecture notes he had taken attending Clebsch's geometry lectures when he edited and revised these note for publication in 1876.

Lindemann also studied at Erlangen and at Munich. At Erlangen he studied for his doctorate and, under Klein's direction, he wrote a dissertation on non-Euclidean line geometry and its connection with non-Euclidean kinematics and statics. The degree was awarded in 1873 for the dissertation Über unendlich kleine Bewegungen und über Kraftsysteme bei allgemeiner projektivischer Massbestimmung Ⓣ.

After the award of his doctorate Lindemann set off to visit important mathematical centres in England and France. In England he made visits to Oxford, Cambridge and London, while in France he spent time at Paris where he was influenced by Chasles, Bertrand, Jordan and Hermite. Returning to Germany Lindemann worked for his habilitation. This was awarded by the University of Würzburg in 1877 and later that year he was appointed as extraordinary professor at the University of Freiburg. He was promoted to ordinary professor at Freiburg in 1879.

Lindemann became professor at the University of Königsberg in 1883. Hurwitz and Hilbert both joined the staff at Königsberg while he was there. While in Königsberg he married Elizabeth Küssner, an actress, and daughter of a local school teacher. In 1893 Lindemann accepted a chair at the University of Munich where he was to remain for the rest of his career.

Lindemann's main work was in geometry and analysis. He is famed for his proof that π is transcendental, that is, π is not the root of any algebraic equation with rational coefficients. The problem of squaring the circle, namely constructing a square with the same area as a given circle using ruler and compasses alone, had been one of the classical problems of Greek mathematics. In 1873, the year in which Lindemann was awarded his doctorate, Hermite published his proof that $e$ is transcendental. Shortly after this Lindemann visited Hermite in Paris and discussed the methods which he had used in his proof. Using methods similar to those of Hermite, Lindemann established in 1882 that π was also transcendental.

In fact his proof is based on the proof that $e$ is transcendental together with the fact that $e^{i\pi} = -1$. Many historians of science regret that Hermite, despite doing most of the hard work, failed to make the final step to prove the result concerning which would have brought him fame outside the world of mathematics. This fame was instead heaped on Lindemann but many feel that he was a mathematician clearly inferior to Hermite who, by good luck, stumbled on a famous result. Although there is some truth in this, it is still true that many people make their own luck and in Lindemann's case one has to give him much credit for spotting the trick which Hermite had failed to see.

Lambert had proved in 1761 that π was irrational but this was not enough to prove the impossibility of squaring the circle with ruler and compass since certain algebraic numbers can be constructed with ruler and compass. Lindemann's proof that π is transcendental finally established that squaring the circle with ruler and compasses is insoluble. He published his proof in the paper Über die Ludolphsehe Zahl Ⓣ in 1882.

Physics was also an area of interest for Lindemann. He worked on the theory of the electron, and came into conflict with Arnold Sommerfeld on this subject. Eckert, in [4], looks at Lindemann's contributions to physics, using manuscript materials, including correspondence with Sommerfeld.

Another research interest of Lindemann was the history of mathematics. He also undertook, in collaboration with his wife, translating work. In particular they translated and revised some of Poincaré's writings.

Lindemann was elected to the Bavarian Academy of Sciences in 1894 as an associate member, becoming a full member in the following year. He given an honorary degree by the University of St Andrews in 1912.

Wussing writes in [1]:-

Lindemann was one of the founders of the modern German educational system. He emphasised the development of the seminar and in his lectures communicated the latest research results. He also supervised more than sixty doctoral students, including David Hilbert.

Hilbert was Lindemann's doctoral student in Königsberg. Another of his doctoral students was Oskar Perron who studied under him in Munich.