Julius Plücker


Born: 16 June 1801 in Elberfeld (now Wuppertal), Duchy of Berg (now Germany)
Died: 22 May 1868 in Bonn, Germany


Julius Plücker's family were descended from merchants who lived in Aachen. This meant that his 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.

Plücker attended the Gymnasium in Düsseldorf then after graduating 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 Bonn, then Heidelberg before going to study in Berlin. His next move was to go to France in 1823 where he attended courses on geometry at the University of Paris. He had completed his doctoral dissertation Generalem analyeseos applicationem ad ea quae geometriae altioris et mechanicae basis et fundamenta sunt e serie Tayloria deducit by 1823 which he submitted the the University of Marburg. His thesis advisor at Marburg was Christian Gerling.

He submitted his habilitation thesis to the University of Bonn in 1824 and was appointed as a dozent. 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. 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. Eccarius [8] sees the competition between the two men for the chair of mathematics at the Polytechnic, and 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 Halle in 1834, then returned to the Rheinische Friedrich-Wilhelms University of Bonn in 1836 to fill the chair of mathematics. In the following year he married Miss Altstätten; they had one son.

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 working on magnetism, electronics and atomic physics. He anticipated Kirchhoff and Bunsen in indicating that spectral lines were characteristic for each chemical substance:-

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 he continued to hold the chair of physics at Bonn until his death, in 1865 his research interests returned to mathematics and 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 Wilson points out in [13] 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. De Vries writes in [9]:-

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 but died before the second part was complete. Klein was 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.

Plücker was awarded the Copley Medal from the Royal Society in 1866.

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

August 2006


MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/Biographies/Plucker.html]