Hermann von Helmholtz

How to use this course

Hermann von Helmholtz worked across more of science than almost anyone of his century. He trained as a doctor, held chairs of physiology for twenty years, and spent his last two decades as Germany’s senior physicist. His results are scattered across textbooks of thermodynamics, fluid mechanics, acoustics, optics, electromagnetism, physiology and geometry. This course gathers them, and follows the life that connected them.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The lessons follow the order of his work: energy (1847), physiology (1850–1867), hydrodynamics (1858), acoustics (1860–1863), geometry (1868–1870), electrodynamics (1870s) and chemical thermodynamics (1882). Lesson 10 is placed last in Part II because it is the most philosophical.

Timeline

Potsdam, 1821–1838

Hermann Ludwig Ferdinand Helmholtz was born in Potsdam on 31 August 1821. His father Ferdinand taught at the Potsdam Gymnasium and cared deeply about philosophy, art and literature. His mother, Caroline Penne, was the daughter of an artillery officer. Hermann was the eldest of four children and a sickly small boy.

At the Gymnasium where his father taught he received a thorough classical education, and at home he absorbed his father’s interest in Kant. His real passion was physics. In a speech of 1891 he recalled working out the paths of light rays through a telescope under his desk while the class read Cicero or Virgil.

The family could not afford a university education in science. There was a way out. The Prussian army trained its doctors free of charge at the Friedrich-Wilhelm Institute in Berlin, in return for years of service as an army surgeon after graduation. In 1837 Helmholtz won one of these places and in 1838 he moved to Berlin. Medicine was the price of an education in science, and he paid it with interest: his medical training gave him the questions that drove his first and most famous physics.

Army surgeon and young physicist, 1838–1848

In Berlin Helmholtz attended university lectures in chemistry and physiology alongside the military curriculum. The decisive teacher was Johannes Müller, the leading physiologist in Germany. Müller still held that living things contain a vital force beyond physics and chemistry, but his students turned against the idea. Helmholtz’s closest friends among them were Emil du Bois-Reymond and Ernst Brücke; with Carl Ludwig they set out to explain life by the same laws that govern inanimate matter.

His 1842 doctoral thesis, De fabrica systematis nervosi evertebratorum, showed that nerve fibres in invertebrates arise from ganglion cells. In 1843 he began his army service as a surgeon with a regiment in Potsdam. He kept a small laboratory in the barracks and studied fermentation, putrefaction and the heat produced by working muscle. That last problem forced him to ask where the heat of an animal comes from. If a vital force could produce work from nothing, an organism would be a perpetual motion machine.

In 1845 the young men around the physicist Gustav Magnus founded the Physical Society of Berlin, and Helmholtz joined. On 23 July 1847 he read to the Society his memoir Über die Erhaltung der Kraft (On the conservation of force). Johann Christian Poggendorff, editor of the Annalen der Physik, declined to publish it, and it appeared as a pamphlet from the Berlin publisher Reimer. The mathematician Carl Gustav Jacob Jacobi was among the first to recognize its value. Lesson 04 treats its content.

The essay made his name. With help from Alexander von Humboldt he was released early from the army in 1848 and became teacher of anatomy at the Berlin Academy of Arts.

Königsberg, Bonn, Heidelberg, 1849–1871

In 1849 Helmholtz became associate professor of physiology at Königsberg, and on 26 August married Olga von Velten. Within two years he made two discoveries that brought him international fame. He measured the speed of the nerve impulse, which Müller had thought too fast to measure. And in late 1850, while preparing a lecture on the eye, he built the ophthalmoscope, the first instrument to show the living retina. Both are the subject of Lesson 05.

In 1855 he moved to Bonn as professor of anatomy and physiology, and in 1858 to Heidelberg, where the state of Baden built him a new physiological institute. The Heidelberg years were his most productive. He completed the Handbuch der physiologischen Optik (1856–1867), published Die Lehre von den Tonempfindungen (1863), wrote the paper on vortex motion (1858) and the papers on the foundations of geometry (1868–1870). His assistants included the future psychologist Wilhelm Wundt. His colleagues included Robert Bunsen and Gustav Kirchhoff, whose spectroscopy was being created in the same town.

Olga, long in poor health, died in December 1859, leaving two young children. On 16 May 1861 he married Anna von Mohl, daughter of a Heidelberg professor and a woman of wide culture, who made their home a meeting place for scientists, writers and musicians. They had three more children. By 1870 Helmholtz was a physiologist only in title. When Berlin sought a professor of physics and Kirchhoff declined, the chair went to Helmholtz, who had never formally studied physics.

The conservation of force (1847)

For the general reader. No machine can produce work from nothing. Helmholtz took this old rule and showed what it implies: there is a single quantity, now called energy, that changes form among motion, heat, electricity, magnetism and chemistry but never changes in total amount. Julius Robert Mayer and James Prescott Joule had reached the same idea in the early 1840s. Helmholtz gave the first general mathematical treatment, across all of physics.

The mechanical core

Helmholtz used “Kraft” in two senses: lebendige Kraft (vis viva, \(mv^2\), or for him \(\tfrac12mv^2\)) and Spannkraft (tension, what we call potential energy). He started from the impossibility of perpetual motion and assumed that matter consists of particles acting on each other by central forces whose strength depends only on distance.

Derivation

The force on particle \(i\) from \(j\) is \(\mathbf F_{ij}=\phi_{ij}(r_{ij})\,\hat{\mathbf r}_{ij}\) with \(\hat{\mathbf r}_{ij}=(\mathbf x_i-\mathbf x_j)/r_{ij}\). Then

The third step uses Newton’s third law, the fourth uses \(\dot r_{ij}=\hat{\mathbf r}_{ij}\cdot(\mathbf v_i-\mathbf v_j)\). Helmholtz also argued the converse: if forces were not central, or depended on more than distance, one could arrange a cycle that returns the particles to their initial positions with a net gain of work. The converse is the weak point of the essay. Lesson 08 shows how it led him into a long dispute with Wilhelm Weber, whose electrical force depended on velocity.

Beyond mechanics

Helmholtz then applied the principle to every branch of physics he knew. Heat is the vis viva of hidden molecular motions plus their tension. In a galvanic circuit the heat released by the current must equal the heat of the chemical reactions in the battery. He derived the energy balance for electromagnetic induction from the forces between currents and magnets, and showed that induction must exist if energy is conserved. He ended with living things: the heat and work of animals must come from the chemical energy of food.

Nerve, eye and colour

For the general reader. In 1850 people assumed thought and sensation travelled instantly. Helmholtz timed a nerve signal in a frog’s leg and found it slower than sound in air. In the same year he built a device to look through the pupil at the back of the living eye. Both were achievements of physical measurement applied to the body.

Timing the nerve impulse

Helmholtz used a frog’s calf muscle with its sciatic nerve attached. He stimulated the nerve at two points, one near the muscle and one farther away, and measured the delay until the muscle contracted. The delay includes the unknown time the muscle takes to respond; subtracting the two delays cancels it. To measure intervals of a few milliseconds he used a method of Claude Pouillet: the stimulus closed a circuit through a galvanometer and the muscle’s twitch broke it, so the swing of the needle measured the time. He later also recorded the twitch directly on a moving drum. His frog values lay roughly between 25 and 40 m/s.

The ophthalmoscope

Light entering the pupil and reflected by the retina leaves the eye along almost the same path, returning toward the light source, because the eye’s optics image the source onto the retina and the retina back onto the source. An observer who is not on the line of illumination therefore sees nothing, and the pupil looks black. Helmholtz placed a few glass plates obliquely in front of the patient’s eye. They reflected light from a lamp into the eye and let part of the returning light pass through to the observer looking along the same axis. A lens corrected for the focus of the two eyes. He described the instrument in 1851, and it transformed ophthalmology.

Three colour receptors

Thomas Young had suggested in 1802 that the eye contains three kinds of sensors. Helmholtz revived and developed the idea. In 1852 he also cleared up a confusion: mixing blue and yellow lights gives white, while mixing blue and yellow pigments gives green. Pigments subtract light and lights add, so the two mixtures obey different rules.

Vortex motion (1858)

For the general reader. Smoke rings hold together and travel far. In 1858 Helmholtz proved that in a fluid without friction, spinning regions keep their spin forever and move with the fluid. The paper founded vortex dynamics. It also inspired William Thomson to propose that atoms were knotted vortex rings in the ether.

The vorticity equation

For an incompressible inviscid fluid of constant density with conservative body force, the equation of Leonhard Euler reads \(\partial_t\mathbf u+(\mathbf u\cdot\nabla)\mathbf u=-\nabla(p/\rho+\Phi)\). Write \((\mathbf u\cdot\nabla)\mathbf u=\nabla(\tfrac12|\mathbf u|^2)-\mathbf u\times\boldsymbol\omega\) with \(\boldsymbol\omega=\nabla\times\mathbf u\), and take the curl. The gradient terms vanish, and with \(\nabla\cdot\mathbf u=0\),

The right side is exactly how a material line element \(\delta\mathbf l\) evolves: \(D\,\delta\mathbf l/Dt=(\delta\mathbf l\cdot\nabla)\mathbf u\). So vortex lines are carried by the fluid and stretched with it.

Statement 2 at a fixed time follows from \(\nabla\cdot\boldsymbol\omega=0\) and Gauss’s theorem applied to a piece of tube; statement 3 follows from the same identity. Constancy in time follows from the transport of vortex lines, or from Thomson’s circulation theorem of 1869, \(\frac{d}{dt}\oint_{C(t)}\mathbf u\cdot d\mathbf l=0\) for a material loop.

Velocity from vorticity

Helmholtz noticed that recovering \(\mathbf u\) from \(\boldsymbol\omega\) with \(\nabla\cdot\mathbf u=0\) is the same mathematical problem as finding the magnetic field of an electric current, and wrote

(plus an irrotational part fixed by the boundaries). A straight line vortex of strength \(\Gamma\) induces circular flow of speed \(\Gamma/(2\pi r)\). He used this to show that a circular vortex ring moves along its axis, carried by its own induced velocity, and analysed how two rings interact: a following ring shrinks, speeds up and passes through the leading one, which widens and slows. Smoke rings do this.

In 1868 he studied flows with surfaces across which velocity jumps, and showed that such surfaces of discontinuity explain jets and wakes. Their instability is now called the Kelvin–Helmholtz instability.

Sensations of tone (1863)

For the general reader. Why does a violin sound different from a flute playing the same note, and why do some chords sound smooth and others rough? Helmholtz answered both with physics. A musical note is a sum of pure tones, the ear separates them, and roughness comes from beats between nearby tones. His book Die Lehre von den Tonempfindungen joined acoustics, physiology and music theory.

Analysing sound

Following Georg Ohm’s acoustic law, Helmholtz held that the ear performs a Fourier analysis: it perceives a periodic sound as a sum of sinusoidal partials. To test this he built hollow glass and brass spheres with a narrow neck, each tuned to one frequency. Held to the ear, a resonator sounds loudly only when the incoming sound contains its frequency. With a set of them he analysed the partials of instruments and of vowels, and he synthesized vowels with electrically driven tuning forks.

Derivation

Treat the air in the neck as a piston of mass \(\rho AL\). Pushing it in by \(x\) reduces the cavity volume by \(Ax\). For adiabatic compression \(\Delta p=\rho c^2\,Ax/V\), since \(\rho c^2\) is the adiabatic bulk modulus. The restoring force is \(A\,\Delta p=(\rho c^2A^2/V)\,x\), so

Consonance and the ear

Two tones of nearby frequencies \(f_1,f_2\) produce beats at \(|f_1-f_2|\). Slow beats are heard as a waver; Helmholtz found that beats at rates of a few tens per second sound harsh. Two complex tones are consonant when few of their partials beat at such rates. For frequency ratios 2:1, 3:2, 4:3 many partials coincide exactly, which explains the ordering of the classical consonances. He proposed that the basilar membrane of the inner ear is a bank of tuned resonators, each responding to a narrow band. Georg von Békésy later showed that the membrane carries a travelling wave with a frequency-dependent peak, a modified form of the same place principle.

He also studied combination tones. If the response of a system is nonlinear, for example \(y=a x+b x^2\), an input with two frequencies produces output at their sum and difference.

The Helmholtz equation and electrodynamics

For the general reader. Any wave of one frequency, whether sound in an organ pipe or light, obeys one equation, which now carries Helmholtz’s name. He met it in acoustics in 1860. In the 1870s he tried to decide between rival theories of electricity, and set the problem whose solution, by his student Heinrich Hertz, proved that light is an electromagnetic wave.

Waves of one frequency

For the wave equation \(\partial_t^2u=c^2\nabla^2u\), a solution \(u=\mathrm{Re}\,[\psi(\mathbf x)e^{-i\omega t}]\) requires

In his 1860 paper on air vibrations in open organ pipes, Helmholtz solved this equation with Green’s methods, using the point-source solution

and proved the acoustic reciprocity theorem: swapping a source and a receiver leaves the measured signal unchanged. Kirchhoff’s diffraction theory of 1882 is built on the same Green’s identity, written for this equation.

Sketch. Use \(\nabla^2(1/4\pi r)=-\delta^3\) to write \(\mathbf F=-\nabla^2\mathbf W\) with \(\mathbf W=\frac1{4\pi}\int\mathbf F(\mathbf x')/|\mathbf x-\mathbf x'|\,d^3x'\), then apply \(\nabla^2=\nabla(\nabla\cdot)-\nabla\times\nabla\times\) and move derivatives onto \(\mathbf F\) under the integral. The vortex paper of Lesson 06 is an application: there \(\nabla\cdot\mathbf u=0\) and \(\nabla\times\mathbf u=\boldsymbol\omega\).

Choosing among theories of electricity

Around 1870 German physicists explained electrodynamics by forces acting at a distance between moving charges, as in the law of Wilhelm Weber, whose force depends on velocities and accelerations. James Clerk Maxwell in Britain had a field theory. From 1870 Helmholtz wrote a general potential for the interaction of current elements containing a free constant, usually written \(k\), so that particular values reproduced the theories of Franz Neumann, Weber and Maxwell. He argued that Weber’s law allowed physically unstable motions in which charges could acquire unlimited kinetic energy, which Weber disputed. When the dielectric medium was given polarizability, his theory admitted transverse waves travelling at the speed of light in the limit that matched Maxwell, along with longitudinal waves for other values of \(k\).

To decide experimentally, Helmholtz had the Berlin Academy set a prize problem in 1879 on the electrodynamic effects of dielectric polarization, and urged Heinrich Hertz, who had taken his doctorate under him in 1880, to attempt it. Hertz did not at first see a feasible method. At Karlsruhe in 1886–1888 he produced and detected electromagnetic waves and showed that they reflect, refract and travel at finite speed, confirming Maxwell’s theory. Helmholtz presented several of Hertz’s results to the Berlin Academy.

Free energy (1882)

For the general reader. How much useful work can a chemical reaction deliver, for instance in a battery? The heat a reaction gives off is the wrong answer. In 1882 Helmholtz split a system’s energy into a “free” part, which can be turned into work at constant temperature, and a “bound” part, which cannot. The free part now bears his name.

Definition

For a system with internal energy \(U\), entropy \(S\) (the quantity of Rudolf Clausius) and absolute temperature \(T\), Helmholtz defined

He called \(F\) the free energy and \(TS\) the bound energy. Josiah Willard Gibbs had introduced equivalent functions in 1873–1878, in a journal little read in Europe; Helmholtz worked independently.

Proof. The first law gives \(W=Q-\Delta U\). The second law, for heat \(Q\) taken from a reservoir at \(T\), gives \(\Delta S\ge Q/T\). Hence \(W\le T\Delta S-\Delta U=-\Delta F\).

Gibbs–Helmholtz equation

Since \(S=-(\partial F/\partial T)_V\),

The same steps at constant pressure relate enthalpy \(H\) and Gibbs energy \(G\). Helmholtz applied the result to galvanic cells. The chemists Julius Thomsen and Marcellin Berthelot held that the heat of a reaction measures its driving force. Helmholtz showed that the electrical work of a reversible cell equals the drop in free energy, and that it differs from the heat of reaction by a term proportional to the temperature coefficient of the cell’s voltage.

The facts behind geometry (1868–1870)

For the general reader. Is Euclid’s geometry true of space by necessity, as Kant had taught, or is it a fact we learn from experience? Helmholtz argued for experience. He asked what geometry follows from one physical fact, that rigid bodies can be moved around freely without changing shape, and found that it allows only a few possibilities, including curved ones.

From physiology to axioms

Helmholtz came to geometry from the study of how we see space. His 1868 paper Über die Thatsachen, die der Geometrie zum Grunde liegen echoed the title of Bernhard Riemann’s habilitation lecture, published the same year after Riemann’s death. Riemann started from a metric; Helmholtz started from motions. He first concluded that only Euclidean and spherical geometry were possible. Eugenio Beltrami wrote to him, and in 1869 he added the hyperbolic case.

Helmholtz assumed the motions were differentiable and used analysis; his argument had gaps. Sophus Lie reformulated the problem in terms of continuous transformation groups and gave the first rigorous proof in 1890, criticizing Helmholtz’s hypotheses. The group of motions in each case has dimension 6.

Imagined worlds

In a popular lecture of 1870, Über den Ursprung und die Bedeutung der geometrischen Axiome, Helmholtz described flat beings living on a sphere, who would find no parallel lines and triangles with angle sums above two right angles. He also imagined our world seen in a convex mirror, where measuring rods shrink as they move away and the inhabitants could not detect it. His conclusion: which geometry holds is a question about the behaviour of rigid bodies and light rays, to be settled by measurement.

Berlin, 1871–1888

Helmholtz took up the Berlin chair of physics in 1871 on unusually favourable terms, including a new physics institute to be built under his direction. He was a poor lecturer for beginners, by many accounts, but a magnet for advanced students. Those who took doctorates under him include Heinrich Hertz (1880), Wilhelm Wien (1886) and Michael Pupin (1889). Albert Michelson worked in his laboratory in 1880–81 and began there the interferometer later used in the Michelson–Morley experiment. Max Planck heard his and Kirchhoff’s lectures in 1877–78, and Ludwig Boltzmann visited in 1871. Planck later succeeded Kirchhoff in Berlin and became Helmholtz’s colleague and friend.

He was rector of the university in 1877–78 and president of the Physical Society of Berlin from 1878 until his death. He was a celebrated public speaker; his popular lectures on vision, music, energy and the axioms of geometry were translated and widely read. With William Thomson, later Lord Kelvin, he formed a lasting friendship across the North Sea. On 27 January 1883 he was raised to the Prussian nobility.

His research turned to electrodynamics (Lesson 08), to the thermodynamics of chemical processes (Lesson 09), to the theory of monocyclic systems, by which he tried to give thermodynamics a mechanical basis (1884), and to the principle of least action, which he believed would unify physics. Meteorology also drew him: he applied the theory of discontinuous fluid motion to waves on surfaces between air layers.

His family life in these years held grief. His son Robert, a gifted young physicist, died in 1889. His daughter Ellen married Arnold von Siemens, son of the industrialist Werner von Siemens.

The Reichsanstalt, death and legacy, 1888–1894

In 1887 the German government founded the Physikalisch-Technische Reichsanstalt in Charlottenburg, near Berlin, with a large gift of land and money from Werner von Siemens. It was meant to combine pure research with the precise standards that German industry needed. In 1888 Helmholtz became its first president, giving up his university teaching. Under him it began the work on electrical standards, thermometry and radiation measurement that, a few years after his death, produced the black-body data from which Planck found his radiation law.

His seventieth birthday in 1891 was celebrated as a national event, with tributes from across Europe. In the summer of 1894 he suffered a stroke, and on 8 September 1894 he died in Charlottenburg after a second stroke, aged 73. He is buried in the Wannsee cemetery in Berlin.

Legacy

• Energy: the 1847 essay made conservation of energy a principle of all physics and of physiology.
• Physiology and psychology: nerve conduction velocity, the ophthalmoscope, the theory of accommodation, the trichromatic theory and the idea of perception as unconscious inference shaped the sciences of the senses.
• Fluid mechanics and acoustics: vortex theorems, Kelvin–Helmholtz instability, Helmholtz resonators and the place theory of hearing.
• Mathematical physics: the Helmholtz equation and decomposition, Helmholtz free energy and the Gibbs–Helmholtz equation.
• Institutions: the model of the national standards laboratory, and a generation of students including Hertz. Germany’s largest association of research centres bears his name.
• Honours: Foreign Member of the Royal Society (1860), Copley Medal (1873), Pour le Mérite (1873), Albert Medal of the Royal Society of Arts (1888).

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Hermann von Helmholtz, MacTutor History of Mathematics.
2. Hermann von Helmholtz, Wikipedia; and the German article Hermann von Helmholtz.
3. Helmholtz-Zentrum Berlin, Hermann von Helmholtz.
4. L. Koenigsberger, Hermann von Helmholtz, 3 vols., Vieweg, 1902–03; abridged English translation, Oxford, 1906.
5. D. Cahan, Helmholtz: A Life in Science, University of Chicago Press, 2018.
6. D. Cahan (ed.), Hermann von Helmholtz and the Foundations of Nineteenth-Century Science, University of California Press, 1993.

Physics

1. H. Helmholtz, Über die Erhaltung der Kraft, Reimer, Berlin, 1847.
2. H. Helmholtz, Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen, J. reine angew. Math. 55 (1858). See also Helmholtz’s theorems.
3. H. Helmholtz, Theorie der Luftschwingungen in Röhren mit offenen Enden, J. reine angew. Math. 57 (1860).
4. H. Helmholtz, Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik, Vieweg, 1863; English translation by A. J. Ellis, On the Sensations of Tone.
5. H. Helmholtz, Handbuch der physiologischen Optik, Voss, 1856–1867; English translation by the Optical Society of America, 1924–25.
6. H. Helmholtz, Über die Thatsachen, die der Geometrie zum Grunde liegen, Nachr. Ges. Wiss. Göttingen (1868).
7. H. Helmholtz, Die Thermodynamik chemischer Vorgänge, Sitzungsber. Preuss. Akad. Wiss. Berlin (1882).
8. H. Helmholtz, Popular Lectures on Scientific Subjects, Longmans, Green, 1873 and later series.
9. S. Lie, Über die Grundlagen der Geometrie, Ber. Verh. Sächs. Ges. Wiss. Leipzig (1890).
10. J. Buchwald, The Creation of Scientific Effects: Heinrich Hertz and Electric Waves, University of Chicago Press, 1994.