卡尔·古斯塔夫·雅各布·雅可比Carl Gustav Jacob Jacobi

卡尔·古斯塔夫·雅各布·雅可比出身于一个犹太家庭，但出生时被赋予法式名字Jacques Simon。他的父亲Simon Jacobi是一位银行家，家境富裕。雅可比是家中次子，长子是Moritz Jacobi，后来成为著名物理学家。Moritz Jacobi在[1]中有独立条目。家中还有一个姐妹Therese 雅可比，以及第三个兄弟Eduard Jacobi，比雅可比年幼。Eduard没有追求学术生涯，而是继承了父亲的职业，成为一名银行家。

雅可比的早期教育由他母亲的一位叔叔提供，随后，就在他十二岁生日前夕，雅可比进入了波茨坦的文理中学。他的叔叔教得很好，而他又有非凡的才能，因此在1817年，还在上学的第一年，他就被安排进了毕业班。这意味着到1816-17学年结束时，他还只有12岁，却已经达到了进入大学所需的标准。然而，柏林大学不接收16岁以下的学生，所以雅可比不得不留在波茨坦文理中学的同一个班级，直到1821年春天。

当然，尽管留在学校的同一个班级，雅可比仍继续推进他的学术研究。他在拉丁语、希腊语和历史方面获得了最高奖项，但他钻研最深的还是数学。到雅可比离开学校时，他已经读过高等数学教材，如莱昂哈德·欧拉的Introductio in analysin infinitorumⓉ（《无穷小分析引论》），并且一直在独自进行研究，试图解决五次方程提出的根式。

雅可比于1821年进入柏林大学，仍不确定自己将专注于哪个学科。他修读了两年哲学、古典学和数学课程，之后才意识到必须在这几门学科之间做出明确决定。他选择了数学，但这并不意味着他能够修读高水平的数学课程，因为当时德国大学数学教育的水平相当差。正如他在文理中学时那样，雅可比不得不自学，阅读约瑟夫·拉格朗日和其他一流数学家的著作。

到1823-24学年结束时，雅可比已经通过了在中学教授数学、希腊语和拉丁语所必需的考试。当然，人们可能会以为他获得教职会有困难，因为正如我们在本文开头所指出的，他是犹太人。他的才华似乎足以克服这一障碍，因为在1825年，他获得了约阿希姆斯塔尔文理中学的教职，这是柏林的一所名校。甚至在收到教职邀请之前，他就已经向柏林大学提交了博士学位论文，并且获准迅速着手撰写他的教授资格论文（Habilitation）论文。

雅可比于1825年向柏林科学院提交了一篇关于迭代函数的论文。然而，审稿人认为这些结果不值得发表，事实上这篇论文并未由柏林科学院发表。这篇论文最终还是发表了，因为1961年它连同评论一起发表在[6]中。[6]的作者Biermann引用了原审稿人的意见，并对他们进行了强烈批评。尽管这对年轻的雅可比来说不是最好的开端，但它并没有长期阻碍他，而他在随后几年的发表记录，无论从数量还是质量来看，都会相当引人注目。

大约在1825年，雅可比从犹太教改信基督教，这使他有可能从事大学教学。到1825-26学年，他已在柏林大学任教。然而柏林的前景并不好，因此在听取同事的建议后，雅可比于1826年5月转往柯尼斯堡大学。在那里，他加入了恩斯特·弗朗茨·诺伊曼和弗里德里希·威廉·贝塞尔的行列，前者也于1825年在柏林获得博士学位，后者则是柯尼斯堡的天文学教授。

雅可比在到达柯尼斯堡之前，已经在数论中做出了重大发现。他现在写信给卡尔·弗里德里希·高斯，告诉他关于三次剩余的结果，这些结果是他受到卡尔·弗里德里希·高斯关于quadratic和biquadratic residues的结果的启发而获得的。卡尔·弗里德里希·高斯对此印象深刻，以至于他写信给弗里德里希·威廉·贝塞尔以获取更多关于这位年轻雅可比的信息。但雅可比在椭圆函数方面也有非凡的新想法（正如尼尔斯·阿贝尔完全独立地、几乎同时所做的那样）。1827年8月5日，雅可比写信给阿德里安-马里·勒让德，后者是该主题的顶尖专家，这封信以及雅可比和阿德里安-马里·勒让德之间的另外22封信，收录在[4]中。

阿德里安-马里·勒让德立刻意识到雅可比在他最喜欢的主题上取得了根本性的进展。不得不说，阿德里安-马里·勒让德对以下事实反应极好：他作为椭圆函数领域顶尖专家的地位一夜之间改变了，因为新理论不仅由雅可比发展，也由尼尔斯·阿贝尔发展。雅可比于1827年12月28日晋升为副教授，主要归功于阿德里安-马里·勒让德对他的高度赞扬。在1828年2月9日写给雅可比的一封信中，阿德里安-马里·勒让德写道：-

看到像您和[尼尔斯·阿贝尔]这样的两位年轻数学家如此成功地耕耘分析的一个分支，我感到非常满意，这个分支长期以来一直是我最喜欢的研究主题，但在我的祖国却没有得到应有的重视。通过你们的著作，你们将自己置于我们时代最优秀分析家的行列。

1829年，雅可比在暑假访问巴黎时遇到了阿德里安-马里·勒让德和其他法国数学家，如约瑟夫·傅里叶和西莫恩·德尼·泊松。在前往巴黎的途中，他在哥廷根拜访了卡尔·弗里德里希·高斯。雅可比关于椭圆函数理论的根本性工作——曾给阿德里安-马里·勒让德留下如此深刻印象——是基于四个theta函数。他于1829年发表的论文Fundamenta nova theoria functionum ellipticarumⓉ（椭圆函数新理论的基础），连同其后来的补充，对椭圆函数理论做出了根本性贡献。然而，尽管雅可比对椭圆函数做出了辉煌贡献，他并未独占这一领域。正如我们上面所指出的，尼尔斯·阿贝尔也在做出根本性贡献，在某种程度上两人之间形成了竞争。阿德里安-马里·勒让德在1829年初写给雅可比的一封信中清楚地表达了这一点：-

先生们，你们在所有这些奇妙的思辨中进展如此迅速，几乎不可能跟上你们——尤其对一个老人来说……我庆幸自己活得够长，得以目睹两位年轻而同样强健的运动员之间这些宽宏大量的竞赛，他们将自己的努力转化为科学的利益，并不断推进科学的边界。

阿德里安-马里·勒让德写下这封信几周后，尼尔斯·阿贝尔去世了。1831年9月11日，雅可比与Marie Schwinck结婚，几个月后，即1832年5月，他在经受了一场长达四小时的拉丁语辩论后被提升为正教授。雅可比作为优秀教师的声誉吸引了许多学生。他引入了讨论班方法来教授学生数学的最新进展。雅可比对这些学生以及他周围的所有其他人[1]产生了重大影响：-

雅可比的个性如此有力，热情如此磅礴，以至于他那些有天赋的学生没有一个能逃脱他的魔力：他们被吸引进他的思想领域，很快代表了一个“学派”。卡尔·威廉·博尔夏特、爱德华·海涅、奥托·黑塞、F J Richelot、J Rosenhain和路德维希·赛德尔属于这个圈子；他们不仅为传播雅可比的数学创造，也为大学教学中新的研究导向态度做出了很大贡献。弗里德里希·威廉·贝塞尔、雅可比和恩斯特·弗朗茨·诺伊曼这三者因此成为德国大学数学复兴的核心。

1833年，雅可比的哥哥莫里茨来到柯尼斯堡与他相聚，并在那里当起了建筑师。莫里茨在那里的两年里对物理学越来越感兴趣，1835年他受聘为多尔帕特土木工程讲席后离开了柯尼斯堡。1834年，雅可比从恩斯特·爱德华·库默尔那里接到了一些工作，后者当时是利格尼茨一所文理中学的教师。文章[20]描述了雅可比如何立即认出了恩斯特·爱德华·库默尔的数学才能。恩斯特·爱德华·库默尔在三阶微分方程方面取得的进展超出了雅可比所达到的水平，雅可比在1836年写信给他的哥哥莫里茨，描述了恩斯特·爱德华·库默尔如何设法解决了他未能解决的问题。

1834年，雅可比证明了如果一个单变量单值函数是双周期的，那么周期的比值是非实的。这一结果推动了该领域的许多进一步工作，特别是约瑟夫·刘维尔和奥古斯丁·路易·柯西的工作。

雅可比在一阶偏微分方程方面进行了重要研究，并将它们应用于动力学的微分方程。他还研究了行列式，并研究了现在称为雅可比行列式的函数行列式。雅可比并不是第一个研究现在以他的名字命名的函数行列式的人，它最早出现在奥古斯丁·路易·柯西1815年的一篇论文中。然而，雅可比在1841年写了一篇长篇回忆录De determinantibus functionalibus Ⓣ（函数行列式），专门讨论这个行列式。他证明了许多事情，其中包括：如果$n$个函数在$n$个变量中函数相关，那么雅可比行列式恒为零；而如果这些函数是独立的，则雅可比行列式不能恒为零。

在[15]中，麦克利里描述了雅可比最令人印象深刻的成果之一：-

曲线整体理论中最漂亮的结果之一是雅可比的定理（1842年）：空间中一条闭可微曲线沿正规方向的球面像将单位球面分成面积相等的区域。这个定理的陈述是雅可比回应托马斯·克劳森（1842年）对雅可比（1836年）早期论文所发表的更正的一篇论文的附言。

1842年7月，雅可比和弗里德里希·威廉·贝塞尔作为普鲁士的代表参加了在曼彻斯特举行的英国协会 for the Advancement of Science会议。雅可比的妻子陪同这两位数学家。他们经巴黎返回柯尼斯堡，雅可比在Académie des Sciences作了演讲。第二年，雅可比身体不适，被诊断出患有糖尿病。他的医生建议他去意大利待一段时间，那里的气候有助于他康复。然而，雅可比并不富有，约翰·彼得·古斯塔夫·勒热纳·狄利克雷在拜访雅可比并发现他的困境后，写信给亚历山大·冯·洪堡，请求他帮助从弗里德里希·威廉四世那里为雅可比争取一些经济援助。

我们应该稍作离题，说明为什么雅可比尽管从富有的父亲那里继承了一小笔财产，却并不富有。整个普鲁士的一场严重商业萧条（事实上是一场全欧洲范围的萧条）导致了一场破产，雅可比在其中失去了所有的钱。现在让我们回到约翰·彼得·古斯塔夫·勒热纳·狄利克雷以及Alexander von Humboldt为帮助雅可比的意大利之行争取支持的尝试。

雅可比曾频繁与Alexander von Humboldt通信。通信始于1828年，但直到1839年之后他们才定期通信，两人之间现存的44封信读来引人入胜（见[5]以及[7]）。约翰·彼得·古斯塔夫·勒热纳·狄利克雷向Friedrich Wilhelm IV提出的请求，在Alexander von Humboldt的大力支持下，获得了成功，雅可比获得了一笔资助，使他能够在意大利待一段时间。他与卡尔·威廉·博尔夏特和约翰·彼得·古斯塔夫·勒热纳·狄利克雷一起出发前往意大利，在几个城镇停留并在卢卡参加了一次数学会议后，他们于1843年11月16日抵达罗马。路德维希·施莱夫利和雅各布·施泰纳也与他们同行，路德维希·施莱夫利担任他们的翻译。

意大利的气候确实帮助雅可比恢复了健康，他开始再次发表作品，此前他的健康曾使他一段时间无法工作。事实上，雅可比对数学的兴趣非常广泛，在罗马期间，他抓住机会满足自己对数学史的兴趣，研究保存在梵蒂冈的丢番图的Arithmetica手稿。尽管他的健康状况有所改善，但人们认为柯尼斯堡的气候对他来说过于极端，不宜返回那里，因此从Friedrich Wilhelm IV那里获得了特许，允许他转到柏林。他获得了薪水的补贴，以帮助抵消柏林较高的生活费用，同时也帮助他支付医疗费用。

1844年6月他已在柏林，尽管健康状况使他无法频繁授课，但他确实在柏林大学讲课。在[21]中，普尔特讨论了他在1847-48年讲授的一门课程：-

雅可比在他关于分析力学的讲座（柏林，1847 - 1848年）中……对约瑟夫·拉格朗日的力学进行了详细而批判性的讨论。约瑟夫·拉格朗日认为力学可以作为一门公理-演绎科学来研究的观点构成了雅可比批评的中心，并基于数学和哲学理由被拒绝。……雅可比的批评源于对数学在经验科学中作用的评价发生了变化。

在22中，Pulte表明雅可比只是到了晚年才持有这些关于分析力学的观点，因为早年他忽视力学的物理解释，而偏爱纯粹公理化和数学化的方法。

到1848年，德意志邦联的情况很糟糕。失业和农作物歉收导致了不满和骚乱。1848年2月巴黎起义推翻Louis-Philippe的消息导致许多州发生革命，柏林发生战斗。共和主义和社会主义情绪意味着君主制陷入困境。雅可比在柏林的宪法俱乐部发表了一次政治演讲，结果既得罪了君主主义者，也得罪了共和主义者。因此，雅可比要求获准加入柏林大学教职的请求被普鲁士政府拒绝。

到1849年夏天，革命被彻底击败。普鲁士政府仍然对雅可比感到不满，取消了他得以在柏林生活的薪水补贴。他不得不搬家，选择了小城哥达。他和家人住在那里，几个月后接受了维也纳大学的讲席。普鲁士政府突然意识到，如果他们迫使雅可比离开普鲁士，将会失去什么，于是他们做出让步，这意味着雅可比可以在柏林大学讲课，而他的家人留在哥达。这对雅可比来说不是一笔好交易，而他接受这一安排的事实意味着他强烈地依恋自己的祖国。

雅可比计划在大学假期与家人共度，1850年夏天他和他们在哥达度过。1851年1月，他感染了流感，随后在体力尚未恢复时又感染了天花。他在感染天花几天后去世。

Scriba在1中将雅可比与莱昂哈德·欧拉作了比较：-

雅可比和莱昂哈德·欧拉在创造数学的方式上是志同道合的。两人都是多产的作家，更是多产的计算者；两人都从大量的算法工作中获得了许多洞见；两人都在数学的许多领域中辛勤耕耘（在这方面，莱昂哈德·欧拉大大超过了雅可比）；两人在任何时刻都能从庞大的数学方法武库中恰好取出那些在攻克给定问题时有望取得最佳结果的武器。

Carl Jacobi came from a Jewish family but he was given the French style name Jacques Simon at birth. His father, Simon Jacobi, was a banker and his family were prosperous. Carl was the second son of the family, the eldest being Moritz Jacobi who eventually became a famous physicist. Moritz Jacobi has an entry in his own right in [1]. There was a sister, Therese Jacobi, and a third brother, Eduard Jacobi, who was younger than Carl. Eduard did not pursue an academic career, but followed instead his father's profession as a banker.

Jacobi's early education was given by an uncle on his mother's side, and then, just before his twelfth birthday, Jacobi entered the Gymnasium in Potsdam. He had been well taught by his uncle and he had remarkable talents so in 1817, while still in his first year of schooling, he was put into the final year class. This meant that by the end of the academic year 1816-17 he was still only 12 years old yet he had reached the necessary standard to enter university. The University of Berlin, however, did not accept students below the age of 16, so Jacobi had to remain in the same class at the Gymnasium in Potsdam until the spring of 1821.

Of course, Jacobi pressed on with his academic studies despite remaining in the same class at school. He received the highest awards for Latin, Greek and history but it was the study of mathematics which he took furthest. By the time Jacobi left school he had read advanced mathematics texts such as Euler's Introductio in analysin infinitorum Ⓣ and had been undertaking research on his own attempting to solve quintic equations by radicals.

Jacobi entered the University of Berlin in 1821 still unsure which topic he would concentrate on. He attended courses in philosophy, classics and mathematics for two years before realising that he had to make a definite decision between these subjects. He chose mathematics, but this did not mean that he could attend high level courses in mathematics for at this time the standard of university education in mathematics in Germany was rather poor. As he had done at the Gymnasium, Jacobi had to study on his own reading the works of Lagrange and other leading mathematicians.

By the end of academic year 1823-24 Jacobi had passed the examinations necessary for him to be able to teach mathematics, Greek, and Latin in secondary schools. Of course, one might have expected him to have problems obtaining a teaching position since, as we noted at the beginning of this article, he was Jewish. His brilliance appears to have been sufficient to allow this hurdle to be overcome for, in 1825, he was offered a teaching post at the Joachimsthalsche Gymnasium, one of the leading schools in Berlin. He had submitted his doctoral dissertation to the University of Berlin even before he received the offer of the teaching post, and he was allowed to move quickly to work on his habilitation thesis.

Jacobi presented a paper concerning iterated functions to the Berlin Academy of Sciences in 1825. However, the referees did not consider the results worth publishing and indeed the paper was not published by the Berlin Academy. The paper was published eventually, for in 1961 it was published with a commentary in [6]. Biermann, the author of [6], quotes the opinions of the original referees and criticises them strongly. Although this was not the best start for the young Jacobi, it did not hold him back for long and his publication record over the following years would be quite remarkable for both the number and quality of the works.

Around 1825 Jacobi changed from the Jewish faith to become a Christian which now made university teaching possible for him. By the academic year 1825-26 he was teaching at the University of Berlin. However prospects in Berlin were not good so, after taking advice from colleagues, Jacobi moved to the University of Königsberg arriving there in May 1826. There he joined Franz Neumann, who had also received his doctorate from Berlin in 1825, and Bessel who was the professor of astronomy at Königsberg.

Jacobi had already made major discoveries in number theory before arriving in Königsberg. He now wrote to Gauss to tell him of the results on cubic residues which he had obtained, having been inspired by Gauss's results on quadratic and biquadratic residues. Gauss was impressed, so much so that he wrote to Bessel to obtain more information about the young Jacobi. But Jacobi also had remarkable new ideas about elliptic functions (as Abel did quite independently and at much the same time). On 5 August 1827 Jacobi wrote to Legendre who was the leading expert on the topic and this letter, together with 22 others between Jacobi and Legendre, is given in [4].

Legendre immediately realised that Jacobi had made fundamental advances in his favourite topic. One would have to say that Legendre reacted extremely well to the realisation that his position as the leading expert on elliptic functions had changed overnight with the new theory being developed not only by Jacobi, but also by Abel. Jacobi's promotion to associate professor on 28 December 1827 was mainly due to the praise heaped on him by Legendre. In a letter, sent to Jacobi on 9 February 1828, Legendre wrote:-

It gives me great satisfaction to see two young mathematicians such as you and [Abel] cultivate with such success a branch of analysis which for such a long time has been my favourite topic of study but which had not been received in my own country as well as it deserves. By your works you place yourselves in the ranks of the best analysts of our era.

In 1829 Jacobi met Legendre and other French mathematicians such as Fourier and Poisson when he made a visit to Paris in the summer vacation. On the journey to Paris he had visited Gauss in Göttingen. Jacobi's fundamental work on the theory of elliptic functions, which had so impressed Legendre, was based on four theta functions. His paper Fundamenta nova theoria functionum ellipticarum Ⓣ published in 1829, together with its later supplements, made fundamental contributions to this theory of elliptic functions. However, despite Jacobi's brilliant contributions to elliptic functions he did not have the field to himself. As we have noted above, Abel was also making fundamental contributions and to some extent a competition had developed between the two. Legendre expressed this clearly in a letter he wrote to Jacobi early in 1829:-

You proceed so rapidly, gentlemen, in all these wonderful speculations that it is nearly impossible to follow you - particularly for an old man ... I congratulate myself that I have lived long enough to witness these magnanimous contests between two young equally strong athletes, who turn their efforts to the profit of the science whose limits they push back further and further.

A few weeks after Legendre wrote this letter Abel died. On 11 September 1831 Jacobi married Marie Schwinck then, a few months later in May 1832, he was promoted to full professor after being subjected to a four hour disputation in Latin. Jacobi's reputation as an excellent teacher attracted many students. He introduced the seminar method to teach students the latest advances in mathematics. Jacobi had a major impact on these students and all others around him [1]:-

Such were Jacobi's forceful personality and sweeping enthusiasm that none of his gifted students could escape his spell: they were drawn into his sphere of thought, and soon represented a "school". C W Borchardt, E Heine, L O Hesse, F J Richelot, J Rosenhain, and P L von Seidel belonged to this circle; they contributed much to the dissemination not only of Jacobi's mathematical creations but also the new research-oriented attitude in university instruction. The triad of Bessel, Jacobi, and Franz Neumann thus became the nucleus of a revival of mathematics at German universities.

In 1833 Jacobi's older brother Moritz joined him in Königsberg where he set himself up as an architect. During the two years Moritz spent there he became more interested in physics and left Königsberg in 1835 when he was appointed to the chair of civil engineering at Dorpat. In 1834 Jacobi received some work from Kummer who was at this time a teacher in a Gymnasium in Liegnitz. The article [20] describes how Jacobi immediately recognised Kummer's mathematical talents. Kummer had made advances beyond what Jacobi had achieved on third-order differential equations and Jacobi wrote to his brother Moritz in 1836 describing how Kummer had managed to solve problems which had defeated him.

In 1834 Jacobi proved that if a single-valued function of one variable is doubly periodic then the ratio of the periods is non-real. This result prompted much further work in this area, in particular by Liouville and Cauchy.

Jacobi carried out important research in partial differential equations of the first order and applied them to the differential equations of dynamics. He also worked on determinants and studied the functional determinant now called the Jacobian. Jacobi was not the first to study the functional determinant which now bears his name, it appears first in a 1815 paper of Cauchy. However Jacobi wrote a long memoir De determinantibus functionalibus Ⓣ in 1841 devoted to this determinant. He proved, among many other things, that if a set of $n$ functions in $n$ variables are functionally related then the Jacobian is identically zero, while if the functions are independent the Jacobian cannot be identically zero.

In [15] McCleary describes one of Jacobi's most impressive results:-

One of the prettiest results in the global theory of curves is a theorem of Jacobi (1842): The spherical image of the normal directions along a closed differentiable curve in space divides the unit sphere into regions of equal area. The statement of this theorem is an afterthought to a paper in which Jacobi responds to the published correction by Thomas Clausen (1842) of an earlier paper by Jacobi (1836).

In July 1842 Jacobi and Bessel attended the meeting of the British Association for the Advancement of Science in Manchester as representatives of Prussia. Jacobi's wife accompanied the two mathematicians. They returned to Königsberg via Paris where Jacobi lectured at the Académie des Sciences. In the following year Jacobi became unwell and diabetes was diagnosed. He was advised by his doctor to spend time in Italy where the climate would help him recover. However, Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and discovering his plight, wrote to Alexander von Humboldt asking him to help obtain some financial assistance for Jacobi from Friedrich Wilhelm IV.

We should make a small digression to say why Jacobi was not a wealthy man despite having inherited a small fortune from his wealthy father. A severe business depression throughout Prussia (in fact it was a Europe wide depression), had led to a bankruptcy in which Jacobi had lost all his money. Let us now return to Dirichlet and Alexander von Humboldt's attempts to help obtain support for Jacobi's trip to Italy.

Jacobi had frequently corresponded with Alexander von Humboldt. The correspondence began in 1828 but only after 1839 did they correspond regularly and the 44 surviving letters between the two men make fascinating reading (see [5] and also [7]). Dirichlet's request to Friedrich Wilhelm IV, supported strongly by Alexander von Humboldt, was successful and Jacobi received a grant to allow him to spend time in Italy. He set off for Italy with Borchardt and Dirichlet and, after stopping in several towns and attending a mathematical meeting in Lucca, they arrived in Rome on 16 November 1843. Schläfli and Steiner were also with them, Schläfli being their interpreter.

The climate in Italy did indeed help Jacobi to recover and he began to publish again, his health having prevented him working for some time before this. In fact Jacobi's interests in mathematics were very wide and while in Rome he took the opportunity to satisfy his interest in the history of mathematics working on manuscripts of Diophantus's Arithmetica which were kept in the Vatican. Although his health had improved it was felt that the climate of Königsberg was too extreme for him to return there, so a dispensation was obtained from Friedrich Wilhelm IV to allow him to transfer to Berlin. He was given a supplement to his salary to help offset the higher costs of living in Berlin, and also to help him with his medical expenses.

He was in Berlin by June 1844 and although his health prevented him from giving frequent lecture courses, he did lecture at the University of Berlin. In [21] a lecture course which he gave in 1847-48 is discussed by Pulte:-

Jacobi in his lectures on analytical mechanics (Berlin, 1847 - 1848) ... gave a detailed and critical discussion of Lagrange's mechanics. Lagrange's view that mechanics could be pursued as an axiomatic-deductive science forms the centre of Jacobi's criticism and is rejected on mathematical and philosophical grounds. ... Jacobi's criticism is motivated by a changed evaluation of the role of mathematics in the empirical sciences.

In [22] Pulte shows that Jacobi only came to hold these views on analytical mechanics only later in his life, for earlier he had ignored the physical interpretation of mechanics in favour of a purely axiomatic and mathematical approach.

By 1848 conditions were bad in the German Confederation. Unemployment and crop failures had led to discontent and disturbances. The news that Louis-Philippe had been overthrown by an uprising in Paris in February 1848 led to revolutions in many states and fighting in Berlin. Republican and socialist feelings meant that the monarchy was in trouble. Jacobi made a political speech in the Constitutional Club in Berlin which managed to upset both the monarchists and the republicans. As a consequence Jacobi's request to be allowed to join the staff of the University of Berlin was refused by the Prussian government.

By the summer of 1849 the revolution was completely defeated. The Prussian government, still feeling aggrieved at Jacobi, took away the supplement to his salary which allowed him to live in Berlin. He had to move, and chose the small town of Gotha. He lived there with his family and a few months later accepted a chair at the University of Vienna. The Prussian government suddenly realised what they would lose if they forced Jacobi to leave Prussia, so they made concessions which meant that Jacobi could lecture at the University of Berlin while his family remained in Gotha. It was not a good deal for Jacobi and the fact that he accepted it means that he was strongly attached to his own country.

Jacobi planned to spend the university vacations with his family and he spent the summer of 1850 with them in Gotha. In January 1851 he contracted influenza, then he contracted smallpox before he had regained his strength. He died a few days after contracting smallpox.

Scriba, in [1], compares Jacobi with Euler:-

Jacobi and Euler were kindred spirits in the way they created their mathematics. Both were prolific writers and even more prolific calculators; both drew a great deal of insight from immense algorithmical work; both laboured in many fields of mathematics (Euler, in this respect, greatly surpassed Jacobi); and both at any moment could draw from the vast armoury of mathematical methods just those weapons which would promise the best results in the attack of a given problem.