尤里乌斯·普吕克Julius Plücker

尤里乌斯·普吕克的家族是商人的后裔，这些商人最初住在亚琛，但在16世纪宗教改革期间定居在埃尔伯费尔德。这意味着普吕克的背景是法国和德国的混合，在他的一生中，很明显他发现两者都很有吸引力。例如，他的大部分数学遵循由加斯帕尔·蒙日发展的法国几何风格。他的父亲Johann Peter Plücker（1771-1844）是埃尔伯费尔德的商人，尽管他后来退休到杜塞尔多夫。普吕克的母亲是Johanna Maria Lüttringhausen（1776-1843），Johannes Lüttringhausen的女儿。Peter和Johanna Plücker都出生在埃尔伯费尔德，他们于1797年9月21日在该镇结婚。他们有三个孩子：普吕克，本传记的主题，Moritz Rudolf Plücker（1804-1876）和Emil Plücker（卒于1871年）。我们注意到，几乎所有资料都给出普吕克的出生日期为1801年6月16日，但他墓碑上的日期是1801年7月16日，我们采用了这个日期。

普吕克首先就读于埃尔伯费尔德的师范学校，由Johann Friederich Wilberg（1766-1846）开办，他研究了不同教学风格对学生性格的影响。普吕克从1806年到1815年在那里学习，他的能力得到了Wilberg的认可，Wilberg找到普吕克的父亲，说服他儿子的才华值得进一步的学术培训。Wilberg认识到几何是培养学生自我创造性思维和独立性的优秀教学工具。他的朋友，数学家William Adolph Diesterweg（1782-1835）在这一想法的技术方面鼓励了他。两人自1807年以来一直是亲密的朋友。Wilberg的另一位朋友是Friedrich Kohlrausch（1780-1865），他于1814年成为杜塞尔多夫Königlichen 文理中学（Gymnasium）的历史教师。这所学校由耶稣会士作为学园创立，并在新校长Karl Wilhelm Kortum（1789-1859）的领导下开始了重大创新发展过程。学校对学习，尤其是科学，采取了热情的态度。

1816年，普吕克听从Wilberg的建议，转至杜塞尔多夫的皇家文理中学，为大学学习做准备。1819年获得文凭毕业后，他遵循当时德国大学生的典型路径，在多所不同大学求学。他首先进入海德堡大学，于1891年夏季学期入学。他在海德堡度过了三个学期，聆听了语文学与古代史教授Georg Friedrich Creuzer（1771-1858）的讲座。随后，他于1820年冬季学期转入波恩大学开始学习。在这里，Karl Wilhelm Gottlob Kastner（1783-1857）教授他物理和化学，Kastner从1818年到1821年在波恩授课。他还跟随天文学、数学和物理学教授Karl Dietrich von Münchow（1778-1836）学习数学和物理，并跟随1819年被任命为数学教授的Wilhelm Adolf Diesterweg学习数学。

他的下一步行动是于1823年3月前往法国，在巴黎大学参加几何学课程。他聆听了让-巴蒂斯特·毕奥、奥古斯丁·路易·柯西、西尔维斯特·佛朗索瓦·拉克鲁瓦和西莫恩·德尼·泊松等人的讲座。他在巴黎期间完成了博士学位论文Generalem analyeseos applicationem ad ea quae geometriae altioris et mechanicae basis et fundamenta sunt e serie Tayloria deducit Ⓣ（一般分析应用于高等几何和基础力学，并从泰勒级数推导），并将其提交给马尔堡大学。他在马尔堡的论文导师是Christian Ludwig Gerling（1788-1864），后者曾师从卡尔·弗里德里希·高斯。他于1823年7月从巴黎将论文寄往马尔堡，并于1823年8月30日被缺席授予博士学位。他留在巴黎，为在波恩大学的教授资格论文（Habilitation）而努力。正是在巴黎学习期间，普吕克认识到了由皮埃尔·西蒙·拉普拉斯和约瑟夫·拉格朗日发展的分析力学以及由路易·普安索发展的几何力学的重要性。几何学与力学之间的这种相互作用，从此成为普吕克整个职业生涯的研究主题。

1825年4月初，他离开巴黎返回波恩，并于4月28日作了就职演讲。院长称赞他的就职演讲是数学与物理学的出色结合，导致了对这两个领域的深入研究。他被任命为讲师（Docent）。两年后，他出版了Analytisch-geometrische Entwickelungen Ⓣ（解析几何的演变）的第一卷。我们将在下文更详细地讨论这部重要著作。他在波恩担任讲师期间开设的课程包括：分析与代数（包括数论）；几何学；数学物理；簿记；以及天文学。他在波恩最初几年讲授的几何学课程基于让-巴蒂斯特·毕奥的Géométrie analytique Ⓣ（解析几何），但从1828-29年冬季学期开始，他讲授自己的几何学成果。他的数学物理课程基于路易·普安索的Elementary Mechanics和西莫恩·德尼·泊松的Mechanics。普吕克所有课程均用德语讲授，但有时偶尔用法语讲课。有时他分别用德语和法语为学生提供辅导。

1828年晋升为波恩大学特聘教授后，他于1833年前往柏林，在柏林大学担任特聘教授一年，同时在弗里德里希·威廉文理中学任教。这种双重角色让普吕克思考如何将教学与研究更紧密地结合起来。特别是，他试图通过使用自己研究中的例子，为教学带来兴奋感，尤其是对最优秀的学生。对大多数德国人来说，柏林的职位是终极目标，人们会期望普吕克在那里度过余下的职业生涯。然而，事情并不那么简单，因为柏林的数学讲席刚刚被雅各布·施泰纳填补。雅各布·施泰纳是德国综合几何学派的领袖，而普吕克则遵循分析方法。如果两人关系良好，这可能是一种优势，但他们的个性意味着他们的关系是持续冲突的。Wolfgang Eccarius [12]认为，两人对理工学院数学讲席的竞争，以及奥古斯都·利奥波德·克雷勒支持普吕克而非雅各布·施泰纳，是他们个人冲突的基础。普吕克很快决定，他必须尽快找到一个远离柏林的职位。他于1833年11月7日成为哈勒大学数学正教授，并在那里待了四个学期。他被任命填补先前由海因里希·谢尔克担任的讲席，后者在任职不到两年后离开哈勒，前往基尔大学担任讲席。在哈勒，普吕克讲授的课程包括：分析与代数；几何学、数学物理；以及物理学。

他于1836年回到波恩莱茵弗里德里希·威廉大学，担任数学讲席。次年，即1837年9月4日，他在波恩Neugasse的Haus Altstätten与Maria Louise Antonie Friederike Altstätten（1813-1880）结婚；他们有一个儿子 Albert Plücker（1838-1901），于1838年8月1日出生在波恩。正如我们已经指出的，普吕克 是一位几何学家，但他坚信数学对物理科学应用的重要性。1847年，他转向物理学，接受波恩的物理学讲席，并研究磁学、电子学和原子物理学。他在指出光谱线对每种化学物质具有特征性方面先于 古斯塔夫·基尔霍夫 和 Bunsen。他与他的学生 Johann Wilhelm Hittorf（1824-1914）一起做出了重大发现。他们共同 [2]：-

……在光谱学中做出了许多重要发现，预见了德国化学家 Robert Bunsen 和德国物理学家古斯塔夫·基尔霍夫，后者后来宣布谱线是每种化学物质的特征。

他做出了其他重要发现，这些发现最终导致了阴极射线管的发明[8]：-

[1847年]他对麦可·法拉第的工作产生了兴趣。麦可·法拉第已开始用气体中的放电进行实验，注意到了火花效应。……普吕克想到，如果气体能被容纳在一个封闭空间内，放电效应应该能在一段时间内被观察到。……1858年，Heinrich Geissler（1814-1879）发明了真空玻璃管。当普吕克产生放电时，一种怪异、神秘而美丽的淡绿色辉光出现了。它持续了很长时间。……普吕克，这位完整的科学家，认识到管内的这种荧光对管壁上的电磁铁有响应；他辨别出这些光的显现是某种电学性质的射线或束。……Geissler和普吕克将麦可·法拉第效应推进为一种视觉显现。

然而，他的学生 Johann Hittorf 写道[2]：-

……Plucker作为实验者从未达到过很高的手工灵巧度。然而，他总是对需要什么有非常清晰的概念，并且高度具备让别人接受他的想法并使他们热情地将这些想法付诸实践的能力。

普吕克继续担任波恩大学物理学讲席直到去世，1865年他的研究兴趣回到数学，菲利克斯·克莱因在1866-1868年担任他的助手。

现在我们简要看一下普吕克对数学做出的极其重要的贡献。他的第一部主要著作是Analytisch-geometrische Entwickelungen Ⓣ（解析几何的发展），分两卷出版，第一卷于1828年，第二卷三年后[1]：-

在每一卷中，他都讨论了直线、圆和圆锥曲线的平面解析几何；许多事实和定理——无论是普吕克发现的还是已知的——都以更优雅的方式得到了证明。两卷中使用的点坐标是非齐次仿射坐标；在第二卷中，使用了平面中的齐次线坐标，以前称为普吕克坐标，并将圆锥曲线视为直线的envelopes。普吕克解析几何的特征已经在这部著作中出现，即圆锥曲线及其束的方程中代数符号的优雅运算。

他的下一部主要著作是System der analytischen Geometrie, auf neue Betrachtungsweisen gegrundet, und insbesondere eine ausführliche Theorie der Kurven dritter Ordnung enthaltend Ⓣ（基于新观点的解析几何体系，特别是包含详细的三阶曲线理论）（1835年），其中讨论了适用于圆锥曲线的点和线坐标。该著作的主要部分讨论了平面三次曲线。1839年，他出版了Theorie der algebraischen Kurven Ⓣ（代数曲线理论），讨论了代数曲线在无穷远点附近的性质，深入研究了平面上的奇点。这部著作还包含了著名的“普吕克方程”，关联了曲线的阶和类。

他开创了对与线丛相关的几何构型的研究。在这种指定坐标的方式中，一个点有一个线性方程，即通过该点的所有直线的方程，而一条线有一对数字，即它截取坐标轴处的$x$和$y$坐标。他还引入了直纹面的概念。他在组合数学方面的工作考虑了雅各布·施泰纳型系统。事实上，Robin Wilson在[24]中指出，普吕克的贡献是块设计的最早参考文献，当时他构造了9阶系统，并陈述了这种系统存在的元素数量的必要条件。Hans Ludwig de Vries在[23]中写道：-

这些笔记的目的是让读者感到有趣的是，早在1835年，普吕克就在他的书《System der analytischen Geometrie, auf neue Betrachtungsweisen gegründet, und insbesondere eine ausführliche Theorie der Curven dritter Ordnung enthaltend》Ⓣ（基于新观点的解析几何体系，特别是包含详细的三阶曲线理论）中发表了S(2, 3, 9)，并在脚注中定义了一般的雅各布·施泰纳三元系统S(2, 3, m) = STS(m)。他陈述了关于STS的第一个定理，该定理说并非每个m都有STS(m)，只有那些形式为m = 6n+3的才有。因此，由于他遗漏了m = 6n+1，关于STS的第一个定理是错误的，或者至少是不完整的。

1868年，普吕克出版了Neue Geometrie des Raumes, gegründet auf die Betrachtung der geraden Linie als Raumelement Ⓣ（基于将直线视为空间元素的新空间几何学）（1868年）的第一部分，但在第二部分完成之前去世了。菲利克斯·克莱因当时是他的助手，并讨论了普吕克打算在第二部分中发展的想法。因此，菲利克斯·克莱因按照普吕克设想的那样执行了计划，于1869年出版了第二卷。Simon Grigorevich Gindikin在[15]中关于这部著作写道：-

……普吕克的想法是考虑这样一个空间，其元素（点！）是通常三维空间中的直线。普吕克在数年间发展了这一想法，最终结果载于他去世后出版的回忆录中，该回忆录由菲利克斯·克莱因和阿尔弗雷德·克莱布什于1868-69年编辑，题为“基于将直线视为空间元素的空间新几何学”。直线空间的维数是四，这很可能是科学中出现的第一个四维空间。奇怪的是，在四维流形出现于相对论并成为时尚的时期，没有人将赫尔曼·闵可夫斯基四重和早50年出现的普吕克四重进行比较。

普吕克于1866年被授予伦敦皇家学会的科普利奖章：-

因其在解析几何、磁学和光谱分析方面的研究。

科普利奖章引文的全文见THIS LINK

这个奖项，以及普吕克的成就在德国缺乏认可，无疑表明他的才华在英国比在德国更被清楚地看到。

普吕克被安葬在波恩的Alter Friedhof（旧公墓）。

Julius Plücker's family were descended from merchants who had originally lived in Aachen but had settled in Elberfeld during the Reformation in the 16th Century. This meant that Julius's background was a mixture of French and German and throughout his life it is evident that he found both attractive. For example, much of his mathematics followed the French style of geometry as developed by Monge. His father, Johann Peter Plücker (1771-1844), was a businessman in Elberfeld although he later retired to Dusseldorf. Julius's mother was Johanna Maria Lüttringhausen (1776-1843), a daughter of Johannes Lüttringhausen. Peter and Johanna Plücker had both been born in Elberfeld and they married in that town on 21 September 1797. They had three children: Julius Plücker, the subject of this biography, Moritz Rudolf Plücker (1804-1876) and Emil Plücker (died 1871). We note that almost all sources give Julius's date of birth as 16 June 1801 but the date on his gravestone is 16 July 1801 and we have adopted this date.

Julius Plücker first attended the Normal School in Elberfeld run by Johann Friederich Wilberg (1766-1846) who had undertaken research on the affects of different styles of teaching on the characters of the pupils. Plücker studied there from 1806 to 1815 and his ability was recognised by Wilberg who approached Plücker's father persuading him that his son's talents merited further scholarly training. Wilberg recognized that geometry was an excellent teaching tool to develop self-creative thinking and independence in his students. His friend, the mathematician William Adolph Diesterweg (1782-1835) encouraged him in the technical aspect of this idea. The two men had been close friends since 1807. Another of Wilberg's friends was Friedrich Kohlrausch (1780-1865) who became a history teacher at the Königlichen Gymnasium in Düsseldorf in 1814. This school had been founded as a Lyceum by the Jesuits and had begun a process of major innovative development under a new director Karl Wilhelm Kortum (1789-1859). The school offered an enthusiastic approach to learning, especially science.

In 1816 Plücker, following Wilberg's advice, moved to the Königlichen Gymnasium in Düsseldorf to prepare for university studies. After graduating with a diploma in 1819 he followed the typical path for German university students of the time, studying at a number of different universities. He first attended the University of Heidelberg which he entered in the summer semester of 1891. He spent three semesters at Heidelberg where he attended the lectures of Georg Friedrich Creuzer (1771-1858), the professor of philology and ancient history. Next he moved to the University of Bonn beginning his studies there in the winter semester of 1820. Here he was taught physics and chemistry by Karl Wilhelm Gottlob Kastner (1783-1857) who lectured at Bonn from 1818 to 1821. He was also taught mathematics and physics by Karl Dietrich von Münchow (1778-1836), the professor of astronomy, mathematics and physics, and mathematics by Wilhelm Adolf Diesterweg who had been appointed professor of mathematics in 1819.

His next move was to go to France in March 1823 where he attended courses on geometry at the University of Paris. He attended lectures by, among others, Jean-Baptiste Biot, Augustin-Louis Cauchy, Sylvestre Lacroix and Siméon Poisson. He completed his doctoral dissertation Generalem analyeseos applicationem ad ea quae geometriae altioris et mechanicae basis et fundamenta sunt e serie Tayloria deducit Ⓣ while he was in Paris which he submitted to the University of Marburg. His thesis advisor at Marburg was Christian Ludwig Gerling (1788-1864) who had studied under Carl Friedrich Gauss. He sent the thesis from Paris to Marburg in July 1823 and was awarded his doctorate 'in absentia' on 30 August 1823. He remained in Paris working towards his habilitation at the University of Bonn. It was while he was studying in Paris that Plücker learned the importance of analytical mechanics as developed by Laplace and Lagrange as well as that of geometric mechanics as developed by Poinsot. This interplay between geometry and mechanics would form the topic of Plücker's research throughout his career from this time onwards.

At the beginning of April 1825 he left Paris and returned to Bonn where he delivered his habilitation lecture on 28 April. The dean praised his habilitation as being an excellent combination of mathematics and physics which led to a deep study of both areas. He was appointed as a docent. Two years later he published the first volume of Analytisch-geometrische Entwickelungen Ⓣ. We say more about this important work below. Lecture courses he gave while a docent at Bonn include: Analysis and algebra including number theory; Geometry; Mathematical Physics; Bookkeeping; and Astronomy. The geometry lectures that he gave in his first years at Bonn were based on Biot's Géométrie analytique Ⓣ but from the winter semester of 1828-29 onwards he lectured on his own geometric results. He based the Mathematical Physics courses on Elementary Mechanics by Louis Poinsot, and Mechanics by Siméon-Denis Poisson. Plücker gave all his courses in German but sometimes gave the occasional lecture in French. Sometimes he offered the students separate tutorials in German and in French.

Promoted to extraordinary professor at Bonn in 1828, he went to Berlin in 1833 and spent a year as an extraordinary professor at the University while at the same time he taught at the Friedrich Wilhelm Gymnasium. This dual role made Plücker think about how to bring his teaching and his research closer together. In particular he sought to bring an excitement to his teaching, particularly for the best students, by using examples from his own research. For most Germans a position in Berlin would be the ultimate goal and one would have expected Plücker to spent the rest of his career there. Things were not so simple, however, for the chair of mathematics in Berlin had just been filled by Jakob Steiner. Steiner was the leader of the German school of synthetic geometry, while Plücker followed the analytical approach. This might have been a strength had the two men been on good terms, but their personalities meant that their relationship was one of continual conflict. Wolfgang Eccarius [12] sees the competition between the two men for the chair of mathematics at the Polytechnic, and August Crelle favouring Plücker over Steiner, as the basis to their personal conflict. Plücker quickly decided that he would have to find a position away from Berlin as soon as possible. He became an ordinary professor of mathematics at the University of Halle on 7 November 1833 and remained there for four semesters. He was appointed to fill the chair previously held by Heinrich Scherk who, after less than two years in post, had left Halle for the chair at the University of Kiel. At Halle, Plücker gave lecture courses on: Analysis and Algebra; Geometry, Mathematical Physics; and Physics.

He returned to the Rheinische Friedrich-Wilhelms University of Bonn in 1836 to fill the chair of mathematics. In the following year, on 4 September 1837, he married Maria Louise Antonie Friederike Altstätten (1813-1880) at Haus Altstätten, Neugasse, Bonn; they had one son Albert Plücker (1838-1901) born in Bonn on 1 August 1838. As we have indicated, Plücker was a geometer yet he firmly believed in the importance of the applications of mathematics to the physical sciences. In 1847 he turned to physics, accepting the chair of physics at Bonn and working on magnetism, electronics and atomic physics. He anticipated Kirchhoff and Bunsen in indicating that spectral lines were characteristic for each chemical substance. He made significant discoveries along with his student Johann Wilhelm Hittorf (1824-1914). Together they [2]:-

... made many important discoveries in spectroscopy, anticipating the German chemist Robert Bunsen and the German physicist Gustav R Kirchhoff, who later announced that spectral lines were characteristic for each chemical substance.

He made other important discoveries which would eventually lead to the invention of a cathode ray tube [8]:-

[In 1847] he became interested in Faraday's work. Faraday had begun to experiment with electrical discharge in gases, noting the spark effect. ... It occurred to Plücker that if gases could be contained in an enclosure, the discharge effect should be observable for a length of time. ... in 1858 Heinrich Geissler (1814-1879) invented the vacuum glass tube. When Plücker generated the discharge, an eerie, mysterious, and beautiful greenish glow appeared. It remained for a persistent time. ... Plücker, the complete scientist, recognised that this fluorescence within the tube responded to an electromagnet on the wall of the tube; he discerned that these expressions of light were rays or beams of some electrical property. ... Geissler and Plücker had propelled Faraday's Effect into a visual manifestation.

However, his student Johann Hittorf wrote [2]:-

... Plucker never attained great manual dexterity as an experimenter. He had always, however, very clear conceptions as to what was wanted, and possessed in a high degree the power of putting others in possession of his ideas and rendering them enthusiastic in carrying them into practice.

Although Plücker continued to hold the chair of physics at Bonn until his death, in 1865 his research interests returned to mathematics and Felix Klein served as his assistant 1866-1868.

Let us now look briefly at the highly significant contributions which Plücker made to mathematics. His first major work was Analytisch-geometrische Entwickelungen Ⓣ published in two volumes, the first in 1828 and the second three years later [1]:-

In each volume he discussed the plane analytic geometry of the line, circle, and conic sections; and many facts and theorems - either discovered or known by Plücker - were demonstrated in a more elegant manner. The point coordinates used in both volumes are nonhomogeneous affine; in volume II the homogeneous line coordinates in a plane, formerly known as Plücker's coordinates, are used and conic sections are treated as envelopes of lines. The characteristic features of Plücker's analytic geometry were already present in this work, namely, the elegant operations with algebraic symbols occurring in the equations of conic sections and their pencils.

His next major work was System der analytischen Geometrie, auf neue Betrachtungsweisen gegrundet, und insbesondere eine ausführliche Theorie der Kurven dritter Ordnung enthaltend Ⓣ (1835) which treats point and line coordinates which apply to conic sections. The main part of the work discusses plane cubic curves. In 1839 he published Theorie der algebraischen Kurven Ⓣ which discussed properties of algebraic curves near infinite points, studying in great depth singular points on the plane. This work also contains the celebrated 'Plücker equations' relating the order and class of a curve.

He initiated the investigation of geometrical configurations associated with line complexes. In this way of specifying coordinates, a point has a linear equation, namely that of all lines through the point while a line has a pair of numbers namely the $x$ and $y$ coordinates of where it cuts the axes. He also introduced the idea of a ruled surface. His work on combinatorics considers Steiner type systems. In fact Robin Wilson points out in [24] that Plücker's contribution is the earliest reference to block designs when he constructed the system of order 9 and stated necessary condition on the number of elements for such a system to exist. Hans Ludwig de Vries writes in [23]:-

It is the aim of these notes to amuse the reader with the remark that already in 1835 Julius Plücker published S(2, 3, 9) in his book 'System der analytischen Geometrie, auf neue Betrachtungsweisen gegründet, und insbesondere eine ausführliche Theorie der Curven dritter Ordnung enthaltend' Ⓣ and defined general Steiner triple systems S(2, 3, m) = STS(m) there in a footnote. He stated the first theorem on STSs, which says that not every m has an STS(m), but only those which are of the form m = 6n+3. So, since he missed m = 6n+1, the first theorem on STSs was wrong, or at least incomplete.

In 1868 Plücker published the first part of Neue Geometrie des Raumes, gegründet auf die Betrachtung der geraden Linie als Raumelement Ⓣ (1868) but died before the second part was complete. Klein was, at this time, his assistant and had discussed the ideas that Plücker intended to develop in this second part. Klein therefore carried out the plan as envisaged by Plücker publishing the second volume in 1869. Simon Grigorevich Gindikin writes about this work in [15]:-

... the idea of Julius Plücker [was] to consider the space whose elements (points!) are lines of the usual three-dimensional space. Plücker developed this idea over several years and the final result is contained in the posthumous memoir, edited in 1868-69 by Klein and Clebsch, entitled "New Geometry of the Space based on the Consideration of a Line as a Space Element". The dimension of the space of lines is four and it is probably the first four-dimensional space that appeared in science. Strangely enough, in the period when four-dimensional manifolds appeared in relativity theory and became fashionable, nobody compared the Minkowski fourfold and the Plücker fourfold which appeared 50 years earlier.

Plücker was awarded the Copley Medal from the Royal Society of London in 1866:-

For his researches in analytical geometry, magnetism, & spectral analysis.

For the full text of the citation for the Copley Medal see THIS LINK

This award, and the lack of recognition of Plücker's achievements in Germany, is certainly an indication that his brilliance was seen more clearly in Britain than it was in Germany.

Plücker is buried in the Alter Friedhof (Old Cemetery) in Bonn.