Hermann Ludwig Ferdinand von Helmholtz

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Helmholtz, Hermann von

(b. Potsdam, Germany, 31 August 1821; d. Berlin. Germany, 8 September 1894)

energetics, physiological acoustics, physiological optics, epistemology, hydrodynamics, electrodynamics

Helmholtz was the oldest of four children. From his mother, Caroline Penn, the daughter of a Hannoverian artillery officer, he inherited the placidity and reserve which marked his character in later life. His father, August Ferdinand Julius Helmholtz, was the typical product of a romantic era. Ferdinand Helmholtz had served with distinction in Prussia’s war of liberation against Napoleon and had studied philology and philosophy at the new University of Berlin before accepting a poorly paid post at the Potsdam Gymnasium. A passionate, romantic figure, he possessed an acute aesthetic sensitivity which he transmitted to his son: a profound concern with music and painting underlay much of Helmholtz’ later work in sensory physiology. Ferdinand also fervently admired the philosophers Kant and J. G. Fichte; Fichte’s son Immanuel Hermann Fichte was his close friend and a frequent visitor. Helmholtz’ own lifelong devotion to epistemological issues was motivated by the intense philosophical discussions to which he had listened as a boy.

At the Potsdam Gymnasium Helmholtz’ interests turned very early to physics, but his father did not have the money to send Helmholtz to the university, and he persuaded his son to turn to medicine, for which there existed the prospect of state financial aid. In 1837 Helmholtz obtained a government stipend for five years’ study at the Königlich Medizinisch-chirurgische Friedrich-Wilhelms-Institut in Berlin. In return he committed himself to eight years’ service as an army surgeon. He passed his Abitur with distinction and left for Berlin in September 1838.

While at the Friedrich Wilhelm Institute, Helmholtz took many courses at the University of Berlin. He studied chemistry under Eilhardt Mitscherlich, clinical medicine under Lucas Schönlein, and physiology under Johannes Müller. Although he took no courses in mathematics, he read privately the works of Laplace, Biot, and Daniel Bernoulli as well as the philosophical works of Kant. During the winter of 1841 Helmholtz began research for his dissertation under Johannes Müller and later moved into the circle of Müller’s students. Chief among these were Ernst Brücke and Emil du Bois-Reymond. Confident and sophisticated, du Bois-Reymond seems to have taken the younger Helmholtz as his protégé. He and Brücke quickly won Helmholtz to their program for the advancement of physiology. With Karl Ludwig the three made up the “1847 school” of physiology. Their program reacted sharply against German physiology of previous decades. Philosophically they rejected any explanation of life processes which appealed to nonphysical vital properties or forces. Methodologically they aimed at founding physiology upon the techniques of physics and chemistry. All of Helmholtz’ minor papers published between 1843 and 1847, most of which treated problems of animal heat and muscle contraction, clearly reflect the mechanistic tenets of the school.

Helmholtz received the M.D. degree in November 1842. After completing the state medical examinations he was appointed surgeon to the regiment at Potsdam. He maintained his Berlin connections, though, and in 1845 du Bois-Reymond brought the shy young doctor into the newly founded Physikalische Gesellschaft. On 23 July 1847 Helmholtz read to the society his epic memoir “Über die Erhaltung der Kraft,” in which he set forth the mathematical principles of the conservation of energy.

In 1848 Brücke resigned his chair of physiology at Königsberg to accept a post at Vienna. When du Bois-Reymond refused the vacant post, Helmholtz was released from his military duty and appointed associate professor of physiology at Königsberg. Before leaving Potsdam he married Olga von Velten on 26 August 1849

From that time on, Helmholtz led a quiet professional life of tireless labor at his research. At Königsberg he measured the velocity of the nerve impulse, published his first papers on physiological optics and acoustics, and won a European reputation with his invention of the ophthalmoscope in 1851. Both scientifically and socially the early 1850’s were a period of widening horizons for Helmholtz. In 1851 he toured the German universities, inspecting physiological institutes on behalf of the Prussian government. In 1853 he made the first of many visits to England, where he formed lasting friendships with various English physicists, especially William Thomson. Despite his success at Königsberg, his situation there was not altogether happy; his wife’s already delicate health was further impaired by the cold climate, and he experienced minor priority conflicts with Franz Neumann. In 1855, with the help of Alexander von Humboldt, Helmholtz obtained a transfer to the vacant chair of anatomy and physiology at Bonn.

At Bonn, Helmholtz continued his research into sensory physiology, publishing in 1856 volume I of his massive Handbuch der physiologischen Optik. His work took a wholly new turn with his seminal paper on the hydrodynamics of vortex motion of 1858. Helmholtz’ philosophical views had begun very early to diverge from his father’s idealist position, and from 1855 he began to develop these views publicly in various popular lectures. Although father and son shared epistemological interests and even held common views on the subjective nature of sensory perception, Ferdinand nevertheless remained intensely suspicious of his son’s physical and empirical methods. During his years at Bonn this divergence created a strain in their frequent correspondence, although Helmholtz’ letters remained dutiful and submissive.

Helmholtz was never satisfied at Bonn. Anatomy was an unfamiliar subject, and there were whispered reports to the minister of education that his anatomy lectures were incompetent. Helmholtz angrily dismissed these reports as the grumblings of medical traditionalists who opposed his mechanistic-physiological approach. At the same time he was becoming the most famous young scientist in Germany. In 1857 the Baden government offered Helmholtz a chair at Heidelberg, then at the peak of its fame as a scientific center. The promise of a new physiology institute convinced Helmholtz to accept in 1858. At the last moment the prince of Prussia intervened to persuade him to stay, but in vain.

The following thirteen years at Heidelberg were among the most productive of Helmholtz’ career. He carried on his research in sensory physiology, publishing in 1862 his influential Die Lehre von den Tonempfindungen als physiologische Gnundlage für die Theorie der Musik. His treatises on physics included “Über Luftschwingungen in Röhren mit offenen Enden” (1859) and his analysis of the motion of violin strings. The Heidelberg years also brought important changes to Helmholtz’ personal life. His wife’s health had declined steadily and she died on 28 December 1859, leaving Helmholtz with two small children. On 16 May 1861 he married Anna von Mohl, the daughter of Heidelberg professor Robert von Mohl. Anna, by whom Helmholtz later had three children, was an attractive, sophisticated woman considerably younger than her husband. The marriage opened a period of broader social contacts for Helmholtz. The community of physicists in England and the rest of Germany began gradually to displace the Berlin circle as the locus of his scientific interaction.

By 1860 Helmholtz had begun research for volume III of his Handbuch der physiologischen Optik in which visual judgments of depth and magnitude were to be treated. The study led him directly into the nativistempiricist controversy and inaugurated a decade of intense concern with epistemological issues. The death of his father in 1858 had eliminated Helmholtz’ reluctance to develop the empiricist position latent in his earlier work. He began that development in volume III of the Handbuch (1867), which was an extended defense of the empirical theory of visual perception. In 1868 and 1869 Helmholtz carried that position still further in his work on the foundations of geometry. He summarized his epistemology in the famous popular lecture of 1878, “Die Thatsachen in der Wahrnehmung.”

By 1866 Helmholtz had completed his great treatises on sensory physiology and was contemplating abandoning physiology for physics. The scope of physiology had already become too great for any individual to encompass, he wrote in 1868, and while a flourishing school of physiology existed in Germany, German physics was stagnating for lack of well-trained young recruits. When Gustav Magnus’ death in 1870 left vacant the prestigious chair of physics at Berlin, Helmholtz and G. R. Kirchhoff, his colleague at Heidelberg, became the primary candidates for the post. The Berlin philosophical faculty preferred Kirchhoff, whom they regarded as the superior teacher. When he refused the post, the nomination went to Helmholtz. Helmholtz’ price was high: 4,000 taler yearly plus the construction of a new physics institute to be under his full control. Prussia readily agreed to his terms, for it was widely recognized that his call possessed great political as well as scientific significance in Prussia’s bid for the leadership of southern Germany. He accepted the Berlin post early in 1871.

Helmholtz inaugurated his new position with a series of papers critically assessing the various competing theories of electrodynamic action. This work first brought Maxwell’s field theory to the attention of Continental physicists and inspired the later reearch of Helmholtz’ pupil Heinrich Hertz, who entered the Berlin institute in 1878. After 1876 Helmholtz contributed papers on the galvanic cell, the thermodynamics of chemical processes, and meteorology. He devoted the last decade before his death in 1894 to an unsuccessful attempt at founding not only mechanics but all of physics on a single universal principle, that of least action.

By 1885 Helmholtz had become the patriarch of German science and the state’s foremost adviser on scientific affairs. This position was recognized in 1887, when Helmholtz assumed the presidency of the newly founded Physikalisch-technische Reichsanstalt for research in the exact sciences and precision technology. Helmholtz’ friend, the industrialist Werner von Siemens, had donated 500,000 marks to the project, and he himself had been among its foremost advocates. Under his administration the Reichsanstalt stressed purely scientific research.

Although Helmholtz’ productivity did not wane, his health began to fail after 1885. He had always suffered from migraine, from which he sought relief in music and mountaineering in the Alps. In old age he began to experience fits of depression which only long vacations could cure. On 12 July 1894 he suffered what appeared to be a paralytic stroke, and he died on 8 September.

Energetics . Before 1847 Helmholtz’ interest in force conversions had been motivated largely by physiological concerns. The mechanistic school to which he belonged demanded that the hypothesis of a unique “vital force” within the animal body be rejected as the first step to refounding physiology on chemical and physical principles. Assuming such a vital force, Helmholtz believed, was tantamount to assuming a perpetuum mobile. Consequently, its refutation necessitated proving that all the body heat and all the muscle force produced by the animal could be derived ultimately from the chemical force released by oxidation of its foodstuffs, with no recourse to a vital force. In this belief Helmholtz had been greatly influenced by the chemist Justus Liebig’s Die Thierchemie (1842), in which Liebig had attempted to argue away experiments of Pierre Dulong and Cáezar Despretz which seemed to refute the chemical theory of animal heat. In 1845 Helmholtz noted that these experiments were invalidated by their assumption that the heats of combustion of complex foodstuffs were equivalent to the summated heats of combustion of their constituent carbon and hydrogen. In 1845 he proved experimentally that chemical changes occur in the working muscle and, in 1848, that heat is generated by muscle contraction.

“Ueber die Erhaltung der Kraft” (1847) set forth the philosophical and physical basis of the conservation of energy. It drew heavily on the works of Sadi Carnot, Clapeyron, Holtzmann, and Joule, although it was far more comprehensive than those previous treatises. The philosophical introduction clearly illustrated the influence of Kantianism on Helmholtz’ thought. Science, he began, views the world in terms of two abstractions, matter and force. The goal of science is to trace phenomena to their ultimate causes in accordance with the law of causality; such ultimate causes are unchangeable forces. We can, Helmholtz implied, know the nature of such forces virtually a priori. If we imagine matter dispersed into its ultimate elements, then the only conceivable change which can occur in the relationship of those elements is spatial. Ultimate forces, then, must be moving forces radially directed. Only the reduction of phenomena to such forces constitutes an explanation to which we may ascribe the status of “objective truth” (from the translation in Richard Taylor’s Scientific Memoirs [London, 1853], p. 118).

That ultimate forces must be of this nature can also be inferred from the impossibility of producing work continually from nothing. That impossibility, Helmholtz demonstrated, is equivalent to the well-known principle of the conservation of vis viva. Assuming that principle to hold for a system of bodies in motion, Helmholtz attempted to prove that the forces under which those bodies move must be functions only of position (and hence not of velocity or acceleration) and also radially directed. If a particle m is acted on by a central force of intensity ϕ emanating from a fixed center of force and moves freely from a distance r to a distance R from that force center, then

where Q is the velocity of m at R and q its velocity at r. The left-hand side of equation (1) is clearly one-half the difference of the vires vivae; Helmholtz calls the right-hand integral the “sum of the tension forces” (Spannkäfte) between the distances R and r. Equation (1) remains valid when summed over the entire system of bodies and hence expresses the most general form of the principle of the conservation of energy.

Helmholtz then demonstrated how the conservation principle could be applied to various physical phenomena. The principle of the conservation of vis viva had already been applied to gravitation, wave motion, and inelastic collision. Previously an absolute loss of force had been assumed in inelastic collision and friction. Helmholtz argued to the contrary that the vis viva apparently lost in such cases is merely converted to tension forces or heat; on the latter assumption Joule had recently measured a mechanical equivalent of heat equal to 521’ meter-kilograms per calorie in mks units. Helmholtz then proceeded to an extended defense of the dynamic theory of heat against the caloric theory, arguing that the free heat of a body consists in the microscopic motion of its particles, its latent heat in the tension forces between its atoms. He then introduced the equations of Clapeyron and Holtzmann for the expansion of gases. The derivation of Clapeyron’s equations, he pointed out, rests upon the untenable assumption that no heat is lost when work is done by a gas in expanding. He concluded by applying the conservation principle to electrostatic, galvanic, and electrodynamic phenomena.

After 1847 Helmholtz’ research interests turned for some time to sensory physiology, and he took no direct part in the subsequent development of the entropy concept or kinetic theory. Late in his career, though, he turned again to research into energy processes, this time those of the galvanic and electrolytic apparatus. In 1872 he showed that convection currents within a polarized electrolytic cell can sustain a feeble current even at voltages too low to sustain electrolytic decomposition. That phenomenon had previously seemed to violate either the conservation of energy or Faraday’s laws of electrolysis. In 1877 Helmholtz attempted to predict theoretically the electromotive force of a galvanic cell for different concentrations of a salt solution. Under certain conditions the cell can be treated as a reversible cycle and the laws of Carnot and Clapeyron applied to it. The theory was in substantial agreement with experimental data by James Moser.

Helmholtz’ research in physical chemistry culminated in his 1882 memoir, “Die Thermodynamik chemiseher Vörgange.” Thermochemistry, especially that of Thomsen and Berthelot, assumed that the heat evolved in reactions is a direct measure of the chemical affinities at work. The occurrence of spontaneous, endothermic reactions had always presented an anomaly in this tradition, for such reactions seemed to act against the forces of chemical affinity. In 1882 Helmholtz distinguished between “bound” and “free” energy in reactions. The former is the portion of the total energy which, in accordance with the entropy principle, is obtainable only as heat; the latter is that which can be freely converted to other forms of energy. From Clasius’ equations Helmholtz derived the “Gibbs-Helmholtz equation,”

where F is the free energy, U the total energy,T the absolute temperature, and where ∂F/∂T yields the entropy. In any spontaneous reaction occurring at constant temperature and volume the free energy must decrease. Hence the free energy, not the total energy change measured by the evolution of heat, determines the direction of any reaction. Helmholtz’ research had been anticipated by J. W. Gibbs, in whose formulation U in equation (2) must be replaced by the enthalpy.

This research led Helmholtz directly to his investigations into the statics of monocyclic systems and the principle of least action. In the former (1884) he demonstrated that it is possible to define certain mechanical systems the internal motions of which can be shown to obey Clausius’ entropy equations. In response to an attack by Clausius he emphasized that the vibrational motion of heat does not rigorously satisfy the conditions of such a system; hence the paper constituted no mechanical derivation of the second law of thermodynamics. In the latter study (1886) he attempted to derive not only all of mechanics, but also thermodynamics and electrodynamics, from the principle of least action as formulated by Sir W. Rowan Hamilton. Although the problem dominated his attention until his death in 1894, Helmholtz achieved no satisfactory derivation. The importance of these studies lies chiefly in their influence upon Heinrich Hertz, who acknowledged his debt to Helmholtz in his Die Principien der Mechanik (1894).

Physiological Acoustics . Helmholtz’ research in sensory physiology began in 1850, when he determined the velocity of the nerve impulse in the sciatic nerve of the frog. In 1852 he obtained more precise results through his invention of the myograph. This device, in which the muscle traces the motion of its contraction upon a rotating drum, permitted more exact measurement of the small time intervals involved than any previous method. Helmholtz’ measurements yielded not only a finite velocity for nerve propagation but also the surprisingly slow one of about ninety feet per second. The result was considered a victory for the mechanistic school, for it seemed to confirm du Bois-Reymond’s hypothesis that the nerve impulse consisted in the progressive rearrangement of ponderable molecules.

At the conclusion of these experiments, Helmholtz’ interest turned immediately to physiological acoustics. Physicists had long known that a vibrating string produces not only a tone of its fundamental frequency f1 but also a series of harmonics 2 f1, 3 f1, and so on. They also knew that two similar tones f1, and f2, when sounded together, would produce beats of frequency f1 - f2 After 1750 a third phenomenon had come to light: Tartini’s tones, or difference tones. If tones f1 and f2 are sounded together, then the acute ear can sometimes hear a third tone f1 - f2 which is not a harmonic. Romieu, Lagrange, and Thomas Young all advanced the obyious “beat theory of difference tones,” which held that the difference tone is a beat frequency so great that it has become a tone in itself. In 1832 G. G. Hällström noted that the harmonics of f1 and f2 should also beat; that is, one should hear beats or difference tones 2 f1 - f2, - 2 f2 - 2 f2, 3 f1f2 and so on. These, however had never been observed. Finally in 1843 G. S. Ohm advanced his law of acoustics, which asserts that the ear perceives only simple harmonic vibrations. The ear, according to Ohm, decomposes the complex sound waves which it receives into the same simple harmonic waves obtainable mathematically from Fourier analysis.

The perception of beats and difference tones seems to contradict Ohm’s law, a fact which perhaps attracted Helmholtz’ attention. In 1856 he demonstrated the use of resonators to isolate and reinforce upper partial tones and thus showed the existence of the higher-order difference tones predicted by Hällström. He also proved that in addition to the difference tones mf1-nf2, there exist also very faint summation tones mf1+ nf2 Because summation tones cannot be predicted from the beat theory, Helmholtz regarded it as decisively disproved and advanced his own transformation theory. In simple harmonic motion the restoring force k on a particle m is proportional to its displacement x; but if the square of the displacement is also sensible, then k = ax + bx2. If two wave trains of force f sin pt and g sin (qt + c) act on m, then the equation of motion can be written and solved by series. Helmholtz showed that the series solution of that equation contains wave functions of all frequencies mp, mq, (mp - nq). (mp + nq). Hence combination tones result from inharmonic distortions of the wave form, either externally in resonators (as Helmholtz showed) or in the ear at the drum-malleus junction (as Helmholtz believed); hence they do not violate Ohm’s law.

Obviously all sounds of the same pitch and intensity do not sound alike. Helmholtz attributed this difference in timbre (Klangfarbe) mainly to the different patterns of upper partial tones, which depend on how the fundamental is produced. He advanced this theory first in connection with his fixed-pitch theory of vowel sounds. The vowel A sung at pitch f1 differs from the vowel E sung at f1 by the same individual, Helmholtz argued, only because the mouth serves as a variable resonator. At the vocal cords, both A and E have the same pattern of upper partials, but the different shapes of the mouth cavity reinforce different ranges of partials, giving A and E different timbres. Helmholtz also argued that timbre is independent of phase differences among the upper partial tones.

Helmholtz’ greatest achievement in physiological acoustics lay in formulating the resonance theory of hearing. Like so much of his physiology, that theory rested upon Johannes Müller’s law of specific nerve energies. Müller taught That the nature of the impulse carried to the sensorium by a given nerve is unique and independent of the nature of the external stimulus. Rigidly interpreted, the law seemed to require each just noticeable difference of any sensory quality to possess its own sensory receptor. Between 1850 and 1855 the microscopic anatomy of the cochlea first became known. Among the structures revealed were the rods of Corti strung out in gradually increasing size along the length of the cochlea—analogous, Helmholtz insisted, to the tuned wires of a piano. In 1857 Helmholtz boldly hypothesized that these rods function as tuned resonators. A complex sound wave transmitted to the cochlea fluid sets in sympathetic vibration those rods tuned to the frequency of its simple harmonic components. These rods in turn excite adjacent nerve endings, which transmit the impulse to the sensorium. Hence the resonance theory satisfies Mõller’s law and provides a physiological explanation of Ohm’s law. In 1869 the experiments of Victor Hensen convinced Helmholtz that the transverse fibers of the basilar membrane, not the rods of Corti, are the cochlea resonators. With this modification the theory survived virtually unchallenged until after 1885.

Helmholtz incorporated all these results in his great work Die Lehre von den Tonempfindungen als physiologische Grundlage für die Theorie der Musik (18631 in which he applied his discoveries to music theory. Musicians had long known that the most perfect consonances are those whose frequencies are small whole-number ratios. Helmholtz explained this by noting that such consonances have the greatest number of coincident upper partial tones. Less perfect consonances have many slightly different upper partials, and these produce beats which are perceived as dissonance. Later editions of Tonempfindungen also incorporated the results of two studies of the ossicular bones carried out in 1867 and 1869, in which Helmholtz evaluated the efficiency and linearity of the ossicular chain as a transformer.

Physiological Optics . Helmholtz inaugurated his study of physiological optics with his invention of the ophthalmoscope in 1851. His friend Ernst Brücke had recently shown how the human eye could be made to glow with diffusely reflected light, like the eyes of many animals. In preparing a lecture demonstration of the phenomenon, Helmholtz realized that by means of a simple optical apparatus this reflected light could be obtained as a magnified, sharply focused image of the subject’s retina. He published the mathematical theory of the ophthalmoscope with an account of the improved instrument in 1851.

Helmholtz turned to the intricate problems of color vision in 1852 with an attack on Sir David Brewster’s new theory of light. Brewster had maintained the objective reality of three primary colors by supposing, in opposition to Newton, that there exist three distinct kinds of light, each of which excites in the eye one of the sensations red, yellow, or blue. Helmholtz regarded the theory as still another confusion of physical stimulus and subjective response. The experiments by which Brewster claimed to have verified his theory, Helmholtz argued, had actually led Brewster astray, for he had failed to obtain pure spectra.

In conjunction with his attack on Brewster’s theory Helmholtz conducted spectrum experiments of his own. To his surprise a mixture of blue and yellow spectral lights yielded a green-tinted white, although a mixture of blue and yellow pigments yields green. From this anomaly Helmholtz elaborated the important distinction between additive and subtractive color mixtures, which he announced in 1852. Yet the same experiments led Helmholtz into serious error. When he attempted to produce white by mixing the pairs of colors which Newton’s color theory predicted to be complementary, he succeeded in obtaining pure white only with yellow and indigo. He concluded that this is the only pair of complementaries and rejected Newton’s color theory. This aspect of his 1852 paper was immediately attacked by H. G. Grassmann, who, like Maxwell, was attempting to develop a mathematical theory of Newton’s color chart. In 1855 Helmholtz acknowledged the experimental error of his earlier paper and announced new experiments which yielded the sets of complementary colors demanded by Newton’s theory.

In his paper of 1852 Helmholtz also revived Thomas Young’s forgotten theory of color vision. In 1801 Young had hypothesized that each retinal nerve ending possesses three distinct color receptors, each primarily sensitive to one frequency of light. When stimulated, each receptor yields one of the subjective color sensations red, green, or violet. Hence all color sensations except the three primary ones are physiological mixtures. Ironically, Helmholtz revived Young’s theory in 1852 only to refute it. He had discovered that spectral colors, when mixed, always yield a duller color of less-than-spectral saturation. Therefore the whole idea that all colors may be obtained from mixtures of three primary colors must be incorrect, he concluded, for the spectral colors, at least, can never be obtained in their full saturation by mixing any three of their number. That fact seemed to refute Young’s theory; for if Young’s physiological primaries are assumed to be spectral red, green, and violet, then that theory cannot explain how other spectral colors are seen in their full prismatic saturation.

Although Helmholtz dismissed Young’s theory in 1852, by 1858 he had changed his mind and become its foremost advocate. In order to save Young’s theory from the objections of the 1852 treatise, Helmholtz assumed that Young’s physiological primaries are not spectral colors at all, but colors of far greaterthan-spectral saturation. Mixtures of the three physiological primaries still undergo the loss of saturation which Helmholtz had noted in 1852, but this lower level of saturation is that of the spectral colors themselves. In this way all the spectral colors which we see can be mixed from three properly chosen physiological primaries, and Young’s theory is saved.

Helmholtz’ assertion that the physiological primaries possess greater-than-spectral saturation followed logically from another amendment to Young’s theory. Helmholtz hypothesized that any wavelength of light, however strongly it excites one set of retinal receptors, always excites simultaneously the other two sets to a much weaker degree. It follows that any physical light, even a single wavelength corresponding to the most saturated color of the spectrum, evokes a color sensation which is not “pure” but a mixture. That mixture, even if it is a spectral color, must necessarily be less highly saturated than the physiological primaries from which it was mixed. In normal vision we never see one physiological primary color alone because there is no obvious way to stimulate one set of retinal receptors without simultaneously stimulating the other two. This fact accounts for the belief that the spectral colors are the most highly saturated which exist. In 1858, however, Helmholtz announced a method by which the pure physiological primaries could be observed approximately. In a paper on afterimages, Helmholtz pointed out that a prismatic color appears far more saturated when viewed after the retina has been fatigued by the complementary color. This fact is easily explained in the Young-Helmholtz theory by assuming that the retinal fatigue briefly inactivates two sets of color receptors. When the third set is then stimulated, we observe its corresponding color less mixed with the other two primaries than usual and hence see it as far more saturated than the spectral colors. Helmholtz regarded this experiment as striking confirmation of his amended version of Young’s theory. In 1859 he further demonstrated the power of the theory by using it to explain red color blindness

Helmholtz incorporated all these results in his Handbuch der physiologischen Optik, a massive work which encompassed all previous research in the field. Volume I, which appeared in 1856, contained a detailed treatment of the dioptrics of the eye which was greatly dependent on J. B. Listing’s previous works. In it Helmholtz treated the various imperfections of the lens system and announced the result that the visual axis of the eye does not correspond to its optical axis. Volume I also elaborated Helmholtz’ theory of accommodation and his invention of the ophthalmometer, both announced in 1855.

In volume II, Helmholtz introduced Young’s theory, calling it a special application of Johannes Müller’s law of specific nerve energies. He also dealt with the complex phenomena of irradiation, afterimages, and contrast, which had dominated the interest of German physiologists since Goethe’s Farbenlehre but could be investigated only through difficult and often dangerous subjective experiments. Helmholtz defended G. T. Fechner’s explanation of afterimages by the fatigue of retinal elements and advanced his own theory that contrast phenomena arise from errors of judgment and have no physiological basis. He took pains to refute all theories which tried to explain these phenomena through synesthesia or retinal induction, particularly that of Joseph Plateau. Helmholtz took no part in the development of the duplicity theory of vision, which lay outside his main area of interest.

Eplstemology . Helmholtz often asserted that the task of modern philosophy is wholly epistemological, and he evinced a dislike for metaphysics. Kantianism exercised a strong influence on his thought as is obvious in his earliest papers. Later, through his physiological study, Helmholtz became convinced that sensory physiology, by revealing the processes of perception, was actually verifying and extending Kant’s epistemological analysis. In Müller’s law of specific nerve energies he recognized the great principle which explained the role of sense organs in transforming abstract, external stimuli into something wholly different: the immediate sensations of consciousness. Helmholtz’ problem was to explain how, despite this radical transformation, we nevertheless have knowledge of the external world.

Helmholtz followed Kant in insisting that the law of causality is transcendental and a priori. In 1855 he asserted that the causal law underlies our belief in external objects, a proposition which prompted charges of plagiarism from Schopenhauer and hence cemented Helmholtz’ distaste for metaphysicians. We have immediate experience, he observed, that changes occur in our sensations independent of our volitions. In order that this effect may have a cause, we postulate objects external to ourselves, which can be further analyzed into two categories: matter and force. Whether such objects actually exist must remain a metaphysical question, for both idealism and realism are wholly consistent systems. That the properties of matter and force are constant depends upon our assumption of the lawfulness of nature, which, in turn, rests upon the a priori status of the causal law. In our perception of the world, though, the conclusions we draw about the existence of external objects and forces and their interrelations do not depend upon reflection; they are instantaneous and unconscious. These highly controversial “unconscious conclusions” (unbewusste Sehlüsse) underlie all of Helmholtz’ epistemology and reveal its debt to English associationist psychology.

Because all external stimuli are mediated and radically transformed by the sense organs, our sensations are not images of external reality but tokens or signs of it. We gain knowledge about the external world by experientially discovering how our volitions can alter our sensations. Visual perception of space and localization, for example, is built up in such a way. By willing to touch an object, walk around it, or merely move our gaze we discover experientially that we can alter the accompanying retinal sensations in very regular ways. Hence, through unconscious conclusions certain retinal patterns become associated with objects localized in space, the idea of space itself having first been built up through the sense of touch. The nature of that retinal pattern matters not at all, Helmholtz insisted. All that matters is that we be capable of distinguishing one retinal point from another (in R. H. Lotze’s terminology, that each point have its own “local sign”) and that the proper eye movements be always able to restore the same retinal pattern.

This empiricist theory of visual perception differed radically from the alternate nativist view. Nativists held that visual perception of space and localization was not wholly learned but was in some sense innate. Johannes Müller, himself a nativist, believed that we are directly aware of the retina’s extension in space and that the local signs have an intrinsic spatial meaning. In the sophisticated theory of Ewald Hering, Helmholtz’ great rival, the nativist theory was extended to depth perception as well. Helmholtz devoted volume III of his Handbuch der physiologischen Optik (1867) to proving that the empiricist theory could explain all the phenomena of visual perception and that nativist theories, especially Hering’s, were incorrect or superfluous. Helmholtz first showed that Donder’s and Listing’s laws, the two basic laws of eye movement, could be easily explained on an empiricist basis in accordance with his own principle of easiest orientation. But Helmholtz organized his primary refutation of nativism around the phenomena of depth perception and binocular vision.

To deal with the problem of single and double vision Johannes Müller had hypothesized that the two retinas must possess paired or “corresponding” points and that each pair of corresponding points contributes to one point of the unified visual field. This hypothesis not only explains why we do not see two visual fields (one corresponding to each eye) but also explains the existence of single and double images. If the eyes fixate on a small object in space, that object will be seen single because its retinal images fall on corresponding retinal points. Many other objects, however, will be seen slightly double, for their retinal images do not correspond. The locus of all points in space which are seen single is called the horopter curve. Its exact determination occupied Helmholtz’ attention, and he and Hering published its general mathematical form independently between 1862 and 1864.

Helmholtz regarded the way in which the disparate images of two corresponding points become united into one as the crux of the nativist–empiricist dispute. Understanding how visual perception of space originates also seemed to hang upon that issue, for Charles Wheatstone’s invention of the stereoscope in 1833 had revealed the dependence of visual depth perception upon binocular double vision. Many nativists maintained that the nerve fibers leading from pairs of corresponding points become anatomically united before entering the sensorium, so that the resulting single image is an organic fusion of the two different images. Helmholtz and the empiricists believed that disparate images from corresponding retinal points enter the sensorium distinct and intact, and that their union into a single image is an unconscious act of judgment dependent upon prior experience. Against the hypothesis of any organic or anatomical union, upon which he believed all nativist theories must rest, Helmholtz marshaled various observations obtained from Wheatstone’s stereoscope. The eyes show surprising ability to fuse the two halves of the steroscopic image, Helmholtz noted, even though these images are different and may fall on noncorresponding points. A still greater problem for nativist theories is the phenomenon of stereoscopic luster. If an area in one half of a stereoscopic drawing is shaded white and the same area in the other half black, the fused image appears not gray but lustrous. Empiricism explains this easily as a learned response to empirical situations in which lustrous objects reflect more light into one eye than into the other. Contrary to nativist principles, the sensations excited in corresponding points clearly yield not a fusion of each but a wholly different sensation. This, Helmholtz claimed, decisively disproved the anatomical union of corresponding points upon which nativist theories depended. The union of binocular images into a single perception of depth, he concluded. must be the result of learning and experience as predicted by the empiricist theory

Helmholtz’ belief in the empirical origin of visual localization did not necessarily conflict with Kant’s doctrine that space in general is a transcendental form of perception. But Helmholtz broke sharply with Kant over his claim that the axioms of geometry were also synthetic, a priori propositions. Motivated by his study of visual perception, throughout the mid-1860’s Helmholtz investigated the most general analytic expressions of spatial relations. He formulated for himself the abstract mathematical concept of the extended n-ply manifold and became convinced that tacit assumptions of congruence and translation underlie the Euclidean axioms. In 1868, before publishing these results, he received a copy of G. F. B. Riemann’s treatise of 1854, Ueber die Hypothesen, welche der Geometrie zu Grunde liegen, and discovered that most of his results had been anticipated. Nevertheless, he published his own work, emphasizing its one aspect that went beyond Riemann’s treatment. Riemann had assumed that in any manifold the distance formula ds must be the square root of a homogeneous function of second degree in dx, dy, dz, and so on. Helmholtz, starting from the assumption of congruence, proved that Riemann’s formula must follow necessarily from that assumption.

Helmholtz’ interest in these problems was never that of the pure mathematician. He sought primarily to demonstrate that the Euclidean axioms presuppose the purely experiential facts of translation and congruence. Since geometries other than Euclidean can be developed from these facts, it follows that the Euclidean axioms cannot be the transcendental conditions for our perception of space, as Kant had claimed. Helmholtz’ contribution to the development of non-Euclidean geometry was therefore a natural extension of his empiricist philosophical position.

Hydrodynamics . In 1858 Helmholtz published his seminal memoir “Ueber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen,” important for both its physical results and its mathematical methods. His motivations for taking up this new research interest remain unclear. One motive seems, however, to have been his interest in frictional phenomena, carried over from his interest in energetics; another was his growing awareness of the power of Green’s theorem.

Previously, Helmholtz began, hydrodynamics had assumed the existence of a velocity potential. Yet Euler had noted that there is fluid motion for which no velocity potential exists, including forms of rotary motion and frictional flow. If there exists a singlevalue velocity potential for a given fluid motion, then (in modern vector notation) ∇ϕ = v and ∇ × = 0. These conditions, Helmholtz showed, exclude the possibility of vortex motion. In cases of fluid motion where rotary motion does occur, then ∇ × v = 2ω where ω is the angular velocity of a given element. From this fact and from the standard Eulerian equations of motion Helmholtz obtained

Hence, if the fluid is ever without rotation, that is, ω = 0, then /dt = 0 and the fluid can never begin to rotate. This principle became known as the conservation of vortices.

Helmholtz defined the vortex line (Wirbellinie) as the locus of the instantaneous axes of rotation of a rotating particle of fluid. A given vortex line, he proved, is always composed of the same particles of fluid and hence shares their motion through the fluid. He defined vortex tubes (Wirbelfinie) as the tubes formed by the vortex lines drawn through all points on the circumference of an infinitely small surface within the fluid. The product of the velocity of rotation and the cross section at any point of a given tube is constant; it follows that such tubes must always be closed within the fluid or terminate on its boundaries.

Helmholtz proceeded to find v in terms of ω, subject to the three conditions ∇· ω = 0, ∇· v = 0, and ∇ × v = 2ω The solution, he asserted, is

Here P functions as a scalar potential while L, M, and N function as the components of the modern vector potential A . Hence Helmholtz had implicitly set out the Helmholtz theorem, that the velocity field is the sum of irrotational and solenoidal parts. But although Maxwell had introduced the vector potential explicitly in 1856, Helmholtz did not regard his L, M, and N as the components of any physical or mathematical entity. He defined each separately on a strict magnetic analogy. L, M, and N are the volume integrals over the fluid space of the magnetic potential exercised on an external point x, y, z by a magnetic fluid distributed with density ω/r. In other words,

P is defined analogously. It follows immediately for Helmholtz that

and from equation (4) that ∇ · A = 0. Hence solution (3) satisfies the necessary conditions.

Equations (3) allowed Helmholtz to calculate easily the velocity induced in a particle a by a rotating particle b at distance r. He obtained the striking result that

This formula, as Helmholtz pointed out, is exactly analogous to the Biot–Savart force law for the magnetic effect of currents. True to the rigorous, starkly mathematical tenor of the entire paper. Helmholtz regarded this as only a heuristic analogy; his own development was strictly kinematic and mathematical.

Although George Stokes had anticipated certain methods and results of Helmholtz, the 1858 memoir was nevertheless a tour de force; yet it seemed to attract little initial attention beyond involving Helmholtz in an insignificant controversy with Joseph Bertrand. In 1866, though, William Thomson (later Lord Kelvin) made it the basis of his theory of the vortex atom. Helmholtz had proved, Thomson, noted, that his vortices share with hard atoms the properties of being conserved, undergoing collision, exerting influence on other vortices at a distance, and possessing well-defined energies. In addition, vortices have suggestive properties not possessed by hard atoms, such as the electrical and magnetic analogies demonstrated by Helmholtz. For a decade Thomson tried to develop a physics based on the assumption that atoms are tiny vortices.

Helmholtz himself never returned to vortex theory, although his previous work did influence his important paper of 1868 on discontinuous fluid motion. Finally, in collaboration with Gustav von Piotrowski, he carried out a series of complex experimental determinations (1860) of the coefficient of internal friction for various fluids.

Electrodynamics . Although Helmholtz had published earlier papers on electrodynamic phenomena, the field began to dominate his research interests only after 1870. This new direction in his research seems to have been motivated chiefly by his desire to bring electrodynamic theory into harmony with the conservation of energy. Concomitant with this purpose, Helmholtz hoped to bring order to a field which he described in 1870 as a “pathless wilderness” of competing mathematical formulas and theories. He undertook his research with three aims in mind: (1) to test the consistency of each contending theory with accepted mechanical and dynamic principles, (2) to derive differing theoretical predictions from each theory, and (3) to carry out experiments in order to decide between competing theories.

In his 1847 memoir Helmholtz had argued that ultimate forces must be conservative. He had also argued that forces cannot be conservative if the force laws expressing them contain terms involving the velocity or acceleration of the ultimate particles between which the forces work. But this argument impugned the fundamental status claimed for Wilhelm Weber’s law expressing the electrodynamic force acting between two charged particles e and e′. According to Weber’s law

where r is the distance between e and e′ and c is a constant. The formula involves not only the distance r but also its time derivatives; hence, according to Helmholtz, the force must violate the conservation of energy.

Helmholtz’ criteria for force laws excited much opposition; his lifelong rival Clausius attacked them in 1853. The proofs upon which they rested were, in fact, incorrect; Helmholtz later acknowledged that force laws involving derivatives of distance can conserve energy, although they cannot be central and obey Newton’s third law. Nevertheless, he continued to believe that the form of Weber’s law implied physical inconsistencies if not explicit violation of the conservation principle. In 1870 he opened his critique of Weber’s law, then the leading Continental formula for the prediction of electrodynamic effects. According to that law, Helmholtz showed, the energy of at least some systems of charges in motion is less than the energy of the same systems at rest. Hence at least some electrostatic equilibriums must be unstable. Furthermore, one can easily show from Weber’s formula that two charges +e and −e can, under certain conditions, continue to accelerate spontaneously until their kinetic energy becomes infinite. Therefore, in both cases Weber’s law predicts physical absurdities.

Helmholtz’ critique provoked a running controversy, conducted with great bitterness by Weber’s pupils, which lasted through the 1870’s. The principals themselves found great difficulty in even understanding each other, for Helmholtz’ conception was entirely of macroscopic phenomena; Weber’s, of microscopic charges. The confrontation ultimately proved indecisive, yet it undermined the confidence of Continental physicists in Weber’s theory and facilitated acceptance of Maxwell’s theory, which replaced Weber’s after 1880.

In cataloguing competing electrodynamic theories, Helmholtz also advanced a theory of his own which he believed would embrace many others as special cases. The intrinsic difficulties of electrodynamic force laws like Weber’s dictated his decision to derive the force from a potential. In 1848 Franz Neumann had successfully derived all electrodynamic effects for closed currents from a potential. In 1870 Helmholtz showed that the most general form of Neumann’s potential must be

In equation (5) p represents the potential which current element d s exercises upon element d s′ when ds carries current i ds′ current j; r represents the distance between elements ds and ds ′, and A = 1/c, where c is an undetermined, constant velocity, In equation (5) k is also an undetermined constant. For k = − 1. equation (5) becomes simply a form of Weber′s law; for k = 1, it becomes Neumann’s potential; and k = 0 corresponds to Maxwell’s theory. The parts of expression (5) which are multiplied by k can be written

This expression becomes zero when integrated around the full circuits S and S’ if either is closed; hence for closed currents all the competing formulas are equivalent. Differences between formulas can arise only for open currents—those in which, according to Helmholtz, changes in the density of the “free electricity” occur. In 1870 there existed little experimental data on open currents.

The difficulties of open currents also arise in the propagation of electrodynamic effects in magnetic and dielectric media. Helmholtz’ discussion of this topic necessitated a comparison of his theory with Maxwell’s. In 1870 Maxwell’s theory was little known on the Continent, for it differed radically from Continental theories. The latter assumed that a body exerted its electrodynamic action on another at a distance, independent of the intervening media. Maxwell’s field theory rejected action at a distance and, as Helmholtz understood it, assumed all electrodynamic action to be propagated through contiguous, progressive polarization of a medium. On the assumption that the luminiferous ether itself is a magnetizable dielectric, Helmholtz noted, Maxwell’s theory yields the striking result that electrodynamic disturbances propagate themselves in transversal waves possessing the velocity of light in free space. Like the English physicists, Helmholtz believed the existence of a dielectric ether to be strongly supported by the experiments of Faraday, especially those on diamagnetism.

Pursuing the comparison of the theories, Helmholtz first demonstrated that the derivation of a wave equation for electromagnetic propagation does not depend upon the particular assumptions of Maxwell’s theory. If the polarization of the medium is taken into account and the polarization expression ∂p/∂t is introduced as one term of the current density, then Helmholtz showed how a wave equation could be derived from his own generalized potential law, even though that law rested upon the initial assumption of action at a distance. The velocity of the waves predicted by Helmho by z’ wave equation depend upon the electrical and magnetic susceptibilities of free space. If these are assumed to be zero (that the ether is not a magnetizable, dielectric medium) then the velocities become infinite. If the susceptibilities are assumed to be large, then the wave velocities become finite. However, the further assumption that k = 0, the condition of Maxwell’s theory, is required in order that the waves be wholly transverse and attain the exact velocity of light in free space. In this sense Maxwell’s theory becomes a special, limiting case of Helmholtz’ more general theory. Continental physicists first became acquainted with Maxwell’s theory in this form, through Helmholtz’ memoirs.

Like most of the Continental school, Helmholtz still distinguished in electrodynamics between inductive forces, those which tend to set in motion the electricity within a conductor, and ponderomotive forces, those which tend to set in motion the conductor itself. In 1874 he devoted a major paper to demonstrating that his generalized potential formula could serve as a potential for ponderomotive as well as for inductive forces. In this attempt he was merely generalizing and extending the earlier work of Franz Neumann. In 1845 Neumann had derived a simple induction law from Lenz’s law and Ampère’s expression for the ponderomotive force between current elements. Later, in 1848, he had published his more famous potential formula, equivalent to the induction law for closed currents. Neumann himself had shown that the potential formula could predict ponderomotive effects and had verified its agreement with Ampére’s law for many simple cases. Helmholtz extended that verification to three-dimensional, deformable conductors and to cases of open currents.

In the course of the 1874 analysis Helmholtz discovered a feasible method through which the various theories could be tested experimentally. Ampère’s law predicted ponderomotive forces only between infinitesimal elements of conductors carrying closed currents. The ponderomotive force law derived from Helmholtz’ potential annexed to Ampère’s expression other terms predicting ponderomotive effects due to the free electricity accumulating at the ends of open circuits. In 1874 Helmholtz and his student N. N. Schiller carried out experiments to determine whether the end of an open current, simulated by an electrostatic discharge, would produce ponderomotive effects. They observed none, and Helmholtz reluctantly concluded that the potential law must be incorrect or that the assumptions underlying it were incomplete. In the experiment Helmholtz noted that charge was continually removed from the discharge point of the electrostatic machine through the convective motion of air particles. The potential law denied that such convection currents produced any electrodynamic effects. But if this assumption were false, he pointed out, then in addition to the ponderomotive effects produced by the open current there would be other electrodynamic effects caused by the convection current. The potential law might not then be strictly false but merely incomplete as long as it failed to take into account that effect. In 1876 Henry Rowland conducted experiments in Helmholtz’ laboratory which proved that convection currents produce electrodynamic effects. Helmholtz immediately pointed out that the results of both experiments can be predicted either from Maxwell’s theory or from the generalized potential law with the dielectric ether. In 1875 Helmholtz had already conducted a different experiment with similar results. He had rotated the plates of a cylindrical capacitor aligned axially in a uniform magnetic field and had observed an induced electromotive force on the plates. This effect could be predicted from the generalized potential law only by assuming that the insulating space between the capacitor plates functioned like Maxwell’s dielectric ether.

Logically, the experimental evidence remained inconclusive at the end of 1876. All the major results, Helmholtz noted, could be explained by Neumann’s induction law without recourse to a dielectric ether. But although Helmholtz in 1875 presented the choice between theories as still open to experimental decision, in practice he had come gradually to regard the dielectric ether as a necessity and Maxwell’s theory as correct. In the Faraday lecture of 1881 he predicted the decline of action at a distance on the Continent and lent full support to Maxwell’s theory. His interest in electrodynamics waned after 1876, and his work was taken up by Heinrich Hertz.

Yet Helmholtz did not accept the Maxwellian view that all current consists of the polarization of media. After 1876 his electrical research turned almost entirely to the galvanic pile, and he became firmly convinced that electricity consisted ultimately of discrete charges. In the Faraday lecture of 1881 Helmholtz set out his theory of “atoms of electricity” and his conviction that chemical forces are ultimately electrical in nature.

Conclusion , Helmholtz exerted incalculable influence on nineteenth-century science, not only through the achievements of his research but also through his brilliant popular lectures and his activity as a teacher and administrator. Helmholtz witnessed the final transition of the German universities from purely pedagogical academies to institutions devoted to organized research. The great laboratories built for him at Heidelberg and Berlin opened to him and his students possibilities for research unavailable anywhere in Europe before 1860. In many respects his career epitomized that of German science itself in his era, for during Helmholtz’ lifetime German science, like the German empire, gained virtual supremacy on the Continent.

Helmholtz belonged to that brilliant and selfconscious generation of German scientists which arose in open reaction to the scientific romanticism of earlier decades. Yet—far more than they cared to admit—Helmholtz and his generation still harbored many of the preconceptions and even the program of the earlier science. Like many of his romantic predecessors, Helmholtz devoted his life to seeking the great unifying principles underlying nature. His career began with one such principle, that of energy, and concluded with another, that of least action. No less than the idealist generation before him, he longed to understand the ultimate, subjective sources of knowledge. That longing found expression in his determination to understand the role of the sense organs, as mediators of experience, in the synthesis of knowledge.

To this continuity with the past Helmholtz and his generation brought two new elements, a profound distaste for metaphysics and an undeviating reliance on mathematics and mechanism. Helmholtz owed the scope and depth characteristic of his greatest work largely to the mathematical and experimental expertise which he brought to his science. Especially in physiology that expertise, shared by few other physiologists of the day, made possible the imposing theoretical and experimental edifices that Helmholtz erected from the simplest of physiological principles. Although the biophysical program of the 1847 school did not prove wholly successful for physiology in general, in Helmholtz’ field of sensory physiology it proved eminently so.

When Helmholtz abandoned physiology for physics in 1871, the former science, he complained, had already grown too complex for any individual to embrace in its entirety. At his death in 1894, that complexity had become true of virtually all fields. Helmholtz was the last scholar whose work, in the tradition of Leibniz, embraced all the sciences, as well as philosophy and the fine arts.


I. Original Works. Helmholtz’ scientific papers have been collected as Wissenschaftliche Abhandlhungen von Hermann Helmholtz, 3 vols. (Leipzig, 1882). His lectures on popular and philosophical subjects are available as Populare wissenschaftliche Vorträge, 3 vols. (Brunswick, 1865–1876); und Vorträge and Reden, 2 vols. (Brunswick, 1884). The treatises on sensory physiology are Handbnch der physiologischen Optik, 3 vols. (Leipzig, 1856–1867), and Die lehre von den Tonempfzndungen als physiologische Grundlage für die Theorie der Musik (Brunswick, 1863); both works went through many later eds. Also available are the lectures delivered by Helmholtz’ was during his years at Berlin: Vorlesungen über die elektroiragnetische Theorie des Lichts, Arthur König and Carl Runge, eds. (Hamburg, 1897); and Vorlesungen über theoretische Physik, 6 vols. (Leipzig, 1897–1907). Most of Helmholtz’ major works have been translated into English, including the treatises on physiological optics, ed. and trans, by James P. C. Southall et al, (Menasha, Wis., 1924–1925); physiological acoustics, trans, by Alexander J. Ellis (London, 1875); and the popu lar scientific lectures, trans. by E. Atkinson (London, 1881). Translations of individual memoirs appeared frequently in contemporary English journals; see the Royal Society Catalogue of Scientific Papers, VII, 946–947; X, 188–189; and XV, 747–748. A complete bibliography of Helmholtz’ works is included in Wissenschaftliche Abhandlungen, III, 605–636.

II Secondary Literature. The standard biography of Helmholtz is Leo Koenigsberger, Hermann von Hehnholtz, 3 vols. (Brunswick, 1902–1903), which contains descriptions of Helmholtz’ papers and lengthy extracts from his correspondence. An abridged trans. into English by Frances A. Welby is Hermann von Helmholtz (Oxford, 1906; New York, 1965). See also Emil du Bois-Reymond, Hermann von Helmholtz, Gedächtnissrede (Leipzig, 1897). On Helmholtz’ role in the discovery of the conservation of energy see Thomas S. Kuhn, “Energy Conservation as an Example of Simultaneous Discovery,” in Marshall Clagett, ed. Critical Problems in the History of Science (Madison, Wis., 1959), pp. 321–356; and Yehuda Elkana, “Helmholtz’ ‘Kraft’: An Illustration of Concepts in Flux,” in Historical Studies in the Physical Sciences, 2 (1970), 263–299. One of the few historical treatments of Helmholtz’ role in sensory physiology is in Edwin G. Boring, Sensation and Perception in the History of Experimental Psychology (New York, 1942), passim. Helmhollz himself gives much excellent historical material in the Handbnch der physiologischen Optik. Ernst Glen Wever and Merle Lawrence, Physiological Acoustics (Princeton, 1954), contains an interesting modern evaluation of Helmholtz’ work in physiological acoustics. Several German works treat Helmholtz’ philosophical views, although they give no adequate account of their development. See Ludwig Goldschmidt, Kant und Helmholtz (Hamburg, 1898); and especially Friedrich Conrat, “Hermann von Helmholtz’ psychologische Anschauungen,” in Abhandlungen zur Philosophie, 18 (1904), which contains an account of his sensory physiology as well. A. E. Woodruff has published two studies of Helmholtz’ electrodynamics: “Action at a Distance in Nineteenth Century Electrodynamics,” in Isis, (1962), 439–459; and “The Contributions of Hermann von Helmholtz to Electrodynamics,” ibid., 59 (1968). 300–311.

R. Steven Turner

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Helmholtz, Hermann Von



From the perspective of posterity, Hermann Ludwig Ferdinand von Helmholtz (1821–1894) made his most significant contributions to the fields of sensory physiology and psychology. In particular, he laid the foundations for the experimental investigation of the sensory processes in audition and vision. His successes resulted from his unusual mathematical and mechanical abilities and from his skill in assembling, sifting, and logically interrelating large quantities of information.

Helmholtz was born in Potsdam, the eldest of five children. The atmosphere of his home, although not scientific, was intellectual. Helmholtz mentioned that he frequently listened to the philosophical discussions between his father, August, a distinguished teacher of languages and philosophy in the Gymnasium, and his colleagues. August Helmholtz was most important in guiding his son’s early education. He taught him Greek and Arabic, encouraged him to study poetry, art, and music, and trained him in the writing of expository essays. Even at an early age, however, Helmholtz was attracted more to physics and mathematics than to the humanities. Speaking of his youth, Helmholtz recalled, in 1891, that while the rest of his class in the Gymnasium read Cicero and Vergil, he himself would work out optical and geometrical problems under his desk.

Helmholtz represented a scientific ideal to his contemporaries. He was a man of remarkable talents, universal interests, and indefatigable energy. He published over 200 papers and books making fundamental contributions to the fields of anatomy, medicine, physics, physiology, and psychology. He invented scientific and medical instruments. A contemplative mind led him to consider the philosophic questions raised by developments in science. A sense of social responsibility led him to deplore the gulf separating those trained in the humanities and those trained in the sciences; this gulf he sought to bridge by delivering popular lectures explaining the developments of science to laymen.

For the social sciences, Helmholtz’ views on the methodology of science and the theory of knowledge are of most immediate relevance. The method of studying and interpreting biological and psychological phenomena was a crucial issue for the sci entists and philosophers of the nineteenth century. Helmholtz was convinced that the same scientific method is appropriate for the analysis of vital and mental phenomena as for the analysis of physical phenomena. However, he did not advocate a reductive materialism. For him, memory and learning are facts, the laws of which can be investigated by experimental procedures, even if they cannot be reduced to known physiological processes (Koenigsberger [1902-1903] 1906, p. 305). Rather, he emphasized that the basic method of all scientific inquiry is to discover facts through systematic experimentation and to go beyond these facts to general laws by analysis and abstraction. His lecture on Goethe (see Helmholtz [1853-1870] 1962, pp. 1-21) contains an interesting presentation of his views. After acknowledging Goethe’s insightful contributions on the morphology of the skull and the plant, Helmholtz brought up the matter of Goethe’s obstinate animosity to Newton’s analysis of color. The difference between them, Helmholtz suggested, lay in their attitudes toward the acquisition and conception of knowledge. Goethe, the artist, believed that direct confrontation with a phenomenon is sufficient to provide an understanding of it. He was “accustomed to look as it were, right into the subject, and to reproduce his intuition…. His success [was] proportionate to the vividness of the intuition” (ibid., p. 16). Helmholtz acknowledged that intuitive insight is the source of the creative idea for both the scientist and the artist but insisted that, for the scientist, careful observation and reflection are an initial step sufficient only to suggest a hypothesis. The establishment of scientific truth depends on experimental verification. Moreover, the artist is satisfied simply to describe and hesitates to abstract. “A step into the region of abstract conceptions which must be necessarily taken if we wish to penetrate into the causes of phenomena, scares the poet away” (ibid., p. 16). Goethe objected to Newton’s theory of color because it substitutes quantitative measurements of the color-mixing relationships for direct naturalistic observation of colors. Helmholtz correctly described how observational intuition led Goethe badly astray when he asserted that it is absurd for white light to be the result of a fusion of colored lights. On the other hand, Goethe’s appeal to experience did reveal important phenomenological facts (the distinction between warm and cold colors, the existence of a psychologically unique red, yellow, green, and blue, and the coupling of the colors red and green and yellow and blue) whose potential physiological significance Helmholtz apparently overlooked. Helmholtz’ intellectual attitude was that of the nineteenth-century physicist: it was the source of his acute and penetrating analyses, although it occasionally led him to miss the point.

Work in physics . At the age of 16, Helmholtz announced to his father that he wanted to study physics. His father, however, could not support the cost of an academic course and enrolled him in the Friedrich Wilhelm Medical Institute in Berlin. This institute gave free instruction on the condition that those enrolled later spend some years as surgeons in the Prussian army. In Berlin, Helmholtz came under the influence of Johannes Miiller, professor of physiology at the University of Berlin, and Miiller’s influence is apparent throughout his physiological research. In 1842, at 21, he received his doctorate for a thesis in anatomy confirming Miiller’s conjecture that the nerve fibers originate in the ganglion cells. Yet Helmholtz was to join Du Bois-Reymond, Brücke, and Virchow, all Müller’s students, in a pact to abolish Miiller’s notion that within the living organism there exists a separate vital force distinct from chemical and physical forces. These men realized that physiology needed to be securely established as a branch of chemistry and physics. Liebig had placed the question of vitalism within the domain of experimental science by transforming the philosophic question into the experimental problem of whether or not the mechanical energy and the heat produced by an organism result entirely from its own metabolism. Helmholtz realized that this problem is closely connected to the law of the conservation of energy and turned his attention to that law. Joule had already established the equivalence between mechanical work and heat, and Helmholtz sought to extend the principle of the conservation of energy to physiology. He began by investigating the relation between muscular activity and heat. For this purpose he devised the myograph, a device which records the twitching of a muscle, a thermal galvanometer sensitive to temperature changes of .001° centigrade, and a means by which a muscle can be stimulated by electric shocks of short duration. He found that a single muscle contraction can give rise to an increase in temperature of from .001° centigrade to .005° centigrade, showing that the chemical processes in muscle contraction produce heat. Helmholtz then turned to determining the equivalence between the heat produced by an animal and the caloric value of the food taken in. He calculated the amounts of heat given off in various ways and the efficiency of an animal’s metabolism and concluded that the total heat given off corresponds to the heat generated by the oxidation of the food consumed. Unknown to Helmholtz, the extension of the law of the conservation of energy from inanimate to animate phenomena had already been suggested by Robert Mayer. Helmholtz next set himself the task of showing the universal application of the conservation of energy principle by theoretically deriving it from more fundamental physical principles. He showed that if all matter consists of particles, and if these interact according to forces acting along the lines joining the particles, and if all forms of energy depend upon the motion and position of the particles, then the conservation of energy must hold. These results he announced in a lecture he gave (at the age of 26) to the Physical Society of Berlin on July 23, 1847.

Work in physiology . Helmholtz’ scientific ability now became generally recognized, and with the aid of Alexander von Humboldt, president of the Academy of Sciences in Berlin, he was released from further military service. After a year as a lecturer in anatomy at the Academy of Arts in Berlin, Helmholtz became professor of physiology at Königsberg. He remained in Königsberg from 1849 to 1855.

Nerve impulses. The first important physiological problem to concern Helmholtz after his move to Königsberg was the rate of nerve transmission. Müller had stated that the speed of the nerve impulse is comparable to the speed of light and unmeasurable. Du Bois-Reymond’s theory of the polarization of molecules in nervous tissue suggested to Helmholtz that the nerve impulse, although electrical in nature, is not simply an electric current. Helmholtz stimulated nerves near and far from a muscle and measured the time it takes for the muscle to contract. The difference in time divided by the distance between the two points showed that the rate of the neural impulse in the motor nerve of a frog is about 27 meters per second. Helmholtz also devised the reaction time experiment to measure the speed of nerve transmission in humans. An electric shock was given to a point on the skin and the observer was directed to move his hand as quickly as he could. Although variable, the results showed that the rate of transmission of the impulse is between 50 and 60 meters per second. The novelty of these studies provoked opposition from physiologists. In fact, both Muller and von Humboldt refused at first to believe the results and it was only through the continued arguments of Du Bois-Reymond that they were convinced of the correctness of Helmholtz’ work.

Optics and visual perception. In 1851 Helmholtz invented the ophthalmoscope, the most celebrated of his inventions. He suggested that the blackness of the pupil results from the fact that the head of an observer looking at the eye of a person blocks the rays of light that would illuminate the retina. The ophthalmoscope is a device that illuminates the retina through the pupil, rendering it visible. In its modern form, constructed by H. Reute in 1852 (M’Kendrick 1899, p. 78), the ophthalmo scope enables one to look at the retina through a small opening in a reflecting plate. In 1852 Helmholtz invented the ophthalmometer, a device for measuring the radii of curvature of the anterior and posterior surfaces of the cornea and the lens. At this time Helmholtz turned to the problem of accommodation. A Dutch physiologist, Cramer, had shown that accommodation consists in an increasing curvature of the lens when the eye changes from a far point to a near point. Helmholtz, unacquainted with the work of Cramer, arrived independently at the same conclusion. Helmholtz’ new contribution to the theory of accommodation was his hypothesis about the way in which the curvature of the lens is altered. Using his anatomical knowledge, he suggested that when the fibers of the ciliary muscles contract, the tension of the lens capsule is reduced, and the lens’s own elasticity causes the anterior surface of the lens to bulge forward.

From 1851 to 1856 Helmholtz was mainly engaged in research in physiological optics. The result of this work, together with a survey of the germane literature, is presented in his monumental three-volume handbook on physiological optics (Helmholtz 1867). The handbook demonstrates Helmholtz’ mastery of the physical, mathematical, and physiological problems involved in vision and visual perception. The worth of the handbook is attested to by its continued scientific pertinence a hundred years later. Volume 1 describes the anatomy and optical properties of the eye. Volume 2 deals with problems of visual sensation. In 1852 Helmholtz undertook to test the laws of color mixture. Newton had failed to distinguish between the mixture of colored lights and of colored pigments. Thomas Young, in his famous paper of 1801, had suggested (in analogy to the mixture of pigments) that red, yellow, and blue are the three primaries from which all colors can be derived by suitable mixtures. But to his surprise, Helmholtz discovered that the combination of yellow and blue lights produces not green, as was supposed, but white. Helmholtz was the first to differentiate clearly between the addition of lights, which produces a color because of the superposition of excitations of different wave lengths stimulating the eye, and the addition of pigments, which produces a color that is a result of the wave lengths reflected by both pigments. Yellow and blue pigments when mixed give green because neither pigment absorbs green light. After clearing up the confusion between light mixtures and pigment mixtures, Helmholtz proceeded to develop a comprehensive theory of color vision. He based his theory on the extension of Müller’s law of specific nervous energies to the sense organs and on the assumption that the facts of color matching accurately mirror underlying physiological processes. Just as the difference between the sensations of light and sound depends on the kind of nerve fibers stimulated, so the sensation of color depends on the kind of nerve fiber excited in the eye. The data on color mixture, moreover, indicate that all hues except the most saturated can be obtained by suitable combinations of three primaries: red, green, and violet. Helmholtz made explicit that part of Young’s theory which dealt with nerve activity, by postulating that the nerve fibers must respond in varying degrees to all wave lengths. Hue depends on the relative frequencies of the impulses set up in the three types of fibers, brightness on the total excitation of the fibers, and saturation on the amount of white produced by fusion. He also extended the theory to account for color blindness, negative afterimages, and successive contrasts. The Young—Helmholtz theory of color vision is still widely accepted, although certain facts of color blindness and color thresholds pose difficulties for it.

Volume 3 deals with problems of space perception and object perception. Central to Helmholtz’ theory of space and object perception is his doctrine of unconscious inference. Helmholtz argued in his inaugural lecture at Köonigsberg in 1852 that the law of specific nerve energies implies that sensations serve only as the signs of external objects. That is, sensations tell us as much about the real nature of external phenomena as the name of a man tells us about the man. Accordingly, perceptual experience involves a construction on the part of the observer. A problem of great concern for Helmholtz was to determine what in the nature of perceptual experience is innate and what is derived from experience. He asserted that what in the perceptual image can be modified by experience must itself be considered a product of experience ([1867] 1924-1925, vol. 3, p. 13). Applying this criterion, he concluded that the only innate properties are the intensity and the quality of a sensation. Thus, conscious experience consists initially of a mass of sight, sound, taste, and smell sensations, each differentiated only with respect to quality and intensity. How does one come to experience coherent objects ordered in space and time? The answer lies in the distinction between sensation and perception. Perception involves a two-stage process: (1) sensation and (2) unconscious inference. Sensations are the immediate products of stimulating sense organs and associated neural systems. Unconscious inference is a central reasoning-like process which utilizes the information provided by sensation to infer the properties of external objects and events. Thus, perception is not simply an aggregate of sensations. Perception goes beyond the sensations provided by the receptor organs and entails a judgment about the environment. The unconscious judgments in perception are automatic and irresistible. The knowledge gained in perception, based on sensations and the memory of sensations, is a knowledge of acquaintance (kennen). In contrast, the conscious judgments in thought are flexible and deliberate, and the knowledge gained, based on concepts, is a knowledge of awareness (wissen). For Helmholtz these differences, however, are only superficial. Both the unconscious judgments in perception and the conscious judgments in thought involve the same mental operations and have the same logical status ([1853-1870] 1962, pp. 178-181). The view that perception inherently involves a judgment has played an important role in psychological theory.

Acoustics. In 1855 Helmholtz was appointed professor of anatomy and physiology at Bonn, where he remained until 1858, when he became professor of physiology at Heidelberg. While still immersed in the preparation of his Physiological Optics Helmholtz had decided to write a similar work on acoustics. In 1856 he reported the discovery of summation tones. These tones, whose frequencies are the sum of two input frequencies, are generated because of the nonlinear response of structures in the external ear. In 1857 Helmholtz formulated his well-known theory of vowel quality. Differences between vowel sounds arise because the mouth cavity when shaped for different vowels serves as an adjustable resonator which differentially strengthens the harmonic components of the fundamental voice tone. Papers quickly followed on sound production, tone quality, harmony, and the musical scales. In 1862 Helmholtz’ classic work, On the Sensations of Tone, appeared. This book brings together his many experimental and historical studies in acoustics and music theory. The aim of the work is to provide connecting links between the fields of physical acoustics, physiological acoustics, and music. The most famous of Helmholtz’ contributions to audition is his theory of pitch perception. Applying the law of specific nerve energies to the ear, Helmholtz suggested that there exist specific fibers in the inner ear that respond by resonance to the different frequencies. Each fiber excites its own nerve so that the neural impulses transmitted to the brain correspond to the component frequencies of a sound wave. Thus, the resonance theory of pitch provides an explanation of Ohm’s law which asserts that it is possible to hear the simple harmonic components into which a complex sound may be analyzed. Although such a simple resonance theory is no longer ten able, experiments have demonstrated that a locus of maximum vibration which changes with sound frequency does occur on the basilar membrane in the way Helmholtz suggested. Helmholtz extended his theory of pitch to providean account of timbre, dissonance, consonance, and melody. Timbre de pends upon the relative intensities of the harmonics present, dissonance on the beats generated when two frequencies affect overlapping segments of the basilar membrane, consonance and melody on the identity of harmonics produced by simultaneous tones or a succession of tones. Helmholtz’ views on the musical relationships have been much discussed and criticized but are still relevant today.

Aesthetics . Since his early student days, Helmholtz had been concerned with problems of aesthetic significance. Although science and art differ in their methods, he believed them to be intimately connected because both seek to acquaint us with reality (Koenigsberger [1902-1903] 1906, p. 172). While the scientist expresses truth by means of abstract concepts, the artist expresses truth by means of images. Since art has a cognitive function, the articulation of meanings that cannot be conveyed by words, it must conform to the laws governing human intelligence. The function of the scientist is not, however, to instruct the artist. This is essentially impossible since neither the artist nor the observer is consciously aware of the laws on whose fulfillment beauty depends ([1862] 1954, p. 366). Rather, the function of the scientist is to provide the artist with the information which may aid him in his artistic invention. The artist can succeed in his work only when he is aware of the subtle relations present within a phenomenon and of their effects upon an observer (Koenigsberger [1902-1903] 1906, p. 106). In a lecture on anatomy and sculpture delivered in 1848, Helmholtz considered the methods and benefits of teaching anatomy to artists (ibid., pp. 51-57), and in a lecture on optics and painting, Helmholtz ([1853-1870] 1962, pp. 250-286) discussed the complex problems faced by the painter who seeks to produce an accurate representation of the forms, sizes, distances, and illuminations of objects in a scene. To cope successfully with these problems, the artist must translate rather than copy nature, taking into careful consideration the operation of the sensory system. For Helmholtz the relations between art and science are particularly clear and striking in music. Musical composition depends more on the immediate sensations than is the case with the other arts, like sculpture, in which the object represented has a much greater importance ([1862] 1954, p. 3). Part 3 of the Sensations of Tone deals with the agreeableness of different chords, the formation of scales, and the elementary rules of musical composition. Helmholtz cautioned that the history of music shows that the construction of scales and musical composition is by no means determined by the properties of the ear but depends on artistic invention and aesthetic principles which change with time (ibid., p. 235).

Epistemology . Helmholtz’ interest in sensory physiology naturally led him to the problems of epistemology. Following the English tradition, he appealed to psychology for an understanding of epistemology. He argued that a knowledge of the sensory mechanisms is necessary for determining the status of objects. In a letter to Carl Ludwig, in 1855, Helmholtz wrote, “I believe that philosophy will only be reinstated when it turns with zeal and energy to the investigation of epistemological processes and of scientific methods…. Most essential of all in this critical investigation is the exact knowledge of the processes of sense-perception” (Koenigsberger [1902-1903] 1906, p. 139). Kant had established 12 categories of a priori concepts. These represented the operation of inherent characteristics of the mind, which condition experience and do not arise from experience. Examples of a priori concepts are space and time, the geometrical axioms, and the law of causality. Helmholtz first questioned the correctness of Kant’s position concerning the inherent nature of space. He concluded that the experience of an ordered space is not innate but is derived from movements of the eye and body ([1853-1870] 1962, p. 156). Helmholtz’ empirical theory of perception led him to an empirical theory of cognition. In 1866 he turned to the epistemology of mathematics. He undertook to show that the geometric axioms are also the products of experience and that they depend upon the notion of congruence, which is based upon mechanical movements and is itself acquired through experience. Thus, the axioms of geometry are not innate but have their origin in an individual’s experience with objects and with the space in which he lives. To show that the axioms of Euclidean geometry are not examples of a priori intuitions, Helmholtz independently developed Riemannian non-Euclidean geometry. In 1888 Helmholtz tried to show that the axioms of arithmetic can also be derived from experience. He accepted only the law of causality as a priori and transcendental and asserted that all thought presupposes the law of causation, which is therefore not derivable from experience (Koenigsberger [1902-1903] 1906, p. 142). For example, the law of causation underlies the unconscious inference by which we pass from sensation to perception, since this process assumes the existence of external objects that cause our sensations. Whether the mechanism of unconscious inference is itself innate or is derived from experience, Helmholtz left as an open question (ibid., p. 110).

In 1871 Helmholtz became professor of physics in Berlin. He spent the last years of his life primarily in physical research. He was concerned with experiments in electrodynamics, physical optics, meteorological physics, and the physical significance of the principle of least action. Helmholtz also encouraged Heinrich Hertz, perhaps his most famous student, who verified Maxwell’s theory by producing electromagnetic waves. In 1887 the Physical-Technical Institute was founded in Berlin, with Helmholtz as director. In 1893 he visited the Chicago World’s Columbian Exposition and toured the United States and Canada. While returning home, he fell down the ship’s stairs and suffered a concussion and severe loss of blood. He recovered, but those who saw him reported that he tired easily and found it difficult to work. In July 1894 he suffered a cerebral hemorrhage and died in September of that year.

Jacob Beck

[See alsoHearing; Perception, article onDepth perception; Psychophysics; Vision, especially the article onColor Vision and Color Blindness.]


(1853–1870) 1962 Popular Scientific Lectures. New York: Dover. → First published in German.

(1862) 1954 On the Sensations of Tone as a Physiological Basis for the Theory of Music, New York: Dover. → First published as Die Lehre von den Tonempfindungen als physiologische Grundlage fiir die Theorie der Musih.

(1867) 1924-1925 Helmholtz’s Treatise on Physiological Optics. Edited by James P. C. Southall. 3 vols. New York: The Optical Society of America. → Translated from the third German edition.


Boring, Edwin G. (1929) 1950 A History of Experimental Psychology. 2d ed. New York: Appleton.

Koenigsberger, Leo (1902–1903) 1906 Hermann von Helmholtz. Oxford: Clarendon. → First published in German.

Mkendrick, JohnG. 1899 Hermann Ludwig Ferdinand von Helmholtz. New York: Longmans.

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HELMHOLTZ, HERMANN VON (1821–1894), German scientist.

Hermann Ludwig Ferdinand von Helmholtz was one of the most versatile scientists of the nineteenth century, endowed with a keen theoretical understanding of physical phenomena, with an unusual ability for putting them into a mathematical form, and with a rare intuition for instrumentation and experimental exploration. Above all, he was able to connect remote scientific fields with each other. During the last decades of his life, he held an outstanding position in European science.


Helmholtz was born in Potsdam, near Berlin, on 31 August 1821. Unable to afford a university education, he entered the Prussian military medical school, which was connected with the university. After taking his doctorate degree with the physiologist Johannes Müller in 1842, Helmholtz served a one-year internship in Berlin's Charité Hospital and then worked as a military surgeon in Potsdam until 1848. Many of Müller's students later came to play important roles in physiology and related subjects, and in 1845 some of them took part in founding the Physical Society, which remains the official German organization of physicists. In July 1847, Helmholtz read his memoir On the Conservation of Force to this group, announcing what was later called the energy conservation principle.

After serving as an anatomy instructor at the Berlin Academy and as an assistant to Müller for a year, Helmholtz began his academic career as a professor of physiology at the University of Königsberg in 1849. There he measured the velocity at which frog muscles conducted nerve impulses and devoted himself to sensory physiology. In 1850 he became famous for inventing the ophthalmoscope, which opened up vast possibilities for ophthalmology. Five years later, Helmholtz moved to the University of Bonn as a professor of anatomy and physiology, and three years later he taught physiology at the University of Heidelberg. During his Heidelberg years, he finished two treatises on sensory physiology, the Handbook of Physiological Optics (1856–1867) and On the Sensation of Tone as a Physiological Basis for the Theory of Music (1863)—presentations that still carry authority in their field and remain standard references in many respects.

The height of Helmholtz's career and fame began when he took up the Berlin chair of physics in 1871. The new German Reich had found in Helmholtz the "Reich's chancellor of science," as he was sometimes called. He began to devote himself almost exclusively to physics, mainly electro-dynamics.

Helmholtz was made a noble by the German emperor in 1883 and four years later became the founding president of the Physikalisch-Technische Reichsanstalt, a nonacademic government institute of physics and technology in Berlin-Charlottenburg. Helmholtz died in Berlin on 8 September 1894. After his death, the rapid development of modern physics away from its mechanistic foundations soon made Helmholtz's worldview outdated but did not overshadow his individual contributions.


The unifying idea of Helmholtz's scientific work can be found in his mechanistic interpretation of nature—that is, in the view that all physical phenomena can be explained in terms of mechanics. This idea served also as a background to Helmholtz's energy conservation principle, to which he was led by asking whether the mechanical equivalent of muscular heat and work could be accounted for solely by the metabolic process. If this were the case, the assumption that a separate life force existed would become obsolete. Helmholtz's energy conservation principle thus marked the beginning of organic physics, which sought to reduce all organic processes to the laws of physics and chemistry, and ultimately to mechanics.

To many contemporaries, Helmholtz's reductionism appeared to be downright materialism, and it is true that at the beginning of his career Helmholtz had strong sympathies with the materialist inclinations of the revolutionary movement of 1848. Helmholtz never claimed, however, that mental phenomena could be reduced to or identified with biological (and thus ultimately with mechanical) processes, as the materialists did. On the contrary, in the famous controversy with the physiologist Ewald Hering (1834–1918) on space perception, Helmholtz argued that visual perception requires a mind that has learned from movements in space and draws unconscious inferences from them. Hering, on the other hand, claimed that perception depends on an innate physiological mechanism that does not require a judging consciousness to turn incoming sensations into perceptions.

Other problems of visual physiology that Helmholtz dealt with included the accommodation of the eye, color vision, contrast phenomena, optical illusions, and eye movements. In physiological acoustics, he developed the resonance theory of hearing, in which the sensations produced by the resonating endings of the acoustic nerves are perceived as a tone. He also solved the problem of consonance and dissonance, which made it possible to explain musical harmony.

From his work with organ pipes, which he needed for his experiments on the physiology of hearing, Helmholtz was led to questions of hydrodynamics. In 1858 he demonstrated the invariance of vortices in ideal fluids and developed the notion of the vortex sheet, which foreshadowed the notion of the boundary layer, which is central to modern fluid mechanics.

In his work on electrodynamics, Helmholtz tried to mediate between James Clark Maxwell's field theory and the Continental approaches, which were based on action at a distance, merging them into a compromise version. The experimental work of his student Heinrich Hertz, which culminated in his detection of electromagnetic waves in 1887–1888, had originally emerged from attempts to confirm Helmholtz's rendering of Maxwell's theory.

Helmholtz is also known as the author of some twenty-five popular and philosophical essays. His interest in philosophy was roused by his father, who was an enthusiast of German idealism, and it gained momentum from his work on perception. In 1855 Helmholtz joined and stimulated the rising neo-Kantian movement with a lecture, "Human Vision," in which he propagated a new epistemology as a cure for speculation in the manner of Hegelian philosophy. He developed an empiricist theory of perception, called the theory of signs, which started from the skeptical claim that human sensations cannot be taken as images of reality, but only as signs, without any similarity to the objects signified. The order humans give to those sensations, however, originates with the outer objects, and this order can be taken as a true image of the world's causal order. One can never know for certain that a sensation is caused by an object, but all human perceptions are hypotheses to this effect and are constantly tested through voluntary manipulation of the world.

From his philosophy of perception, Helmholtz was led to the foundations of geometry, where he defended an empiricist interpretation of the axioms. Space has to be taken as the epitome of the movability of rigid bodies or, equivalently, of the congruence of relations between bodies and surfaces. He found that such movements and relations were possible not only in Euclidean space but also in non-Euclidean space. He concluded that non-Euclidean spaces are conceivable and that the Euclidean axioms are not necessarily true of space as claimed by Kant.

See alsoEinstein, Albert; Hertz, Heinrich; Maxwell, James Clerk; Physics; Science and Technology.


Primary Sources

Helmholtz, Hermann von. Wissenschaftliche Abhandlungen. 3 vols. Leipzig, Germany, 1882–1895. An almost complete collection of Helmholtz's scientific papers.

——. Vorlesungen über theoretische Physik. 6 vols. Leipzig, Germany, 1897–1907. Lectures on theoretical physics, given during his time as president of the Physikalisch-Technische Reichsanstalt.

——. Epistemological Writings: The Paul Hertz/Moritz Schlick Centenary Edition of 1921. Edited by Robert S. Cohen and Yehuda Elkana. Translated by Malcolm F. Lowe. Dordrecht, Netherlands, and Boston, 1977. This edition has become famous for its extensive and congenial annotation in the context of rising logical empiricism.

——. Helmholtz's Treatise on Physiological Optics. Edited by James P. C. Southall. 3 vols. Rochester, N.Y., 1924–1925. Reprint, Mineola, N.Y., 2005. Translation of Handbuch der physiologischen Optik.

——. Science and Culture: Popular and Philosophical Essays. Edited by David Cahan. Chicago, 1995. Selected popular and philosophical writings.

Secondary Sources

Cahan, David, ed. Hermann von Helmholtz and the Foundations of Nineteenth-Century Science. Berkeley, Calif., 1993. An excellent collection, written by historians and philosophers of science and covering virtually all areas of Helmholtz's work. Includes a fairly complete bibliography of secondary works.

Cranefield, Paul F. "The Organic Physics of 1847 and the Biophysics of Today." Journal of the History of Medicine and Allied Sciences 12 (1957): 407–423. Vivid account of the program of organic physics, its context, and its limitations.

Darrigol, Olivier. "From Organ Pipes to Atmospheric Motions: Helmholtz on Fluid Mechanics." Historical Studies in the Physical and Biological Sciences 29 (1998): 1–51. Profound account of Helmholtz's contributions to fluid mechanics, exemplifying Helmholtz's unity of thought across many disciplines.

Hatfield, Gary. The Natural and the Normative: Theories of Spatial Perception from Kant to Helmholtz. Cambridge, Mass., 1990. Profound history of spatial perception in the nineteenth century, concentrating on the philosophical context.

Heidelberger, Michael. "Hermann von Helmholtz (1821–1894)." In Die großen Physiker. Vol. 1: Von Aristoteles bis Kelvin, edited by Karl von Meÿenn, 396–415, 474–475, and 528–531. Munich, Germany, 1997. Attempt at a short general account of Helmholtz's life and work.

Koenigsberger, Leo. Hermann von Helmholtz. 3 vols. Translated by Francis Welby. Oxford, U.K., 1906. Rich and authoritative, yet hagiographic and somewhat Whiggish account of Helmholtz's work and life by one of his students.

Rechenberg, Helmut. Hermann von Helmholtz: Bilder seines Lebens und Wirkens. Weinheim, Germany, 1994. Heavily illustrated, very readable scientific biography.

Richards, Joan L. "The Evolution of Empiricism: Hermann von Helmholtz and the Foundations of Geometry." British Journal for the Philosophy of Science 28 (1977): 235–253. Account of Helmholtz's foundational work on geometry and its later impact; still valid.

Schiemann, Gregor. Wahrheitsgew ißheitsverlust: Hermann von Helmholtz' Mechanismus im Anbruch der Moderne. Darmstadt, Germany, 1997. Thorough exploration of Helmholtz's mechanical interpretation of nature and its contemporary context in philosophy, science and society.

Michael Heidelberger

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Hermann Ludwig Ferdinand von Helmholtz

The German physicist and physiologist Hermann Ludwig Ferdinand von Helmholtz (1821-1894) made the first mathematical analysis of the principle of the conservation of energy and invented the ophthalmoscope. He also investigated the physics of tone and color perception.

Born on Aug. 31, 1821, in Potsdam, Hermann von Helmholtz was the eldest of six children of August Helmholtz, a teacher of philosophy and classics in the local gymnasium, and Caroline Penne Helmholtz, a descendant of the family which took prominent part in the founding of Pennsylvania. At the age of 17 he entered the Friedrich Wilhelm Institute as a scholarship student of medicine. Happily for science, Helmholtz had the genius to absorb medical and physiological training with the mind of a physicist. A great help in this respect was a teacher of his at the institute, Johannes Müller, the foremost German physiologist at that time, who insisted on carrying as far as possible the physical and chemical explanation in all problems of physiology. For Helmholtz this also meant the utmost use of mathematics, which he learned on his own to a very advanced degree by studying in his free time such mathematical classics as the works of Euler, Bernoulli, D'Alembert, and Lagrange. His doctoral dissertation (1842) described and analyzed the connections between nerve fibers and nerve cells, a discovery which he made by skillful use of the microscope and which forms the histological basis of the physiology and pathology of the nervous system.

Helmholtz's scholarship status implied the obligation to serve for 8 years as army doctor. His first post (1842-1843) was that of a house surgeon at the Charité Hospital in Berlin, followed by a stint of 5 years as assistant surgeon to the Royal Hussars at Potsdam. Despite time-consuming duties, Helmholtz found ways of developing his scientific interests. While at the Charité, he published a demonstration of the strictly chemical nature of fermentation and noted that a vitalistic account would be equivalent to assuming a perpetual-motion process. His papers on metabolism during muscular activity (1845) and on physiological and animal heat (1846, 1847) clearly indicated the great goal toward which his creative mind inevitably tended. In February 1847 he sent the first draft of the introduction of "The Conservation of Force" to Emil Du Bois-Reymond, who immediately declared that it was "an historical document of great scientific import for all time." The 26-year-old Helmholtz read the paper on July 23 before the Physical Society of Berlin. He was not the first to enunciate the idea that physical force (energy) was conserved in its various transformations, but his originality consisted in giving the principle a generalized mathematical form, which readily yielded expressions for kinetic and potential energy in mechanics, thermodynamics, electricity, and magnetism.

Academic World

The impact of the lecture and the availability of a teaching post in anatomy at the Academy of Arts in Berlin made it possible for Helmholtz to transfer to the academic world. In 1849 he accepted the invitation to serve as professor of physiology at the University of Königsberg, where he married Olga von Velten the same year. Among his first papers at Königsberg was "On the Method of Measuring Very Small Intervals of Time and Their Application to Physiological Purposes" (1851). The purpose was the determination of the rate of transmission of sense impressions along the nerves. The value he found shortly afterward was approximately 30 meters per second, in good agreement with subsequent determinations. The next year Helmholtz wrote of a marvelous new device to investigate the eye, the ophthalmoscope, which made its inventor world famous almost overnight.

In 1855 Helmholtz went to Bonn as professor of physiology and anatomy. During the following 3 years he started his incisive analysis of the mathematical relationships underlying tone perception and the esthetical judgment about various tonalities. At the same time he began publishing in a mature form his previous studies of the physiology of vision. The first volume of his famous Handbook of Physiological Optics appeared in 1856. The massive volume was a systematic application of physics to the phenomenon of vision. "On the Integrals of the Hydrodynamic Equations Which Express Vortex Motions" (1858) represented not only a brilliant solution of seemingly insoluble equations, but it also became the foundation for late-19th-century physicists who tried to devise a so-called vortex model of the ether.

Helmholtz spent the next 13 years at the University of Heidelberg. During his first year there his wife died. In 1861 Helmholtz married Anna von Mohl, who became mother of three children. He spent 13 years at Heidelberg, officially a physiologist but in reality a physicist.

The second part of Helmholtz's Handbook of Physiological Optics was published in 1862. The next year saw the publication of another systematic application of physics in physiology, Theory of the Sensations of Tone as Physiological Basis of the Theory of Music. In 1867 followed the third, and final, volume of the Handbook. Helmholtz's last 3 years at Heidelberg were marked by portentous investigations in theoretical physics. His interest was now in fluid mechanics, the foundations of geometry and electricity. Characteristically enough, when the chair of physics became vacant in Bonn, the university tried to get Helmholtz back as professor of physics. Helmholtz declined the opportunity, but in his reply he made no secret of his lifelong ambition: "Physics was really from the outset the science which principally attracted my interests. I was mainly led to medicine and thereby to physiology by the force of external circumstances. What I have accomplished in physiology rests mainly upon a physical foundation."

Life in Berlin

The same letter also gives a clue to the inspiration that animated Helmholtz during the last two major assignments of his life. The first of these was the professorship of physics at the University of Berlin (1871-1888). Following the Franco-Prussian war of 1870-1871, Berlin emerged as the capital of the First Reich, and as a professor of physics at the University of Berlin Helmholtz could be most influential in promoting the cause of scientific instruction in German schools and universities. There was much to be done if, in a predominantly humanistic and classical educational system, science courses were to gain a rightful place. It was the professorship of physics at the University of Berlin which he finally found most promising to reward him "for taking the new work of a new post" upon himself. His request was immediately granted that a chair in physics be complemented by an Institute of Physics reserved exclusively for advanced research. Shortly after his arrival in Berlin, Helmholtz gave evidence of the high level in physical research which he had in mind. His memoir read in 1872 before the Berlin Academy of Sciences, "On the Theory of Electrodynamics," represented a remarkable effort to provide a generalized form of electrodynamics, of which even James Clerk Maxwell's theory was but a particular case. In 1873 he was invited to lecture in the United States, but he felt that acceptance would hinder his researches at his institute, which had begun to attract the most promising young physicists in Germany. The foremost among them was Heinrich Hertz, who developed under Helmholtz's guidance the interest to test experimentally a chief consequence of Maxwell's theory, the propagation of electromagnetic oscillations in space. A high point in Helmholtz's own reflections on the problems of electricity was his Faraday lecture in 1881, in which he emphasized the essentially atomic structure of electricity, a contention fully confirmed a few years later by J. J. Thomson's work with cathode rays.

During this period Helmholtz began to achieve the status of a chief adviser to the state in scientific matters. He played a prominent part in the establishment of the Imperial Physico-Technical Institute (Physikalisch-Technische Reichsanstalt), of which he became the first director in 1888. Although a chief aim of the new institute was to carry out high-precision measurements, Helmholtz wanted the new institute to serve pure science as well as technology. During his last years acclaim and honors poured in from every side. His seventieth birthday became an occasion for nationwide celebrations. He died on Sept. 8, 1894, after months of struggle with paralysis. In no small measure, through his inspiration physics in Germany rose to unparalleled heights by the end of the 19th century, a position which remained unchallenged for another generation.

Further Reading

Leo Koenigsberger's classic three-volume biography is available in English in a one-volume abridgment by Frances A. Welby as Hermann von Helmholtz (1906). It covers equally well the personal and scientific aspects of Helmholtz's career and discusses his influence on science in the second half of the 19th century. A specialized study of Helmholtz is Richard M. Warren and Roslyn P. Warren, Helmholtz on Perception, Its Physiology and Development (1968). A detailed biographical account of Helmholtz's life is in Bessie Zaban Jones, ed., The Golden Age of Science: Thirty Portraits of the Giants of 19th-Century Science by Their Scientific Contemporaries (1966). Good discussions of Helmholtz are also in Henry Ernst Sigerist, The Great Doctors: A Biographical History of Medicine (1933), and Ralph Hermon Major, A History of Medicine (vol. 2, 1954). □

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Hermann von Helmholtz

German scientist who conducted breakthrough research on the nervous system.

Hermann Helmholtz was one of the few scientists to master two disciplines: medicine and physics. He conducted breakthrough research on the nervous system , as well as the functions of the eye and ear. In physics, he is recognized (along with two other scientists) as the author of the concept of conservation of energy.

Helmholtz was born into a poor but scholarly family ; his father was an instructor of philosophy and literature at a gymnasium in his hometown of Potsdam, Germany. At home, his father taught him Latin, Greek, French, Italian, Hebrew, and Arabic, as well as the philosophical ideas of Immanuel Kant and J. G. Fichte (who was a friend of the family). With this background, Helmholtz entered school with a wide scope of knowledge. Though he expressed an interest in the sciences, his father could not afford to send him to a university; instead, he was persuaded to study medicine, an area that would provide him with government aid. In return, Helmholtz was expected to use his medical skills for the good of the governmentparticularly in army hospitals.

Helmholtz entered the Friedrich Wilhelm Institute in Berlin in 1898, receiving his M.D. four years later. Upon graduation he was immediately assigned to military duty, practicing as a surgeon for the Prussian army. After several years of active duty he was discharged, free to pursue a career in academia. In 1848 he secured a position as lecturer at the Berlin Academy of Arts. Just a year later he was offered a professorship at the University of Konigsberg, teaching physiology. Over the next 22 years he moved to the universities at Bonn and Heidelberg, and it was during this time that he conducted his major works in the field of medicine.

Helmholtz began to study the human eye, a task that was all the more difficult for the lack of precise medical equipment. In order to better understand the function of the eye he invented the ophthalmoscope, a device used to observe the retina. Invented in 1851, the ophthalmoscopein a slightly modified formis still used by modern eye specialists. Helmholtz also designed a device used to measure the curvature of the eye called an ophthalmometer. Using these devices he advanced the theory of three-color vision first proposed by Thomas Young. This theory, now called the Young-Helmholtz theory, helps ophthalmologists to understand the nature of color blindness and other afflictions.

Intrigued by the inner workings of the sense organs, Helmholtz went on to study the human ear. Being an expert pianist, he was particularly concerned with the way the ear distinguished pitch and tone. He suggested that the inner ear is structured in such a way as to cause resonations at certain frequencies. This allowed the ear to discern similar tones, overtones, and timbres, such as an identical note played by two different instruments.

In 1852 Helmholtz conducted what was probably his most important work as a physician: the measurement of the speed of a nerve impulse. It had been assumed that such a measurement could never be obtained by science, since the speed was far too great for instruments to catch. Some physicians even used this as proof that living organisms were powered by an innate "vital force" rather than energy. Helmholtz disproved this by stimulating a frog's nerve first near a muscle and then farther away; when the stimulus was farther from the muscle, it contracted just a little slower. After a few simple calculations Helmholtz announced the impulse velocity within the nervous system to be about one-tenth the speed of sound.

After completing much of the work on sensory physiology that had interested him, Helmholtz found himself bored with medicine. In 1868 he decided to return to his first lovephysical science. However, it was not until 1870 that he was offered the physics chair at the University of Berlin and only after it had been turned down by Gustav Kirchhoff. By that time, Helmholtz had already completed his groundbreaking research on energetics.

The concept of conservation of energy was introduced by Julius Mayer in 1842, but Helmholtz was unaware of Mayer's work. Helmholtz conducted his own research on energy, basing his theories upon his previous experience with muscles. It could be observed that animal heat was generated by muscle action, as well as chemical reactions within a working muscle. Helmholtz believed that this energy was derived from food and that food got its energy from the Sun. He proposed that energy could not be created spontaneously, nor could it vanishit was either used or released as heat. This explanation was much clearer and more detailed than the one offered by Mayer, and Helmholtz is often considered the true originator of the concept of conservation of energy.

Helmholtz had been a sickly child; even throughout his adult life he was plagued by migraine headaches and dizzy spells. In 1894, shortly after a lecture tour of the United States, he fainted and fell, suffering a concussion. He never completely recovered, dying of complications several months later.

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Hermann Ludwig Ferdinand von Helmholtz


German Physicist, Philosopher and Physiologist

Hermann von Helmholtz's scientific career marked him as one of the foremost scientific minds of the nineteenth century. His landmark theories and ideas encompassed electrodynamics, fluid and thermo dynamics, mathematics, optics, and physiology.

Though interested in many aspects of knowledge in his youth, Helmholtz was not distinguished by any early scientific gifts. He leaned toward physics in his university years but began his scientific work in the field of medicine with medical school in Berlin in 1838. Lectures included attendance at the University of Berlin, where he found mentors in J. P. Müller in physiology and H. G. Magnus in physics. Helmholtz's medical thesis (1842) on the relation of animal nerve fibers and cells and heat led to his interest in the theory of the conservation of energy. Thus, early on this personal research netted the writing of a fundamental paper in classical physics titled "On the Conservation of Force" (1847), which would find application in several areas of experimental science. Yet his primary career was as a professor of physiology, which included several important academic positions: Königsberg (1849), Bonn (1855), and Heidelberg (1858).

Starting in 1850 Helmholtz would invent instruments for studying the human eye, leading to contributions to the understanding of the structure and mechanism of the eye. This was expanded between 1857 and 1859 into investigations of physiological acoustics, the study of tone (combination and vowel), and important work on the mechanism of the bones of the middle ear, emphasizing the physical nature of the eardrum. He had also returned to nerve studies earlier (1850), successfully measuring the speed of the propagation of nerve impulses.

In regard to his physical study Helmholtz's 1847 paper proved that the work or energy done by a force was conserved, and that the total energy of particles interacting through some central force was dependent upon only the mass of the particles and their spatial separation. In this he had philosophically and physically defined the theory by just two concepts: matter and force (Helmholtz called it an "ultimate" force), and his theory was more comprehensive than that of other scientists studying the conservation of energy, such as Sadi Carnot (1796-1832), Julius Mayer (1814-1878), and James Joule (1818-1888). Helmholtz also did studies in energetics, or the transformation of energy of one type into another, which were important to the prevailing concept of a mechanical or dynamic theory of heat. Among the applications in which he proved the conservation of energy were electrostatics, electrodynamics, and thermodynamics.

Helmholtz provided seminal contributive papers to the fields of electricity and magnetism (1858) and the physics of fluid flow (also 1858). He also worked out two basic, or primitive, equations based on the mathematics of fluid flow, showing the stability of vortex motion and the constancy of velocities of that motion. Later, he produced an important paper ("On Atmospheric Motion," 1888) dealing with the limited radiation in the polar regions, which meant sinking air over that area in relation to the general circulation of the atmosphere.

Helmholtz's renown as a theoretical physicist grew as his life progressed. He held the physics professorship of his late professor Magnus at Berlin (1871). By 1888 he had become the foremost scientific advisor in Germany, the proof of which was his appointment as the director of the new Physico-Technical Institute at Berlin, which was the first national scientific laboratory in Europe.


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Helmholtz, Hermann Ludwig Ferdinand von (1821–94) German anatomist, physicist, and physiologist. He made contributions in acoustics and optics, expanding Thomas Young's three-colour theory of vision. His experiments on the speed of nerve impulses led him to formulate a principle of conservation of energy.

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Helmholtz, Hermann Ludwig Ferdinand von (1821–94) German physiologist and physicist. In 1850 he measured the speed of a nerve impulse and in 1851 invented the ophthalmoscope. In physics, he discovered the conservation of energy (1847) and introduced the concept of free energy.