卡尔·龙格Carl David Tolmé Runge

卡尔·龙格的父母是Julius Runge和范妮·龙格。Julius Runge出身于商人家庭，但大约在1836年前往古巴哈瓦那。他经常回到不来梅，但大约20年间主要住在哈瓦那。范妮·龙格是住在哈瓦那的英国商人Charles David·龙格的女儿。龙格听起来不太像英国名字，确实如此，因为这个家族有胡格诺派血统。朱利叶斯和范妮于1846年5月16日在哈瓦那结婚。龙格是他父母四个儿子中的第三个。这段婚姻还育有四个女儿。尽管龙格出生在不来梅，但他的早年是在哈瓦那度过的。龙格一直与母亲非常亲近，像典型的英国孩子一样被抚养长大。事实上，朱利叶斯和范妮在家中说英语，因此孩子们成长过程中以英语为第一语言。龙格的四个年长兄弟姐妹中有三个最终定居英国。朱利叶斯退休后，全家回到不来梅永久居住，但朱利叶斯的退休生活很短暂，他于1864年1月18日去世。范妮独自一人抚养八个孩子。福曼写道[1]：-

因此，在他的成长过程中有很强的英国元素，尤其是对体育、自立和公平竞争的强调，这与汉萨城镇的公民传统相结合，影响了他的政治和社会观点。

龙格在不来梅上了中学，并于1875年通过了大学入学考试。1875年离开学校后，龙格与母亲一起花了六个月时间游览意大利的文化中心。回到德国后，他于1876年复活节进入慕尼黑大学学习文学和哲学。他的三个哥哥对追求学术生涯不感兴趣，他们都选择进入商界。然而，龙格作为文学学生的生涯很短暂，因为上了六周课后他就转到了数学和物理。龙格与同学马克斯·普朗克一起上课，他们成了亲密的朋友，并终生保持这种友谊。

1877年秋天，马克斯·普朗克和龙格都去了柏林，但龙格在听了卡尔·魏尔斯特拉斯的讲座后转向了纯数学。他觉得古斯塔夫·基尔霍夫和赫尔曼·冯·亥姆霍兹讲授的数学物理讲座不是特别有吸引力。另一方面，他喜欢恩斯特·爱德华·库默尔开设的课程以及卡尔·魏尔斯特拉斯开设的课程。尽管龙格已投身于纯数学的研究，但他并未失去对哲学的最初兴趣。他在1878-79年冬季学期参加了Friedrich Paulsen关于弗洛伦斯·南丁格尔·大卫休谟的课程。Paulsen刚被任命为柏林大学的编外教授，龙格对他评价极高。他写道，Paulsen是两位[1]之一：-

……我最好的知识和能力都归功于他。

他的博士学位论文于1880年6月23日提交给柏林大学，内容涉及微分几何。论文题目是Über die Krümmung, Torsion und geodätische Krümmung der auf einer Fläche gezogenen Curven Ⓣ（论实体上曲面曲线的曲率、挠率和测地曲率），由Ferdinand Rudio、W Sachse和Adolf Piltz审阅，后者是恩斯特·爱德华·库默尔和卡尔·魏尔斯特拉斯的博士生。虽然卡尔·魏尔斯特拉斯是他的导师，但他并未建议龙格的论文题目；这个题目反而来自他与数学协会中其他学生的讨论，他积极参与了该协会。

在1880-81学年取得文理中学教师资格后，他完成了必要的考试并回到柏林，开始与利奥波德·克罗内克合作。龙格随后研究了一种代数方程数值求解的方法，其中根表示为系数的有理函数的无穷级数。这类方程有三种标准的数值解法，即艾萨克·牛顿、约翰·伯努利和格拉夫的方法，而龙格发现的方法将三种标准方法都作为特例包含在内。他将这些结果收入1883年2月提交给柏林的教授资格论文（Habilitation）论文中。这使他获得了在柏林大学授课的资格，并在那里继续从事代数和函数论的研究，成为围绕利奥波德·克罗内克形成的数学家群体的一员。

龙格在职业生涯的这个阶段发表的作品很少，但在1884年9月于斯德哥尔摩拜访约斯塔·米塔格-莱弗勒之后，他被约斯塔·米塔格-莱弗勒说服，将自己得出的结果写出来。在这种鼓励下，他写了一些论文，于1885年发表在约斯塔·米塔格-莱弗勒的期刊Acta mathematica上。龙格不仅作为数学家崭露头角，他还进入了柏林的社会生活[1]：-

龙格——身材高瘦，头颅硕大而轮廓精致——年轻时练就了非凡的滑冰技巧；19世纪80年代初在柏林，当这项活动变得极为时髦时，他显得格外引人注目。

柏林最令人印象深刻的教授之一是埃米尔·杜布瓦-雷蒙，数学家Paul du Bois-Reymond的哥哥。埃米尔·杜布瓦-雷蒙对生理学、医学和哲学感兴趣，并于1880年在柏林科学院发表了一篇著名演讲，其中他列出了七个他宣称科学无法解释的谜题。这些谜题包括：物质与力的终极本质；运动的起源；生命的起源；以及自由意志问题。龙格与埃米尔·杜布瓦-雷蒙一家变得友好，尤其是与他的孩子们。1885年，他与艾梅·杜布瓦-雷蒙订婚，但她的父亲自幼受虔敬派教养，持有严格的观点，说在龙格获得教授职位之前不允许他们结婚。幸运的是，这并没有成为长久的障碍，因为1886年3月龙格获得了汉诺威技术高等学校的讲席。他于1887年8月与艾梅结婚，并在汉诺威待了18年。帕申（与龙格合作了七年，见下文）这样描述龙格及其在汉诺威的家庭生活[4]：-

他们有四个女儿和两个儿子，其中一个儿子在战争[第一次世界大战]中阵亡。龙格在汉诺威的家……那些有幸踏入其中的人永远不会忘记。这个家庭培养了许多科学与艺术。龙格本人弹钢琴，他和孩子们常常演奏《马太受难曲》等音乐经典。龙格是一位善于处理事务的人，极具个人魅力。他喜爱各种运动，并练习骑自行车、体操和游泳。在汉诺威，他常常每天四次骑自行车从家到技术高等学校，距离约八公里。在他的一切活动中，他都把科学事务放在首位，并愿意为它们的进步牺牲一切。

在汉诺威获得教授职位后的一年内，龙格就从纯数学转向研究氢以外元素光谱线的波长。（J J 约翰·雅各布·巴耳末曾为氦的光谱线构造了一个公式。）事实上，龙格对这一主题的兴趣是通过埃米尔·杜布瓦-雷蒙产生的，那是在龙格娶他女儿的前一年。杜布瓦-雷蒙曾在柏林科学院听过海因里希·凯泽关于这一主题的讲座，这让他很感兴趣，而在凯泽于1885年秋被任命为汉诺威技术高等学校物理学教授后，三人开始共同研究这个问题。凯泽和龙格在七年间于柏林科学院的学报上发表了七篇联合论文。这些论文总计超过350页，做出了重要贡献。在其中一篇发表于1888年的论文中，他们写道，这个关于氢以外元素光谱线波长的问题[1]：-

……比任何其他物理过程都能更深入地洞察原子的组成和性质。

龙格做了大量实验工作，并发表了大量结果。他成功地将氦的光谱线排列成两个光谱系，直到1897年，这被认为是氢是两种元素混合物的证据。与凯泽合作七年后，凯泽于1894年离开汉诺威，去担任波恩大学物理学讲席。这一职位因海因里希·鲁道夫·赫兹在36岁时因血液中毒不幸去世而空缺。龙格独自继续光谱学研究六个月，但随后说服弗里德里希·帕申加入他。帕申是一位实验家，他们在汉诺威一起工作了七年。龙格于1895年访问英国，并与瑞利变得友好。两年后，他前往美国，在那里与A A 迈克耳孙成为朋友。在美国期间，他参观了叶凯士天文台，并得到乔治·E·黑尔提供的教授职位，但他拒绝了。帕申这样描述他们在汉诺威的合作工作[4]：-

当这种气体最初由Ramsay发现时，我们研究了氦的光谱。随后我们转向氧、硫和硒的光谱。在这些光谱中，我们首次发现了两种不同多重度的系列系统。

1901年，Paschen离开汉诺威，被任命为图宾根大学物理学教授。在这个阶段，龙格继续与Julius Precht合作，后者此前在海德堡担任理论物理学特聘讲席。龙格和Precht合作[4]：-

……关于镭的光谱以及埃里克·克里斯托弗·齐曼对其谱线的影响。他们发现了弧光谱和火花光谱中的主线群，并将其归类为碱土金属的谱线。

有趣的是，为什么龙格在汉诺威技术高等学校待了这么久，而其他人却被任命到更有声望的讲席。原因几乎可以肯定是因为龙格不合群。一方面，许多人认为他是数学的叛徒，因为他离开了数学领域去从事物理学研究。然而，物理学家们认为龙格是数学家，不会认为他完全适合在顶尖机构担任物理学讲席。值得在此指出的是，龙格一直认为自己是个数学家。1904年，Klein说服哥廷根大学向龙格提供一个应用数学讲席，龙格于同年10月就任此职，直到1925年退休[4]：-

在哥廷根，他被任命从事应用数学的教学和研究。他发展了许多数值和图形方法，给出了微分方程的数值解等。他是将这类数学引入德国的先驱。

龙格深入参与教学，研究工作减少。他于1908年出版了Analytische Geometrie der Ebene，并于1924年与H约翰·萨穆埃尔·柯尼希合著了Vorlesungen über numerisches RechnenⓉ（数值计算讲义）。1909年10月至1910年1月，他在纽约哥伦比亚大学讲授Graphical methods，并于1909-1910年担任德皇威廉德国历史与制度教授。

这里是Introduction to Graphical methods。

龙格是哥廷根大学[1]中一位极具影响力的人物：-

尽管他持有自由派政治观点——公开反对第一次世界大战期间的兼并，并在战后加入民主党——龙格仍保有同事的信任和他在大学中的影响力。

龙格在1923年达到68岁的退休年龄，但他继续管理他的研究所，直到他的继任者古斯塔夫·赫格洛茨于1925年到来。然而，我们应该注意到，当古斯塔夫·赫格洛茨接任时，该讲席的名称不再是“应用数学”。

我们上面已经注意到，龙格非常热爱运动，即使年岁渐长也依然健康而活跃，在他70岁生日时，他还通过做倒立来逗孙辈们开心。他仍在数学上活跃，有几个雄心勃勃的项目正在进行中。然而六个月后，他心脏病发作并去世了。他的母亲Fanny已于1910年去世。

Carl Runge's parents were Julius Runge and Fanny Tolmé. Julius Runge was from a family of merchants but he went to Havana, Cuba, around 1836. He returned frequently to Bremen but lived mostly in Havana for around 20 years. Fanny Tolmé was the daughter of the English merchant Charles David Tolmé who was living in Havana. Tolmé does not sound very English, and indeed it is not for the family was of Huguenot descent. Julius and Fanny were married in Havana on 16 May 1846. Carl was the third of his parents' four sons. There were also four daughters from the marriage. Although Carl was born in Bremen, he spent his early years in Havana. Carl, who was always very close to his mother, was brought up like a typically British child. In fact Julius and Fanny spoke English at home so the children grew up with English as their first language. Three of Carl's four older siblings eventually settled in England. After Julius retired, the family returned to live permanently in Bremen, but Julius had only a short retirement for he died on 18 January 1864. Fanny was left on her own to bring up the eight children. Forman writes [1]:-

There was thus a strong British element in his upbringing, particularly an emphasis upon sport, self-reliance, and fair play that, in combination with the civic traditions of the Hanseatic town, influenced his political and social views.

Carl attended the Lyceum in Bremen and, in 1875, passed the examinations required for university entrance. After leaving school in 1875, Carl Runge spent six months with his mother visiting the cultural centres of Italy. On his return to Germany he enrolled at the University of Munich at Easter 1876 to study literature and philosophy. His three elder brothers had not been interested in pursuing academic careers and they had all opted to enter the commercial world. Runge's career as a student of literature was, however, short lived for after six weeks of the course he changed to mathematics and physics. Runge attended courses with fellow student Max Planck and they became close friends, remaining so for the rest of their lives.

In the autumn of 1877 Planck and Runge both went to Berlin but Runge turned to pure mathematics after attending Karl Weierstrass's lectures. He had not found the lectures on mathematical physics given by Kirchhoff and by Helmholtz particularly attractive. On the other hand, he loved the courses given by Ernst Kummer as well as those given by Weierstrass. Although Runge had committed himself to the study of pure mathematics he did not lose his initial interest in philosophy. He attended Friedrich Paulsen's course on David Hume in the winter semester of 1878-79. Paulsen had just been appointed as an extraordinary professor at Berlin and Runge rated him extremely highly. He wrote that Paulsen was one of the two [1]:-

... to whom I owe the best of my knowledge and ability.

His doctoral dissertation, submitted to the University of Berlin on 23 June 1880, dealt with differential geometry. The title of the thesis was Über die Krümmung, Torsion und geodätische Krümmung der auf einer Fläche gezogenen Curven Ⓣ and it was examined by Ferdinand Rudio, W Sachse, and Adolf Piltz who was a doctoral student of Kummer and Weierstrass. Now, although Weierstrass was his advisor, he had not suggested the topic of Runge's thesis; rather this had come out of discussions he had with other students in the Mathematischer Verein in which he was actively engaged.

After qualifying to be a Gymnasium teacher during session 1880-81, he completed the necessary examinations and returned to Berlin where he began to collaborate with Kronecker. Runge then worked on a procedure for the numerical solution of algebraic equations in which the roots were expressed as infinite series of rational functions of the coefficients. There were three standard methods for the numerical solution of such equations, namely by Newton, Bernoulli and Gräffe, and the method found by Runge had all three of the standard methods as special cases. He included these results in his Habilitation thesis which he submitted to Berlin in February 1883. This entitled him to lecture at the University of Berlin, and there he continued to undertake research on algebra and function theory as part of the group of mathematicians which had built up around Kronecker.

Runge had published little at this stage in his career but after visiting Mittag-Leffler in Stockholm in September 1884 he was persuaded by Mittag-Leffler to write up the results he had produced. With this encouragement, he wrote a number of papers which were printed in Mittag-Leffler's journal Acta mathematica in 1885. Runge was not only making his mark as a mathematician, he was also entering into the social life of Berlin [1]:-

Runge - tall, lean, with a large finely sculptured head - had developed exceptional skill as an ice skater in his youth; and in Berlin in the early 1880s when that activity was becoming extremely fashionable, he cut a striking figure.

One of the most impressive professors at Berlin was Émile du Bois-Reymond, older brother of the mathematician Paul du Bois-Reymond. Émile du Bois-Reymond was interested in physiology, medicine and philosophy and gave a famous speech at the Berlin Academy of Science in 1880 in which he listed seven riddles which, he declared, science could not explain. These included: the ultimate nature of matter and force; the origin of motion; the origin of life; and the question of freewill. Runge became friendly with Émile du Bois-Reymond's family, particularly with his children. He became engaged to Aimée du Bois-Reymond in 1885 but her father, who had been brought up a Pietist and had strict views, said that they would not be allowed to marry until Runge had obtained a professorship. Fortunately this did not prove to be an obstacle for long for in March 1886 Runge obtained a chair at the Technische Hochschule at Hannover. He married Aimée in August 1887 and remained in Hannover for 18 years. Paschen (who collaborated with Runge for seven years, see below) writes about Runge and his family life in Hannover [4]:-

They had four daughters and two sons, one of whom was killed in the war [World War I]. Runge's home at Hannover ... will never be forgotten by those who had the privilege of entering it. The family cultivated many sciences and arts. Runge himself played the piano, and he and his children would often render musical classics such as the 'Matthäus Passion'. Runge was a man of affairs and of great personal charm. He was fond of all kinds of sports and practised bicycling, gymnastics, and swimming. At Hannover he used to ride his bicycle a distance of about eight kilometres from his house top the Technische Hochschule four times a day. In all his activities he placed scientific things foremost and was willing to sacrifice everything to their advancement.

Within a year of taking up the professorship at Hannover, Runge had moved away from pure mathematics to study the wavelengths of the spectral lines of elements other than hydrogen. (J J Balmer had constructed a formula for the spectral lines of helium.) In fact Runge's interest in this topic came though Émile du Bois-Reymond in the year before Runge had married his daughter. Du Bois-Reymond had attended a lecture at the Berlin Academy by Heinrich Kayser on the topic, which had interested him, and after Kayser was appointed as Professor of Physics at the Technische Hochschule of Hannover in the autumn of 1885, all three men began to work on the problem together. Kayser and Runge published seven joint papers in the Proceedings of the Berlin Academy of Science over a period of seven years. These papers, covering over 350 pages, made important contributions. In one of these papers, published in 1888, they write that this problem of the wavelengths of the spectral lines of elements other than hydrogen was [1]:-

... affording a much deeper insight into the composition and nature of atoms than any other physical process.

Runge did a great deal of experimental work and published a great quantity of results. He succeeded in arranging the spectral lines of helium in two spectral series and, until 1897, this was thought to be evidence that hydrogen was a mixture of two elements. After working with Kayser for seven years, Kayser left Hannover in 1894 to take up the chair of physics at the University of Bonn. This had become vacant through the tragic death of Heinrich Hertz at age 36 from blood poisoning. Runge continued his work on spectroscopy alone for six months but then persuaded Friedrich Paschen to join him. Paschen was an experimentalist and they worked together at Hannover for seven years. Runge visited England in 1895 and became friendly with Lord Rayleigh. Two years later he travelled to the United States where he became friends with A A Michelson. While in the United States he visited Yerkes Observatory and was offered a professorship by George E Hale but he declined. Paschen writes about their joint work in Hannover [4]:-

We investigated the spectrum of helium when this gas was first discovered by Ramsay. We then turned to the spectra of oxygen, sulphur, and selenium. In these spectra we found for the first time series systems of two different multiplicities.

In 1901 Paschen left Hannover when he was appointed professor of physics at the University of Tübingen. At this stage Runge continued his work with Julius Precht who had previously held the Extraordinary Chair of Theoretical Physics at Heidelberg. Runge and Precht worked [4]:-

... on the spectrum of radium and on the Zeeman effects on its lines. They found the principal groups of lines in the arc and in the spark spectrum and classified them as belonging to that of an alkali-earth.

It is interesting to consider why Runge remained so long at the Technische Hochschule at Hannover while others were appointed to more prestigious chairs. The reason almost certainly is that Runge did not fit in. On the one hand he was considered a traitor to mathematics by many having left the area to undertake research in physics. Physicists, however, considered Runge a mathematician and would have not considered him entirely suitable for a chair of physics at one of the leading institutions. It is worth remarking at this point that Runge always considered himself a mathematician. In 1904 Klein persuaded Göttingen to offer Runge a chair of Applied Mathematics, a post which Runge took up in the October of that year and held until he retired in 1925 [4]:-

At Göttingen he was appointed for teaching and research in applied mathematics. He worked out many numerical and graphical methods, gave numerical solutions of differential equations, etc. He was the pioneer in introducing this kind of mathematics into Germany.

At Göttingen, Runge became highly involved in teaching and undertook less research. He published Analytische Geometrie der Ebene in 1908 and the book Vorlesungen über numerisches Rechnen Ⓣ, co-authored with H König, in 1924. He lectured on Graphical methods at Columbia University in New York from October 1909 to January 1910, being Kaiser Wilhelm Professor of German History and Institutions for the year 1909-1910.

Here is the Introduction to Graphical methods.

Runge was a highly influential figure in the University of Göttingen [1]:-

Despite his liberal political views - open opposition to the annexations during World War I and membership in the Democratic Party afterwards - Runge retained the confidence of his colleagues and his influence in the university.

Runge reached the retirement age of 68 in 1923 but he continued to run his Institute until his successor, Gustav Herglotz, arrived in 1925. We should note, however, that the name of the chair ceased to be 'Applied Mathematics' when Herglotz took over.

We have noted above that Runge was very sporting, a fit and active man even as he grew older, and on his 70th birthday he entertained his grandchildren by doing handstands. He was still mathematically active with several ambitious projects underway. However six months later he had a heart attack and died. His mother Fanny had died in 1910.