Julius Robert Mayer

How to use this course

Julius Robert Mayer was a country doctor with little training in physics. In 1842 he published a short paper claiming that “forces”, in the sense of causes of motion and heat, can change form but never be created or destroyed. He gave a number for the exchange rate between heat and mechanical work, derived from data on air. A year later James Prescott Joule announced a number obtained by experiment, and in 1847 Hermann von Helmholtz gave the principle its general mathematical form. Mayer’s priority was ignored, then disputed, then recognized. He also applied the principle to living bodies, to the Sun and to the tides, years ahead of others.

Each technical lesson opens with a paragraph for the general reader, then states the physics in modern form, follows Mayer’s own argument, and closes with exercises with folded solutions. Mayer wrote in words, not equations. Where we write formulas, we translate his reasoning; where his reasoning was wrong or incomplete, we say so.

Timeline

Heilbronn and Tübingen, 1814–1839

Julius Robert Mayer was born on 25 November 1814 in Heilbronn, a trading town on the Neckar in the kingdom of Württemberg. He was the youngest of three sons of Christian Mayer, who kept the Rose Pharmacy in the town. The pharmacy gave the boy an early acquaintance with chemicals, balances and recipes. He attended the Heilbronn Gymnasium and from 1828 the Protestant seminary at Schöntal, and passed the Abitur in Stuttgart.

At Easter 1832 he entered the University of Tübingen to study medicine. He spent ten semesters there and preferred anatomy and physiology. He helped found a student corps, Guestphalia. In 1837, after a farewell celebration, he and his friend Wilhelm Griesinger, later a leading psychiatrist, were punished for wearing the colours of the forbidden association. Mayer was expelled for a year and briefly held in custody. He went on to study in Munich and Vienna, returned, and in 1838 passed his medical examinations and took his doctorate. His dissertation, Über das Santonin, reported case histories of children treated with santonin, a drug against roundworms.

A practice in Heilbronn brought him few patients. He used the time to learn French and Dutch and decided to see the world as a ship’s doctor. In 1839 he travelled through Paris to Holland, passed the Dutch examination for medical officers, and signed on with a merchant ship bound for the East Indies.

The voyage to Java, 1840–1841

On 22 February 1840 the three-masted Java sailed from Rotterdam. After about 98 days at sea it reached Java in June. The voyage was uneventful for a doctor, but Mayer reported two observations that he later said set him thinking. A helmsman told him that the sea is warmer after a storm. And in the roads of Java, where sickness broke out among the crew, Mayer bled the sailors, as medicine then required, and saw that the blood from their veins was nearly as bright red as arterial blood.

The interpretation

Venous blood is darker because the tissues have taken oxygen from it. Mayer followed the chemist Antoine Lavoisier, who had treated respiration as slow combustion. In a hot climate, he reasoned, the body needs to produce less heat to keep its temperature, so it burns less, so less oxygen is removed and venous blood stays red. The heat of the body comes from the chemical oxidation of food.

The next step was the one that made him a physicist. The body also does mechanical work. If heat and work both come from the same combustion, then the total heat produced, plus the heat equivalent of the work done, must equal the heat of the chemical reaction. Work and heat must therefore be measurable in one currency. On the return voyage, and after his arrival in Heilbronn in February 1841, he tried to write down the general law that would make this true.

From blood to physics, 1841–1842

Back in Heilbronn, Mayer became district surgeon (Oberamtswundarzt) and later town physician. On 16 June 1841 he sent a paper, Über die quantitative und qualitative Bestimmung der Kräfte, to Johann Christian Poggendorff’s Annalen der Physik und Chemie, the leading physics journal in Germany. It received no reply. The manuscript was found among Poggendorff’s papers after his death in 1877.

The rejection was understandable. Mayer measured the “force” of a moving body by its momentum, mass times velocity, not by its kinetic energy, and his mechanics was confused. He spent the following months learning physics and mathematics, and discussing his ideas with physicists, who were mostly unconvinced.

His second attempt was shorter and clearer. Bemerkungen über die Kräfte der unbelebten Natur (Remarks on the forces of inanimate nature) appeared in the May 1842 issue of Justus von Liebig’s Annalen der Chemie und Pharmacie. It stated that forces are indestructible and convertible, and that the fall of a weight from a height of about 365 metres corresponds to warming an equal weight of water from 0 to 1 °C. It said little about how he obtained the number. The paper attracted almost no attention.

The same year Mayer married Wilhelmine Closs (1816–1899), a merchant’s daughter from Winnenden. They lived in Heilbronn, where his practice grew large.

Forces as causes: the 1842 paper

For the general reader. Mayer’s starting point was philosophical: every effect has a cause equal to it, and nothing comes from nothing. He called the causes of change “forces” and argued that they can turn into one another, like motion into heat, but never disappear. In modern language, he stated the conservation of energy.

Mayer’s axioms

Mayer summarized his guiding idea in a Latin maxim: causa aequat effectum, the cause equals the effect. Forces, he wrote, are causes. A cause that produces an effect does not vanish but turns into the effect. So forces are quantitatively indestructible and qualitatively convertible. He named three kinds of force in inanimate nature: “falling force” (the separation of a weight from the Earth, our gravitational potential energy), motion, and heat. A raised weight falls and acquires motion; the motion, when the weight strikes the ground, becomes heat.

What is conserved in an impact

The 1841 manuscript used momentum \(mv\) as the measure of motion. Momentum is conserved in every collision, so it cannot be the quantity that turns into heat when motion is destroyed. The 1842 paper uses the product of weight and height of fall, equivalent to \(\tfrac12mv^2\), and is correct on this point. Mayer also pointed out that if motion stops without producing heat, something has been destroyed, which his principle forbids; so the heating of bodies by friction and impact is a consequence of the principle, not an accident.

Why it was resisted

In 1842 heat was widely regarded as a substance, caloric, and Sadi Carnot’s theory of engines assumed it was conserved. Mayer’s principle denies this: heat can be created from motion and destroyed in producing work. His argument was general and philosophical, with almost no experiment of his own. Physicists trained in the experimental tradition found it vague, and his use of “force” for what we call energy, while standard in German, added confusion.

The equivalent from the gases

For the general reader. Mayer had no laboratory. He found his number in tables of measurements made by others. Warming air while letting it expand takes more heat than warming it in a closed box. The extra heat, Mayer argued, is used up in pushing back the atmosphere, and the ratio of that work to the extra heat is the exchange rate between work and heat.

The argument

Take a mass \(m\) of air at pressure \(p\) and temperature \(T\). Warm it by \(\Delta T\) in two ways. At constant volume the heat needed is \(Q_V=mc_V\Delta T\). At constant pressure it is \(Q_p=mc_p\Delta T\), and the gas expands by \(\Delta V\), lifting the load on it and doing work \(p\,\Delta V\). If the internal state of the gas is the same at the end of both processes, the extra heat has become this work:

For an ideal gas \(p\,\Delta V=\frac{m}{M}R\,\Delta T\), so this becomes

Mayer’s numbers

The 1842 paper gave only the result. In 1845 Mayer laid out the calculation. He used the specific heat of air at constant pressure, \(c_p=0.267\) cal g\(^{-1}\) K\(^{-1}\), and the ratio \(\gamma=1.421\) derived from measurements of the speed of sound. Then \(c_V=0.188\). He considered a column of air of 1 cm\(^2\) cross-section under the 76 cm mercury column of the atmosphere, which weighs about 1033 g. Warming 1 cm\(^3\) of air by 1 °C at constant pressure raises the column by \(1/274\) cm. The result was about 365 kilogram-metres per kilocalorie. He later revised it to about 425. The modern value is 426.9.

The hidden assumption

The argument assumes that the heat \(Q_V\) and the heat \(Q_p-p\Delta V/J\) leave the gas in the same internal state, that is, that the internal energy of the gas depends on temperature only and not on its volume. Otherwise some of the extra heat could go into separating the molecules. The justification is the experiment of Joseph Louis Gay-Lussac, in which a gas expanding into an evacuated vessel showed no net change in temperature. Joule repeated it more carefully in 1845. For real gases the assumption holds to a fraction of a percent near room conditions, which is why Mayer’s method works when good data are used.

Sound and the ratio of specific heats

For the general reader. Mayer’s calculation needed the ratio of the two specific heats of air, and nobody could measure the specific heat at constant volume directly. The ratio came instead from the speed of sound. Sound travels faster than Isaac Newton had predicted because air warms slightly when compressed by a sound wave. The size of the correction gives the ratio, and through Mayer’s argument it gives the mechanical equivalent of heat. Tyndall later stressed that Mayer had in effect computed the equivalent from the velocity of sound.

Newton and Laplace

A small compression of a fluid with pressure \(p\) and density \(\rho\) travels at speed \(c=\sqrt{(\partial p/\partial\rho)}\), where the derivative is taken along the process the fluid actually follows. Newton assumed the temperature stays constant, so \(p\propto\rho\) and \(c=\sqrt{p/\rho}\), about 280 m/s in air at 0 °C. Measurements gave about 331 m/s. In 1816 Pierre-Simon Laplace explained the gap: the compressions are too fast for heat to flow, so they are adiabatic, and the pressure rises more steeply.

Derivation of the adiabat. For an ideal gas with \(U=U(T)\), no heat flow means \(nC_{V,m}\,dT=-p\,dV\). With \(pV=nRT\), \(\frac{C_{V,m}}{R}\,d(pV)=-p\,dV\), which rearranges to \(\frac{dp}{p}=-\frac{C_{V,m}+R}{C_{V,m}}\frac{dV}{V}\). Mayer’s relation \(C_{p,m}=C_{V,m}+R\) turns the factor into \(\gamma\), so \(pV^\gamma\) is constant. Then \(\partial p/\partial\rho=\gamma p/\rho\).

The chain of reasoning is circular only in appearance. The speed of sound and the density give \(\gamma\); calorimetry gives \(c_p\); then \(c_p-c_V=c_p(1-1/\gamma)\) is the extra heat at constant pressure, and the gas law gives the work. Mayer’s relation turns acoustics and calorimetry into a measurement of \(J\).

Organic motion and metabolism (1845)

For the general reader. Liebig declined to publish Mayer’s long second essay, so Mayer printed it himself in Heilbronn in 1845. It applied the principle to life. Plants capture the energy of sunlight and store it in chemical form. Animals release it by oxidation, as heat and muscular work. No special “vital force” is needed. It was the first energy budget of the living world.

The forms of force

Die organische Bewegung in ihrem Zusammenhange mit dem Stoffwechsel (Organic motion in its connection with metabolism) listed the forms of force in nature: mechanical (as potential and as motion), heat, magnetism and electricity, and chemical separation. It gave examples of conversion between each pair. A steam engine turns heat into motion; friction turns motion into heat; an electrical machine turns work into electricity; a battery turns chemical separation into electricity. The essay then turned to the earth as a whole. The Sun is the source of almost every force on the surface: it drives the winds and the water cycle, and through plants it supplies the chemical force of food and fuel.

Against vital force

Many physiologists of the 1840s held that living matter obeys a special vital force. Mayer argued that muscle is a machine that converts chemical energy into work, not a source of force. He argued that the fuel for muscular work is carried in by the blood and oxidized, rather than drawn from the substance of the muscle itself. He also compared the body’s heat with that of the blood’s oxidation, connecting the observation from Java to the general principle.

The heat of the Sun (1848)

For the general reader. If energy is conserved, the Sun must be paying for its light from some store. Mayer showed that no chemical fuel could keep it shining for long. He proposed that meteors falling onto the Sun at enormous speed turn their motion into heat. The theory was wrong, but the question he posed, what powers the Sun, was right, and it shaped astrophysics for the next ninety years.

Mayer’s argument

Mayer worked out the idea from 1841 onward and published it at his own expense in 1848 as Beiträge zur Dynamik des Himmels in populärer Darstellung (Contributions to celestial dynamics, in popular form). He used Claude Pouillet’s 1838 measurement of the solar constant. Chemical combustion, he showed, could keep the Sun shining for only a few thousand years. He rejected the Sun’s rotation as a source too. Bodies falling from far away onto the Sun arrive at a speed of several hundred kilometres per second, and he estimated that each kilogram then releases thousands of times more heat than a kilogram of coal burns.

The meteoric theory was found again, independently, by John James Waterston in 1853 and developed by William Thomson in 1854, before Thomson recognized Mayer’s priority. It failed because the required mass of infalling matter would increase the Sun’s gravity and shorten the year measurably, and because so much matter is not seen near the Sun. Mayer noticed the first problem himself and tried to avoid it. In 1854 Hermann von Helmholtz proposed instead that the Sun shrinks slowly, releasing gravitational energy; this gave a lifetime of some tens of millions of years. The answer came in the 1930s, when Hans Bethe and others showed that the Sun shines by nuclear fusion.

Tides and the rotation of the Earth

For the general reader. The tides rub against the sea floor and coastlines, and friction turns motion into heat. Mayer asked where that motion comes from. His answer: from the spin of the Earth. Tidal friction must slowly lengthen the day. He was right, and today the effect is measured to a fraction of a millisecond per century.

The argument of 1848

The Dynamik des Himmels also treated the tides. The flood and ebb of the oceans dissipate energy continuously. That energy must come from somewhere, and Mayer identified the rotation of the Earth as its source. Immanuel Kant had suggested in 1754 that the tides slow the Earth, but without the energy argument. Mayer faced a difficulty: Pierre-Simon Laplace had concluded, from ancient eclipse records, that the length of the day had not changed measurably in about 2500 years. Mayer suggested that the Earth’s contraction as it cools, speeding its spin, might offset the tidal effect. The modern explanation of the discrepancy is different. In the 1850s John Couch Adams found an error in Laplace’s theory of the Moon’s motion, and in the 1860s Charles Delaunay and William Ferrel showed that tidal braking resolves the remaining discrepancy.

Lunar laser ranging shows the Moon receding by about 3.8 cm per year. The astronomical record shows the day lengthening by about 1.7 ms per century on average over the past 2700 years. Tides alone would give about 2.4 ms per century; the difference comes from the slow rebound of the Earth since the last ice age, which reduces its moment of inertia. Mayer’s proposed compensation by contraction was the right kind of idea, though for the wrong cause.

Release and trigger (Auslösung)

For the general reader. A spark sets off a barrel of gunpowder. The spark’s energy is tiny and the explosion’s enormous, so here cause and effect are clearly not equal. Mayer, who had built his physics on the rule that the cause equals the effect, had to explain such cases. He called them release processes: a small cause opens the way for a large store of energy already present. The idea became important in physiology, where a nerve signal releases the much larger energy of a muscle.

Conversion and release

Mayer returned to the subject several times and treated it at length in his last publication, Die Torricellische Leere und über Auslösung (1876). In a conversion, the force of the cause becomes the effect, and the two are quantitatively equal. In a release, the cause only removes an obstacle, and the effect is paid for from a separate store. The amount released bears no fixed ratio to the trigger. Energy is still conserved, but only when the stored energy is included in the account.

Typical examples are gunpowder ignited by a spark, an avalanche loosed by a small disturbance, and the action of nerves on muscles. In each, a correct energy account has two parts: the trigger, which is converted, and the store, which is released.

Crisis, 1846–1854

The years after 1845 brought Mayer almost no recognition. His essays of 1845 and 1848 were printed at his own expense and sold few copies. In 1847 Hermann von Helmholtz published Über die Erhaltung der Kraft (On the conservation of force), written without knowledge of Mayer’s work, and gave the principle a mathematical form that physicists could use. Joule’s experiments were becoming known in Britain. Mayer felt his priority slipping away.

In 1849 he wrote to the Paris Academy to assert his priority over Joule, and Joule replied in the Comptes rendus that Mayer’s number was a deduction, not an experimental proof. In 1848 two of his young children had died in quick succession, and the political upheavals of 1848–49 added to the strain.

He slept badly and fell into depression. On the night of 18 May 1850, in a fever, he jumped from an upper window of his house. He survived with serious injuries. In 1851 he published Bemerkungen über das mechanische Aequivalent der Wärme, to defend his priority, and a paper on the work of the heart. In 1852 and 1853 he spent months in mental institutions, at Göppingen, Kennenburg near Esslingen and Winnenthal. At Winnenthal he was subjected to the restraint chair and the straitjacket. He returned to Heilbronn in 1854 but for some years withdrew from science and from most of his practice.

Recognition and legacy, 1858–1878

Recognition began slowly in the late 1850s. In 1859 the philosophical faculty at Tübingen gave him an honorary doctorate, and the Bavarian Academy of Sciences elected him a corresponding member. Rudolf Clausius, who had cited the 1842 paper in 1850, and Helmholtz, who from 1854 acknowledged Mayer’s priority in print, lent their authority.

The English controversy

The decisive turn came from London. On 6 June 1862 the Irish physicist John Tyndall gave a Friday evening lecture On Force at the Royal Institution, in which he presented the whole conservation of energy as the work of a German physician named Mayer. Clausius helped Tyndall obtain Mayer’s papers. William Thomson and Peter Guthrie Tait answered in an article, Energy, published late in 1862, which argued that Joule’s experiments, not Mayer’s speculation, established the principle. Joule wrote to defend his own claims. The quarrel continued through the 1860s and took on a national colouring, British against German science.

Honours

In 1867 Mayer published his collected papers as Die Mechanik der Wärme. In the same year King Karl of Württemberg awarded him the Knight’s Cross of the Order of the Württemberg Crown, which carried personal nobility; he became Julius Robert von Mayer. In 1869 he spoke at the meeting of German naturalists and physicians in Innsbruck, a sign of his rehabilitation, and the Paris Academy awarded him the Prix Poncelet. He became a corresponding member of the Paris Academy in 1870. In 1871 the Royal Society gave him the Copley Medal, a year after Joule. He published popular lectures and, in 1876, his essay on release.

Mayer died of pneumonia, after a long illness, in Heilbronn on 20 March 1878, aged 63. His son, a physician in the town, cared for him at the end. A bronze statue by Wilhelm von Rümann was unveiled on the Heilbronn market square in 1892.

Legacy

• Energy: the first published statement, in 1842, of the conservation of energy as a universal principle linking heat and work, with a numerical equivalent.
• Thermodynamics: the relation \(C_{p,m}-C_{V,m}=R\), still called Mayer’s relation.
• Biology: the energy budget of living things, from sunlight through plants to muscle and heat, and the rejection of vital force as a source of energy.
• Astronomy and geophysics: the question of the Sun’s energy source, and the identification of tidal friction as a brake on the Earth’s rotation.
• Physiology and psychology: the concept of release, taken up in later theories of nerve action and stimulus.

Sources and further reading

Biography

1. Julius von Mayer, Wikipedia; Robert Mayer, German Wikipedia.
2. H. Munk, Mayer, Robert, Allgemeine Deutsche Biographie 21 (1885).
3. Mayer, Julius Robert von, Neue Deutsche Biographie and Deutsche Biographie.
4. K. L. Caneva, Robert Mayer and the Conservation of Energy, Princeton University Press, 1993.
5. John Tyndall rescues Julius Mayer, 1862, Skulls in the Stars.

Physics

1. J. R. Mayer, Bemerkungen über die Kräfte der unbelebten Natur, Annalen der Chemie und Pharmacie 42 (1842) 233–240. English translation in Phil. Mag. 24 (1862).
2. J. R. Mayer, Die organische Bewegung in ihrem Zusammenhange mit dem Stoffwechsel, Heilbronn, 1845.
3. J. R. Mayer, Beiträge zur Dynamik des Himmels in populärer Darstellung, Heilbronn, 1848.
4. J. R. Mayer, Bemerkungen über das mechanische Aequivalent der Wärme, Heilbronn, 1851.
5. J. R. Mayer, Die Mechanik der Wärme in gesammelten Schriften, Stuttgart, 1867; 2nd ed. 1874.
6. R. B. Lindsay, Julius Robert Mayer: Prophet of Energy, Pergamon, 1973 (selected writings in translation).
7. M. Aquilini et al., on Mayer’s calculation of the mechanical equivalent, Accademia Nazionale delle Scienze detta dei XL, Memorie (2019).
8. H. Kragh, The source of solar energy, ca. 1840–1910: from meteoric hypothesis to radioactive speculations, arXiv:1609.02834.
9. T. S. Kuhn, Energy conservation as an example of simultaneous discovery, in M. Clagett (ed.), Critical Problems in the History of Science, University of Wisconsin Press, 1959.
10. Mayer’s relation and Tidal acceleration, Wikipedia.