雅各布·施泰纳Jakob Steiner

雅各布·施泰纳的父母是安娜·芭芭拉·韦伯（1757-1832）和尼克劳斯·施泰纳（1752-1826）。安娜和尼克劳斯于1780年1月28日结婚，育有八个孩子。施泰纳是孩子中最小的，早年帮助父母打理他们在乌岑斯托夫村附近经营的小农场和生意，该村位于伯尔尼以北约24公里处。他直到14岁才学会读写，但随后证明自己极为宝贵[1]：-

童年时他不得不在农场和生意上帮助父母：他的计算技能帮了大忙。

施泰纳，然而，他想为自己谋求更好的出路，但他的父母却很高兴他能帮忙打理生意。18岁时，他违背父母的意愿，离家前往纳沙泰尔湖东南端的伊韦尔东，就读于约翰·海因里希·裴斯泰洛齐的学校。裴斯泰洛齐从1805年到1825年在该镇经营着他那所创新的学校，施泰纳于1814年春季入学。施泰纳无力为在该校的教育支付任何费用，这并非问题，因为裴斯泰洛齐想在穷人身上试验他的教育方法。裴斯泰洛齐的学校对施泰纳对待数学教学的态度以及他从事数学研究时的哲学都产生了非常显著的影响。他在1826年写道（例如见[1]）：-

裴斯泰洛齐学校所使用的方法，把数学真理视为独立反思的对象，这使我在那里求学时，去为课程中提出的定理寻找教师所提供的根据之外的其它根据。只要可能，我就寻找更深层的基础，而且我常常成功，以至于我的教师宁愿要我的证明而不要他们自己的。结果，我在那里一年半之后，就被认为可以教授数学了。……在我不知情或并不希望的情况下，对教学的持续关注因追求科学统一性和连贯性而得到加强。正如数学某一分支中的相关定理在不同类中彼此衍生一样，我相信数学本身的分支也是如此。我瞥见了数学所有对象的有机统一这一观念；我当时相信，我能在某所大学里找到这种统一；即使不是作为一门独立学科，至少也以具体建议的形式。

1818年秋，施泰纳离开伊韦尔东，前往海德堡，在那里靠私人数学课谋生。他在海德堡大学听组合分析、微分与积分以及代数的讲座。也正是在这时，他对力学产生了兴趣，并在1821年、1824年和1825年就此写了三份未发表的手稿。1821年复活节，他离开海德堡前往柏林，在那里再次靠家教所得的非常微薄收入维持生活。他没有正式的教学资格，因此他决定需要参加必要的考试，以便能成为文理中学（Gymnasium）的数学教师。他并不完全成功，因为在柏林参加必要考试后，他只获得了有限的教学许可。他的问题不在数学，而在历史、文学等其它被考查科目。然而，这一有限许可足以让他被任命到柏林的韦尔德文理中学。

施泰纳在韦尔德文理中学担任数学教师的时期证明是困难的。起初，他的教学收到了良好的报告，但他与学校校长Zimmermann博士发生了争执。也许可以理解，Zimmermann希望施泰纳使用Zimmermann自己编写的教科书教授他的课程。施泰纳是Pestalozzi教学方法的坚定信徒，在课堂上使用了这些方法。Zimmermann声称这些只适合初级课程，施泰纳于1822年秋天被解雇。官方原因是他的教学受到批评，但真正原因显然是他希望使用他认为最好的方法，而不是校长Zimmermann博士所要求的方法。他再次从事私人辅导以赚取足够的钱，以便参加柏林大学的课程，他从1822年11月到1824年8月这样做了。卡尔·古斯塔夫·雅各布·雅可比虽然比施泰纳小八岁，但此时是柏林大学的学生，很快施泰纳和卡尔·古斯塔夫·雅各布·雅可比成为了朋友。1825年，施泰纳被任命为柏林技术学校的助理教师。

施泰纳在韦尔德文理中学经历过的那类困难，在技术学校再次出现。人们期望他毫无疑问地服从校长K F von Klöden的命令。Klöden几乎可以肯定正确地认为，施泰纳没有给予他应得的尊重。他以严厉的方式报复，对施泰纳提出不合理的要求，[1]：-

……即使是受军纪约束的士兵，也很难指望会接受。

尽管气氛恶劣，施泰纳在技术学校任教期间仍设法进行了一些杰出的数学研究。他于1829年被提升为高级教师。我们已经提到，施泰纳与卡尔·古斯塔夫·雅各布·雅可比变得友好，但他也与柏林其他有影响力的数学家变得友好。其中也许最重要的是奥古斯都·利奥波德·克雷勒，但他于1826年抵达柏林后与尼尔斯·阿贝尔的友谊也很重要。施泰纳成为奥古斯都·利奥波德·克雷勒的期刊Journal für die reine und angewandte Mathematik的早期投稿人，该期刊是创办的第一本完全致力于数学的期刊。该期刊第一卷于1826年出版，其中包含施泰纳的第一篇长作，Einige geometrische Betrachtungen Ⓣ（一些几何考虑）。这篇论文很重要，因为它是首次发表的关于点相对于圆的幂的理论以及圆的相似点的系统论述。它也很重要，因为施泰纳在许多证明中使用了反演原理。这篇论文是施泰纳在奥古斯都·利奥波德·克雷勒的期刊上发表的62篇论文中的第一篇。在他的论文《关于平面和空间分割的若干规律》中，该文也出现在奥古斯都·利奥波德·克雷勒的期刊第一卷中，他考虑了这个问题：空间被$n$个平面分割，最多能分成多少部分？这是一个优美的问题，其解为$(n^{3} + 5n + 6)/6$。解法见[3]。

1832年，施泰纳出版了他的第一本书Systematische Entwicklung der Abhangigkeit geometrischer Gestalten voneinanderⓉ（几何图形相互依赖关系的系统发展）。其中许多材料在此前六年已出现在施泰纳的论文中。该书的序言对施泰纳研究数学的一般方法以及书中几何材料的处理方式给出了一个有趣的视角：-

本著作试图发现使最丰富多样的空间现象彼此联系起来的有机体。存在有限数量的非常基本的关系，它们共同构成了一个图式，借助它，其余定理可以被逻辑地且毫无困难地展开。通过适当采用少数基本关系，人便成为整个领域的主人。秩序取代了混乱：人们看到所有部分如何自然地啮合，以最美的秩序排列，并形成定义明确的群。以这种方式，人们同时获得了自然在尽可能经济且以最简单的方式赋予图形无穷多性质时所由出发的元素。这里主要的事情既不是综合方法也不是分析方法，而是发现图形之间的相互依赖关系，以及它们的性质从简单图形传递到更复杂图形的方式。这种联系与过渡是几何学所有其余个别命题的真正来源。那些其存在本身以前必须通过巧妙证明才能使人相信、并且一旦被发现便作为某种奇妙之物而矗立的图形性质，现在被揭示为这些新发现的基本元素的共同性质的必然结果，而前者由后者先验地建立起来。

施泰纳因其卓越成就而受到表彰。1833年4月20日，经卡尔·古斯塔夫·雅各布·雅可比推荐，他被柯尼斯堡大学授予荣誉博士学位，随后于1834年6月5日当选为柏林科学院。1834年10月8日，他被任命为柏林大学新设立的几何学编外教授。这一职位是由亚历山大·冯·洪堡和威廉·冯·洪堡专门为他设立的；他担任该职位直至去世。1844年他在罗马，在这次访问中，他花时间研究一个现在被称为“罗马曲面”或“施泰纳曲面”的三类四次曲面。1854—55年冬天他在巴黎，逗留期间当选为Académie des Sciences。

他是射影几何的最伟大贡献者之一。他发现了“施泰纳曲面”，其上有一个双无限的conic sections。“施泰纳定理”指出，从圆锥曲线的两个点投影它的两个 pencil 是射影相关的。另一个著名结果是“让-维克托·彭赛列-施泰纳定理”，它表明欧几里得作图只需要一个给定的圆和一把直尺。这基本上是他的第二本书Die geometrischen Konstructionen ausgefuhrt mittelst der geraden Linie and eines festen Kreises Ⓣ（通过直线和圆执行的几何作图）（1833）的主题。证明，基本上如施泰纳所给出的，在[3]中重现。他的许多出版物涉及圆锥曲线和曲面的研究。例如，他考虑了这个问题：在所有可以外接于（内接于）给定三角形的椭圆中，哪一个具有最小（最大）面积？今天这些椭圆被称为“施泰纳椭圆”。

在1848年写的一篇具有根本重要性的短文中，题为Allgemeine Eigenschaften algebraischer CurvenⓉ（代数曲线的一般性质），他讨论了一个点相对于给定曲线的极曲线。他还引入了施泰纳曲线，讨论了拐点处的切线、二重切线、尖点和二重点。特别地，他指出了四次曲线的二十八条二重切线的相应关系。然而，这些丰富的材料被呈现出来时，完全没有给出施泰纳所找到的证明的任何提示。奥托·黑塞描述这些结果时说：-

……它们像皮埃尔·德·费马的定理一样，是当代和未来世代的谜。

本文中所有结果的完整证明由安东尼奥·路易吉·高登齐奥·朱塞佩·克雷莫纳找到，并发表在他关于代数曲线的书中。

施泰纳不喜欢代数和分析，认为计算取代思考，而几何激发思考。托马斯·阿彻·赫斯特这样描述他：-

他是个中年人，身材相当粗壮，有一张长长的富有智慧的脸，留着胡须和八字胡，前额突出而饱满，头发深色，偏向于变灰。他脸上首先让你注意到的是有一丝忧虑和焦虑，几乎是痛苦，仿佛源于身体上的痛苦——他患有风湿病。他从不事先准备讲座。因此他常常在当下卡住或无法证明他想证明的东西，而每次这样的失败，他肯定会说些有特色的评论。

卡尔·古斯塔夫·雅各布·雅可比这样写他的朋友施泰纳：-

从少数空间性质出发，施泰纳试图借助简单的图式，对大量被割裂的几何定理获得一个全面的看法。他寻求为每个定理指定其相对于其他定理的特殊位置，为混乱带来秩序，按照自然将所有部分相互连接，并将它们组装成定义明确的群。在发现连接最多样空间现象的有机体时，他不仅促进了几何综合的发展；还为数学所有其他分支提供了一个完整方法和执行的模型。

尽管是数学天才，在其他方面施泰纳是个难相处的人。约翰·卡尔·布尔克哈特写道[1]：-

学生和同时代人写到施泰纳几何研究的辉煌，以及他在引导他人进入他发现的新领域时展现的火热气质。与此相结合的是非常自由的政治观点。此外，他常常行为粗鲁，说话直率，从而疏远了一些人。

施泰纳生命中的最后十年因疾病而日益艰难。肾脏问题使他一年中大部分时间都待在故乡瑞士，只在冬季前往柏林授课。最终他完全卧床不起，无法履行任何教学职责。施泰纳终身未婚，也许正因为如此，他去世时留下了一笔财富。这笔财富的三分之一捐给了柏林科学院，用以设立施泰纳奖。其余的钱则分给了他的亲属和他故乡乌岑斯托夫村的学校。他最后的愿望是，家乡的贫困儿童能够获得比他本人更好的教育机会。

Jakob Steiner's parents were Anna Barbara Weber (1757-1832) and Niklaus Steiner (1752-1826). Anna and Niklaus were married on 28 January 1780 and they had eight children. Jakob was the youngest of the children and spent his early years helping his parents with the small farm and business that they ran near the village of Utzenstorf, about 24 km north of Bern. He did not learn to read and write until he was 14 but he then proved invaluable [1]:-

As a child he had to help his parents on the farm and in their business: his skill in calculation was of great assistance.

Jakob, however, wanted something better for himself but his parents were delighted to have his help with their business. At the age of 18, against the wishes of his parents, he left home to attend Johann Heinrich Pestalozzi's school at Yverdon at the south-east end of the Lake of Neuchâtel. Pestalozzi ran his innovative school in the town from 1805 to 1825 and Steiner entered in the spring of 1814. The fact that Steiner was unable to pay anything towards his education at the school was not a problem, for Pestalozzi wanted to try out his educational methods on the poor. Pestalozzi's school had a very significant effect on Steiner's attitude both to the teaching of mathematics and also to his philosophy when undertaking research in mathematics. He wrote in 1826 (see for example [1]):-

The method used in Pestalozzi's school, treating the truths of mathematics as objects of independent reflection, led me, as a student there, to seek other grounds for the theorems presented in the courses than those provided by my teachers. Where possible I looked for deeper bases, and I succeeded so often that my teachers preferred my proofs to their own. As a result, after I had been there for a year and a half, it was thought that I could give instruction in mathematics. ... Without my knowledge or wishing it, continuous concern with teaching has intensified by striving after scientific unity and coherence. Just as related theorems in a single branch of mathematics grow out of one another in distinct classes, so, I believed, do the branches of mathematics itself. I glimpsed the idea of the organic unity of all the objects of mathematics; and I believed at that time that I could find this unity in some university; if not as an independent subject, at least in the form of specific suggestions.

In the autumn of 1818, Steiner left Yverdom and travelled to Heidelberg where he earned his living giving private mathematics lessons. He attended lectures at the Universities of Heidelberg on combinatorial analysis, differential and integral calculus and algebra. Also at this time he became interested in mechanics and he wrote three unpublished manuscripts on the topic in 1821, 1824 and 1825. At Easter 1821 he left Heidelberg and travelled to Berlin, where again he supported himself with a very modest income from tutoring. He had no formal teaching qualifications so he decided that he needed to sit the necessary examinations to allow him to become a mathematics master in a gymnasium. He was not completely successful for after taking the necessary examinations in Berlin he was only awarded a restricted license to teach. His problem was not in mathematics but in the other subjects which were examined such as history and literature. This restricted license was, however, sufficient to allow him to be appointed to the Werder Gymnasium in Berlin.

Steiner's time as a mathematics teacher at the Werder Gymnasium proved a difficult one. At first he received good reports on his teaching but he fell out with the director of the school, Dr Zimmermann. Perhaps, understandably, Zimmermann wanted Steiner to teach his courses using a textbook written by Zimmermann himself. Steiner, who was a firm believer in Pestalozzi's methods of teaching, used those methods in the classroom. Zimmermann claimed that these were only suitable for elementary courses, and Steiner was dismissed in the autumn of 1822. The official reason was that his teaching was receiving criticism but the real reason was clearly his desire to use the methods he thought best rather than those required by the director, Dr Zimmermann. Again he took up private tutoring to earn enough money to allow him to attend courses at the University of Berlin, which he did from November 1822 to August 1824. Carl Jacobi, although eight years younger than Steiner, was a student at the University of Berlin at this time and soon Steiner and Jacobi became friends. In 1825 Steiner was appointed as an assistant master at the Technical School of Berlin.

The type of difficulties that Steiner had experienced at the Werder Gymnasium again arose at the Technical School. He was expected to follow the orders of the director, K F von Klöden, without question. Klöden, almost certainly correctly, believed that Steiner was not giving him the respect that he deserved. He retaliated in a severe manner, making unreasonable demands of Steiner that [1]:-

... even a soldier subject to military discipline could hardly be expected to accept.

Despite the bad atmosphere, Steiner managed to carry out some outstanding mathematical research while teaching at the Technical School. He was promoted to senior master in 1829. We have already mentions that Steiner became friendly with Jacobi, but he also became friendly with other influential mathematicians in Berlin. Perhaps most important of these was August Crelle but his friendship with Niels Abel after he arrived in Berlin in 1826 was also significant. Steiner became an early contributor to Crelle's Journal, the Journal für die reine und angewandte Mathematik, which was the first journal devoted entirely to mathematics founded. The first volume of the journal appeared in 1826 and contains Steiner's first long work, Einige geometrische Betrachtungen Ⓣ. This paper is important as being the first published systematic account of the theory of the power of a point with respect to a circle, and the points of similitude of circles. It is also important for Steiner's use of the principle of inversion in many of the proofs. This paper was the first of 62 papers which Steiner published in Crelle's Journal. In his paper Several laws governing the division of planes and space, which also appeared in the first volume of Crelle's Journal, he considers the problem: What is the maximum number of parts into which a space can be divided by $n$ planes? It is a beautiful problem and has the solution $(n^{3} + 5n + 6)/6$. See [3] for a solution.

In 1832 Steiner published his first book Systematische Entwicklung der Abhangigkeit geometrischer Gestalten voneinander Ⓣ. Much of the material had already appeared in Steiner's papers over the preceding six years. The Preface of this book gives an interesting view of Steiner's approach to mathematics in general and the geometric material of book in particular:-

The present work is an attempt to discover the organism through which the most varied spatial phenomena are linked with one another. There exist a limited number of very simple fundamental relationships that together constitute the schema by means of which the remaining theorems can be developed logically and without difficulty. Through the proper adoption of the few basic relations one becomes master of the entire field. Order replaces chaos: and one sees how all the parts mesh naturally, arrange themselves in the most beautiful order, and form well-defined groups. In this manner one obtains, simultaneously, the elements from which nature starts when, with the greatest possible economy and in the simplest way, it endows the figures with infinitely many properties. Here the main thing is neither the synthetic nor the analytic method, but the discovery of the mutual dependence of the figures and of the way in which their properties are carried over from the simple to the more complex ones. This connection and transition is the real source of all the remaining individual propositions of geometry. Properties of figures the very existence of which one previously had to be convinced through ingenious demonstrations and which, when found, stood as something marvellous, are now revealed as necessary consequences of the common properties of these newly discovered basic elements, and the former are established a priori by the latter.

Soon Steiner was being honoured for his remarkable achievements. He was awarded an honorary doctorate by the University of Königsberg on 20 April 1833 on the recommendation of Jacobi, then elected to the Berlin Academy of Sciences on 5 June 1834. He was appointed to a new extraordinary professorship of geometry at the University of Berlin on 8 October 1834. The post had been specially created for him by Alexander and Wilhelm von Humboldt; he held it until his death. He was in Rome in 1844 and on this visit he spent his time investigating a fourth order surface of the third class now called the 'Roman surface' or 'Steiner surface'. He spent the winter of 1854-55 in Paris and during his stay there was elected to the Académie des Sciences.

He was one of the greatest contributors to projective geometry. He discovered the 'Steiner surface' which has a double infinity of conic sections on it. The 'Steiner theorem' states that the two pencils by which a conic is projected from two of its points are projectively related. Another famous result is the 'Poncelet-Steiner theorem' which shows that only one given circle and a straight edge are required for Euclidean constructions. This was basically the topic of his second book Die geometrischen Konstructionen ausgefuhrt mittelst der geraden Linie and eines festen Kreises Ⓣ (1833). The proof, essentially as given by Steiner, is reproduced in [3]. Many of his publications involved an investigation of conic sections and surfaces. For example he considered the problem: Of all ellipses that can be circumscribed about (inscribed in) a given triangle, which one has the smallest (largest) area? Today these ellipses are called the 'Steiner ellipses'.

In a short paper of fundamental importance written in 1848 entitled Allgemeine Eigenschaften algebraischer Curven Ⓣ he discussed polar curves of a point with respect to a given curve. He also introduced Steiner curves, discussed tangents at points of inflection, double tangents, cusps and double points. In particular he indicated the resulting relationships for the twenty-eight double tangents of the fourth degree curve. This wealth of material is presented, however, without any indication of the proofs which Steiner had found. Otto Hesse described these results saying:-

... they are, like Fermat's theorems, riddles for the present and future generations.

Complete proofs of all the results in this paper were found by Luigi Cremona and published in his book on algebraic curves.

Steiner disliked algebra and analysis and believed that calculation replaces thinking while geometry stimulates thinking. He was described by Thomas Hirst as follows:-

He is a middle-aged man, of pretty stout proportions, has a long intellectual face, with beard and moustache and a fine prominent forehead, hair dark rather inclining to turn grey. The first thing that strikes you on his face is a dash of care and anxiety, almost pain, as if arising from physical suffering - he has rheumatism. He never prepares his lectures beforehand. He thus often stumbles or fails to prove what he wishes at the moment, and at every such failure he is sure to make some characteristic remark.

Jacobi wrote of his friend Steiner:-

Starting from a few spatial properties Steiner attempted, by means of simple schema, to attain a comprehensive view of the multitude of geometric theorems that had been rent asunder. He sought to assign each its special position in relation to the others, to bring order to chaos, to interlock all parts according to nature, and to assemble them into well-defined groups. In discovering the organism through which the most varied phenomena of space are linked, he not only furthered the development of a geometric synthesis; he also provided a model of a complete method and execution for all other branches of mathematics.

Despite being a mathematical genius, in other ways Steiner was a difficult person. Burckhardt writes [1]:-

Students and contemporaries wrote of the brilliance of Steiner's geometric research and of the fiery temperament he displayed in leading others into the new territory he had discovered. Combined with this were very liberal political views. Moreover, he often behaved crudely and spoke bluntly, thereby alienating a number of people.

The last ten years of Steiner's life were increasingly difficult through illness. Kidney problems caused him to spend most of the year in his native Switzerland, only going to Berlin in the winter to deliver his lectures. Eventually he became totally bedridden and was unable to carry out any teaching duties. Steiner never married and, perhaps as a consequence, left a fortune on his death. One third of this fortune went to the Berlin Academy to found the Steiner Prize. The rest of the money was divided between his relatives and the school in his native village of Utzenstorf. His last wish was that poor children in his home town could have a better educational opportunity than he himself had.