卡尔·魏尔斯特拉斯Karl Weierstraß

卡尔·魏尔斯特拉斯的父亲Wilhelm Weierstrass在魏尔斯特拉斯出生时是Ostenfelde市长的秘书。Wilhelm Weierstrass是一个受过良好教育的人，对艺术和科学有广泛的了解。他当然有能力获得比他所获得的更高的职位，这种态度可能是魏尔斯特拉斯早期职业生涯中职位远低于其杰出能力的原因之一。魏尔斯特拉斯的母亲是Theodora Vonderforst，魏尔斯特拉斯是Theodora和Wilhelm四个孩子中的长子，他们都没有结婚。

魏尔斯特拉斯八岁时，Wilhelm Weierstrass成为一名税务稽查员。这份工作使他只能在任何一个地方短暂停留，因此随着全家在普鲁士各地搬迁，魏尔斯特拉斯频繁地转学。1827年，魏尔斯特拉斯的母亲Theodora去世，一年后他的父亲Wilhelm再婚。到1829年，Wilhelm Weierstrass已成为Paderborn主要税务办公室的助理，魏尔斯特拉斯则进入了那里的天主教文理中学。尽管魏尔斯特拉斯不得不兼职做簿记员以帮助家庭财务，他在文理中学仍然表现出色。

在文理中学期间，魏尔斯特拉斯无疑达到了远超预期的数学能力水平。他经常阅读奥古斯都·利奥波德·克雷勒的Journal，并给他的一个兄弟辅导数学。然而，魏尔斯特拉斯的父亲希望他学习金融，因此，在1834年从文理中学毕业后，他进入了波恩大学，为他规划了包括法律、金融和经济学在内的课程。考虑到他父亲为他规划的普鲁士行政部门的职业生涯，这确实是一门精心设计的课程。然而，魏尔斯特拉斯陷入了要么遵从父亲的意愿，要么学习他所热爱的学科即数学的冲突之中。

魏尔斯特拉斯内心冲突的结果是，他既没有参加数学讲座，也没有参加他计划课程的讲座。他对内心冲突的反应是假装不在乎自己的学业，并花了四年时间 intensive fencing and drinking。正如比曼在[1]中所写：-

……责任与志趣之间的冲突导致了身心俱疲。他试图通过参与无忧无虑的学生生活来克服自己的问题，但徒劳无功……

然而，他确实自学了数学，阅读了皮埃尔·西蒙·拉普拉斯的Mécanique célesteⓉ（天体力学），然后又读了卡尔·古斯塔夫·雅各布·雅可比关于椭圆函数的著作。他通过研读克里斯托夫·古德曼的讲课记录，逐渐理解了椭圆函数论中必要的方法。在大约50年后写给索菲斯·李的一封信中，他解释了大约在这个时候，尽管父亲反对，他如何最终下定决心学习数学（见[1]）：-

……当我在学生时代于奥古斯都·利奥波德·克雷勒的期刊中注意到[尼尔斯·阿贝尔写给阿德里安-马里·勒让德的一封信]时，[它]至关重要。从定义该函数的微分方程直接推导出尼尔斯·阿贝尔所给出的函数表示形式……，是我为自己设定的第一个数学任务；其幸运的解决使我决心全身心投入数学；我在第七个学期做出了这个决定……

魏尔斯特拉斯已经决定成为一名数学家，但他本应还在学习公共财政与行政管理的课程。做出决定后，他又在波恩大学度过了一个学期，他的第八个学期于1838年结束，由于未能学习他所注册的科目，他没有参加考试就离开了大学。魏尔斯特拉斯的父亲对儿子放弃学业感到极度不安。他被一位家族朋友，帕德博恩法院院长，说服允许魏尔斯特拉斯在明斯特神学与哲学学院学习，以便他能参加必要的考试，成为一名中学教师。

1839年5月22日，魏尔斯特拉斯在明斯特的学院注册。克里斯托夫·古德曼在明斯特授课，这就是魏尔斯特拉斯如此热衷于在那里学习的原因。魏尔斯特拉斯参加了克里斯托夫·古德曼关于椭圆函数的讲座，这是该主题最早的讲座之一，克里斯托夫·古德曼大力鼓励魏尔斯特拉斯进行数学研究。1839年秋天离开明斯特后，魏尔斯特拉斯为1840年3月报名的教师资格考试而学习。然而，此时魏尔斯特拉斯的父亲再次换了工作，于1840年1月成为一家盐厂的厂长，全家现在住在利珀河畔利普施塔特附近的Westernkotten，位于帕德博恩以西。

应魏尔斯特拉斯的要求，他在1840年5月收到的试卷上得到了一道关于椭圆函数表示的题目，他提交了自己的重要研究作为答案。克里斯托夫·古德曼评阅了试卷，并评价了魏尔斯特拉斯的贡献：-

……与那些被冠以荣耀的发现者同等地位。

晚年时，魏尔斯特拉斯得知克里斯托夫·古德曼的评论后说，如果当时知道，他会发表自己的结果。魏尔斯特拉斯也评论说克里斯托夫·古德曼在赞扬时多么慷慨，尤其是因为他曾对克里斯托夫·古德曼的方法提出过严厉批评。

到1841年4月，魏尔斯特拉斯已通过必要的口试，并开始在明斯特的文理中学担任教师，为期一年试用期。尽管此时他没有发表任何数学作品，但他在1841年和1842年写了三篇短文，这些在[3]中有描述：-

魏尔斯特拉斯在1857年之后晚年基于其复变量函数理论的概念，明确见于他1841年至1842年在明斯特所写的未发表作品中，当时他仍受克里斯托夫·古德曼的影响。他将解析函数的概念从可微函数转变为可展开为收敛幂级数的函数，这一转变发生在魏尔斯特拉斯数学活动的早期阶段。

魏尔斯特拉斯于1842年在西普鲁士（今波兰）的德意志克罗内高级文理中学开始其作为合格数学教师的职业生涯，直到1848年他搬到布劳恩斯贝格的霍西恩学院。作为数学教师，他还被要求教授其他科目，魏尔斯特拉斯教授物理、植物学、地理、历史、德语、书法甚至体操。晚年时，魏尔斯特拉斯描述这些悲惨岁月为“无尽的沉闷和无聊”，其中[1]：-

……他既没有同事进行数学讨论，也无法访问数学图书馆，而科学信件的交流是他负担不起的奢侈。

大约从1850年起，魏尔斯特拉斯开始遭受眩晕发作，这些发作非常严重，大约一小时后以剧烈呕吐结束。在大约12年期间频繁发作使他难以工作，人们认为这些问题很可能源于他作为学生时所遭受的精神冲突，以及他在从事繁重教学工作的同时，利用每一分钟空闲时间致力于数学所带来的压力。

毫不奇怪，当魏尔斯特拉斯在布劳恩斯贝格学校简介上发表关于abelian functions的论文时，它们并未引起数学家的注意。然而，1854年，他在Crelle's Journal上发表了Zur Theorie der Abelschen FunctionenⓉ（《论阿贝尔函数理论》），这无疑引起了关注。这篇论文并未给出魏尔斯特拉斯所发展的超椭圆积分反演的全套理论，而是初步描述了他的方法，涉及将阿贝尔函数表示为恒收敛的幂级数。

随着这篇论文魏尔斯特拉斯一举成名。柯尼斯堡大学于1854年3月31日授予他荣誉博士学位。1855年，魏尔斯特拉斯申请了布雷斯劳大学的讲席，该职位因恩斯特·爱德华·库默尔迁往柏林而空缺。然而，恩斯特·爱德华·库默尔试图施加影响，使魏尔斯特拉斯去柏林而非布雷斯劳，因此魏尔斯特拉斯未获任命。约翰·彼得·古斯塔夫·勒热纳·狄利克雷于1855年写给普鲁士文化大臣的一封信强烈支持给予魏尔斯特拉斯一个大学职位。详情见[10]。

在布劳恩斯贝格被提升为高级讲师后，魏尔斯特拉斯获得了一年休假，以便专心从事高等数学研究。然而，他已经决定，永远不会再回到学校教书。

魏尔斯特拉斯 在下一篇论文 Theorie der Abelschen Functionen Ⓣ（阿贝尔函数论）中发表了他的超椭圆积分反演理论的完整版本，该文于 1856 年发表在 Crelle's Journal 上。有多所大学有意为他提供讲席。当奥地利的大学还在讨论这一前景时，柏林工业学院（后来的技术高等学校）向他发出了讲席邀请。尽管他更愿意去柏林大学，但 魏尔斯特拉斯 肯定不想回到布劳恩斯贝格的 Collegium Hoseanum，因此他于 1856 年 6 月 14 日接受了该学院的邀请。

不断有人向魏尔斯特拉斯发出邀请，因此当他在1856年9月参加维也纳的一次会议时，有人向他提供了奥地利任何一所大学他自选的讲席。在他决定如何对待这一邀请之前，柏林大学于10月向他提供了一个教授职位。这正是他长久以来想要的工作，他迅速接受了，尽管由于年初已接受了工业学院的邀请，他若干年内无法正式就任柏林大学的讲席。

魏尔斯特拉斯成功的数学讲座吸引了来自世界各地的学生。他讲座的主题包括：Fourier series和积分在数学物理中的应用（1856/57），解析函数论导论（他在其中阐述了他1841年获得但从未发表的结果），椭圆函数论（他的主要研究课题），以及在几何和力学问题中的应用。

在1859/60年的讲座中，魏尔斯特拉斯讲授了Introduction to analysis，他首次处理了该学科的基础。1860/61年，他讲授了Integral calculus。

我们在上文描述了魏尔斯特拉斯从1850年起遭受的健康问题。尽管他已经获得了他梦寐以求的职位，但他的健康在1861年12月彻底崩溃。他花了大约一年时间才恢复到足以再次讲课，而且他再也没有完全恢复健康。从那时起，他坐着讲课，由一名学生在黑板上为他书写。他从1850年起遭受的发作停止了，取而代之的是胸部问题。

在1863/64年关于The general theory of analytic functions的课程中，魏尔斯特拉斯开始阐述他的实数理论。在1863年的讲座中，他证明了复数是实数的唯一交换代数扩张。卡尔·弗里德里希·高斯曾在1831年承诺给出这一证明，但未能给出。

1872年，他对严格性的强调使他发现了一个函数，该函数虽然连续，但在任何点都没有导数。那些严重依赖直觉进行发现的分析学家对这个反直觉的函数相当沮丧。波恩哈德·黎曼曾在1861年提出可以找到这样的函数，但他的例子未能在所有点上都不可微。

魏尔斯特拉斯的讲座发展成了一门四个学期的课程，他一直讲授到1890年。这四门课程是

1. 《解析函数论导引》，
2. 《椭圆函数》，
3. 《阿贝尔函数》，
4. 变分法或椭圆函数的应用。

多年来，这些课程不断发展，已出版了若干版本，例如威廉·基灵在1868年所作的笔记以及阿道夫·赫维兹在1878年所作的笔记。魏尔斯特拉斯的方法至今仍主导着分析教学，这一点从这些讲座的内容和风格中可以清楚地看出，尤其是Introduction课程。其内容包括：数、带有魏尔斯特拉斯幂级数方法的函数概念、连续性与可微性、解析延拓、奇点、多变量解析函数，特别是魏尔斯特拉斯的“预备定理”，以及围道积分。

在柏林，魏尔斯特拉斯有两位同事恩斯特·爱德华·库默尔和利奥波德·克罗内克，三人共同使柏林获得了作为学习数学的首选大学的声誉。利奥波德·克罗内克多年来是魏尔斯特拉斯的密友，但在1877年，利奥波德·克罗内克对格奥尔格·康托尔工作的反对导致两人之间产生了裂痕。这变得如此严重，以至于在1885年的某个阶段，魏尔斯特拉斯决定离开柏林去瑞士。然而，他改变了主意，留在了柏林。

大量学生受益于魏尔斯特拉斯的教学。我们列举几位在我们档案中其他地方提到过的：保罗·巴赫曼、Bolza、格奥尔格·康托尔、Engel、费迪南德·格奥尔格·弗罗贝尼乌斯、Gegenbauer、库尔特·亨泽尔、奥托·赫尔德、阿道夫·赫维兹、威廉·基灵、菲利克斯·克莱因、Kneser、莱奥·柯尼希斯贝格尔、马蒂亚斯·莱尔希、索菲斯·李、雅各布·吕罗特、弗朗茨·梅滕斯、赫尔曼·闵可夫斯基、约斯塔·米塔格-莱弗勒、欧根·尼托、弗里德里希·赫曼·肖特基、赫尔曼·阿曼杜斯·施瓦茨和奥托·施托尔茨。然而，有一位学生特别值得单独提及。

1870年，柯瓦列夫斯卡娅来到柏林，魏尔斯特拉斯私下教授她，因为她不被允许进入大学。就魏尔斯特拉斯而言，她显然是一个非常特殊的学生，因为他写信给她说：-

……梦想并陶醉于如此多的谜题，这些谜题留待我们解决，关于有限和无限空间，关于世界体系的稳定性，以及关于未来数学和物理学的所有其他重大问题。……你一直亲近……在我整个一生中……我从未发现任何人能像你一样带给我对科学最高目标的如此理解，以及对我的意图和基本原则的如此愉快的认同。

正是通过魏尔斯特拉斯的努力，柯瓦列夫斯卡娅获得了哥廷根大学的荣誉博士学位，他还利用自己的影响力帮助她在1883年获得了斯德哥尔摩的职位。魏尔斯特拉斯和柯瓦列夫斯卡娅在1871年至1890年间通信了20年。交换了160多封信（见[5]、[7]等），但魏尔斯特拉斯在她死后烧毁了柯瓦列夫斯卡娅的信件。

魏尔斯特拉斯设定的严格标准，例如将无理数数定义为收敛级数的极限，强烈影响了数学的未来。他还研究了整函数、一致收敛的概念以及由无限乘积定义的函数。他的努力在[2]中总结如下：-

被称为现代分析之父的魏尔斯特拉斯设计了级数收敛的检验方法，并为周期函数、实变量函数、椭圆函数、阿贝尔函数、收敛无限乘积和变分法理论做出了贡献。他还推进了双线性和二次型的理论。

魏尔斯特拉斯发表的[1]很少：-

……因为他那种批判性的判断力总是迫使他把任何分析都建立在牢固的基础上，从新的进路出发，并不断修正和扩展。

然而，他确实编辑了雅各布·施泰纳的全集和卡尔·古斯塔夫·雅各布·雅可比的全集。他决定主持出版自己的全集，就他而言，这将涉及大量来自他讲课课程中未发表的材料，而魏尔斯特拉斯意识到，没有他的帮助，这将是一项困难的任务。前两卷于1894年和1895年出版，是在他1897年去世之前出版的仅有的两卷。他最后几年很艰难[1]：-

在他生命的最后三年里，他困于轮椅，无法行动，依赖他人。他死于肺炎。

他的Complete Works剩余各卷出版缓慢；第3卷于1903年，第4卷于1902年，第5卷和第6卷于1915年，第7卷于1927年。这七卷于1967年重印。更多作品至今仍在继续出版，特别是根据听课者所做的笔记整理而成的他的讲课课程版本。

Karl Weierstrass's father, Wilhelm Weierstrass, was secretary to the mayor of Ostenfelde at the time of Karl's birth. Wilhelm Weierstrass was a well educated man who had a broad knowledge of the arts and of the sciences. He certainly was well capable of attaining higher positions than he did, and this attitude may have been one of the reasons that Karl Weierstrass's early career was in posts well below his outstanding ability. Weierstrass's mother was Theodora Vonderforst and Karl was the eldest of Theodora and Wilhelm's four children, none of whom married.

Wilhelm Weierstrass became a tax inspector when Karl was eight years old. This job involved him in only spending short periods in any one place so Karl frequently moved from school to school as the family moved around Prussia. In 1827 Karl's mother Theodora died and one year later his father Wilhelm remarried. By 1829 Wilhelm Weierstrass had become an assistant at the main tax office in Paderborn, and Karl entered the Catholic Gymnasium there. Weierstrass excelled at the Gymnasium despite having to take on a part-time job as a bookkeeper to help out the family finances.

While at the Gymnasium Weierstrass certainly reached a level of mathematical competence far beyond what would have been expected. He regularly read Crelle's Journal and gave mathematical tuition to one of his brothers. However Weierstrass's father wished him to study finance and so, after graduating from the Gymnasium in 1834, he entered the University of Bonn with a course planned out for him which included the study of law, finance and economics. With the career in the Prussian administration that was planned for him by his father, this was indeed a well designed course. However, Weierstrass suffered from the conflict of either obeying his father's wishes or studying the subject he loved, namely mathematics.

The result of the conflict which went on inside Weierstrass was that he did not attend either the mathematics lectures or the lectures of his planned course. He reacted to the conflict inside him by pretending that he did not care about his studies, and he spent four years of intensive fencing and drinking. As Biermann writes in [1]:-

... the conflict between duty and inclination led to physical and mental strain. He tried, in vain, to overcome his problems by participating in carefree student life ...

He did study mathematics on his own, however, reading Laplace's Mécanique céleste Ⓣ and then a work by Jacobi on elliptic functions. He came to understand the necessary methods in elliptic function theory by studying transcripts of lectures by Gudermann. In a letter to Lie, written nearly 50 years later, he explained how he came to make the definite decision to study mathematics despite his father's wishes around this time (see [1]):-

... when I became aware of [a letter from Abel to Legendre] in Crelle's Journal during my student years, [it] was of the utmost importance. The immediate derivation of the form of the representation of the function given by Abel ..., from the differential equation defining this function, was the first mathematical task I set myself; and its fortunate solution made me determined to devote myself wholly to mathematics; I made this decision in my seventh semester ...

Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration. After his decision, he spent one further semester at the University of Bonn, his eighth semester ending in 1838, and having failed to study the subjects he was enrolled for he simply left the University without taking the examinations. Weierstrass's father was desperately upset by his son giving up his studies. He was persuaded by a family friend, the president of the law courts at Paderborn, to allow Karl to study at the Theological and Philosophical Academy of Münster so that he could take the necessary examinations to become a secondary school teacher.

On 22 May 1839 Weierstrass enrolled at the Academy in Münster. Gudermann lectured in Münster and this was the reason that Weierstrass was so keen to study there. Weierstrass attended Gudermann's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies. Leaving Münster in the autumn of 1839, Weierstrass studied for the teacher's examination which he registered for in March 1840. By this time, however, Weierstrass's father had moved jobs yet again, becoming director of a salt works in January 1840, and the family was now living in Westernkotten near Lippstadt on the Lippe River, west of Paderborn.

At Weierstrass's request he was given a question on the paper he received in May 1840 on the representation of elliptic functions and he presented his own important research as an answer. Gudermann assessed the paper and rated Weierstrass's contribution:-

... of equal rank with the discoverers who were crowned with glory.

When, in later life, Weierstrass learnt of Gudermann's comments he said that he would have published his results had he known. Weierstrass also commented on how generous Gudermann had been in his praise, particularly since he had been highly critical of Gudermann's methods.

By April 1841 Weierstrass had taken the necessary oral examinations and he began one year probation as a teacher at the Gymnasium in Münster. Although he did not publish any mathematics at this time, he wrote three short papers in 1841 and 1842 which are described in [3]:-

The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann. The transformation of his conception of an analytic function from a differentiable function to a function expansible into a convergent power series was made during this early period of Weierstrass's mathematical activity.

Weierstrass began his career as a qualified teacher of mathematics at the Pro-Gymnasium in Deutsch Krone in West Prussia (now Poland) in 1842 where he remained until he moved to the Collegium Hoseanum in Braunsberg in 1848. As a teacher of mathematics he was required to teach other topics too, and Weierstrass taught physics, botany, geography, history, German, calligraphy and even gymnastics. In later life Weierstrass described the "unending dreariness and boredom" of these miserable years in which [1]:-

... he had neither a colleague for mathematical discussions nor access to a mathematical library, and that the exchange of scientific letters was a luxury that he could not afford.

From around 1850 Weierstrass began to suffer from attacks of dizziness which were very severe and which ended after about an hour in violent sickness. Frequent attacks over a period of about 12 years made it difficult for him to work and it is thought that these problems may well have been caused by the mental conflicts he had suffered as a student, together with the stress of applying himself to mathematics in every free minute of his time while undertaking the demanding teaching job.

It is not surprising that when Weierstrass published papers on abelian functions in the Braunsberg school prospectus they went unnoticed by mathematicians. However, in 1854 he published Zur Theorie der Abelschen Functionen Ⓣ in Crelle's Journal and this was certainly noticed. This paper did not give the full theory of inversion of hyperelliptic integrals that Weierstrass had developed but rather gave a preliminary description of his methods involving representing abelian functions as constantly converging power series.

With this paper Weierstrass burst from obscurity. The University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1855 Weierstrass applied for the chair at the University of Breslau left vacant when Kummer moved to Berlin. Kummer, however, tried to influence things so that Weierstrass would go to Berlin, not Breslau, so Weierstrass was not appointed. A letter from Dirichlet to the Prussian Minister of Culture written in 1855 strongly supported Weierstrass being given a university appointment. Details are given in [10].

After being promoted to senior lecturer at Braunsberg, Weierstrass obtained a year's leave of absence to devote himself to advanced mathematical study. He had already decided, however, that he would never return to school teaching.

Weierstrass published a full version of his theory of inversion of hyperelliptic integrals in his next paper Theorie der Abelschen Functionen Ⓣ in Crelle's Journal in 1856. There was a move from a number of universities to offer him a chair. While universities in Austria were discussing the prospect, an offer of a chair came from the Industry Institute in Berlin (later the Technische Hochschule). Although he would have prefered to go to the University of Berlin, Weierstrass certainly did not want to return to the Collegium Hoseanum in Braunsberg so he accepted the offer from the Institute on 14 June 1856.

Offers continued to be made to Weierstrass so that when he attended a conference in Vienna in September 1856 he was offered a chair at any Austrian university of his choice. Before he had decided what to do about this offer, the University of Berlin offered him a professorship in October. This was the job he had long wanted and he accepted quickly, although having accepted the offer from the Industry Institute earlier in the year he was not able to formally occupy the University of Berlin chair for some years.

Weierstrass's successful lectures in mathematics attracted students from all over the world. The topics of his lectures included:- the application of Fourier series and integrals to mathematical physics (1856/57), an introduction to the theory of analytic functions (where he set out results he had obtained in 1841 but never published), the theory of elliptic functions (his main research topic), and applications to problems in geometry and mechanics.

In his lectures of 1859/60 Weierstrass gave Introduction to analysis where he tackled the foundations of the subject for the first time. In 1860/61 he lectured on the Integral calculus.

We described above the health problems that Weierstrass suffered from 1850 onwards. Although he had achieved the positions that he had dreamed of, his health gave out in December 1861 when he collapsed completely. It took him about a year to recover sufficiently to lecture again and he was never to regain his health completely. From this time on he lectured sitting down while a student wrote on the blackboard for him. The attacks that he had suffered from 1850 stopped and were replaced by chest problems.

In his 1863/64 course on The general theory of analytic functions Weierstrass began to formulate his theory of the real numbers. In his 1863 lectures he proved that the complex numbers are the only commutative algebraic extension of the real numbers. Gauss had promised a proof of this in 1831 but had failed to give one.

In 1872 his emphasis on rigour led him to discover a function that, although continuous, had no derivative at any point. Analysts who depended heavily upon intuition for their discoveries were rather dismayed at this counter-intuitive function. Riemann had suggested in 1861 that such a function could be found, but his example failed to be non-differentiable at all points.

Weierstrass's lectures developed into a four-semester course which he continued to give until 1890. The four courses were

1. Introduction to the theory of analytic functions,
2. Elliptic functions,
3. Abelian functions,
4. Calculus of variations or applications of elliptic functions.

Through the years the courses developed and a number of versions have been published such as the notes by Killing made in 1868 and those by Hurwitz from 1878. Weierstrass's approach still dominates teaching analysis today and this is clearly seen from the contents and style of these lectures, particularly the Introduction course. Its contents were: numbers, the function concept with Weierstrass's power series approach, continuity and differentiability, analytic continuation, points of singularity, analytic functions of several variables, in particular Weierstrass's "preparation theorem", and contour integrals.

At Berlin, Weierstrass had two colleagues Kummer and Kronecker and together the three gave Berlin a reputation as the leading university at which to study mathematics. Kronecker was a close friend of Weierstrass's for many years but in 1877 Kronecker's opposition to Cantor's work cause a rift between the two men. This became so bad that at one stage, in 1885, Weierstrass decided to leave Berlin and go to Switzerland. However, he changed his mind and remained in Berlin.

A large number of students benefited from Weierstrass's teaching. We name a few who are mentioned elsewhere in our archive: Bachmann, Bolza, Cantor, Engel, Frobenius, Gegenbauer, Hensel, Hölder, Hurwitz, Killing, Klein, Kneser, Königsberger, Lerch, Lie, Lüroth, Mertens, Minkowski, Mittag-Leffler, Netto, Schottky, Schwarz and Stolz. One student in particular, however, deserves special mention.

In 1870 Sofia Kovalevskaya came to Berlin and Weierstrass taught her privately since she was not allowed admission to the university. Clearly she was a very special student as far as Weierstrass was concerned for he wrote to her that he:-

... dreamed and been enraptured of so many riddles that remain for us to solve, on finite and infinite spaces, on the stability of the world system, and on all the other major problems of the mathematics and the physics of the future. ... you have been close ...throughout my entire life ... and never have I found anyone who could bring me such understanding of the highest aims of science and such joyful accord with my intentions and basic principles as you.

It was through Weierstrass's efforts that Kovalevskaya received an honorary doctorate from Göttingen, and he also used his influence to help her obtain the post in Stockholm in 1883. Weierstrass and Kovalevskaya corresponded for 20 years between 1871 to 1890. More than 160 letters were exchanged (see [5], [7] etc.), but Weierstrass burnt Kovalevskaya's letters after her death.

The standards of rigour that Weierstrass set, defining, for example, irrational numbers as limits of convergent series, strongly affected the future of mathematics. He also studied entire functions, the notion of uniform convergence and functions defined by infinite products. His effort are summed up in [2] as follows:-

Known as the father of modern analysis, Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations. He also advanced the theory of bilinear and quadratic forms.

Weierstrass published little [1]:-

... because his critical sense invariably compelled him to base any analysis on a firm foundation, starting from a fresh approach and continually revising and expanding.

However, he did edit the complete works of Steiner and those of Jacobi. He decided to supervise the publication of his own complete works, in his case this would involve a great deal of unpublished material from his lecture courses and Weierstrass realised that without his help this would be a difficult task. The first two volumes appeared in 1894 and 1895, being the only ones to appear before his death in 1897. His last years were difficult [1]:-

During his last three years he was confined to a wheelchair, immobile and dependent. He died of pneumonia.

The remaining volumes of his Complete Works appeared slowly; volume 3 in 1903, volume 4 in 1902, volumes 5 and 6 in 1915, and volume 7 in 1927. The seven volumes were reprinted in 1967. More work continues to be published today, particularly versions of his lecture courses taken from the notes made by those who attended the lectures.