Julius Plücker

How to use this course

Julius Plücker had two careers. From 1825 to 1846 he was a geometer who showed that algebra, used cleverly, could do everything the synthetic geometers of his day did with figures. From 1847 he held the chair of physics at Bonn and did pioneering experiments on magnetism in crystals, on electric discharges in rarefied gases and on the spectra of gases. In his last years he returned to geometry and founded line geometry, which his assistant Felix Klein carried forward.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. Lessons 04–06 follow his geometry of 1828–1839. Lessons 07–09 follow his physics of 1847–1865. Lesson 10 returns to geometry and his last work. Where modern physics explains something Plücker could only observe, the lesson says so explicitly.

Timeline

Elberfeld, Bonn, Paris, 1801–1825

Julius Plücker was born in Elberfeld, a textile town east of Düsseldorf, in 1801. His father, Johann Peter Plücker, was a businessman. Most reference works give his birthday as 16 June. The gravestone in Bonn gives 16 July, and the MacTutor biography follows it.

He attended school in Elberfeld and then the gymnasium in Düsseldorf, graduating in 1819. He spent three semesters at Heidelberg and moved to Bonn in the winter of 1820, where he studied mathematics and physics. The Prussian university at Bonn was barely two years old.

Paris

In March 1823 he went to Paris, then the capital of mathematics. He heard Jean-Baptiste Biot, Augustin-Louis Cauchy, Sylvestre François Lacroix and Siméon Denis Poisson. More important for his own work was the French school of geometry founded by Gaspard Monge. Its students had revived pure geometry, and Jean-Victor Poncelet had just published his treatise on projective properties of figures (1822). Plücker absorbed the problems of this school but chose to attack them with algebra.

While in Paris he sent a Latin dissertation on analytic methods in geometry and mechanics to the University of Marburg, where the astronomer and mathematician Christian Ludwig Gerling examined it. Marburg awarded him the doctorate in absentia on 30 August 1823. He returned to Bonn and qualified as a university lecturer with a habilitation lecture on 28 April 1825.

The analytic geometer, 1825–1836

Plücker’s first book, the Analytisch-geometrische Entwicklungen, appeared in two volumes in 1828 and 1831. In 1828 Bonn made him an extraordinary professor. The book develops what he called abridged notation (Lesson 04): instead of writing out the equation of a line or conic, one names it by a single letter and reasons with linear combinations of these letters. The second volume introduced coordinates for lines and put the principle of duality on an algebraic footing (Lesson 05).

Duality was then the subject of an angry priority quarrel between Poncelet and Joseph Gergonne, and Plücker’s papers were drawn into it. His algebraic approach made the dispute look less important. Once a line has coordinates just as a point does, every theorem about points and lines has a dual obtained by exchanging the roles of the coordinates.

Berlin and Halle

In 1833 he moved to Berlin as extraordinary professor at the university, also teaching at the Friedrich Wilhelm Gymnasium. Berlin was not a comfortable place for him. Jakob Steiner, the leader of the synthetic school that rejected coordinates, was working there. The two men were rivals in method and in the competition for posts, and the relationship was hostile from the start. Plücker left within months. From 7 November 1833 he was ordinary professor at Halle, where he stayed for four semesters.

In 1835 he published System der analytischen Geometrie, which contains a detailed classification of plane cubic curves. It also contains the configuration of the nine inflection points of a cubic, which lie by threes on twelve lines; this is an early example of what combinatorialists now call a Steiner triple system (Exercise 6.3). In 1836 he returned to Bonn as ordinary professor of mathematics.

From geometry to physics, 1836–1847

Back in Bonn, Plücker married Antonie Altstätten in 1837; their son Albert was born the next year. In 1839 he published Theorie der algebraischen Curven, his deepest work on plane curves. It contains the relations between the degree, class and singularities of a curve now called the Plücker formulas (Lesson 06). The book also gives the first explicit quartic curve all of whose 28 bitangents are real. In 1846 a final geometrical treatise, System der Geometrie des Raumes, treated surfaces of the second order by the same methods.

His reputation in Germany remained modest. The synthetic geometers around Steiner dominated, and analytic methods were considered less pure. He was read more warmly in Britain, where Arthur Cayley and George Salmon built directly on his work on curves. The Royal Society made him a Foreign Member in 1855.

The physics chair

In 1847 Plücker became professor of physics at Bonn and took charge of its physical cabinet. Why a leading geometer took this step is not documented. He had always valued applications of mathematics to physics, and the new experiments of Michael Faraday on magnetism attracted him; his first physical work began by repeating some of them. For the next eighteen years he published almost no mathematics.

His student and collaborator Johann Wilhelm Hittorf later wrote that Plücker never acquired great manual dexterity as an experimenter. What he had, Hittorf said, was a very clear idea of what an experiment needed to show, and the ability to make others eager to carry it out. The Bonn glassblower and instrument maker Heinrich Geissler was one of those others.

Abridged notation and pencils

For the general reader. To find where two circles cross, you could solve equations. Plücker preferred to give each curve a name, say \(S\) and \(S'\), and to notice that every combination \(S+\lambda S'\) passes through all their common points. This simple observation, used systematically, proves many classical theorems in a few lines.

Homogeneous coordinates

A point of the projective plane is a ratio \((x:y:z)\) of three numbers, not all zero. Ordinary points have \(z\ne0\) and correspond to \((x/z,y/z)\); points with \(z=0\) are at infinity. A curve of degree \(n\) is the zero set of a homogeneous polynomial \(F(x,y,z)\) of degree \(n\). August Ferdinand Möbius (with barycentric coordinates, 1827) and Plücker introduced such coordinates independently, and Plücker used them systematically.

A homogeneous polynomial of degree \(n\) in three variables has \(\binom{n+2}{2}\) coefficients. Up to a scalar factor, the curves of degree \(n\) form a projective space of dimension

Requiring the curve to pass through a point is one linear condition on the coefficients, so \(N_n\) points in general position determine a unique curve of degree \(n\): 2 points a line, 5 a conic, 9 a cubic.

The idea of combining equations goes back at least to Gabriel Lamé (1818). Plücker made it a method. A classic application is Pascal’s theorem.

Proof. Let the sides, in order, be the lines \(\ell_1,\dots,\ell_6\). The cubics \(A=\ell_1\ell_3\ell_5\) and \(B=\ell_2\ell_4\ell_6\) meet in nine points: the six vertices and the three points \(P_{14}=\ell_1\cap\ell_4\), \(P_{25}\), \(P_{36}\). Choose a further point \(Q\) on the conic \(C\), and choose \(\lambda\) so that \(A+\lambda B\) vanishes at \(Q\). This cubic meets the conic in seven points (six vertices and \(Q\)), more than the \(2\cdot3=6\) allowed by Bézout’s theorem, so it contains \(C\) as a component: \(A+\lambda B=C\cdot\ell\) for a line \(\ell\). The points \(P_{14},P_{25},P_{36}\) lie on \(A+\lambda B\) but not on \(C\), since each side meets the conic only at its two vertices. So they lie on \(\ell\). ∎

Cramer’s paradox

Two cubics meet in \(3\cdot3=9\) points, and 9 points seem to determine a cubic. So how can two different cubics pass through the same nine points? Gabriel Cramer and Leonhard Euler discussed this in letters of 1744–45. Plücker gave the explanation that became standard: the nine intersection points of two cubics are never in general position. Any cubic through eight of them passes through the ninth, and they impose only eight independent conditions. The pencil \(A+\lambda B\) is the whole one-parameter family of cubics through them. The same reasoning appears in the proof of Pascal’s theorem above, and in the group law on a cubic curve.

Line coordinates and duality

For the general reader. A point in the plane can be described by numbers. Plücker saw that a line can be too: the line \(ux+vy+wz=0\) is fixed by the ratio \((u:v:w)\). Then points and lines become two equal partners, and every theorem about one becomes a theorem about the other.

Points and lines as equals

In the projective plane, a line is the set of points \((x:y:z)\) satisfying \(ux+vy+wz=0\). Plücker took \((u:v:w)\) as the coordinates of the line. The incidence relation is symmetric. Fixing \((x:y:z)\) and letting \((u:v:w)\) vary, the same equation describes the pencil of lines through a point. The line through two points is their cross product, and the point on two lines is the cross product of their coordinates.

Dual curves and class

A curve can be regarded as the set of its points or as the envelope of its tangent lines. At a smooth point \(p\) of \(F=0\), the tangent line has coordinates \((u:v:w)=\nabla F(p)\). The closure of the set of tangent lines is the dual curve \(C^*\) in the dual plane. Its degree is the class of \(C\): the number of tangents to \(C\) through a general point.

Proof. The tangent at \(p\in C\) is \(u=Ap\) (up to scale), so \(p=A^{-1}u\) and \(p^{\mathsf T}Ap=u^{\mathsf T}A^{-1}AA^{-1}u=u^{\mathsf T}A^{-1}u=0\). ∎

Biduality \((C^*)^*=C\) holds for all plane curves in characteristic 0. Under duality, a double tangent (bitangent) of \(C\) becomes a node of \(C^*\), and an inflectional tangent becomes a cusp of \(C^*\). This is the dictionary behind the Plücker formulas.

The Plücker formulas

For the general reader. A smooth curve of degree 3 has exactly 9 points where it is flattest (inflection points). A smooth curve of degree 4 has exactly 28 lines that touch it twice. Plücker found the general rule behind such numbers: a small set of equations that link the degree of a curve, the number of tangents through a point, and the curve’s special points.

The invariants

Let \(C\) be an irreducible plane curve of degree \(d\) over \(\mathbb C\) whose only singularities are \(\delta\) ordinary nodes and \(\kappa\) ordinary cusps, and suppose its dual \(C^*\) has the same property. Let \(d^*\) be the class, \(\delta^*\) the number of bitangents and \(\kappa^*\) the number of inflection points (the nodes and cusps of \(C^*\)).

Only three of these relations among \(d,d^*,\delta,\delta^*,\kappa,\kappa^*\) are independent, so three suitably chosen numbers determine the rest. Plücker stated six equations of this kind. Reading the common quantity as the genus of the curve came later, with the work of Bernhard Riemann and Alfred Clebsch.

Derivation of the class formula

The tangents from a general point \(q\) touch \(C\) where \(F=0\) and \(\sum_i q_i\,\partial F/\partial x_i=0\). The second curve, the first polar of \(q\), has degree \(d-1\), so by Bézout it meets \(C\) in \(d(d-1)\) points counted with multiplicity. The polar passes through every singular point of \(C\). A careful local computation shows that a node absorbs 2 of the intersections and a cusp absorbs 3, and these do not correspond to tangents from \(q\). Hence \(d^*=d(d-1)-2\delta-3\kappa\).

Flexes and the Hessian

A smooth point of \(C\) is an inflection point exactly when the Hessian determinant \(H=\det(\partial^2F/\partial x_i\partial x_j)\) vanishes there. \(H\) has degree \(3(d-2)\), so it meets \(C\) in \(3d(d-2)\) points. Again a local analysis shows that a node absorbs 6 and a cusp 8, giving \(\kappa^*=3d(d-2)-6\delta-8\kappa\). The other two formulas follow by applying these to \(C^*\) and using biduality.

Magnecrystallic action

For the general reader. Faraday showed in 1845 that all matter responds to a strong magnet: iron is attracted, while bismuth and many other substances are weakly repelled. Soon after taking the physics chair, Plücker found that a crystal also turns in a magnetic field, in a way set by its internal crystal axes rather than its outer shape. This is magnetic anisotropy, and he spent a decade sorting out its laws.

What Plücker observed

Suspending small crystals between the poles of a strong electromagnet, Plücker found that the optic axis of certain uniaxial crystals set itself in the equatorial position, perpendicular to the line joining the poles. Further experiments showed that in other crystals the optic axis went to the axial position instead. In the paper “On the Magnetic Induction of Crystals”, read to the Royal Society in 1857 and published in 1858, he proposed that induction in a crystal is governed by an auxiliary ellipsoid, using Poisson’s theory of induced magnetism and a construction of his student August Beer. He found that each crystal has directions along which it behaves like an amorphous body. He stressed that the optic axes exert no special force; they merely mark directions of the crystal’s structure.

The modern description

For a weakly magnetic material, \(|\chi|\ll1\), the magnetization is \(\mathbf M=\chi\,\mathbf H\approx\chi\,\mathbf B/\mu_0\), where \(\chi\) is a symmetric \(3\times3\) tensor. Its principal axes are tied to the crystal lattice. A sample of volume \(V\) in a uniform field has energy

This explains why the same experiment gave “equatorial” for some crystals and “axial” for others. The ellipsoid with semi-axes related to the principal susceptibilities is the geometric form of the tensor, close in spirit to Plücker’s auxiliary ellipsoid. Plücker himself later accepted William Thomson’s (Lord Kelvin’s) theory of magnetic induction as the right foundation for these laws.

The glow that a magnet moves

For the general reader. Pass a high voltage through a glass tube from which most of the air has been pumped, and the tube glows. In 1858 Plücker noticed a green glow on the glass near the negative electrode, and found that a magnet pushed it around. Something was leaving the negative electrode and could be steered by magnetism. Forty years later it was identified as a stream of electrons.

The experiments

From about 1856 Plücker studied how strong magnets act on the luminous discharge in glass tubes of highly rarefied gas. Heinrich Geissler, working in Bonn, made tubes with sealed-in platinum electrodes, evacuated with an improved pump. In an 1858 paper in Poggendorff’s Annalen der Physik, Plücker reported that the glass wall near the cathode glowed with a greenish fluorescence, and that an electromagnet displaced this glow. The Royal Society’s Copley Medal address summarized his finding that a magnet gathers the bluish negative light from each point of the cathode along a line of magnetic force. Plücker gave mathematical definitions of these curves.

Hittorf continued the work. In 1869 he showed that an object between the cathode and the glass casts a sharp shadow, so that whatever leaves the cathode travels in straight lines. Eugen Goldstein named the agent Kathodenstrahlen, cathode rays, in 1876. William Crookes studied them in better vacua, and in 1897 J. J. Thomson measured their charge-to-mass ratio and identified them as particles far lighter than any atom.

Charges in a magnetic field

A particle of charge \(q\) and mass \(m\) in a uniform field \(\mathbf B\) obeys \(m\dot{\mathbf v}=q\,\mathbf v\times\mathbf B\).

Proof. Dotting the equation of motion with \(\mathbf B\) gives \(\dot v_\parallel=0\). Dotting with \(\mathbf v\) gives \(\frac{d}{dt}\tfrac12 m|\mathbf v|^2=0\), so \(|\mathbf v_\perp|\) is constant too. In the plane perpendicular to \(\mathbf B\), the force has constant magnitude \(|q|v_\perp B\) and is perpendicular to the velocity, so the motion is uniform circular with \(m v_\perp^2/r_L=|q|v_\perp B\). ∎

For electrons of a few hundred to a few thousand electron volts and laboratory fields, \(r_L\) is a millimetre or less. The electrons are tied to field lines, which is why the glow traced out the lines of force that Plücker described. The direction of deflection follows from the sign: for an electron, \(\mathbf F=-e\,\mathbf v\times\mathbf B\).

Why the vacuum matters

In a gas at temperature \(T\) and pressure \(p\), with molecular diameter \(d\), the mean free path is

Electrons travel freely only over distances of order \(\lambda\) (larger by a modest factor, since electrons are small and fast, but of the same scaling). At the pressures of the early Geissler tubes most of the energy goes into collisions and a glowing column of gas. As the pressure falls, the dark space near the cathode grows, and the beam reaches the glass, where it excites fluorescence.

Spectra of gases

For the general reader. Light from a glowing gas, passed through a prism, breaks into bright lines whose pattern identifies the gas. Plücker made this observation with his discharge tubes around 1859, just as Kirchhoff and Bunsen were establishing spectrum analysis with flames. He was among the first to see the visible lines of hydrogen. With Hittorf he discovered that one element can show two quite different spectra.

The capillary tube

In a wide tube the light of the discharge was too faint for a spectroscope. Plücker had Geissler make tubes with a narrow capillary section. The current density there was high, and the capillary glowed brightly enough to give clean spectra. Each gas gave its own spectrum, even in quantities too small for chemical detection. Plücker thus showed, independently of Gustav Kirchhoff and Robert Bunsen, that spectral lines are characteristic of the emitting substance. According to Hittorf, he was the first to see the three bright visible lines of hydrogen, now called H\(\alpha\), H\(\beta\), H\(\gamma\).

Two spectra of one element

The joint memoir of Plücker and Hittorf, read to the Royal Society in 1864 and printed in the Philosophical Transactions for 1865, reported that nitrogen, sulphur and several other elements each show two spectra with no lines in common. They called them spectra of the first order (shaded bands, at lower temperature) and of the second order (bright lines, at higher temperature, for example with a Leyden jar in the circuit). The change was abrupt and reversible. They attributed it to different states of the gas. They also saw the hydrogen lines broaden at high temperature until the spectrum approached a continuum.

Quantum mechanics explains both findings. Band spectra come from molecules such as \(\mathrm N_2\), whose vibrational and rotational levels give closely spaced lines that merge into shaded bands. Line spectra come from atoms and ions, produced in the hotter, more violent discharge. Broadening in a dense discharge is mainly due to the electric fields of nearby charged particles (pressure or Stark broadening); thermal Doppler broadening is much smaller.

Johann Balmer found this formula in 1885 from four measured wavelengths. Plücker died seventeen years before, without knowing that the lines he had seen would become the key to atomic structure.

Line geometry

For the general reader. Ordinary geometry builds space out of points. Plücker proposed building it out of straight lines. Space has a four-dimensional family of lines, so families defined by one equation (complexes) are three-dimensional, and their study is a geometry in its own right. The idea came from optics and mechanics, where rays of light and axes of rotation are lines, and it later fed into robotics and kinematics.

Coordinates of a line

Let \(x=(x_0:x_1:x_2:x_3)\) and \(y\) be two distinct points of a line \(L\) in projective 3-space \(\mathbb P^3\). Plücker took the six \(2\times2\) minors

Replacing \(x,y\) by two other points of \(L\) multiplies all \(p_{ij}\) by the same nonzero determinant, so \((p_{01}:p_{02}:p_{03}:p_{12}:p_{13}:p_{23})\) is a point of \(\mathbb P^5\) determined by \(L\).

Proof sketch. Expand \(\det(x,y,x,y)=0\) along the first two rows (Laplace expansion): the result is \(2(p_{01}p_{23}-p_{02}p_{13}+p_{03}p_{12})\). For the converse, given \(p\) on the quadric, one of the coordinates is nonzero, say \(p_{01}\ne0\); then the points \((0,p_{01},p_{02},p_{03})\) and \((-p_{01},0,p_{12},p_{13})\) span a line with the given coordinates. ∎

The two lines meet exactly when the four points lie in a plane, that is, when the determinant vanishes. In Euclidean language, with \(x_0=1\), a line has direction \(\mathbf d=(p_{01},p_{02},p_{03})\) and moment \(\mathbf m=(p_{23},-p_{13},p_{12})=\mathbf x\times\mathbf y\), with \(\mathbf d\cdot\mathbf m=0\). Two lines meet when \(\mathbf d\cdot\mathbf m'+\mathbf d'\cdot\mathbf m=0\).

Complexes

A line complex is a three-parameter family of lines given by one equation in the \(p_{ij}\); a congruence is a two-parameter family, and a ruled surface a one-parameter family. In a linear complex \(\sum a_{ij}p_{ij}=0\), the lines of the complex through any point lie in one plane. This correspondence of points and planes is the null system studied earlier by Möbius in statics. Linear complexes describe instantaneous screw motions of a rigid body: the lines along which the motion has no component form such a complex. Klein’s dissertation (1868), written under Plücker, classified quadratic line complexes.

Last years and Klein, 1864–1868

In 1864 Plücker sent the Royal Society two memoirs, one on gas spectra with Hittorf and one titled “On a New Geometry of Space”. Both appeared in the Philosophical Transactions for 1865. From then on his research was mainly geometry again, though he kept the chair of physics. A paper on the foundations of mechanics followed in 1866.

On 30 November 1866 the Royal Society awarded him its Copley Medal, its highest honour, for his researches in analytical geometry, magnetism and spectral analysis. The same year the seventeen-year-old Felix Klein became his assistant in the physics laboratory. Klein helped him prepare the Neue Geometrie des Raumes, gegründet auf die Betrachtung der geraden Linie als Raumelement, whose first part appeared in 1868.

Plücker died in Bonn on 22 May 1868 and is buried in the Alter Friedhof there. The second part of his book was unfinished. Klein, who had discussed its plan with him, completed it and published it in 1869, and in doing so came into contact with Alfred Clebsch at Göttingen. Klein took his doctorate at Bonn in 1868 with a dissertation on line geometry; Rudolf Lipschitz is also named as an advisor. Among Plücker’s other students were Hittorf, who became professor at Münster, and August Beer, remembered for the Beer–Lambert law of absorption.

Legacy

Plücker’s geometry won through. Within a generation, coordinates and algebra had absorbed the synthetic geometry of Poncelet and Steiner, and the questions he asked about curves became the starting point of algebraic geometry.

• Projective algebraic geometry: homogeneous coordinates, the systematic use of duality, and the Plücker formulas, later generalized by Zeuthen, Max Noether and many others, and now understood through the genus and the Riemann–Hurwitz formula.
• Grassmannians: his line coordinates generalize to the Plücker embedding of the Grassmannian of \(k\)-planes in an \(n\)-dimensional space, cut out by the Plücker relations. These are central in Schubert calculus, representation theory, integrable systems (the KP hierarchy) and the theory of positive Grassmannians in physics.
• Klein and Lie: line geometry was the starting point of Felix Klein’s work and, through Klein, of Sophus Lie’s line–sphere correspondence and his theory of continuous groups. Klein’s Erlangen programme grew from these studies.
• Kinematics and robotics: Plücker coordinates and screw theory describe the motions of rigid bodies and are standard in robot kinematics and computer graphics.
• Physics: his discharge experiments began the line of research that led through Hittorf, Goldstein and Crookes to J. J. Thomson’s electron (1897). His gas spectra were part of the founding evidence for spectrum analysis, and the hydrogen lines he saw led to Balmer’s formula and to Bohr’s atom.
• Honours: Foreign Member of the Royal Society (1855) and Copley Medal (1866). His collected mathematical papers were edited by Arthur Schoenflies and his physical papers by Friedrich Pockels (1895–96).

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Julius Plücker, MacTutor History of Mathematics.
2. Julius Plücker, Wikipedia.
3. Obituary notice, Proceedings of the Royal Society (1869), via MacTutor.
4. Julius Plücker receives the Copley Medal, address of 30 November 1866, Proc. Roy. Soc. 15 (1867).
5. Plücker’s correspondence with Faraday, Epsilon: Faraday letters.

Mathematics

1. J. Plücker, Analytisch-geometrische Entwicklungen, 2 vols., Essen, 1828–1831.
2. J. Plücker, System der analytischen Geometrie, Berlin, 1835.
3. J. Plücker, Theorie der algebraischen Curven, Bonn, 1839.
4. J. Plücker, On a new geometry of space, Phil. Trans. R. Soc. 155 (1865).
5. J. Plücker, Neue Geometrie des Raumes, gegründet auf die Betrachtung der geraden Linie als Raumelement, Leipzig, 1868–1869 (part II edited by F. Klein).
6. Plücker formula and Plücker coordinates, Wikipedia.
7. C. T. C. Wall, Singular Points of Plane Curves, Cambridge University Press, 2004 (modern treatment of the Plücker formulas).
8. H. Pottmann and J. Wallner, Computational Line Geometry, Springer, 2001.

Physics

1. J. Plücker, On the magnetic induction of crystals, Phil. Trans. R. Soc. 148 (1858), text.
2. J. Plücker, Ueber die Einwirkung des Magneten auf die elektrischen Entladungen in verdünnten Gasen, Ann. Phys. (1858), doi:10.1002/andp.18581790106.
3. J. Plücker and J. W. Hittorf, On the spectra of ignited gases and vapours, with especial regard to the different spectra of the same elementary gaseous substance, Phil. Trans. R. Soc. 155 (1865), text.
4. Cathode ray, Wikipedia (history section).