**Julius Plücker**'s family were descended from merchants who had originally lived in Aachen but had settled in Elberfeld during the Reformation in the 16

^{th}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]:-

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

[However, his student Johann Hittorf wrote [2]:-In1847]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. ... in1858Heinrich 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.

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.... 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.

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]:-

His next major work wasIn 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.

*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]:-

In 1868 Plücker published the first part ofIt is the aim of these notes to amuse the reader with the remark that already in1835Julius 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.

*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]:-

Plücker was awarded the Copley Medal from the Royal Society of London in 1866:-... 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 in1868-69by 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 appeared50years earlier.

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

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.

**Article by:** *J J O'Connor* and *E F Robertson*