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Hermann von Helmholtz 














IN the historical record of science the name of Helm- 
holtz stands unique in grandeur, as a master and leader 
in mathematics, and in biology, and in physics. His 
admirable theory of vortex rings is one of the most 
beautiful of all the beautiful pieces of mathematical 
work hitherto done in the dynamics of incompressible 
fluids. In 1843, when he was only twenty-two years 
old, and barely emerged from his undergraduate course 
as a medical student, he showed, in his first published 
scientific paper, a clear appreciation of the necessity 
of distinguishing vital from non-vital phenomena : and 
he gave experimental evidence tending to prove that 
putrefaction and fermentation are essentially vital 
actions : and thus led the way to Pasteur's splendid 
and beneficent discoveries. His Erhaltung der Kraft, 
published in 1847, was a guide to his own countrymen, 
and to the rest of the world, in the doctrine of energy 
through the whole range of dynamic action in dead 
and living matter, then despised and rejected by nearly 
all the high priests of science ; now cherished by all 
as a most fruitful result of modern research. His 
Tonempfindungen, and his Physiological Optics, are 
not mere textbooks : they are ever memorable Prineipia 
of the perception of sound, and of light, by living 



The professional career of Helmholtz was unparalleled 
in the history of professions. He was Military Surgeon 
in the Prussian army five years ; Teacher of Anatomy 
in the Academy of Arts in Berlin one year ; Professor 
of Pathology and Physiology in KCnigsberg six years ; 
Professor of Anatomy in Bonn three years ; Professor 
of Physiology in Heidelberg thirteen years ; Professor 
of Physics in the University of Berlin about twenty 
years, till he became Director of the new ' Physikalisch- 
Technische Reichsanstalt '. He occupied this post during 
the last years of his life, still continuing to give lectures 
as Professor of Physics. 

Beginning with the generous aid and co-operation of 
Werner von Siemens, and ultimately supported by the 
financial resources of their country, Helmholtz created 
the Reichsanstalt, which has already conferred inestim- 
able benefits, not only on Germany, but on the whole 
world. It is an example, tardily and imperfectly fol- 
lowed by Great Britain and other countries only now 
beginning to learn that scientific research yields results 
which are valuable, not merely for the discovery of 
truths appreciated only by scientific workers, but for 
contributing in many ways to the welfare of the whole 

The Faraday Lecture, delivered by Helmholtz before 
the Fellows of the Chemical Society in the theatre 
of the Royal Institution on Tuesday, April 5, 1881, 
was an epoch-making monument of the progress of 
Natural Philosophy in the nineteenth century, in 
virtue of the declaration, then first made, that elec- 
tricity consists of atoms. Before that time atomic 


theories of electricity had been noticed and rejected 
by Faraday and Maxwell, and probably by many 
other philosophers and workers ; but certainly accepted 
by none. Now in the beginning of the twentieth 
century we all believe that electricity consists of atoms. 
How far-reaching is this theory, and how much science 
is enriched by it, is splendidly illustrated by Becquerel's 
discovery of radio-activity, and the magnificent harvests 
of new and astonishing truth which have been 
gathered by the numerous and brilliant workers in 
the field of investigation thus opened to the world. 

I cannot conclude this short preface without re- 
ferring to the great debt which the world owes to 
Helmholtz, in having given to Hertz the inspiration 
to find experimental proof of Maxwell's electric waves ; 
and giving him, in the Physical Institute of the 
University of Berlin, the apparatus and appliances by 
means of which he carried out the investigation. To 
this we owe the first practical demonstration of pro- 
gressive electric waves, and of stationary waves, in 
air, and therefore inferentially in ether undisturbed 
by ponderable matter. Thus in Helmholtz we find 
a prime factor in the grand series of theoretical and 
experimental researches through which wireless tele- 
graphy has been achieved. 

The Oxford University Press has earned the grati- 
tude of all English-speaking scientific workers in 
giving to them this English version of the very 
valuable and interesting Life of Helmholtz, by 
Dr. Konigsberger. 



I DECIDED at the end of last year, in consequence 
of my long personal and scientific connexion with 
Hermann von Helmholtz, and at the repeated wish 
of his widow, the late Frau Anna von Helmholtz, to 
compile a Biography of the great investigator. 

Thanks to the letters and other communications 
received from his family and from a vast number of 
distinguished scholars and friends, to the permission 
accorded by the Prussian Government to avail myself 
of the Official Papers relating to the career of Helm- 
holtz, and above all to the active co-operation of his 
daughter, Frau Ellen von Siemens, I have been 
enabled to give a connected account of his life; for 
this help I beg to express my cordial thanks. 

It is for the indulgent reader to decide whether it 
lies within the power of a mathematician to present 
the epoch-making contributions of von Helmholtz to 
the most various departments of human knowledge 
in a form that shall be universally intelligible. 


October, 1902. 

In the English edition the Life has been slightly abridged, with 
the permission of the author and German publishers. 





His BOYHOOD, 1821-1838 6 




1842. De Fabrica Systematis Nervosi Evertebratorum. 
Inaugural Dissertation 24 

1843. 'On the Nature of Fermentation and Putrefaction.' 
Muller's Archiv 27 


1845. 'On Metabolism during Muscular Activity/ Mailer's 
Archiv, 1845 . 32 

1846. 'Physiological Heat.' Encyklop. Worterbuch d. med. 
Wissenschaften ."" 32 

1847. 'Report on Work done on the Theory of Animal 
Heat in 1845.' Fortschritte d. Physik 34 

'On the Conservation of Energy.' (Berlin Physical 
Society) published by G. Reimer, Berlin. (Trans.) 'On 
the Conservation of Force: a Physical Memoir/ in Taylor, 
Scientific Memoirs (Nat. Phil.\ 1853 39 

' Development of Heat in Muscular Activity.' Mailer's 
Archiv, 1848 . . 50 

1848. Probationary lecture to the Academy of Arts, Berlin, on 

' Some Aspects of the Teaching of Anatomy to Art-Students/ 51 




1848. ' Report on Work done on the Theory of Animal Heat 

in 1846.* Fortschr. d. Physik 59 

1849. ' Principles of Construction of a Tangent Galva- 
nometer.* (Berlin Phys. Soc.) 59 


1849. Marriage with FrSulein Olga von Velten . . .61 

1850. ' Rate of Transmission of the Excitatory Process 
in Nerve/ Berlin Monatsber., Comptes Rendus, XXX, 
XXXIII . . 63 

' Measurements of the Time-Relations in the Contraction 
of Animal Muscles, and Rate of Propagation of the Exci- 
tatory Process in Nerve.' Mutter's Arch. ... 66 

'On the Methods of Measuring very small Intervals of 
Xime, and their Application to Physiological Purposes/ 
Lecture in Kftnigsberg. (Trans.) Phil. Mag. [4], VI . 69 

Discovery of the Ophthalmoscope. (Berlin Phys. Soc.) 73 

1851. Description of an Ophthalmoscope for the Investigation 

of the Retina in the Living Eye. (Berlin, Forster) . . 74 
* On the Duration and Course of the Electrical Currents 

induced by Variation of an Inducing Current.' Berl. 

Monatsber., Poggend. Annal., LXXXIII .... 79 
Appointed Ordinary Professor at Ko"nigsberg ... 88 

1852. ' Measurement of the Rate of Transmission of the 
Excitatory Process in Nerve/ Part II. Mutter's 
Arch 88 

'Results of Recent Discoveries in Animal Electricity/ 
Kiel Allg. Monatsschr. 7" . . 89 

'On the Nature of Human Perceptions/ Inaugural 
Lecture T~ .... 92 

'Theory of Compound Colours/ Poggend. Annal., 
LXXXVI ; (trans.) Phil. Mag\\ IV .... 93 

'On Brewster's New Analysis of Solar Light.' Berl. 
Monatsber., Poggend. Annal., LXXXIX; (trans.) Phil. 
Mag. [4], IV . . 93 

' A Theorem of the Distribution of Electrical Currents in 
Material Conductors/ Berl. Monatsber 99 

1853. ' On Some Laws of the Distribution of Electrical 
Currents in Material Conductors with Application to 
Experiments in Animal Electricity/ Poggend. Annal., 



1853. ' Report on Work done in Acoustics in 1848.' Fortschr. 
d. Physik. 

'On the Scientific Researches of Goethe.' (Lecture at 
Konigsberg.) (Trans.) Helmholtz, Pop. Set. Lect., Series I 103 

' On a Hitherto Unknown Alteration in the Human Eye 
during Accommodation.' Berl. Monatsber. . . . 107 

First Journey to England 109 

1854. ' Reply to the Observations of Dr. Clausius.' Poggend. 
Annal., XCI 115 

' On the Interaction of Natural Forces.' Lecture, 
(trans.) Phil. Mag. [4], XI ; Pop. Set. Lect., Series I . . 120 

'On the Rate of Certain Processes in Muscle and Nerve.' 
Berl. Monatsber 125 

1855. ' On the Composition of Spectral Colours.' Poggend. 
Annal.,XClV 131 

' On the Sensibility of the Human Retina to the Most 
Refrangible Rays of Solar Light.' Poggend. Annal., XCIV 133 

'Addendum to a Paper by E. Esselbach on the Measure- 
ment of the Wave-Length of Ultra-Violet Light.' Berl. 
Monatsber. 134 

' On the Accommodation of the Eye.' Graefe's Archiv 
fur Ophthalmologie 134 

' On Human Vision.' (Konigsberg, Lecture) . . . 138 

First meeting with W. Thomson (Lord Kelvin) . . 145, 

1856. 'On the Movements of the Thorax.' (Bonn, Nieder- 
Rheinische Gesellschaft) 148 

'Contraction -Curves of Frogs' Muscles.' Nieder-Rh. 

Sitzungsber. 149 

' On the Explanation of Lustre.' Nieder-Rh. Sitzungsber. 149 
' On Combination Tones.' Poggend. Annal., XCIX . 151 
' The Action of the Muscles of the Arm.' (Lecture, Bonn) 
' The Telestereoscope.' Pogg. Annal. 
Textbook of Physiological Optics. Part I . . . 155 

1857. ' On the Vowel-Tones.' Letter to Bonders . '.158 

1858. ' On Integrals of the Hydrodynamic Equations which 
express Vortex-Motions.' Crelle's Journ., LV, (trans.) Phil. 
Mag. [4], XXXIII . . . . . . . .167 

' Sur le Mouvement le plus general d'un Fluide.' ' Sur 
le Mouvement des Fluides.' ' Reponse a la Note de 
M. J. Bertrand du 19 Octobre 1868' 170 



1858. 'On the Subjective After-images of the Eye.' 
Nieder-Rh. Sitzungsber. 171 

'On After-images/ (Karlsruhe, Naturf.-Versammlung) 172 
' On the Physical Causes of Harmony and Dissonance/ 

(Karlsruhe, Naturforscher-Versammlung) . . .172 

'On the Physiological Causes of Harmony in Music/ 

(Bonn, Lecture). Pop. Set. Lect., Series I . . . .172 


1859. 'On the Quality of Vowel-Sounds/ Poggend. 
Anna!., CVIII ; (trans.) Phil. Mag. [4], XIX . . .178 

' On Timbre/ (Heidelberg Scientific and Medical Society, 
1860) 180 

'On Aerial Vibrations in Tubes with Open Ends/ 
Crelle's Journal, LVII 180 

'On Colour- Blindness/ (Heidelberg Society) ... .183 

Death of Helmholtz's first wife 185 

1860. ' On Friction in Liquids/ Vienna, Akad. d. Wissensch. 186 
' On Contrast Phenomena in the Eye/ (Heidelberg Soc.) 187 
Textbook of Physiological Optics. Part II . . .191 
' On Musical Temperament/ (Heidelberg Society) . 194 
' On the Persian and Arabian Scales/ (Heidelberg Soc.) 196 
'On the Motion of the Strings of a Violin/ Glasgow, 

Proc. Phil. Soc., V 196 

1861. ' On the Application of the Law of the Conservation of 
Force to Organic Nature/ Proc. Roy. Inst., Ill . . 199 

Marriage with Fraulein Anna von Mohl . . . . 200 
'Contribution to the Theory of Reed-pipes/ Pogg. 

Ann., CXIV 205 

' On a General Method of Transformation of the Problems 

concerning Electrical Distribution/ (Heidelberg, Dec. 8) . 206 

1862. Pro- Rector of Heidelberg University . . . 211 
' On the Relation of the Natural Sciences to Science in 

general/ Pro-Rectorial Address, (trans.) Popular Scientific 

Lectures, Series I 211 

'On the Horopter/ Graefe's Archiv, 1864. 'Remarks 
on the Form of the Horopter/ Pogg. Ann., 1864 . . 213 

1863. The Theory of the Sensations of Tone as a Physio- 
logical Basis of the Theory of Music. (Translated by 

A. J. Ellis) . 213 

Note on the Signification of Historical Researches in 
Branches of the Natural Sciences. 



'On the Motions of the Human Eye.' Heidelberg, 
Lecture 218 

1864. Journey to England 22 L 

' On the Normal Motions of the Human Eye in relation 

to Binocular Vision/ (Croonian Lecture.) Proc. Roy. Soc., 
XIII . ... .224 

' Experiments on Muscular Sounds.' Berlin Academy. 
1 On Muscular Sound/ (Lecture, Heidelberg.) 

' On the Influence of Rotation of the Eye on the Out- 
ward Projection of Retinal Images.' (Lecture, Heidelberg.) 

1865. 'On Ocular Movements.' (Lecture, Heidelberg.) 
' On Stereoscopic Vision.' (Lecture, Heidelberg.) 

'On the Properties of Ice.' Lecture, Heidelberg. 'On 
the Regelation of Ice.' Phil. Mag. [4], XXXII . . 228 

1866. Journey to Paris 232 

1867. Handbook of Physiological Optics. Third (final) part 235 
' The more recent Developments in the Theory of Vision.' 

Preuss. Jahrb., XXI, 1868 ; Pop. Set. Lect., Series II . 237 

' The relation of Optics to Painting.' Lectures translated 
in Pop. Sci. Lect., Series II 240 

' Communication concerning Experiments on the Rate of 
Transmission of Irritation in the Motor Nerves of Man 
carried out by Mr. N. Baxt of St. Petersburg in the Physio- 
logical Laboratory at Heidelberg.' (Berlin Academy) . 244 

; On the Period necessary for becoming conscious of a 
Visual Impression.' Results of an experiment carried out 
by Mr. N. Baxt in the Heidelberg Laboratory. (Berlin 
Academy) 245 

'On the Mechanics of the Auditory Ossicles.' (Berlin 
Academy) 246 

'The Mechanics of the Auditory Ossicles and of the 
Membrane of the Tympanum.' Pfluger's Arch., 1869 . 246 

Ophthalmological Congress in Paris. Lecture. ' Sur 
la production de la sensation du relief dans 1'acte de la 
vision binoculaire ' 247 

Commencement of the translations of Tyndall's lectures 
and of Thomson and Tait's Natural Philosophy . . . 248 ^_ 

1868. 'On Discontinuous Movements of Fluids.' Berlin 
Monatsber. ; (trans.) Phil. Mag. [4], XXXVI . . .254 

' Contribution to the Theory of Stationary Currents in 
Frictional Fluids.' (Lecture, Heidelberg.) 



'On the actual Foundations of Geometry/ (Lecture, 

Heidelberg, Gottingen Nachrichten, 1868) .... 254 

' On the Origin and Significance of Geometric Axioms/ 
Lecture at Heidelberg, translated in Pop. Scient. Lect., 
Series II 254 

Note on the Fundamental Notions of Mathematics and 
Physics 256 

1869. Correspondence with Beltrami 263 

'The Facts of Perception/ Address delivered in 1878 

at the Commemoration of the Foundation of the University 

of Berlin 267 

'On the Origin and Signification of Geometrical Pro- 
positions/ Reply to Prof. Land, 1878. 

' On the Physiological Action of brief Electrical Shocks 
in the Interior of extended conducting Masses/ (Heidel- 
berg Scientific and Medical Society) . . . . 268 
' On Electrical Oscillations/ (Heidelberg Society) . 268 
'On the Aim and the Developments of the Natural 
Sciences/ (Innsbruck Naturforscher-Versammlung) . 268 
' On Hay^ever/ Vir chow's Archiv .... 268 
'On Sound Vibrations in the Cochlea/ (Heidelberg 
Society) 269 

1870. 'On the Laws of Inconstant Electrical Currents in 
Material Conductors/ (Heidelberg Society) . . . 269 

'On the Theory of Electrodynamics/ Part I : 'On the 
Motional Equations of Electricity for resting conducting 
Bodies/ Crelle Journ., LXXI I 269 

Call to the Professorship of Physics in Berlin . . 275 

1871. ' On the Origin of the JPlanetary System/ (Lecture, 
Heidelberg.) Translated in Pop. Set. Lect., Series II . 280 

PROFESSOR OF PHYSICS IN BERLIN, 1871-1888 . . . 281 

1871. 'On the Rate of Transmission of Electrodynamic 
Effects/ (Berlin Academy) 282 

'In memory of Gustav Magnus/ (Address to the 
Academy of Sciences, Berlin, translated in Pop. Sci. Lect., 
Series II) 282 

' Preface and Critical Supplement to the German trans- 
lation of " J. Tyndall, Fragments of Science," ' 1874 . . 284 

Journey to England 285 

1872. 'On the Theory of Electrodynamics/ (Berlin Academy) 288 



'On the Theory of Electrodynamics.' Part II : Critical 
notes. Crelle's Journal, LXXV, 1873 .... 289 

1 Comparison of Ampere's and Neumann's Law regarding 
Electrodynamic Forces.' (Berlin Academy) . . . 289 

'On the Theory of Electrodynamics.' Part III : Electro- 
dynamic Forces in Moving Conductors. Crelle's Journal, 
LXXVIII, 1874 . . ... 289 

' Experiments on Electromotive Forces induced by 
Motion in an Open Circuit.' (Berlin Academy) . . 291 

'On Galvanic _ Polarization of Platinum.' (Leipzig 
Naturforscher-Versammlung) 294 

' On Galvanic Polarization in Fluids free from Gas.' 
(Berlin Academy.) (Trans.) Phil. Mag. [4], XLVII . . 294 

' Report on the Experiments of Dr. E. Root of Boston 
concerning the Penetration of Platinum by Electrolytic 
Gases.' (Berlin Academy.) (Trans.) Phil. Mag., 1876 . 295 

'Report on Experiments on the Electromagnetic Effect 
of Electrical Convection, carried out~by M r. "Henry A. 
Rowland.' (Berlin Academy.) (Trans.) Phil. Mag., 1876 . 295 

Note on the Theory of Electrodynamics .... 295 

1873. 'On a Theorem concerning geometrically similar 
Movements of Fluid Bodies, and its Application to the 
Problem of steering Air-balloons.' (Berlin Academy) . 296 

' The Theoretical Limit of the Efficiency of Microscopes.' 
Pogg. Ann., 1874 . . 299 

1874. 'Contribution to the Theory of Anomalous Dis- 
persion.' (Berlin Academy.) Pogg. Ann., CLIV, 1875 . 300 

1875. 'Whirlwinds and Thunderstorms.' (Lecture, Ham- 
burg.) Deutsche Rundschau, 1876 301 

Death of Robert von Mohl 302 

1877. Appointed Professor of Physics at the Military 
Institute for Medicine and Surgery, Berlin . . . 304 

' On Thought in Medicine.' Lecture translated in Pop. 
Sci. Lect., Series II 304^ 

Death of his daughter Kathe 305 

'On Academical Freedom in the German Universities.' 
Rectorial Address, translated in Pop. Sci. Lect., Series II . 306 

' On Galvanic Currents caused by Differences in Con- 
centration : Deductions from the Mechanical Theory of 
Heat.' Wiedemann's Ann., Ill; Phil. Mag., 1878 . . 309 



1878. 'The Facts in Perception/ Address on Foundation 
Day of the University of Berlin 312 

'Telephone and Timbre.' (Berlin Academy.) Wiede- 
mann's Ann., V 314 

' On the Signification of the Convergent Position of the 
Eyes for the Purpose of determining the Distance of 
Objects seen binocularly/ (Berlin Physiol. Soc.) 
.. Journey to Italy 314 

' Prefatory Note to a posthumous Treatise of Franz Boll : 
Theses and Hypotheses concerning Light and Colour 
Sensation.' Du Bois-Reymond's Archiv, 1881. 

1879. 'Studies on Electrical Boundary Layers.' Wiede- 
mann's Ann., VII . . . . . . . . 317 

Heinrich Hertz at Helmholtz's Physical Institute . . 321 

1880. 'On the Motional Currents on Polarized Platinum.' 
(Berlin Academy.) Wtedemann's Ann., XL 

Journey to Spain 323 

1881. 'On the Forces acting on the Interior of magnetically 
or dielectrically polarized Bodies/ (Berlin Academy.) 
Wiedemanris Ann., XIII 329 

Note on the Theory of Attractions within Magnetizable 
or Dielectric Media 329 

'An Electrodynamic Balance/ Wiedemann's Ann., XIV. 

Journey to England 330 

' On the Modern Development of Faraday's Conception 
of Electricity/ Faraday Lecture to the Chemical Society, 
London 330 

Note: Observations supplementary to the Faraday 
Lecture 331 

Note : ' On the Electrodynamic Theory of Optical 

Journey to Paris to take part in the Electrical Congress . 332 

Journey to Florence and to the Electrical Exhibition in 
Vienna. 333 

' On the Deliberations of the Paris Congress concerning 
Electrical Units/ (Lecture, Berlin Electro-technical Society) 333 

'On Galvanic Polarization of Mercury and on Recent 
Experiments by Mr. Arthur Konig relating to it/ (Berlin 
Academy) .... .333 

1882. Appearance of the Wissenschaftliche Abhandlungen, 
Vol.1; 1883, Vol. II 334 



' The Thermodynamics of Chemical Processes.' (Berlin 
Academy) . . . 335 

1883. Note on an Introduction to Thermodynamics . . 340 
' The Determination of Magnetic Moments by means of 

the Balance/ (Berlin Academy.) Note thereon. 

Correspondence with Heinrich Hertz .... 344 
Journey to Rome to take part in the Geodetic Congress . 347 

1884. Journey to England 348 

Marriage of his daughter Ellen to Arnold Wilhelm von 

Siemens 349 

' Studies on the Statics of Monocyclic Systems/ (Berlin 
Academy) . . 349 

'Generalization of the Theorems on the Statics of 
Monocyclic Systems/ (Berlin Academy) .... 350 

' Principles of the Statics of Monocyclic Systems/ 
Crelle's Journ., XCVII ... ... 350 

' On the Physical Signification of the Principle of Least 
Action/ Crelle's Journ., C, 1886 350 

'On the History of the Discovery of the Principle of 
Least Action/ (Address to the Berlin Academy) . . 350 

1885. Handbook of Physiological Optics. Second edition. 
1885-1895 . ... 363 

'Report on Sir William Thomson's Mathematical and 
Physical Papers, Vols. I and II.' Nature, XXXII . . 364 

1886. The Quincentenary of the University at Heidelberg . 365 

Conferment of the Graefe Medal 365^ 

'On the Formation of Clouds and Thunderstorms/ 

(Physical Society) 367 

Appointed Vice- Chancellor of the ' Peace ' Class of the 

Order ' Pour le merite ' 368 

Correspondence with Heinrich Hertz .... 368 

1887. Foundation of the Imperial Physico-Technical Institute, 
Berlin 369 

'Experiment to demonstrate the Cohesion of Fluids/ 
(Physical Society.) 

' Joseph Fraunhofer : Address on the occasion of the 
Centenary of his Birth/ 

' Further Researches concerning the Electrolysis of 
Water/ (Berlin Academy.) Wiedemanrts Ann., XXXIV 375 

'On an Ice Machine working with Pictet's Fluid/ 
(Berlin Phys. Soc.) 



Note: ' Thermodynamical Observations on Chemical 
Processes' 375 

Essay on ' Numbers and Measurements/ dedicated to 
E. Zeller . 377 



1888. 'On the Intrinsic Light of the Retina/ (Physical 
Society) 381 

Correspondence with Heinrich Hertz .... 383 

1889. ' In memory of R. Clausius.' (Berlin Physical Society) 383 

Death of Bonders 384 

' On Atmospherjc_Mx3ynints.' (Berlin Academy) . . 385 
' On Atmospheric-Movements.' Second communication. 

(Berlin Academy, and Physical Society) .... 385 

Death of his son Robert 387 

Congress of Naturalists at Heidelberg .... 388 

1890. ' The Energy of the Waves and of the Wind.' (Berlin 
Academy.) Wiedemanris Ann., XLI 390 

Representative of the University of Berlin at the Sex- 
centennial Jubilee of the Foundation of the University of 

'On the Work of the Imperial Physico -Technical 
Institute* 391 

1891. ' Observations on Preparatory Training for Academical 
Studies.' Discussions on Questions concerning Higher 
Instruction, Berlin, Dec. 4-17 392 

' An Attempt to extend the Application of Fechner's Law 
in the Colour System.' Zeitsch. f. Psycho!, u. Physio/, d. 
Sinnesorgane, II 395 

'An Attempt to apply the Psycho-physical Law to the 
Colour Differences of Trichromatic Eyes.' Zeitsch. f. 
Psycho!, u. Physio!, d. Sinnesorgane, III . . . 396 

'Shortest Lines in the Colour System.' (Berlin 
Academy) 397 

Celebration of his Seventieth Birthday on Nov. 2 . . 399 

1892. 'Autobiographical Notes. Address delivered at the 
Banquet to celebrate his Seventieth Birthday.' (Berlin: 

A. Hirschwald) ......... 400 

'The Principle of Least Action in Electrodynamics.' 
(Berlin Academy) 401 



' Goethe's Anticipations of Coming Scientific Ideas.' 
(Address to the Goethe Society at Weimar on June n.) 

Deutsche Rundschau, LXXII 403 

Fiftieth Anniversary of his degree 404 

Correspondence with Heinrich Hertz .... 405 

Death of Werner von Siemens 406 

' Electromagnetic Theory of Colour Dispersion/ (Berlin 
Academy.) Wiedemann* s Ann., XLVIII .... 407 

'Additions and Corrections to the Essay: Electro- 
magnetic Theory of Colour Dispersion/ Wiedemann's 
Ann., XLVIII 407 

1893. 'Address to E. du Bois-Reymond drawn up on behalf 

of the Royal Academy of Sciences ' 409 

'Conclusions from Maxwell's Theory as to the Move- 
ments of pure Ether.' (Berlin Academy) . . . .411 

Note : ' How one may imagine the Movement of the 
Ether in Maxwell's Theory of Electrodynamics to take 

place' . . .411 

Journey to the Exhibition at Chicago .... 412 
Accident on his return 418 

1894. Death of Heinrich Hertz on Jan. i 419 
' Supplement to the Essay : On the Principle of Least 

Action in Electrodynamics.' (Berlin Academy) . . . 425 
Note : ' Further Researches on the Completeness of the 

unknown Electrodynamic Forces.' 

' On the Origin of the Correct Interpretation of our 

Sensual Impressions.' Zeitsch. f. Psycho/, u. Physiol. d. 

Sinnesorgane, VII . . . . . . . . 426 

Hertz proposed by Helmholtz for the Prize of the Peter 

Muller Foundation ........ 428 

Commencement of illness on July 12 .... 429 

Death on Sept. 8 430 

Note: 'Address to Naturalists' 430 

Commemoration at the Academy of Music on Dec. 14 . 437 

1899. Unveiling of the Statue in front of the University on 

June 6 . 438 

Death of Mrs. von Helmholtz on Dec. i . . . . 439 

1901. Death of his son Fritz on Nov. 17 .... 440 


Portrait by von Lenbach, 1876 . . . Frontispiece 

Daguerreotype, 1848 To face page 58 

Pastel by von Lenbach, 1894 378 



HERMANN VON HELMHOLTZ was the son of August Ferdi^ 
nand Julius Helmholtz, who was born on December 21; 
1792, in Berlin, and was educated at the Friedrichs-Gymna- 
sium; he matriculated on October 15, 1811, in the Theological 
Faculty of the University. Notwithstanding a feeble consti- 
tution, he took part in the campaign of 1813-1814, was sworn 
in as a volunteer at Breslau immediately after the Royal Pro- 
clamation on March 30, 1813, and was promoted to be second 
lieutenant on September 8, after the battle of Dresden. 

After the Peace of Paris, 1814, he obtained his discharge, 
and returned to Berlin, but felt himself obliged to give up 
his theological studies from conscientious motives, since he 
was unable to reconcile himself to the hyper-orthodox views 
that prevailed at the time. He therefore chose the study of 
the classical languages as his profession, although his inclina- 
tions would have led him to prefer philosophy. 

A protracted nervous fever obliged him to relinquish the 
campaign of 1815, when he accepted a temporary engagement 
as private tutor to a couple of talented and industrious lads 
with whom he was happy and contented ; and he only parted 
from them reluctantly in order to provide for his future, and 
secure himself a permanent position. 

After passing a qualifying examination in Berlin he was 
appointed form-master at the Potsdam Gymnasium in 1820, 
and became Professor by Royal Patent in 1828. 

Directly after his appointment to the Gymnasium he married 
Fraulein Caroline Penne, the daughter of a Hanoverian 
artillery officer, who was born on May 22, 1797. She was 
descended in the male line from the famous American colonist 
William Penn, the founder of Pennsylvania, and on her mother's 


side from a family of French refugees named Sauvage. This 
happy union lasted till 1854, Frau Helmholtz doing much by 
her faithful fulfilment of her domestic duties to lighten the 
heavy calling of her husband, who was weighed down by his 
sense of duty and scrupulous conscientiousness. 

Caroline Helmholtz is described as being excessively simple 
in appearance, and was profoundly emotional and of quick 
intellect. Everything she said was incisive, and her homely 
judgements were clear and luminous. She seemed to pene- 
trate obscure points by intuition, ' without any deep reflection/ 
expressing her conclusions in simple language. 'A refined 
officers daughter/ says her younger son, Otto, ' she was 
compelled by the straitened circumstances which were all my 
father could provide for her, to consecrate her whole life to 
the maintenance of the household, and the education of 
her children, particularly of the two daughters, since my 
father was physically much enfeebled by the effects of his 
campaigns/ His profession, moreover, was onerous, for the 
strictest discipline prevailed at that time among Prussian 
officials, greatly to the advantage of the country, and to the 
ultimate weal of the whole of Germany, 

Thus when the young teacher expressed his desire to be 
associated with the general insurance fund for widows of 
officials, the Consistory made the following characteristic reply, 
which is illustrative of the rigid discipline then maintained in 
Prussian offices, and of the temper in which the rising genera- 
tion were brought up : 

' Your memorial is incomplete in its contents and most repre- 
hensible in its form. It was your duty definitely to explain 
that you could, and to engage that you would, provide a pension 
of at least 100 thaler at the General Institute for the Relief 
of Widows on behalf of your future wife, so that the declaration 
that you had decided to provide for your wife is obviously not 
sufficiently definite. We shall expect to receive the amended 
statement within eight days. . . . With regard to the form, you 
ought to know, or your sense of propriety should have informed 
you, that an official statement or memorial to the Board should 
not be drawn up upon a single page, but should occupy an entire 
sheet. The leaf you have handed in testifies to the greatest 
inattention and neglect of the respect due to the Board, and 


to its President, by whom the communication of the 3rd inst. 
was issued. We are compelled to call your attention to this 
carelessness on account of its consequences, and to recommend 
you to observe the claims of propriety and duty/ 

Such reprimands, however, were quietly accepted by the 
young Prussian official, in whom respect for authority was in- 
born. He devoted himself whole-heartedly to his profession, 
giving instruction in German and philosophy, translating and 
interpreting Plato, The Odyssey, Ovid and Virgil with his pupils, 
while for four years he was further responsible for the teaching 
of mathematics and physics in the higher forms. Notwithstand- 
ing this press of work he found time for painting (in which he 
was self-taught), for philosophical study, and for the publica- 
tion in the annual School Report of essays, such as ' The Early 
Development of the Hellenes', 'Historical Problems of the 
Coming Century ', and * The Arabs, as described in the 
Hamaseh', the merits of which were recognized at a later 
time by eminent authorities. Thanks to his wide literary 
studies, he had acquired a considerable knowledge of 
aesthetics ; his scientific interests were comprehensive ; and 
he was a stimulating and capable teacher, with a pronounced 
individuality as shown by the official report of the Head- 
master, which emphasizes his admirable influence upon the 
character and achievements of his pupils no less strongly 
than it is attested by such of the latter as still survive. 

In the words spoken fifty years later at the Commemoration 
Festival of the University of Berlin, by one who has earned 
undying fame, 'the older among us can remember the men 
of that period, who had been the foremost volunteers in our 
army, who were always ready to plunge into a metaphysical 
discussion, who were well read in the works of Germany's 
great poets, who burned with wrath at the name of the First 
Napoleon, and glowed with pride and inspiration in relating 
the deeds of the war of liberation.' 

In respect of Latin, his profession seems to have been 
merely the ' bread-study ' that he was wont to call it, but he 
was an enthusiastic Hellenist, and had a great influence over 
his pupils, endeavouring to give them a feeling for poetic 
beauty, instead of merely providing them with grammatical 
instruction. As a schoolboy he was unable to acquire the 

B 2 


Ciceronian style, and at a later time was wont to explain his 
predilection for the Greek language by saying with his greater 
son, that 'linguistic talent is not one thing, but like all other 
intellectual functions the sum of different factors'. 

He was one of the most distinguished teachers of the 
Gymnasium, and, with the mathematician Meyer, received 
frequent ovations from his pupils. A prominent member of 
the Prussian Civil Service writes : ' We reckoned it one of 
our happiest hours when we could persuade him to read us 
poems, dramas, ballads, and the like. Once, for instance, 
I remember his giving us the first monologue in Faust, and 
another time Burger's " Lied vom braven Mann" with so much 
force and feeling that we sat silent and deeply moved ; in later 
life, and even to the present day, his voice and his expressive 
countenance have often come back to me/ 

Despite the conscientiousness of the pedagogue, however, 
he was consumed with the enthusiasm and fire of the patriot. 
Once at the request of his pupils he devoted the three hours 
of German instruction which he gave the second class as 
form-master to an account of the feeling that inspired the 
Prussian people prior to 1813, and they applauded him enthu- 
siastically. But the Headmaster got wind of it, and the 
favourite teacher received an intimation that any repetition 
of the indiscretion would be punished by dismissal. This 
was in the middle of the forties, when the pressure of 
reaction in Church and State bore heavily upon Prussia, 
and resulted in the so-called Treubund (league of faith) 
to which the Headmaster and others of the teaching staff 

Hence the discipline of silence was imposed upon Helm- 
holtz for the sake of his family, though his discontent with 
the political condition of Germany occasionally broke out in 
private. His eldest son, Hermann, was born in 1821, the 
daughters Marie and Julie soon after, and twelve years later 
the second son, Otto. Later again there were two others, 
Ferdinand and Heinrich, who died in infancy in the years 
1836 and 1839. His income was inadequate, his salary only 
being raised to 160 at the close of his teaching career, and 
as a good husband and prudent father there was nothing for 
him to do but to keep his political ideas to himself, and avoid 


political conversation even in his own house. Henceforward 
he took his solitary walks to the mill of Sans Souci, and buried 
himself in philosophical reflections. 

But if Ferdinand Helmholtz thus avoided any open ex- 
pression of political opinion, his philosophical views would 
not permit him to keep silence in questions of ecclesiastical 
orthodoxy. An array of sketches and notes for speeches still 
bears witness to his profound philosophical ideas and noble 
religious convictions. His relations to his wife and his plans 
for the education of his children were based on genuine reli- 
gious and ethical feelings. But he abhorred all ecclesias- 
tical bigotry, and subscribed unhesitatingly to a declaration 
published on August 15, 1845, by such men as Alschefski, 
Bellermann, Bonnel, Jonas, Lisco, Meinecke, &c., which began 
with the words, 'A party has organized itself within the 
Protestant Church which clings tenaciously to the conception 
of Christianity inherited from the earliest traditions of the 
Reformation. This formula is its Pope. Faithful are such 
as submit themselves to it unconditionally, unfaithful and 
politically suspect all who have not joined it ' ; adding, ' We 
declare that we believe a healthy issue of this contest to be 
possible only if the right of free development is maintained 
intact on all sides/ 

We are thus able to form some picture of Ferdinand 
Helmholtz, and to understand the almost exaggerated ap- 
preciation of the friend with whom he maintained a life-long 
correspondence, and made many a journey. This was Imanuel 
Hermann Fichte, son of Gottlieb Fichte, and Professor of 
Philosophy in Tubingen from 1842, who writes of ' unalterable, 
and ever increasing affection a reciprocal attachment that was 
of the weightiest consequence in both our lives '. 

Helmholtz pursued his vocation as teacher faithfully till 
1857, with the utmost devotion to his duties. He followed 
the later career of his children with affection and interest, 
but was always, in virtue of his serious philosophical tempera- 
ment, a somewhat exacting critic. Finally, when his energies 
began to fail, he applied for a pension, which was granted with 
a gratifying recognition of his long and faithful service. 

BOYHOOD: 1821-1838 

August 31, 1821, at Potsdam, in the house known as No. 8 
Hoditzstrasse, and was baptized on October 7 in the Lutheran 
Church of the Holy Spirit. 

For the first seven years of his life he was an ailing child, 
confined to his room for long periods, and often to his bed, 
but he was energetic in work and play : his time was occupied 
with picture-books and games, more especially with wooden 
blocks, and his mental powers were carefully fostered by his 
parents. Each infantile disorder from which he suffered re- 
newed the anxiety of his tenderly attached family. ' I heard/ 
writes Frau von Bernuth, his father's cousin (and the daughter 
of Surgeon-General Mursinna of Berlin), 'that your son had 
scarlet fever ; and feared the worst as he is so delicate. Thank 
God that he has recovered! You must not be distressed 
because he has learned little so far. I am sure it will be for 
his good not to begin before his eighth year. Alexander von 
Humboldt learned nothing before he was eight, and now the 
King has made him President of the Academy of Sciences, 
with the title of Excellency, and a big yearly stipend and 
this is what I predict for your son/ 

When he was seven years old, Hermann, though still in 
delicate health, was sent to the Normal School of Potsdam, 
and even there astonished his masters in the Geometry Class, 
because (thanks to his toy blocks) he knew all the facts which 
they expected him to learn. His health improved by degrees 
with gymnastics and daily bathing, and his great love of Nature 
was developed at the same time by frequent walks with his 
father in the beautiful environs of his native town. In the 
spring of 1832, he was admitted to the lowest form of the 
Gymnasium, where he followed the teaching easily enough, 


and gave satisfaction to his masters. His handwriting indeed 
was criticized, and his mathematical home-tasks were in- 
adequately performed : but his power of working by himself, 
and the attention, zeal, and thought which he bestowed upon 
his studies, were highly commended. At the outset, in the 
lower classes, he was hampered by the want of a good memory 
for disconnected facts : l this showed itself/ he says fifty years 
later, 'in the difficulty which I still distinctly remember of 
distinguishing between right and left ; later on, when I got 
to languages in my school-work, it was harder for me to 
learn the vocabularies, grammatical irregularities, and idio- 
matical expressions, than for the others. History, in particular, 
as it was taught in those days, was quite beyond me. It was 
a real torture to learn prose extracts by heart. This defect 
has of course increased, and is a nuisance in my old age. 
I found no difficulty in learning the poems of the great masters, 
but the more laboured verses of second-rate poets were far 
less easy/ 

The father's influence was the most important factor in the 
boy's intellectual development. At home he occupied himself 
in arousing his children (with whom he was always on cordial, 
if not affectionate, terms) to a ^sense of the ideal in poetry, art, 
and music, while at the same time he strove to make them 
good patriots. As a keen teacher of Greek, he read Homer 
with his pupils, and as their instructor in German he gave 
them great facility of expression by means of prose essays 
and metrical exercises. 

The first three years of school-life were thus devoted mainly 
to grammatical studies, and to the aesthetic side of young 
Helmholtz's education, but with his entry into the second 
class the curriculum was widened to include mathematics and 
physics. The teaching of Prof. C. Meyer, Helmholtz's first 
mathematical tutor, is still praised by his surviving students. 
His treatise on ' The Caustic Curves produced by the Reflection 
of Light from Curves of the Second Order', which was pub- 
lished in the School Report for 1838, proves that Meyer 
combined scientific with pedagogic interests, and it may have 
been thanks to him that young Helmholtz, while his class 
were reading Cicero or Virgil, which did not interest him, 
would often be engaged beneath the table in working out the 


passage of ra}^s of light through the telescope, or in learning 
some of the optical theorems that served him in good stead 
later on in the construction of the ophthalmoscope. 

He was more fascinated by the elements of physics as 
taught in the Gymnasium, than by his purely geometrical and 
algebraic studies. And when he began to follow the physical 
and chemical experiments which Professor Meyer demonstrated 
in the laboratory to his students, and listened to the scientific 
discussions between his father and his mathematical colleague 
(in which, among other matters, the question of a perpetuum 
mobile, and the futile attempts to realize it, was continually 
cropping up), the boy's desire to immerse himself in these 
problems waxed stronger and stronger, and he burned to 
enlarge his mental horizon by independent and original ex- 
periments. It was at this time indeed (as Helmholtz often 
attested in later days) that he conceived the idea which in- 
creasingly dominated him, that the knowledge of natural laws 
should give us not only a spiritual mastery over Nature, but 
an actual material control of her processes. The vigorous 
young scholar was consciously outgrowing the narrow circle 
of his home and school relations. 

With no other appliances than some spectacle glasses and 
a little botanical lens belonging to his father, young Helmholtz 
and a friend contrived to make up optical instruments, modi- 
fying the construction again and again until he hit off some 
practicable arrangement. The necessary knowledge had to 
be acquired from a few antiquated textbooks on physics and 
chemistry possessed by his father, ' to which the discoveries 
of Lavoisier and Humphry Davy had not yet penetrated, while 
phlogiston still played its part and galvanism ended with the 
voltaic pile/ 

At fifteen, Helmholtz was described by his fellow-students 
as reserved and self-contained, showing invariable kindness 
to those weaker than himself. As regards his studies, he was by 
no means devoted exclusively to the exact sciences, for his first 
school-report in the first class testifies to a fairly level interest 
in all branches of his studies, his progress in Latin, Greek, 
Hebrew, religious instruction, mathematics, and physics being 
characterized as good, and history and geography as excellent : 
while the same appears from the decision of the masters' 


meeting in August, 1837, that at the Michaelmas speeches 
Helmholtz should reply with a Latin ode to the farewell 
oration delivered in German by one of the Abiturienten. 

While still in the second class Helmholtz announced to his 
father that he wished to devote himself to science, but 
when the worthy man, who had the education of four 
children on his hands, explained that he could not afford 
to provide him with instruction in physics unless he took 
up the study of medicine as well, he accepted the situation 

As early as 1835 his father applied for his admission to the 
Royal Friedrich-Wilhelm Institute of Medicine and Surgery 
in Berlin, which gave considerable assistance to medical 
students, inasmuch as it guaranteed them a complete course 
of study and means of livelihood, in return for a certain number 
of years' service as army surgeons. 

The competition for entry to this Institute was, however, 
too keen for it to be promised to Helmholtz's father two years 
in advance, and the application had to be renewed when 
Hermann had reached the first class. It was then successful, 
and he was summoned to an examination in Berlin during the 
Easter vacation of 1837. 

1 Dear Father/ writes the lad of sixteen, on March 30, from 
Berlin, ' I arrived the day before yesterday in a raging snow- 
storm. Tell mother, however, that I hardly felt the cold, 
except in my hands, which were quite numb. Yesterday 
morning I went to the Pepiniere at 9.30, and was called up at 
10.30. Surgeon-General Schulz was very kind. He inquired 
after you, tested my eyes to see if I were short-sighted, and 
asked three of the staff-doctors to guess my height. They 
decided that I was about four inches high (sic). He questioned 
me about my health, admonished me to emulate my quasi- 
ancestor Mursinna, and prove myself worthy of him, and not 
be afraid of the examination : even if I could not answer all 
the questions exhaustively it would not matter, as this was only 
to be a general test of my acquirements. 

' He then gave me a note for Dr. Figulus, who was to examine 
me. The doctor was not at home then, or an hour later, and 
I only found him after dinner. He gave me an appointment 
for this morning. Yesterday afternoon I went about in the 


town, and got pretty tired. This morning at eight I commenced 
my doctor's examination. I had to do my curriculum vitae 
in German and Latin. There was not time to make a fair 
copy of the Latin. The examination is all in writing. I don't 
know how long it will last/ 

Helmholtz returned with the news of his success to his 
delighted parents at Potsdam, and then gave himself up once 
more to severe and regular study of the most heterogeneous 
subjects, devoting himself to each in turn with the same interest 
and enthusiasm, though he could not forbear to say in his 
curriculum vitae a year later : ( quorum (veterum scriptorum) 
cognitio quantum valeat ad conformandum animum, nemo est, 
qui ignoret; deinde maxima atque plurima debeo Schmidtio 
professori, quum aliis in disciplinis, turn in historiis, quibus 
nihil est praestantius ad cognoscendam naturam hominum et 
populorum. Pater meus artis poeticae et oratoriae praecepta 
mihi dedit, quarum ilia et iucundissima est et utilissima ad 
elocutionem elegantem et copiosam. Omnium disciplinarum 
autem maxime iam a pueritia me delectavit physice et mathe- 
matice, quibus eruditus sum a Meierio, viro harum rerum 

After a full year's work, in which he not only prepared 
for the Abiturienten examination, but also, in view of his 
medical career, embarked on the scientific studies that had 
hitherto been outside his curriculum such as botany and 
zoology, with the elements of anatomy from Oken's Natur- 
geschichte fur alle Stdnde, and physiology from Magendie's 
Textbook his father sent him off with some other boys of 
his class to the Harz Mountains. They took long walks, 
to the great benefit of his none too robust constitution, absorb- 
ing the influences of Nature and Art at the various places 
where they halted. 

Refreshed in mind and body by this expedition, Helmholtz 
and one fellow-student embarked upon the written part of the 
Abiturienten examination, which lasted from August 20-25. 
His translation of sixty lines of the Hecuba of Euripides was 
marked ' very satisfactory ' ; his French version of a piece of 
two columns called Die Katakomben was ' excellent ' ; while the 
Hebrew professor gave him the highest praise for his Latin 
commentary on Deut. ix. 1-3, which was not a compulsory 


item for the medical student. As might have been expected, 
his father's judgement of his German essay on ' The Ideas and 
Art in Lessing's Nathan der Weise* was a little severe, although 
he could not refrain from praising its simple and expressive 

Helmholtz solved his four mathematical tests, two being 
geometrical and two arithmetical, correctly, his treatment of 
them showing ' great lucidity and grip ' of the elements of 
mathematics. In addition, he presented a fifth and voluntary 
exercise on ' The Laws of the Free Fall of Bodies '. This essay, 
as it now lies before us, shows unusual precision of thought 
and expression, and it is obvious that the author had pondered 
deeply and often over physical problems. 

The viva voce examination took place on September 12, 1838, 
and young Helmholtz left the Gymnasium with brilliant 

In later years the staff of the Potsdam Gymnasium might 
well be proud of the education for which the young man was 
in part at least indebted to it. * Our teachers encouraged us to 
read a great deal, and we were eventually able to peruse the 
authors to whom they introduced us with comparative ease, 
and did so at home after school hours, in addition to the study 
of foreign languages. I took up English and Italian privately 
at school, as well as Hebrew, and got a very good mark in 
Hebrew. I even began Arabic in the first class with a master 
who knew it, and found plenty of time for all these things/ 
Later on he read the Fables of Lokman in the original, in his 
leisure moments. 

As soon as he had received the Abiturienten certificate, his 
father wrote on September 16 to Surgeon-General v. Wiebel : 
' I recommend this good boy, my dearest treasure, on whose 
education I have expended my best energies, to the fatherly 
care of one who is so valued for his goodness/ The father 
had to guarantee a monthly allowance of 185. during his son's 
term of study, to be paid quarterly, in advance, to the account 
of the Institute, while the student was bound after his five 
years' education at the King's expense to serve as surgeon 
to a company or squadron for eight consecutive years. 

And so Hermann Helmholtz, permeated with a thirst for 
knowledge, and inspired with a deep love for natural science, 


to which his future was to be consecrated, was launched into 
a new life. Happily for himself, and to the great advantage of 
the world of science, his education had not been one-sided. 
By reason of his natural endowment, and thanks to the cease- 
less efforts of his parents, whose intellectual standard was ever 
set to high ideals, he was filled with passionate enthusiasm for 
music and poetry, as well as for art and science. 




IN BERLIN: 1838-1842 

ON his arrival at the Royal Friedrich-Wilhelm Institute 
(October, 1838), Helmholtz gave his parents a brief description 
of the strictly regulated conditions of his new life : 

' I got here safely on Friday. My things arrived shortly after. 
The servant and the porter made difficulties at first on account 
of the piano, as there was no place for it in my quarters. 
The room next this is intended for two, and has ample space. 
Accordingly I deposited it there, and said that Surgeon-Major 
Grimm had given me leave to bring it. The place is fairly roomy 
for the two of us ; it is up two flights of stairs at the end of 
the building opposite the entrance, so that I have to go half the 
length of the Hoditzstrasse to reach the street. The room has 
one inconvenience the three students who live in the next 
rooms invariably pass through it, although this is forbidden, and 
they ought to go across the yard ; but it can't be helped. It 
would be hard for them if they wanted to call the servant to 
have to go down two long flights of stairs, and then all the way 
up again. In order to make this plainer I will draw you a little 
plan. . . . My room-fellow is the son of a Silesian engineer ; 
he has already been one half-year at the Academy, that is to 
say, has attended classes and lectures, but did not lodge or take 
food there. He has extraordinary execution on the piano, 
but only cares for florid pieces and for modern Italian music. 
A few other fellows have also been coming to our room, as 
they had sent away their hired instruments during the 
vacation, but we hope this will stop now. Frau v. Bernuth 
has so far fed me sumptuously, so much so indeed that often 
I can hardly get up the two flights of stairs to my room. Each 


time I leave the table she tells me everything I have done 
amiss, and flatters herself I have improved a little already. 
There are a few new pictures at the Exhibition, but they 

are not worth much, the only one I care for being a Jephtha 

We have not yet got our plan of studies. As soon as I know 
a little more of the real life here I will write to you again. 
I am so far unpleasantly conscious of separation from you, 
as everything has to be paid for, and the senior students who 
come in pretty frequently to inspect the freshmen (Fiichse) rob 
us of nearly all our leisure moments. . . .' 

On November 2 he received a letter from his father, full of 
good advice, and anxiety for the hope and joy of the family : 

'Dear Son! We were very glad to learn from your letters 
that you have arrived safely, and have received your things; 
your mother could hardly wait until a letter came from you; 
she was positively ill from her anxiety for news. Your room 
is not far from that in which I passed my own years at the 
University; my windows looked on to the Friedrichstrasse, 
and were above the gateway nearest to you. May you be as 
happy in your abode, and enjoy as many happy moments of 
a higher life there, as fell to my lot ! Your first disagreeable 
reception as a "fox" was only to be expected no one is let 
off but you may comfort yourself with the reflection that it 
is the last time you will have to go through it, and if you 
take it wisely, and hold your own, it will soon be over. I only 
hope your comrade is a stout-hearted, industrious lad ; if he is, 
it will be great luck for you. His playing the piano so well 
is your best chance of improving yourself, and do not be so 
accommodating as to leave all the playing to him because he 
does it better than you, for it was under similar circumstances 
that I forgot all I ever learned : and, above all, don't let your 
taste for the solid inspiration of German and classical music 
be vitiated by the sparkle and dash of the new Italian extrava- 
gancesthese are only a distraction, the other is an education. 
Be thankful for Cousin Bernuth's lessons, even if they are given 
somewhat crudely ; behind these conventional social forms 
there lies in reality a deeper meaning, which is forgotten 
though it is still there, so that the forms help people to get 
on in society ; to give them life so that they cease to be empty 
form and convention is the task of the individual. . We are 


all well, and all love you dearly, and hope you will still, as ever, 
be our delight and pride. Be good, and devote yourself 
seriously and whole-heartedly to your profession, to science, 
and to virtue. Write as soon as possible to describe your 
studies and your every-day life. We shall be delighted to 
see you, provided your work allows it. If your companion 
is a good fellow, and you think it would improve your rela- 
tions with him, you can bring him with you later on.' 

The young student soon accommodated himself to his new 
surroundings, and affectionately reassures his parents, who 
were anxious about his food and lodging on account of his 
weak health : 

' I am well. The classes have commenced to-day, and 
regular work, which we hope will bring more quiet to our 
room. So far these visitors have been rather unwelcome, 
especially when I was practising, as they often expected 
me to play valses, &c., for them. At last I would not do it 
any more, and got my chum to play, while they sometimes 
danced till, as Dr. Knapp told me, the company-surgeon below 
complained of them. I have not sought their company very 
much, so that K. tells me I am called unsociable. He advises 
me to be patient, and says he also had to put up with the 
seniors coming to his room and playing there (which is for- 
bidden in the Institute), though he and his friend did not play 
with them. My room-mate is good-natured, but not exactly 
clever, as I see by his note-books, from which I wanted to 
fill in my own, as I was not able to take notes myself to-day, 
and also missed the first lecture on Splanchnology. I went 
with him, as he has already been studying a whole half-year, 
and he took me to the lecture-room in the University, where 
Professor Schlemm generally lectures, and where on the time- 
table at the door we saw " Mondays, 9-10, Prof. Schlemm ", 
and a number of students already waiting. However, as we 
did not find the others from the Institute, he went round to 
make inquiries, and I waited for him near the notice-board, 
but he lost himself in the crowd, and never reappeared. I went 
back to the lecture-room, where more students had collected 
meantime, but the Professor did not turn up. At last we went 
to the anatomical theatre, behind the Garrison Church, and 
heard that Professor Schlemm was really lecturing there. As 


the lecture was almost over it was no use going in, so I looked 
at the subjects which had been brought for dissection, and had 
been partly cut up. I did not experience any disagreeable 

' We have forty-eight lectures in the week : six on Chemistry 
in Mitscherlich's house, six on General Anatomy, four on 
Splanchnology, three on Osteology, three on the Anatomy 
of the Sense-organs. All these in addition to the Osteology 
at the anatomical theatre. Then four on Physics by Turte, 
two on General Medicine with Hecker at the University, two 
on Logic by Wolf in the anatomical theatre! three on History 
by Preuss, two on Latin by Hecker, one on French by Pastor 
Gosshauer in the Institute. Besides these we have twelve hours 
of revision classes, but these only begin in a fortnight's time. 

' You must not be afraid that I shall give up my music, for 
the new style my comrade admires so much does not satisfy 
me, and I am obliged to play myself to hear anything better. 
Besides which, other people's expression and execution seldom 
satisfy me; I always care much more for music when I am 
playing it myself. The food in the Institute is not so bad 
as most people make out, though less good than in a private 
home. We can have two helpings of soup and vegetables, 
but only one of meat. Or instead of vegetables we may have 
sauce over the meat, with potatoes. Dr. Grimm has been in 
to see if everything was right ; he came to me, and asked about 
the classes, what I thought of the food, and so on.' 

In spite of these good accounts the anxious mother sent her 
husband off to Berlin to make sure of their son's welfare, and 
only after that does she write cheerily to her boy that 'all 
beginnings are hard', and crack jokes with him in her usual 
merry fashion. 'Wilhelm Wilkens,' she says, 'was here the 
other day, to ask what father found when he went to see 
you. He hurried along to school with him without speaking; 
but father understood, and gave him all the news of you. 
Oh you dumb, self-contained, reserved creatures! Unless you 
alter, people will have nothing to say to you. . . . Write to me 
of your classes, your chilblains, your tempers, your discontent, 
and the good things that happen to you. God grant you may 
do the right, and leave the wrong undone/ 

Young Helmholtz devoted himself impartially to the study 


of physics, chemistry, and anatomy, and worked hard to acquire 
the necessary knowledge of these subjects from books and 
lectures, but in leisure moments his thoughts always turned 
to his home, and notwithstanding his occasional Sunday visits 
to Potsdam, he was wont in any passing fit of depression to 
disburden himself of his thoughts and feelings by writing 
to his anxious parents : 

'Since I was with you work has begun in earnest. The 
revision classes, including two in osteology, have all started, 
and we often have to sit through the evening learning one 
muscle after another till our heads split. It is easier to me 
than to the others, but even I have had an attack of chagrin 
against God and the world, such as every one here is subject 
to occasionally. But it generally goes off in a few hours, and 
our youthful ardour reasserts itself. Any spare time I have 
during the day is devoted to music, and so far, even on the 
worst days, I have put in about an hour, and more on Friday, 
Saturday, and Sunday. By myself I play sonatas of Mozart 
and Beethoven, and often with my chum the new things he 
gets hold of, which we run through at sight. In the evenings 
I have been reading Goethe and Byron, which K. borrowed 
for me, and sometimes for a change the integral calculus. 

'The day after I went to Potsdam I received an invitation 
from Geheimrath Langner, to whom Mrs. Wilkens gave me 
an introduction. I met several young people there, mostly 
law students, but they made us play whist! Happily one of 
the players in my rubber knew as little as I, and the others 
hardly more. It was a fine game, and a fine mistake too, to 
set us down to whist. It lost me the chance of making friends 
with the young people, among whom was a sailor, just back 
from North America. Aunt Bernuth was much amused at it ; 
she has presented me with a pair of gloves, which come in very 
handy this weather. Every morning we have an anatomical 
revision class in an unwarmed room, and going across to the 
dissecting-room is a treat without one's cloak! Our rooms 
have been rather better during the last few days, as we have 
twice had a fire ; before that it was so cold that one could not 
write at all, and could hardly play/ 

After spending Christmas at home, and working industriously 
through the second half of the first term, Helmholtz returned 


to his parents and brothers and sisters for a longer holiday at 
the close of the session. 

' We are through with all our lectures now except Mitscher- 
lich's, as he only breaks up next Saturday. So I must stay 
here all this week, and see how I can pass the time ; till now 
I have employed it in reading Homer, Byron and Biot, and Kant ; 
I am a little out of touch with all these studies, especially the 
last, and need to apply myself to them again. That once done, 
they fascinate me only the more. In particular I could hardly 
tear myself away from Homer, and devoured two or three 
books one after the other in an evening. I shall be with 
you next Saturday or Sunday/ 

In the second term Helmholtz began to feel more at home 
in his quarters; his studies assumed a serious aspect, and 
he became especially engrossed in Johannes Muller's lectures 
on physiology. In his leisure moments he studied Kant, and 
the Second Part of Faust, and having been appointed assistant- 
librarian of the Institute, found opportunities of enlarging his 
knowledge by means of the more recondite works contained 
in the library. In April, 1839, he writes to his parents : 

' Two important changes have been made in our section : my 
comrade has moved into the next division above, where some one 
has left, and another, who is bored with the Institute, has asked 
for and obtained his discharge ; so that there were two vacancies 
for University students. Meantime I have become acquainted 
with several good fellows, and did not intend to trust to luck 
a second time in the choice of a companion, so I proposed 
to the worthy Konigsberger, who has improved very much 
during the term, that we should join forces. We could either 
have occupied my room or his. In order to escape the passing 
through, and to have more space, I went to him, his room 
being really intended for three men. So now I am living in 
the third room of the wing in which I formerly inhabited 
the first, and can use the right of way. My present chum 
is a lanky fellow, unskilful and untidy in all mundane things, 
but good-natured, conscientious, and talented. He has a 
vast memory, e.g. last term he amused himself by learning 
the Hecuba of Euripides by heart in odd quarters of an 
hour at anatomy; he makes metrical translations from the 
English, and from Euripides his favourite tragic author, and 


paints moonlight landscapes with body-colour; in fact he is 
altogether rather sentimental, especially when he is reading 
aloud and playing the flute, in which, however, he does not 
excel, since he has no idea of time. He is the one in the 
division who has his work most at heart, and gets into dis- 
cussions, although he is tolerably orthodox and he has some 
curious notions about art. Another advantage is that my room 
is no longer crowded with the people who were attracted by 
the playing of my former comrade. 

' I am one of the assistant-librarians for this term. It loses me 
some two hours weekly, but is the only way to find out what 
there is of any value in the library, among the endless heaps 
of antiquated literature. 

1 We are expected to go to forty-two lectures a week in the 
summer term. According to the time-table (which, however, 
only accounts for thirty-nine, as they left out History) we have 
only one lecture from four to five, or five to six, on the three 
first afternoons of the week ; the three last are free. But most 
mornings we go straight on from six till one. Mitscherlich's 
Zoo-chemistry is a new lecture. We have six hours Botany 
and six of Natural History with Link, six of Physiology with 
Miiller, six of Chemistry, six of Zoo-chemistry with Mitscherlich. 
The house lectures are three on History with Preuss, two 
Latin with Hecker, one French with Gosshauer. Revisions 
four in Chemistry, three Physiology (with Herr v. Besser, who 
sits opposite Klotz at Muller's lectures), two Osteology, one 
Botany. No Logic or Psychology, nor does Link ever lecture 
on Mineralogy, although we have to take all these subjects in 
the first examination/ 

In spite of the many lectures and necessary study which 
these involved, Helmholtz found time to enjoy a splendid per- 
formance of Euryanthe, and to admire Seydemann's Mephisto- 
pheles, and Clara Stich as Gretchen in a representation such as 
he had never before seen of Faust. His time became more and 
more filled up, for all the free afternoons were struck out on 
account of the many lectures. Muller's physiology pleased 
him immensely, and Mitscherlich's zoo-chemistry also interested 
him, the experimental chemistry being, as he says, ' chock-full, 
but the least bit tedious/ Link, however, seemed to him to 
suffer ' from a superabundance of intellect ; after two months' 

c 2 


lectures on Natural History he is still at the philosophical 
introduction (good heavens !) '. And along with all this, 
Helmholtz was taking fencing and swimming lessons, so as 
to come up to the standard required from a student, hoping 
thereby to be rid of his savings, ' since otherwise the confounded 
spring will strew them to the winds/ 

In December Helmholtz passed his tentamen philosophicum. 
He writes : 

1 1 got through the examination for the philosophicum all 
right yesterday, and received a good certificate. The report 
in chemistry was excellent, in physics, psychology, zoology, 
and botany very good, in mineralogy pretty good. This last 
is the best Weiss gives as a rule ; at least I have been told that 
he said no more for examinees who knew a great deal of 
mineralogy. Of us four I had the best certificate, and Kunth 
congratulated me on it as he handed it to me. Even if the 
examination requires not so much special knowledge as a 
bird's-eye view of the whole, it has its uses as an incentive 
for going deeper into the sciences, and becoming interested 
in them/ 

Directly after this, on December 12, he demonstrated one 
of his own anatomical preparations of the peritoneum at 
a students' meeting. It was neatly done, and explained in a 
capital lecture, which was highly commended by Dr. Frost, 
a London botanist who happened to be present. 

The end of the year brought anxiety for the health of his 
dearly-loved mother, but she was able to join in the family 
festivities at Christmas, and he utilized the rest of the winter 
term and the Easter holidays in preparation for the clinical 
lectures given during the summer. 

The hospital work again tried his health severely, and on 
August 25, 1840, he received ten weeks' leave for a journey to 
Silesia, Prague, and Dresden. 

At the beginning of the winter term of 1840 Helmholtz took 
his anatomical examination, the prospect of which was alarming, 
though he was well prepared for it. On October 30, however, 
he announces the successful issue of the two days' examination 
to his anxious parents, with the news that both his tests had 
been got through without comment from the professors : 

' I still have to make an anatomical preparation, but this can 


only mean a higher or lower mark. W. and F., who were 
a term ahead of me, have also been in now; and had the 
same good fortune. Although none of us ever went up to an 
examination with better consciences than to this anatomy, we 
were all on thorns over the demonstrations, especially in the 
first public one, where we each had to give a full description 
of the tissues in one of the body-cavities, selected by lot, before 
a crowd of other students, who all arrived on the scene owing 
to some alteration of arrangements in the examination. The 
examiners, M tiller and Gurlt, sat there they yawned and 
looked horribly bored. The second demonstration, on a pre- 
paration of bone, and one of intestine preserved in spirit, 
came off before the examiners alone, when they were even 
more bored, and only too pleased if the candidate in his haste 
omitted something. At the end we only regretted that we had 
spent so much labour over our anatomy, and tried to reassure 
the crowd who are still trembling at the idea of it.' 

As soon as the second part of the examination was over, 
Helmholtz was free to plunge into the independent scientific 
work he thirsted for. His visits to Potsdam became less 
frequent, the letters to his parents fewer, and he was already 
considering the subject of his doctor's thesis. The winter 
session of 1840-1, and the summer of 1841, were devoted to the 
extension of his knowledge on all sides, more particularly in 
mathematics and recondite branches of mechanics. Still he 
always found time and opportunity to take part in amateur 
theatricals with his friends, and he followed the growth of 
national feeling and consequent political developments with 
keen interest. The witty lampoons and satires with which the 
Berliners revenged themselves for their deluded hopes, after 
the accession of Frederick William IV, were a perpetual source 
of amusement to him. 

Hardly, however, had he embarked on the anatomical and 
physiological researches that were to serve for his doctor's 
thesis, when he was prostrated for several months by a severe 
attack of typhoid fever. He was able to return to Berlin in the 
winter of 1841, and once more took up the question suggested 
to him (that is, in a general sense) by his master Johannes Miiller. 
From this time he lived entirely in the circle of M tiller's pupils, 
since he had already formed a friendship with the physiologists 


Briicke and du Bois-Reymond, who were two years senior to 
himself, and like him devotedly attached to their teacher. 
Friendly intercourse with each other, and daily exchange of 
ideas with the great investigator, who showed them ' the 
working of the mind of an independent thinker', ennobled 
their lives and efforts. ' Whoever/ said Helmholtz half a cen- 
tury later, ' comes into contact with men of the first rank has 
an altered scale of values in life. Such intellectual contact is 
the most interesting event that life can offer/ 

Miiller's pupils were united in a common attempt to connect 
physiology with physics, and to place its conclusions upon 
a more exact basis, but, as they often confessed at a later period, 
Helmholtz had a decided advantage over the rest, since in 
mathematics he gained a tremendous power in the clear for- 
mulating of problems and the precise determination of appro- 
priate methods of solution. Yet his wealth of mathematical 
knowledge had been won by private study of the works of the 
great mathematicians; for among all the different lectures he 
attended there was strangely enough not one on this subject, 
while he said so little about his mathematical learning that even 
his close friends Briicke and du Bois-Reymond were unaware 
of it. The time had not yet come when he was to dominate the 
problems of physiology and physics as one of the greatest of 

Miiller had indeed emancipated himself from the earlier 
and essentially metaphysical views of the nature of life, and 
demanded an empirical foundation for all scientific concepts, 
but, as his pupils recognized, he had been unable to free him- 
self entirely from the traditions of nature-philosophy and from 
metaphysical conceptions. Under his influence Helmholtz 
endeavoured to lay the foundations of a strict physical science 
by ascertaining the facts in certain definite problems, thereby 
co-operating with the ceaseless efforts of his master. 

And with how modest an equipment Helmholtz set about 
his colossal investigations! During his illness in the Charite 
Hospital, where he was nursed free of charge as a student, 
he saved enough of his little monthly allowance to procure 
a small and very ordinary microscope, and it was with this 
instrument, supplemented by a few antiquated textbooks of 
physics and chemistry, that he attacked his task. 


He passed his oral examination at the end of June, but his 
hopes of receiving the doctor's degree in the summer term 
were disappointed. 

On August i, 1842, he writes to his father : ' I went to-day 
to Professor M tiller with my thesis. He received me very 
kindly, and after inquiring what my conclusions were, and 
on what evidence they were based, declared the subject to 
be of the greatest possible interest, since it proves the origin 
for nerve-fibres that was conjectured in the higher animals, 
but had never been determined. But he advised me to work 
it out upon a more complete series of animals than I have at 
present, so as to get more cogent proof than is possible from 
the examination of three or four only. He mentioned several 
animals as being the most likely to yield good results, and 
invited me to use his instruments at the Anatomical Museum 
if my own were inadequate. If I were not obliged to hurry 
in taking my degree, he advised me to employ the vacation 
for further work, so as to produce a fully-developed thesis 
that need fear no future attack. As I had nothing reasonable 
to urge to the contrary, and had said most of the same things 
to myself already, you will have to give up your twenty-year- 
old doctor, and content yourself with having him at twenty-one. 
If this distresses you too much, send me a line, and I will 
translate the discourse I delivered here at Easter, at the 
Institute, and shall be doctor next week. The Potsdam 
worthies will presumably conclude that I have failed in my 
examination, and those in Berlin that I wanted to get oft 
the doctor's banquet, but they shall all be satisfied sooner 
or later. I was rather surprised myself, and did not like 
the delay, but, as I said, I can find no good reason against it.' 

After a four weeks' tour in the Harz Mountains he returned 
to Berlin on September 30, and was appointed house-surgeon 
at the Charite. 

It was an arduous post, unsatisfactory 'on account of the 
tedious and for the most part incurable diseases ', and occupied 
him from 7 a. m. to 8 p. m., with only short breaks of an hour, 
or even a half-hour ; but the work was congenial and instructive, 
and Helmholtz found time to follow the advice of his teacher, 
verifying and extending his previous researches with the aid 
of the instruments provided at the Anatomical Museum. 


' I am working hard at my thesis. Once I thought I had 
arrived at a conclusion of the utmost importance ; on closer 
observation the day before yesterday I found the contrary, 
and then on looking into it again more carefully yesterday 
discovered that the first idea was right with certain limita- 
tions. To-day I am going for the point more carefully/ His 
only recreation from this labour was the Art Exhibition, and 
he greatly admired Lessing's Huss, l a picture that is perhaps 
worth more than all the earlier exhibitions put together; at 
any rate here in Berlin we have no picture so profound, so 
inspired, and so characteristic. Every one is charmed with 
it, except the Berlin professors/ 

M tiller ultimately declared the thesis to be satisfactory, and 
Helmholtz took his doctor's degree on November 2, 1842 ; his 
Inaugural Dissertation, De Fabrica Systematic Nervosi Everte- 
braforum, or 'The Structure of the Nervous System in 
Invertebrates', being dedicated to Johannes Miiller. His 
microscopical discovery, that the nerve-fibres originate in the 
ganglion cells discovered by Von Ehrenberg in 1833, has been 
recognized by all physiologists as the histological basis of 
nervous physiology and pathology: the connexion till then 
sought in vain between nerve-fibres and nerve-cells, and there- 
with the proof of the central character of these cells, was 
established by him for invertebrates in this first-rate contribution 
to minute anatomy. 

As soon as Helmholtz had overcome the initial difficulties of 
his post, he devoted himself gladly to his calling, but managed 
to find time to pursue the many-sided studies of the previous 
year, and develop them profitably. He became so absorbed 
in his work that he could not tear himself away, and for the 
first time failed to convey his birthday congratulations in person 
to his father. In writing, he refers the older man (who cared 
little for practical realities) to the ideals of a brighter future, 
but his own scientific thoughts were turning more and more 
away from the metaphysical, idealistic views by which science 
was then held in bondage. He directed his hopes entirely 
to the real world, the world of fact, thus laying the foundation 
of the mighty structure which was to be erected in the second 
half of the nineteenth century. 

The theoretical methods which obtained in that period of 


the development of medicine were fast approaching their end. 
The methods were rejected, and the facts as well, and it was 
seen that medical science, like all the other sciences, had to 
be reconstructed. After physics and chemistry first shaped 
themselves on scientific lines in France in consequence of the 
epoch-making labours of Coulomb and Lavoisier, Mitscherlich 
and Liebig established chemistry as a science in Germany, 
while Ohm, Franz Neumann, Gauss, and Wilhelm Weber 
built up methods of experimental and mathematical physics 
upon a solid foundation. Yet it required a titanic labour 
to transfer these principles of methodic investigation from 
inorganic to organic Nature. 

After Ernst Heinrich Weber had demanded that vital 
phenomena should be explained in terms of physical processes, 
Johannes Miiller endeavoured in all his physiological work 
to clear the way for inductive methods of investigation, and 
to push deductive reasoning and metaphysical conceptions 
more and more into the background. But he could not 
emancipate himself from the notion of a separate, individual, 
vital force, distinct from the chemical and physical forces 
working within the organism, and capable of binding and 
loosing the action of the same. This is only abolished by 
death ; the forces which it restrained are then set free, and 
produce corruption and putrefaction; the vital force has vanished, 
and is not replaced, nor converted into any other perceptible 
form of energy. Miiller did not attempt to disguise the in- 
consistency of his position, and as a result the four gifted 
young investigators, Briicke, du Bois-Reymond, Helmholtz, 
and Virchow, were all striving to develop a logical and unified 
physiology according to the principles of exact investigation. 
Each sought to abolish the notion of vital force from the 
department of physiology which he regarded as his own, and 
to cultivate physiology as a branch of physics and chemistry. 

But, in the mind of Helmholtz, the conflict between realistic 
and metaphysical principles had become a resolute fight 
against the dominating ideas in a wider field than physiology : 
a vanishing vital force for which nothing was substituted 
appeared to him a physical paradox a disappearance of 
energy and matter was unthinkable. He had never heard 
a mathematical lecture, but the Pepiniere possessed the works 


of Euler, Daniel Bernoulli!, d'Alembert, and Lagrange, mathe- 
maticians of the previous century, and arming himself with 
a small textbook of higher analysis he had plunged even as a 
student, during the time of his assistant-librarianship at the 
Institute, into the fundamental investigations of these great 
mathematicians penetrating into the significance of the prin- 
ciples of mechanics, as laid down by these immortal thinkers, 
with which such metaphysical views were incompatible. 

The time was not yet ripe for bringing forward these 
universal and comprehensive ideas ; Helmholtz had learned 
from the stern methods of Johannes Muller that only definite 
and methodical experiment could make the general principles 
of science intelligible, and set them on a sure foundation. As 
soon as his doctor's examination was over, he applied himself 
in M tiller's laboratory to a problem which, on account of 
Liebig's work, was then in the forefront of interest. Helmholtz 
attacked it on far wider grounds, with the intention of bringing 
the so-called 'vital forces' within the scope of scientific study. 

Liebig was engaged in a fierce campaign against the organ- 
ized nature of yeast, as discovered by Schwann and Cagniard- 
Latour, and its role in alcoholic fermentation, and upheld the 
essentially chemical theory of fermentation and putrefaction, 
based on Gay-Lussac's experiments. Helmholtz immediately 
recognized the crucial importance of this question, and its 
close connexion with the possibility of perpetual motion, and 
set himself to decide the point. The economies of his winter's 
illness had provided him not only with a microscope for the 
morphological investigations described in his thesis, but also 
with the recently published Organic Chemistry of Mitscher- 
lich ; and in the early months of the year 1843, when he was 
not much hampered by his duties in the children's ward, he 
plunged into, extended physico-chemical investigations which 
henceforward absorbed him. 

In this arduous and exhausting work he had to trust entirely 
to himself, as he knew little of even the most important 
publications on the subject. On July 25 he applies to his 
father to procure him the necessary literature: 'Could you 
be so good as to borrow of Professor Meyer, or through his 
intermediation, by next Sunday afternoon, that treatise of 
Mitscherlich on Fermentation which you mentioned the other 


day, or to ask him where I could meet him to hear the results 
of these experiments, if he is unable to lend it?' and in spite 
of the heavy work of the summer months on the out-patient 
post at the Charite, he adds, ' I am now so far advanced with 
my experiments that I am ready to begin writing, and it is 
essential to acquaint myself with the work of Mitscherlich/ 

The main object of this paper, published in Mullens Archiv 
in 1843, with the title 'On the Nature of Fermentation and 
Putrefaction ', was to support Liebig in his attack on vitalism 
by proving that there can be no such thing as spontaneous 
generation. Helmholtz found, however, that the transforma- 
tions known as fermentation and putrefaction are not the result 
of chemical action, as supposed by Liebig, and therefore due 
to the action of oxygen, or the introduction of residual dis- 
integration products of the putrefying substances. He showed 
(and the clear and precise wording of his results is of especial 
interest in view of the great discoveries of Pasteur at a later 
time) that putrefaction can occur independently of life, but 
that it offers a fertile soil for the development and nutrition 
of living germs, and is modified in its aspects by them. 
Fermentation is one such putrefactive process modified by 
organisms, and correlated with them ; it strikingly resembles 
the vital process in the similarity of the substances it attacks, 
in its rate of growth, and the similarity of the conditions 
determining its continuance or check. 

These observations, which Helmholtz was unable to pursue 
from the inadequacy of the means at his command, seemed 
actually to give fresh support to vitalism, and his experiments 
were accordingly regarded with suspicion, above all by the 
physicists. Magnus, indeed, whose generous liberality knew 
no scientific jealousy, invited him to make use of his private 
laboratory, and to take advantage of ' methods of investigation 
that would throw more light on the subject than such as a 
young army surgeon living on his pay could provide for him- 
self; but it was not until two years later that Helmholtz was 
able to convince him by a series of new experiments of the 
accuracy of his previous work. Nor did he publish anything 
further on the subject, which had occupied him almost daily 
for three months. He was already engaged on other and far 
profounder problems, the solution of which was to condemn 


that aspect of physiology against which he and Liebig had 
been contending, and to found an entirely new era in science. 

In the meantime Helmholtz, after receiving a prize in May 
for his excellent hospital work, was on duty in the eye wards 
during August; after which, on the recommendation of his 
superiors (who had long since recognized his merits), he was 
appointed assistant-surgeon to the Royal Hussars at Potsdam, 
under the regimental doctor Branco. His Government Ex- 
amination was not due for two years, so that he could reckon 
on a protracted period of leisure, not unduly interrupted by 
official duties, for the development of all the weighty thoughts 
that had pressed on him since the beginning of his student 
life, of which the investigations already accomplished were 
merely the preliminary verification. 

2 9 


As Hussar-Surgeon, Helmholtz was now obliged to forgo 
the scientific atmosphere in which he had lived, under the 
inspiration of Johannes Mtiller, in constant intercourse with 
such congenial intellects as his chosen friends du Bois- 
Reymond and Briicke. His peace-loving, retiring nature was 
harassed at first by the rude five o'clock awakening, when 
the bugler blew the reveille at his door to rouse the barracks, 
and interrupted his slumbers, but he soon grew accustomed 
to this, and attended with alacrity to his official duties. He 
arranged a small laboratory for physics and physiology in the 
barracks, where he was frequently visited by du Bois-Reymond 
and Brucke, who came out from Berlin to discuss their plans 
for the future reconstitution of physiology. Though restricted 
to the most elementary instruments (he constructed an elec- 
trical machine for himself, which he presented later on to his 
brother), Helmholtz was always backed up by advice and help 
from du Bois-Reymond, who, he writes, 'tended me like 
a mother, to enable me to attain a scientific position/ He 
plunged at once into his projected investigations on the meta- 
bolism in muscular activity, and embarked on a series of 
laborious experiments on the conduction of heat in muscle, and 
the rate of transmission of the nervous impulse. 

All was well with his parents ; he himself lived a quiet and 
retired life, wholly immersed in his work. His brother's 
friends have not yet forgotten the impression made upon them 
by the young man of twenty-three, who, with a contour of head 
and a manner inherited from his mother, was distinguished by 
an expression of absolute calm and intellectuality. All that he 
said gave an impression of truth and vigour. His extra- 
ordinary gift of observation excited the admiration of his 


friends; on every walk he discovered new things they had 
not seen. When watching the play and splashing jets of 
a fountain at Sans Souci he heard melodies and chords in 
the murmur which they were unable to perceive, even when 
he drew attention to them. His duties with the squadron left 
him much spare time, but he devoted every moment to science ; 
his brother tells us that he made use of the midday recreation to 
study the Fundamenta Nova Functionum Ellipticarum of Jacobi, 
and his much-read copy still shows traces of industrious work, 
and endeavour to render these difficult matters (with which 
even the mathematicians were unfamiliar at that time) clear to 
himself and applicable to physiology. 

But the family life in Potsdam reacted unfavourably, though 
not for long, upon the relation between father and son. The 
more the young man's thoughts, the direction of his labours, 
and his whole scientific attitude (which was so soon to be 
adopted by the entire world of science) took him away from 
metaphysical speculation, the stronger and for some time the 
more irreconcilable became the contrast with the wholly specu- 
lative philosophy of his father. While Ferdinand Helmholtz 
admitted only the deductive method in science, and held in- 
ductive reasoning to be inimical to it, Hermann on the contrary 
bore the latter upon his shield, proclaiming it to the end of his 
life the salvation of science in general and not merely of the 
physical sciences. The father (secure in the consciousness 
that he must be the better able as a philosopher to appreciate 
the relation in which man stands to experience, and with the 
best intentions in the world of directing his son, his 'dearest 
treasure ', into the right paths of scientific discovery) missed 
no opportunity in their daily intercourse of bringing his general 
philosophical convictions and metaphysical conceptions to bear 
upon the young man, doing all he could to shake him in his 
methods of thought and experiment. 

Helmholtz, who was already concentrating himself upon the 
experimental evidence that was to establish his world-famed 
Law of the Conservation of Energy, saw that no agreement 
in scientific investigation and method was possible upon such 
divergent grounds, and that it was wiser not to discuss his 
work with his father. The old man naturally felt this keenly, 
but at least it kept their domestic relations in good train, and 


the future soon gave him cause to rejoice over the famous son, 
by whom the family name was to become known throughout 
the world. 

Since Helmholtz was entirely thrown upon his own re- 
sources during these years in Potsdam, the need of scientific 
intercourse often drew him to Berlin to forgather with his 
great teacher Johannes Muller, and his devoted friends du 
Bois-Reymond, Briicke, and Ludwig (who was his senior by 
five years). The young men were all striking out new paths 
in their chosen sciences, but they willingly and ungrudgingly 
gave the palm to Helmholtz, as du Bois-Reymond and Brucke 
loved to relate in after years. 

Nor was it long before Helmholtz made his mark in a wider 
circle. Mailer's distinguished pupils at Berlin had become 
acquainted with other students in physics and chemistry, at 
the informal classes of their master, Gustav Magnus, and in 
1845 tne Physical Society had been founded by du Bois-Rey- 
mond, Brucke, Karsten, Beetz, Heintz, and Knoblauch. Du 
Bois introduced young Helmholtz to the Society, where he was 
warmly welcomed as its greatest ornament, and for over ten 
years contributed reports to the Fortschritte der Physik on 
certain departments of physics and physiology. 

The riddle of the existence and nature of vital force the 
question whether the life of organisms was the effect of one 
special, self-engendered, definitely directed force, or merely the 
sum of the forces that are effective in inorganic nature also, 
modified only by the manner of their concurrence such were 
the questions raised again and again by Muller, and trans- 
formed by Liebig into the far more concrete problem ot 
whether the mechanical energy and the heat produced in an 
organism could result entirely from its own metabolism, or not. 
Helmholtz soon perceived that all these questions were inti- 
mately connected with the validity of that law of the Conserva- 
tion of Energy which had for years seemed incontestable 
to him; but it was necessary to prove the accuracy of his 
mathematico-physical propositions by a vast number of experi- 
ments in different regions of physiology and physics, before he 
could hope to see the principle admitted by science. He 
began, therefore, in 1845, by testing the accuracy of his 
physical conceptions upon a highly complex physiological 


problem, in a paper published in Mutter's Archiv with the 
title, ' Metabolism during Muscular Activity/ 

Lavoisier had already shown that a man who is doing work 
requires more oxygen than a man at rest, but while it was 
admitted that certain ponderable or imponderable substances 
were consumed in the production of mechanical effects, and 
were perpetually renewed by the vegetative vital processes, 
and also that the amount of certain excreted nitrogenous 
matters was increased by muscular activity, all data in regard 
to the initial and intermediate steps of the process and the 
seat of its occurrence were wanting. Helmholtz accordingly 
set himself to investigate the modifications produced in 
the chemical constitution of muscle by its own activity. 
Resorting to the frog, 'that ancient martyr to science/ he 
succeeded by means of the little electrical machine constructed 
by himself, and of a Leyden jar, in showing that the compo- 
nents within a muscle undergo chemical transformation during 
its activity in virtue of the chemical processes described in his 
account of fermentation and putrefaction ; and these conclu- 
sions led him after protracted experiment to a series of 
important results, which were for a long time the only exact 
data in connexion with this subject. Yet, as he was per- 
petually studying the inter-connexion of all these problems, 
and their relation with the great law that governed his imagina- 
tion, Helmholtz soon perceived that before he could obtain 
exact data as to metabolism, it would be necessary to ascertain 
the relations between muscular action and the heat therein 
developed. This, however, entailed a fresh series of experi- 
ments, which had to be left over for a later period. 

At Michaelmas he was to go up for the Government Examina- 
tion at Berlin, and only had time before it to write an article 
for the Encyclopaedia of Medical Science (issued by some 
members of the Medical Faculty of Berlin), entitled ' Heat, 
physiological/ which comprised the most recent observations 
on animal heat. In this he strove to bring order into the 
distorted conceptions then prevailing as to the nature of heat, 
and the essay is distinguished by that grasp of the historical 
development of the most heterogeneous branches of natural 
science, for which Helmholtz was afterwards so generally 


After giving an historical and critical review of the facts that 
relate to the regulated high temperature of the more perfectly 
organized animals, which persists throughout life, and only 
disappears with its extinction, and a table of the differences 
of temperature in different kinds of animals, he investigates 
the origin of animal heat. In a very interesting discussion 
of the different views that have been held as to its nature, 
Helmholtz states the most important results of the prevailing 
theories in the form of the law that is fundamental to organic 
heat, viz. that the total heat liberated by the union of two 
or more elements to form the same compound must be the 
same, whatever the intermediate processes through which 
the system may have passed. But since quantity of sub- 
stance can neither be increased nor diminished in nature, it 
follows from the accepted theories of heat that its quantity 
in nature must be an absolute constant, whence he concludes 
that the actual temperature of an organism can only be due 
to a supply of heat from without, either free or latent, and, 
inasmuch as sources of free heat exist only in exceptional 
cases, organic heat must necessarily be derived from the latent 
heat of the food. The work that had been done on pro- 
duction and loss of heat, and on animal metabolism, justified 
the conclusion that the materials supplied to the body in 
respiration and digestion provide the entire sum of vital warmth 
during the successive stages of their combination within the 
body. According to Helmholtz the only alternative is to admit 
that organisms are the seat of a special force (the so-called vital 
energy), by which the forces of nature can be engendered ad 
infinitum a hypothesis that contradicts all known laws of 
mechanical science, but cannot theoretically be objected to, 
if physiologists choose to assume that such an incomprehensible 
phenomenon is the distinguishing characteristic of the life- 

In October, 1845, Helmholtz received six months' leave, in 
order to pass the State Examination in medicine and surgery, 
during which time he was attached as surgeon to the Friedrich- 
Wilhelm Institute. By the terms of his engagement, he was 
subsequently bound to serve twice this period as surgeon to 
the King's army. On February 7, 1846, he returned to his 
official duties with full credentials. 


The five months of his stay in Berlin were spent in hard 
work, and in close scientific intercourse with his friends du 
Bois-Reymond and Briicke, who, when separated later on, 
communicated all their scientific projects, and the results of 
their work, to each other in writing. This was the first 
Christmas that Helmholtz had not spent with his parents. 
He was doing his clinical tests at the Infirmary, and at 
the same time working steadily in the laboratory of Magnus 
at his researches on fermentation and putrefaction, while he 
regularly attended the meetings of the Physical Society. 

In January, 1846, he announces in a letter to his parents 
that he had done well in the examination as physician and 
surgeon, but did not pass as an * operator '. 

Immediately after his return to his military duties at Potsdam 
we find him busy again with his experiments on the heat 
evolved in muscular activity, and from this time he exchanged 
ideas regularly with du Bois-Reymond, partly in letters written 
about once a fortnight, partly at meetings between the two 
friends in Berlin or Potsdam. Up to July i, he was on duty 
in the field hospital, after which he was t happy again with 
leisure to experiment*. After satisfying himself 'with great 
difficulty as to the constancy of the frog-current between 
copper electrodes in a solution of copper sulphate*, he pro- 
ceeded to investigate the nature of the chemical processes 
which he had discovered in muscle. 

At the beginning of October, 1846, Helmholtz sent a ' Report 
on Work done on the Theory of Animal Heat for 1845', 
at du Bois' request, to the Fortschritte der Physik, issued 
by the Physical Society. This was merely an abstract from 
the article in the Encyclopaedic Dictionary above mentioned, 
but it anticipates more definitely the conclusions of his great 
work. He states without hesitation that the material theory 
of heat is no longer tenable, and that a kinetic theory must 
be substituted for it, since heat originates in mechanical forces, 
either directly by friction, or indirectly from an electrical 
current produced by the motion of magnets. This conception 
of heat as motion involves the conclusion that mechanical, 
electrical, and chemical forces must always be the definite 
equivalent of one and the same energy, whatever the mode 
by which one force is transformed into another. The empirical 


confirmation of this law must be the imperative duty of physicists 
and physiologists. 

The last months of 1846 were wholly taken up with experi- 
ments on the heat evolved during muscular action. After 
much discussion with du Bois-Reymond by word of mouth 
and letter as to the conversion of a thermo-multiplier of 
extreme sensitivity, by empirical graduation, into a thermo- 
meter for thousandths of a degree, Helmholtz begs the loan 
of the portable balance made for du Bois by Halske, 'in 
order to experiment on the ash of muscle and the composition 
of nerve with regard to their possible alteration by muscular 
contraction/ But though du Bois himself brought the balance 
to Potsdam, Helmholtz was compelled by stress of official 
duties to break off his experiments for a long time at the 
end of 1846. The New Year diverted his scientific enter- 
prises into another and wider field, and was also destined 
to be the most important of his life in aspects other than 

After the death of Surgeon-Major von Velten, his widow had 
moved from Riesenburg to Potsdam with her two daughters, 
in order to profit by the advantages of the intellectual society 
to which she was there introduced by her brother (a surgeon- 
major in the Hussars), and to obtain a good education for her 
children. Her husband was the son of that Cornet Velten of 
the Ziethen Hussars, who during the retreat at the battle 
of Kunersdorf came upon the King, standing alone on an 
eminence in the field, with his sword driven into the ground 
in front of him, facing death or captivity. Velten cut his way 
through, along with Captain von Prittwitz, and helped the King 
to escape on his own horse, for which he was ennobled, and 
received the order pour le merite. Frau von Velten was the 
daughter of the late Hofrath Puhlmann, Court Painter and 
Director of the Picture-Gallery founded by Frederick the 
Great. Helmholtz soon obtained an introduction to this dis- 
tinguished family, though ' at first he was somewhat of a foreign 
element '. 

His sister-in-law relates that he was ' grave and reserved, a 
little awkward and shy among other more lively and sociable 
young men. Some one made the characteristic remark, when 
Helmholtz was introduced to us, that he was a very clever man 

D 2 


if only you could dig him out, and that proved treasure-trove 
indeed '. Before long he was accepted as one of the household, 
which, he declared, seemed to him less a part of ordinary life 
than of some beautiful romance ; while his judgement in all 
matters was soon accepted as conclusive. He played the piano 
a great deal with the younger sister, Olga, who sang exceedingly 
well, displayed his talent for reading aloud, made pretty verses 
for the young ladies, and acted comedy almost like a professional, 
taking the humorous parts for choice, especially those with 
a touch of the grotesque. A still existing play-bill tells us that 
he took the chief role of Herr Petermann in a piece called 
Lodgings to Let, performed on December 27, 1846, at the house 
of Rigler, the Director of the Gymnasium. It is on record 
that Helmholtz gave up his time most amiably, and worked 
hard at the performance, although it was obvious from his 
acting that his mind was occupied with other and higher 
thoughts ; in fact, he was just then writing the Introduction to 
his memoir on the * Conservation of Energy '. l He was 
becoming/ writes his sister-in-law, * an inseparable part of our 
existence, and there was a ripening sense between him and my 
sister that their lives were bound up together. Olga was not 
beautiful, but she was refined and agreeable; she never put 
herself forward, but listened with attention and keen observa- 
tion. Her mind was alert, amusing, witty, almost sarcastic ; but 
there was about her a breath of femininity and simple purity 
that was irresistible/ 

The betrothal took place on March n, 1847, an d Helmholtz 
writes a characteristic letter to his bride-elect on a day when he 
had vainly expected her at a symphony-concert at the Sing- 
Akademie in Berlin. ' You did not come, and so my ear went 
wrong also. I fancied it must always have been your soul, 
with your deep musical perceptions, that had governed the 
harmonies in my brain. My ear heard only musical figures, 
and my soul heard naught. Needless to say this was in the 
Mozart Symphony, one of his finest, with which every one else 
was delighted. But forlorn as I was, bereft of the better half 
of my soul, I might as well have been listening to scales upon 
the piano. I only recovered myself in the Coriolanus Overture. 
That is a jewel so short, so convincing, so decided, and proud 
amid a host of restless and entangled motifs, while it dies off 


so sadly in melancholy strains ... an unsurpassable master- 

The wedding could not take place until Helmholtz received 
some permanent appointment, so the young man embarked 
courageously upon his great scientific enterprise. By the 
middle of February, 1847, ne nac * sent du Bois-Reymond the 
sketch of his Introduction to the ' Conservation of Energy'. ' Not 
because I think it is ready, for even in reading it over I see 
that most likely none of it can stand, but because I do not 
yet see how many times I shall have to rewrite it before it is 
done, and I want to know if you think the style in which it 
is written one that will go down with the physicists. I pulled 
myself together at the last reading, and threw everything over- 
board that savoured of philosophy, wherever it was not abso- 
lutely essential, so this may have made some gaps in my logic. 
Still you will be able to see the nature of the argument from 
it. Don't put yourself out to read it ; do it at your leisure, and 
then write to me : where you find obscurities or lacunae in the 
details note them on the margin. I may come to Berlin myself 
after a while to talk to you about it.' 

Du Bois received the Introduction with enthusiasm, and 
declared that it must remain as it was, f an historical document 
of great scientific import for all time.' 

It was during the first quarter of 1847, while he was on 
field-hospital duty, that the young investigator found oppor- 
tunity to formulate the ideas which he had cherished since 
the beginning of his studies, and had tested by experiments 
in the most disparate branches of physiology and physics with 
a view to publishing the results. Neither he nor his friends 
had any notion that other workers were engaged on the same 
problems. When free of the hospital post, Helmholtz at once 
resumed his experimental work, and reconstructed the apparatus 
for his thermal experiments on muscle, giving much good 
advice from his own experiences to du Bois about his experi- 
ments, while he ' waited impatiently for the spring and the frogs'. 
But once more he had to interrupt his experiments, since Halske 
kept him waiting too long over the construction of a Neefs 
interrupter for his electrical apparatus, till finally the moment 
for producing the ' Conservation of Energy ' before the world 


On July 21 he announced to du Bois-Reymond that he would 
bring forward his ' Conservation of Energy ' on the 23rd at the 
Physical Society. The meeting was one of the most memorable 
in the annals of the Society ; as du Bois tells us, Helmholtz 
revealed himself at one bound, to the surprise of all his friends, 
as a master of mathematical physics. The members of the 
Physical Society were acquainted with the Law of the Con- 
servation of Energy when it was still unknown to all the rest 
of the world. 

As soon as the meeting was over Helmholtz sent the manu- 
script to Magnus, with whom he was on very friendly terms, 
begging him to arrange for its publication in Poggendorff J s 
Annalen. But Magnus, though he willingly and cordially 
recognized the merits of Helmholtz's essay, took exception to 
the character of his work. He regarded experimental and 
mathematical physics as separate departments, and warned him 
repeatedly against undue partiality for mathematics, and the 
attempt to bring remote provinces of physics together by 
its means. Accordingly he only sent the memoir with 
a few words of general recommendation to Poggendorff, and 
this apparently not till he was constrained to it by du Bois- 
Reymond, who, with Brucke, had all the younger physicists 
and physiologists of the Physical Society upon his side. 
Poggendorff replied that the subject-matter was not in his 
opinion sufficiently experimental to justify him in publishing it 
in the Annalen, though he acknowledged its importance as 
a theoretical treatise. Both he and Magnus therefore advised 
the author to bring it out as an independent publication. Du 
Bois forwarded these letters to Helmholtz with forcible ex- 
pressions of his annoyance with Poggendorff and Magnus, 
urging him if possible to get the essay published by Reimer 
in Berlin as a pamphlet, and recommending him to restore 
the Philosophical Introduction. Helmholtz did not allow 
his satisfaction in the work to be destroyed by the reser- 
vations of the older physicists, but he altered certain 
parts of the Introduction in order still further to emphasize 
his position in regard to the prevailing scientific concep- 
tions. This Introduction was the foreword of modern 
science in the second half of the nineteenth century, while 
its fine and simple style already proclaim its author a 


master of language : hence it may be fitly cited in this 
connexion : 

'The principal contents of the present memoir show it to 
be addressed to physicists chiefly, and I have therefore thought 
it judicious to lay down its fundamental principles purely in the 
form of a physical premise, and, independent of metaphysical 
considerations, to develop the consequences of these principles, 
and to submit them to a comparison with what experience has 
established in the various branches of physics. The deduction 
of the propositions contained in the memoir may be based on 
either of two maxims ; either on the maxim that it is not 
possible by any combination whatever of natural bodies to 
derive an unlimited amount of mechanical force, or on the 
assumption that all actions in nature can be ultimately referred 
to attractive or repulsive forces, the intensity of which depends 
solely on the distances between the points at which the forces 
are exerted. That both these propositions are identical is 
shown at the commencement of the memoir itself. Meanwhile 
the important bearing which they have upon the final aim of 
the physical sciences may with propriety be made the subject 
of a special introduction. 

' The problem of the sciences just alluded to is, in the first 
place, to seek the laws by which the particular processes of 
nature may be referred to, and deduced from, general rules. 
These rulesfor example, the law of the reflection and re- 
fraction of light, the law of Mariotte and Gay-Lussac regarding 
the volumes of gases are evidently nothing more than general 
ideas by which the various phenomena which belong to them 
are connected together. The finding out of these is the office 
of the experimental portion of our science. The theoretic 
portion seeks, on the contrary, to evolve the unknown causes 
of the processes from the visible actions which they present ; 
it seeks to comprehend these processes according to the laws 
of causality. We are justified, and indeed impelled in this 
proceeding, by the conviction that every change in nature 
must have a sufficient cause. The proximate causes to which 
we refer phenomena may, in themselves, be either variable 
or invariable; in the former case the above conviction impels 
us to seek for causes to account for the change, and thus 
we proceed until we at length arrive at final causes which 


are unchangeable, and which therefore must, in all cases 
where the exterior conditions are the same, produce the same 
invariable effects. The final aim of the theoretic natural 
sciences is therefore to discover the ultimate and unchange- 
able causes of natural phenomena. Whether all the processes 
of nature be actually referable to such, or whether nature is 
capable of being completely comprehended, or whether changes 
occur which are not subject to the laws of necessary causation, 
but spring from spontaneity or freedom, this is not the place 
to decide; it is at all events clear that the science whose 
object it is to comprehend nature must proceed from the 
assumption that it is comprehensible, and in accordance with 
this assumption investigate and conclude until, perhaps, she 
is at length admonished by irrefragable facts that there are 
limits beyond which she cannot proceed. 

' Science regards the phenomena of the exterior world ac- 
cording to two processes of abstraction ; in the first place it 
looks upon them as simple existences, without regard to their 
action upon our organs of sense or upon each other ; in this 
aspect they are termed matter. The existence of matter in 
itself is to us something tranquil and devoid of action : in it 
we distinguish merely the relations of space and of quantity 
(mass), which is assumed to be eternally unchangeable. To 
matter, thus regarded, we must not ascribe qualitative differ- 
ences, for when we speak of different kinds of matter we refer 
to differences of action, that is, to differences in the forces of 
matter. Matter in itself can therefore partake of one change 
only, a change which has reference to space, that is, motion. 
Natural objects are not, however, thus passive ; in fact we 
come to a knowledge of their existence solely from their 
actions upon our organs of sense, and infer from these actions 
a something which acts. When, therefore, we wish to make 
actual application of our idea of matter, we can only do it by 
means of a second abstraction, and ascribe to it properties 
which in the first case were excluded from our idea, namely, 
the capability of producing effects, or, in other words, of 
exerting force. It is evident that in the application of the 
ideas of matter and force to nature the two former should 
never be separated: a mass of pure matter would as far as 
we and nature are concerned be a nullity, inasmuch as no 


action could be wrought by it either upon our organs of 
sense or upon the remaining portion of nature. A pure force 
would be something which must have a basis, and yet which 
has no basis, for the basis we name matter. It would be 
just as erroneous to define matter as something which has 
an actual existence, and force as an idea which has no corre- 
sponding reality. Both, on the contrary, are abstractions 
from the actual, formed in precisely similar ways. Matter 
is only discernible by its forces, and not by itself. 

'We have seen above that the problem before us is not to 
refer back the phenomena of nature to unchangeable final 
causes. This requirement may now be expressed by saying 
that for final causes unchangeable forces must be found. 
Bodies with unchangeable forces have been named in science 
(chemistry) elements. Let us suppose the universe decom- 
posed into elements possessing unchangeable qualities, the 
only alteration possible to such a system is an alteration of 
position, that is, motion ; hence the forces can only be moving 
forces dependent in their action upon conditions of space. 

' To speak more particularly : the phenomena of nature are 
to be referred back to motions of material particles possessing 
unchangeable moving forces, which are dependent upon condi- 
tions of space alone. Motion is the alteration of the conditions 
of space. Motion, as a matter of experience, can only appear 
as a change in the relative position of at least two material 
bodies. Force, which originates motion, can only be conceived 
of as referring to the relation of at least two material bodies 
towards each other; it is therefore to be defined as the 
endeavour of two masses to alter their relative position. But the 
force which two masses exert upon each other must be resolved 
into those exerted by all their particles upon each other ; hence 
in mechanics we go back to forces exerted by material points. 
The relation of one point to another, as regards space, has 
reference solely to their distance apart : a moving force, there- 
fore, exerted by each upon the other, can only act so as to 
cause an alteration of their distance, that is, it must be either 
attractive or repulsive. 

1 Finally, therefore, we discover the problem of physical 
natural science to be, to refer natural phenomena back to 
unchangeable attractive and repulsive forces, whose intensity 


depends solely upon distance. The solvability of this problem 
is the condition of the complete comprehensibility of nature. 
In mechanical calculations this limitation of the idea of moving 
force has not yet been assumed : a great number, however, of 
general principles referring to the motion of compound systems 
of bodies are only valid for the case that these bodies operate 
upon each other by unchangeable attractive or repulsive forces ; 
for example, the principle of virtual velocities ; the conservation 
of the motion of the centre of gravity ; the conservation of the 
principal plane of rotation ; of the moment of rotation of free 
systems; and the conservation of vis viva. In terrestrial matters 
application is made chiefly of the first and last of these princi- 
ples, inasmuch as the others refer to systems which are supposed 
to be completely free; we shall, however, show that the first 
is only a special case of the last, which therefore must b^ 
regarded as the most general and important consequence o. 
the deduction which we have made. 

' Theoretical natural science therefore, if she does not rest 
contented with half-views of things, must bring her notions 
into harmony with the expressed requirements as to the 
nature of simple forces, and with the consequences which 
flow from them. Her vocation will be ended as soon as 
the reduction of natural phenomena to simple forces is com- 
plete, and the proof given that this is the only reduction of 
which the phenomena are capable/ x 

Helmholtz accepted du Bois-Reymond's advice and wrote to 
G. A. Reimer, who replied that he was only too glad, since 
du Bois answered for the value of the treatise, to undertake 
its publication. He brought it out in 1847, and presented 
Helmholtz, to his great surprise, with an honorarium. 

The Law of the Conservation of Energy as put forth by 
Helmholtz suffered the vicissitudes incident to the birth ol 
all great thoughts. However much a generalization may be 
foreshadowed by experiment in different directions, and sug- 
gested and discussed by speculative thinkers, yet when it 
finally appears in concrete form it is sure to encounter doubts 
as to its accuracy, or, if the magnitude and worth of the dis- 
covery are recognized, suspicions of its originality, and 

1 J. Tyndall, 'The Conservation of Force,' Scientific Memoirs [Natural 
Philosophy], I, pp. ii4seqq. 


disputes as to priority. While the memoir was enthusiasti- 
cally welcomed by the younger physicists and physiologists 
of Berlin, who were led by du Bois-Reymond, and Helmholtz 
to his high delight was praised by the military authorities 
' for the splendid practical turn that he had given to his 
studies ', the older scientists with hardly an exception rejected 
the ideas which the work expressed, fearing, strangely enough, 
that such speculations would revive the phantasm of Hegel's 
'nature-philosophy', against which they had fought so long, 
and in the end so successfully. There was but one, after 
Johannes Mttller the most gifted scientific thinker of the day, 
the mathematician Joh. Jac. Jacobi, who from his profound 
studies of the principles of mechanics clearly recognized the 
close connexion between the work of Helmholtz and that 
of the great French mathematicians of the preceding century. 
Notwithstanding the doubts of his distinguished colleagues 
Lejeune-Dirichlet and Eisenstein, he unhesitatingly proclaimed 
the significance of Helmholtz's work, and by this gave confi- 
dence and self-assurance to its author. In the original 
treatise Helmholtz had only attempted to give a critical 
survey and arrangement of the facts in the interests of 
physiology, expecting the physicists at most to reproach him 
for having, as a young doctor, brought forward as new what 
was well known to them, but he now realized from the general 
opposition that he had been the first to set forth a universal 
law of experimental science, and to purify and free it from 
vague philosophical and speculative reflections. 

The elementary scientific discussions about perpetual motion 
in his parents' house had never proved its impossibility con- 
clusively for Helmholtz, and while still a student at the 
Friedrich-Wilhelm Institute he resorted to the works of Daniel 
Bernoulli, d'Alembert, and other mathematicians of the eigh- 
teenth century which he found in its library. From these he 
obtained the strictest and most convincing proof that a per- 
petuum mobile cannot be produced by purely mechanical forces. 
Just as the works of a clock have no energy of their own, and 
can only distribute evenly, over a considerable period, what 
is supplied to them from without, so, as these great thinkers 
showed by rigid mathematical proof for all pure motive forces, 
our machines and apparatus have no intrinsic energy, but 


merely give out in other forms what is communicated to them 
from the store of energy in the universe. 

Yet it still remained an open question whether perpetual 
motion might not be possible in the great field of the other 
natural forces, which cannot be reckoned as purely motive 
heat, electricity, magnetism, light, chemical affinity but are 
all, nevertheless, in manifold relations with the mechanical 
processes, since in almost every natural process mechanical 
effects are produced, and mechanical work is performed. 

Helmholtz's medical studies, and his knowledge of the 
biological side of natural phenomena, led him in the first 
instance to consider the possibility of perpetual motion in these 
processes, which he had studied since 1841. 

After his physiological observations had led him time after 
time to the conclusion that there could be no perpetuum mobile 
for the natural forces that here come under consideration, and 
when he had convinced himself that such a thing was alto- 
gether impossible, he inverted the problem hitherto pro- 
pounded by the scientists, as to how the relations between 
natural energies could be utilized to construct a perpetuum 
mobile, and asked himself what the relations between the 
forces of nature must be, if perpetual motion were indeed 

As a matter of fact, this reversal of the problem had been 
previously made for heat, though in less general terms, and 
with less conscious intention, by Robert Mayer and Colding, 
with whose investigations Helmholtz was not acquainted, and 
by Joule, whose experiments he heard of first when his own 
work was completed. Helmholtz found that all known rela- 
tions of forces lead to the conclusion that perpetual motion 
is impossible : he plotted out a further series as yet unknown, 
the actual existence of which had to be ascertained, and 
endeavoured to formulate all the relations between the dif- 
ferent processes of nature which could be deduced from his 
assumption. The result proved that there is no cycle 
throughout the entire range of natural processes by which 
mechanical energy can be generated without corresponding 
expenditure ; the quantity of working force may indeed be 
lost to the particular machine, but not to the universe as 
a whole. ' Nature as a whole possesses a store of energy 


which cannot in any wise be added to or subtracted from: 
the quantity of energy in inorganic nature is as eternal and 
unalterable as the quantity of matter/ the constancy of which 
had been established as a fundamental law of chemistry by- 
Lavoisier half a century before. 

Helmholtz termed this universal principle, now known as 
the Law of the Conservation of Energy, the Law of the 
Conservation of Force; it asserts that each transformation 
of energy takes place under exactly measurable quantitative 
relations, whether the form of energy be the vis viva of motion, 
or electrical and magnetic energy, or heat, whence again the 
impossibility of perpetual motion follows. 

In order to include within the scope of his considerations 
such natural forces as may be still unknown, he affirms with 
the care of a great investigator that the validity of the law of 
the constancy of the sum of vis viva and of what he called the 
'tensional forces', i.e. of actual and potential energy, is in the 
highest degree probable, since it contradicts none of the known 
facts of science, and is on the contrary confirmed by many of 
these in the most striking manner. He tests the energy- 
equivalents of heat, of electrical action, of magnetism and 
electro-magnetism, and after finding the law to be universally 
valid, turns as physiologist to the natural processes of organic 
existence, and shows that the problem of the conservation 
of energy is here a question of whether the oxidation and 
metabolism of the nutritive substances generate an equivalent 
quantity of heat to that given off by animals, a problem which 
had already occupied him for some months in Potsdam. 

' I have endeavoured/ he says at the close of this masterly 
treatise, l to state in the most complete manner possible the 
inferences which flow from a combination of the law with 
other known laws of natural phenomena, and which still await 
their experimental proof. The object of this investigation was 
to lay before physicists as fully as possible the theoretic and 
practical importance of a law whose complete corroboration 
must be regarded as one of the principal problems of the 
natural philosophy of the future/ 

At the time when Helmholtz began his analytical study of 
the natural sciences, the law of the persistence of matter 
(by which the elements may alter in regard to the mode of 


their distribution in space, but are unalterable in their pro- 
perties) was admitted by all physicists to obtain in every 
change of organic and inorganic nature. The great prin- 
ciple of the conservation of energy, which Helmholtz placed 
beside this law, postulates that all forces are to be measured 
in terms of mechanical force, and that all forms of energy are 
ultimately kinetic, so that the final aim of natural science 
must be to reduce itself to mechanics. 

Equipped only with such literary matter as the library of 
the Gymnasium could afford him during his residence in 
Potsdam, unaware of Robert Mayer's nine-page note ' On 
the Forces of Inorganic Nature', published in Wohler and 
Liebig's Annalen der Chemie in 1842, after it also had been 
rejected by Poggendorff, and of the same author's treatise pub- 
lished in 1845, on 'Organic Motion in relation to Metabolism', 
Helmholtz had by 1843-4 completed the essentials of the 
work which Kirchhoff estimated twenty years later as the 
most important contribution to natural science made in our 
era, while Hertz, Helmholtz's great pupil, says of it that 
' Physical research had been diverted by the close of the 
century into an entirely new channel. Under the over- 
mastering influence of Helmholtz's discovery of the conser- 
vation of energy, its object was henceforward to refer all 
phenomena in last resort to the laws which govern the 
transformation of energy J . 

But while the great significance of Helmholtz's work was 
immediately recognized by the younger generation of scientific 
men, the older physicists still held aloof from it, on the ground 
that it was a relapse into the 'nature-philosophy' of Hegel. 
In other quarters, again, where the importance of the great 
law was admitted, the honour of its discovery was withheld 
from Helmholtz. It was said that he had borrowed the idea 
from Dr. Julius Robert Mayer, a Heilbronn physician, who 
had published a thesis on the same subject, and had even 
determined the mechanical equivalent of heat. ' This report,' 
says du Bois-Reymond, ' has lasted, like the fame of Helmholtz's 
treatise, to the present day, and has been greedily accepted 
by those who love to trail shadows across the sunlight.' 

As regards priority, Helmholtz, after he had become 
acquainted with the writings of Robert Mayer, took every 


opportunity, in discussing the discovery of the Law of the 
Conservation of Energy, of insisting that it was Mayer who 
had first expressed his conviction that the sum of energy 
in the universe could neither be destroyed nor added to, and 
who formulated this view in his law of 'the equivalence of 
heat and work '. The English physicist Joule had also under- 
taken an extensive series of experiments, independent of 
Mayer, with the object of determining the equivalence between 
heat and work empirically, and a lively controversy as to the 
priority of Mayer's work had therefore already been raised 
by those 'who attach more weight to the collection of data 
than to the formulation of general principles'. 

Helmholtz himself, both in his original treatise and in 
discussing the subject afterwards, used to say that the work 
which he then undertook was one of pure criticism and 
arrangement, since its principal aim could only be to test 
the earlier conclusions derived from inductive methods, upon 
the newly-acquired material. If a law is to hold good through- 
out the universe for the vast complex of natural processes, 
it was not in his estimation sufficient merely to state this as 
Mayer had done ; evidence sufficient to enforce conviction 
of its probability must be produced, so that scientific men may 
bear it in mind for future confirmation. 

1 In those days it was far more important than might possibly 
now be the case, to make clear from beginning to end that 
the law was a law of facts, abstracted from the facts, and to 
be confirmed again by facts/ 

Helmholtz consistently recognized that when the Law 
of the Conservation of Energy had made its way later on, 
and its accuracy was generally accepted, every one would 
admit that Mayer had in 1842 reached a perception of its 
meaning and universal significance, just as Faraday must have 
had a presentiment of the same law long before Joule gave 
definite scientific expression to it, and filled up the most 
important gap in the empirical evidence in its favour. 

But Mayer had not arrived at this perception by scientific 
methods. After the data (which were familiar enough to many 
of his predecessors) had arranged themselves in his conscious- 
ness, ' the creative idea presented itself suddenly, not as a 
demonstrated truth, but as a problem, for the solution of which 


empirical facts must be investigated/ Unlike Helmholtz, he 
did not test the accuracy of the law (conceived as it were 
by inspiration, and by a certain creative activity of his brain), 
or rather its consequences, upon all the natural processes 
known at that time: the recognition of the principle in fact 
involved other, and deeper mathematical, knowledge than any 
Mayer could command. 

Helmholtz by his discovery gave an impulse to the whole 
later development of mathematical physics, and showed by 
rigid mathematical proof that whenever natural bodies act 
upon each other by attractive or repulsive forces, which are 
independent of time and velocity, the sum of their vires vivae 
and 'tensional forces' must be constant; but if these forces 
depend upon time and velocity, or act in other directions than 
the straight lines which unite the two active material points, 
then (provided the action and reaction are equal) combinations 
of such bodies will be possible in which energy may be either 
lost or gained ad infinitum. When in this sense the above 
forces are described as ' conservative ', the Law of the Conser- 
vation of Energy says no more than that all elementary natural 
forces are conservative. 

The work which the great French mathematicians had done 
in mechanics was familiar to Helmholtz, and to him it was 
no new induction, but merely the definite statement and 
complete generalization of an established conviction, to say 
that all elementary forces are conservative. In Helmholtz's 
opinion those great thinkers must have made the same con- 
jecture, but did not state it in terms, since they could not 
prove it, having ' set themselves the particular task of educating 
men from the false rationalism of scholasticism to the strict 
appreciation of experimental data'. Helmholtz termed his 
theorem the law of the 'Conservation of Energy' (Erhaltung 
der Kraff), to mark that it was an extension of the already 
known law of the * Conservation of Vis Viva * (Erhaltung 
der lebendigen Kraff), and to make its relation clear with the 
old question of the possibility of a perpetuum mobile. 

Mayer, by trying to get rid of the conception of force in 
mechanics, and defining as Kraft (force) what had previously 
been defined as work, i. e. the product of force into the dis- 
tance through which it acts, obscured the meaning of the 


well-known law of the conservation of vis viva, and delayed 
the strict mathematical expression of the law which he divined. 
Helmholtz, on the other hand (by analogy with the name 
1 quantity of vis viva ', which was used by Leibniz to express 
the work-equivalent of the velocity of the moving masses), 
gave the name 'quantity of tensional force* to this product, 
and in thus expressing the work-value of those forces which 
are actually engaged in the stress of producing motion, he 
established a connexion between actual and potential energy, 
the sum of which is constant for all transformations. 

This conception of a definite store of energy in the universe 
was quite new and due to Helmholtz alone; it was defined 
'as a quantity which can no more be destroyed nor added to 
than a substance, which acts in space and yet cannot be sub- 
divided with space like a material substance, because each 
division of space would not involve the portion of " tensional 
energy " which exists between the particles of matter on either 
side of the dividing surface/ a conception familiar to modern 
science, and founded solely upon the mighty work of Helm- 
holtz and the splendid pioneering achievements of Lord 

Twenty years later Helmholtz again took occasion to ascribe 
priority in the conception of the conservation of energy to 
Robert Mayer, as against the claims of Joule. In a letter 
to Tait on the occasion of a dispute about priority in the 
question of absorption and radiation, he puts Kirchhoff for- 
ward, since he was the first to formulate the law, and thus 
made the great discoveries that are involved in it possible. 
Helmholtz maintained that Kirchhoff 's work in this field 
represented one of the most instructive cases in the history 
of science. Many investigators had been on the verge of 
the same discoveries, but the development of spectral analysis 
as a whole became possible only after Kirchhoff had theoreti- 
cally determined those general properties of heat which are 
its fundamental basis. He clearly enunciates the relation of 
Mayer to Joule, and thus indirectly to himself also : 

1 Robert Mayer was not in a position to carry out experi- 
ments. He was repulsed by the physicists with whom he 
was acquainted, and could hardly find acceptance for his 
first condensed statement. While no one can deny that Joule 


did more than Mayer, and that many particulars are confused 
in Mayer's first note, we must, I think, regard him as a man 
who of and for himself conceived the idea which has rendered 
the great recent advance of natural science pqssible, nor is 
his merit in any way lessened by the fact that another man, 
in another country and different sphere of activity, had 
simultaneously made the same discovery, and worked it out 
afterwards with greater completeness.' 

Robert Mayer himself was far from claiming priority over 
Helmholtz in his epoch-making work, and the Naturforscher- 
Versammlung at Innsbruck in 1868, at which Helmholtz ac- 
knowledged Mayer's priority clearly and without ambiguity 
wherever it was due, left the two distinguished men on the 
best possible understanding. 

With this work on the Conservation of Energy, Helmholtz 
took first rank not only among physicists, but among physiolo- 
gists also, who recognized that his law afforded them an 
invaluable weapon for the attack upon vitalism; and he now 
went on to verify it for the natural processes of living 
organisms, by continuing his earlier experiments on the de- 
velopment of heat during muscular activity. His results were 
given to the Physical Society in November, 1847, and published 
the next year in M tiller's Archiv. 

This research, which is a model of the application of the most 
delicate physical methods to physiological problems, was in- 
tended to determine whether the rise of temperature in a working 
muscle takes place in the substance of the muscle itself, in conse- 
quence of internal processes brought about during contraction 
by a disturbance of equilibrium, or whether it is merely the 
result of increased flow of arterial blood. While the previous 
thermo-electric determinations of temperature in animals had 
been made with only a Becquerel couple, Helmholtz employed 
a triple junction of iron and German-silver which trebled the 
electromotive force, and found with the finest measurements, 
and ingenious contrivances for the exclusion of every other 
increase of temperature, that in excised and isolated thighs 
of the frog, there was a rise of temperature derived solely 
from molecular processes, when the muscle was caused to 
contract by stimulating the spinal cord with a Neefs inter- 
rupter modified for the purpose. The heat in contraction was 


therefore actually produced in the muscle substance, while 
attempts to demonstrate a rise of temperature in nerve during 
the transmission of excitation from spinal cord to muscle 
yielded negative results. These data all made for the verifi- 
cation of Helmholtz's great law, though the investigations could 
not then be held conclusive. 

The close of 1847 was devoted to severely theoretical 
studies, as appears from the many notes on classical pro- 
blems in pure mathematics found among Helmholtz's papers, 
but his medical duties hindered him more than was desir- 
able in the free disposal of his time, until at the begin- 
ning of 1848 the current of his life was altered by a fortunate 

His friend Brucke, who was teacher of anatomy at the 
Academy of Arts and assistant at the Museum of Anatomy 
and Zoology, was appointed Professor of Physiology and 
General Pathology at Konigsberg, and the reversion of the 
posts thus vacated in Berlin devolved on his intimate friend 
and contemporary du Bois-Reymond, who was two years older 
than Helmholtz. But since du Bois' private means made it 
possible for him to devote himself entirely to his investigations 
in animal electricity, without taking up professional duties, he 
retired from the competition for the Academy post in favour of 
his younger friend, arranging with Brucke that Helmholtz 
should be brought forward. 

On the strength of an excellent testimonial received from 
Johannes Miiller, Helmholtz was invited (August 19, 1848) 
to give a trial lecture before the Senate and Professors of the 
Academy. This was found among his papers, and has never 
before been published: 

'I shall endeavour, in the lecture which I have the honour 
of giving before you, to develop those points which seem to 
me the most essential in the teaching of anatomy to art- 
students, and the methods that should be employed. I must 
from the outset claim the forbearance of this distinguished 
assembly, since I am well aware what varied capacities and 
kinds of knowledge should be combined in any one who endea- 
vours to fulfil the duties of this post successfully, and how 
difficult is the handling of this science, if it is to be raised from 
the dry and often barren forms of a colossal effort of memory to 
a living thing that can be applied to Art. 

E 2 


1 The end to be aimed at is one so special, so totally different 
from what is usually expected from a teacher of anatomy, that 
I venture to say that but few of its postulates can be theoreti- 
cally laid down beforehand; the majority must follow from 
practice and experience. The new lecturer must indeed 
draw up some system of instruction, but little may possibly 
remain of his scheme by the time it has been carried into 

1 Anatomy, taught as an exact science to the medical student, 
has quite other aims, other proportions, and a widely different 
method. It starts with the necessity of giving the sharpest and 
most abstract definitions that are possible; for the physician 
and surgeon cannot limit himself to the appearance of the parts 
as they are in the sound body. His principal business is to 
discover sharp and simple characteristics that will not leave 
him in doubt, even where illness or lesion has so distorted 
the appearance of the parts, that the untrained eye could no 
longer find its way among them ; and medical anatomy is thus 
in its essentials a collection of dry concepts, very difficult to 
realize, of evil repute even for the long-suffering memory of the 
medical student, and the obvious appearance is hardly ever 
called in as more than an aid to the memory, while on the 
other hand it is all-essential to the artist. 

' For the doctor, for instance, what is of importance in 
any particular musck besides the point of attachment which 
determines its action, is the situation of the vessels and 
nerves upon or beneath it, the lie of the fascia that sur- 
round it, and control the flow of pus, and so on. What 
the muscle looks like, whether thin or thick, round or flat; 
to what extent it consists of flesh, and where its tendons 
begin ; whether it can be seen through the skin to these 
and similar questions he is for the most part quite indifferent, 
while it is just these points that make the muscle interesting 
to the artist. 

1 Art anatomy is therefore distinct from medical anatomy not 
merely in its content (since it embraces a portion of the latter, 
but has to work it up more specially), but still more essentially 
in its methods. 

' How anatomy should be taught for the artist is best decided 
by determining wherein and why it can help him. The artists 
of antiquity were ignorant of the internal aspects of the human 
body. The ancients had a natural, unconquerable aversion to 
the dissection of corpses, and were also hampered by their 
religious convictions, which made all desecration of the dead 
an unpardonable trespass against the most awful and sacred 
laws of the gods. Even late into the Middle Ages human 
bodies were never dissected by physicians those of apes at 


most. A medical student may have obtained some essential 
knowledge from the dismemberment of these man-like animals, 
but the works of even the most renowned medical authors of 
antiquity, e.g. Galen, contain anatomical observations which 
are incorrect for man, and true of apes only. This substitute 
for human anatomy could have been of no use to artists ; they 
were restricted to careful observation of the surface of the 
body, and at most could only learn the connexions of the bones, 
muscles, and tendons from animals, and compare these as well 
as might be through the skin by eye and touch in man, and 
endeavour to guess at the form of them. 

1 And yet despite these limitations, how marvellous a perfec- 
tion is exhibited by the art of antiquity not only in the most 
accurate knowledge of the resting form, with a delicate sense of 
beauty in all its proportions, but in the finest observation of the 
play of living muscles. This knowledge of the human form 
is so perfect in the ancient masters that they were able to 
dominate their subject with admirable inspiration and freedom, 
the freedom that modern art strives after, too often vainly, and 
which is only attained by a few favourites of genius. 

' We are tempted to inquire the need of anatomy, when the 
acme of sculpture was reached in ignorance of it. Why study 
below the surface, when it is the surface only that art has to 
render? To this we must reply that even in these works of 
inimitable talent, exquisite beauty, and laborious industry, there 
are some not inconsiderable errors which a good anatomist 
would have been able to avoid, though possessing far less skill 
than these sculptors. A muscle, e. g., is often yisible only as 
a little swelling, but the slightest increase or displacement of 
this swelling is sufficient in many cases to produce an anatom- 
ical absurdity, into which the most skilful copyist would readily 
fall if he were ignorant of the meaning of the protrusion, while 
any one familiar with the lie of the muscles in the figure would 
avoid it. It would be useless to multiply examples unless we 
had the statues here to illustrate them ; I will only, to make 
myself intelligible, adduce one instance, taken from a well-known 
and not ignoble statue of a Greek orator, usually known as the 
Germamcus, which was the work of the younger Cleomenes, 
in the post-Alexandrian period of Greek Art. The curve of 
the leg that stands free is so exaggerated, that the extensor 
muscles of the lower thigh (m. rectus femoris and sartorius\ 
which lie below it, and are felt in the living subject close 
beneath the skin, or even protrude a little, are altogether 
obliterated. In a " Shooting Apollo " in the Berlin Museum 
the posterior part of the deltoid muscle is constructed as if its 
insertion lay at right angles to the arch of the shoulder-blade, 
whereas it is parallel with it. 


' You may say that such an error, since it is patent only to 
the expert's eye (had it been more obvious, the Greek artist 
would never have perpetrated it), is of no consequence to the 
general effect of the statue, and that it is splitting hairs to dwell 
upon it. The creative artist produces the form which he has 
conceived without troubling about particulars, he is led on only 
by the sense of ideal beauty which hovers before his brain and 
eyes ; and with the same unconsciousness of details and their 
causes, the connoisseur revels in the spectacle of living har- 
mony afforded him in the work of the artist. Yet the artist's 
genius lies in the mysterious power of forming an original 
conception, and expressing it in a form that deliberate reflec- 
tion subsequently acknowledges to be true and perfect. And 
just as it is certain that the spectator will be elevated in 
proportion with the splendour and fidelity of the artist's con- 
formity to and interpretation of the ideal content of his work, 
so surely will every failure in this respect detract from the 
living beauty of the figure, even when the critic is unable to 
say wherein the fault consists, and what has caused it. 

' It cannot be denied that the lack of anatomical knowledge 
among the ancients is often perceptible as a defect in their 
productions, however much their marvellous talent for the 
representation of truth and beauty may have obviated its con- 
sequences. Then again it must be remembered that the 
ancients had far more abundant opportunity of observing 
the human form than is possible in modern times, and that the 
curriculum of the art-school has to supplement this want as far 
as possible. The modern, who can only study the human 
form in the model-room, or at best in a bathing establishment 
(where indeed he seldom finds a wholly pleasing subject under 
the one-sided and distorting conditions of our civilization), is at 
a great disadvantage as compared with the ancients, and would 
be on very unequal terms of competition if he were not 
equipped with accessory instruction. He further has to reckon 
with the factor that, for the same reason, the public know much 
less of the human body. Anatomy can no more than any other 
branch of study be a substitute for genius in the artist, whether 
in regard to capacity for reproduction or to sense of beauty, 
but it can set him forward on his way, and sharpen his powers 
of observation. 

'The question of the benefit of anatomy to the artist may 
therefore be reduced to this : What more can the knowledge 
of the internal structure of the body give him than he has 
acquired from the external study of it on the living model ? 
In reply I would submit the following considerations : 

' i. It facilitates appreciation of the different forms of various 
parts of the body in different postures, since these forms can 


all be referred to the same underlying anatomical mechanism ; 
it therefore makes the work of the artist easier when he has to 
do without a model. (Instance the different curves of the 
upper arm, and the different positions of the hand in turning 
the fore-arm.) 

'2. It teaches the distinction between the essential and non- 
essential parts of the human form. The prominences and 
depressions on the surface of the human body are of very 
different importance according as they correspond with bones, 
muscles, or folds of skin. Even when the sculptor only wants 
to reproduce a given model, he may err, as we pointed out, in 
the reproduction of some swelling which is essential to the 
anatomical structure. But the sculptor never should imitate 
slavishly, since his model is always that of a man who has 
grown up with human imperfections, and falls short of the 
ideal : he must modify the individual form till he obtains 
the most perfect expression of its spirituality. Given a mus- 
cular man for the model, with the intention of making a 
figure as strong and tense as possible, a sort of Hercules, 
he must not thicken the folds of skin and the muscles 
equally to produce his effect, but, on the contrary, must rather 
reduce the skin to bring out the muscularity; or conversely, 
if he is planning a Silenus. It is further to be noted that 
the emphasizing of the more important anatomical features at 
the cost of the less essential, adds clearness and simplicity to 
the figure. 

'3. Finally, it is impossible to study the gestures of the 
moving figure upon models, which must always be supported 
in order to maintain their posture without effort. This brings 
us to the important study of the variation in shape of active 
muscles. The model stands with relaxed muscles, even if he 
be successfully propped up in the required attitude. The 
artist must know what muscles swell and protrude in move- 
ment, unless his work is to give the effect of standing still, 
like the model. And even if he makes the model perform 
the action sometimes, and strives to fix the gestures in his 
memory, he still cannot entirely forego his knowledge of 
synthetic muscular action, since a noble, well-formed body 
moves differently from one that is less well developed. The 
former at each movement utilizes only so many limbs and 
muscles, with so much force, as are indispensable to the 
motion, giving an impression of grace and ease, while the less 
skilled and weakly individual works harder and uses more 
muscles simultaneously. 

' Still, we must never forget that Anatomy is but an instru- 
ment to further the more exact knowledge of the human form, 
and that, like all other helps to artistic study, it can never 


replace the immediate perception of the forms themselves, nor 
the artistic sense of beauty. To the artist it is a means of 
lightening the mental conquest of the ever-changing com- 
plexities of his material object, the human form, sharpening his 
perception of what is essential, making the whole form trans- 
parent to him, and arming him with the instruments of a 
searching criticism of the work he has accomplished. Art, 
however, begins where anatomy ends ; the spirit of the artist 
is shown in the wise application of the forms whose con- 
nexions and simple outlines have been taught by anatomy, and 
in the distinguishing characteristics of his figures. Here the 
artist, as in a Hercules, suggests the muscles lying in hard 
lumps beneath the skin ; there in the female figure they are 
merely indicated by slight changes in the curvature of the 
limb; in children again they are entirely concealed by the 
plump rolls of fat ; he alters the normal magnitudes of 
the parts according to his subject, and determines their 
position and motion. The display made by the artist who puts 
too much anatomical science into his figures, as has so often 
been alleged of Michael Angelo, and with more justice of his 
less inspired followers, is as unpleasant and false as the neglect 
of anatomical accuracy which produces lifeless or distorted 

' It is important for the student to study forms with very 
decided development of muscle, but he must not subsequently 
reproduce them with absolute fidelity on all occasions. It is 
interesting in this connexion to compare the frequent blunders 
of modern art with the work of the old masters : e. g. the 
Discobolos of Myron, from the zenith of Greek Art, who is in 
an attitude of the most violent exertion. He has checked his 
run to fling the discus, while, with the finest observation of 
actual movement, the spectator is only shown the great and 
almost continuous bulging of the limbs, although an impression 
of great vitality is imparted. Many another would have over- 
laden such a figure, since even in a simple resting form it 
appears impossible to some to show enough muscles. 

4 It is essential for the artist, and therefore a principal aim 
in instruction, that anatomy shall give him as clear and com- 
plete a picture as possible. He must not only bear the 
position, attachment, and function of the different muscles 
in mind, so that when he thinks of them he can form a correct 
notion, as is perhaps sufficient for medical studies, but he must 
be accustomed to see the underlying parts clearly through 
the intervening veil of skin, and never to picture the arm 
without realizing the bundles of muscles that lie within it. 
Nor must he fail in knowledge of the positive anatomical 
details, since they will serve as a criterion in the searching 


criticism which he must apply to the figures he has created, in 
order to facilitate the discovery of errors. The cardinal point 
in a lecture on anatomical details must therefore be its applica- 
tion to the living and unblemished form. The student must 
be trained to compare the appearance of a dead subject, that 
has all its anatomical parts exposed, with living forms, and to 
recognize anatomical details in life-models and works of art, 
where they are more or less concealed, in order by such 
exercises to sharpen his perception of anatomical errors. It 
follows as a matter of course that anatomy for art-students 
should only treat of such parts of the body as are of signifi- 
cance for its external form. Instruction in anatomy must 
therefore embrace : 

' i. Bones and such cartilages as are externally visible, form- 
ing the fixed skeleton of the body, which determine the per- 
manent relations of the several elements. This section, with 
the exception of the cranial bones, must be treated in as much 
detail as for medical anatomy, since even the smaller pro- 
minences on a bone are important as points of insertion for 
the muscles. 

1 2. Joints and ligaments, also treated in detail, notably in 
regard to limitations of movements. 

'3. Muscles, briefly as regards the deeper, in detail for the 
more superficial, with particular reference to their appearance 
through the skin. Besides the functions of single muscles, 
systems of composite movements must be included. 

' In addition, some instruction in animal anatomy should be 
given, say on the horse, so far as our teaching apparatus will 

The lecture gave complete satisfaction to the authorities, 
and on September 30, 1848, Helmholtz was released from the 
three years' military service for which he was still inden- 
tured, and entered on a civil career. The transfer was 
accomplished with little trouble, at the instance of ' the good 
genius who then presided over science in Berlin Alexander 
von Humboldt'. 

The post at the Academy of Arts carried a salary of 60 per 
annum, and he was further appointed assistant at the Anatomi- 
cal Museum, with a salary of 30, on the recommendation of 
Johannes Miiller, who testified to his being as skilful in 
anatomy as he had proved himself in physiological experi- 

Thus, in 1848, Helmholtz left the military service, to which 
he had belonged since October, 1838, and he also ceased to 


practise as a doctor from the moment he was emancipated 
from his official obligations. But he always kept up his con- 
nexion with medical science, and frequently asserted later on 
that to a certain extent he felt more at home in it than in 
any other department, while he looked back to his education 
at the Military College with affection, and owed to it, as 
Ludwig justly remarks, 'the care he invariably bestowed 
upon his personal appearance, and the general decorum of 
his attitude.' 





IN BERLIN : 1848-1849 

ALTHOUGH Helmholtz was now set free from his duties as 
army surgeon, the preparing of his winter's lectures to the 
Academy, and still more the task of making preparations in 
comparative anatomy at the Anatomical Museum during the 
summer, took up so much time that, save for a short report 
to the Fortschritte der Physik on the theory of physiological 
heat, he was unable to bring any fresh work to completion. 

By the beginning of the winter session, in which, besides 
his lectures on osteology and myology (given to an audience 
of five), he had to make preparations of the human subject 
for the lectures and for the Museum, his work grew a little 
oppressive, since his brain was teeming with scientific ideas, 
and urged him on to original research. Accordingly he wrote 
to his friend Briicke, to ask if he were taking the duties of 
his new post too seriously. Brucke advised him to abate his 
pedagogic ardour, and in the following January he writes to 
his brother Otto, ' Tell the parents that all goes well ; I have 
less to do with my artists now, because I let them draw a good 
deal from specimens, and can generally leave them after a 
short lecture/ 

Helmholtz was, moreover, eager to justify the expectations 
of Brucke and his other friends. On March 16, 1849, he read 
a paper to the Physical Society, entitled ' Principles of Con- 
struction of a Tangent Galvanometer', in which he suggested 
precisely the same construction of galvanometer as was sub- 
sequently communicated to the Academic in Paris by Gaugain 
in 1853. Helmholtz was unable to claim priority for it, 
since the minutes of the meeting of the Physical Society had 


been lost. In the next place he sketched out the plan of the 
epoch-making researches by which he opened fresh paths in 
the physiology and pathology of nerve and muscle, and created 
new methods of investigation. But he had scarcely embarked 
on this work when his life took another and decisive turn, 
which made it possible for him to follow his great aims 

Briicke had been called to the University of Vienna, and 
the Medical Faculty of Konigsberg had nominated du Bois- 
Reymond, Helmholtz, and Ludwig as candidates for the vacant 
post (April i, 1849). Du Bois was not inclined to leave Berlin 
until he had completed his work on Animal Electricity, and 
Ludwig, though senior, was passed over on political grounds, 
so that on May 19, 1849, Helmholtz was by order of the 
Cabinet appointed Extraordinary Professor of Physiology at 
Konigsberg, with a salary of 120. He was commanded to 
go at once to Konigsberg to commence his lectures on physio- 
logy in the summer term, while the Academy of Arts was 
desired by the Minister to release Helmholtz from his post 
as teacher of anatomy, in which he was succeeded by du 





THE fact that Helmholtz should, at such an early age, be 
appointed Extraordinary Professor and Director of the Physio- 
logical Institute, with a salary of 120 (which his father had 
only obtained after years of painstaking activity), completely 
revolutionized the old man's views as to the value of his 
son's achievements, and he often remarked that his Hermann 
had advanced much farther than he, who was only Professor 
at a Gymnasium. For the last two years the relations between 
father and son had rarely permitted any exchange of ideas, 
owing to the wide divergence of their scientific views, but from 
this time the elder Helmholtz was keenly desirous of becoming 
acquainted with all his son's work, and, whenever possible, of 
taking part in it. With Hermann's call to Konigsberg, accord- 
ingly, begins a most interesting correspondence between father 
and son, which extended over a period of ten years, and affords 
us many glimpses into the development of the great thinker's 

Now that Helmholtz had a settled position he was able to 
put an end to his long engagement, and bring home his 
beloved bride. The marriage took place on August 26, 1849, 
at Dahlem, near Berlin, in the house of the bride's sister. 
1 The ceremony was performed in the little, old village church, 
which was filled with a festive procession of friends, parents, 
and relatives on either side.' The young couple set out for 
Konigsberg directly after the ceremony. Helmholtz's parents 
were overjoyed, and looked forward hopefully to the future. 
1 Dear children/ writes the father on September 16, ' I wish 
I knew what and how to write, to give you as much satisfaction 


from my letter as yours has given me! Here our quiet life 
goes on as usual; your present is so beautiful that you can 
scarcely appreciate the happiness of recollecting what is past. 
Or should I as a wise parent check your transports by sad 
legends of the envious gods who leave nothing perfect to us 
poor mortals? advise you like Polycrates to sacrifice your 
dearest jewel, and commend you to bitter renunciation in order 
that you may remember that you are mortals called to sorrow ? 
Olga, make your Hermann tidy, for that is his weakest point, 
and when he is a father, he must set his children a better 
example than I have given him. Now I have quite done ; 
for what to me is so important, that you should be assured 
of my love, will seem to you in the fullness of your own like 
a drop in the ocean/ 

The new house was soon got ready. ' As soon as we had 
put our house in order/ Helmholtz writes in the middle of 
October to du Bois, ' everything was very comfortable, and 
we were able to enjoy the best part of our life without let or 
hindrance. I can only recommend you with the best con- 
science in the world to provide yourself at the first oppor- 
tunity with just such a dear wife as I have found. Marriage 
makes one so fully contented with the present, so certain of 
one's portion, that my working power has substantially 
increased/ And in fact he employed the vacation in sketch- 
ing out new enterprises, and continuing his former work, 
bringing an entirely new method to bear on the experiments 
already commenced upon the nature of muscular movement 
during a single twitch. His young wife gave him valuable 
assistance by observing the divisions of the galvanometer 
scale, and long series of tables in her handwriting still exist 
among his papers. 

In the first half of the winter session he was almost 
entirely absorbed in the preparation of his lectures, and 
could only undertake such minor experiments as were neces- 
sary for purposes of demonstration. 

1 A larger bit of work/ he writes to his father in December, 
* from which I was getting a good many results in the October 
vacation, must now be put off till Christmas. Seven students 
have entered their names for my lecture, three to five of 
whom generally appear, according to weather. I am still 


much handicapped in physiological experiments because the 
proper equipment of my laboratory is hindered from want 
of funds. But now I have been allotted 15 for this year, 
and the same for next, for expenditure on instruments and ex- 
periments ; so I shall be, and am, better off in this respect 
than I was in Berlin/ 

The father's reply, at once admonitory and soothing, was sent 
on December 28 as a Christmas and New Year greeting : 

' May 1850 bring you as much of happiness and of God's 
blessing as 1849. Above all may you both be kept in health : 
and to you may it bring good results in your scientific labours. 
I am sorry that you have so small an audience, for nothing 
inspires a teacher so much as applause, and the response of 
numbers to what he offers. You have all the more reason 
to cultivate a fluent style and popular manner, as well as 
depth and solidity; in this way you will gratify the wishes 
of the authorities, who have doubtless sent so young a man, 
with this remarkable salary for an Extraordinary Professor, 
to Konigsberg in order to awaken a keener interest for what 
is really profound and scientific, as one might say fundamental, 
in Medicine, and thus initiate further developments of this 
practical art in Konigsberg also. Physiology is so closely 
related to Philosophy, and has such important general 
interests, that you will doubtless discover a form of lecture 
and choice of subject that will attract many from other Facul- 
ties, notably from the Philosophical, especially if you make 
friends with Rosenkranz, whose fame attracts many philoso- 
phers to Konigsberg. Professor Meyer says that the Konigs- 
berg students of his day were distinguished by keen activity 
and industry, especially in science and mathematics. So we 
must hope that you will be more successful in this respect 
next year, for the sake of your own affairs as well, for you 
will soon find that, even while you are only two, your stipend 
is small enough in the circle in which you and your wife are 
moving ; and you decline to practise, which is really the most 
lucrative, though I grant you the most disturbing and fatigu- 
ing profession/ 

Helmholtz used the Christmas holidays as he had planned, 
to complete the experiments broken off in October, and was 
able by January 15, 1850, to send du Bois a short report ' On 


the Rate of Transmission of Excitation in Nerve', with the 
request that he would communicate it to the Physical Society, 
1 to secure priority in their Proceedings.' He sent the notice 
at the same time to Johannes Mtiller for the Academy in 
Berlin, and to Alexander von Humboldt for Paris, contenting 
himself for the moment with the statement of his discovery, 
that during the excitation of nerve with the current induced in 
a coil by the opening of the circuit in another coil, a measur- 
able time (some 0-0014-0-0020 second) elapses, before the 
stimulus of an instantaneous electrical current applied to 
the sciatic plexus of large frogs, with nerves 50-60 millimeters 
in length, is transmitted to the point at which the nerve enters 
the gastrocnemius muscle. He tells du Bois-Reymond how, 
'after a severe struggle, I have converted a bold mathemati- 
cian, who gets somewhat confused over non-mathematical 
logic, and is himself lecturer on mechanics, to the doctrine of 
the conservation of energy, so that it is now official doctrine in 
this University. Neumann is rather difficult to get at; he is 
hypochondriacal, shy, but a thinker of the first order/ 

Du Bois was again the only man who understood the brief 
note thus published by Helmholtz merely to establish his 
priority. 'Your work, I say with pride and grief, is under- 
stood and recognized in Berlin by myself alone. You really 
have, begging your pardon, expressed the subject so obscurely 
that your report could at best only be an introduction to the 
rediscovery of the method. The consequence was that Muller 
failed to rediscover it, and the Academicians decided after he 
had spoken that you had not eliminated the time lost during 
the contraction of the muscle. I had to explain it separately to 
one after the other to Dove, to Magnus, to Muller himself, 
who would have nothing to do with it. I brought it forward 
at the Physical Society, where at any rate we did not have 
the same difficulty. Humboldt was quite mystified, and 
refused to send your note to Paris, so I offered to make it 
plainer. I have done this on my own responsibility ; you will 
observe that I have not altered a single detail, but kept rigidly 
to what you gave me, while I have developed it inductively. 
The note on rapidity per second is not mine, but Humboldt's.' 
In conclusion du Bois-Reymond expresses his wish that 
Helmholtz would continue these researches: 'The lay of 


experiment you are on is wonderfully lucky do oblige me by 
keeping to it ; we could work into each other's hands, and get 
something out of it.' 

It was not surprising that this brief communication from 
Helmholtz should again arouse questioning and contradiction 
on the part of the older physiologists and physicists. Johannes 
M tiller had expressly stated six years before that we never 
should be able to determine the rate of the nervous impulse, 
since we had no means of experimenting over enormous 
distances, by which the velocity of light, in this respect analo- 
gous with the activity of nerve, had been calculated. He 
held that the time occupied by the passage of a sensation 
from periphery to brain and cord, and the efferent reflex that 
produces a contraction, is too infinitesimal to be measured. 
And, indeed, so long as the nervous impulse was referred by 
physiologists to the diffusion of an imponderable agent, or to 
a psychical principle, it necessarily appeared incredible that 
the velocity of this current could be measurable within the 
short compass of the animal body. Du Bois' work had, how- 
ever, made it more than probable to Helmholtz that the propa- 
gation of excitation in nerve is essentially conditioned by 
altered arrangement of the molecules, whence he conjectured 
that rate of propagation is a measurable, and even a moderate 
magnitude, since it is a case of molecular action in ponder- 
able bodies. 

These investigations all fell into place in the chain of 
Helmholtz's thoughts and opinions, which were directed, to 
the exclusion of any metaphysical speculation, towards the 
discovery of facts alone. It is interesting to find that while 
the opponents of 'nature-philosophy' had set themselves 
against the Law of the Conservation of Energy, because they 
saw in it merely a philosophical play of ideas with no strong 
scientific basis, the next, strictly physico-physiological, work 
of Helmholtz provoked doubt and remonstrance not only from 
the physiologists, but from the philosophers also, as they were 
unable to admit a time-interval between the idea and the con- 
comitant physiological action. 

In order to explain this antagonism it is only necessary to 
remember the views that prevailed at that time in regard to the 
connexion of the sciences, more particularly of physiology 


and physics. Helmholtz himself tells how a Professor of Physio- 
logy, celebrated for his literary activity, and a witty speaker, 
replied with annoyance when invited by a physicist, during 
a discussion upon the images in the eye, to accompany him 
to his home to see the experiment, that 'a physiologist had 
nothing to do with experiments, though they might be well 
enough for the physicists ' : while a Professor of Pharmacology 
and academic reformer, who tried to persuade Helmholtz to 
divide his physiology, taking the intellectual part himself, and 
leaving the lower experimental side to a colleague, gave up all 
hopes of him when he explained that he regarded experiment 
as the true basis of science. 

When at last Johannes Muller and A. von Humboldt were 
convinced (before the full publication of the experiments) of 
the correctness of his work, Helmholtz sent a short account 
of it on March 29 to his father : 

' I have another six weeks' vacation, and am using this time 
to prosecute my discoveries in regard to the transmission of 
nervous activity, extending it to as many cases as possible, 
and getting it ready for publication. Since my first note to 
the Academies of Paris and Berlin I have been studying the 
point in man also, and here too have found it possible to 
demonstrate that the time required for a message from any 
part of the body to reach the brain (e. g. ^V second from the 
great toe) is longer in proportion to the distance it has to 
travel, while another interval of time is required before the 
process that excites contraction can be transmitted from the 
brain through the nerves to a muscle. I expect to finish 
the experiments, and get them worked up these holidays. My 
first communication has been published in the Monatsberichte 
of the Academy in Berlin, and the Comptes Rendus of the 
Academic in Paris, and I have had two very appreciative letters 
about it from J. Miiller and A. von Humboldt. I call this 
work a bit of good luck, for it will not fail to excite attention. 
That it will be noticed in Paris, though perhaps not with a 
very good grace, is shown by a scoffing article in the National, 
by the reporter who has already been heckling du Bois- 
Reymond. Unluckily I have not been able to get hold of 
the article here. Don't let this distress you : one cannot 
expect the French to take such things kindly from a German, 


and I have got all I want for the moment if they are alive 
to it. Du Bois is in Paris these holidays to give his own 
things at the Academic, and writes that he will bring mine 
forward also. He is very clever at that job, and I have no 
doubt that he will present the German scientists in quite an 
imposing light to the Frenchmen. Konigsberg is a splendid 
place to work in, because it does not tempt one to do much 
else, and yet there is plenty of intellectual life about it. The 
apparatus which I used in my work was made quite well for 
me here/ 

In a few days, however, Helmholtz received a letter of 
affectionate criticism from his father, intimating that with all 
deference to his son's authority, he found difficulty in under- 
standing the result of his investigations : 

I As regards your work, the results at first appeared to me 
surprising, since I regard the idea and its bodily expression 
not as successive, but as simultaneous, a single living act, that 
only becomes bodily and mental on reflection : and I could as 
little reconcile myself to your view, as I could admit that 
a star that had disappeared in Abraham's time should still be 

Helmholtz was far from wishing that his father should accept 
his results upon the scientific appreciation of others, contrary 
to his own convictions, and lost no time in sending him the 
following lucid explanation of the meaning of his work : 

I 1 am adding a note intended so far as may be to remove 
your doubts about the rate of propagation in nerve. You must 
remember that the interaction of mental and bodily processes 
is initiated in the brain, and that consciousness, intellectual 
activity, has nothing to do with the transmission of the message 
from the skin, from the retina, or from the ear, to the brain. 
In relation to intelligence this transmission within the body 
is as external as the propagation of sound from the place at 
which it takes origin, to the ear. Just as in this case it is 
the elasticity of the air that conveys the concussion of the 
resonant body to the nervous apparatus of the ear, so it is 
here the motions of the minute material particles of the nervous 
substance which are propagated from the end of the nerve 
to its origin in the brain, where they are first recognized as 
a message to consciousness. That the velocity of this trans- 

F 2 


mission in nerve is by no means so enormous as that of light 
or electricity may be .conjectured from du Bois' discovery that 
electricity is developed in nerve during the propagation of a 
message or stimulus; whence we must conclude that the 
material particles of the nerve are altered in position during 
the process. Transmission is in this case, as a matter of fact, 
extremely slow, slower than that of sound. The reason why 
the time occupied by this propagation seems to us so infini- 
tesimal lies in the fact that we are unable to perceive more 
quickly than our nervous system can act, and thus the intervals 
required for its operations appear to us imperceptibly small. 
The inaccuracy of our time-perceptions when based upon a 
comparison between our perceptions by two different sense- 
organs has recently been demonstrated in the most striking 
manner. Astronomers vary in their estimation of the moment 
at which a star crosses the web of their telescopes by more than 
a whole second, while the estimates of any individual taken by 
himself agree within one-tenth of a second if frequently re- 
peated. Still more surprising is the difficulty of determining 
whether the beats of two gently-ticking watches coincide, or 
fall between each other, if held to either ear, while nothing 
is easier than the same determination if both are held to the 
same ear. I picture the matter to myself in this way: two 
perceptions of different organs can only be estimated as regards 
their time-relations, when there is a sufficient interval between 
to reflect "now you have perceived one, but not as yet the 
other". Our thought is not so rapid as we usually believe, 
as I have proved from my experiment of taking an electric 
shock at any point on my skin, and then trying to move my 
hand as quickly as might be, measuring the time between 
the shock and the first commencement of the movement. With 
great attention, when the will is ready to act the instant it 
receives the message, the message is only delayed about one- 
tenth of a second in the brain, and is carried on with such 
mechanical regularity to the motor nerve as a motor stimulus, 
that I think the delay must here be caused only by the neces- 
sary mechanical molecular processes. When, on the contrary, 
the attention is fatigued, so that on receiving the message it 
becomes necessary to think what is to be done, a much longer 
and quite irregular interval is required. 


1 1 have not yet completed my work on the frog, as there 
are still various experiments to make, diagrams of apparatus to 
be drawn, &c., but I expect to finish in the Whitsuntide holidays. 
The experiments on man must be varied and repeated before 
I can publish them later on/ 

While working out his experiments on the frog, Helmholtz 
was actually taking time-measurements on himself and other 
men, which seemed to establish the rate of transmission in 
the motor and sensory nerves of man at fifty to sixty metres 
per second. At the end of April he announces the completion 
of the first part of his paper, for Mutter's Archiv, to du Bois- 
Reymond, and dispatches it on July 26 along with the news 
that he is 'father of a well-formed healthy girl'. As regards 
new discoveries, he announces a theorem on the form of the 
rise of electrical currents in a coil, which act inductively either 
upon this or any connected system of other coils a task which, 
however, took him nearly a year more before he was able to 
deduce a conclusion. 

In the meanwhile (July 19, 1850) du Bois presented Helm- 
holtz's comprehensive work on ' Measurements of the Time- 
relations in the Contraction of Animal Muscles, and Rate of 
Propagation in Nerve/ to the Physical Society. It was at 
once published in Mutter's Archiv under du Bois* supervision, 
Helmholtz consenting, for the benefit of 'those who are only 
half acquainted with Ohm's Law', to certain alterations advised 
by du Bois-Reymond. He now realized from the report of 
the latter, who had returned in a somewhat exasperated mood 
from Paris, where his lecture on Helmholtz's method of mea- 
suring the propagation of the nervous impulse was not very 
favourably received at the Academic des Sciences, that the 
novelty of the work demanded a thorough exposition. The 
article was through the press by December, and in the same 
month Helmholtz gave a lecture to the Society of Physics and 
Economics in Konigsberg, of which he was this year Director 
(being elected President two years later), which dealt in a 
generally comprehensible manner with the subject, and was 
entitled, 'On Methods of measuring very small Intervals of 
Time, and their Application to Physiological Purposes/ 

In Part I of this great work (Part II only made its appearance 
two years later), 'which opened a new and unbounded field 


of investigation to physiologists/ Helmholtz set himself in the 
first place to study the processes that take place in a simple 
muscle-twitch, consequent on a stimulus of infinitesimal dura- 
tion, when the muscle in order to do work must pass from 
a state of rest to a state of motion, and the quantit}^ of work 
done depends essentially upon the rapidity of this transition. 
From these facts he went on to the question of the rate at 
which excitation is propagated in nerve. From his first ex- 
periments at the beginning of 1849 he had concluded, from the 
curves obtained by plotting the height to which a weight is 
lifted against time, that the energy of the muscle was not at 
its maximum immediately after the excitation, but that it rose 
for some time, and then dropped again. In order to show 
these facts plainly, and at the same time to determine the time- 
relations and the stages in which the mechanical activity, the 
energy of the muscle, rises and falls after an instantaneous 
excitation, Helmholtz, at the outset of his experiments, con- 
trived a very ingenious piece of apparatus. He attached a 
metal ring to the muscle, which carried a scale-pan of light 
weight ; the upper part of the ring was supported in such a 
way by a metal pin, that it could not drop lower when the 
load was increased. On then closing a current, part of which 
passed through the muscle, part through a galvanometer, 
the pin, and the ring, the galvanometer circuit was broken 
by the contraction of the muscle and consequent lifting of the 
ring off the pin, and by placing weights in the pan it became 
possible to compare the elastic force of the muscle in the 
resting state with that after excitation. In order to follow 
the very rapid twitch of the muscle in its successive stages, 
and to investigate the propagation of excitation in the nerve, 
new methods for the measurement of infinitesimal fractions 
of time were devised by Helmholtz, who thus provided fresh 
appliances for the delicate and complicated investigations of 
physiological processes. 

The need of some method by which it should be as possible 
to measure minute fractions of a second, as it is with a powerful 
microscope to estimate fractions of length, had long been felt 
in a variety of astronomical and physical observations. Two 
such methods, invented mainly for the exigencies of artillery, 
had already been devised on widely different principles. In 


the one, perfected by Werner Siemens, the intervals of time 
are measured in terms of intervals of space ; in the other, the 
mechanical effect produced by a force of known intensity 
during the interval is measured, and the time of action is 
subsequently calculated from it. This second method, dis- 
covered by Pouillet in 1844, consisted in the measurement of 
very small intervals of time by the deflection of a galvanometer 
needle after the passage of a very short electric current, and was 
elaborated by Helmholtz for physiological purposes. He made 
the electro-magnetic determinations by means of a mirror attached 
to the magnet, as introduced by Gauss and Weber, and estab- 
lished the constant factor necessary to convert the differences of 
oscillation into the corresponding time-differences, by a strictly 
accurate method. 

Starting from the simplest cases, he next attacked the more 
complicated problem, whether there is any appreciable lost 
time during the propagation of a message from the remote 
ends of the sensory cutaneous nerves, or from the nerve- 
endings in the sense-organs, to the brain, or from the brain 
to the muscles via the motor nerve trunks. He first deter- 
mined in the frog, by a long series of most delicate experi- 
ments, that when the muscle or nerve of an animal is excited 
by an instantaneous electrical shock, a short time, about 
one-hundredth of a second, elapses, during which the elastic 
tension of the muscle is not appreciably altered the so-called 
latent period of excitation after which the muscular tension 
gradually rises to a maximum, and then as gradually falls 
again. If, further, two different points of a motor nerve 
are excited by an instantaneous stimulus, and if the magni- 
tude of the excitation is alike for both, the time-relations of 
the corresponding contractions will also be the same, but 
the total effect makes its appearance later when the stimulus 
is applied to a more distant point of the nerve. The 
transmission of excitation through nerve to muscle there- 
fore occupies a measurable time, and its speed is actually 
found to be more than ten times less than the velocity of 
sound in air. ' Happily,' says Helmholtz, ' the distances our 
sense-perceptions have to traverse before they reach the brain 
are short, otherwise our consciousness would always lag far 
behind the present, and even behind our perceptions of sound/ 


Two points in this lecture gave rise to an interesting corre- 
spondence with du Bois-Reymond. Helmholtz, using what 
is now a familiar figure, compared the nerve-fibres with the 
wires of the electric telegraph, which in an instant transmit 
intelligence from the outposts to the controlling centre, and 
then convey its orders back to the outlying parts to be executed 
there; and at the close of his discourse he illustrated the 
rapidity of nervous conduction by saying that the whale 
probably feels a wound near its tail in about one second, and 
requires another second to send back orders to the tail to 
defend itself. On March 18, 1851, du Bois sends Helmholtz 
a lecture which he had given at the Sing-Akademie before 
the ' familiar audience of non-working classes ', and remarks : 
'Your surprise on reading this will be little less than my 
own on reading your lecture. Apparently we have hit on the 
same thing in two illustrations, and in various other points, 
even to the expressions we make use of, so that no stranger 
would absolve me from plagiarism. I feel greatly flattered 
at this conformity in the motions of our brain molecules. Give 
my lecture to your good wife to read. The ladies are angry 
because I made myself comprehensible to them what do I 
take them for ? they expected something more scientific from 
me/ He then describes the difficulty he finds in devising an 
instrument which shall be set going by a muscle so as to close 
the circuit during a given fraction of the twitch, and ends with 
the words : ' Write and tell me what avalanche of ideas this 
tinkling of a mule-bell rouses in your brain/ After congratu- 
lating du Bois in his answer of April n on his election to 
the full membership of the Academy, Helmholtz replies : 

'As regards the coincidence in our lectures I concede you 
priority in the matter of the electric telegraph, since you long 
ago suggested the hypothesis that the ganglia represent the 
intermediate stations of the electric telegraph in the nerve 
circuit. But in the story of the whale the truth is so romantic 
that no one will believe it. The moral is that people can easily 
be mistaken about plagiarism. . . . My wife joins forces with 
those who say that you have made yourself too intelligible. 
It is impossible to please every one on these occasions, but 
one generally gets more thanks for not making one's stuff too 
plain to the audience, and for leaving the majority a few 


riddles, which are probably only understood by a handful 
of one's hearers/ 

His father, to whom he had sent his lecture on February 27, 
was by no means of this opinion, and writes on April 19 to 
du Bois-Reymond : 

1 Dear Doctor, heartiest thanks for the copy of your interest- 
ing lecture to the scientific society. I rejoiced in the clearness 
of your style, which enables even the uninitiated to glance 
into the secrets of your science, and by its wit and poetic 
taste redeems it from its usual dryness. It is admirable 
in you amid your heavy labours thus to find time to refresh 
yourself with the poets, and to round the realism of your 
nature-studies with art and poetry. I wish my son had some- 
thing of your spirit: he is so little able to escape from his 
scientific rigidity of expression, even in an essay read before 
a Society in Ko"nigsberg, that I am filled with respect for an 
audience that could understand and thank him for it. I confess 
that when I read it much remained very obscure to me/ 

While Helmholtz was thus engaged in fundamental researches 
which had the common aim of building up a mechanical con- 
ception of the universe (in the best sense), he lighted casually, 
and as the fruit of his lectures at the end of 1850, upon his 
discovery of the ophthalmoscope, which ' revealed a new 
world 1 to the ophthalmologists, and which, with the doctrine 
oi the Conservation of Energy, did most to establish and extend 
his reputation. 'The excellent discipline every University 
teacher is subject to, in being obliged each year to treat the 
whole of his subject so as to convince and satisfy the best 
of his students/ resulted on his own avowal in this splendid 

After communicating his invention to the Physical Society 
in Berlin, on December 6, he wrote on December 17, 1850, to 
his father: 

' In regard to time- measurements I have at present no new 
results, but have devoted myself to the construction of fresh 
apparatus and necessary preliminaries. But I have made a 
discovery during my lectures on the Physiology of the Sense- 
organs, which may be of the utmost importance in ophthalmo- 
logy. It was so obvious, requiring, moreover, no knowledge 
beyond the optics I learned at the Gymnasium, that it seems 


almost ludicrous that I and others should have been so slow 
as not to see it. It is, namely, a combination of glasses, by 
means of which it is possible to illuminate the dark back- 
ground of the eye, through the pupil, without employing any 
dazzling light, and to obtain a view of all the elements of the 
retina at once, more exactly than one can see the external 
parts of the eye without magnification, because the transparent 
media of the eye act like a lens with a magnifying power of 
twenty. The blood-vessels are displayed in the neatest way, 
with the branching arteries and veins, the entrance of the 
optic nerve into the eye, &c. Till now a whole series of most 
important eye-diseases, known collectively as black cataract, 
have been terra incognita, because the changes in the eye were 
practically unknown, both during life, and, generally speaking, 
after death. My discovery makes the minute investigation 
of the internal structures of the eye a possibility. I have 
announced this very precious egg of Columbus to the Physical 
Society at Berlin, as my property, and am now having 
an improved and more convenient instrument constructed to 
replace my pasteboard affair. I shall examine as many patients 
as possible with the chief oculist here, and then publish the 

The ophthalmoscope was, however, some time in making its 
way, on account of the mathematical and physical knowledge 
presupposed by the ' Description of an Ophthalmoscope for the 
Investigation of the Retina in the Living Eye', published in 
the autumn of 1851, and people were at first very shy of 
employing it. One distinguished surgical colleague told Helm- 
holtz he should never use the instrument it would be too 
dangerous to admit the naked light into a diseased eye; 
another was of opinion that the mirror might be of service 
to oculists with defective eyesight he himself had good eyes 
and wanted none of it. But by December 16 of the same year 
Helmholtz was able to write to his father : 

'Eighteen orders for the ophthalmoscope have dropped in, 
one after the other, so that my mechanician is doing a good 
trade. The world is getting to hear of it.' 

Forty years later he tells the story of its discovery : 

'While preparing my lectures I hit upon the invention of 
the ophthalmoscope, and then on the method of measuring the 


velocity of nervous impulses. The ophthalmoscope became 
the most popular of my scientific achievements, but I have 
already pointed out to the oculists that good fortune had more 
to do with it than merit. I had to explain the theory of the 
emission of reflected light from the eye, as discovered by 
Briicke, to my students. Briicke himself was but a hair's 
breadth off the discovery of the ophthalmoscope. He had only 
neglected to ask himself what optical image was formed by the 
rays reflected from the luminous eye. For his purpose it was 
not necessary to put this question. Had it occurred to him, 
he was just the man to answer it as quickly as I, and to invent 
the ophthalmoscope. I was turning the problem over and over, 
and pondering the simplest way of making it clear to my 
audience, when I came on the further issue. 

' The oculist's perplexity in dealing with the condition known 
at that time as black cataract was familiar to me from my 
medical studies, and I at once set to work to manufacture the 
instrument out of spectacle lenses and the cover-glasses used 
for microscopical objects. At first, however, it was very diffi- 
cult to use. I might not have persevered save for my convic- 
tion that it must succeed; but after about eight days I had 
the great joy of being the first to see a living human retina 
exposed before me/ 

As a matter of fact the discovery of the ophthalmoscope had 
not been quite such a simple invention as Helmholtz describes 
it. The principle underlying the apparatus was difficult to 
grasp without considerable knowledge of optics, and its intro- 
duction was therefore a comparatively slow matter, and was 
delayed until improved mechanical conditions rendered the 
handling of it much simpler although Bonders, the cele- 
brated physiologist at Utrecht, held the original form of 
Helmholtz's instrument to be optically perfect. 

The familiar fact that the eyes of certain animals, such as 
cats and owls, glisten in the dark, had already been correctly 
interpreted by Johannes Muller to mean that these so-called 
'glowing' eyes do not really glow but only reflect light, 
and that the retina of the eyes that glisten most are pro- 
vided with a background specially adapted for the reflection 
of light. Brucke had shown that the eyes of animals seem 
to glisten most when the beam of a dark-lantern is thrown 


into the eye that is to be examined, the observer looking 
past it. All eyes can be made to glisten, both in animals and 
in man. The first human eye purposely made to shine in the 
dark was du Bois-Reymond's, illuminated by Brucke. Brucke's 
subsequent attempts at constructing an instrument for the 
illumination of the retina failed (according to Graefe) on account 
of the mode of illumination he adopted. 

Helmholtz asked himself in the first place, as he related 
in the monograph published in Berlin, 1851, why all that we 
can see of the background of the uninjured eye appears 
absolutely dark; he ascribed this to the refractive media of 
the eye, which under normal circumstances prevent us from 
seeing illuminated points of the retina behind the pupil. The 
first requisite therefore was to find a source of illumination 
by which just that portion of the retina that we see through 
the pupil may be adequately illuminated. By means of a little 
camera obscura blackened inside he proved by calculation and 
experiment that for any system of refractive surfaces, the 
reflected rays, even when they have passed through the re- 
fracting media and left the eye, must be wholly congruent 
with the incident rays, and that they all return ultimately to 
the original point of illumination. 

Since the observer's eye cannot be brought into line with 
the reflected light without intercepting the incident rays, no 
light can be returned to his pupil from the recesses of the 
observed eye that has not emanated from his own. Only 
those points of the retina will accordingly be visible on which 
the dark image of his own eye is projected. Further, if the 
observed eye is not a perfect refracting system, part of the 
light reflected from it will indeed return to the luminous 
point, but part passes by it, and it becomes possible for an 
observer, placed as nearly as possible in the line of the incident 
light, to perceive a little of the light that emerges; and this 
produces the luminosity of the human eye as discovered by 
Brucke. In order to obtain an exact image, some method is 
required which makes it possible to look into the eye not only 
in the approximate, but in the exact line of the incident light, 
and Helmholtz found, in trying to make the image as bright 
as possible, that this can be done by superposing three parallel 
glass plates, in which the light from a source of illumination 


placed at one side of the observing eye is reflected. A portion 
of this light will then enter the subject's eye, and illuminate 
its fundus; the rays of light reflected from the illuminated 
background on to the glass plates will now return partially 
from the surface of these to the source of light, but another 
part of the reflected rays will pass through the glass plates, 
and reach the eye of the observer. But the visual field of 
the latter, limited by the pupil of the observed eye, is so small, 
on account of the relatively considerable distance of one eye 
from the other, that it would be impossible to combine the 
observed details into a whole ; it is therefore imperative to bring 
the two eyes as close to one another as possible. This causes 
the image as a rule to fall behind the observer, who cannot 
distinguish it plainly, and since a normal eye can only combine 
parallel and diverging rays upon its retina, and not those that 
are convergent, Helmholtz (as the readiest means of making 
the converging beams diverge) inserted a concave lens between 
the mirror and the eye of the observer, and by this simple 
contrivance provided the essentials of his ophthalmoscope. 
He also suggested a number of other practical methods of 
constructing the instrument. 

'The construction of the ophthalmoscope, 1 says Helmholtz 
at a later time, in discussing his medical and physiological 
work, 'was a turning-point in my position in the eyes of the 
world. From that moment I found favour with authorities 
and colleagues, and was left free to follow the promptings 
of my scientific curiosity. I attribute my subsequent success 
to the fact that circumstances had fortunately planted me with 
some knowledge of geometry and training in physics among 
the doctors, where physiology presented a virgin soil of the 
utmost fertility, while on the other hand I was led by my 
acquaintance with the phenomena of life to problems and 
points of view that are beyond the scope of pure mathe- 
matics and physics/ 

The invention of the ophthalmoscope, with the mechanical 
constructions and modifications which it entailed, as well as 
a series of experiments in physiological optics undertaken 
with the instrument, took up no more than the early weeks 
of 1851. Helmholtz discovered among other points that light 
which impinges directly on the optic nerve does not give rise 


to any sensation, but that it must fall on a nerve-ending in 
the retina before it can be perceived. He then returned to 
the work that had been so happily interrupted, on the stimula- 
tion of nerve. In his earlier experiments he had assumed the 
induction shock by which the nerve was excited to be instan- 
taneous, and he now proceeded, in order to justify this view 
for the minute time-intervals involved, to answer the question 
previously raised as to the exact time at which an induction 
current produces its physiological effect. It was important 
not to ascribe to some action in the nerve a loss of time 
which might have occurred in the electrical apparatus. This 
is especially true of experiments on man, when the currents 
are of such a strength that serious errors might creep in. In 
the middle of April he sent du Bois a short note on this 
physical preliminary to his further work on nerve, for the 
Academy, together with the longer paper for Poggendorff's 
Annalen. Du Bois-Reymond replies : 

' My brain reels at your appalling industry and encyclopaedic 
knowledge. How can you get up new lectures and still carry 
on all this work ? All the same I am not quite satisfied with 
your exposition. I have read your essay and the abstract 
several times without understanding what you did, or how 
you did it. At last I discovered the method for myself, and 
then I saw what you are driving at. Don't be vexed if I say 
that you must take more pains to get away from your own 
standpoint, and place yourself on the level of those who do 
not yet know what it is all about, and what you want to 
tell them/ 

This reproach, however, was unjustified, for in the nature 
of the case the treatise presupposed that its readers would 
be trained physicists and mathematicians. Both then and 
later Helmholtz was in the habit of writing and re-writing 
many parts of his papers four and even six times, altering 
the arrangement before he was satisfied, and never holding 
an investigation to be finished till it presented itself to him 
in logical completeness, correctly formulated. Accordingly he 
replies : l As regards the form of the essay, I took particular 
pains with it, and finally satisfied myself. But it is true that 
the more one elaborates a thing the harder it often becomes 
to understand. It is a very difficult subject to deal with.' 


In 'The Duration and Nature of the Electrical Currents 
induced by Variations of Current' (Poggendorff's Annalen) he 
begins by stating a mathematical law, which he had verified 
by a long and difficult series of experiments. By means of 
this law F. Neumann was enabled to solve the problem he 
had previously laid aside, as to the distribution of current 
in a copper disk rotating below the two poles of a magnet 
carrying out the integrations without neglecting secondary 
inductive actions, and propounding theorems that could be 
experimentally tested. If an electric circuit contains voltaic 
cells and a coil, and if / be the intensity of the battery 
current, W the resistance of the circuit in absolute units, 
/ the time, and P the self-induction of the coil which depends 
only upon geometrical relations, then a stationary magnet will 
be deflected by the inducted current alone through an angle 
directly proportional to P and /, and inversely proportioned 
to W\ but if the battery current alone acts on the magnet 
for the very brief time /, a deflection proportional to the 
products of 7 and t will result ; hence it follows directly 
that the battery current in the time represented by the quotient 
of P and W produces the same effect as the whole of the 
induced current. Dove had already pointed out that the 
intensity of the counter-current induced on closing the circuit 
is always less than that of the inducing current, while the 
weaker current requires more time to produce the same effect 
than the stronger, whence it follows that the minimum duration 
of the counter-current at closure is the quotient P/W. Now, 
since this minimum can be augmented at will, by reducing the 
resistance W of the circuit, and increasing the self-induction P 
of the coil by increasing its mass, it is experimentally possible 
(and this is the cardinal point of the theorem) to make the 
time required by the current to reach the same value at all 
parts of the circuit very small by comparison with the above- 
mentioned interval, and conditions can be brought about under 
which the rate of transmission of the electric current in the 
circuit is imperceptibly small as compared with the fractions 
of time, in which the intensity of the current is not perceptibly 
altered. This, however, gives the necessary conditions for 
the application of Ohm's Law, i. e. the equalizing of current- 
strength throughout the circuit ; for the alterations in intensity 


of induced currents are effected so slowly that the strength 
of current is equalized in the entire circuit, and the intensity of 
current in a simple circuit is then determined by Ohm's Law, in 
the form of an exponential function of time, on the assumption 
that the induced electromotive force lags by a very small interval 
behind the variations of the inducing current. After Helmholtz 
had extended this exponential law mathematically to divided 
circuits, he tested it experimentally by means of a new type 
of galvanic contact-key, which made it possible to vary the 
interval between the opening and closing of any current, as 
required. Instead of measuring the time directly, he calculated it 
from the action of the currents upon a magnet, by a modification 
of Pouillet's method. Thus he proved that, subject to the 
assumption that the induced electromotive force coincided in 
time with the inducing variations of current, no inducing 
action was present y^^o- second after breaking the current 
in a coil. By this means the time-intervals due to the electric 
current were distinguished with mathematical accuracy from 
those due to nerve-action, and Helmholtz was free, after this 
absolute verification of his methods, to devote himself to the 
complicated experiments on the excitation of nerve that 
occupied him during the remainder of 1851. 

In the autumn holidays of 1851 Helmholtz carried out his 
project of inspecting some other physiological laboratories. 
After taking his wife and child to Dahlem, he went in the 
first place for a few days' rest to his Berlin friends, Professor 
Heintz and his family, at Halle. Then after a short stay in 
Kassel he went on to Gottingen. His first visit was to 
Professor Ritter, one of his former teachers at the Gymnasium 
of Potsdam, and his father's faithful friend, in whose company 
he 'saw the little town, rather better built than Halle and 
Konigsberg, with the University as its fulcrum ' ; after which 
he called on the professors, though somewhat hampered by 
the presence of the King. He describes them in a letter to 
his wife as 'the aristocracy of the town; one can see that 
they are alive to their own merits, and a little inclined to 
over-estimate the accomplishments of their circle not specially 
in regard to myself, but obviously in referring to any third 
person'. His lively descriptions show that he much enjoyed 
this visit to the Gottingen doctors : 


' I have found even more people to visit here than I ex- 
pected, and Institutes where no money has been grudged 
on the equipment. The physiologist Wagner, an old man, 
who is conscious of his own importance, and evidently appre- 
ciates the notice taken of him by the King (the title Hofrath 
counts for more here than Professor), is not quite up to the 
level of the physical knowledge required nowadays, but he 
feels that, and is careful not to give himself away. The 
physicist Weber, who, next to Neumann, is certainly the 
greatest mathematical physicist in Germany, showed me a 
great deal of very interesting and very perfect physical 
apparatus, but with less apparent cordiality than his brother 
in Leipzig. I also met a young anatomist and physiologist 
Bergmann ; an oculist Ruete, who has done important work 
on the physiology of the eye; an accomplished surgeon 
Baum, recently imported from Greifswald; a mathematical 
optician Listing, whom I had not heard of before, but who 
certainly deserves to be known ; and lastly a philosopher 
Lotze, who has worked a great deal at the principles of 
pathology and physiology, but is unfortunately too hypochon- 
driacal and self-centred for one to get any exchange of ideas 
out of him in such a short time. All these people received 
me with the greatest sympathy and cordiality, and gave me 
all the time they could spare; it was agreeable to find that 
they were in touch with my somewhat intricate nerve work, 
and approved of it, or at least had apparently sufficient 
confidence in my physical knowledge (for which Weber is 
responsible) to accept my results. The ophthalmoscope is 
a splendid toy to travel with ; I demonstrated it this morning, 
and it is making quite a sensation here. This evening several 
of them are going to take me out walking, if they are not 
summoned to the King. On the other hand, I was surprised 
to find that they do not take kindly to du Bois-Reymond's 
conclusions: they query here and there, and do not see the 
importance of the work, and I have had to stand up for it. 
Their objections are based on the opinion of the Weber who 
is here, and on the Paris Commission, and are also due to 
imperfect comprehension of the subject. Little inventions 
like the ophthalmoscope make a better impression. I am 
demonstrating my frog-curves everywhere.' 


From Gottingen, Helmholtz went on by Marburg, where he 
called on Knoblauch and the physiologist Nasse, to Giessen, 
in the hope of making acquaintance with Liebig, for whom 
he had a great admiration. 

* Liebig, the king of chemists, as he himself and his scholars 
think him, was unfortunately away; he has gone to London 
to see the Exhibition, and be feted by the English. I much 
wanted to meet him. All I could do was to see his empty 
laboratory, to which students flock from the whole of Europe 
and America, for practical work, and which was shown me 
by his son, a young doctor who studied physiology with du 
Bois-Reymond at one time, but will probably go into practice. 
I was surprised to find no remarkable appliances ; on the con- 
trary everything was covered with dirt; there were very few 
people working. It presented an extraordinary contrast to 
the laboratories of Heintz and others, w r hich are at least as 
convenient, far better equipped, clean, and tidy. But externals 
matter little. For in spite of his vanity Liebig is the greatest 
of living chemists, and his renown as a teacher has spread 
far and wide/ 

He next went by Frankfurt, where he revived his old 
admiration of Lessing's Huss, the Ezzelino, and two small 
landscapes by the same master, to Heidelberg, where he 
found Henle, and comments on his laboratory as being ex- 
cellent for anatomy, but poorly equipped for physiology. 
Henle explained that he had only taken on physiology and 
general pathology in addition to anatomy, as a temporary 
arrangement, in consequence of a disagreement with Tiede- 
mann, and hoped he would soon be given a physiological 
colleague. ' He suggested to me/ writes Helmholtz to his 
wife, ' what may make a great difference to our future : viz. 
he and the younger professors of the Medical Faculty are 
desirous that I should be offered the Chair at Heidelberg. 
We must consider this. Heidelberg would not be a bad 
place to work in : the Germans have rather left it, because 
there is a dearth of teachers for the moment, but students 
are still coming from North America, Brazil, England, France, 
Greece, and Russia/ 

After staying some hours in Baden-Baden, Helmholtz 
went to Kehl, and crossed the bridge over the Rhine into 


the French Republic. 'There was plenty to amuse one. 
Liberty, Fraternity, and Equality were flaunted everywhere ; 
Property of the Nation was posted on all the public buildings, 
and many private houses displayed other frightfully demo- 
cratic devices. The country people and the lower classes 
in the town looked just the same as in Baden, only they seemed 
to be more stupid, but the better parts of the town appeared 
thoroughly French/ 

After admiring the Cathedral of Freiburg, he visited the 
1 World's Wonder ' at Schaffhausen, and was greatly impressed 
with it. 

1 1 went down the hill to the bank : it certainly looked rather 
bigger, more like a waterfall, but still I went to bed somewhat 
disappointed. Next day, however, I got a different impression 
of it. One does not appreciate its proportions in the evening, 
because it is surrounded by rocks 200 to 300 feet in height, so 
that the 6o-foot waterfall looks small beside them, and one 
cannot see the chief beauty, the wonderful dark-green colour 
of the water, which makes a magnificent effect as it mixes 
with the white foam. The effect, however, is overwhelming 
when one goes to a stage that has been erected at the edge 
of the Fall, where the appalling mass of water dissolved in 
foam and mist plunges down close to one. At first the sight 
is hardly bearable, one loses breath, and feels drawn after it. 
Afterwards, in spite of the constant shower of spray, I revelled 
in the spectacle of this force and motion, and could hardly 
tear myself away. . . . On Friday evening I reached Zurich, 
and looked up Ludwig, who received me with great cordiality. 
He has a noble and delightful nature, and is greatly improved 
since he has got rid of that Bohemian manner. This is ap- 
parently thanks to his wife, whom I have only learned to 
know at present by her quiet, sensible ways. . . . He is a man 
of the utmost kind-heartedness, and has formed an extravagant 
opinion of my excellences, related to him partly by du Bois. 
If you heard all the praises he showered upon me, you 
would certainly have been satisfied with him. He is extra- 
ordinarily industrious, works continuously in the right direc- 
tion, and is adored by his students, as many of them said 
and showed me, so that besides all the good work he has 
done I hope still better from him. But he is somewhat 

G 2 


weary and hypochondriacal, perhaps from his arduous work. 
He was ceaselessly engaged in finding amusement for me, 
and kept every one else away, so that he might talk to me 
alone. And talk we did, about every conceivable subject 
in physiology and physics. ... In the morning I generally 
went to the Anatomy Department with Ludwig, and looked 
at experiments, instruments, and collections ; in the afternoon 
we went out into the country, except on one day when it 

From here Helmholtz set out on his first Swiss journey, 
and went to all the places he afterwards revisited so many 
times. In the letters to his wife he gives his impressions 
with youthful vigour and enthusiasm, along with many a 
scientific and aesthetic appreciation, such as we find so 
frequently later in his papers and lectures. 

After climbing the Rigi, he wandered on by Fluelen over 
the Gothard and the Furka Passes to the Rhone Glacier, 
whose blue ice-slopes made a deep impression on him. ' By 
glacier you must not picture the snow-covered tops of moun- 
tains, but masses of ice that have slidden down into the valleys, 
to melt there while they are perpetually renewed from above. 
Picture the Brauhausberg in Potsdam made of ice, and packed 
into a narrow valley between gigantic peaks of rock, above 
that another precipice of 1,000 feet, on which the ice-blocks 
are piled up, and tumble thence to renew the lower masses, 
while the whole is pierced with innumerable sky-blue rifts, 
and then you have a picture of the Rhone Glacier/ He 
rejoiced in the enchanting loveliness of the Rosenlaui Glacier, 
where the sun streams through the ice into heavenly blue 
caverns and fissures ; climbed the Faulhorn ; visited the 
upper Glacier at Grindelwald; and then stayed a few days 
in Interlaken, where he resumed the scientific correspon- 
dence with his friends du Bois-Reymond, Briicke, and 
more particularly Ludwig, with whom his stay in Zurich 
had united him more closely than ever. One of his letters 
to Ludwig is interesting from its reference to a candidate 
for the Chair of Physics at Zurich : ' On the contrary I 
think you might do great things by joining forces with 
Kirchhoff; he is extraordinarily clear-headed and perspica- 
cious in the most complicated questions ; I much wish for 


your sake and that of physiology that Kirchhoff might 
go to Zurich/ 

After crossing the Gemmi he went to Leukerbad, thence 
partly on foot, partly on horseback, to Lago Maggiore, and 
on by Como to Milan, 'a great and splendid town with all 
the brilliancy of Italian life. In beauty of form the Cathedral, 
Milan's pride, is far behind the Gothic cathedrals of Germany. 
Its Gothic forms are mere arbitrary decoration ; but they 
are tastefully applied, and the innumerable pillars and arches, 
and well-carved statues, all standing out in white marble 
against the blue sky, are a sight that cannot be imagined. . . . 
We went, too, to the ruins of Leonardo da Vinci's master- 
piece, "The Last Supper," and to the Picture Gallery in the 
Brera Palace/ 

At length he reached the city that he had so desired to 
see from boyhood. 

' Venice is the city of wonders, a living fairy-tale. In spite 
of all one has seen in pictures, or heard described, the reality 
surpasses everything. The Piazza, of St. Mark, with its rich 
mosque-like church, enclosed within the rows of palaces, the 
countless lights above the deep-blue moonlit heavens, and 
a few paces off the deep-blue sea, with crowds of people as 
though it were a festa, all make up an indescribable picture. 
To-day and yesterday we went round with a great crowd to 
see all the wonders ; but one becomes almost wearied with 
these impressions. The historical memories, the extraordinary 
wealth which Venice has harvested from half the earth, the 
art treasures, for the most part still in their pristine freshness 
of colour, cannot be overlooked. ... In Germany we can only 
form a poor idea of Italian Art ; here one can drink it in fully. 
I went alone to the Accademia, to enjoy the masterpieces to 
the full, and did not repent me, but found great satisfaction, 
such as one cannot obtain in Germany. It is a collection of 
the masterpieces of the Venetian school, including Titian's 
great " Assumption of the Virgin ", which I knew already from 
engravings. But engravings are even a worse substitute for 
this than a piano score for a symphony, since the indescribable 
beauty of the work consists in its miraculous light and colour. 
I have never seen the like, nor can one imagine it till one 
has seen it, because this kind of beauty is of another order 


from our German pictures. Moreover this one work is 
unparalleled among the rest of the Italian paintings that I 
have seen here, although many give this feast of colour 
in an extraordinary degree, and one sees the greatest 
number of inspired and ideal heads that you could possibly 

'When I had seen the Accademia, I wanted no more, but 
prepared for departure, strolled a little in the streets, heard 
the band in the evening on the Piazza of St. Mark, and then 
at 10 p.m. went off in a gondola to the steamboat. As we 
sailed away the moon had risen ; we left the lights and palaces 
of beautiful Venice, and fared out through the openings of the 
canals in the Lagoon, to the still blue Adriatic/ 

From Venice, Helmholtz went by Trieste to Vienna, attracted 
thither by his old friendship with Briicke, to whom he wished 
to bring the ophthalmoscope in person. 'We arrived so early 
that I went first with Herr R. to his hotel, washed, breakfasted, 
and then, about nine o'clock, went on to Briicke. He was 
much pleased to see me, and I at once took up my abode with 
him. Then directly after appeared Professor Wagner from 
Gottingen, and the next day Professor Bunsen from Breslau, 
one of our most gifted chemists, so that we are quite a learned 
society. Briicke is just the same : he looks rather better, and 
is as cheerful, calm, and friendly as usual ; his wife is pretty, 
and has the same pleasing, cheerful manner. ... As for Vienna, 
I have so far seen only scientific things because it is generally 
raining. On Friday, Briicke showed us his Physiological Insti- 
tute first of all, and we admired living chameleons, strange 
creatures with strikingly Egyptian characteristics. In the 
afternoon we were able to take a short walk, and discussed 
how it would be possible to help du Bois, but could not hit 
off anything. In the evening, the ophthalmoscope for Briicke's 
benefit. On Saturday morning I went to the mortuary in 
the Hospital to see the celebrated pathological anatomist, 
Rokitansky. Briicke and Wagner had a competition 
with their splendid microscopes, and both won. Afterwards 
I demonstrated the ophthalmoscope to Wagner and his 
friends, and to Bunsen, and in the afternoon showed Briicke 
my induction work. In the evening there was company 
at the philosopher Lott's, where I found Wagner and 


many of the Vienna professors. The atmosphere was 
pleasant and cordial, but they told a good many rather trivial 

' Sunday. In the morning Rokitansky demonstrated his 
splendid collection of specimens in pathological anatomy to 
us, and we saw the famous museum of wax models. In the 
afternoon a projected expedition to SchCnbrunn was frustrated 
by the weather. Brucke, Wagner, and I therefore went to 
look at two famous statues by Canova a monument of a 
princess in the Augustinian Church, and the statue of Theseus 
in the Public Gardens. Neither of them was the least com- 
parable with what I had seen in Italy. Then we walked round 
the city on the walls, which are rather pretty, fled to Wagner's 
hotel in a storm, and had some wise talk with him/ 

Apropos of this conversation he writes a few days later to 
Ludwig : ' Rudolph Wagner was there too, and wanted to know 
what we thought about the relation between soul and body, 
and other obscure points of physiology. He seems much 
concerned with these points, about which one can hardly say 
anything. Bunsen was also present, and urged me to go with 
him to Breslau.' 

After fetching his family from Dahlem, Helmholtz returned 
refreshed in mind and body with them to Konigsberg, and 
at once resumed his experiments on the excitation of nerve, 
the importance of which was being more and more recognized 
by physiologists. On his father's birthday he sends him the 
welcome news : 

'The French Academic inform me in a very courteous 
manner that they have appointed a Committee to draw up a 
report on my communications on the measurement of time- 
relations. For the moment the Committee will not be in a 
position to complete its report, since it will not be possible 
to repeat the experiments ; but it shows that they are alive 
to the thing. My official relations here are unchanged. 
Only I hear privately, and beg you not to repeat, that my 
Faculty have petitioned the Ministry to make me Ordinary 
Professor. I hear no more of Heidelberg. A second 
paper on time-relations is ready. In the Christmas holidays 
I am to draw up a report on du Bois J work for the Kieler 


After sending the account of his experiments on the graphic 
record of the rate of nervous transmission in the frog to 
Johannes Miiller for the Archiv, he employed the Christmas 
holidays, and the first weeks of the New Year, in writing 
a report (originally intended to be a popular lecture), at the 
request of Karsten, upon ' Recent Developments in Animal 
Electricity '. On February 2 he tells du Bois that in reading 
his book with the aim of summarizing the work on animal 
electricity, he has discovered a theorem which seems to him 
completely to resolve the difficulties as to the co-ordination 
of the different elements of a muscle; by a combination of 
the laws of electric potential and of the superposition of 
currents, he had succeeded in proving that when electromotive 
forces are distributed in any way in any conductor, all external 
action exerted by the conductor, i. e. all the derived currents 
which it excites in any linear or non-linear circuit, may 
be replaced by a distribution of electromotive forces upon 
its surface, just as the external action of a magnet can be 
stated in terms of the distribution of magnetic fluid upon its 

In the meantime the Prussian Ministry were not slow to 
accept the recommendation of the Medical Faculty, and to 
appoint Helmholtz, who was now recognized everywhere as 
a physiologist and physicist of the first rank, to be Ordinary 
Professor of Physiology, which he became by Royal Brevet 
on December 17, 1851. 

While he was busying himself with the required Inaugural 
Dissertation, for the subject of which he turned to the Physio- 
logical Theory of Colour, Part II of his great work on the 
physiology of nerve and muscle appeared in Mullers Archiv 
under the title ' Measurements of the Rate at which Excitation 
is transmitted in Nerve'. In Part I he had already proved, 
by means of the electro-magnetic method of time-measurement, 
that the mechanical response of a muscle made its appearance 
later after excitation of the nerve, when the excitation had 
to travel through a longer portion of nerve before reaching 
the muscle, but the application of this method involved pro- 
tracted experiments, and necessitated a favourable condition 
of the frog's tissues, on account of the long duration of the 
experiments. He now endeavoured, with a graphic method 


of time-measurement previously described in Part I by which 
a contracting muscle records the magnitude of its twitch upon 
a travelling surface to find a simpler method of confirming his 
determination of the rate of nervous transmission. After Ludwig, 
with the kymograph, had succeeded in recording the variations 
of blood-pressure in the vessels of a living animal, Helmholtz 
constructed his myograph for the graphic record of the con- 
traction of a muscle its principle being that a lever lifted 
by the twitching muscle traces a curve upon a surface moving 
at uniform speed, the vertical co-ordinates of which are pro- 
portional to the shortening of the muscle, the horizontal to 
the time. If two curves are recorded one after the other in 
such a way that the writing-point is always at the same place 
on the travelling surface at the moment of excitation, then both 
curves will start from the same point, and it can be seen from 
their congruence or non-congruence whether the different 
stages of the mechanical response of the muscle occur in both 
cases at the same interval after excitation. Two years later 
Helmholtz published important additions to these experiments 
on the frog, as recorded with the myograph. 

Simultaneously with the above he published in the Kieler 
Monatsschrift, for April, 1852, the essay previously announced 
to du Bois on the ' Results of Recent Researches in Animal 
Electricity'. He gave a masterly sketch of the development 
of nerve physiology, and described the interest attaching to 
the 'investigation into the nature of the mysterious agent 
which, acting along barely visible nerve-threads, produces 
such fine gradations, such mighty energies, such complicated 
exchanges of sensation and motion an agent that is the first 
link in the chain of processes which connect the mind that 
feels and wills with the material outer world, enabling it to 
receive and give out impressions '. As early as 1743 the Leipzig 
mathematician Hausen had expressed the opinion that this 
agent might be identical with electricity. Helmholtz now ex- 
pounded the opposite theories of Galvani and Volta, the first 
of whom regarded animal electricity as the source of the 
electrical phenomena in all his experiments, while the latter 
by his theory of contact electricity, which led him to the most 
brilliant discoveries, had pushed the experiments relating to 
animal electricity proper completely into the background. He 


cites the long roll of Matteucci's painstaking experiments (to 
which he was obliged to return many years later in another 
connexion), and finally embarks upon a masterly exposition 
of du Bois-Reymond's discoveries : ' the fruits of assiduous 
study, and of ten years' labour consistently concentrated upon 
one aim, during which the frog and the divisions of the 
galvanometer were his world a rare example of methodical 
observation, of rich knowledge, and of that perspicacity of 
conception which is learned in the school of mathematics/ 
After communicating the most important results of du Bois' 
investigations, experiments which he himself demonstrated to 
his audience by the now familiar method of throwing a beam 
of light from a mirror connected with an astatic system of 
magnets upon a graduated scale, he goes on to say that certain 
physiologists assume what is propagated in the nerve during 
excitation to be some definite form of motion like the undula- 
tions that are propagated as sound-waves in the air, and as 
light-waves in the ether. He submits that electrical phenomena 
also lead to the idea of such a motion, since the extra- 
ordinary rapidity of the variations of electromotive force, both 
in magnitude and direction, makes it probable that this force 
affects very mobile particles, and that the orientation of these 
particles is temporarily altered by excitation, from the excited 
point of the nerve onwards to the muscle, and within the 
muscle itself. He considers the unexpectedly low rate of 
propagation in nervous excitation as determined by himself 
to be incompatible with the older view of an immaterial or 
imponderable principle as the nervous agent, but quite in 
harmony with the theory of the motion of material particles 
in the nerve substance. 

This publication concluded the series of the closely-con- 
nected physiological investigations which Helmholtz had begun 
immediately after the publication of his thesis, and he now 
turned to physiological optics. In this subject he worked 
out new physical principles, upon which, as physicist, physio- 
logist, philosopher, and aesthetician, he erected a structure 
of such extent and security as had never been dreamed of, 
which to this day arouses wonder and astonishment. 

He had already instituted comprehensive experiments on 
the law of colour-mixture, in order to correct an error of 


Newton, which had been perpetuated in the following centuries ; 
and had since his visit to Vienna been in correspondence with 
Brucke, who was much occupied with such investigations. 
On December 22, 1851, Brucke writes to Helmholtz: 'As 
you know, Goethe in his unfortunate theory of colours 
explains all colours by the overlapping of light and dark, 
basing his argument on the well-known fact that transparent 
media in front of a dark ground may look violet-grey, blue- 
grey, and blue, while in transmitted light they look brown, 
yellow, or red. I have also noticed in my experiments on 
chamaeleons, how often very distinct colours are produced in 
this way in the animal kingdom, and have not found this 
phenomenon explained by the undulatory theory, although 
the explanation seems obvious enough. . . . Pray tell me if the 
different colouration of the reflected and refracted rays has 
been considered in optics/ 

Brucke had no idea that Helmholtz had already made funda- 
mental discoveries in this direction, which were to astonish 
physicists and physiologists alike in the dissertation he was 
about to publish. His father, to whom Helmholtz made a 
short communication of the contents of the paper, writes to 
him enthusiastically on April 5 : ' Your letter of the 2ist gave 
us as much pleasure as all its predecessors, and there was the 
usual murmur of impatience till every one had read it. May 
God fulfil you ever more and more as a prophet of truth 
and fountain of wisdom, so that you may not have lived in 
vain for eternal Humanity, but may ever continue one of its 
corner-stones on earth ; then I shall find consolation for the 
lack of results in my own life. God have your health in His 
keeping, and grant you increasingly such a position in externals 
that your intellectual life may find its full development. I am 
most curious about the interesting work that you have chosen 
for your thesis ; you will clash with Goethe there ! ' 

He is delighted when Helmholtz tells him that the coming 
summer may bring great changes in his external position : 

'There are three physiological vacancies in prospect. So 
there may be a regular migration, in which each must try 
for the best new place. Not that there are many candidates 
in the field, du Bois, Eduard Weber in Leipzig, and I being 
almost the only ones.' 


While Helmholtz was engaged upon his inaugural disserta- 
tion (the scope of which grew wider and deeper, until it 
eventually took shape as a searching critique of Brewster's 
contributions to optics), du Bois-Reymond, who had lately 
returned in such good spirits from England that he warns 
Helmholtz 'not to go to England, it spoils one's taste for 
Germany', advises his friend on June 15, that he had told 
Sir David Brewster about the ophthalmoscope, and that if 
a copy of Helmholtz's paper on it were sent, Sir David would 
be responsible for an English translation. Helmholtz replies 
in a few days : 

' I should imagine that you would like England under such 
conditions. I can hardly avail myself of Brewster's proposal 
to assist in the preparation of an English translation of The 
Ophthalmoscope, because the second part of the essay on 
physiological optics which I intend to use for my disserta- 
tion, and which I shall give as a lecture, and then send to 
Poggendorff, is intended as a contradiction of Brewster's 
analysis of solar light, a theory which he has much at heart, 
and has defended with some heat. His observations on this 
subject are perfectly correct, but the alteration of the colours 
of the spectrum by absorbing media depends mostly upon 
subjective phenomena, contrast and the like, as may be 
proved convincingly. My treatise is of course written as 
cautiously as possible, but still I fear that Brewster may 
take it amiss.' 

On June 28, 1852, Helmholtz delivered his Inaugural Lecture 
'On the Nature of Human Sense- Perceptions ', in which he 
not only displayed his unique gift for making the stiffest 
scientific problems intelligible by a lucid and exceptionally 
beautiful exposition, but once more opened up new fields 
of inquiry, lying ' nearer the limits of human knowledge '. It 
was not until a much later time, after the publication of his 
Physiological Optics, that the full import of the physical, 
physiological, and epistemological discoveries which he had 
even then made could be recognized. It was this lecture 
that at last won him the complete approval of his critical 
father : 

'Thanks for the Inaugural Lecture sent me by Dr. Fried- 
lander, which pleased me greatly by its clearness and its 


unaffected popular style. You make the conclusions of 
science comprehensible even to the laity, showing where the 
details are leading to, and what is the way and aim. It 
almost seems to me that this mathematical-empirical method 
of investigation, when once it develops into a definite art, and 
no longer depends upon individual genius, may inaugurate 
a new, perhaps slow, but certain way to philosophy, which 
will at any rate define exactly the objective substratum of all 
knowledge, rendering its nature indubitably clear, and thus 
establishing the ego-doctrine of Fichte as the only possible 
mode of philosophical thought/ 

In a letter written in September, to announce the birth of 
his son Richard, Helmholtz gratified his father by admitting 
that it had (as the latter surmised) been his intention in the 
lecture to give an empirical statement of Fichte's fundamental 
views of sense-perception, and expresses pleasure that his 
father is content with the form of the dissertation, and approves 
of his philosophical opinions. 

His general philosophical and epistemological views were, 
however, quite unlike Fichte's, since they were based upon 
exact investigations, the results of which he set forth in the 
inaugural thesis ' On the Theory of Compound Colours ', and 
in the paper published in Poggendorffs Annalen, 'On Sir 
David Brewster's New Analysis of Solar Light/ which laid 
the foundation of the whole of the modern theory of colour. 
After Newton's discovery of the composition of white light, 
he assumed the existence of seven principal colours in the 
spectrum, apparently taking this number from the analogy 
which he sought to establish between these colours and the 
intervals of the musical scale. But while two tones of different 
vibration-frequency and musical pitch produce sensations of 
harmony or dissonance when struck together, but can still 
be distinguished separately by the ear, luminous rays of 
different wave-length and colour give rise to impressions 
which fuse into a single new colour-sensation. This combina- 
tion is due to a purely physiological phenomenon involved 
in the specific mode of reaction of the optic nerve. Before 
the discovery that white light is due to a mixture of coloured 
lights, the mixture of pigments had led to the theory of 
the three elementary colours, red, yellow, blue, from which 


all other colours must be derived by combination ; Newton 
assumed (without proving it by any prolonged experiments) 
that the same results must obtain for the composition of 
coloured light also : and on this supposition of three primitive 
colours, Thomas Young built up his hypothesis that the 
particles which lie upon the surface of the retina are capable 
of specific vibrations ; that, at any spot, particles of three 
different vibration-rates, corresponding with the oscillation- 
frequencies of the three primary colours, are in immediate 
juxtaposition ; and that, lastly, mixed sensations are produced 
by excitation of the ends of the specifically reacting nerve- 
fibres. Helmholtz now discovered that the mixture of colouring 
substances gave quite different results from the blending of 
spectral colours. It is only when the two colours happen to 
lie near each other in the spectrum, that the fusion of coloured 
light yields almost the same results as the mixture of pigments, 
because the resulting colour then resembles the intermediate 
colour-tones of the spectrum ; the disparity is most marked 
in the combination of blue and yellow, which yield green 
when pigments are mixed and white with the corresponding 
mixture of spectral colours. Now green is among the colours 
most imperfectly produced by the fusion of spectral colours ; 
it was therefore necessary for Helmholtz, if the hypothesis 
of the composition of all colours out of three elementary 
colours was to be maintained, to select for these three 
primaries, not red, yellow, and blue, but red, green, and 
violet; by the mixture of these three all the weaker com- 
pound colours can be obtained, while if the saturated colours 
of the spectrum are to be imitated, at least five primaries, 
red, yellow, green, blue, and violet, are required. In order 
to settle, in the first place, whether there are three elementary 
colours, from which all possible colours are, or at any rate 
may be, built up, this hypothesis had to be tested upon 
the prismatic colours, as the purest and most saturated. By 
observing through a prism set vertically the spectra of the 
two limbs of a X'-shaped slit (letting the bands of colour 
run in the one from left above to right below, and in the 
other from right above to left below, so that each coloured 
band from the one spectrum intersected all the bands of the 
other in the common field), he obtained all the combinations 


which could be produced from any two simple colours. The 
relative intensity of the mixed colours could be altered by 
shifting the prism from its vertical position to one more or 
less oblique, and care was taken to examine the points of 
which the colour was to be determined through a small 
diaphragm (since it is impossible to judge the colour of the 
field so long as it is surrounded by equally saturated colours) ; 
thus all the combinations could be investigated at all degrees 
of their relative intensity by means of a telescope, the cross- 
wires of which were set parallel with the colour-bands of 
the spectra. Helmholtz then discovered the surprising fact 
that contrary to the views hitherto entertained there are 
only two among the colours of the spectrum, yellow and 
indigo-blue, which together yield pure white, that is, are 
complementary to each other, whereas their combination 
had always been supposed till then to produce green. The 
whole width of the spectrum is divided into three sections 
by the rays which produce white: the colours of the first 
and second sections combine to tones of yellow, with transi- 
tions to red, flesh-colour, white, and green ; colours of the 
second and third to blue, with transitions to green, white, 
and violet; colours of the first and third to purple-red, with 
transitions to flesh-colour, rose, and violet. Now since the 
mixture of yellow and blue does not produce green, but 
at most a faint greenish-white, Helmholtz had to conclude 
that the hypothesis of the composition of all colours from 
red, yellow, and blue was erroneous. But he further investi- 
gated the composition of pigment colours, and was thereby 
enabled to explain the obtained results. If a yellow and a 
blue powder are mixed, the blue particles which lie upon 
the surface will yield blue, and the yellow particles on the 
surface yellow light, which combine to form white or greenish- 
white. From within the mixture, only such light will return as 
can penetrate the blue as well as the yellow particles, and 
since blue substances can only let green, blue, and violet 
light through, and yellow substances only red, yellow, and 
green light, green light alone is able to return from within the 
pigment ; this combines with the whitish light reflected from the 
surface, so that green predominates. Thus Helmholtz estab- 
lished the composition of pigmentary colours on purely physical 


principles. If the theory of three elementary colours is to 
go (Young proposed red, green, and violet), then Young's 
theory of the three fundamental colours as the three funda- 
mental qualities of sensation must also be given up. For 
if e. g. the sensation of yellow is only aroused by the yellow 
rays of the spectrum because they simultaneously excite the 
sensations of red and green, which in combination give yellow, 
then it would follow that the same sensation must be excited 
by the simultaneous action of the red and green rays, which, 
however, never produce so bright and vivid a yellow as that 
due to the yellow rays. 

Helmholtz was led to these investigations in following up 
the phenomena described by Brewster, which were in apparent 
contradiction with Newton's theory, and involved a more 
searching analysis of coloured light than had been made by 
Newton, Goethe, or Brewster. Brewster, like Goethe, had 
stated (and built up his entire theory of colour on that state- 
ment) that it was not the differing refrangibility of the rays 
that determined the colours of the prismatic image, but that 
there were three different kinds of light, red, yellow, and 
blue, exhibiting every degree of refrangibility, and so arranged 
that the red light contains a preponderance of rays of less 
refrangibility, the yellow more rays of mean refrangibility, and 
the blue more of greater refrangibility; hence the first pre- 
dominates at the less refrangible end of the spectrum, the 
second in the middle, the third at the most refrangible end. 
The remaining colours of the spectrum would be produced 
by the mixture of the three primitive colours. The object of 
Helmholtz's inaugural dissertation was to prove this view 
untenable, but he now went farther. Brewster had recog- 
nized that if different coloured rays of equal refrangibility 
exist, the compound light formed by them must behave as 
simple light in prismatic analysis, but declared that such 
rays might be separated by taking advantage of their difference 
of absorption in coloured media ; thus there would be rays of all 
three descriptions in all portions of the spectrum, and conse- 
quently the white light due to their union : this is in direct 
contradiction with Newton's theory, according to which homo- 
geneous light passing through coloured media may indeed be 
weakened or extinguished, but can never exhibit changes of 


colour. In order to refute Brewster's theory, or to answer 
the question whether the colour of homogeneous light is 
altered by coloured media or no, the validity of his experi- 
ments had in the first instance to be tested. Helmholtz 
found that Brewster had overlooked the false light cast 
over the observer's field of vision by the slight turbidity 
inevitable in transparent bodies. He shows that the altera- 
tions of colour which Brewster had remarked are due partly 
to impurities in the glass of the prism and to irregularities 
in the polishing of its faces, partly to multiple reflections 
at the surfaces of the prism and of the coloured media, 
as well as to dispersion of light in the eye itself; further, 
such alterations of colour may depend on contrast effects, 
excited by the luminosity of the spectral colours; while, 
finally, the colours of the spectrum excite a different im- 
pression with different intensities of light. Thus he refuted 
the theories of Brewster by a long series of conclusive experi- 

These investigations were just concluded when Helmholtz 
delivered his Inaugural Lecture, for which he selected a 
subject standing in the closest relation with his previous 
work, namely, a general discussion of the mode in which 
our sense-perceptions correspond with the objects perceived: 
a question that led him far into problems of the theory of 
knowledge, and at the same time struck the note of his 
further researches in physiology and physics. 

After a masterly exposition of the undulatory theory of 
sound and light, and a defence of Newton's theory of colour 
against that of Brewster on the grounds above stated, he 
emphasizes the simplicity and apparent unity of a compound 
colour-sensation. He refers the opposition of Goethe (and 
after him of the whole of the Hegelian school) to the idea 
of the composite nature of white light to the idiosyncracies 
of the poet-genius, who held it to be his highest function to 
insist on the adequacy of sense phenomena, and assumed 
that the same directness of perception should be possible 
in the intellectual world. 

In order to obtain a less superficial view of optical pheno- 
mena, it is necessary to recognize the relation borne by 
luminous sensations to the objects sensed; and Helmholtz 


discusses two propositions that were of great importance in 
the development of the theory of knowledge, that, on the 
one hand, all is not light that is perceived as light (a dictum 
enforced by Johannes Muller), while on the other, there 
is also light to which we are not sensible, i. e. invisible light, 
such as the chemical rays, which exert a chemical action 
beyond the visible spectrum. It is highly probable, he says, 
that light-rays and heat-rays are identical within the luminous 
portion of the spectrum, although the intensest heat lies beyond 
the red end, so that radiant heat and light may be regarded 
as identical ; the reason that luminosity is confined to so 
small a group of the long series of vibrations appears from 
Briicke's theory that the transparent media of the eye admit 
these only to the retina, while all the rest are excluded. 
From the fact that sensibility to light and heat do not exactly 
correspond in their limits, Johannes Muller had previously 
concluded that the specific character of luminous sensation 
is conditioned by the specific activity of the optic nerve, 
which, excite it as you will, can only yield the one sensation 
of light. The radiation which we term now light, now radiant 
heat, impinges on two different kinds of nerve end-organs, 
in the eye and in the skin, and the disparity in quality of 
the sensation is due not to the nature of the object sensed, 
but to the kind of nervous apparatus that is thrown into 

From this simple and obvious truth, Helmholtz developed 
his entire theory of knowledge. Which colour combinations 
appear the same, depends only upon the physiological law 
of their composition ; equality of colour arising from different 
mixtures of coloured light has only a subjective value, and the 
groups of isochromic combinations of colour correspond with 
no objective relations, independent of the nature of the seeing 
eye. But if this be true for colour as a property of light, it 
must necessarily be true for colour as a property of bodies 
also. A body that only gives out orange light must have 
a different internal structure from a body that gives out only 
red and yellow, or a third which gives red, orange, and 
yellow. Yet the colour of these three bodies during white 
illumination must be the same ; the similarity has no 
objective, but merely a subjective value. At the end of the 


lecture, he gathers up all these propositions in a form that 
anticipates his later conclusions: 

' Sensations of light and colour are only symbols for relations 
of reality. They have as much and as little connexion or 
relation with it, as the name of a man, or the letters of his 
name, have to do with the man himself. They inform us 
by the equality or inequality of their appearance whether 
we are dealing with the same, or with different, objects and 
properties of objects . . . beyond this they tell us nothing. As 
to the real nature of the external phenomena to which we 
refer them, we learn nothing, as little as we know of a man 
from his name/ 

This lecture again attracted the attention of the philosophers 
to his views on physics and physiology, without, however, 
gaining their approval. 

As soon as Helmholtz had finished his Inaugural Thesis 
he proceeded to develop the theorem of current distribution, 
previously communicated to du Bois-Reymond, of the impor- 
tance of which he was well aware. At the end of April he 
writes to Ludwig : ' I had the luck to discover a mathematical 
theorem as to the distribution of current in bodies, which gave 
du Bois so much trouble ; this greatly simplifies the matter, 
but involves a few minor alterations in the hypotheses he 
suggested/ In the middle of July, 1852, he sends du Bois 
a note for the Academy, entitled, ' A Theorem of the Distribu- 
tion of Electrical Currents in Material Conductors/ while at 
the beginning of the following year, 1853, he sent the full 
exposition of the subject, ' Upon Certain Laws of the Distri- 
bution of Electrical Currents in Material Conductors, with 
Application to Experiments in Animal Electricity/ to Poggen- 
dorff for his journal. In this work Helmholtz for the first 
time enters the field of mathematical physics and physiology, 
with the full equipment of the higher mathematical analysis, 
of which he was the only master in its application to the latter 
science. Even here, and much more in his later works, we 
feel that, as he himself insists, his juvenile penchant for 
geometry had developed into a kind of special mechanical sense, 
'by means of which I almost feel how the stress and strain 
are distributed in a mechanical contrivance ': while, on the other 
hand, it is plain that he strives to make complex and important 

H 2 


mechanical relations plain to himself and others by theoretical 
analysis. On July 22, du Bois-Reymond presented the abstract 
of the theorem to the Academy, and wrote on August 3 to 
Helmholtz : ' What a cornucopia of communications you shower 
on us ; such fertility is unheard of. But your comparison with 
Gauss's law of the compensation of internal magnetic forces 
by surface-distribution does not please me : why should your 
theorem not appear sui generis ? For the rest it consoles me 
that Kirchhoff, with whom I often discussed the problems that 
are so easily handled in the light of the new theorem, also 
failed to solve them. The theory of nerve and muscle 
currents is at last demonstrable, and that hideous Chapter III 
of my book can be reduced to a short and elegant exposition/ 

But the elaboration of the memoir designed for Poggendorff 
involved many difficulties, since in addition to the theorem so 
greatly admired by du Bois, which was at least comprehensible 
without recourse to higher mathematics, it also comprised the 
deduction and application of the most difficult propositions of 
the Theory of Potential, of which almost all the German physi- 
cists of the day, with the exception of Neumann, Weber, and 
Kirchhoff, were ignorant. 

In the middle of November Helmholtz writes to Kirchhoff: 
4 1 have not yet elaborated my theory of current-distribution 
in regard to animal electricity, because new problems are per- 
petually cropping up. I am hampered by the want of Green's 
works, and by the fact that Neumann has not yet published 
anything on these questions. I cannot talk to him freely 
about it, because the propositions which I invent and use are 
either in his unpublished notes, or are so much like his, that 
after each discussion with him I am left doubting whether to 
publish any given point or no. Accordingly I am debarred 
from learning Green's theorems out of the notebooks of 
Neumann's pupils/ 

Nor had the difficulties all been got over even by the end of 
January, 1853. Helmholtz tells du Bois that he has been 
unable to find a general proof of an important theorem which 
can be easily proved for conductors in which the resistance 
is equal in all parts, viz. that an electromotive force applied 
to any given surface-element a of a conductor will produce in 
any other given surface-element ft the same component of 


current, normal to the surface, as that electromotive force 
applied to ft would produce in a. 

He expects that he will have to leave the proof of this 
theorem to the future, but at last he overcomes the difficulty, 
and is able to announce to Ludwig early in March, 1853 : 
1 1 have meantime discovered and worked out some new 
theorems on the distribution of galvanic currents in material 
conductors, by which the theory of currents of animal electricity 
can be demonstrated with strict accuracy, and by a very 
simple method, while du Bois-Reymond had to make shift with 
exceedingly complex approximations. My results of course 
agree in essentials with those of du Bois.' 

Du Bois-Reymond himself affirmed at a later time that he 
had been helpless in face of these great difficulties until 
Helmholtz came to his aid with the conception of electro- 
motive surfaces, and the theorem of the equal and opposite 
action of two electromotive surface-elements, by means of 
which the previously insuperable difficulties became almost 
elementary. This very interesting and fundamental work on 
the distribution of electrical currents in material conductors is 
purely mathematical in character, owing to Helmholtz's method 
of proving the theorems, which are intelligible enough from the 
physical point of view. It is essentially connected with the 
treatise on the Conservation of Energy, since Helmholtz merely 
substitutes for the expression 'free tension' there employed, 
the identical concept of Gauss's potential, or Green's potential 

In his inquiry he starts from the three equations which 
Kirchhoff had laid down for dynamic equilibrium in the distri- 
bution of currents in systems of material conductors, and had 
shown to be necessary and adequate for the expression of 
potential as a function of the co-ordinates. He has no diffi- 
culty in devising a quite general proof of the law of the 
superposition of electrical currents, which had been already 
recognized as valid for individual cases. He expresses it very 
simply by saying, that if in any system of conductors constant 
electromotive forces are introduced at different points, the 
electrical potential at any point of the system will be equal 
to the algebraic sum of the potentials which are due to each of 
the forces independently of the others. With this he associates 


the law of electromotive surfaces, which states that if electro- 
motive forces existing anywhere within a conductor produce 
certain currents in an attached conductor, it must be possible 
to devise a distribution of electromotive forces on the surface 
of the first conductor which would produce the same currents. 
He arrives at the distribution on the surface by assuming 
the conductor to be insulated and determining the electrical 
potential at any point of its surface due to the currents excited 
by the internal forces ; the required surface electromotive force 
(taken from within outwards) is then equal to this difference 
of electrical potential. He terms the surface, thus conceived 
as electromotive, the positive effective surface ; and from these 
two theorems (deduced by strict mathematics) derives a series 
of important conclusions. These again yield the theorem that 
the potentials within the attached conductor are equal to the 
sum of the potentials existing in it antecedently, and of those 
produced by the positive effective surface. It is obvious that 
different modes of distribution of e. m. f. at the surface of 
a conductor (if they are to give the same derived currents as 
the internal electromotive forces) can only vary by a difference 
of potential that is constant for all points of the surface ; and 
from equally simple considerations (based entirely on Ohm's 
Law) results the general and important theorem, that when 
any two points on the surface of an extended conductor con- 
taining constant forces are connected with a given system of 
linear conductors, it is always possible to substitute for it 
a linear conductor of definite e. m. f. and resistance, which 
will give rise to precisely the same current in all linear con- 
ductors connected with it, as the material conductor. He 
introduces the conception of electrical double layers, which 
assumes that at opposite sides of a surface and at infinitesimal 
distances from it, there will be exactly the same quantity of 
positive electricity on the one side as there is of negative on 
the other. By this device he transforms the Poisson-Gauss 
equation for non-equilibrium states on which Kirchhoff founded 
his equation for equilibrium so that the potential shall not (as 
in that equation) be uniform, and the force components on 
either side of the surface non-uniform, but conversely, with 
uniformity of force the difference of potential function shall 
be a quantity that is different from zero, which he terms the 


electrical moment of the surface : he thus succeeds in reducing 
problems of current distribution to a question of the potential 
of electrical surfaces and bodies. Lastly, with the aid of 
Green's propositions, he develops the theorem communicated 
in January to du Bois-Reymond of the equal mutual action of 
two electromotive surface-elements, and thus clears the way 
for the experimental confirmation of the law and its important 
applications. For since each single element of an electromotive 
surface discharges as much electricity into a galvanometer 
circuit as would flow through itself if its e. m. f. were situated 
in the galvanometer circuit, the total effect of all the electro- 
motive surface-elements must be equal to the whole current 
passing through the galvanometer. In all experiments on animal 
electricity, in which nerve and muscle represent extended 
material conductors, with electromotive forces distributed in 
them, it now becomes possible to test and correct the theo- 
retical conclusions of du Bois-Reymond and other physiologists 
as to arrangement of electromotive elements within the 
nerve or muscle, by means of laws that have been ascertained 

Meantime, the interest of the various medical and literary 
circles in KGnigsberg had been aroused by the effect of 
Helmholtz's Physiological Theory of Colour upon the scientific 
world ; and he responded to their inquiries by giving a lecture 
on ' Goethe's Scientific Researches ', delivered on January 18 
(Coronation Day) to the Deutsche Gesellschaft. 

Helmholtz had been induced by various considerations to 
protest against the attack made by Goethe upon the optical 
work of Newton, and he accordingly desired, in order to avert 
misunderstanding, to show that although Goethe's physical 
conclusions were often erroneous, he had done indisputable 
service in botany and osteology, and must always be reckoned 
among the great men of science. In his Sketch of a General 
Introduction to Comparative Anatomy^ Goethe expresses the 
idea (which was never better nor more clearly stated, and 
has subsequently been but little altered) that all differences 
in the structure of animal species must be looked upon as 
variations of a common type, brought about by the coalescence, 
alteration, increase, atrophy or total loss of single parts. A 
similar analogy between the different parts of one and the 


same organism, as seen in animal species, exists in the 
manifold repetition of the same parts in plants, and again 
in the transition from the leaves of the stem to those of the 
calyx and petals from which follows Goethe's theory of 
the ' Metamorphosis of Plants ', which has been accepted, at 
any rate in its essential features, by the botanists. The repeti- 
tion of homogeneous parts in animals, which Goethe noticed 
accidentally, led him to extend his doctrine to animals also, 
but in Helmholtz's opinion these osteological conclusions 
have been less favoured by science. 

According to Helmholtz it was wonderful that Goethe should 
divine the existence of such a law, and follow out its indi- 
cations so acutely, although he neither saw what law it was, 
nor even tried to find out; since he always held the view 
that ' Nature must yield up her own secrets, inasmuch as she 
is the transparent symbol of her ideal significance'. 

But after emphasizing the great services which Goethe had 
rendered to the natural sciences, Helmholtz claimed the right 
of criticizing the physical conclusions arrived at by this great 
genius, and pointed out the errors into which Goethe fell, when 
he attempted to repeat Newton's experiments with the prism, 
in order to investigate the aesthetic laws of colour in painting. 
He was ignorant of the most elementary principles of optics, 
and held all Newton's facts and deductions to be an absurdity. 
Helmholtz discusses the very interesting question why this 
master-mind should have attacked Newton and the physicists 
in general with such unparalleled animosity, imputing nothing 
but ill will to his antagonists, and why this greatest of poets 
should have assumed his achievements in science to be far 
more valuable than all he had accomplished in poetry. Helm- 
holtz attacks his subject with a brilliancy of comprehension, 
a depth of aesthetic feeling, an appreciation of the poetic and 
scientific qualities of the man, and of the inductive and deductive 
methods of reasoning, possible only in a great and contem- 
plative thinker, expressing himself in the language of inimitable 
charm and vigour that characterizes all his writings, and is 
unique as a model for the popular treatment of scientific 

For the poet, he says, as for every other artistic genius, 
an idea is not expressed as the product of a slowly matured 


intellectual concept, but the material of his art becomes the 
direct vehicle of the idea. With Goethe the phenomenal is 
the immediate expression of the ideal, in which he is the 
forerunner of Hegel's 'nature-philosophy', and he therefore 
appreciates experiments that can be carried out in clear sun- 
shine, under the open heavens, in contrast to Newton's slits 
and glasses. Goethe could not and would not grasp the fact 
that the pure tone of white light is due to a fusion of colours ; 
he endeavoured by a consistent observation of facts to deter- 
mine their connexion, so as to discover the causes of the 
phenomena of nature, without trespassing into the realm of 
concepts : while the physicist denies all authority to sensation, 
and is increasingly aware that the nature of sense-perception 
depends less upon the properties of the objects perceived than 
upon those of the sense-organs by which he obtains his in- 
telligence. All Goethe's conclusions and explanations are 
accordingly fallacious. 

' Goethe is only content when he can stamp reality itself 
with poetry. In this lies the peculiar beauty of his poems, 
and it accounts for his resolute hostility to the mechanism that 
threatened to disturb his poetic repose, and his determination 
to attack the enemy in his own camp. Yet we cannot 
conquer the mechanical laws of matter by ignoring them : we 
can only subordinate them to the aims of moral intelligence. 
We must understand its levers and pulleys if we are to 
control them by our will, and herein lies the great significance, 
and full justification, of physical research in the advance of 

Forty years later, in a lecture given at Weimar to the 
Goethe-Gesellschaft, on 'Goethe's Anticipations of Coming 
Scientific Ideas', Helmholtz found a fresh opportunity of in- 
sisting on the great importance of Goethe's work for the general 
development of science. While his judgement of the optical 
part of it remained unshaken, he now interprets Goethe's errors 
and prejudices by his aversion to the abstractions of intangible 
concepts, in which the theoretical physics of the day was wont 
to reckon. He holds Goethe's protest against the abstractions 
of matter and force to be not unjustified, since ' though they 
were used by the great theoretical physicists of the seventeenth 
and eighteenth centuries in a coherent and definite sense, 


they contained the germs of the wildest misunderstandings, 
which now run riot in perverted and superstitious minds'. 
Helmholtz recognized that the seed which Goethe sowed in 
the field of natural science had developed in rich and full 
abundance, since Darwin's theory of the modifications of 
organic form rests confessedly upon those very analogies and 
homologies of structure in plants and animals which Goethe, 
the first discoverer, presented to his contemporaries in the 
form of anticipations only, while Darwin developed this poetic 
forecast into a mature concept. He finds the reign of law 
among physical phenomena, which Goethe sought to discover, 
expressed with the greatest precision and lucidity in Kirchhoff s 
lectures on mathematical physics, which enrolled mechanics 
among the ' descriptive sciences'. For Helmholtz, science and 
art are intimately connected, since both express and enunciate 
truth. The artist can only succeed in his work when he has 
a subtle knowledge of the natural relations of the phenomena 
which it expresses, and of their effect upon the auditor or 
spectator. 'When the task can be fulfilled by expression in 
the tangible images of poetic divination, the poet proves him- 
self capable of the highest achievement ; where the strict in- 
ductive method alone can avail, he founders. But again, where 
cardinal points of the relation between reason and empirical 
fact are involved, his firm grasp of reality preserves him from 
error, and gives him a sure insight which extends to the very 
limits of human understanding.' 

The commencement of the New Year found Helmholtz in 
depressed conditions. It had been a sad Christmas, for his 
wife was ill of nervous gastritis, from which she only recovered 
after some weeks of unremitting attention from her mother and 
sister. His mother had been laid by with a serious operation. 
He himself was suffering from frequent attacks of migraine, 
which kept him in bed for days together. He proposed to take 
his wife to Marienbad on his way to England, but his father 
disapproved, on economical grounds, though this difficulty was 
partly removed by a rise of salary in April, 1853, which brought 
his income up to 150. 

From the beginning of 1853 to the summer vacation, Helm- 
holtz was engaged in the continuation of his measurements of the 
rate of transmission in nerve and muscle, which necessitated 


the contrivance of a special instrument for measuring the very 
small electrical currents which du Bois had observed in muscles. 
In addition to this he embarked on a protracted study of the 
adaptation of the eye for different distances, which was, however, 
interrupted by the discoveries of a young Dutch physiologist, 
Cramer. As early as January 23, Helmholtz sent du Bois 
a short preliminary notice for the Academy 'On a hitherto 
unknown Alteration in the Human Eye, during Altered Accom- 
modation '. In this he describes an observation made as early 
as the winter of 1852, that in accommodation for near objects, the 
image reflected from the front surface of the lens is diminished 
to nearly half its size, a considerable alteration which cannot 
be explained by a change of position of the lens, but only on 
the supposition that there is an alteration of its form by the 
increased curvature of the anterior surface. He had previously 
determined by many exact observations that the edge of the 
pupil bulges outwards in near vision. 

In March he joyfully communicates his ' little discovery in 
regard to the accommodation of the eye ' to Ludwig, as pub- 
lished in the monthly reports. But on July 3 he informs him : 
' Bonders has written to tell me that a Dr. Cramer has been 
before me, his paper having been "crowned" in 1851 by the 
Haarlem Society, though it is only now being published : I 
shall receive a copy of it shortly. Lately there have been 
a good many coincidences between my work and that of others : 
(i) on Brewster's Theory ; a portion of my results were also 
discovered by a young physicist, Felix Bernard, and published 
in the Annales de Phys. et Chim. in the same month in which 
mine appeared in Poggendorff, but he had communicated the 
work some time before to the French Faculty : (2) in January 
Gaugain brought out a tangent galvanometer, on the principle 
of the one I had made in 1849 for du Bois-Reymond's ex- 
periments ; mine, however, is more convenient and better : 
(3) Foucault describes a method for the uniform lighting of 
large surfaces with homogeneous or mixed light; I received 
his paper after myself inventing and constructing the ap- 
paratus : (4) Cramer on Accommodation ; I am most curious 
to see this paper. Your textbook, so far as it goes at present, 
is my faithful friend when I am preparing my lectures/ 

The promised copy of Cramer's paper was so long delayed 


that Helmholtz writes to Bonders after his autumn journey 
of 1853:- 

'I have not yet received Dr. Cramer's treatise on the Ac- 
commodation of the Eye, and confess that I am very curious 
to see how much room he has left me for my own observations. 
Your letter was the first I had heard of his work, and though 
I regret the time lost on investigations that turn out to be 
the property of another, I am of course only too glad to clear 
the way as much as I can for a young man who makes his 
debut in science with such a striking piece of work, and to 
help him towards recognition. I had equally arrived on theo- 
retical grounds at the idea of a simultaneous tension of the 
radial and circular fibres of the iris, but am inclined to ascribe 
a considerable part to the tensor choroidae also. 1 

Helmholtz employed the enforced leisure of summer to con- 
tinue his experiments on the mixture of homogeneous colours 
by other methods than those hitherto employed. He found, in 
agreement with Grassmann, that besides indigo-blue and yellow, 
another pair of complementary colours existed in the spectrum, 
although he had not previously been able to demonstrate 
them, and further, that all colours, with the exception of 
green, yield simple complementary colours. In order to deter- 
mine the breadth of the colours in the spectrum, he defined 
white as the sensation due to the sum of the light-components 
simultaneously perceived by the eye, together with the vivid 
memory of such as were perceived immediately before. 

He also carried on his experiments on the time-relations 
of excitation in man throughout the summer, arriving at the 
conclusion that the rate of the nervous impulse in man is 
about three times as great as in the frog. As regards the 
initiation of electrical processes during the excitation of nerve 
and muscle, he was able to determine 'that the electrotonic 
condition of the nerve begins with the entry of the primary 
current, whereas the negative variation of the muscle begins 
appreciably later than the excitation, but precedes the first 
trace of contraction*. 

At the beginning of August, Helmholtz left KSnigsberg, and 
took his wife and the two children to his mother- and sister- 
in-law at Dahlem. Du Bois-Reymond was not in Berlin, but 
he had the satisfaction of seeing Johannes Muller, who was 


just starting for Sicily. He visited his parents in Potsdam, 
and found his father well and cheerful, but his mother greatly 
altered. At a dinner given by Magnus, at which H. Rose 
was also present, he met Tyndall, the English translator of 
his works : ' he is a very talented young man, and interested 
me more than any of the other strangers; unfortunately he 
will not be in England when I am there/ He also went to 
see Dr. Graefe, 'who was still attending to his clinique, and 
showed me some cases with the ophthalmoscope, and a mass 
of drawings made with the instrument, saying many kind 
things about the great utility of this invention/ After providing 
himself with letters of introduction from Magnus, Dove, and 
H. Rose, to distinguished men of science in England, and to 
the chemist Hofmann, Helmholtz went on to Bonn, where he 
had the pleasure of spending a few hours with Plucker. ' At 
first he seemed to me a little exclusive, like most of the Bonn 
professors, but afterwards he became quite genial, and com- 
memorated my visit in a remarkably good bottle of wine.' 
Helmholtz crossed by Ostend to London, putting up at an hotel 
which Tyndall had recommended to him. He gave himself 
up entirely to English life, and sent daily letters to his wife 
with vigorous sketches of his varied and original observations. 
Space forbids the quotation of more than a few of these, which 
have special reference to the English scientists with whom 
Helmholtz now for the first time became acquainted, and with 
whom he formed a lifelong connexion. 

' And now you shall hear about this great Babylon. Berlin, 
both in size and civilization, is a village compared to London. 
Everything here is on such a gigantic scale, that one ceases to 
wonder at anything. The disadvantages of an enormous city 
are pleasantly counterbalanced by the wonderful Parks within 
the town, and the green of its suburbs. But I had better go 
on with my diary, so as to tell you everything. ... In the first 
place I went to Bence Jones, physician, physiologist, and 
chemist, hoping to get news of du Bois-Reymond, and of the 
chemist Hofmann. But he had gone off to du Bois' wedding. 
The Embassy was in the same direction, so I went there to 
present my letters to Bunsen. Bunsen was engaged, and 
invited me to visit him next day. This errand showed me 
something of the Park, which extends unbroken from the 


West End nearly to the centre of London. In the afternoon 
I explored it further; imagine enormous smooth spaces of 
short, fine grass, dotted over with fine old trees or groups 
of trees, a few paths cut through them which are only used 
in wet weather (for when it is dry every one walks as he 
pleases over the grass), and there is the ideal that you wanted 
for our garden. Huge sheep, as fat as stuffed wool-sacks, graze 
everywhere on the grass and keep it short. . . . The break- 
fast with Bunsen was just a way of receiving one's visit at a 
leisure moment. Milady and two daughters, a Professor Larso 
from Berlin, and Privatdocent Bottiger from Halle, both 
Oriental scholars, were there too. The meal was refined 
without being luxurious, but was swallowed post-haste. Each 
helped himself as he pleased without waiting for the others. 
I was last, because I had to talk so much. Bunsen somewhat 
resembles S., interested in everything, lively, but a little con- 
ceited. He was most affable and officious, and wrote me 
a letter of introduction to the zoologist Richard Owen which 
I did not want. For the rest, everything was on a very grand 
scale in the house. British Museum. Here were Layard's 
monuments, Elgin's Marbles from the Parthenon, those from 
the Lycian tombs, &c., all in real life. The Assyrian bulls 
with human heads are enormous monsters. The reliefs are 
far more vigorous than in the drawings; they are very clear 
and sharply worked out, and parts of them look as if they were 
quite new. In England they excite more interest than in other 
places, because they are supposed to confirm certain passages 
in the Old Testament. As regards style, they are infinitely 
finer than anything in Egyptian art, and are parallel with the 
best productions of the ancient Greeks. Bunsen tells me that 
much progress has been made in deciphering the inscriptions. 
' My attempts to see Professor Owen were in vain, but 
I succeeded in finding the first physicist of England and 
Europe, Faraday perhaps, unfortunately, for the first and last 
time, since he leaves town on Monday, and does not know 
if he is coming to Hull. Those were splendid moments. He 
is as simple, charming, and unaffected as a child ; I have never 
seen a man with such winning ways. He was, moreover, 
extremely kind, and showed me all there was to see. That, 
indeed, was little enough, for a few wires and some old 


bits of wood and iron seem to serve him for the greatest 

'From there I went to the National Gallery. There are 
some beautiful Rembrandts, and fair examples of Rubens 
and the Italian masters, and two marvellous Murillos. In 
the afternoon I went by omnibus to Hammersmith, a suburb 
with villas on the Thames, to see Professor Wheatstone, the 
physicist, and inventor of the first practicable electrical tele- 
graph. He had left, but they gave me hopes of finding him 
in Hull. In the evening dined at seven with Dr. Bence Jones : 
only he, du Bois and his wife, and I. Bence Jones is a 
charming man. Simple, harmless, cordial as a child, and 
extraordinarily kind to me. He appointed a second meeting 
for the next day, to see the ophthalmoscope, and took me to 
a mechanician where Faraday's instruments for the detection of 
table-turning are on view. On Thursday morning I worked at 
the lecture I am to give (" On the Mixture of Homogeneous 
Colours "). At noon, when I was going out for lunch, I met 
Professor Plucker from Bonn in the street, who joined me, 
and said that Professor Sommer from Konigsberg was staying 
in the same house with him. Afterwards I went to Bence 
Jones, to keep my appointment. 

'On Friday Airy invited me to go to Greenwich and dine 
with him. He had been rather stiff the first time, and can 
be very disagreeable, but on this occasion he was charming, 
and as I inspected all his appliances, praising much, and 
criticizing some things, he was quite unable to stop perambu- 
lating, so that I have probably seen more of the Observatory 
than any one else. Besides the regular observatory, of which 
I understood little, there are remarkably fine contrivances for 
magnetic and meteorological observations, in which the state 
of the instruments perpetually daguerreotypes itself, so that 
the series of observations is more exactly and completely 
recorded than it could be by the most accurate observer. 
Then we saw apparatus for the electro-magnetic measurement 
of time in star-transits, and electric clocks, which indicate 
the time simultaneously in London and at the mouth of the 
Thames, and at all the London railway stations. Airy's house 
and family life were arranged, as we should say, in style, but 
it is so with most of the English professors. His wife was 


rather formal, well preserved, with pleasant manners. The 
English ladies are all very interested in their husbands' work, 
and she was familiar with everything. He has a splendid 
position. What he writes goes out to the world not under 
his own name, but in that of Astronomer Royal, and he is 
superior to the rest from his training in methods: most of 
the English physicists do great things purely from instinct, 
not like the French from training in the best methods, so 
that their work is often spoiled by ignorance of the most 
ordinary matters. The afternoon at Greenwich was one of 
the most interesting and delightful of my journey. 

'On Tuesday I looked up Wittich, and went about with 
him. In the morning we explored Westminster Abbey: its 
architecture is not nearly so beautiful as that of the best 
German cathedrals. It is too narrow, and the vaulting is not 
very intricate, but the array of monuments to the famous 
dead is extraordinarily imposing, and must stimulate the pride 
of Englishmen in the highest degree. To have had such men, 
and to see them so honoured, is grand. There lie professors of 
physics and chemistry between the kings, generals, and artists ; 
even tragedians of the first rank have found their place and 
their monument here : Newton, James Watt, Humphry Davy, 
Thomas Young, Shakespeare, Milton, Garrick, Mrs. Siddons, 
Henry V, Richard II, Edward's sons, Warren Hastings, the 
two Pitts, Mary Stuart, and Elizabeth. 

'On Wednesday I packed up, and went to Hull by train 
of course, not steamer. I met Dr. Plucker at the station 
and travelled with him. The journey is uninteresting. 
Farther north, the country is not so exquisitely green as it 
is near London, and is mostly hilly. Here in Hull we are 
quartered on various people, I with a physician, Dr. Cooper, 
where I live "very fashionably", and am well taken care of. 
The foreigners (besides myself and Plucker there is only 
a Russian, du Hamel, here) are treated with exquisite courtesy. 
Yesterday evening at eight was the first General Meeting, when 
the President gave a survey of the progress of science during 
the last year : 600 persons were present, each of whom had 
paid at least i t and 175 of them were ladies. We strangers 
were named in the report which the Secretary gave at the 
end; I was mentioned as Professor H. from Konigsberg, who 


had contributed one of the most important advances in con- 
tinental science. My ' Conservation of Energy ' is better known 
here than in Germany, and more than my other works. 

' Early this morning, Thursday, I was invited to breakfast 
by Mr. Frost, a wealthy private individual and a geologist. 
At his house I met Professor Stokes of Cambridge, a young 
but most distinguished man, whom I had not expected to 
see, because he had been in Switzerland. . . . 

'The British Association at Hull was, as I have already 
told you, remarkably well attended ; there were 850 members 
and 236 ladies. Here in England the ladies seem to be very well 
up in science, though of course many of them come to show 
themselves, or from curiosity, to listen to the discussions, 
and amuse themselves with them. Still on the whole they are 
attentive, and don't go to sleep, even under provocation. The 
six sections of the Society sit every day from eleven to three. 
From ten to eleven is occupied by the committees; I was 
taken to the committee of the Physics Section. The public 
generally wander from one section to another to hear the most 
distinguished speakers. The communications naturally varied 
greatly in quality : some were important scientific contributions, 
some the tomfoolery of crack-brained persons who imagine 
they have got hold of startling discoveries. But the presidents 
generally knew how to suppress these people. I was most 
interested in the arrangements for scientific investigations by 
committees, and the way in which the English attack these 
questions. Now, for instance, they are engaged upon a geo- 
logical comparison of the surface of the earth with that of 
the moon, by means of their splendid telescopes, a number 
of astronomers and amateurs having joined together for this 
purpose. Further, they are preparing to send a gigantic 
telescope to the Southern Peninsula at Government cost, to 
explore the southern heavens. The most popular departments 
were geology, geography, and ethnology. These, too, attracted 
the most distinguished speakers ; it is important to engage 
a great number of people upon common work in these de- 
partments, and the Association is very well adapted for this. 
On the other hand, many of the best chemists, physicists, 
and astronomers were absent, e. g. Airy, Faraday, Wheatstone. 
Others were there whom I much wanted to meet. Grove, 


a jurist and distinguished physicist from London, Andrews, 
Professor of Chemistry in Belfast, Stokes, a physicist from 
Cambridge; there was no one who could properly be called 
a physiologist. The clearest and most popular, as well as 
most valuable communieations, were those of the geologists 
Phillips and Hopkins, and the ethnographer Dr. Latham, 
but many of them were tedious, and many to my surprise 
were mumbled, and so badly delivered that they were unin- 
telligible. I joined in one discussion ex tempore, and explained 
a point in the optics of the eye that had been worked out in 
Germany. I got through all right, though I made plenty of 
mistakes, but the English people praised me, and said it was 
quite clear and easy to understand, although I used certain 
words in a different sense from that which they usually 
convey. I read my lecture on "The Mixture of Colours" 
aloud to Dr. Francis, who corrected the mistakes, and was much 
commended for it. The style of course was not entirely my 
own, but they were pleased with the delivery, and I received 
many compliments at the expense of Professor Plucker, who, 
considering how often he comes to England, speaks very badly/ 

Helmholtz intended to visit Utrecht on the return journey, 
to make acquaintance in person with Donders, but was sum- 
moned home by his wife's illness. This time, however, she 
soon recovered from the attack of the malady to which she 
was becoming increasingly liable, and he was able to talk 
over his travelling experiences with her, invigorated in mind 
and body. This journey to England made a deep and abiding 
impression upon him, and he took every subsequent oppor- 
tunity of revisiting his scientific friends there. 

' England/ he writes to Ludwig, ' is a great country, and one 
feels what a splendid thing civilization is, when it penetrates 
into all the least relations of life. Berlin and Vienna are 
mere villages in comparison with London. London is quite 
indescribable; one must see its traffic with one's own eyes, 
before one can realize it; it is an event in one's life to see 
it; one learns to judge the ways of man by other standards/ 

At length, in the middle of October, 1853, Helmholtz received 
Cramer's memoir for which he had been waiting so many 
months, and was enabled to continue the work on accom- 
modation that had been interrupted by Donders's letter. After 


declining to write a textbook on Physiological Physics at 
the invitation of Vieweg, who in consequence gave the com- 
mission to Kick, Helmholtz, at Karsten's request, undertook 
the section on Physiological Optics in his great Encyclopaedia 
of Physics (a task which unexpectedly required ten years for its 
completion), and then busied himself in the first place with 
a new method of determining in the living eye the forms and 
distances of the refracting surfaces, the cornea and the anterior 
and posterior surfaces of the lens, in order to define the path 
of the rays of light in the eye. By April, 1854, he had got 
so far that after studying Cramer's paper, he could write to 
Bonders hopefully of speedily determining the curvature of 
the iris and displacement of the border of the pupil as it occurs 
in adaptation : 

' I received Dr. Cramer's treatise directly after I had written 
my first letter to you. I have studied the book, for which 
again my best thanks, although I found it rather troublesome, 
as I first had to learn Dutch to read it. Happily your language 
is so much akin to ours that it is not difficult to understand. 
Dr. Cramer's work is interesting, and very satisfactory. I did 
not succeed in experimenting with fresh-killed eyes, because 
I used rabbits. Cramer's experiments on such eyes show 
that the iris is necessary to adaptation for near vision, as 
I had previously surmised. But it still seems to me doubtful 
whether the iris alone is involved. When the accommodation 
is for near vision, the edge of the pupil itself bulges forward, 
while a contraction of the iris alone, i.e. of its radial and 
circular fibres, which Cramer rightly assumes to occur, would 
be apt to produce the contrary effect/ 

These difficult optical investigations were now pushed aside 
by another task, which was forced upon him by an unfortunate 
incident. At the close of 1853, Clausius published an unjustifi- 
able attack in the Annalen upon Helmholtz's memoir on the 
Conservation of Energy, which was a source of great annoyance 
and distress to Helmholtz, since it emanated from a contemporary 
and distinguished member of the Physical Society, whom he 
had known intimately since 1848, and whom he had for a long 
time been in the habit of meeting almost daily. At the be- 
ginning of 1854, he refuted the attack in the same journal, 
under the title ' Reply to the Observations of Dr. Clausius ', 

I 2 


with such success that any doubt as to the correctness of 
his statements was henceforth impossible. 

This assault might have been prejudicial to Helmholtz, 
inasmuch as it conveyed the impression to non-mathematical 
physicists that his conclusions were erroneous, and as he was 
not a professed mathematician, the allegations of Clausius 
might have been accepted. At the time when Helmholtz 
published 'The Conservation of Energy ', he had already 
done a great deal of work in the direction of a mechanical 
theory of heat, but in the printed essay omitted everything that 
savoured of hypothesis, ' in order to facilitate the reception of 
the work by the physicists.' At a later period he had entirely 
left the matter aside, in the belief that the mechanical theory 
of heat could only be promoted by avoiding all presumptions 
as to the constitution of the molecules, and examining generally 
how the motions within the complex molecules affected the 
position of adjacent molecules. But he had been engaged 
on far wider problems prior to the publication of * The 
Conservation of Energy '. When Carnot (on the presumption 
that heat was material, and as such could neither be destroyed 
nor added to) investigated the processes by which heat is 
able to perform mechanical work, he found that this can 
occur only when heat is passing from a warmer to a colder 
body. Perpetual motion would then be an impossibility 
only if the return of heat from the colder to the warmer 
body required an amount of work to be performed equal to 
that done by the previous and opposite process, besides which 
this expenditure of energy would have to be independent of the 
nature of the transmitting substance. Subsequent work upon 
the conservation of energy, however, made it impossible to 
maintain the material nature of heat, which had been an 
essential postulate in Carnot's deduction. Helmholtz had 
already attempted to formulate proofs, based on mechanical 
principles, for certain of Carnot's conclusions which seemed 
to him to hold good in the theory of heat, but he was forced 
for the time to leave over any decision as to the validity of 
these propositions. He had, therefore, gone much farther 
than Clausius had detected from the published memoir on the 
Conservation of Energy ; farther indeed than Clausius himself 
had advanced at a much later period. 


Clausius in the first instance attacked Helmholtz's derivation 
of the law of the development of heat in electrical discharges 
from the law of the conservation of energy. As stated in 
a letter to du Bois-Reymond, he had not noticed that the 
definition given by Helmholtz of the ' potential of a mass-in- 
itself ' differed from the ordinary definitions. Helmholtz gives 
the particulars of this dispute some years later in a letter to 
Tait of March 17, 1867: 

'As to my discussion with Clausius, there was no essential 
difference between us as to the mechanical equivalent, except 
that Clausius takes the heat of the spark into account, while 
I believed it might be neglected, and that I took the potential 
of a body in itself as the sum of m a m b /r ab without excluding 
the repetitions of the indices (ab) and (ba\ while Clausius fol- 
lowed the other mathematicians in excluding these repetitions, 
so that what he terms potential was only half as large as what 
I defined as such. Substantially both were equally correct/ 

Helmholtz was able to refute the second objection raised 
to his work with equal ease, that, namely, criticizing the 
conclusions he deduced from the law of Riess, to the effect 
that with different charges, and a varying number of similar 
Leyden jars, the heat developed in each individual part of 
the wire closing the circuit must be proportional to the 
square of the quantity of electricity, and inversely proportional 
to the surface of the jars. This charge does not really touch 
Helmholtz, or the conclusions which he deduced under the 
assumption of this law, since Clausius attacked the correctness 
of the law of Riess in itself, and disputed its universal validity, 
while Helmholtz said in his paper that the law was in need 
of experimental confirmation. In regard to a misunderstanding 
of a passage in Holtzmann's book, Helmholtz candidly admitted 
his mistake, as appears from a letter to Ludwig. Clausius's 
main attack on the work of Helmholtz was directed against the 
proof of the proposition, that the principle of the conservation 
of vis viva holds good only where the working forces can 
be resolved into forces due to material points, acting in the 
direction of the lines joining the points, while their intensity 
depends solely upon the distance. Upon this Helmholtz 
founds a long and important argument with which his epoch- 
making thermodynamic work, the greatest achievement of the 


last decade of his life, is intimately connected, though the 
latter was of course considered from an altogether different 
point of view. 

Helmholtz fully recognized the importance of an attack 
upon this particular part of his treatise on the Conservation 
of Energy, because this was his main advance on the investiga- 
tions of Robert Mayer, and the chief significance of his own 
work rests upon the same considerations. Both engineers and 
physicists had for a long time defined the product of the 
mass of a weight raised, and the height it is raised to, as 
the measure of work done ; this conception of quantity of work, 
as the product of force into a distance, had to be transferred 
from the case in which there is a force of constant magnitude 
acting in a constant direction, viz. gravitation, to the cases 
where a large or even infinite number of particles, acting 
upon each other, undergo relative displacement, so that work 
is done along the path of each individual particle by the 
forces exerted by the other particles. 

Green had defined this amount of work as potential, for attrac- 
tive and repulsive forces, the intensity of which is inversely 
proportional to the square of the distance of the interacting 
masses, and applied its mathematical properties to the explana- 
tion of electrical and magnetic phenomena. It was then seen 
that this same quantity of work, taken negatively, is a factor 
to be considered in all problems of mechanics and physics : 
it was named potential energy to distinguish it from the product 
of half the masses into the squares of the velocity, which was 
termed vis viva, or actual energy. By this conception Helmholtz 
was able to proceed from the earlier law of the * Conservation 
of Vis Viva', as laid down in the mechanics of ponderable 
masses, to the great law of the ' Conservation of Energy ', 
which, in addition to asserting that matter can neither be 
destroyed nor added to, affirms the constancy of energy as 
the sum of actual and potential energy. The old formula, 
the so-called 'law of the conservation of vis viva\ only dealt 
with cases in which the potential energy was unchanged, and 
therefore disappeared in the final result. 

Clausius protested against the derivation of the law of the 
' conservation of vis viva ', as given by Helmholtz in his memoir, 
where he takes it as the point of departure for his own great 


law. He objected that Helmholtz had, even in the simple case 
in which two particles act on one another, assumed, in addition 
to his assumption of the law of the conservation of vis viva, 
that the magnitude of the force was a function of the dis- 
tance, concluding therefrom that the direction of the force 
coincided with the line connecting those points. Helmholtz 
showed this objection also to be ill-founded, and embraced the 
opportunity of giving a further and more complete discussion 
of this point, suggesting a new and interesting treatment of 
the subject. Starting with the definition that movable points 
have the same relative position to each other, whenever 
a system of co-ordinates can be constructed in which all the 
co-ordinates shall have relatively the same values, Helmholtz 
expresses the law of the conservation of vis viva in this form : 
'When any number of particles in motion are only moving 
under the influence of such forces as they are themselves 
exerting upon one another, then the sum of the vis viva of 
all particles at any moment in which all the particles recover 
their same relative position, is constant whatever their direction 
and velocity at intermediate times * ; and in virtue of this law 
he again refutes Clausius's objection that in certain cases the 
vis viva may be a purely arbitrary function of the co-ordinates 
of the system. He states expressly that he made the assump- 
tion in his treatise that the force exerted by one particle upon 
another is independent of any other forces that may be acting 
upon it, a principle which has always been accepted in 
mechanics. He concludes by saying that he had expected 
criticism from Clausius in regard to his Theory of Galvanism : 

4 The chapter on electro-dynamics in my treatise was written 
under great difficulties. At that time I scarcely had access to 
any mathematical and physical literature, and was almost 
wholly confined to what I could discover for myself. It can 
only be a gain if the ideas which I endeavoured to bring 
forward in my paper at a time when they were finding 
little response from the physicists, are taken up afresh by 
another thinker, and handled with the same thorough criticism 
as Dr. Clausius has bestowed on other chapters of the Theory 
of the Conservation of Energy/ 

He then enumerates the results obtained at a later period 
under more favourable circumstances, and lays down the 


principle, among others, that if a magnet is brought from infinity 
to a body magnetized by induction, mechanical work will be 
done, the value of which will eventually equal half the potential 
of the magnetized, in respect of the magnetizing body. But ' in 
order not to forestall Clausius ', he did not cite all the results 
which he had already arrived at. At the time when he pub- 
lished the ' Conservation of Energy ' he had access only to a few 
isolated portions (apart from their context) of the works of 
Poisson, Green, and Gauss, and therefore confined himself to 
the case in which the iron magnetized by induction was per- 
fectly soft, and so offered no resistance to magnetization (the 
distribution of the magnetism thus being similar to that of 
electricity in conductors electrified by induction). It is, how- 
ever, obvious from a fragmentary note that he had worked 
out the mathematical aspects of the problems involved, starting 
with the assumption that the magnetization of any element of a 
body is proportional to the magnetizing force. 

The memoir thus designed by Helmholtz to refute the 
attacks of Clausius is of the greatest interest, since on the one 
hand it gives the first clear indication of the extent and depth 
of the work already accomplished by Helmholtz in mathe- 
matics and physics previous to his twenty-fourth year, and on 
the other it foreshadows the deductions of the marvellous 
achievements of his later life. 

The most brilliant and the best known of Helmholtz's popular 
scientific lectures, i.e. that 'On the Interaction of Natural 
Forces, and recent Physical Discoveries bearing on the same ', 
was written as the direct consequence of his renewed pre- 
occupation with the Law of the Conservation of Energy after 
the appearance of Clausius's criticism, and of the demand that 
reached him on all sides in KOnigsberg for some more popular 
account of the great principle which was to underlie the 
science of the future. His stern fathers opinion is interesting 
and characteristic : 

' It has given me the greatest pleasure, partly from its 
lucidity and wealth of facts, its easy wit, its hold on true 
science amid all the difficulties of interesting a non-scientific 
audience, partly from the high ideal relation it establishes 
between investigations that would otherwise appear totally 
independent of one another. 


'The view that all phenomena of sensation, from the least 
of the infusoria to the most stupendous solar system, are 
transitory, follows of course from a philosophic conception 
of Time and Space, and of an eternally creating Idea; but 
I rejoice in this thought, already familiar to me in M tiller's 
physiology, since even the much-abused natural science is 
finding its way by physical experiments to the same goal as 
that which the philosophical development of the idea has 
reached already; and thus to those for whom the reality of 
the spiritual has no meaning, the Eternal Idea is revealed in 
the external creation. It is only when we perceive that nature 
and history are the expressions of the divine life, objectively 
immovable, dependent on no subjectivity of the individual or 
epoch of development, laid down for each in every age as the 
sacred tables of revelation in imperishable bronze, that we enter 
upon the sure and never to be abandoned way that leads us 
to the knowledge of God. 

' The only thing I do not like in your lecture, though I quite 
appreciate your motive, is the introduction of the Mosaic 
Creation. That is fundamentally untrue, and a weak concession 
of science which we should not make to our opponents, who 
are idly or childishly clinging to the letter of their beliefs. 
I found fault with Fichte for this when he sent me his last 
great philosophical work ; he admitted my point, and promised 
not to do it again. No enemy is converted by it, while the 
weaker minds are confused, either as to the meaning of the 
Bible, or of the conclusions reached by Science/ 

Even du Bois-Reymond, whose style is incomparable, writes 
to him about this lecture: 'I find it unique, especially at the 
beginning and close, and wonder at the way your style has 
developed. It has been welcomed in all quarters/ It did 
indeed fulfil all the conditions Helmholtz elsewhere laid down 
as essential to a popular treatment of science. 

Nor did Helmholtz content himself with giving a masterly 
exposition of the Law of the Conservation of Energy, in a style 
that made it intelligible to all, along with the historical develop- 
ment of the mechanical principles involved in it, and a generous 
recognition of the fact that ' the first who conceived and stated 
this universal law of nature correctly was a German physician, 
J. R. Mayer, of Heilbronn, in 1842*. Taking his stand upon 


this great law (to which he leads his audience by comparing 
the development of energy in natural processes in relation 
to its utility to man, with the driving energy of machines), 
he proceeds to the question whether the total quantity of 
working energy, which cannot be augmented without a corre- 
sponding consumption, can be either lost or diminished, and 
replies that ' it certainly can for the purposes of our machines, 
but not for nature as a whole*. He then passes on to the 
Carnot-Clausius law, according to which heat can only be 
converted into mechanical work when it passes from a warmer 
to a cooler body, even then its conversion is only partial, 
so that we cannot transform the heat of any body that cannot 
be further cooled into another form of energy, whether 
mechanical, electrical, or chemical and develops the conse- 
quences of this law of nature for the universe : ' these physico- 
mechanical laws are, as it were, the telescopes of our spiritual 
eye, and penetrate into the farthest night of past and future ' 
thence deducing results which du Bois-Reymond aptly reckons 
among his ' most brilliant discoveries '. 

If all bodies in nature had the same temperature, it would 
be impossible to convert any portion of their heat into 
mechanical work. The potential store of energy in the 
universe can thus be divided into two portions, one of which 
is heat and continues as such, while the other (to which 
a portion of the heat of the warmer bodies, and the total 
supply of chemical, electrical, and magnetic energy belong) 
is the source of all the countless interacting changes in 
Nature. Now since the heat of warmer bodies is perpetually 
striving to pass into those that are cooler, so as to establish 
an equilibrium of temperature, and since in every chemical 
or electrical process, and at each motion of a terrestrial body 
subject to collision or friction, a portion of the mechanical energy 
passes into heat, of which a part only can be reconverted, 
it follows that while the first portion of the store of energy 
(the unaltered heat) increases constantly in every natural 
process, the second portion, the mechanical, chemical, and 
electrical energies, is constantly diminished. And thus, as 
all the energy of the world must eventually be transformed 
into heat, and all heat will attain an equilibrium of temperature, 
there will come, as Lord Kelvin predicted, a total arrestation 


of all natural processes, and the universe will be condemned 
to a state of eternal rest. 

But there remains the great mystery of the origin of the 
sun's heat, which keeps up the circulation of water on the 
earth by means of cloud, and rain, and streams, which governs 
all inorganic movement, and preserves the cycle of life by 
the metabolism of plant and animal. The actual heat of the 
sun, and the number of calories it gives out incessantly, could 
be computed, but there was no valid hypothesis as to the 
origin of this heat. Helmholtz set out from the Kant- Laplace 
hypothesis, that the materials now distributed in the sun and 
planets had originally occupied space in the form of a cir- 
culating nebula, which acquired the multiform aspect of the 
planetary system in virtue of its centrifugal and gravitational 
forces. He assumed that the density of the nebulous mass 
was at first a vanishing quantity in comparison with the present 
density of sun and planets, and then calculated how much 
work had been expended on this condensation, and how 
much of this work still exists in the form of mechanical 
energy, as the attraction of the planets towards the sun, and 
the vis viva of their motions, after which he estimated by 
means of the mechanical heat-equivalent, how much of that 
work has been converted into heat. Helmholtz found that 
only some 454th part of the original mechanical energy remains 
as such, while the remainder transformed into heat suffices 
to heat a mass of water equal to the mass of the sun and 
planets taken together to 28,611,000 degrees of the centigrade 
thermometer. 'The enormous quantity of heat lost from our 
planetary system without compensation is not, however, lost 
to the universe ; it has radiated, and is daily radiating, into 
infinite space, and we know not whether the medium, through 
which the vibrations of light and heat are conducted, has any 
limits at which the rays are compelled to turn back, or whether 
they pursue their way for ever to Infinity/ 

It follows from the conclusions of Helmholtz, that even the 
mighty primaeval dowry of the sun's heat, by whose shining 
and heat-giving rays the immense wealth of organic and 
inorganic processes upon the earth is constantly replenished, 
must one day be exhausted, and that Humanity is threatened 
with an eternal ice-age, even if, as Lord Kelvin suggests, the 


sun perpetually receives a certain increment of heat from 
the contraction incident on its cooling. This fundamental law 
of nature, indeed, leaves a long, but by no means an eternal 
existence to the human race: 'Just as the individual has to 
face the thought of death, so too the race ; but it rises above 
the forms of life gone by, in having higher moral problems 
before it, in the consummation of which it finds its destiny/ 

How great an effect, both ethical and scientific, this lecture 
produced in the scientific world, is shown by a letter of Ludwig 
to the Prussian Minister. Ludwig had embarked on a scientific 
dispute with Rudolph Wagner at Gottingen, which speedily 
degenerated, thanks to the retrograde party in science, into 
a war of religious views. ' What have you been up to in 
Gsttingen with R. Wagner?' writes Helmholtz to Ludwig. 
' Dark rumours have reached us here, which sound as though 
you, like Dr. Eck and Dr. Luther of yore, had held, or wanted 
to hold, a public disputation on the nature of the soul, in which 
Wagner of course would have fought with the Bible in his hand, 
and you with the weapons of the Devil, atheism and such like.' 

At this time, notwithstanding the efforts of Helmholtz and 
du Bois-Reymond, Ludwig was in fact in no good odour in 
Prussia. When he was again passed over a year later on 
the occasion of a vacancy in the physiological chair at Bonn, 
although a long way senior to the other candidates, he ad- 
dressed a letter to the Prussian Minister which was of real 
benefit from its distinguished sentiments, and in which, as 
the event was now past, and no unworthy motives could be 
ascribed to his communication, he pointed out the untenability 
of confusing scientific progress with religious principles. With 
reference to the foregoing lecture by Helmholtz, he says: ' How 
remote these religious ideas are from physical physiology is 
apparent from the fact that the physiologist Volkmann of 
Halle, a prominent supporter of the orthodox party, and 
our very dear friend, is not merely a stanch Christian, but 
has lately been busying himself in the attempt to deduce 
a proof for the personality of God from this very lecture of 

On June i, Helmholtz tells his father that a second edition 
of the lecture has already been asked for, adding : * I have 
read several very flattering notices of it, but it was evident 


that the reviewers had no conception of the scientific point of 
view. The general trend of education in Germany is still quite 
aloof from natural science/ 

The summer of 1854 brought Helmholtz many pleasant 
distractions : the best of all being a long-projected four weeks' 
visit from his father, whose dearest wish had been to see 
his son in his home life, and the distinguished scientist in 
the learned circles of Konigsberg. With the latter the father 
also hoped to enter into relations, since he had been greatly 
pleased to learn from his son that the chief librarian and 
Orientalist Olshausen had come upon his treatise on Arabic 
literature in the library, and was much pleased with it, as 
he had long been engaged upon a similar subject. The King, 
too, came to the old Coronation city, and Helmholtz, as Dean 
of the Medical Faculty, had to appear ' three days running 
at Court in a scarlet mantle : at the reception, the banquet, 
and the departure'; while lastly, the second marriage of his 
widowed sister-in-law Betty took place at his house, so that 
during the summer months his time was fully occupied. 
Despite these interruptions he pursued his experiments on 
the excitatory process in nerve, together with some difficult 
optical problems, during the summer, and on July 3 sent 
du Bois a short notice for the Academy, ' On the Rate of 
Certain Processes in Muscle and Nerve/ recorded with his 
frog-tracing apparatus, or as he 'will pompously term it in 
future', the Myographion. The instrument, however, made its 
way slowly, and was little used in the physiological institutes ; 
even du Bois did not venture, on account of its high price, 
to suggest to Joh. Miiller to purchase one for the Anatomical 
Institute. Helmholtz had also constructed new appliances for 
time-measurements on man as early as the winter of 1853-4, 
but owing to the removal of the laboratory in the summer 
of 1854 to the anatomical buildings, was as yet unable to make 
any such experiments, using his spare time instead 'for some 
miscellaneous experiments in physiological optics, which have 
the advantage of not exceeding the comprehension of the scien- 
tific public, so that these worthies may perhaps be inclined to 
believe in my time-measurements, even if they cannot under- 
stand them '. After recalling the definition he had previously 
given in which the period of latent excitation is that in which 


the mechanical properties of the muscle show no alteration, the 
period of rising energy that in which the tension of the muscle 
increases till it reaches a maximum, and, lastly, the period of 
falling energy that in which the tension falls rapidly at first, 
and afterwards very gradually, until finally the initial state 
of rest is re-established he deduces a series of important 
theorems, by means of the myograph, by simple inspection of 
the fully or partially coinciding curves of contraction. In 
these he states that the negative variation of the muscle 
current which induces secondary contraction appears before 
the contraction of the muscle, while the electrotonus of the 
nerve on the contrary coincides with the electrical current 
that excites it. The most important result, however, was the 
proof that two instantaneous excitations produce the strongest 
contraction of the muscle when the interval between them is 
equal in length to the period of rising energy, while, on the 
contrary, two stimuli are not stronger than a single stimulus 
when the interval between them is so small that the first 
contraction has not reached any perceptible height before 
the second begins. He notes provisionally a result to which 
he came back later, pointing out its importance for the 
mechanics of the spinal cord, since it is a means of dis- 
tinguishing between direct and reflex twitches, viz. that the 
tracing of the contraction of the thigh-muscle in strychninized 
frogs excited from the sensory nerve shows that as com- 
pared with rate of propagation in the nerve, the reflex 
twitch is evoked after a comparatively long interval, and that 
in reflexes the passage of the excitation in the cord takes 
more than twelve times as long as its transmission in the 
afferent and efferent nerves. 

Helmholtz now became more and more immersed in optical 
problems, and was hoping, after a number of publications on 
this subject in the journals during the summer of 1854, to 
finish his great work on Accommodation, when on October i 
he received the news of his mothers sudden death on 
September 30: the long distance made it impossible for him 
to arrive in time for her funeral. 

1 For the departed such a swift decease can only be regarded 
as a blessing/ he writes to his father; ' she had suffered enough,, 
and more, in her lifetime for the readiness with which she 


ever sacrificed her health and energies to her dear ones. We 
can only take comfort in remembering that she spent the 
last years of her life in a comparatively peaceful if not too 
happy state, and was taken from us to her reward by a quick 
and painless death/ 

The aged father, bereaved of his faithful companion, was left 
with two daughters and a son. 

His eldest daughter, Marie, born on July 16, 1823, who was 
the more attractive, and intellectually the more gifted of the 
two sisters, gave promise of being a clever artist, but had to 
forgo the exercise of her undoubted talent on account of 
her eyesight. Her wish to make use of her talents led her 
at a later time to seek an independent sphere. She went 
to Russia with the family of Count Bareschnikow, and never 
returned to her own country. She died at Federowska in 
Smolensk, of a nervous fever, on December 17, 1867. The 
sunny charm of her personality was a sacred memory to her 
famous brother. 

The younger daughter, Julie, born September 2, 1827, re- 
mained at Potsdam to take care of her father. In spite of much 
ill health she devoted her whole life to others with fidelity 
and self-sacrifice. The happiest times she knew, though they 
occurred at long intervals, of years sometimes, were spent 
in the house of her brother Hermann, sharing in the develop- 
ment of his richly gifted life. She died after much suffering 
from an attack of apoplexy on July 21, 1894. 

The second son, Otto, born on January 27, 1834, was at the 
time of his mother's death attending the Industrial Institute 
in Berlin, where he had been since he left the Gymnasium 
at Potsdam, with the intention of becoming an engineer (sorely 
against the wish of his father and teachers, whose prejudices 
were very generally shared in those days, but with the full 
approval of his brother Hermann). His brother writes to him : 
1 As to the dispute about " trade " and " not trade ", it is obvious 
from your accounts that you do not see the thing in such a 
light that I need enroll myself on the side of R. and his learned 
contempt for these low employments. The value of work 
depends not upon the material handled, whether in inorganic 
things or in mental products, but upon the amount of intel- 
lectual energy that is put into it, and on whether the work 


is merely a bread-earning industry, or a matter of independent 
intellectual interest. The man who only works on as he was 
taught by his teacher or master in bygone days, and merely 
cares to earn the means of his subsistence or pleasure, will 
be crushed by the mechanical side of his work, but any one 
who works from pleasure in the thing, and tries to help the 
subject forward, will be ennobled by his work, let it be what 
it may/ 

Otto Helmholtz accordingly went into metallurgy, and soon 
became a distinguished engineer, the present Director of the 
great Rhenish Steel Works at Ruhrort. The brothers were 
united in the most intimate friendship until the death of the 
great scientist. 

All the interests of the bereaved father now gravitated to- 
wards Konigsberg. He was greatly pleased, and encouraged 
to look to the future without bitterness, and with confidence 
in his own powers, when his son Hermann (to whom he had 
sent some copies of the papers formerly published in the 
Reports of the Potsdam Gymnasium, for distribution among 
his friends) wrote : l Lobeck told me the other day that he 
had been astonished to hear that the philologist Helmholtz 
was such a near relation of mine. It had not occurred to him 
to connect us, because our subjects are so very different/ He 
goes on, ' I wonder what you would have said if I had become 
a Peer of Prussia ; as our two famous politicians Simson and 
Schubert declined, they asked me among others if I would not 
go up for election. Of course I refused decidedly, because 
that career needs quite a different sort of ambition from any 
I am prone to/ 

The infirm old father heartily approved of his rejection of 
the membership of the Upper House, ' because your scientific 
work in science will be the most profitable to you/ 

Five years of prolific academic work and splendid achieve- 
ment in the different branches of science thus went by in 
Konigsberg; Helmholtz and his wife were happy and settled 
there. Cheerful and contented, serious and industrious, averse 
to no social pleasures, they gradually formed an agreeable 
society of friends, who shared the interests of both wife and 

'When/ writes his sister-in-law, 'I look back at the style 


of the domestic and social life of those days as com- 
pared with that at the close of Helmholtz's life, it makes 
me sad to think of the indescribable modesty of those earlier 
wants and pretensions, although my chief feeling is that 
never did he appear more truly great than at this time, when 
his marvellous genius was developing and growing, along 
with his sincere and noble nature. The man who ranked 
among the elite of the intellectual heroes of Europe, and was 
feted by kings and princes, never seemed to me worthier of 
regard than the modest, indefatigable young investigator, 
who used to construct his bits of apparatus for optical ex- 
periments from his wife's reels, and his children's bricks, 
with ends of wax tapers and scraps of string/ This homely 
apparatus for his intricate and delicate experiments was, how- 
ever, by no means a drawback in the eyes of Helmholtz. 
He was wont to say at a later time, when he had magnificent 
Institutes under his control, ' I was in the habit (and found 
the habit a very useful one) when I wanted to invent some 
totally new method, of making myself models of the required 
instruments, which although they were very fragile, and put 
together as a stop-gap out of the poorest materials, served 
me at least in so far that I could detect the first signs of 
the results expected, and learned the most important of the 
obstacles on which I might founder. This taught me by 
experience the difficulties which hamper the mechanician in 
such new experiments. And it was only when I had made my 
own theoretical conjectures and provisional experiments, that 
I took counsel with the mechanician who was to work out my 
models in brass and steel. And then the difficulties began/ 

He often said in jest to his wife, ' Lend me your eyes for 
half an hour, and you'll be worth something in my optical 
experiments/ His wife was indeed all that he had hoped 
for and counted on, his faithful helpmeet and his true comrade. 
She worked and wrote for him ; he read the lectures he was 
going to publish aloud to her before delivering them, that she 
might judge by her estimate how they would appeal to an 
educated audience. 

But in the meanwhile her health, which had long been a 
cause of anxiety, was steadily growing worse, and the last visit 
to the sea, usually so productive of good, had failed in its 


effects : she had suffered from a cough since the birth of her 
children, and did not spare herself enough in her invincible 
loyalty to her duties. The doctors thought that the cold 
climate of Konigsberg was one cause of her frequent illnesses, 
and when the physiological post in Bonn fell vacant, Helmholtz 
naturally endeavoured, if only in his wife's interests, to get 
himself transferred from Konigsberg. 

He took no steps, however, without ascertaining the wishes 
of his old friends Ludwig and du Bois-Reymond, in case the 
former wished to return to Germany, and the latter to become 
a regular professor at last. It was not until he heard that 
Ludwig's political attitude at Marburg, and the erroneous 
reports as to his atheism, gave him no prospect of a call to 
Prussia, and that du Bois hesitated to take this post because 
his appointment to the chair at Berlin was almost a certainty, 
that he wrote on November 5, 1854, from Konigsberg to du 
Bois-Reymond : 

1 If you have decided not to take the vacant post at 
Bonn, I should be obliged if you would let me hear definitely, 
because at an equivalent salary I should prefer the post at 
Bonn, and should like to approach the Ministry on the subject. 
My reasons are that I should have a wider circle of activity at 
Bonn, a slight though not at first important increase in fees, 
and, lastly, there is my wife's health, which seems to be 
seriously endangered in this climate. I myself lose no small 
portion of my energies for work through the inevitable chills. 
You see that my reasons are not so pressing as to prevent my 
leaving the post to you with the best possible grace, but I 
should grudge it to any one else/ 

Du Bois did not reply till December 6, when he writes : 
' I was unable to send a definite answer to your letter before, 
and cannot do so even now. Strictly speaking, I have never 

approached the subject of Bonn with the Ministry With the 

name you have made for yourself as a teacher you cannot 
fail to get another appointment before long. I shall probably 
keep out of the running. At present it makes me furious to 
see my neglected apparatus and manuscripts, and how you 
managed in your first years at Konigsberg to make such 
colossal researches is a mystery to me. But, to be sure, Du 
gleichst dem Geist, den Du begreifst? 


Johannes Schulze of Bonn, whom Helmholtz consulted, 
replied that the Ministry intended to appoint an anatomist to 
Bonn, since the existing physiological lectures appeared to 
give satisfaction, but that Helmholtz should have his support 
if he would promise to give most of his time, for the present 
at any rate, to anatomy. His wife's health appeared to him 
such a serious consideration that he again expressed his wish 
to undertake the post on these terms, with certain reserva- 
tions, but the answer was so long delayed that he gave up all 
hopes of the appointment. 

During the summer of 1854, Helmholtz, who now devoted 
himself almost entirely to physiological optics, had sent a 
paper to Poggendorff, which was published in the following 
year with the title ' On the Composition of Spectral Colours '. 
In this he returns to the observation (erroneously stated in his 
earlier work on compound colours, and subsequently corrected) 
that indigo and yellow are the only complementary colours 
in the spectrum, an assertion legitimately attacked by Grass- 
mann in favour of Newton's earlier theories of colour-mixture. 
The special physiological properties of the human eye, to which 
the erroneous conclusion was due, were now submitted by 
Helmholtz to a thorough analysis. 

Owing to the dispersion of colours in the eye, it cannot 
be simultaneously accommodated for two kinds of rays ; if a 
luminous point sends out red light and blue light at the same 
time, and if the eye is accommodated for the distance of the 
point with red illumination, blue light gives a diffusion- 
circle, and there is either a red point in a blue circle, or 
with reversed accomm dation a blue point in a red circle. 
The eye can indeed be accommodated so that red and blue 
light form diffusion-circles of equal magnitude, and there is 
a minute speck of the mixed colour, yet it is scarcely 
possible to fix this position of the eye when there is any 
considerable difference in the refrangibility of the two kinds 
of light ; whereas in the complementary colours previously ex- 
amined by Helmholtz the difference of refrangibility is minimal, 
and accommodation accordingly is more easily fixed. Find- 
ing that his earlier methods only resulted in a minute area 
covered with the mixed colour, he adopted an arrangement 
similar to that of Foucault, in which now one and now the 

K 2 


other of the two colours flashes out at the edge of the field, 
while the remaining area (kept as large as possible) exhibits 
the complementary colour. If a colour mixture is obtained that 
can be taken as white, white daylight must be admitted from 
some other part of the room, and allowed to fall on white 
paper in order that its colour may be compared with that 
of the mixed light. Helmholtz then found that the mixed 
colour altered somewhat according to the position of the image 
on the retina. When he combined red and greenish-blue, so 
that the common field of illumination appeared as nearly white 
as possible, with red predominating slightly, the image appeared 
distinctly green on fixing a point on the paper lying near 
the bright area; and the same occurred when the eye was 
brought so near that the area of mixed colour covered a large 
enough portion of the field of vision for many elements of 
the retina, in addition to the yellow spot, to receive the 

After defining more precisely his use of the different names 
of colours he succeeded, under the above conditions, in pro- 
ducing white from a mixture of indigo-blue and yellow, of 
cyano-blue and golden-yellow, of violet and greenish-yellow, 
and of greenish-blue and red. Green alone failed to give 
any simple complementary colour; in order to produce white 
it had to be mixed with purple, that is with at least two 
other colours, red and violet. He then examined the sensi- 
bility of the eye for the individual elements of the violet 
end of the spectrum, and found that the human eye could 
detect all the refrangible rays of this region which were able 
to pass through the prisms, and he accordingly altered the 
name ' invisible rays ' to that of ' ultra-violet rays '. The ob- 
jective intensity of these is by no means a vanishing quantity, 
as is apparent from the fact that while we perceive nothing 
of the ultra-violet rays of a spectrum thrown on to a sheet 
of plain white paper, because they are masked by the ordinary 
diffuse light, the same parts of a spectrum thrown on to paper 
soaked in quinine solution will, owing to the less refrangible 
light from the fiuorescing quinine, affect the retina with suffi- 
cient energy to be visible. The comparison of colour-tones 
at different points of the ultra-violet spectrum implied the use 
of equal light-intensities, since the colour-tone of this band 


alters more rapidly with intensity of illumination than any other 
portion of the spectrum, and Helmholtz was able to detect a 
whole series of distinct tones of purple. He could not then 
extend his investigations of the sensibility of the retina to the 
ultra-violet rays, since the glass prisms he was using did not 
show sufficient of the ultra-violet spectrum. But he raised 
two other very interesting questions as to the relations of the 
wave-lengths of complementary colours, and the relations of 
intensity required if the mixture of simple complementary 
colours is to produce white. He succeeded in answering these 
questions quantitatively, and concluded on the ground of his 
measurements of the brightness of the colours whose mixture 
produces white, that there must be differences of saturation 
in the various simple colours, violet being the most, and yellow 
the least saturated. The treatise, which has been fundamental 
for all later work of the same kind, concludes with an inquiry 
into the validity of Newton's Colour Circle, which Helmholtz 
designates as one of the most brilliant inspirations of that great 

After a long delay he received two rock-crystal prisms from 
Oertling in Berlin, which he had ordered through du Bois 
for his earlier experiments, and from which he obtained an 
ultra-violet spectrum, more than twice as long as that given 
by the glass prisms. In the paper which he sent immediately 
to Poggendorff, 'On the Sensibility of the Human Retina to 
the most Refrangible Rays of Solar Light/ he propounds the 
important, but highly complicated problem, whether the retina 
sees the ultra-violet rays directly, like the other colours of 
the spectrum, or fluoresces under their influence, and whether 
the blue colour of the ultra-violet rays is light of lower 
refrangibility, which is first developed in the retina under 
the influence of the violet rays. By varying the methods 
hitherto employed, he showed that the human retina is capable 
of directly perceiving all the rays of the sun's light, the re- 
frangibility of which exceeds that of the ultra-red rays ; while 
further under the action of the ultra-violet rays the retinal 
substance scatters mixed light of lower refrangibility, the total 
colour of which is not pure white ; and lastly, the fluorescence 
of the retina is inadequate to explain the perception of the 
ultra-violet rays in general. He found that the tolerably 


saturated blue colour of the ultra-violet rays was absolutely 
different from the almost totally white hue of the light dis- 
persed by the dead retina. The wave-length of ultra-violet 
light was also measured by Esselbach in Helmholtz's laboratory 
at Konigsberg, and under his supervision. This research was 
communicated by Magnus to the Berlin Academy in December, 
1855, under the title, ' Measurement of the Wave-length of 
Ultra-violet Light, by E. Esselbach,' with an appendix by 
Helmholtzon the physiological-optical results of these measure- 
ments. He gives an extended comparison of the relations of 
the length of light-waves with that of musical intervals, accord- 
ing to which the entire visible portion of the solar spectrum 
comprises an octave and a fourth ; his tables show how little 
analogy there is between sensations of tone and of colour, 
since the whole of the intermediate degrees between yellow 
and green are compressed into the breadth of a small semi- 
tone, while at the end of the spectrum there are intervals as 
large as a major or minor third, in which the eye is unable to 
perceive any alteration of colour-tone. 

His great treatise on Accommodation was now approaching 
its conclusion. It was published in 1855, in Graefe's Archiv 
f. Ophthalmologie, and contributed an extraordinary mass of 
new points of view, methods, and results to physiological 

The priority of one fundamental discovery, as previously an- 
nounced in the Monatsberichte of the Academy, had indeed to 
be forgone in favour of Cramer, i. e. that the lens in the resting 
condition of the eye, when it is accommodated for far vision, 
is not in its natural form, but is flattened by the surrounding 
structures, but that the pull of Briicke's muscle enables it 
to resume its natural form of marked curvature and greater 
thickness, in virtue of its elasticity results which he obtained 
not by watching the alteration of form, or displacement of 
the media of the eye in accommodation, but by investigating 
the changes of the weak light-reflexes first observed by Sanson 
within the pupil, which take place at both surfaces of the 
crystalline lens, and are sufficient to account for accommo- 
dation. But many difficult questions still remained, which 
could only be solved by a gifted mathematical physicist. Such 
were the exact determination of the inner and outer surfaces 


of the cornea, the alterations of the iris in accommoda- 
tion, and lastly the curvature of the anterior and posterior 
surfaces of the lens, which he determined with astonishing 

Starting from the presumption that a convex mirror-surface 
gives smaller images of the surrounding objects in proportion 
as its radius of curvature is smaller, so that the radius of 
curvature may be calculated from the size of the images, he 
attempts to measure the size of the minute image on the 
cornea, but is at once pulled up by the difficulty that the living 
eye cannot be fixed as immovably as such an exact measure- 
ment requires. In order to measure the free corneal image, 
while the eye itself is in motion, he therefore applied the 
principle of the heliometer (by which astronomers can estimate 
the least distances of the stars in the moving heavens, notwith- 
standing their apparent motions, so exactly that they can plumb 
the profundities of the firmament of the fixed stars), applying 
it in an altered form to the moving eye. He constructed the 
ophthalmometer, by which he succeeded in measuring the 
curvature of the cornea and other phenomena of the living 
eye with greater accuracy than had hitherto been possible on 
the dead eye. The principle of the ophthalmometer, which was 
to play so great a part in physiological optics, depends on 
the fact that objects observed through a glass plate with per- 
fectly even and parallel surfaces, held obliquely to the line 
of vision, seem to be displaced laterally, and that this dis- 
placement increases with the increasing angle of incidence of 
the rays of light upon the plate. When two plane-parallel 
glass plates are rotated in opposite directions in front of a 
telescope obliquely to its axis, two images of any object that 
is within the field of the telescope appear simultaneously: if 
the two glass plates are then rotated till the two images overlap, 
Helmholtz showed that the size of the object observed can 
be estimated from the magnitude of the angle of rotation, irre- 
spective of the distance of the object from the telescope, because 
the ophthalmometer shows the same linear displacement at all 
distances. The limitations of the Institute compelled Helmholtz 
to construct his telescope from materials which he happened to 
possess, and the entire instrument, except the plane-parallel 
glass plates, was made in Konigsberg ; but he soon suggested 


to Bonders a more practical construction, in order to obtain the 
maximum of brilliancy in the images. 

When this instrument is used to measure the curvature of 
the cornea, the latter must reflect the image of some external 
object of known size and distance, and the magnitude of the 
image can then be measured in the ophthalmometer. Helmholtz 
made the important discovery that in all diseases of the eye 
that are associated with alterations of pressure in the fluid 
media, these changes can be detected on the cornea. His 
measurements of the radius of curvature at different points of 
the cornea showed that it corresponds in form with an ellip- 
soid, produced by the revolution of an ellipse about its major 
axis, so that the base of the cornea forms a plane vertical to 
the major axis of the ellipse, and the central point of the cornea 
coincides with the vertex of the ellipse. During accommodation 
there is not the slightest alteration of curvature in the cornea. 
This method could not be employed in determining the form 
of the inner surface of the cornea, because the image from the 
anterior corneal surface is so much stronger than that from 
the posterior that the latter cannot be observed, if the two 
occur very close together. Experiments with the cornea of 
dead eyes showed, however, that the thickness of the cornea 
scarcely alters at all in the two central quarters, but increases 
fairly rapidly towards the margin. From this it may be 
assumed, in calculating refraction of the eye, that the aqueous 
humour extends to the anterior surface of the cornea. And 
since the lens extends close to the iris, it is only necessary, 
in order to determine the distance of lens from cornea, to 
measure that of the pupillar border of the iris from the cornea, 
which Helmholtz again accomplished by means of the ophthal- 
mometer. It was also shown by a series of extremely delicate 
observations that since the lens is always situated close to 
the pupillar border of the iris, while the form of the cornea and 
the volume of the aqueous humour are not altered in accommo- 
dation, displacement of the middle portion of the iris and lens 
cannot occur without such a backward movement of the iris 
at its periphery, that the anterior chamber gains as much in 
volume there as it loses in the centre. 

The curvature of the anterior part of the lens cannot be 
measured directly by the images, because the reflected image 


is not sharp. The size of the image has to be compared with 
that of a corneal image close to it, by means of two reflected 
objects, one of which must be of varying magnitude, in order 
that the corneal image of one may be made equal to the Sanson's 
image of the other. It was then ascertained by means of the 
ophthalmometer that in near accommodation the anterior surface 
of the lens is more strongly curved, the radius of curvature 
is accordingly smaller, and its vertex is pushed forward. When 
Helmholtz applied the method to the posterior surface of the 
lens, he discovered, in determining the position of the latter, 
and the question whether cornea and crystalline lens are 
symmetrical to the same axis, that, in the eyes under examina- 
tion, there was a slight but perceptible defect of centring, 
which produced the so-called astigmatism of the eye : the 
effect of which is that we cannot clearly see horizontal and 
vertical lines at the same distance simultaneously. This he 
characteristically expresses by saying that the eye, in spite of 
its wonderful powers, is an instrument so full of serious defects 
that if a mechanician turned out anything so imperfect he 
would show him to the door. In regard to the actual curvature 
of the posterior surface of the lens, he found that it became 
a little more convex during accommodation, and did not alter 
its position perceptibly. Lastly, in regard to the question how 
the observed changes of form in the lens are produced, he 
inclines to the view that the ciliary structures must be admitted 
to participate in some way or other in the movements of 

Writing before the publication of this work, Helmholtz 
informs du Bois that the article on Accommodation in 
Graefe's Archiv fur Ophthalmologie is in the press, but not 
out yet: 

' I have determined the measurements of the curvature of 
the cornea and anterior and posterior surfaces of the lens, and 
their distances in the living eye, by new methods not indeed 
with the greatest attainable accuracy, but only so as to 
show people that it can be done ; for I realized during the under- 
taking that it would be useless to expend great pains upon 
it. The human eye is not even properly centred, the magni- 
tude of the corneal excentricity appears to be quite irregular 
and adventitious, and so on. You must judge the paper from 


these points of view, when you receive it, as should I think 
be the case shortly/ And du Bois-Reymond judged it as 
follows : ' Never had any one blended the fullest knowledge of 
physical and mathematical optics with such a vivid and exact 
idea of the anatomical conditions of vision as did Helmholtz/ 

Just as Helmholtz's treatise on the Law of the Conservation 
of Energy had been epoch-making in the development of 
the physical sciences, so his experiments on Accommodation in 
conjunction with the Ophthalmoscope brought about a com- 
plete revolution in ophthalmology. As his great lecture on 
' The Interaction of Natural Forces ' had made the principles of 
the colossal work of his youth accessible to the scientific world 
as a whole, so now the opportunity presented itself of bringing 
the physiological-optical discoveries that had occupied him in 
the past year before wider circles. On February 27, 1855, 
at KGnigsberg, he gave a popular scientific lecture in aid of 
the Kant Memorial, which treated of the subjectivity of the 
sensations, and of their analogy with Kant's theory, and the 
psychical processes that underlie the interpretation of our 
sensations. ' Last Tuesday/ he writes to his father, ' I gave 
another lecture upon " Human Vision ", in which I tried to 
put forward the correspondence between the empirical facts 
of the physiology of the sense-organs and the philosophical 
attitude of Kant, and also of Fichte, although I was somewhat 
hindered in my philosophical exposition by the need of making 
it popular/ 

He sends Ludwig the following interesting account of the 
philosophical views which then prevailed in Konigsberg : 

* In the early years of my stay, " nature-philosophy " was 
still rampant among the students, and the scientific circles of 
the city often took up the cudgels against my attitude. I never 
set myself aggressively in opposition to Rosenkrantz, who had 
once been the demi-god of the city, though now he has only 
a very limited and half-incredulous public ; but left the weight 
of facts to speak for itself. . . . The more intelligent portion of 
the scientific public will only attend, as a rule, to speculative 
investigations when they issue from men whose sound and 
original experimental work has proved them to be firmly 
grounded on the rock of facts/ 

But while he believed that philosophy, when it has been 


purged of metaphysics, will still remain as the vast field of 
knowledge of mental and psychical processes, and the laws 
that govern them, and that it alone can provide the scientific 
worker with the necessary insight into the potentialities of the 
instrument with which he works the human mind a letter 
written twenty years later to Kick shows that the development 
of philosophy, for which Helmholtz longed, and strove on the 
lines of his theory of knowledge, was slow to the last degree 
in its evolution : 

' I believe that philosophy will only be reinstated when it 
turns with zeal and energy to the investigation of epistemo- 
logical processes and of scientific methods. There it has a 
real and a legitimate task. The construction of metaphysical 
hypotheses is vanity. Most essential of all in this critical 
investigation is the exact knowledge of the processes of sense- 
perception. . . . Philosophy has been at a standstill because 
it was exclusively in the hands of the philologists and theo- 
logians, and has so far imbibed no new life from the vigorous 
development of the natural sciences. Hence it has been 
almost entirely confined to the history of philosophy. I 
believe that any German University that had courage to 
appoint a scientific man with an inclination for philosophy to 
its Chair of Philosophy would confer a lasting benefit on 
German science.' 

The lecture on 'The Interaction of Natural Forces' had 
not merely treated of the Law of the Conservation of Energy 
in a generally intelligible form, but had developed a series 
of totally new consequences for the constitution of the uni- 
verse from this standpoint, so that it represented another 
distinct scientific achievement. In like manner the lecture on 
' Human Vision ', which set out with the review and interpre- 
tation of the laws he had discovered in physiological optics, 
went on, in bringing ( an offering of respect and veneration ' 
to Kant, to develop the philosophical consequences of his 
discoveries, which were recognized ere long as the first 
principles of the modern theory of knowledge. Helmholtz's 
interest in epistemological questions had been awakened in 
early days, when his father, who had imbibed a deep impres- 
sion of Fichte's idealism, held discussions upon the profoundest 
problems of speculative philosophy with his colleagues, who 


venerated Kant or Hegel. He had long been convinced that 
if the physicist tests the galvanometer and telescope he intends 
to work with to the limits of their efficacy, it is no less in- 
cumbent on a scientific man to include the intellect in the 
sphere of his investigations, in order to ascertain what he can 
arrive at by its means, and where it is likely to fail him. Helm- 
holtz was fully aware that he had on the one hand 'all the 
metaphysicians, including the materialists, and all minds with 
lurking metaphysical tendencies ', to reckon with, while on the 
other, the scientific world would be impelled by the excrescences 
of Hegel's 'nature-philosophy 1 to extreme suspicion of all 
speculative explanations of natural phenomena, and would 
extend this legitimate prejudice to the epistemological and 
psychological investigations in which the attempt to penetrate 
the laws of mental activity is both valid and necessary. 

After pointing out in the lecture that physical science still 
professes the principles of Kant (whose philosophy does not 
add to the content of cognition by pure thought, but derives 
all perception of reality from experience, and makes the 
sources of our knowledge and the degree of its justification 
the sole objects of investigation), he proposes the Theory of 
Sense-Perception in man as the real theme of his lecture, 
since it is here that philosophy and natural science are most 
in touch. He inquires how the empirical data for the organ 
of the eye stand in relation to the philosophical theory of 
knowledge. After a full account of the construction of the 
eye, and of his theory of accommodation, he gives an explana- 
tion of Joh. Miiller's fundamental doctrine of specific senses, 
' Light is only light when it falls on the seeing eye/ The 
discussion of the theory of colours, the facts on which the 
construction of the stereoscope is based, and other optical 
phenomena, show us more and more plainly how little we 
reflect in the daily, practical use of our sense-organs on the 
part these have to play, how exclusively we interest ourselves 
in such perceptions as bring us intelligence from the outer 
world, and how little we attend to other perceptions not 
adapted to this end. Now as consciousness (contrary to the 
earlier theories) does not perceive sensations locally, at their 
seat in the body, it can only know by unconscious inference 
whatever we do not perceive directly. This inference is 


mechanical in character, and comes under the category of 
involuntary combinations of ideas, arising when two percepts 
are frequently associated together. Thus in optical delusions, 
the mechanism of which is evident, we are aware that the 
idea called up by the sense-impression is false, but the idea 
nevertheless persists in full vigour. When, given a certain 
position of the eyes, any object excites a sensation of light 
in certain nerve-fibres of our two eyes, our past experience 
that it is necessary to stretch the arm out a certain distance, 
or to take a certain number of steps in order to reach it, 
has established an involuntary relation between the given 
visual impression and its distance and direction : the judgement 
of distance by the eyes is learned empirically. ' I distinctly 
remember the moment at which I became aware of the law 
of perspective, that distant objects look smaller. I was taken 
past a high tower, on the topmost gallery of which people 
were standing, and begged my mother to lift down the little 
puppets, as I quite thought she could reach the gallery of 
the tower by stretching out her arm. Afterwards I often 
looked up at the gallery of the same tower when people 
were standing on it, but nevermore to the eye of experience 
did they look like pretty dolls/ When once we have learned 
to see, i. e. to associate the idea of a certain object with 
certain sensations, which we perceive, the seat of the optical 
image upon the retina is a matter of indifference, since it 
is only the fibres of the optic nerve which are excited that 
are in question. 

Helmholtz does not here attempt to decide the question of 
how far acquired associations of ideas, or such as are innate, 
i. e. implicate in the actual organization of man, are involved 
in the interpretation of our sense-organs. For him, sensations 
are mere signals to consciousness, the meaning of which we 
learn by a mental process, signals of the existence of changes 
in the external world, but not images of these changes except 
in so far as they represent the sequence of the same in 
time, and may therefore be considered to give us a direct 
representation of the temporal course of natural phenomena. 
It was only at a later time that he came forward as the 
champion of the empiricist, as against the nativistic theory, 
though he had already made a great stride in this direction. 


Since we never can perceive the objects of the external 
world directly, but only from their action upon our nervous 
mechanism, the question obviously presents itself, how in 
the first instance we ever got into touch with the real world 
by means of our nervous sensations? We must postulate 
the presence of external objects as the cause of our nervous 
excitation, since there can be no effect without a cause : but 
this dictum can be no law of experience, we already need 
it for the knowledge that there are any objects at all in the 
space around us. Yet it cannot come from the internal ex- 
perience of our self-consciousness, since we regard the self- 
conscious acts of our will as free. Hence we must fall back 
on Kant's conclusion that all our thoughts and acts, the greatest 
as the least, are founded on our confidence in the unalterable 
uniformity of nature, and that the axiom, 'no effect without 
a cause/ is a law of our thought prior to all experience. Among 
the papers left by Helmholtz is the following interesting note 
on this subject: 

'The Law of Causation (the presupposed uniformity of 
nature) is a mere hypothesis, and not otherwise demonstrable. 
No previous uniformity can give proof of future uniformity. 
The sole test of any hypothesis is, try if it be so, and you 
will find out (best by experiment, where possible). In com- 
parison with other hypotheses which enunciate special laws 
of nature, the Law of Causality is exceptional in the following 
ways: (i) all others presuppose it; (2) it gives us our sole 
possibility of knowing something we have not observed ; (3) it 
is the necessary foundation of premeditated action ; (4) we 
are reduced to it by the natural mechanics of our combinations 
of ideas. Hence we are induced by the strongest motives to 
desire its validity. It is the groundwork of all our thoughts 
and acts. Until we have it we cannot test it; therefore we 
can but believe in it, act upon it, and find it justified by fair 
tests. We must anticipate the consequences; then the con- 
sequences will be its confirmation. We must be aware that 
we anticipated the result ; then we shall be aware of the law. 
Thinking means seeking for uniformity ; judging, that we have 
found it. Hence, without the law of causation there can be 
no thought. No thought without acceptance of the law of 
causation is tautology ; query, are we justified in thinking, and 


has our thought any meaning? this meaning can only be 
expressed in action (internal or external)/ 

The Bonn appointment had not yet been decided, although 
Helmholtz's call was rumoured in all the German papers. On 
March 15, 1855, du Bois-Reymond sends a few lines from a 
letter of von Humboldt: 'They beg me to bestir myself for 
Helmholtz, whom I love and esteem as much as you. I cannot 
say a word until I know your wishes. If you are not in haste, 
and can wait on, do not quit the capital, where you should 
have a great future/ adding that he had replied : ' I beg that 
you will exert yourself for Helmholtz as if there were no 
question of me/ 

On March 24 Humboldt writes to Helmholtz : 

1 Dear Professor, I was agitating on your behalf long before 
you honoured me with your confidence. The deplorable state 
of your wife's health makes a move from that raw climate most 
desirable. When Herr von L. first spoke to me of your fresh 
request, I ascertained from our mutual friend du Bois-Reymond 
that he did not wish to leave Berlin ; so that my earlier 
friendship with du Bois does not prevent my having a free 
hand. Why should we seek abroad what lies so brilliantly 
to our hand ? . . . Any one who knows the history of science 
is aware that no individual, especially in the present state of 
knowledge, could possibly be equally strong in anatomy and 
in physiology, and the greater the renown of a man in one 
of these two branches, the more he is open to charges of 
weakness and negligence in the other. I have with great effort 
written a long and enthusiastic letter based on the materials 
you sent me, as I have only done once in my life before for 
Dr. Brugsch's Egyptian expedition, refuted the opinion of 

, without mentioning his name, and based my proposal 
on our friendship, your domestic trouble, your splendid talents, 
and extraordinary industry. I hope much good will come 
from this well-deliberated step. I am glad to have found an 
opportunity of offering you this poor proof of my sincere 

Humboldt had no doubt that the gifted investigator would 
soon rise to be a first-class teacher and authority in the ranks 
of the anatomists, for he was familiar with his admirable 
anatomical dissertation, and was interested in many of Helm- 


holtz's anatomical observations as reported to him by du Bois. 
In Berlin, for instance, the young surgeon had amused himself, 
in the intervals of keen mental activity, by watching the move- 
ments of the people going in and out through the Branden- 
burg Gate, with a telescope, from his little laboratory in a tower 
at the corner of the Dorotheen-strasse and Sommer-strasse, 
comparing them with the descriptions and figures given by 
Weber in his work on the human locomotor apparatus. In this 
way he discovered, as du Bois relates, that there was an error 
of some practical importance in Weber's figures, in consequence 
of which thousands of recruits had been compelled into an 
unnatural position of the foot during their march on parade. 
His observations were long afterwards confirmed by instan- 
taneous photography. 

On March 27, his appointment as Professor of Anatomy and 
Physiology in Bonn, from Michaelmas, 1855, was announced. 

During the summer Helmholtz devoted himself almost ex- 
clusively to his Handbook of Physiological Optics, which was 
to be given to the printers early in the winter, and he writes 
of the finished portion to Ludwig : ' The only really new bit 
of mathematics in Part I of Physiological Optics is the proof 
of Gauss's laws of principal points and nodal points by 
means of an accessory theorem (p. 50), which finds a useful 
application in the theory of the ophthalmoscope also/ 

During his last days at Konigsberg he received an invitation 
from William Thomson, now Lord Kelvin, from Kreuznach, 
to attend the British Association in September. Thomson 
wrote that his presence would be one of the most interesting 
events of the meeting, so that he hoped to see him on this 
ground, but also looked forward with the greatest pleasure 
to such an opportunity of making his acquaintance, as he had 
desired this ever since the ( Conservation of Energy ' had come 
into his hands ; he ended with expressing his deep regret that 
he had not been present at the Hull Meeting, having only 
heard later that Helmholtz had been there. 

On July 29, Helmholtz left Konigsberg, and went, after a short 
visit to his relations in Dahlem and Potsdam, to Bonn, where 
he found a suitable and healthy dwelling for his family in the 
building that had formerly been the summer residence of 
the Ecclesiastical Elector of Cologne, and was accordingly 


known as the Vinea Domini. He then proceeded by Bingen 
to Kreuznach, in order to make acquaintance with Thomson 
before his projected journey to England. He writes to his 
wife on August 6, 1855, that Thomson had made a deep 
impression on him : 

' I expected to find the man, who is one of the first mathe- 
matical physicists of Europe, somewhat older than myself, 
and was not a little astonished when a very juvenile and 
exceedingly fair youth, who looked quite girlish, came forward. 
He had taken a room for me close by, and made me fetch 
my things from the hotel, and put up there. He is at Kreuz- 
nach for his wife's health. She appeared for a short time 
in the evening, and is a charming and intellectual lady, but 
in very bad health. He far exceeds all the great men of science 
with whom I have made personal acquaintance, in intelligence 
and lucidity and mobility of thought, so that I felt quite wooden 
beside him sometimes. As we did not get through nearly all 
we wanted to say yesterday, I hope you will let me stay over 
to-day in Kreuznach/ 

The closest friendship and mutual esteem connected these 
two great men for nearly forty years, until death separated 

The last report from Konigsberg, on 'Work bearing on 
the Theory of Heat in the year 1852,' had dealt with the 
famous publications of William Thomson, and these were now 
discussed by word of mouth at Kreuznach by the two great 
legislators in the field of science. Thomson, after establishing 
the already known law that the heat produced by animals, 
together with the work done by them, is equivalent to the 
chemical energies of their food and of the inspired oxygen, 
had arranged the different sources from which mechanical 
effect can be derived according to their origin, and came to 
the conclusion that the heat radiating from the sun, including 
its own light, is the principal source of all terrestrial processes, 
and that the motions of the earth, moon, sun, with their mutual 
attractions, are a potent source of kinetic energy, while an 
exceedingly small portion only is of purely terrestrial origin. 
The conclusion deduced by Thomson from Carnot's Law, to 
the effect that the heat of the coolest body in the universe 
will always persist as a work-equivalent, but cannot be 


reconverted into any other form of energy, had been briefly 
discussed by Helmholtz in the report, while he indicates the 
conclusions which follow from the considerations laid down by 
Thomson, and which he himself had developed so brilliantly 
in his lecture on ' The Interaction of Natural Forces '. 

In the middle of September Helmholtz fetched his family 
from Dahlem, and settled them at Bonn, the move being ac- 
complished without difficulty, as his wife was in fairly good 


AT BONN: 1855-1858 

HELMHOLTZ soon accustomed himself to his new surround- 
ings. He writes indeed to Bonders in October : ' I could only 
take a very few of my instruments away from Konigsberg 
as my own property, and find practically nothing here. It 
is a case of beginning over again to collect apparatus, and 
that with uncommonly small funds. Our Ministry still clings 
to the fiction that it will be involved in the war in the East, 
and declines to make any outlay of money/ 

But in December he informs his father : ' All goes well here 
on the whole. While the thermometer in Konigsberg was 
already below zero, and the freight wagons are crossing the 
Vistula on the ice, we have had alternately mild frost and wet 
weather, and are more inclined to moderate our stoves than 
to keep them up. The effect on Olga's health has been all one 
hoped; she has left off coughing since we arrived in Bonn. 
As regards my official position, the prospects for the winter are 
favourable. I have forty-five students at my lectures, and it 
is altogether quite different from Konigsberg. The Anatomy 
Lectures are very troublesome this first time, especially in 
certain subjects, but it will go much better next year. I find 
anatomy more interesting too than I had expected, because the 
teaching of this science has hitherto conspicuously neglected 
the functions of the organs, so that interesting questions and 
points of view crop up on all sides as soon as one looks at 
them with the eyes of the physiologist. My success in physio- 
logy this summer is a little uncertain on account of the rivalry 
with Professor B. . . . The Faculty proposed that we should 
each give a course in physiology (instead of dividing it, and 
each taking a six hours' course, or half of it) after the Dean had 
asked me if I should not prefer that. I replied that I could 

L 2 


not myself go back on my promise to B., but should be 
justified in breaking it, if the demand were made by the Faculty. 
B. himself did not seem to object/ 

At first Helmholtz was fully occupied with his lectures in 
anatomy, and as the immediate result he made a short com- 
munication to the Nieder-Rheinische Gesellschaft on March 12, 
1856, ' On the Movements of the Thorax/ in which he threw 
himself into a controversy as to the intercostal muscles. 
He concluded that the leverage of the upper ribs was the 
strongest, while it becomes weaker from above downwards, 
so that the thorax must be regarded as a basket of elastic 
hoops, each of which has its position of equilibrium, from 
which it is shifted during inspiration by the pull of the muscles, 
and which it recovers by its own elasticity in expiration, 
since expiration in quiet breathing seems to be effected merely 
I by the relaxation of the inspiratory muscles. 

He found great satisfaction in his scientific and pedagogic 
functions in this new field, and writes at the close of the first 
winter session, March 6, 1856, to his father : 

' All has gone well so far in my official relations. To-morrow 
will be the last of my lectures. The audience has kept fairly 
up to the mark, and the older students, who are taking Anatomy 
for the second time, have told me repeatedly that I have 
shown them and told them much that had escaped them before. 
So I am justified in hoping that I shall succeed with the 
Anatomy Lectures, and things will go better when I have got 
the Museum into order. It has been frightfully neglected.' 

But while Helmholtz believed that he had been successful 
in the anatomy lectures, du Bois writes to him on April 27, 
1856, on Lehnert's authority, that it had been reported to the 
Ministry that his lectures in anatomy were inadequate. Du 
Bois replied to Lehnert that while all things are possible, 
and stupidity probable, this was not only improbable, but 
also impossible; whereupon Lehnert, after giving him the 
source of the mischievous report, begged him ' to reassure the 
Minister personally, since he was suffering pangs of conscience 
for having made such bad provision for Anatomy at Bonn'. 
Helmholtz replies to du Bois on May 3 : 

1 The report made to the Minister annoyed me ; since it is 
not even an exaggeration of facts, but is a pure invention, 


which shows up the intention of its author in no amiable light 

I was told that people said I brought a good deal of physiology 
and chemistry into my anatomy, which restricted the amount 
of anatomy proper, and they made jokes at the introduction 
of a cosine in physiological optics. But I received many 
indications of interest and appreciation of my lectures from 
the older students, and from my colleagues also/ At the close 
of this letter Helmholtz sends heartiest congratulations 'to 
the young sucking-philosopher, who has taken up his abode 
with you, and is doubtless already occupied with such difficult 
questions as the formation of concepts of time and space of 
which he knows more now than all the learned physiologists 
in the world'. 

Nor was it as a teacher alone that Helmholtz had found 
a congenial sphere of activity. He sought to acquaint the medical 
world of Bonn, who were somewhat remote from his stand- 
point, with his nerve-work, by reading a paper on the 'Con- 
traction Curves of the Muscles of the Frog ' recorded with the 
myograph, to the Nieder-Rheinische Gesellschaft on May 14. 
He also endeavoured, while preparing his Physiological Optics, 
to interest his scientific colleagues in these new and difficult 

On March 6 he made a short, but important and interesting 
communication to the same Society 'On the Explanation of 
Lustre'. Helmholtz started from the fact that in looking at 
dull surfaces, they appear equally illuminated, and equally 
coloured, to both eyes, while for shining surfaces the contrary 
is the case, since one eye may be affected by the more or 
less regularly reflected light from the smooth surface, and 
the other not. The surfaces then appear brighter to the one 
eye, and if the reflected light differs in colour from that of 
the surfaces, of a different colour also, although these differ- 
ences of colour as presented in daily experience to both eyes 
by shining surfaces are usually insignificant. Now if the 
observer looks with the stereoscope at any surface that appears 
brighter or somewhat differently coloured to one eye than 
to the other, he will conclude from the analogy of everyday 
experience that this surface is lustrous, a phenomenon that 
had long been known, but had found no adequate explanation. 
With greater differences of colour, empirical analogy is totally 


wanting, and different persons form a different judgement, 
some seeing a mixed colour, others irregular specks of colour. 
From this Helmholtz deduces the all-important conclusion that 
the sensation from each eye'comes separately to consciousness, 
so that simple vision with both eyes is not the consequence 
of an anatomical junction of the nerve-fibres, but the result 
of an act of judgement. 

In September, 1854, Helmholtz writes from Konigsberg to 

1 1 have been busy for some time over my Physiological Optics. 
I cannot make it all as popular as the doctors would like, but 
I have tried to arrange so that what they do not care for is 
put together, leaving the rest for them. I have not covered 
much of the ground yet, because I began with the hardest 
parts, refraction, accommodation, &c., and was tempted into 
making new and systematic measurements on the living eye, 
which have only resulted in the conclusion that the human 
eye is so irregular that exact measurements do not repay one. 

I have also made a number of experiments on colour There 

are so many vexed questions in physiological optics that can 
be settled by a couple of accurate experiments, that one is 
ashamed to bandy words about them, without making the 
experiments, so that I really see no prospect of getting on 
any faster/ 

Part I of the Handbook appeared in 1856. Meantime, with a 
view of establishing the subjectivity of sensation for the other 
senses also, and of determining the psychical processes by which 
we understand these sensations, Helmholtz had been turning 
his attention to physiological acoustics, in which he again 
opened up an entirely new field of physiological discovery, 
and in which his results were as admirable as those he obtained 
in physiological optics. 

On May 21 he writes to Wittich : 

1 During the winter I reinvestigated the connexions of the 
auditory ossicles, and worked at Tartini's tones. I am so far 
ready that I have sent a resume to the Academy at Berlin, 
and am now working at the longer paper. I hope to derive 
the whole theory of harmony from the fundamental fact that 
the ear perceives movements that are regularly repeated at 
given intervals as a continuous sensation of tone, and that 


a continuous sensation of tone is felt to be a consonance, a 
discontinuous sensation to be a dissonance. Tell Richelot that 
I am now endeavouring to establish thorough bass upon an 
integration of partial differential equations of the second order 
and second degree. I hope this may interest him more than 
the subjects of my earlier work/ 

On June 18, he writes to William Thomson : ' I have busied 
myself with certain observations in acoustics during the winter, 
especially on combination tones, which have shown me that 
these tones, which have hitherto always been supposed to 
originate within the ear, can arise externally to it also, when- 
ever the vibrations of the air or of any other elastic body, 
including the tympanum of the ear, are so strong that the 
second power of the elongation has influence on the motion, so 
that the law of the superposition of small vibrations ceases 
to be valid. If m and n are the vibration numbers of two 
simultaneously sounding tones, I have, in addition to the long- 
recognized tone of (m n) beats, discovered another tone of 
(m + n) beats/ 

His paper on Combination Tones appeared the same year 
in Poggendorff's Annalen. It was known that there is on 
the one hand undisturbed superposition of various sound- 
waves in the air, and that on the other the ear, when simul- 
taneously affected by several such waves of sound, has the 
power of perceiving and recognizing each of them separately. 
But in such cases the ear not only hears the different tones 
excited by the resonant bodies, but other additional, if feebler, 
tones, the combinational tones, which are not primarily pro- 
duced by one of the sounding bodies, but are of secondary 
origin from the concurrence of two primary tones. These 
were formerly held to be subjective phenomena, dependent on 
the special nature of the sensation of the vibrations of sound 
through the auditory nerve. Helmholtz, however, submitted 
this question, as well as the possible existence of other than 
the known combinational tones, to a searching examination, 
supplemented by mathematical analysis. 

He names any oscillation of an elastic body, in which the 
distance of each vibrating particle from the position of equi- 
librium can be represented as a simple sine-function, with 
a constant factor, of a linear expression of time, a simple 


vibratory motion ; if the oscillations are transmitted through an 
elastic medium, a simple wave-motion; he calls all other 
oscillations that can be expressed, as was already known, as 
a sum of such sine-functions with arguments, which again are 
linear functions of time, a compound vibratory or wave motion. 
Starting with the fact that wherever investigation by mathe- 
matics and mechanics establishes the existence of compound 
wave-movements, the trained ear is able to distinguish tones 
which correspond with the simple wave-movements contained 
in them, he next propounds the same question for simple wave- 
motions, and tries to discover means of producing simple wave- 
motions in the air. But since all resonant elastic bodies assume 
various vibrational forms in which they can give out tones of 
different pitch, Helmholtz selected a tone-producer, which 
imparts its vibrations as little as possible to the air, while 
another, the resonator, was so arranged that it was set in 
sympathetic vibration with the first, and gave out its vibrations 
easily and forcibly to the air. If the prime tone of the two 
bodies is exactly the same, while all the higher partial tones 
of the one are different from those of the other, the resonator 
will only be excited by the prime tone, and will only give out 
the vibrations of the prime tone to the air. Helmholtz chose 
a tuning-fork, and as resonator took the string of a monochord, 
or an air-chamber formed of cylindrical tubes made of paste- 
board, closed at both ends with a round opening in the centre 
r"*of one end. With the help of this arrangement it was found 
that simple tones, as Helmholtz calls tones produced by simple 
vibrations, on the analogy of the simple colours of the spectrum, 
only give out clearly such deeper combination tones as have 
a vibration number equal to the difference of the vibrational 
numbers of the generating tones, and that when combination 
tones of another order exist along with these, they are too 
weak to be audible with generating tones of moderate strength. 
When, therefore, combination tones of the higher order are 
very perceptible in compound tones, they must be the com- 
bination tones of the higher partials. Helmholtz also finds 
a second class of combination tones, the vibration frequency 
of which is equal to the sum of the generating tones, which 
he calls summational tones, while he designates the others as 


Starting with the assumption made by Ohm in 1843, tnat 
in auditory sensation the ear analyses the motions of the air 
into simple vibrations, in the same way that Fourier's series 
for each periodic function is composed of the sum of periodic 
sine-functions, or that any wave-form may be composed of 
a number of simple waves of different length, of which the 
longest has the same length as the given wave-form, while 
the others are a half, a third, or a fourth, &c. of this length, 
Helmholtz gives the name of compound tone (Klang) to the 
composite tone of a musical instrument, while he confines 
the term tone to simple tones. A compound tone is really 
a chord with a predominating prime tone ; its strength will be 
the sum of the strengths of the individual tones which it con- 
tains, its pitch the pitch of its prime tone. The ear analyses 
all sound-waves according to Fourier's theorem, by resolving 
the wave-form into a sum of simple waves. It perceives the 
proper tone of each simple wave, whether the waves in the 
first instance issued as such from the source of tone, or have 
united together on the way, and by listening attentively it is 
possible to detect the over-tones corresponding to the separate 
simple waves. 

These considerations reinforced the views which Helmholtz 
deduced from optics in regard to our sensations. A certain 
compound tone is the adequate sensuous token of the presence 
of a certain resonating body. In analysing this sound, we 
must give the same artificial support to our attention before 
we can perceive the over-tones, as is required in the case of 
double images and the blind spot, just as we do not normally 
realize that the sensuous apperception of an object corporeally 
extended in space is built up from the two distinct retinal 
images of our two eyes. Helmholtz further determined that 
combination tones appear only with strong generating tones, 
that their intensity grows much faster than that of the prime 
tones, and that the latter may almost entirely disappear when 
the intensity is very great. 

He now proceeded to attack the problem as a whole from 
its mathematical aspects. He found the generally accepted 
view of the simple superposition without mutual disturbance 
of a system of tone-waves excited simultaneously in the air, 
to be contrary to the laws of mechanics, and proved by strict 


mathematical deduction that the different simple oscillatory 
motions of an elastic body are superposed without disturbance, 
as long as the amplitude of the oscillations is so small that 
the motive forces excited by the displacements are sensibly 
proportional to the latter. When, however, the amplitude of the 
vibrations is so great that the squares of the displacements exert 
a perceptible influence on the magnitude of the motive forces, 
new systems of simple oscillatory motions arise, the vibration- 
period of which corresponds with that of the combinational 
tones. The vibratory motions of the air, produced by various 
sources of sound in simultaneous action, correspond with the 
exact sum of the motions produced by the separate sources 
of sound only when the vibrations are infinitesimal, i. e. when 
the alterations of density are so small that they do not come 
into play as compared with the total density, and when there- 
fore the displacements of the oscillating particles are vanish- 
ingly small in comparison with the dimensions of the entire 
mass; if the law does not hold good, combination tones are 
produced. It followed from these considerations that the 
origin of combinational tones is not necessarily dependent on 
the sensations of the auditory nerve. With two simultaneous 
tones of the right strength, the combination tones may corre- 
spond with actual vibrations of the tympanum and auditory 
ossicles, received by the nervous apparatus in the usual way. 
But, as Helmholtz pointed out, conditions similar to those 
affecting the movements of the apparatus of the tympanic 
cavity may also occur outside the ear, so that vibrations 
corresponding to combination tones may be produced quite 
independent of the human ear, and external to it, and he made 
experiments to prove the objective existence of combination 
tones. The nature of combination tones has therefore nothing 
to do with the marvellous property by which the ear analyses 
a confused group of sound-waves into the single tones of which 
it is made up, and distinguishes the voices of separate indi- 
viduals, and the quality of the different musical instruments. In 
regard to the origin of this property of analysing the aerial 
motion produced by the joint effect of a number of resonant 
bodies into the elements corresponding with the particular 
effects, Helmholtz had indeed formed a special hypothesis, 
but felt it necessary to test it upon the various phenomena. 


Hence, while still immersed in the preparation of Physiological 
Optics, Helmholtz was planning his great work on the Theory 
of the Sensations of Tone, busying himself in the first place 
with the physiological questions it involved, the interest of 
which is increased ' by the antiquity they have attained, un- 
solved ', and with their significance for music and phonetics. 

Before setting out on his summer journey, he was gratified 
by a visit from Bonders, and then left for Schwalbach on 
August 15, 'in order,' as he writes to his father, 'to meet 
Prof. Thomson from Glasgow, whom I visited last year in 
Kreuznach, and who has principally concerned himself with 
the Theory of the Conservation of Energy in England. He 
is certainly one of the first mathematical physicists of the day, 
with powers of rapid invention such as I have seen in no 
other man.' 

After spending a day there, and the next morning trying 
some new experiments with the siren, which had occurred 
to Thomson in the night, and which 'if they succeed must 
yield the most striking results ', he joined his travelling com- 
panion for Switzerland, Dr. Otto Weber, Privatdocent of 
surgery in Bonn, 'a talented young man who had previously 
done a good deal of work in geology,' in Frankfurt. From 
Heidelberg, where he 'found Kirchhoff already gone, and 
Bunsen packing', he went to Basle, whence he sent his wife 
an enthusiastic description of the Holbein drawings : ' They are 
marvellously finished ; it is rare to find such a combination of 
force, character, and dramatic vigour, though there is little 

From there he went to Chamounix, and made several long 
excursions on the mountains and glaciers, the beauties and 
dangers of which he describes to his wife in glowing colours. 
Fatigue and longing for work, however, took the 'thirty-five 
year old dotard' back to Bonn by September i. A few days 
later he received specimen copies of the first edition of the 
Handbook of Physiological Optics, in which he gathers the results 
of years of work into a harmonious whole. 

Besides the papers on physiological optics which have already 
been mentioned, and an admirable review of all previous work 
comprised under this heading, the book contained a store 
of new and most important results, which provided a firm 


mathematical basis for the whole structure of physiological 
optics. After a masterly anatomical description of the eye, 
Helmholtz divides the theory of visual sensation into three 
sections, dealing with the path of light in the eye, the sensa- 
tions of the optic nerve, and the interpretation of visual 
sensations or visual perception. In the part first published, 
he is principally concerned with the problem of the refraction 
of the light-rays, or the dioptrics of the eye. He begins 
with a simpler and more comprehensive account of refraction 
in centric systems of refracting and reflecting spherical sur- 
faces, than that given by Gauss, and then applies the theorems 
to the refraction of light-rays in the media of the eye, where 
he makes the interesting point that the distance between the 
principal points in the crystalline lens is less than it would be in 
a lens of the same form with the refrangibility of the nucleus : 
at the same time he is led by measurements carried out upon 
the living eye, to doubt whether the form and focal length of 
a dead lens are the same as in the living eye accommodated 
for far vision. He examines the different reduction methods 
of Listing, and, after defining accommodation, discusses its 
mechanism, and the theory of diffusion-images upon the retina, 
with the aid of all the measurements previously made by others 
and by himself with different optometers. He makes some 
excellent observations on astigmatism and the entoptic phe- 
nomena of the eye, but in this first section touches the question 
of colour only in so far as it relates to the dispersion of colours 
in the eye. His estimation of brightness in the diffusion-area 
produced by dispersion of a single luminous point, as also at 
the edge of an evenly illuminated surface, is interesting, and 
he goes on to explain why the chromatic dispersion of images 
in the eye interferes so little with the acuteness of vision; 
a combination of lenses designed to make the eye achromatic 
had no perceptible effect on the clearness of vision. Lastly, 
he works out the refraction at the vertex of an ellipsoid with 
unequal axes, and investigates pencils of rays falling obliquely 
upon a spherical surface. 

In order to establish the mathematical theory of the lumi- 
nosity of the eye, and of the ophthalmoscope, Helmholtz develops 
some more general theorems than those already published 
in his work on the ophthalmoscope, the following of which 


may be cited : (i) when two rays of light pass in opposite 
directions through any number of simple refracting media, 
and coincide in one of these media in a straight line, the}' must 
coincide in all; (2) if the pupil of the observed eye is to be 
luminous, the, image of the source of light on its retina must 
wholly or partially coincide with the image of the observer's 
pupil ; (3) if in a centric system of refracting spherical surfaces, 
the refractive index of the first refracting medium be n v and 
that of the last n. 2 , and there be in the first, vertical to the 
axis of the system, and close to the axis, a surface-element 
a, and in the last a similar element ft, then when a has the 
illumination nf . H, and ft has the illumination n% . H, as much 
light falls from a upon ft as from ft upon a; this law, applied 
to the problem of luminosity in the eye, tells us that the 
quantity of light which falls from any surface-element of the 
retina of the observed eye into the eye of the observer is 
equal to the illumination of the retinal element by the source 
of light, multiplied by the quantity of light that would fall 
upon that retinal element from the pupil of the observer, if 
its illumination were unity. By these laws he succeeded in 
establishing a general method for determining the illumination 
of any spot of the observed retina, as seen by the observer 
with the ophthalmoscope, and on this again he founds his 
comparison of the various forms of the ophthalmoscope. The 
historical evolution is everywhere traced out in its smallest 
details, with a careful summary of the literature of the 

Meantime his acoustic observations widened in import, and 
a most valuable series of physiological discoveries in optics 
and acoustics followed in quick succession. 

On May 18, 1857, he writes to du Bois : 

'I have gradually accumulated a considerable amount of 
material for the reform of physiological acoustics, and am waiting 
for instruments to carry it out. I will mention one fact that has 
interesting bearings on nerve physiology, i. e. that the fibres 
of N. acitsticus, " which perceive the higher tones, must be 
capable of distinguishing as many as 150 alternations of rest 
and excitation (150 vibrations) per second from a continuous 
excitation, whereas in the optic nerve and in muscle, a rhythm 
of 10-15 P er second acts as a continuous excitation. This 


agrees with the rapid alternations of electrical distribution in 
nerve, and it looks as if the lag in the above effects must occur 
in the fibres of the muscle and the portions of the retina that 
are sensitive to light.' 

By the end of the year his Theory of Vowel Tones, with 
which he had been occupied for months, was so far advanced 
that he was able on November 4, 1857, to tell Bonders that 
they were distinguished by the higher partial tones which 
accompany the prime tone. By singing into a piano, it is easy 
in pronouncing a, o, e to set the strings corresponding to the 
upper partial tones into vibration ; it is only necessary to sing 
the tone exactly, and keep it on; the experiment succeeds 
best with practised singers, 'with my wife better than myself.' 
If the fundamental tone is called the first tone, and the higher 
partials, with two, three, four, &c. times as many vibrations, are 
termed the second, third, fourth, &c., then when a is sung the 
third and fifth tones are heard plainly with the first, while 
the second, fourth, and seventh are weaker ; with o the third 
is rather weaker than with a, while the second and fifth are 
very feeble ; with u the prime is almost the only tone audible, 
the third is weak ; with e the second is very marked, the upper 
tones are scarcely audible; and with i the clear character of 
the vowel seems to depend on second and third tones pre- 
ponderating over the weak ground-tone, while the fifth is heard 

There were still, however, great difficulties to overcome 
before the Vowel Theory could be completely established, and 
Helmholtz was occupied with these till the beginning of 1859. 
' In the next place I must attack the problems relating to the 
origin of timbre (Klangfarbe), since these will solve the funda- 
mental problem of physiological acoustics discussed by Ohm 
and Seebeck : what kind of vibration corresponds with a single 
audible tone ? I believe Ohm to be right in his view that the 
ear analyses and hears the motions of the air in exact corre- 
spondence with Fourier's theorem/ he writes to Bonders in 
the letter above quoted. 

It was many years since Helmholtz had discussed his philo- 
sophical position with his father. On Becember 17, he 
writes : 

'All goes well here, we are all flourishing. My official 


position has improved since Prof. Budge went to Greifswald. 
I am now the only official representative of Physiology, and 
the Ministry can no longer come down on me for comparative 
and microscopic anatomy, for which I might have been held 
responsible. For if I lecture on human anatomy in the winter, 
and take physiology as my principal subject in the summer, 
my time is full, and no one can reasonably expect more 
of me/ 

He then supplements these few lines, on December 31, by 
a long letter, in reply to one received from his father with 
the news that he had retired from his post at the Gymnasium : 

' I am delighted at what you write about your present life ; 
I think you will be more and more interested in philosophy, 
the more you give yourself up to it. It seems to me a favour- 
able moment for voices of the old school of Kant and the 
elder Fichte to obtain a hearing once more. The philosophical 
vapouring and consequent hysteria of the " nature-systems " of 
Hegel and Schelling seem to have exploded, and people are 
beginning to interest themselves in philosophy again. I have 
only read a little of the Anthropologie of the younger Fichte ; 
I found much that was interesting, but as a whole the book 
gave me the impression of a series of plausible but unfounded 
hypotheses, and I laid it aside, as I saw that one would have 
to discover his main argument from his other writings. The 
younger Fichte, indeed, appears to me not to be free from the 
reproach which has brought philosophy into disrepute, thanks 
to Hegel and Schelling. He introduces a number of matters 
into his discussion, which he thinks he is obliged to talk about, 
though they do not really belong to philosophy at all, but either 
come into the scope of experimental science, or are matters of 
purely religious faith. Philosophy finds its great significance 
among the sciences as the theory of the source and functions 
of knowledge, in the sense in which Kant, and, so far as 
I have understood him, the elder Fichte, took it. Hegel, how- 
ever, wanted it to replace all the other sciences, and to find 
out by its means what is perhaps denied to man, by which he 
diverted philosophy from its proper scope, and gave it tasks 
it can never accomplish. The majority of educated men at 
first believed in him, and then rejected philosophy altogether, 
seeing that nothing came of it. The popularity of Schopen- 


hauer at the present time seems really to be due to the fact 
that he goes back to the sound old Kantian standpoint/ 

And on March 4 (after a long letter from his father) he 
resumes : 

'We who approach natural science from the mathematical 
point of view are disciplined to a painful exactitude in the 
testing of facts and consequences, and compel each other 
to proceed by very short and safe steps in the hypotheses 
with which we endeavour to sound what is still an unexplored 
ocean, so that we are perhaps too much afraid of a bolder 
application of the facts of science, which may very well be 
justified upon other occasions. 

' Your letter implies that you suspect me of believing in the 
trivial tirades of Vogt and Moleschott. Not in the very least. 
And I must protest vigorously against your taking these two 
men as representatives of natural science. Neither has so far 
shown by any special scientific achievement that he possesses 
either the respect for facts, or the discretion in accepting 
conclusions, that is acquired in the discipline of science. 
A sober investigator knows right well that the fact of his 
having gained a little insight into the complexities of natural 
processes in no way justifies him in concluding more than 
other men as to the nature of the soul. And for this reason 
I do not think you are right in supposing the majority of sober 
men of science to be inimical to philosophy. Indifferent indeed 
they are, but that I put down solely to the exaggerations of 
Hegel and Schelling, who have been presented to them as 
typical philosophers. Lotze, for instance, has a fair following 
among the naturalists. Personally I get no satisfaction out of 
him. He is not clear or strict enough for me. I feel the 
crying want of a special treatment of certain questions, which 
have not, so far as I know, been attacked by any modern 
philosopher, and which lie wholly within the field of a priori 
concepts which Kant investigated, e. g. the derivation of the 
principles of geometry and mechanics, the reason why we are 
logically bound to reduce reality to two abstractions matter and 
energy, &c., or again, the laws of the unconscious arguments 
from analogy, by which we pass from sensations to sense- 
perceptions. I see plainly that these can only be solved by 
philosophical investigation, and are resolvable by it, so that 


I feel the need of more profound philosophical knowledge. 
Schopenhauer I deliver over to you; I disliked what I have 
read of him/ 

Helmholtz had hardly settled down at Bonn, when proposals 
were made to him to move to another sphere of action. In 
April, 1857, Bunsen tells him ' the Baden Ministry are willing 
to make considerable sacrifices, in order to attract a good 
physiologist to Heidelberg'. The selected candidates were all 
of first rank; Brttcke, Ludwig, du Bois-Reymond, Helmholtz. 
The Faculty desired one of the two last, and as a member of the 
Senate Bunsen invited Helmholtz to state his present income, 
and the conditions on which he would consent to be called 
to Heidelberg. In his reply, dated May 16, 1857, Helmholtz 
points out that for the time being he is under certain obligations 
of personal gratitude to the Prussian Ministry for sending him 
to Bonn on account of his wife's health, and that the situation 
at Bonn is at present so little developed that its temporary 
disadvantages would not justify him in disregarding these 
obligations. Helmholtz also believed that du Bois-Reymond 
was almost certain to accept the Chair at Heidelberg, which 
was an additional reason to him for refusing to consider it. 

On June 20, Kirchhoff writes to his old friend Helmholtz, 
to the effect that he was the only candidate selected by the 
Faculty, and begging him, if he really declined it, to recommend 
the appointment of du Bois to the Ministry. On July 14, 
Helmholtz informs du Bois that after putting the case to the 
Prussian Ministry, he had been promised an increase of 60 
salary, with fresh promises of reconstruction of the Anatomy 
Buildings, so that he had decided definitely to remain at Bonn. 

In July, Helmholtz had the satisfaction of receiving a visit 
from his father and his sister Julie, after which he went off 
for several weeks to Switzerland, his farthest point being 
the Gornergrat. His letters to his wife are as usual filled 
with beautiful and enthusiastic descriptions, but by the end 
of August he is back at Bonn, to make the necessary prepara- 
tions for the Congress of Natural Science to be held there 
at the end of September, which he describes to his father 
as follows on October 3 : 

'We were all rather upset till the day before yesterday; 
People have been here without intermission ever since you 


left, all expecting to see more or less of us, and we had no 
peace. The Congress itself was very successful (about 1,000 
members), and there were many interesting and distinguished 
visitors although the most important did not turn up more 
in fact than one could profit by, so that I did not see a good 
many I wanted to forgather with. The meeting was very 
interesting, but a fearful rush for me, and I avoided most of 
the social things. Olga was quite upset, because she had to 
receive a number of visits in my absence. Only two of our 
rooms were occupied. Prof. Dove of Berlin and v. Wittich 
from Konigsberg came to us. They were both out all day 
like myself. Wittich and I usually came in for meals. I could 
not have stood public dinners on the top of the mental excite- 
ment. I did go to the first, but it upset me for the whole 
of the next day. We did not entertain much either, only having 
a few men to tea one evening. 

' I gave an address on the Telestereoscope at a general meeting 
in the Riding School, and put up two different instruments 
first in the Hall, then in the Anatomy School, and eventually 
in my own house, and had enough to do in demonstrating 
them to every one. At the Anatomical Section, which was 
well arranged, and well attended, I put up and demonstrated 
experiments with the myograph on Time Measurements in 
Nerve, because these experiments have been too little seen 
and repeated. I also gave a short lecture on the movement 
of the auditory ossicles, illustrated by a few preparations. 

' In the Physical Section I spoke on Combination Tones. 
Besides this I joined in the debate on Expenditure, in which 
I was in the minority, as you will have seen by the papers, 
but my minority had the most authority all the same. And 
I had another unexpected pleasure. I received a letter im- 
mediately after from Munich, from the King of Bavaria, who 
is willing to apply a considerable annual sum to the very 
objects I proposed to the Congress, i. e. to endow such 
scientific undertakings as exceed the powers of private persons, 
and asks my advice as to expenditure. Of the social things 
I only took the expedition to Coblentz, and went to the concert 
with Olga. It was given by the city of Bonn, and crowned 
the functions. Beethoven's works alone were given/ 

Helmholtz's experiment in Acoustics had been reported to 


the King of Bavaria, and he was invited to draw up a brief 
report on the result of his discoveries in the Theory of Tone 
for the King's edification. On April 15, 1858, he writes to 
du Bois-Reymond : 

' I have now put together a complicated apparatus at the 
King of Bavaria's expense, by which one is able to control 
the vibrations of a tuning-fork at will by an electro-magnet, with 
complete command of intensity and difference of phase. This 
is in order to regulate the production of timbre (Klangfarbe). 1 
The cost of this apparatus was 400 gulden, paid by the King. 

After he had commenced his lectures and scientific work for the 
winter session, Bunsen again approached him on December 15, 
1857, stating that the Baden Ministry had not given up hopes 
of persuading Helmholtz to come to Heidelberg, since the 
postponement of the building of a new Institute at Bonn 
seemed to cancel some of his reasons for refusing, and that 
they had therefore taken no steps as yet in other directions. 

Helmholtz consulted his father, who urged him strongly to 
go to Heidelberg: 'Your scientific life and satisfaction in your 
official career will have widely different prospects in Heidel- 
berg as compared with Bonn; the mere fact of your being 
able to confine yourself to physiology will be valuable to your 
own scientific projects. Your obligations to science are greater 
than to the State.' 

Helmholtz finally accepted the invitation from Heidelberg; 
and on February 27, 1858, Kirchhoff writes to him: 'All 
Heidelberg is rejoicing at your decision, and I hope you will 
find a congenial atmosphere here.' On March 5, Helmholtz 
informs du Bois that he had forwarded his resignation to 
v. Raumer, as the Senate and Ministry had taken no steps 
since his first refusal of the call to Heidelberg to carry out their 
promises. On the same day he writes to his father : 

'At last I have accepted the call to Heidelberg. I have 
already sent in my resignation to the Minister, v. Raumer. 
The correspondence over the rebuilding of the Anatomy Depart- 
ment was left four months before the Prussian Ministry 
attended to it, at the end of which time an undecided, pro- 
crastinating answer was sent, and the feeling of the Academic 
Senate, who were to cover a portion of the cost by selling 
some of the land belonging to the University, is quite uncertain, 

M 2 


so that while there is some probability of the construction of 
the Anatomical Buildings, nothing is decided. In Heidelberg, 
on the other hand, I have a position that is in every way 
convenient for my scientific enterprises. I went to Heidelberg 
myself, at Carnival time, when they take a few days' holiday 
here upon the Rhine, to inspect the situation, . and also to 
Carlsruhe. The present accommodation in Heidelberg is im- 
possible, and I was obliged to ask for a new Physiological 
Institute, to which the Minister agreed without much difficulty/ 

His father replies with great satisfaction on March 9 : 

' I am rejoiced at your good fortune, and that you have 
made a reputation which secures you a most distinguished 
position antf an academic position while you are still so 
young. I told you before that the post at Heidelberg was, 
in my opinion, in every way more profitable to you than the 
Chair at Bonn, where I was little pleased with the temper of 
the learned gentlemen, and thought it very unlike Konigsberg. 
The proximity of the Court and the fashionable world is not 
good for men of letters, who are too prone to idealism from 
their preoccupation with science, and too little versed in worldly 
matters. I was grieved to hear of your beloved Olga's illness. 
Doubtless the first days of summer will relieve her of her 
suffering, which one may hope will not return in the more 
southern and much milder climate of Heidelberg, with its 
sunny valleys, near to so many famous health resorts. You 
must work harder than ever in the future, to live up to your 
great renown, and overcome all envy at your appointment/ 

But as Helmholtz had not obtained his discharge from the 
Prussian Government by April, he was obliged to postpone 
his departure for Heidelberg till the autumn. On April 28, 
on his return from a visit to Bonders at Utrecht, he received 
a letter from du Bois-Reymond announcing the death of 
Johannes Muller, with the information on good authority that 
the Prince of Prussia had made very sharp inquiries into the 
reasons for Helmholtz's departure from Bonn, and had declared 
his intention of going himself to Baden, to release Helmholtz 
in person from his contract, and set him at liberty to enter 
into new relations with the Prussian Government. Helmholtz 
accordingly writes to Bonders: 

/The Prince of Prussia, who is at present carrying on the 


Government, had already expressed his displeasure at the way 
the Prussian Government have subordinated scientific interests 
to those of Church and State on several occasions, when my 
resignation was laid before him. He took this opportunity of 
once more fulminating against the Minister, and proposed to 
settle the matter in person with the Grand Duke of Baden. 
The Minister did not at first agree to this, as it was a reproach 
to himself. Finally, when J. Miiller was dead, and there was' a 
want of good candidates, came a request from the Ministry that 
I would remain here under the same conditions as were offered 
me at Heidelberg/ 

He gives the following account of these negotiations to his 
father : 

' I now know from reliable sources that the negotiations 
of which Olga has told you were really undertaken in con- 
sequence of the Prince of Prussia's action, although this has 
since been expressly denied. I had already promised -the 
Baden Government to accept the post in Heidelberg, and had 
no very pressing interest to make me give Bonn the pre- 
ference. Although, generally speaking, it is my rule to stay 
where I am well off in many respects, and not to exchange 
what is known and endurable for the unknown, because it 
has a more seductive aspect, and though Olga's health makes 
a move undesirable, and the pecuniary emoluments may, at 
the outset, be larger here than in Heidelberg still it must 
be admitted that the people there seem most anxious to do 
whatever is necessary, in order to promote the scientific success 
of the position offered me. In Prussia they are promising 
me what I asked, only for personal reasons, and would 
probably carry it out literally, refusing later on what would 
by then be absolutely necessary, in the way they always 
stint everything connected with the University. So that I 
really had no reason for pulling the chestnuts out of the fire, 
and compromising myself with the Baden Government for the 
sake of Prussia. 

' At first they simply tried to make me break my word, and 
quoted a string of stories about other Professors who had 
broken theirs. I declined firmly. Next they wanted me to 
appeal to the Prussian Government to back up a petition 
to Baden to absolve me from my promise; a statement to 


this effect was laid before me which I was to sign there and 
then, as it had to be telegraphed to Berlin in a desperate hurry 
(which turned out to be mere invention). I rejected this and 
drew up another document, which I signed, in which everything 
relating to my own wishes was left out, and it was thrown 
entirely on the Government to work Baden so that I should 
be released from my promise. Then, I said, I was prepared 
to remain in Bonn. So that I made my position circum- 
stantially plain. I did not for a moment suppose that any 
one would negotiate, and openly expressed my conviction that 
I could not see why the Baden Ministry should consent ; but 
our Prussian officials have far too high an opinion of the 
importance of their Ministers not to think that one of the lesser 
German States would not at once acquiesce in their wishes.' 

The negotiations dragged on for some time, as the Prussian 
Government gave it to be understood that Helmholtz wished 
to be released from the appointment at Heidelberg, and he 
was obliged to circulate the drafts of his letters to prove that 
he was not playing fast and loose with Baden. Finally, he 
received his conge, and, after three years' connexion, was free 
to depart from Bonn. 

' These three years/ says his sister-in-law, ' were a con- 
tinuation of the life at Konigsberg, save that external relations 
had broadened, and that the indescribable charm of the land- 
scape made a most poetic background for their daily life. The 
two children were growing in mind and body, and Helmholtz 
was a devoted father. They had plenty of friends and social 
intercourse. Their circle included the families of Heine, Busch, 
Naumann, Otto Jahn, the biographer of Mozart, the elder 
Arndt, who was particularly attached to Olga, the surgeon 
Weber, several English families, and for a short and much 
appreciated time, Prof. Bonders from Utrecht, who was a great 
friend of both Hermann and Olga. The old terrace on the 
Rhine, with its view of the Drachenfels, where they lived in 
the ancient Vinea Domini, has seen many a gathering of clever 
and congenial people, and when the garden was illuminated in 
Donders's honour with coloured lamps, and the children ran 
about in their merry play, it gladdened one's heart to see this 
sunny family happiness.' 

Frau Geheimrath Busch, who was a daughter of Mitscherlich, 


writes that 'after forty years the impression made on me by 
that noble head, with its deep clear gaze, its classic and 
dignified expression, is indelible. Helmholtz was generally 
cheerful and sympathetic, even playful, and delighted of an 
evening in reading plays aloud ; he- preferred a character-part 
in Shakespeare, or some other classic. It was an intimate 
little circle in which the Helmholtz couple took the lead. 
Often enough Helmholtz would sit still, plunged in his own 
thoughts, but I never saw him out of temper, or anything but 
cordial '. ^ 

The last year of Helmholtz's stay at Bonn was marked by 
a series of important publications. The complexities of 
acoustics had induced him two years previously to occupy 
himself with the application of Green's Theorems to hydro- 
dynamic and aerodynamic problems. In 1857, m a work of 
genius that proved him to be a mathematician of first rank, 
4 On the Integrals of the Hydrodynamic Equations which 
express Vortex-motion ' (published in Crelle's Journal f. reine 
u. angew. Mathematik), he gave the solution of some ex- 
tremely difficult hydrodynamical problems. He rejected the 
earlier hypotheses, and followed up the analogies between the 
motion of fluids and the electromagnetic action of electrical 
currents, which were of so much importance for his subsequent 
work on the Theory of Electricity and Magnetism. Up to that 
time the integrals of hydrodynamic equations had been deter- 
mined almost exclusively on the assumption that the rectangular 
components of the velocity of each element of the fluid are 
the differential co-efficients, with reference to the co-ordinates, 
of a certain function, which Helmholtz termed the velocity- 
potential an assumption which was lawful so long as the 
motion of the fluid resulted from the action of forces which 
had a potential of their own. Helmholtz abolished this limi- 
tation, and took into account the friction between the elements 
of the fluid, and against fixed bodies, the effect of which on 
fluids had not till then been defined mathematically, and 
endeavoured to determine the forms of the motion which 
friction produces in fluids. Starting with the equations of 
motion for the interior particles of a liquid, he pictures the 
changes undergone by an indefinitely small volume of the 
fluid in an indefinite fraction of time as composed of three 


separate motions : a translation of the element of the fluid 
through space, an expansion or contraction of the elements 
in three principal directions of dilatation (in which any rect- 
angular parallelepiped, whose edges are parallel with the 
principal directions of dilatation, remains rectangular), and, 
lastly, a revolution round some instantaneous axis of rotation 
in any direction. In the first place he proves by rigid mathe- 
matical deductions that the existence of a velocity-potential 
is incompatible with the existence of a rotation of the fluid 
elements, but when there is no velocity-potential some fluid 
elements at least can rotate. On the assumption that all forces 
acting on these fluids have a ' force-potential ', it follows neces- 
sarily that such particles of water as have no initial rotary 
motion cannot be thrown into rotation at a later period. If 
lines drawn through the fluid so that their direction coincides 
everywhere with the direction of the instantaneous axis of 
rotation of the element of fluid lying there, are termed vortex- 
lines, it follows again from the equations of hydrodynamics 
that each vortex-line remains permanently composed of the 
same elements of fluid, and swims along with them in the 
fluid. Helmholtz terms a filament of the fluid with an 
indefinitely small cross-section, a vortex-filament, when it is 
produced by drawing vortex-lines through every point in the 
circumference of any indefinitely small surface. Since, further, 
the expressions for rotary velocity show that the magnitude 
of the latter varies in any given element, in the same propor- 
tion as the distance between this element and its neighbours 
on the axis of rotation, it follows that the product of the angular 
velocity and the cross-section in any portion of a vortex-filament 
containing the same particles of fluid remains constant during 
the motion of the filament, and further that this product does not 
vary throughout the whole length of any given vortex-filament. 
The vortex-filaments must accordingly return upon themselves 
within the fluid, or end at its boundaries. But from this it 
follows directly, that if the motion of the vortex-filaments in the 
fluid can be determined, the velocity of rotation can be ascer- 
tained, and that the velocities of the elements of the fluid are 
determined for any given moment of time, when the angular 
velocities are given, with the exception of an arbitrary function, 
which covers the limiting conditions. This determination 


of velocities connotes the important law that each rotating 
element of fluid implies in every other element of the same 
fluid mass a velocity whose direction is perpendicular to the 
plane through the second and the rotation axis of the first 
element. The magnitude of this velocity is directly propor- 
tional to the volume of the first particle, its angular velocity, 
and the sine of the angle between the line that unites the two 
elements and that axis of rotation, and inversely proportional 
to the square of the distance between the two elements. But 
since the same law holds for the force exerted by an electrical 
current in the first element, parallel with its axis of rotation, 
upon a magnetic particle in the second element, Helmholtz, 
by means of the definition of w-dimensional space (i.e. one 
which can be traversed by n i, but not more, surfaces, 
without being separated into two detached portions), formu- 
lates a law that has acquired great importance in electrical 
theory. When, that is, a velocity-potential exists in a simply 
connected space full of moving fluid, the velocities of the 
fluid elements are equal to, and in the same direction as, 
the forces exerted on a magnetic particle in the interior of 
the space by a certain distribution of magnetic masses at its 
surface. But if vortex-filaments exist in such a space, the 
velocities of the fluid elements are represented by the forces 
exerted on a magnetic particle by closed electrical currents, 
which flow partly through the vortex-filaments in the interior 
of the fluid mass, partly on its surface, their intensity being 
proportional to the product of the cross-section of the vortex- 
filament and the angular velocity. Since the first motion 
implies a velocity-potential with only one value, while the 
second in the non-rotating particles of water implies a velocity- 
potential with more values than one, it is sufficient in hydro- 
dynamic integrals of the first class to know the motion of the 
surface ; in those of the second class, we must further deter- 
mine the motion of the vortex-filaments in the interior of 
the fluid, with reference to their mutual action, and having 
regard to the limiting conditions. Helmholtz succeeded in 
doing this for certain simple cases, in which the rotation of 
the elements occurs only in given lines or surfaces, and the 
form of these lines or areas remains unaltered during motion, 
e.g. in straight, parallel, or circular vortex-filaments which 


theorems and conclusions, taken as pure mathematics, are 
fundamental laws of the modern Theory of Functions. His 
conclusions for ring-shaped vortex -filaments are very inte- 
resting. When two such rings of small section have the same 
axis and the same direction of rotation, they travel in the same 
direction; the first ring widens and travels more slowly, the 
second shrinks and travels faster, till finally, if their velocities 
are not too different, it overtakes the first and penetrates it, 
the same process is then repeated for the next ring, and so 
on. An analogous statement holds good when the directions 
of rotation are opposite. 

When Tait, some ten years later, proposed to translate 
the work of Helmholtz on Vortex- Motion, and wrote to him 
on the subject, Helmholtz replied : ' If you find quaternions 
useful in this connexion, it would be highly desirable to draw 
up a brief introductory explanation of them, so far as is 
necessary in order to make their application to vortex-motion 
intelligible. Up to the present time I have found no mathe- 
matician, in Germany at any rate, who was able to state what 
quaternions are, and personally I must confess that I have 
always been too lazy to form a connected idea of them from 
Hamilton's innumerable little notes on the subject." 

In 1868 Bertrand published some criticisms in regard to the 
universality of Helmholtz's methods. Helmholtz had assumed 
(as above) that the motion of an indefinitely small volume of 
water is due to the propagation of an element of fluid through 
a space, the expansion or contraction of the element in three 
principal directions of dilatation (so that a rectangular parallele- 
piped constructed of water, whose sides are parallel with the 
principal directions of dilatation, remains rectangular, while 
its sides alter in length, but remain parallel with their previous 
direction), and to revolution round a definite instantaneous 
axis of rotation. Bertrand contended that in a great number 
of cases oblique parallelepipeds might also be constructed 
with an arbitrary direction of the edges, which could be 
transformed into other parallelepipeds, whose edges should 
remain parallel with those of the former. Helmholtz replied 
in three Notes published in the Comptes Rendus for 1868, 
'Sur le mouvement le plus general d'un fluide/ 'Sur le 
mouvement des fluides,' and 'Reponse a la note de M. J. 


Bertrand du 19 octobre,' in which he showed that the motion 
defined by Bertrand can be due to the combination of a rotation 
and three rectangular dilatations, and that 'he did not by 
dilatations mean translations'. 

Lord Kelvin has associated his Theory of the Constitution 
of Matter with Helmholtz's law that a vortex in a frictionless 
fluid persists as an invariable quantity. Kelvin sees a funda- 
mental analogy between the indestructibility of the vortex and 
the indestructibility of matter. He conceives an atom as a 
whirl or vortex in the ether, and accounts for the chemical 
disparity of the atoms on the supposition that we have in them 
different combinations of vortex rings. 

After Helmholtz had finished this paper, which was intelligible 
in the first instance only to mathematical physicists, and which, 
with that on Aerial Vibrations in Tubes, published the following 
year, was always regarded by Kirchhoff as the author's most 
important contribution to the subject of mathematical physics, 
he busied himself in the remaining months of his residence 
at Bonn with optical and acoustic investigations. 

On July 3, 1858, he read a paper to the Nieder.-Rhein. 
Gesellschaft, ' On Subjective After-images of the Eye/ which 
was subsequently extended in Physiological Optics. He had 
already, in his previous work on Colour- Mixture (under- 
taken in support of Young's theory of the red-, green-, and 
violet-perceiving elements of the fibres of the optic nerve) 
come to the conclusion that the spectral colours are not the 
most saturated that can occur in visual sensation. In order 
to settle this last fact he first of all examined Fechner's theory 
of the subjective after-images of the eye. After looking at 
a bright object, and then exposing the eye to complete 
darkness, a positive after-image first appears, i.e. the bright 
parts of the object appear bright, and the dark are dark ; 
with uniformly illuminated surfaces, on the contrary, the after- 
image is mostly negative, i. e. the bright spots of the image 
appear dark, and the dark, bright. Fechner's explanation is 
that positive after-images result from persistent excitation of 
the points of the retina that had been excited by light, negative 
after-images from fatigue of the same points rendering them 
less sensitive to new impacts of light ; the strength of illumina- 
tion of any surface required in order to turn the positive 


after-image that appears on a dark ground into a negative 
image, diminishes with the time. 

After confirming Fechner's theory in detail, Helmholtz 
produced after-images of pure prismatic colours in his eye, 
observed them upon a field covered with another prismatic 
colour, and found that the phenomena in no way differed 
from those which arise on regarding the colours of natural 
bodies and pigments. There was, however, one interesting 
case, in which he looked at a round spot brightly illuminated 
by a spectral colour, then observed its after-image upon a field 
covered with the complementary colour, and completely purified 
from diffuse white light. The complementary colour then 
appeared purer and more saturated within the after-image, 
than around it. Helmholtz thence concluded that although 
the prismatic colours are the purest and most saturated, that 
is the most free from a mixture of white, presented to us in 
nature, yet, by the above means, the sensation of even more 
highly saturated colours may be excited, in comparison with 
which the purest prismatic colours will appear whitish. 

Towards the end of August Helmholtz moved with his 
family to Heidelberg, where he received a great ovation. The 
newly-formed Ophthalmological Society presented him with 
a cup, inscribed 'To the Creator of Modern Science, the 
Benefactor of Mankind, in grateful remembrance of the dis- 
covery of the Ophthalmoscope/ 

In September he attended the meeting of the British 
Association at Aberdeen, and the Naturforscher-Versammlung 
at Carlsruhe, the capital of the State to which he was to 
consecrate his mighty energies for the next thirteen years. 
His address 'On After-images ' gave a summary of the 
experiments and conclusions described above, and another 
discourse 'On the Physical Causes of Harmony and Dis- 
sonance* was the epitome of a lecture on 'The Physiological 
Causes of Harmony in Music', delivered to a large and 
enthusiastic audience the year before at Bonn Beethoven's 

'Of all the subjects at which I have worked,' he says forty 
years later, ' I have chiefly felt myself a dilettante in Music. 
Art and Science are essentially distinct in their external 
aspects and technique ; but I am none the less convinced 


of the profound internal relations between them. Art, too, 
strives to acquaint us with reality, with psychological truths, 
though it expresses them in the wholly different form of 
sensual manifestation, and not in that of concepts. Eventually, 
however, the complete phenomenon connotes the conceptual 
idea, and the two are ultimately united in the whole/ 

Such was the standpoint from which Helmholtz connected 
physical and physiological optics with aesthetics, in a form 
that was epoch-making for future generations. 

Starting from the well-known observation that we can feel 
the vibrations of the air in deep tones through our skin, he 
shows in his lecture how the aerial vibrations first become 
sound, when they impinge on the hearing ear, and then 
develops his views as already enunciated upon the correspon- 
dence of quality of tone (timbre Klangfarbe) and wave-form. 
He discusses his theory of prime tones and over-tones, and 
suggests that the over-tones give rise to the indefinable 
peculiarity of tone that is known as timbre. And thus, 
as the existence of over-tones depends on wave-form, he 
identifies timbre with wave-form. Since the cochlea of the 
ear is separated by membranes into three chambers, the middle 
one containing innumerable microscopic lamellae, which lie in 
regular apposition like the keys of a piano, and are connected 
at one end with the fibres of the auditory nerve, and at the 
other with the extended membrane, and since elastic appen- 
dages to the end of the nerves had been discovered in the 
form of stiff hairs, Helmholtz regards it as probable that each 
of these appendages is tuned like the strings of the piano to 
a single tone, and that each tone which reaches the ear not 
only sets the lamella in the organ of Corti that corresponds 
with its prime tone in sympathetic vibration, with excitation of 
its associated nerve-fibres, but affects the lamellae corresponding 
with the upper partials also, so that the over-tones are perceived 
as well as the prime tone. Strictly speaking, therefore, in 
relation to sensation, the tones of musical instruments may 
all be looked on as chords with a predominating fundamental 
tone. It is true that a certain measure of attention is needed 
in order to detect the over-tones, but Helmholtz succeeded 
in hearing the partials of the human voice, and in making 
other people recognize them. 


The discovery made by Pythagoras, that vibrations which 
bear to each other simple ratios of number, e.g. the octave, 
fifth, twelfth, major third, produce a pleasing impression, 
while tones with more complex ratios of vibration-frequency 
are dissonant, had not been adequately explained by the 
assumption that the contemplation of simple ratios of vibration 
affords a pleasurable sensation to the mind. 

1 Mathematics and music, the sharpest antithesis of intellectual 
activity that can be found, are yet interrelated, mutually helpful, 
as if to show the secret consistency that is implicated in all 
the activities of our mind, and which leads us to surmise 
unconscious expressions of a mysterious law of reason in the 
revelations of artistic genius.' 

It had been proved in the earlier acoustic work of Helmholtz 
and others, that when two tones have only an approximately 
equal vibration-period, and their crests coincide at the outset, so 
as to reinforce each other, the undulations of the one will 
gradually outrun those of the other, and produce an alternating 
ebb and flow in the tone, which are known as beats, and 
which, if they become more rapid, are converted into a con- 
tinuous tone-sensation. When the ratio of the prime tones 
is 2 to 3, the two over-tones of six vibrations (whose existence 
was previously ascertained) are exactly equal, and do not 
disturb the harmony of the fundamentals, but when the ratio 
is only approximately that of 2 to 3, the two partials are not 
exactly equal, but produce beats with one another, and the 
tone becomes harsh. Consonance and dissonance are there- 
fore distinguished by the even flow of tones in the former, as 
undisturbed as if each tone were sounding alone, while in 
dissonance there is an incompatibility, and the tones are broken 
up by their mutual action into separate impacts, which disturb the 
listener and make him wish for harmony. In conclusion, Helm- 
holtz drew a contrast between eye and ear, pointing out that 
the eye cannot analyse a compound system of light waves, i. e. 
composite colours, and is indifferent, in a mixed colour, whether 
the component colours are or are not in simple ratios of 
vibration-frequency. The eye has no harmony in the same 
sense as the ear ; it has no music. ' The phenomena of purely 
sensual harmony are indeed only the lowest grade of musical 
beauty. Consonance and dissonance are but the means, albeit 


an essential and powerful means, to the higher and more 
spiritual beauties of music/ 

At the end of September, 1858, Helmholtz took up his 
abode in beautiful Heidelberg, and there with Bunsen and 
Kirchhoff inaugurated an era of brilliancy, ' such as has seldom 
existed for any University, and will not readily be seen 





THE move to Heidelberg involved a temporary interruption 
of Helmholtz's important experimental researches. While 
waiting eagerly in the hope that the new Institute might 
be ready for his apparatus by the beginning of the session, 
he occupied himself in endeavouring to finish Part II of 
Physiological Optics by the middle of October. He writes to 
Wittich that he is ' sticking over the after-images, and cannot 
get to the end of them ' ; while after commencing his lectures 
he was ' experimenting in acoustics on Sundays, and in spare 
moments '. 

He soon settled down at Heidelberg with his family, and 
writes on December 1 1 to his father : ' So far all goes well 
with my official concerns in Heidelberg. I have as large 
an audience, in spite of the reduced numbers of the medical 
students, as I had in Bonn for physiology. Indeed, the number of 
students is too large for the place, and we are rather crowded ; 
but the plans for a new building are to be made out at once, 
and then we can arrange ourselves better. 

1 In November I was elected a corresponding member of 
the Academy of Sciences at Munich, and have to-day received 
my first Order, from Holland, of the Dutch Lion. Prof. Donders 
writes from Utrecht that a new hospital for diseases of the 
eye has been founded there under his direction, and opened 
with much ceremony, on which occasion they thought it 
becoming to celebrate the discovery of the ophthalmoscope 
in this way. You see Heidelberg is bringing me luck as 
regards outside recognition.' 

The father's answer expresses a lively satisfaction at the 
happiness of his children. It was the last letter written by 
the old man (now an invalid of sixty-seven) to his son ; and 
he ends with a vigorous criticism of a new work on the nature 


of the soul by his friend Fichte, offering at the same time 
to draw up his own solution of the problem : 

' For although your task be the rigid investigation of the 
physical, its coherence, and the significance of particulars in 
and for the body, it seems to me that this necessarily involves 
some conception of what the body is and means for the soul, 
and of the life that develops in it. Both Anthropology and 
Psychology may suggest much that would be of value to your 
material researches/ 

At the close of the winter session Helmholtz went to the 
Festival of the Bavarian Academy, at Munich, in March, 1859, 
and on the 3oth describes it as follows to his wife : 

' I have been getting on famously. Early on Sunday Eisenlohr 
appeared with Jolly, the physicist here, to fetch me. After 
writing our names down at the Academy, Jolly took us to 
Kaulbach's studio, which was filled with an enormous picture 
of the Battle of Salamis, a powerful and impressive work. 
Kaulbach himself, whom I met again at the first banquet, is 
a most charming and refined artist, with a keen interest in 
everything that has even the remotest bearing on art. He 
is justly beloved by every one here. . . . Afterwards we 
wandered about a little in the streets, and paid visits to a few 
of the Academicians : both midday and evening I was bidden 
to Jolly's. His wife comes from Heidelberg, and is a sister- 
in-law of Weber. . . . Monday was the great function. At 9 a.m., 
Service with a good solid sermon in the Protestant church ; 
at n, the meeting, at which King Ludwig appeared, and 
presentations were made. At the meeting, an Old-Bavarian 
Catholic, an Orientalist named M tiller, made a speech on 
the history of the Academy, in which he let out with such 
bitterness against the Jesuits that I could hardly believe my 
ears. As we dined later, I breakfasted with many others on 
a glass of beer, which as a matter of fact far surpasses all 
foreign imitations of Bavarian beer. At three there was a great 
banquet, here at the Bayrischer Hof. I sat between SchOnbein 
and Bischoff, opposite Liebig and Kaulbach; it was very 
amusing. In the evening one of Terence's plays was given 
in the small theatre. 

' Yesterday, early, I went with Eisenlohr to the optical 
works of Steinheil outside the city, and saw much that was 


excellent. Then a rather dull sitting with speeches, more 
Bavarian beer, and a rest at mid-day. Then dinner with His 
Majesty, preceded by a very long and elaborate reception. 
The King is very friendly, and talks sensibly, but seems to 
have inherited his father's bad constitution. He congratulated 
himself on making personal acquaintance with me ; I thanked 
him for graciously permitting me to do so. He hoped that 
I would make some acoustic discoveries that would benefit 
the architecture of public halls; but I could hold out small 
prospect of that. The banquet in the Barbarossa Hall was 
most brilliant: the food very delicate and not substantial, as 
I like it. Subsequently Oedipus Colonus at the Theatre, with 
Mendelssohn's music, but it is less inspired than his Antigone. 

1 To-day the saloons of the Castle are thrown open to us ; 
in the evening a great May Festival at the Rathhaus, when 
the beer is tapped solemnly. ... I must conclude, for R. Wagner 
has just come to fetch me/ 

During this Academic Festival, Helmholtz gave a lecture on 
April 2, ' On the Quality of Vowel Sounds ' ; the important parts 
of it were subsequently published in Poggendorff, after the 
Vowel Theory had been completed by verbal and written 
discussion with Bonders. 

On June 13, 1859, he writes to Ludwig that the necessary 
preliminary study of the motion of air in tubes had led him 
to a definite theory of timbre (Klangfarbe). His detailed 
explanation shows how chords of different timbre and equal 
pitch of fundamental tone are distinguished by the ear because 
of the different frequencies and strengths of the harmonic over- 
tones, i. e. timbre results from the combination of the prime 
tone with different intensities of over-tones. He defines as 
the musical quality of tone that part of it which is independent 
of the irregular murmur that cannot properly be reckoned 
with the musical constituents of the tone, e. g. the scraping 
of the violin bow, the whistling of the stream of air blown 
over a flute, the varying intermittency of the expired breath 
in the pronunciation of consonants ; and then proceeds to the 
question of whether the distinction of musical timbre depends 
only on the perception of over-tones of different intensity, or 
whether the ear can distinguish differences of phase also. 
Quite different wave-forms obtain for a wave composed of a 


prime tone and its first higher octave, according to whether 
the condensation-maximum of the fundamental coincides with 
that of the octave or not. Helmholtz endeavoured to decide 
these questions by building up tones of different timbre by 
the direct combination of simple tones produced according to 
his own method with a tuning-fork. He selected the different 
vowels of the human speech as a suitable object for imitation 
because these can be produced as evenly sustained musical 
tones. He had characterized these vowels (in writing to Bon- 
ders) as sounds in which it is not the fundamental tone, but 
one of the over-tones that is the strongest. He now adds the 
more exact determination that o arises when the fundamental is 
strongly accompanied by the higher octave, a weak accompani- 
ment of the second and third tones producing an improvement 
in the sound, while e is characterized by the third tone, with 
moderate strength of the second, and the transition of o to e is 
produced by diminishing the second tone, and letting the third 
swell out, so that when both partials mentioned are given 
strongly, o modified (o) arises. Thus he shows that the results 
produced with the tuning-fork are confirmed by the investi- 
gations of the tones of the human voice, at least when the 
vowels are sung to a definite note. Since the vowel, as 
pronounced, is a sound produced by the vibration of the 
vocal cords, and the mouth, according to Helmholtz's theory, 
acts as a resonator, which intensifies a given over-tone, 
corresponding with a given vowel- sound, alteration in the 
position of the mouth will produce given vowel-sounds from 
the same musical sound. In order to demonstrate his vowel 
theory, Helmholtz constructed little glass bulbs as resonators 
with two openings, one of which was prolonged into a short 
funnel-shaped neck to be inserted into the ear. Then, on 
sounding the proper tone outside, the mass of air within the 
sphere vibrated in S3^mpathy, and thus acted on the ear. With 
these resonators it was easy not only to demonstrate most 
of the acoustic phenomena, such as objective combinational 
tones, over-tones, and their beats, but also to establish the 
accuracy of the vowel theory. Further, it was proved that 
musical timbre depends solely on the presence and intensity 
of the partial tones contained in the musical tone, and not in 
their different phases, although this is only certain where the 

N 2 


investigation extends to the sixth or eighth partial tone. By 
establishing this last theorem, that difference of phase does 
not come into the question, Helmholtz confirmed his previous 
assumption that our sensation of different qualities of tone 
is reduced to the fact that other nerve-fibres, corresponding 
with the partials, are simultaneously excited along with the 
fibres that respond to the fundamental tone. This simple 
explanation would not suffice, if the difference in phase of the 
deeper harmonics had to be considered. 

Helmholtz gave an enlarged account of his work in the 
following year to the Nat. Hist. Med. Verein, at Heidel- 
berg, in a lecture ' On Timbre '. He removed the restriction 
that the vowel-sounds should be sung upon a single note (that 
of a man's voice at B), and investigated all pitches of sung 
vowels, finding that certain vowels are characterized by still 
higher over-tones. 

In the paper laid before the Bavarian Academy, Helmholtz 
refers to the great work, which he had termed a preliminary 
study, published that year in the Reine u. Angew. Mathematik 
'The Theory of Aerial Vibrations in Tubes with Open 
Ends/ the contents of which he had already communicated 
to the above Society at Heidelberg on March 15. This research, 
with that mentioned above on Vortex Motion, must be reckoned 
among the most brilliant of Helmholtz's mathematical achieve- 
ments, only rivalled, and perhaps surpassed, by the work of the 
last ten years of his life. 

' In 1891,' he writes, ' I have been able to solve a few 
problems in mathematics and physics, including some that 
the great mathematicians had puzzled over in vain from 
Euler onwards : e. g. the question of vortex motion, and 
the discontinuity of motions in fluids, that of the motions of 
sound at the open ends of organ pipes, &c. But any pride 
I might have felt in my conclusions was perceptibly lessened 
by the fact that I knew that the solution of these problems 
had almost always come to me as the gradual generalization of 
favourable examples, by a series of fortunate conjectures, after 
many errors. I am fain to compare myself with a wanderer 
on the mountains, who, not knowing the path, climbs slowly 
and painfully upwards, and often has to retrace his steps 
because he can go no farther then, whether by taking thought 


or from luck, discovers a new track that leads him on a little, 
till at length when he reaches the summit he finds to his shame 
that there is a royal way, by which he might have ascended, 
had he only had the wits to find the right approach to it. In 
my works I naturally said nothing about my mistakes to the 
reader, but only described the made track by which he may 
now reach the same heights without difficulty/ 

The Theory of Organ Pipes had till then been treated on 
the assumption that the motion of the aerial particles within 
the tubes was everywhere parallel to their axis, and that both 
velocity and pressure were equal at all points of the same cross- 
section of the tube, a view that was valid for the parts of a 
cylindrical or prismatic tube more remote from the open ends, 
but was inadmissible near the open ends where the waves 
which are plane in the tube spread out from it in the form of 
spherical waves ; for such a transition could not come about 
suddenly. The view of Bernouilli, Euler, and Lagrange that 
the condensation of the air at the open end of the tube was 
nil was equally inaccurate, since the density there cannot be 
taken as equal to that of the undisturbed air, but only to the 
altered density of the adjacent air that is itself thrown into 
vibration in the free space. Helmholtz followed up his earlier 
work in acoustics by an exact theoretical inquiry into the 
question as to the manner in which plane sound-waves, pro- 
duced in the depth of a cylindrical tube, behave on their escape 
into free space. He settled this very difficult problem mathe- 
matically, without resort to hypothesis, by setting himself to 
discover what form of vibration is permanently set up, when 
the cause of the vibrations is allowed to act continuously and 
uninterruptedly. In accordance with his earlier theory, he 
assumed that the vibrations correspond with those of a simple 
tone, since all complex vibrational forms can be considered as 
due to the summation of a number of such simple tones. 

After applying the most important general laws of the Func- 
tions of Electrical Potential to the Theory of Sound-waves, he 
goes on to his particular problem of determining the motions 
of air at the open end of a cylindrical tube, when plane waves, 
corresponding to a simple tone, are produced within the tube 
from any cause, and communicate their motion from the mouth 
of the tube to the external air when it is affected by no other 


sound-producing agency. By means of the repeated applica- 
tion of Green's theorem to four distinct spaces, he was enabled, 
under certain assumptions as to the magnitudes involved, with- 
out knowing the special form of opening or the motion of the 
air within the opening, to deduce certain relations between 
the plane and the hemispherical waves that spread out into 
the remoter space ; and thus the unsolved problem of the influ- 
ence of the open end upon plane waves was determined. 

In the first place it was found for the form of the waves 
in the tube, that the maxima and minima of the vibrations, 
i.e. their nodes and internodes, occurred at quarter wave- 
lengths from each other, and that the phases of motion differed 
by a quarter of a period at the maximal and minimal points. 
Helmholtz termed the distance of the cross-section from a 
point on the axis at a given definite distance from the mouth, 
the reduced length of the tube, and found that the maxima 
of vibration occurred throughout where the reduced length 
was equal to an even multiple of the quarter wave-length, 
while the surfaces of least motion, or nodal surfaces, occurred 
on the contrary wherever the reduced length of the tube 
equalled an uneven multiple of the quarter wave-length. 

After deducing this general law, by which the problem is 
referred to the determination of the reduced length in the 
different forms of tube, Helmholtz next proposes to discover 
in what forms of tubes the aerial motion at the mouth, and the 
reduced length, may be fully determined for sound-waves of 
such great wave-length, that the dimensions of the opening 
of the tube, its cross-section, and that of the part of the tube 
that deviates from the cylindrical, vanish. 

Helmholtz contributed a supplement to these inquiries in the 
lecture given on Feb. 27, 1863, to the Nat. Hist. Med. Verein, 
'On the Influence of Friction in the Air upon the Motions 
of Sound/ He returns in this to the theoretical differences 
between real and reduced lengths in particular forms of the 
mouth of tubes, since the theory for narrow tubes shows far 
smaller differences than are actually found by experiment; 
the correspondence was much closer when the friction of the 
air is taken into consideration, which Helmholtz was able to do 
on the basis of Stokes's investigations, as he had previously in 
the case of fluids. 


The universal recognition of Helmholtz's acoustic achieve- 
ments by the scientific world, and his election to the Corre- 
sponding Membership of the Academy at Vienna, and the 
Scientific Society at Erlangen, had filled his aged father with 
pride and delight, and as the old man ' had felt much better of 
late, and the indications of brain trouble were quite insignificant', 
it was an unexpected shock to Helmholtz to receive the news 
on June 4 that his father had had a stroke, and lay at death's 
door. He started at once for Potsdam, leaving his sick wife 
with a heavy heart, but his father died before his arrival. ' The 
circumstances,' he writes to his wife, ' were much the same as 
in my mother's case, only the stroke was less rapidly fatal.' 

Helmholtz returned from the funeral to find little comfort 
in his own house ; his wife's health was failing irrevocably, 
slowly at first, but afterwards very rapidly. 'Nothing did good,' 
writes her sister, ' and at length we gave up all hope ; Heidel- 
berg knew only the shadow of her former self.' Helmholtz 
suffered severely from all this agitation ; his frequent migraines, 
which were becoming more frequent and serious, obliged him 
at the doctor's orders to go off to Switzerland at the beginning 
of the autumn holidays, a change that was always beneficial. 
But he was shortly recalled to Heidelberg by disquieting news 
of his wife's health, and came back to sad and heavy months, 
in which his only comfort lay in the severest intellectual 

At the outset he went on with the work commenced during 
the previous summer, on friction in fluids : ' I have just begun 
some work on friction in fluids with Piotrowski, in which he 
will do the experimental part. I hope we shall get the funda- 
mental hydrodynamic equations in reference to friction out of 
it. After that, any special work on the motion of fluids would 
be reduced to a mathematical problem, although it would only 
be resolvable in a very few cases/ 

But this most arduous mathematical work, which required 
the greatest mental concentration, proved impossible in his 
perpetual preoccupation and anxiety over his wife's illness, 
and he turned for distraction to easier experimental questions 
in optics and acoustics, as an appendix to his earlier work. 

On Nov. n, 1859, he gave a lecture to the Nat. Hist. 
Med. Verein, on ' Colour-Blindness ', which led on from his 


lecture at Carlsbad on 'After- Images' (September, 1858), 
part of which had already been given to the Nieder-Rhein- 
ische Gesellschaft He pointed out in the first place that 
the Theory of the Three Fundamental Colours could not be 
retained, in the sense of deriving all actual objective colours 
from any given three such objective colours, since it is im- 
possible, if we select any three spectral colours (as the most 
saturated colour we know), to derive all remaining spectral 
colours from them, as the resulting mixture is always more 
or less white. Young's theory is, however, independent of this, 
when it states that there are three principal colour sensa- 
tions, distributed to three systems of nerve fibres, which 
can be excited collectively, but in different degrees of in- 
tensity, by all kinds of light, so that they yield qualitatively 
different sensations ; here the choice of fundamental colours 
is arbitrary, to a certain extent. In any case spectral colours 
will not excite the separate fundamental colour sensations pure, 
and distinct from the other two; this would agree with the 
view of Helmholtz set forth in his Theory of After-images, 
that there are more highly saturated sensations of colour than 
those which are aroused by spectral colours. In support of 
Young's hypothesis, Helmholtz examined a colour-blind subject 
with the help of Clerk Maxwell's colour-tops (which in sound 
eyes produce any given colour by the mixture of three 
suitable fundamental colours, with the addition of white, 
exhibited on sectors of variable breadth), and found Maxwell's 
results confirmed, since his patient could match all colours 
by mixtures of yellow and blue; thus for his colour-blind 
eyes one of the fundamental sensations was wanting. He 
found the colours which the colour-blind confuse with 
neutral grey to be red and green-blue, the red of which 
appeared to them dark-grey, and the complementary greenish- 
blue a very light grey, since the colour-blind eye was found 
to be very insensitive to red. By this means red was proved 
to be one of the fundamental colours. Helmholtz gives 
the name of red-blindness to this kind of colour-blindness, in 
which, according to Young's theory, there is a paralysis of the 
red-perceiving sensory nerves. He regarded it as probable 
that the other class hitherto denoted as colour-blind are green- 
blind, although the experiments had not at that time been 


carried out, which, by the aid of Maxwell's tops, establish in 
the case of colours which appear approximately the same to 
the colour-blind, whether the difference lies in the tone of 
colour or the degree of saturation. 

Helmholtz's working capacity, however, became gradually 
exhausted, since the condition of his beloved wife was growing 
more and more serious ; her relatives took charge of the family, 
and gave ceaseless attention to the invalid and care to her 
children. ' It was my privilege/ writes her sister, 'to be with 
her to the end. She died conscious, in simple strength as she 
had lived, fearless, with her friend beside her, ever turning 
towards the highest, on Dec. 28, 1859.' 

Her husband wrote of her : ' I enjoyed the purest and highest 
happiness that marriage can give one; it was too beautiful 
for this world/ 

A simple stone marks the grave in Heidelberg churchyard, 
with the inscription, ' Blessed be the rich seed that Love 
scatters round it.' 

For many months Helmholtz was totally incapacitated for 
work by the heavy blow that had befallen him. On April 9, 
1860, he writes to Donders : ' My warmest thanks to you and 
yours for your sympathy with my heavy loss. I have been 
unable to write before, because I have been ill for some time. 
I had got into a state of nervous irritation from the disturbed 
nights and agitation of the last few months, so that I could not 
do any continuous writing without severe headache or attacks 
of fever. It has been a sad time. In not being able to work, 
I lost the best means of defence against the feeling that one 
is alone and has no interests in the world. And I spent two 
months thus in sleepless nights and weary days. Since the 
beginning of March I have been able to get a little comfort 
in work/ 

The fainting fits to which he had been subject of late years 
increased in consequence of all this trouble. ' Before Whitsun- 
tide/ he writes on June 27, 'the attacks came on sometimes twice 
a day; now they are less frequent and severe, so that I can 
hold up against them if needs be/ But he constrained himself 
to work, since that alone could fortify him. During March he 
concluded the paper which he had announced to Ludwig 
and Thomson the previous summer, in co-operation with 


Piotrowski, the latter having done the experimental work under 
Helmholtz's direction, and presented it on April 12, 1860, 
under the title, ' The Friction of Liquids/ to the Academy of 

The equations of motion within a non-viscous fluid mass 
subject to friction had been developed earlier by Poisson, 
Navier, and Stokes, and confirmed by experiments conducted 
in very long, narrow tubes, but it had proved impracticable to 
reconcile theory with experiment when the tubes were wide. 
Helmholtz now undertook to investigate a second case of 
motion in fluids (the theory of which can be derived com- 
pletely from the hydrodynamic equations for fluids exerting 
friction), in order to obtain a new determination of the constant 
for the internal friction of water, so far derived only from 
Poiseuille's observations, and to compare the same with the 
observations. He succeeded in proving this for the movement 
of water in a sphere, polished and gilded inside, by throwing 
the spherical vessel into vibration, round a perpendicular axis, 
by means of a special apparatus, while the lag in the vibrations 
in the fluid was measured with a reflecting mirror and tele- 
scope. In this case, the force exerted by the fluid within the 
vessel upon its walls was experimentally determined, and com- 
pared with the force calculated from the mathematical theory 
of the motions of fluids. 

He simplified the hydrodynamic equations by making the 
vibrations of the sphere so small that the squares of the velo- 
city vanished as compared with its first power, and thereby 
succeeded, on the assumption that gravity was the sole external 
force, in finding particular integral equations, by which the 
components of the velocity at any moment of the water, present 
at a given point, could be expressed as the product of the co- 
ordinates, multiplied by a function of the time and of the 
distance of the point from the origin of the co-ordinates. This 
function satisfies a differential equation, analogous to the 
known potential equation of a sphere, which here contains 
a further factor, involving the friction-constant for the interior 
of the fluid ; the form of the motion corresponding with this 
integral equation may be described by saying that the mass 
of water splits into concentric spherical layers, each of which 
performs a rotary movement like that of a thin hollow sphere 


round the direction of gravity. He analyses this function, 
which characterizes the angular velocity of the rotation, into 
an exponential function lineally dependent upon the time, 
and another that depends only upon the distance, and is 
independent of the time, and thence deduces the general 
integral of the normal differential equation, characteristic of 
this second factor; after this it is a simple matter to deter- 
mine the integrals of the equations of motion for a fluid 
mass, subject to friction, within a hollow sphere. Since it 
was assumed that no force, beyond gravity, was acting inside 
the mass of water, the forces which set it in motion can 
only act upon the outermost layer, and this is actually set 
in motion by the friction of the vessel with which it is in 
contact. It does not adhere to the inner wall of the vessel, 
but glides along it. 

As the analytical function for the components of the force 
with which a fluid in motion acts upon a superficial layer was 
known, it was evident that the force which the moving water 
exerts upon its outermost layer must be balanced by the force 
exerted by the wall of the vessel on the outermost layer of 
water ; hence it appeared in the first place that the motion in the 
vessel described by the integrals arrived at fulfilled the condi- 
tions of Piotrowski's experiments. Helmholtz was next able, by 
comparison of these experiments and the theoretical functions, 
to calculate the constants for the internal friction of different 
fluids, their value differing according to the nature of the fluid 
and its temperature. The experiments, however, presented 
great difficulties, since they seemed to show that the chemical 
composition of the wall of the tube was not in every case with- 
out influence upon the motion of the fluids. 

Immediately after this, Helmholtz gave a lecture to the Med. 
Nat. Hist. Verein, on 'Contrast Phenomena in the Eye', in 
which he endeavoured to distinguish contrast phenomena 
from after-images, and to demonstrate a method by which the 
true simultaneous contrast-images could be investigated apart 
from after-images. In trying to determine the most favourable 
conditions for the appearance of the familiar phenomena of 
contrast, he finds that all the conditions are fulfilled in the 
phenomenon of coloured shadows. When these are observed 
through a blackened tube, the eye retains an impression of 


a colour once established, even when the conditions which 
produced it are removed; in homogeneous red illumination 
the parts that are poorly lighted take on the complementary 
green in consequence of retinal fatigue. Helmholtz conjec- 
tured that the appearance of true contrast phenomena depends 
upon an error of judgement : we can compare correctly when 
the points to be compared are adjacent in the field of vision ; 
spatial separation and succession in time on the contrary 
weaken the positiveness of the impression. He opposes this 
view to the earlier explanations which assumed actual alteration 
of the nervous excitation. 

Starting from Fechner's Theory of After-images, which 
only fails to give a positive explanation of phenomena in 
cases where the circumstances are very complicated, Helmholtz 
attempts to give a theoretical exposition of the temporal 
sequence of visual impressions. Since Fechner gives two 
grounds of explanation, to which he refers the complexity 
of the phenomena relating to this subject, i. e. survival of 
excitation, and fatigue of the nervous mechanism of the 
eye owing to previous excitation, it is obvious that in the 
colour phenomena of after-images, each of these processes 
must come into play for each of the three kinds of nerve-fibres 
assumed by Young's Colour Theory. Accordingly there must 
be six quantities of alterable magnitude, on which depend the 
brilliancy and colour of the after-image observed under certain 
given external conditions of illumination. As an after-image 
is positive when the after-excitation more than counteracts the 
fatigue, negative in the opposite case, an explanation of the 
complex processes with several colours is only possible under 
definite quantitative assumptions as to the time phenomena 
of excitation and fatigue in the nerve apparatus. As at the 
time of his investigation there were very few real quantitative 
determinations, Helmholtz confined himself to finding the mathe- 
matical functions, the variation of which with time corresponds, 
at least in direction, with the course of the phenomena, even 
if no exact correspondence of actually measured magnitudes 
be demonstrable. 

'We find/ he says in a note on these observations, 'that 
two kinds of alterations are brought about in the living eye 
by light, apart from any distinctions of colours, i. e. excitation 


and fatigue. Neither of these processes is in its time-relations 
directly conditioned by the action of light. For when the light 
is cut off, excitation of the points of the retina previously 
stimulated still persists for a recognizable time in the dark 
field, and on testing with renewed and equal illumination of 
the field traces of fatigue are visible for a long time as negative 
after-images. We can also see how these conditions gradually 
disappear while the eye is resting in the dark, when they 
decline very fast and perceptibly at the outset, but the residue 
subsequently vanishes very slowly. As a rule, indeed, excita- 
tion dies out more quickly than fatigue. My conclusion is 
that persistent processes obtain in the living eye, even during 
the action of light, which tend to abolish both excitation and 
fatigue; the simplest mathematical expression of this fact is 
that the velocity with which the excitation 5 disappears, or, if 
the time be denoted /, the negative differential quotient of s 
with respect to the time, is proportional to the total strength 
of excitation at the moment, provided there be no simultaneous 
action of light. In the same way I assume for the alteration 
of fatigue / that so long as there is no augmentation by 
simultaneous excitation we have a differential equation of the 
same form between / and /. On the other hand, the sensation 
may of course be reinforced by a new impression of light. 
This increase as a rule is not sudden, since the fresh exciting 
impression is added at each moment to the residue of the 
previous excitation. We may take the consequent incre- 
ment of excitation as proportional to the luminous intensity 
of the impression. Further, this rise of excitation is conditioned 
by the concomitant fatigue, and the increment is less in pro- 
portion as the fatigue is greater. If we take/= i as the 
maximum value of the fatigue when the new impression 
fails to produce any effect, we may take that portion of 
the differential quotient of s with respect to /, produced by 
the new light of intensity /, as mi (i /), so that this differen- 
tial quotient = as + mi (i /). Fatigue is correspondingly 
augmented by excitation in proportion to the magnitude 
of stimulus, and this increment may be taken as proportional 
to the excitation ; hence the complete expression of the 
alteration of fatigue will be the differential quotient of / 
with respect to /, viz. bf+ns. The two equations then 


exactly determine the course of the two processes. Since the 
four constants a, b, m, and n can only be determined from 
the results of the experiments, and the values / and 5 are 
constantly changing, it is certain that within narrow limits of 
the values / and 5, the above equations must be correct. 
Whether they hold good for wider limits, or whether the 
four quantities here given as constants are really independent 
of 5 and f t can at present be determined only by experi- 
ment, i.e. comparison of the results of our equations with 

Taking the intensity i of the light that impinges on the eye 
during the observed time as a constant, and forming the uni- 
versal integrals of the two linear differential equations for / 
and 5 with constant co-efficients, Helmholtz finds the value 
to which the magnitude of excitation approximates increasingly 
with protracted illumination i t and thence deduces the maximal 
value of the persistent illumination of the eye as the quantity 

-. If /'and 5 be taken as the limits to which the magnitude 

of fatigue or of excitation approximates gradually with long 
illumination t, it follows from the given integral functions that, 
if 5 and / are both at the outset larger or smaller than the 
values F and S, which they finally arrive at, 5 must at first 
exceed or fall short of S, then reach a maximum or minimum, 
and finally rise or fall once more to the value S, while / con- 
stantly rises or falls to the value F; and the same is true 
whether at the outset s>S,f<F or s<S,f>F. 

The first case explains directly 'the alternation of posi- 
tive and negative after-images, when the eye looks steadily 
at a constantly illuminated field, or even at the retinal 
field illuminated with its intrinsic light, on which a lighter 
or darker object has become temporarily visible, and then 
disappeared. In the former case excitation and fatigue are 
simultaneously augmented at the point on the retina that is 
covered by the image of this object ; in the latter they are 
simultaneously diminished. Upon the disappearance of the 
object, excitation and fatigue of the spots involved return 
gradually to their final value, at which the other parts of the 
retina have remained. The positive after-image corresponds 
With the period at which the excitation has not yet reached 


this final value, the negative with the period when it has 
been exceeded, as always occurs in this case. The moment 
of alternation between negative and positive images is found 
by determining that value of /, at which s exceeds S. The 
greater the excitation, and the less the fatigue deviates from 
its final value, the longer will be the duration of the positive 
image. A very brief period of excitation favours this. Further, 
with otherwise similar conditions, it is favourable to the dura- 
tion t of the positive image that the magnitude of intensity 
of the persistent illumination shall be low, as is actually seen 
in experiments in which the positive image is visible for the 
greatest length of time upon the perfectly dark field. The 
longer the time /, the smaller J as follows from the exponential 
quantities of the integrals ' will be the intensity of the nega- 
tive image. In a wholly darkened field of vision, where the 
retina is excited only by entoptic stimulation, the negative 
image is only visible when the ratio between the final and 
initial value of/ and the final value has become fairly large 
in consequence of very strong illumination, or its prolonged 
action. If we neglect the feeble intrinsic light of the retina 
the course of the excitation will be quite independent of the 
concomitant fatigue/ 

Helmholtz only began to develop the ' rise of excitation in 
the recuperated eye* from the integral formulae. The ap- 
plication of the relations found to the problem of intermittent 
illumination leads to an expression for the strength of excita- 
tion with persistent illumination of a given intensity, from 
which Helmholtz concludes that the hypothesis made in 
framing the differential equations is in agreement with the 
well-known law, 'by which the apparently uniform brilliancy 
of a periodically alternating illumination is equal to that which 
would be obtained if the whole quantity of light in each period 
were evenly distributed over the entire period/ 

During the summer, Part II of the Textbook of Physiological 
Optics appeared. It was indeed ready before the death of his 
wife, and on August 6 he sent a copy to Fechner, with the 
following words : 

1 You will find the same subjects in this second part that you 
have dealt with lately in your own work. I had written the 
chapter on Intensity of Light, in all essentials, before I received 


your treatise on it. I therefore introduced some modifications 
afterwards. In the After-images, as you will see, I have 
modelled myself on you throughout. Contrast gave me the 
most trouble; I have tried to clear up this chapter, but have 
not yet got it right/ 

Part II of Physiological Optics deals with the Theory of 
Visual Sensation, and treats in the first place of the various 
forms of stimulation of the optic nerve, and then of its 
excitation by light in particular, after which Helmholtz gives 
a connected development of the theories previously published 
by himself and others on simple and compound colours. In 
connexion with the intensity and duration of visual sensation, 
he gives a number of experimental methods and results, some 
of which, e. g. the Psycho- Physical Law of Fechner, are sub- 
stantially completed by some of his own later works, and lastly 
he deals with the Theory of After-images and Contrast Pheno- 
mena, on the lines indicated above, and illustrated by new and 
interesting experiments. 

Two points may be selected from the wealth of new results 
that had not previously appeared, either in his earlier works 
or in the long series of profound theoretical deductions and 
delicate experiments which combined his own with the work 
of other investigators. 

For the purposes of physiological investigation it was neces- 
sary to make a much more precise analysis of simple light than 
was required by physical work in general, and in the first place 
to investigate the theory of refraction in prisms, in so far as 
this is essential to the production of pure spectra. While 
formerly only the refraction of single rays of light in prisms, 
not the position and character of the prismatic images, had been 
determined, Helmholtz now investigated the prismatic images 
formed by any kind of homogeneous light, when the eye looks 
through a prism, or examines the light issuing from a prism 
with lens or telescope, since these images must be regarded 
as objects for the further optical images produced by the media 
and lenses of the eye. 

If a ray is passed through different refractive media, and the 
length of its path in each medium multiplied by the refractive 
index of that medium, the sum of all these particular quan- 
tities being termed the optical length of the ray, then the 


optical length is proportional to the time in which the light 
traverses the length of the ray, and equal to the distance which 
the light would have traversed in the same time in empty 
space. The law of refraction of light -rays may accordingly 
be expressed by saying that the optic length of the ray between 
given points in the first and last medium must be a limiting 
value (maximal or minimal) when the refracting media are 
limited by surfaces of continuous curvature. In following up 
the analogy with the potential function, Helmholtz finds that 
if the rays have started from any point, and are broken by an 
indefinite number of surfaces of continuous curvature, they 
will after the last refraction be perpendicular to that curved 
surface for the points of which collectively the optical length 
of the ray is of constant value. This surface contains all points 
at which the same phase of ether vibration occurs, and is 
accordingly a wave surface. After laying down this theorem he 
applies the known properties of the normal and of the curva- 
ture of a surface to the determination of the course of the rays 
in an infinitesimally thin bundle of rays. But this further gives 
the laws of the refraction of bundles of rays in prisms. Helm- 
holtz finds that an infinitesimally thin bundle of homocentric 
rays, starting from a point at infinite distance, will only remain 
homocentric after its passage through a prism if it has passed 
through at an angle of minimal deviation, i. e. if it is in a plane 
perpendicular to the refracting edge, and makes an equal 
angle with both surfaces of the prism. Since a luminous 
point can only form a clear image when the refracted light 
is homocentric, the accuracy of the image of a line of light is 
evidently not affected by deviations of the rays, provided 
they lie in the direction of its image. From this he arrives at 
the images of luminous objects, when these consist of vertical 
bright lines of different, monochromatic lights ; and is able to 
determine the brilliancy of the spectrum, and to prove that 
its brightness, apart from loss by reflection and absorption, 
is directly proportional to the brightness of the spectral colours 
involved, and the apparent breadth of the slit inversely pro- 
portional to the apparent length of the part of the spectrum 
that is in question. 

Meantime Helmholtz had determined, while still engaged 
on his Physiological Optics, to write a similar work on Sensa- 


tions of Tone, in consequence of the radical discoveries he 
was making. In 1860 he writes to Bonders : ' I have decided 
to put my acoustic work together in a book. It will be 
a small volume, as popular, in style as possible, so as to make 
it available to lovers of music. I think I shall be able to 
expound the physico-physiological basis of the theory of 

Helmholtz sought comfort and distraction in hard mental 
work : his home, despite the devoted and tender ministration 
of his mother-in-law, who looked after the two little children, 
was empty and desolate. All the external honours that poured 
in on him his appointment as Corresponding Member of the 
Academy of Gottingen, the Sommering Prize given him by the 
Senkenberg Naturforschende Gesellschaft at Frankfurt-a.-M., 
and so on affected him little, though in former days he would 
have welcomed them for the pleasure they gave his father and 
his beloved wife. In the summer of 1860 he betook his sorrow, 
and the fatigue engendered by the term's work and his cease- 
less study of the deepest problems of human knowledge, to his 
friend W. Thomson (Lord Kelvin) in the island of Arran, 
returning after some weeks, refreshed in body and mind, via 
Edinburgh and Hamburg, to Heidelberg. 

He now occupied himself almost exclusively with acoustics, 
and writes to his brother Otto : 

* The physiological basis of consonance and dissonance may 
be thus simply expressed : consonance is a continuous sensa- 
tion of tone, dissonance is discontinuous. Two tones that are 
near each other give coincident beats, i. e. intermittent excita- 
tion of the nerve. The whole theory of Harmony, and of 
our modern System of Tone, follows directly from the beats 
of harmonic over-tones, combinational tones, &c/ 

On Nov. 23 he gave a lecture to the Nat. Hist. Med. Verein 
on 'Musical Temperament*, in which he dealt with the dis- 
advantages of tempered intonation for various instruments, and 
in which the breadth of his historical studies, which in itself 
makes his later theory of the sensations of tone such a mar- 
vellous achievement, is obvious. In any given major scale, the 
major third and the fifth are always tuned so that their vibra- 
tion numbers are as 4 : 5 and 2:3; the three chords contained 
in the scale are then pure. On passing into another key, the 


new chord of the final tone now gives a fifth, which is no longer 
identical with the third of the original key; hence in keyed 
instruments it is usual to substitute for these two slightly 
different tones (one of which is the third of the original tonic), 
one single tone, since an impure fifth is more readily perceived 
than an impure third. On further progression by fifths the tonic 
is not recovered ; in order to distribute the error evenly, all the 
fifths must be slightly altered ; the deviation of the fifths in the 
now general system of intonation will however be exceedingly 
small, since the ratio of the pure to the tempered fifths will be 
as 886 : 885. But since this produces errors in the thirds, and 
modern music is harmonic throughout, the discord of the 
false intervals of intonation makes itself unpleasantly heard 
in the beats of their combination tones and harmonic over- 
tones. In the instruments best adapted for artistic music, the 
disadvantage of tempered music is least felt, because the 
singing voice is independent of it, while the harshness can be 
modified on bowed instruments, and the piano, the tones of 
which soon die away, does not favour dissonances. The want 
of true intonation is, on the other hand, apparent in all long- 
sustained tones, particularly in the harmonium, where the beats 
are too obvious when the instrument is played slowly, and the 
difference between pure and tempered chords is so marked, 
that the latter sound like dissonances in contrast with the former. 
In order to obtain pure harmony, Helmholtz gives two distinct 
values to each note of the scale, according as it is the third or 
fifth, in relation to the tonic of a major chord, and gives the series 
of major chords which satisfy the conditions. In practice, either 
two keyboards must be introduced, or the instrument tuned 
correctly for each key that occurs in the course of the 
piece, by arranging the notes in eight groups, and providing 
all the notes of each group with a separate supply of 
wind from the bellows, when almost pure intervals can be 

In this connexion we may refer to a later work of Helm- 
holtz. Starting from the view that scales originated in the 
desire to distinguish single tones clearly and sharply from one 
another (whole tones only being recognized by the unpractised 
ear of uncivilized peoples to this day), and that semitones 
were only introduced into music as the ear became gradually 

o 2 


developed, Helmholtz thought it well to undertake historical 
studies of the development of the scale in different nations, our 
major and minor scales having been developed very late. In 
July, 1862, he read a paper on 'The Persian and Arabian 
Scales ' to the Nat. Hist. Med. Verein. In the system he pro- 
poses for the construction and tuning of musical instruments, 
in which all the keys can be played in pure consonant chords, 
twice as many intervals were required as usual. These 
historical studies led him to the conclusion that in Greek 
scales, the fifth fifth from C upwards was used as the third 
of C, from which it differs only by the minute interval 
81/80; while if eight fifths are taken downwards from C the 
tone F flat is reached, which only differs from the third of 
C by something like the tenth part of the interval 81/80, so 
that it can be substituted for it. The musicians of Persia and 
Arabia took advantage of these substitutions to obtain pure 
natural scales ; their system was one of seventeen fifths, from 
which scales with Pythagorean, or natural, thirds and sixths 
were derived. Helmholtz followed a common physical prin- 
ciple throughout, i. e. that of the relationship of musical tones. 
The definition chosen by him, according to which two com- 
pound tones are related when they have some common 
over-tones, shows that the notes most nearly related to the 
fundamental are the octave, and then the fifth and fourth ; 
these related musical tones occur in all ' modes '. 

After Helmholtz had ascertained from his experiments on 
timbre that differences in quality of musical tone depend 
principally upon the number and strength of the harmonic 
partials that accompany the prime tone, he next had to 
investigate the forms of the elastic vibrations executed by 
variously sounding bodies. He published his observations on 
the vibrations in strings produced by the bow of a violin, in 
the Proceedings of the Glasgow Philosophical Society, on 
December 19, 1860, with the title ' On the Motion of the Strings 
of a Violin '. It was at once obvious that the string set in 
motion by the bow can only vibrate in the same plane as 
the string and the hair of the bow. He powdered the string 
of a fine instrument with starch, illuminated it strongly, and 
examined its motions by means of a vibration microscope 
constructed for the purpose, the object-glass of which was 


carried by a tuning-fork which vibrated under the influence of 
an electro-magnet once to four vibrations of the string. In 
this way he discovered that the starch granules described a 
shining curve, the horizontal abscissae of which corresponded 
with the displacements of the tuning-fork, and the vertical 
ordinates with the displacements of the string. This motion 
may therefore be imagined as consisting of two different kinds 
of vibration, the first of which greatly preponderates in regard 
to amplitude, while its period corresponds with the period of 
the fundamental tone of the strings, independent of the point 
to which the bow is applied ; the second weaker motion, on the 
contrary, makes very minute deflexions in the curve, since its 
vibration-period corresponds with one of the higher partials of 
the string ; at all nodes of the partial the principal motion alone 
appears. Experiment showed in regard to the principal motion 
that every point of it first advances with constant velocity in 
one direction, and then returns to its first position with another 
constant velocity, from which, in view of the fact that the 
vibrations of a string occur in a plane, the analytic expression 
of the displacement of any point may be stated with the aid of 
Fourier's series as a function of the distance of the point 
from one end of the string, and of the time. Along with this 
principal form of vibration, other lesser vibrations, which 
may be expressed in precisely the same way, are produced 
if the bow touches any point of which the distance from the 
nearest end of the string is the reciprocal value of a whole 
number of lengths of the string ; in such a case the 
over-tones (on the analogy of Young's investigations for the 
strings of a harp) of which the multiples correspond to that 
whole number will not be heard, although the ear can plainly 
distinguish all other over-tones. Helmholtz concludes from the 
simple analytical consideration of this combination, that during 
the motion of the string the base of the abscissa of its point 
of greatest displacement moves to and fro along the line of 
equilibrium with constant velocity, while the apex itself 
describes two parabolic curves that run above and below the 
position of equilibrium and through the ends of the string, and 
the actual form of the string at any instant is given by the two 
straight lines that join a point on the parabolic curves with the 
ends of the string. 


A year of work and of the richest scientific discoveries lay 
behind him of sad memories also. At the Christmas season 
of 1859, the beloved wife lay dying, who had watched him 
grow up from the modest assistant-physician at Potsdam into 
the most famous physicist and physiologist of the day, and who 
had beautified his home and life with the most self-sacrificing 
love and discerning judgement. And now the Christmas of 
1860 found him weary in body, overtasked in mind, desolate 
in his affections. Notwithstanding the love and devotion of 
his mother-in-law, he felt that the education of his two little 
children could never be all that he and his wife had planned, 
and further, that his deep interest in science and art, and 
the love of aesthetics which had never been eclipsed by his 
profoundest abstractions, were in danger of atrophy, whereby 
he would lose the marvellous fertility and productivity that 
elevated him above all the other scientific men of his day. 

On February 13, 1861, he writes to Lord Kelvin : 

'On this account I had seriously to think of introducing 
a new order of things, and if this had to be done, it was better 
on all accounts that it should come soon. At the end it did 
come about more rapidly than I had expected, for when love 
has once obtained permission to germinate, it grows without 
further appeal to reason. My fiancee is a gifted maiden, young 
in comparison with myself, and is I think one of the beauties 
of Heidelberg. She is very keen-witted and intelligent, accus- 
tomed to society, as she received a good deal of her education 
in Paris and London, in the charge of an English lady, the 
wife of her uncle Dr. Mohl, Professor of Persian at the College 
de France, in Paris. She therefore speaks French fluently, 
and is decidedly better than I in English. For the rest her 
"fashionable" (sic) education has in no way interfered with 
her straightforward, simple nature/ 

Du Bois writes to inquire further details, since all that 
concerned Helmholtz had a lively interest for him. 

1 My bride-elect/ replies Helmholtz on March 2, 1861, ' is 
the daughter of Robert v. Mohl; she appealed to me from 
the outset of my life here as a most intelligent young lady, 
but I saw very little of her. She was away for a long time 
in Paris, staying with her uncle Julius v. Mohl, Professor of 
Persian at the College de France. His wife is an English^ 


woman, and Anna went to her several times for long visits 
to Paris and England, where she imbibed the best sides of 
French and English manners and customs. I must confess 
that I rather avoided than sought Anna v. Mohl last summer, 
for I felt that a girl like her would be dangerous for me, and 
I should never have presumed, as a widower with two children, 
and no longer in my first youth, to seek the hand of so young 
a lady, who had every qualification for playing a prominent 
part in society. However, it all came about very quickly, 
and now I can once more face the future happily. The 
wedding is to be at Whitsuntide/ 

In the Easter vacation Helmholtz went to England to give 
two lectures on 'The Physiological Theory of Music', but found 
he had to deliver a third without any preparation, since Bence 
Jones and Faraday insisted on his giving an evening discourse 
on the Conservation of Energy. In this lecture, delivered April 
12, 1861, * On the Application of the Law of the Conservation 
of Force to Organic Nature/ he first, as in all his earlier 
lectures, gives an account of the principle of the conservation 
of force, which, with Rankine, he prefers (since it bears no 
relation to amount of force) to term the conservation of energy, 
and which he designates as the most important advance 
of science in the century because it embraces all laws of 
physics and chemistry. He now proceeds to apply this 
law to organic nature ; he points out that continuation of life 
is bound up with continued supply of means of sustenance, 
which after complete digestion pass into the blood, are slowly 
consumed in the lungs, and finally produce almost the same 
compounds with the oxygen of the air, as those which 
would be produced by burning the food in an open fire. 
Now since the amount of heat produced by oxidation is 
independent of the time occupied in consumption and of 
the intermediate stages, it can be calculated from the 
mass of the materials consumed, how much heat, or its 
equivalent work, can be produced by any animal body 
experiments that involve great difficulties. On a later occa- 
sion, when a criticism of some scientific book was demanded 
from him, he pointed out that the difficulty which arises in con- 
sidering these very important and very recondite physiological 
problems (the question of heat production in animal bodies, 


and its connexion with metabolism) is principally due to the 
fact that the heat to be measured is not evolved suddenly, but 
in the course of hours, so that it cannot be even approximately 
collected in an apparatus, while further a considerable quantity 
of air has to be led through such an apparatus, which carries 
off a good portion of the heat produced, apart from the aug- 
mentation of work consequent on the necessary size of the 
apparatus. Within limits, however, experiment shows that 
the heat actually produced in animal bodies corresponds with 
that given off in chemical processes. He makes it intelligible 
that the animal body does not differ from a steam-engine in the 
way in which it obtains warmth and energy, but only in the 
objects, and the way and means, for which and by which 
it uses the acquired energy. In conclusion he points out 
how the hypothesis of a vital energy has been gradually 
eliminated by all these experiments, so that the younger 
scientific workers, who are seeking the true causes for all 
these processes, no longer admit any distinction in chemical 
and mechanical work within and without the living body. 
The law of the conservation of energy points the way in 
which these fundamental questions, which have given rise to 
so many speculations, can be really and adequately solved 
by experiment. 

On May 16, 1861, Helmholtz was married to Fraulein Anna 
von Mohl. ' Even Helmholtz's closest friends/ writes the sister 
of his first wife, ' found it difficult to reconcile themselves to his 
marrying again after only one year. After the ideal happiness 
of the former marriage such a step, taken so quickly, appeared 
almost inconceivable. They did him injustice. He did not 
really lose his wife at the time of her bodily death . . . she 
had gone from him before, owing to the terrible nature of her 
illness. For more than a year, her inner life had been dying out 
step by step, paralysing all her interests and sympathies. 
Only in death did she regain her old intellectual and moral 
eminence. Thus Helmholtz had long been a solitary man 
when she died, and the outlook for the future with two 
small children and their grandmother, who in spite of all 
devotion and self-sacrifice was an old woman, was a sad one. 
For Helmholtz, who was accustomed to the most active mental 
companionship, it was absolutely impossible. He chose a wife 


who responded to all his needs. Anna von Mohl, a person 
of great force of character, talented, with wide views and 
high aspirations, clever in society, and brought up in a circle 
in which intelligence and character were equally well developed, 
was to the time of his death an admirable companion, while her 
judgement was always a law to him.' 



AT HEIDELBERG: 1858-1871 (continued). 

AFTER a few days' absence at Whitsuntide, Helmholtz 
returned to Heidelberg with his young wife. ' To the end 
of her life she spoke with emotion of this first journey from 
Baden to Schloss Eberstein with her husband, whose eminence 
and noble character were just dawning on her/ On reaching 
home, he found warm congratulations from his old friend 
Ludwig, who at the same time expressed ' unbounded astonish- 
ment at the increasing significance of the discoveries made 
by Helmholtz ' receiving the modest answer : 

' I wish you did not think so disproportionately well of my 
work, and so little of your own. We all have our special 
capacities, and I know very well that I could not have 
discovered the dependence of the salivary secretions upon 
nervous control, or carried out any of your other investigations/ 

Under these altered conditions Helmholtz appeared in a new 
light ; the dark shadows that had saddened his life for so many 
years were dispersed, and his home was brightened by the 
charming and universally popular wife, who shed sunshine 
around her. 

He had already become recognized as the first authority 
in the scientific world, where he was looked up to on all 
sides with admiration and astonishment, while his work in 
optics and acoustics had attracted the attention of the world 
of art as well; and now his influence extended still further, 
among other ranks of educated society. In Konigsberg and 
Bonn his public lectures had introduced his vast and com- 
prehensive scientific views to the wider circles of the world 
of science: at Heidelberg, though his circumstances were 
naturally somewhat restricted, his house became the centre 
of scientific and artistic activity, to a degree which ordinarily 
is only possible under more favourable conditions. 


'Journeys to England/ writes his sister-in-law, Freifrau v. 
Schmidt-Zabierow, the elder daughter of Robert v. Mohl, ' as 
well as long and frequent visits to our relatives in Paris, where 
the intellectual atmosphere of our aunt's salon at 120 Rue du 
Bac, which was the focus of the best society, had developed 
my sister's rich gifts to the utmost, and made it a necessity for 
her to live in intercourse with distinguished people. She had 
ample opportunities of forming such connexions, both in our 
parents' house, and among the many intellectual foreigners of 
good social standing who were at that time living in Heidelberg. 
A widened outlook on life and greater demands upon it were 
the natural result of these international relations. My sister 
had as perfect command of French and English as of her 
mother tongue, and all restriction to any particular set of 
society was abhorrent to her from her earliest youth. Her 
fresh and merry temperament, her sense of humour, her rapid 
grasp of things and people, may have had a directly beneficial 
influence upon Helmholtz.' 

In spite, however, of these extended social relations, Helm- 
holtz's intellectual and thoroughly genial life expended itself 
mostly in his own house, where his incomparable wife 
succeeded in maintaining her environment at an unusually 
high level, and in respecting the limits enforced by her 
husband's ceaseless activity in work and thought. Order 
now began to reign in his library and workroom, of which 
she had written a few months before their marriage : 'How 
I shall have to struggle with myself and subdue my natural 
inclinations before I can become a really useful wife! Do 
not lose patience with me, Hermann, I am easily discouraged ; 
but I must tell you that your writing table is frightfully untidy. 
If I were not far too well brought up in regard to learned 
confusion, I should take the liberty of sorting out all the 
written papers from the blank sheets, with energetic hand, 
and putting away all the letters in a drawer N.B. unread and 
then go over everything with a damp cloth, on Miss Nightingale's 
principle. But as it is I must leave things as they are, and 
am only thankful to have discovered one human failing in youJ' 

His correspondence with his scientific friends at this time 
was even more extensive than before. If there was less 
science in his letters to du Bois-Reymond, because Helmholtz 


was now working at subjects with which du Bois was not 
much concerned, this was replaced by an increasingly 
intimate correspondence by word and letter with W. Thomson 
(Lord Kelvin), in which they not only discussed the epoch- 
making work they were themselves engaged upon, but com- 
municated to one another the most important researches 
and discoveries of other observers during the long period 
of nearly fifty years. In this way Helmholtz was the first 
to inform Lord Kelvin of KirchhofFs discovery of metals in 
the solar atmosphere. Although the letter in question can 
no longer be found by Lord Kelvin, he addressed the following 
very interesting lines to the author on September 26, 1902 : 

' There should be several others between that date and 1856, 
when I first had the great pleasure of making personal 
acquaintance with Helmholtz in Kreutznach when he came 
to see me, and in Bonn where I returned his visit. 

' There should be a letter of November or December, 1859, 
telling me of Kirchhoffs discovery of metals in the solar 
atmosphere by spectrum analysis. You may possibly find 
my answer which I wrote immediately on receiving it, telling 
him that as chanced two or three days before, I had, in 
a lecture to my students in Glasgow University, told them 
that I had learned from Stokes that the double dark line D 
in the spectrum of sunlight proves that there is sodium vapour 
in the sun's atmosphere, and that other metals might be found 
there by the comparison of the Fraunhofer dark lines in 
the solar spectrum with the dark lines produced in flames 
by metals. I am sure I must also have told him that I had 
been giving this doctrine regularly in my lectures for several 

' I well remember that at that time I was making " Properties 
of Matter" the subject of my Friday morning lecture. On 
one Friday morning I had been telling my students that we 
must expect the definite discovery of other metals in the sun 
besides sodium by the comparison of Fraunhofer's solar dark 
lines with artificial bright lines. The next Friday morning 
I brought Helmholtz's letter with me into my lecture and read 
it, by which they were told that the thing had actually been 
done with splendid success by Kirchhoff.' 

After the publication of the Second Part of Physiological 


Optics Helmholtz had devoted all his energies to the preparation 
of his great work on Acoustics, and by the beginning of 1861 
was feeling that after years of preparation he might soon hope 
to lay the results of his profound discoveries in acoustics and 
the art of music before the educated world. Soon after the 
heavy misfortune that befell Lord Kelvin, Helmholtz wrote on 
January 16, 1861, to Thomson's wife : 

1 1 have been working all the winter at my physiological 
theory of music, and have only two chapters left to write ; 
then the first draft of it will be ready, though much will doubtless 
have to be worked up in detail, and improved. I hope to give 
the book to the printers after Easter. Mr. Thomson will find 
a great deal that is new since we discussed it last summer, 
which I have put in while I was working out the details. 
I have penetrated a long way into the Theory of Music with 
my physical theories, much farther than I dared to hope at 
the outset, and the work has amused me considerably. In 
developing the consequences of any valid general principle 
in individual cases, one constantly comes on new and quite 
unexpected surprises. And as the consequences are not 
arbitrary, and contingent on the caprice of the author, but 
develop according to their own laws, I often have the im- 
pression that it is not my own work that I am writing out, but 
some one else's. Mr. Thomson must have found the same 
thing in his own work on the mechanical theory of heat. 
I have also had to look through a great deal of music, and to 
study the history of music. The Scotch Ballads have been 
of great use for this, as they have preserved many of the 
ancient forms/ 

A paper on 'The Theory of Reed Pipes' (July, 1861) con- 
cluded the publication of Helmholtz's detailed acoustic obser- 
vations, and he went on to formulate a physiological acoustic 
as previously announced to Thomson. 

General happiness, and satisfaction with his new circum- 
stances, had restored his mental energy and inexhaustible 
powers of work, and with these he had recovered his old 
delight in art and nature, and conceived the notion of building 
the bridge to lead from physics and physiology to aesthetics. 

At the close of the session, after making a cure at Kissingen, 
he took a long journey in Switzerland and Italy with his young 


wife, returning with restored health and rejuvenated mind and 
body. He was now able to participate cheerfully in all that 
life offered him in the beautiful city on the Neckar, where for 
the first time he felt himself at home. In September he 
brought his children Kathe and Richard (who since April had 
been at Dahlem with their grandmother) to Heidelberg, where 
he now had a commodious dwelling on the Anlage which he 
shared with Frau von Velten. 

Helmholtz took up his book on Acoustics with fresh energy, 
plunged into arduous optical problems (the solution of which 
was to form Part III of the Physiological Optics), elaborated 
the structure and detail of his Theory of Knowledge, and at 
the same time continued the electrical investigations to which 
he had been led by du Bois-Reymond and by his own 
physiological researches. 

In a lecture given to the Nat. Hist. Med. Verein at Heidelberg 
(December 8, 1861) on 'A Universal Method of the Trans- 
formation of Problems of Electrical Distribution ' a number of 
interesting and important propositions were brought forward 
by Helmholtz, who was not acquainted with the work done by 
others in this department. 

Immediately after the publication of this paper, he was 
informed that its essential results were already contained in 
two letters from W. Thomson to Liouville, and at once acknow- 
ledged this in the Heidelberg Transactions of May 30, 1862. 
He also wrote on May 27 to W. Thomson : 

'I have to beg you to answer a scientific question. Last 
autumn I fell back on potential functions again. I was troubled 
by the difficulties that remained unsolved in my work on sound 
vibrations in an open cylindrical tube. The difficulty of attacking 
the question lay in the fact that the aerial motions are discon- 
tinuous at the edge of the open end of the tube. This led me 
to investigate the distribution of electricity at a circular edge. 
I found that I could derive this in certain cases from the 
distribution along the straight edge in which two infinite 
planes intersect one another, and I solved the problem for this 
case. Afterwards, however, I noticed that you had already 
stated in the Cambridge Math. Journ. that you had solved this 
question, and I want to know whether you have published the 
solution, or intend to publish it, in which case it is not worth 


my while to work out my own solution for the press. The 
principle of reflection from a spherical surface, by which 
a straight edge can be converted into a circular one, was 
also discovered (as he supposed) by another and very capable 
young mathematician, Lipschitz, but happily we came on it in 
time in your previous work. Unfortunately I have already 
published it in a short Note in the Transactions of our own 
Scientific Society, for which I must beg your pardon ; but in 
its fuller exposition by Lipschitz, the priority will be given 
to you/ 

Thomson at once gave him full information upon these 
mathemical points. 

Meantime, the great work on Acoustics was nearing its 
completion; on April 29, 1862, Helmholtz writes to Bonders 
(after telling him that a son had been born on March 3, who 
received the names of Robert Julius, and whose life nearly cost 
that of his mother) : 

' As to my work on acoustics, A Physiological Basis for the 
Theory of Music, the blocks are made, the text is being set up, 
and two-thirds of the manuscript have been sent off: there is 
a good deal still to alter and patch up in the last third, but the 
most important parts are already written. I shall be thankful 
when I have finished the last words of this long-winded under- 
taking, for I have been working on it for seven years, which 
certainly won't be seen from the size of the book. And then 
very likely the philosophers and musicians will regard it as 
trespassing on their domain, while there are not many musical 
people like yourself, for instance, among the physicists and 
physiologists. You will be my most intelligent critic, and 
I shall be very curious to hear whether my bold attempt to 
bring scientific methods into aesthetics will meet with your 

The busiest and most productive period of Helmholtz's life 
in Heidelberg opened with the year 1862. His Theory of the 
Sensations of Tone and Physiological Optics were both near 
their completion ; his epistemological views were shaping 
themselves into a consistent system of philosophy; he was 
incessantly occupied with problems in hydrodynamics and 
electrodynamics, and his thoughts were already turning to the 
investigations of the axioms of geometry, which in a few 


years' time were to reveal the depth of his mathematical and 
philosophical conceptions to the scientific world. During the 
next ten years, Helmholtz displayed an illumination in his 
view of scientific problems, an elevation of philosophical 
conception, a purposeful attitude in regard to the riddles and 
mysteries of Nature, a grasp of all the resources of thought 
and feeling available for the investigation of the whole field of 
human knowledge, such as are seldom met with in the history 
of the sciences, and can only be appreciated in their full extent 
and significance by those who had the privilege of personal 
contact with this extraordinary genius. 

In his youth his friends du Bois-Reymond, Brttcke and 
Ludwig had applauded his marvellous discoveries, and now it 
was Bunsen and KirchhofT who were amazed at his scientific 
achievements. Long after KirchhofT had won immortal fame 
by his discovery of spectrum analysis, he used to say, modestly 
indeed, but none the less truly, ' I am content if I can even 
understand a single work of Helmholtz, but there are still 
many points in his great book on acoustics that I cannot unravel/ 

It was to this time of intellectual activity that Helmholtz 
referred when he said thirty years later in his celebrated speech 
on the commemoration of his 7oth birthday, November 2, 1891 : 

'There are many narrow-minded people who admire them- 
selves enormously if they have one stroke of luck, or think 
that they have had one. A pioneer in science, or an artist, 
who has a repeated run of happy accidents, is indubitably 
a privileged character, and is recognized as a benefactor of 
mankind. But who can count or weigh such lightning flashes 
of the mind ? Who can trace out the secret threads by which 
our conceptions are united ? For 

Was vom Menschen nicht gewusst, 
Oder nicht bedacht, 
Durch das Labyrinth der Brust 
Wandelt in der Nacht. 

' I must confess that the departments in which one has not to 
trust to lucky accidents and inspirations have always had the 
greatest attraction for me. Yet as I have often been in the 
predicament of having to wait on inspiration, I have had some 
few experiences as to when or how it came to me, which may 
perhaps be of use to others. Often enough it steals quietly 


into one's thoughts and at first one does not appreciate its 
significance; it is only sometimes that another fortuitous 
circumstance helps one to recognize when, and under what 
conditions, it occurred to one ; otherwise it is there, one knows 
not whence. In other cases it comes quite suddenly, without 
effort, like a flash of thought. So far as my experience goes it 
never comes to a wearied brain, or at the writing-table. I must 
first have turned my problem over and over in all directions, 
till I can see its twists and windings in my mind's eye, and run 
through it freely, without writing it down ; and it is never possible 
to get to this point without a long period of preliminary work. 
And then, when the consequent fatigue has been recovered 
from, there must be an hour of perfect bodily recuperation and 
peaceful comfort, before the kindly inspiration rewards one. 
Often it comes in the morning on waking up, according to the 
lines I have quoted from Goethe (as Gauss also noticed, 
Works, v. p. 609 : Law of Induction discovered January 23, 
1835, at 7 a.m. before rising). It came most readily, as I ex- 
perienced at Heidelberg, when I went out to climb the wooded 
hills in sunny weather. The least trace of alcohol, however, 
sufficed to banish it. Such moments of fertile thought were 
truly gratifying, but the obverse was less pleasant when the 
inspiration would not come. Then I might worry at my 
problem for weeks and months, till I felt like the creature on 
the barren heath 

Von einem bosen Geist im Kreis herumgefuhrt, 
Und ringsumher ist schone grune Weide. 

Sometimes nothing but a severe attack of headache could 
release me from my spell, and set me free again for other 

To all these great scientific labours and projects were now 
added no less arduous official duties but in Heidelberg his 
lectures on physiology and on the general results of science, 
as well as the direction of the work in the Laboratory, were no 
mere tasks to be reluctantly fulfilled. Nor did he regard the 
University lectures simply as an obligation laid upon him by 
the State, ' which provided him with sustenance, with scientific 
instruments, and with a good proportion of spare time,' and 
therewith had the right to claim from him that whatever he 


discovered by its aid should be freely communicated -to his 
students and his fellow citizens; he always appreciated the 
fact that lecturing compelled him to test each isolated pro- 
position strictly, to formulate each conclusion correctly, and, 
since he could only assume a limited amount of previous know- 
ledge in his hearers, to state the evidence for the views he was 
maintaining in as simple a manner as possible. His audience 
took the place of his friends, whom he always imagined as 
present at his scientific lectures. ' I always pictured the most 
intelligent of my friends before me, as my conscience ; I asked 
myself if they would sanction what I was saying. They haunted 
me as the embodiment of the scientific spirit of an ideal 
humanity, and set my standard/ 

' As a student in Heidelberg/ says Engelmann, ' I followed his 
lectures on physiology, and the public lectures on the general 
results of natural science, which he gave every winter at that 
time. In intellectual and social life there are two forms of 
energy, and it is the sum of these which determines the value of 
the whole. With Helmholtz only a small part of the enormous 
supply of energy stored up in him was actually in evidence at 
any given moment. The conversion of his potential energy 
into kinetic was slow, unlike what happens with those whom 
people are wont to describe as geniuses. As he never worked 
out the details of his lectures, but composed them as he went 
along, he spoke slowly, deliberately, and at times a little 
haltingly. His eyes looked away beyond his audience as 
though he were seeking the solution of a problem at an infinite 
distance. In the physiology classes he assumed no more know- 
ledge and insight in his medical students than did any other 
teacher of the same department. He seldom gave the names 
of any experimenters, and least of all his own.' 

He was a keen teacher in the laboratory, and every earnest 
student became one of his friends in science. As free from 
professional jealousy as Magnus, whom he had so often com- 
mended, he frequently supplied the fundamental ideas for the 
splendid work that issued from his laboratory, and provided 
a wealth of suggestions for the overcoming of new experimental 
problems, in which more or less ingenuity was required. 
1 Whoever had the luck/ says Bernstein, who for years was his 
assistant at the Physiological Institute, 'to watch Helmholtz 


experimenting, will never forget the impression which he gave 
of the purposeful activity of a master-mind when confronted 
with difficulties. He turned out models of ingenious con- 
trivances with the simplest materials, corks, glass rods, bits of 
wood, cardboard boxes, and the like, before putting them into 
the hands of the mechanician. No accident ever disturbed the 
wonderful serenity and equanimity of Helmholtz's tempera- 
ment ; he was never upset by the clumsiness of others. Men 
who had worked for him for years never saw him excited 
under such circumstances/ 

He was respected and admired by the Government of Baden, 
by his colleagues, by the students of every faculty, and it was 
but a slight token of this feeling that made him Pro-Rector of 
Heidelberg University as early as 1862. 

The discourse which he delivered on this occasion (November 
22, 1862), ' On the Relation of the Natural Sciences to Science 
in General/ was a model of style, and contained a wealth of 
ideas and points of view which he enlarged on and enriched 
on various subsequent occasions, and which were frequently 
utilized by others as the foundation of their efforts at organiza- 
tion. In contrast with the one-sided view of many scholars, 
knowledge does not seem to him the sole aim of mankind upon 
this earth. Even if the sciences evoke and educate the finer 
energies of man, it is in action alone that he finds a worthy 
destiny; his goal must be the practical application of his 
knowledge, or the enlargement of science itself, which again 
is an act that promotes the welfare of mankind. But it is not 
enough to have a knowledge of facts in order to collaborate 
in the progress of science : science consists in the unveiling 
of laws and the discovery of causes. If science aims at the 
predominance of mind over matter, it is none the less the duty 
of educated men to recognize the equality of both, and to 
distinguish them only by their content. If the physical 
sciences have been more perfected as regards their scientific 
form, the mental sciences which resolve the human mind itself 
into its different activities and impulses treat of richer material, 
more closely knit with the interests and emotions of man. 

Such knowledge, however, is slow to make its way ; before 
his death, Helmholtz was lamenting in his congratulatory 
address to the Academy at Berlin on the Jubilee for the fiftieth 

p 2 


year of his friend du Bois-Reymond's doctorate, that a great 
gulf still divides the philosophical and historical interests in our 
nation (as in all civilized Europe) from those of natural science 
and mathematics; the two worlds hardly understand each 
other's aims in thought and work, and this is a serious 
hindrance to salutary co-operation, and to the concordant de- 
velopment of mankind. It is for this reason that Helmholtz 
advocates the increase of popular scientific lectures in the 
best sense, as a means of harmonizing the different scientific 
views, since it is not so much information about the results 
of these discoveries that is demanded by the more intelligent 
and cultured of the laity, but rather ' some idea of the mental 
activities of the scientific investigator, of the particular character 
of his scientific methods, his aims, and the new solutions which 
his work offers for the great mysteries of human existence '. 

Helmholtz only alludes in passing to the question of in- 
struction, which afterwards became of such burning impor- 
tance ; he gives the preference to classics over modern 
languages in the education of the young, on account of its fine 
aesthetic and logical training, and in discussing the question 
whether mathematics, as 'the representative of self-conscious 
logical activity ', should not be made more important in school 
studies, he expresses himself in favour of this, since the 
individual will presently have to graduate in sterner schools 
of thought than that of the grammarian. 

Helmholtz tried to define the characteristic difference between 
the physical and the mental sciences more particularly, by 
saying that the physical sciences for the most part reduce 
their inductions to definite and universal laws and theorems, 
while the mental sciences are chiefly concerned with inferences 
from the psychological sense of touch. In the preface to his 
translation of Tyndall's Fragments of Science he describes in 
clear and beautiful language the importance of the classics in 
the development of a moral and aesthetic sense, and in the 
evolution of an intuitive knowledge of human sensations, ideas, 
and conditions of civilization : but he denies that the exclusively 
literary method of education is the most important function 
in the methodical training of that faculty ' by which we subject 
the unorganized material, governed as it would seem more by 
chance than by reason, which we encounter in real life, to the 


systematizing concept, whereby it is rendered capable of being 

expressed in words*. He finds in the simpler relations of 

inorganic nature an instrument for the systematic development 

of a train of ideas, comparable with ' no other human invention 

in respect of its congruity, certainty, exactitude, and fecundity '. 

In the Academic Discourse he insists on the incontrovertible 

fact that even if the antithesis between the moral and the 

physical sciences had been unduly emphasized by Hegel and 

Schelling, it had a real basis in the nature of things, and must 

be taken into consideration. In comparing the different 

physical sciences one with another, he points out the great 

advantage which the experimental sciences have over those 

which depend on observation in the investigation of the 

universal laws of nature, since they can arbitrarily modify the 

conditions under which the effect ensues, and may therefore 

limit themselves to quite a small number of characteristic 

observations, in establishing the validity of any law. He 

demands of experimental and mathematical science that it 

shall strive after the attainment of laws, to which there are 

no exceptions, ' since it is under this form alone that our 

knowledge prevails over space and time, and the forces of 

nature/ Thus he regards the Law of Gravitation as the 

greatest logical achievement of the human mind, but finds 

absolute certainty of inference in mathematics alone; there 

no authority is paramount other than pure reason, and the 

whole of science is constructed from the fewest axioms. 

' In mathematics we see the conscious, logical activity of our 
mind in its purest and most complete form; we can here 
appreciate its travail as a whole the precaution with which 
it must advance, the accuracy that is necessary in order to 
determine the exact import of the acquired general propositions, 
the difficulty of forming and of understanding abstract con- 
cepts while at the same time we learn to put confidence in the 
certainty, the scope, and the profit of such intellectual labour/ 
At this time Helmholtz was busying himself with difficult 
questions in physiological optics, and more particularly with 
the construction of the Horopter, while the issue of the Lehre 
von den Tonempfindungen (Theory of the Sensations of Tone) 
was completed by the end of 1862, and on December 14 
Helmholtz writes to Thomson : ' I am much interested in the 


two papers you announce on the cooling of the earth, and the 
alterations of form in elastic spherical shells, which may con- 
ceivably have reference to the earth also, since I am now 
engaged on some lectures to the students of all faculties on 
the general results of natural science, in which I want to give 
a popular exposition of the law of the conservation of energy 
and its consequences, and to use it as a connecting thread to 
draw the different branches of the physical sciences together. 
I have so far given the history of the planetary system of the 
sun and earth, and convinced myself that many problems have 
been neglected by the astronomers and geologists, which might 
well be attacked now, under the present conditions of know- 
ledge, though only by those who are trained physicists and 
mathematicians. Your undertaking to write a Textbook of 
Natural Philosophy is very praiseworthy, but will be exceed- 
ingly tedious. At the same time I hope it will suggest ideas 
to you for much valuable work. It is in writing a book like 
that, that one best appreciates the gaps still left in science. 

' My book on acoustics is just out, with the title Die Lehre 
von den Tonempfindungen, als physiologische Grundlage fur die 
Theorie der Musik (On the Sensations of Tone, as a Physiological 
Basis for the Theory of Music). The publisher has already 
informed me that the copy I intended for you has been sent 
to your address in Glasgow. The issue of this book and the 
pro-rectorial business that has devolved on me this year have 
taken up so much of my time that I have not been able to 
attend to anything else. Now I have gone back to the com- 
pletion of my Physiological Optics, one section of which is still 

In his Sensations of Tone Helmholtz set himself the task 
of connecting the border-land of physical and physiological 
acoustics with that of the science of music and aesthetics, so 
that he leaves out of consideration pure physical acoustics, 
which is merely a part of the theory of elastic bodies. 
Following the principle he had adopted for the work on 
optics, he divides his physiological acoustics into three parts, 
the first of which is occupied with determining the mode in 
which sound is conducted to the sensory nerve within the ear, 
and contains the physical part of the corresponding physio- 
logical investigation of the sensations ; the second, and more 


especially physiological part, treats of the excitation of the nerve 
itself, in correspondence with different sensations; and the 
third, and more essentially psychological, section endeavours 
to lay down the laws by which the ideas of definite external 
objects, or percepts, result from these sensations. The physical 
and mathematical basis of auditory sensation forms the subject 
of a later very interesting work, which is, however, intelligible 
only to mathematicians, while the physiological factors in 
audition, along with the psychological and aesthetic considera- 
tions, are submitted to a remarkable, and, for the most part, 
easily intelligible analysis in the work itself. 

Part I, which treats of the composition of vibrations, the 
theory of over-tones, and that of timbre or quality of sound, 
presents, along with Part II, which deals with interference in 
harmony, combinational tones and beats, consonance and 
dissonance, an admirable and masterly treatise, popular in the 
best sense, with a penetrating analysis, and an exposition, 
illustrated by a wealth of new experiments, of the results 
already published by Helmholtz in his separate papers. He 
gives a precise account of the theory and construction of his 
harmonium with naturally just intonation, and shows how the 
siren of Cagniard-Latour, perfected by Dove, had now de- 
veloped into his polyphonic siren. The first two parts of this 
unique work accordingly deal with such phenomena as are 
mechanically determined by the construction of the ear, and 
are therefore independent of volition, so that it is possible to 
determine the exact laws by which they are governed. Part 
III, which is eminently original, magnificently planned, and 
admirably carried out, deals with the relationship of musical 
tones, and with scales and tonality, and enters the region of 
aesthetics in order to establish the elementary rules of musical 

1 The relations between the physiology of audition and the 
theory of music/ he said on a later occasion, ' are particularly 
clear and striking, because the elementary forms of musical 
composition depend far more essentially on the nature and 
individuality of our sensations than is the case with the other 
arts, in which the kind of material utilized and the objects to 
be represented have a much greater influence.' 

Starting from the conviction acquired in the course of his 


historical studies of the development of music, that the system 
of scales and modes, and the harmonies built up from them, 
do not rest merely on unalterable laws of nature, but are at 
least in part the consequence of aesthetic principles, which 
have been modified with the progressive development of 
Humanity, he shows that music, like architecture, has evolved 
along essentially distinct lines. He distinguishes three principal 
periods in the art of music the homophonous (univocal) music 
of Antiquity, with which we may connect the existing music 
of Oriental and Asiatic peoples ; the polyphonous music of the 
Middle Ages, multivocal, but as yet having no regard to the 
independent musical significance of harmony, which persisted 
from the Tenth to the Seventeenth Century ; and lastly modern 
or harmonic music, characterized by the independent signi- 
ficance of harmony as such, which originated in the Sixteenth 
Century. The demonstration and deduction of this classification 
for the history of music in all nations is not only one of 
Helmholtz's greatest titles to fame, but is for all time an 
admirable instance of the way in which it is possible to 
connect historical with scientific investigation. 

Even in his earlier works he had shown that the over-tones 
play a great part in musical composition as regards harmony, 
but that the law which determines the melody of harmonious 
combinations of tones is unconscious, since, though the over- 
tones are perceived by the nerves, they do not come under the 
heading of conscious representations, although their consonance 
or dissonance is none the less felt. Harmony and discord 
are but the means to the higher spiritual beauty of music, 
while the melody expresses a movement of which the character 
is plain and obvious to the direct perception of the hearer. 
Since the degrees of the motion must be exactly measurable 
to direct sense-perception by their velocity and magnitude, 
melodious movement is for Helmholtz merely alteration of 
pitch in time, and since the eye may follow a continuous 
movement, while the ear fails to do so, inasmuch as it has 
no capacity for retracing the path correctly and comprehending 
it as a whole, melodic progression must advance by easily 
estimated and definite stages. In the music of all peoples the 
alteration of pitch in melodies takes place by successive steps, 
and not by a continuous progression : in both melodic and 


harmonic music sounds with harmonic over-tones are preferred, 
and to produce a good musical effect there must be a certain 
moderate intensity of the five to six lowest partial tones along 
with a low intensity of the higher partials, which confirms the 
significance of the over-tones for melody also. He sums up 
his results by saying that in music the more or less harmonic 
effect of the intervals in melody and harmony is connected 
with the special sensible phenomena of the over-tones, which 
limit the harmonic intervals the more plainly and exactly in 
proportion as they are just and simple. 

In treating of the difficult question of scales, for which he 
had previously developed the essential principles, he proposes 
the law of the relationship of musical sounds. He defines 
sounds as related in the first degree when they have two 
common partial tones, and in the second degree when they 
are both related in the first degree to the same third tone, 
so that the strength of the relation depends upon the strength 
of the common over-tones, and on this ground of the natural 
relationship of tones to one another he develops the scales, 
although he admits that they were not exclusively derived from 
the law of relations of tone in all epochs, so that it was to some 
extent an arbitrary principle of style. 

Helmholtz's theory of scales, and of harmony and melody, 
threw light on some of the darkest and most difficult points 
of general aesthetics, and showed that these considerations 
were closely allied to the doctrine of sense-perception, i.e. 
to physiology, while the aesthetic analysis of complete musical 
works of art, and the comprehension of the reasons of their 
beauty, seemed to him still to be stopped by apparently insuper- 
able obstacles. He subsequently affirmed on various occasions, 
as in his Goethe Lecture at Weimar, that it was a mistake to 
suppose that any aesthetic investigations could lead to the 
discovery of rules for the guidance of artists. 

'The real difficulty lies in the complexity of the psychical 
motives that here come into play. It is indeed at this point 
that we reach the most interesting part of musical aesthetics . . . 
since in the last resort we are seeking to explain the marvel of 
the great works of art, the expressions and impulses of the 
different psychical temperaments. Attractive, however, as the 
goal may be, I would sooner leave these inquiries, in which 


I feel myself too much of a dilettante, to others, and remain on 
the firm ground of natural science to which I am accustomed/ 

This splendid work, which was intended for the instruction 
of an extensive public in the literary world, was widely read, 
but could be understood as a whole only by a chosen few, 
since not a little training in physics and even mathematics 
was indispensable for its real appreciation. On February 27, 
1864, Helmholtz writes to Ludwig, in answer to his expressions 
of amazement at the stupendous production : 

' I am delighted that you are satisfied with my Sensations of 
Tone, because you are one of the few musical men of science 
who I can hope will succeed in understanding the whole. The 
book appears to me so far to have had rather a succes destime, 
than any real effect in convincing people. Not that I had ever 
cherished any illusions to the contrary. At all events I see 
that it has made an impression, and venture to hope that it will 
gradually win its footing/ 

While his Sensations of Tone was passing through the press, 
Helmholtz had occupied himself almost exclusively with prob- 
lems in physiological optics, and was now engaged, in connexion 
with his work on the horopter, upon a series of ingenious ex- 
periments, and a profound mathematical analysis of the very 
difficult subject of the movements of the eye, and their relation 
to binocular vision, in which he referred all the theorems 
previously discovered by himself and others to one single law, 
that of the simplest orientation in space. He published these 
researches at length in Graefe's Archiv f. Ophthalmologie, with 
the title, 'On the Normal Movements of the Human Eye/ 
After it had been proved by Bonders' experiments that the 
purpose of single binocular vision is not promoted by the move- 
ments of the eye, Helmholtz endeavoured to find an optical 
law for eye-movements, starting from the conviction that an 
organ so well adapted to its functions as the eye, must fulfil 
some optical aim in these movements also. He arrived at this 
law by a further development of that of simplest orientation. He 
connected the proposition that every given position of the line 
of vision corresponds with a given degree of rotation of the 
eyeball, with the question how the latter is retained during its 
movements, or how it has become possible for the eye to 
remain accurate in its orientation when the fixation point in the 


visual field is moving. To obtain a definite answer to this, the 
further question (of great importance to optics and the theory 
of knowledge) must be considered : how it comes about that in 
the movement of the eye, owing to which the light impression 
varies constantly at each point of the retina, our inference 
should be that in spite of this variation of all the luminous im- 
pressions there is no displacement nor alteration of the object, 
but only a motion of the eye; it is obviously sufficient if this 
inference obtains for infinitely small displacements of the eye. 
But in order to convince ourselves that every alteration of the 
image upon all points of the retina collectively depends on the 
altered position of the eye alone, and not on any change of 
the object in the field of vision, we must fulfil the condition in 
virtue of which the transition of any point of the image from 
the retinal fovea to a definite point on the retina at infinitesimal 
distance, can only occur by rotation round a given axis, which 
is unalterable relatively to the eye. 

It follows from the law of simplest orientation and from the 
well-known theorem of mechanics, according to which the 
axial directions of infinitely small revolutions are compounded 
by the law of the parallelogram of forces, that the movement 
of the point of fixation to any second point of the visual field 
at infinitely small distance, must be caused by its revolution 
round an axis lying in a given plane, which is unalterable in 
relation to the eye. Now since the axes of rotation for all the 
movements which occur lie in one plane, no infinitely small 
rotation of the eye can produce a rotation of the same round 
the line perpendicular to this axial plane, which Helmholtz terms 
the atropic line of the eye. But since rotations of finite mag- 
nitude can obviously not be compounded by this mechanical 
law, the necessary condition for the maintenance of orientation 
in the visual field is not altogether fulfilled in the movements of 
the eye. Hence we must look for a law of eye-movements 
which makes the sum of all the deviations from this principle 
a minimum. 

Helmholtz now arrived quite unexpectedly at the law which 
Listing had already expressed without giving any reason for it. 

If we call that position of the eye, from which all infinitely 
small movements of the eye occur without rotation round the 
visual line, the primary position, and all others secondary posi- 


tions, then the position of the eye may be found at any given 
secondary position if it is moved from the primary to the 
secondary position by rotation round an axis, which is per- 
pendicular to the primary and secondary directions of the line 
of vision. The great importance of this axiom only became 
apparent through the interpretation given by Helmholtz of 
Listing's law as the solution for a minimum of the given 

In the closing words of this fundamental paper he says, 
i I therefore believe that the law of the movements of the eye, 
as explained above, is acquired by the use of the eyes, in which 
we are continually proving the need of the most exact orienta- 
tion possible, and that the deduction I have made from this 
need is in the last resort the origin of the law. We should 
expect that the development of the muscle would eventually 
enable these movements of the eye, as required by the need 
of orientation, to be effected with the least possible exertion. 
The movements of the eye are controlled by the habit arising 
from the need of orientation, and I do not see the necessity of 
seeking for anatomical contrivances to account for the law of 
these movements/ 

The fatigues of his year of office, his lectures, the laboratory, 
and above all his unbroken scientific work had so affected 
Helmholtz, that his physician Friedreich urged him at the 
beginning of the summer holidays to travel, in order to re- 
cuperate himself as soon as possible. On August 29, 1863, 
he writes from Heiden (Appenzell) to Bonders: 'As last year's 
cure at Kissingen was not much good, Friedreich ordered me 
this summer to drink whey, which I have been doing here at 
Heiden, and am now going off to the mountains with my 
colleague Bunsen. I am to meet him on September 3 at 
Amsteg, and we intend to go round about the Gotthard to 
Disentis, Airolo, the Tosa Falls, and the Eggischhorn. It is 
a sad moment when a man is first compelled to become a 
hypochondriac, and to pay so much attention to his health/ 

The scientific work of the ensuing winter was again devoted 
entirely to the Physiological Optics, Part III of which was once 
more to include a number of highly complicated problems ; at 
the same time he was occupied with numerous public lectures. 

'This winter,' he writes on February 27, 1864, to Ludwig, 


' I have had to serve the Public and Mammon, and to treat the 
Conservation of Energy as the milch cow. I have given eight 
lectures on it in Karlsruhe, and am preparing to do the same 
in London at Easter in English. I always look on a journey 
to England as a kind of intellectual " cure ", which shakes one 
out of the comfortable indolence of dear old Germany into 
more active life, and lectures such as I gave there once before 
are a good means of establishing closer working relations with 
the English men of science.' 

In 1863 the families of Helmholtz and Kirchhoff moved into 
the new ' Friedrichsbau ', which for those days was a fine and 
roomy group of buildings, containing laboratories, lecture-rooms, 
and dwellings for the staff. Helmholtz's house was always the 
centre of a delightful society, where plain living and high 
thinking were the order of the day. He had formed close 
friendships with his colleagues, Kirchhoff, Bunsen, and Zeller, 
while the Helmholtz family was in intimate social relations 
with those of von Vangerow, Haeusser, Gervinus, Friedreich, 
Kopp, Wattenbach, and others. Under these improved con- 
ditions he was able to attend more to his children, and personally 
superintended the education of his son Richard, who had entered 
the Heidelberg Gymnasium in 1862. 

The latter writes : ' With regard to the intercourse between 
my father and his children, it was chiefly at meals and out 
walking that we saw him. In bad weather we went by the 
Rohrbacher Landstrasse, otherwise generally to the Wolf's 
Hohle, Gaisberg, Sprung, Philosophenweg, &c. It gave him 
keen pleasure to show us any natural phenomenon ; I shall 
never forget one autumn morning of thick fog, when he saw 
there would be sunshine up above, and took us by the Sprung- 
weg, to show us the rolling, sharply defined sea of mist, with 
only a few spires rising out of it. In the winter of 1862 my 
father taught me to draw with mathematical instruments, and 
in 1863 essayed to teach us the elements of thorough-bass, 
which succeeded very well with my sister at any rate/ 

In the Easter holidays Helmholtz spent some weeks in 
England, staying on the way in Utrecht with his friend 
Bonders, whom he found 'as blooming, affectionate, and 
poetical as ever', and in whose house he passed several very 
pleasant days. 


On March 14 he writes to his wife : ' We went to a smoking- 
concert, i. e. to the rehearsal for the great orchestral concerts, 
where they give certain soli that are left out in the concert 
proper, and which the gentlemen of Utrecht listen to over their 
wine and cigars. I heard the Symphonic Preludes of Lizst, 
which are effective and extraordinary enough, but hardly beau- 
tiful ; the Oberon Overture was very good, and as a piano solo 
in between we had the Variations Serieuses of Mendelssohn, 
in the style of church music, which were very fine, and which 
I recommend you to study. Bonders had been giving public 
lectures here on Acoustics, so that my book on the Sensations 
of Tone is known to every one, even to the musicians. O. Jahn 
could not understand it, but hoped to study it with G., and told 
me he had had an enthusiastic letter about it from Claus 

He went on through Brussels to London, and received a 
warm welcome from his friend Bence Jones. He sent a full 
report of his doings to his wife, some of which are interesting 
enough to transcribe. 

' I have cast myself into the whirlpool of the great Babylon, 
and so far am swimming merrily. After writing to you at the 
Royal Institution, where I waited in vain for Tyndall, I went 
up to see Faraday, who lives there. He was as charming as 
ever, but has given up his lectures, as his memory is failing 
him ; and the general impression that he makes on one is less 
acute than it was formerly. . . . Then I went on to the meeting 
of the Royal Society, where Tyndall was giving an address on 
some new and very ingenious experiments he had made, the 
interpretation of which, however, gave rise to much discussion. 
After arranging with Prof. Stokes to be in Cambridge the 
Friday after Easter I went at eleven o'clock to a party at 
Mr. Gladstone's, the Minister. . . . 

' Yesterday I did more work ; in the morning I wrote part 
of my Croonian Lecture ; at twelve o'clock I met Prof. Tyndall 
at the Royal Institution, to get things together for my first two 
lectures ; for the second of these I have made an original draw- 
ing in water colours, which represents a sunbeam seen from the 
side, and vies with Turner in its clouds and the boldness of 
the colour. . . . On Wednesday I worked in the morning, and 
then went to the College of Surgeons, to see Mr. Huxley, 


Professor of Zoology, who is just now the chief partisan of 
Rationalism against the Biblical view of Science, a most 
intelligent young man, whom I had met before. . . . 

1 Yesterday morning I went to Oxford, and am staying with 
Max Mullen He is a clever young man of the world, whose 
like I have never yet seen in a professor of philology, and grasps 
everything, even the scientific matters with which he is less 
familiar, with extraordinary rapidity. His wife is an English 
lady, who is also most attractive, well-informed, and pretty, so 
that I spent two very pleasant days there. Oxford is probably 
unique of its kind in the world ; its many old, and characteristic- 
ally beautiful, and well-preserved buildings, with trim grass 
lawns and handsome trees, are all stately to a degree, and 
very magnificent. It is quite impossible to picture it at home 
until one has seen it, and I now understand the devotion of an 
Englishman to his University. The system works admirably 
for the education of " gentlemen " but it cannot lead to much 
in science, and it needs an extraordinary interest in science to 
prevent a Fellow from sinking into indolence. My journey to 
Glasgow went off very well. The Thomsons have lately moved 
to live in the University Buildings; formerly they spent more time 
in the country. He takes no holiday at Easter, but his brother 
James, Professor of Engineering at Belfast, and a nephew who 
is a student there, were with him. The former is a level-headed 
fellow, full of good ideas, but cares for nothing except engineer- 
ing, and talks about it ceaselessly all day and all night, so that 
nothing else can be got in when he is present. It is really 
comic to see how both brothers talk at one another, and neither 
listens, and each holds forth about quite different matters. 
But the engineer is the most stubborn, and generally gets 
through with his subject. In the intervals I have seen a 
quantity of new and most ingenious apparatus, and experi- 
ments, of W. Thomson, which made the two days very interest- 
ing. He thinks so rapidly, however, that one has to get at the 
necessary information about the make of the instruments, &c., 
by a long string of questions, which he shies at. How his 
students understand him, without keeping him as strictly to 
the subject as I ventured to do, is a puzzle to me ; still, there 
were numbers of students in the laboratory, hard at work, and 
apparently quite understanding what they were about. 


' Thomson's experiments, however, did for my new hat. He 
had thrown a heavy metal disk into very rapid rotation ; and it 
was revolving on a point. In order to show me how rigid it 
became in its rotation, he hit it with an iron hammer, but the 
disk resented this, and it flew off in one direction, and the iron 
foot on which it was revolving in another, carrying my hat away 
with it and ripping it up. 

1 1 got to Manchester on April 4 ; the Roscoes live outside in 
a charming cottage at the edge of a great park. Roscoe had 
two friends to dinner Mr. Joule, a brewer and the chief dis- 
coverer of the conservation of energy, and his colleague, Clifton, 
a physicist, who were both very pleasant, lively individuals, so 
we spent a most interesting evening. On Sunday morning we 
were alone after breakfast, and boldly planned out new ventures 
in physical chemistry: we discussed the English Universities, 
and were both of the same mind. 

1 Yesterday, in London, I went to see Mr. Graham, the Master 
of the Mint, one of the first English chemists, who took me 
round himself and explained everything to me. I was the most 
interested in Graham's own laboratory, where he showed me 
a quantity of marvellous new experiments, and presented me 
with coins, instruments, and chemicals. Then I went with an 
old Berlin friend to Kensington, to see Prof. Clerk Maxwell, 
the physicist at King's College, a keen mathematician, who 
showed me some fine apparatus for the Theory of Colours which 
I used to work at ; he had invited a colour-blind colleague, on 
whom we experimented.' 

These many-sided interests, and the absorbing work of 
preparing his Croonian and other lectures, were darkened by 
the first shadow of the fatal illness of his son Robert. But his 
wife worded her letters so as to keep Helmholtz from any 
immediate return to Heidelberg, and he hoped that his own 
advice and that of the friendly physicians attending the boy 
might avert the danger. 

On April 14, 1864, Helmholtz gave his Croonian Lecture to 
the Royal Society, 'On the Normal Motions of the Human Eye 
in relation to Binocular Vision/ in which he sketched out his 
conclusions in regard to the horopter, and the movements of 
the eye. 

' It was ten before I had finished the first part of my lecture. 


I broke off and left the tribune. But it was decided, at the 
motion of General Sabine as President, that I should go on 
speaking, and so I held forth on the movements of the human 
eye, and their relation to visual perception, till half-past ten, when 
I had pretty well done. It comforted me, however, to see that 
several gentlemen rose after me, and made some confirmatory 
observations. Sabine proposed a vote of thanks, in which he 
praised my facility in English. I fear it flowed rather like a 
mountain torrent from my lips, but I could hardly speak at all 
at the end/ 

During his four-weeks' stay in England he also delivered six 
popular lectures in London, ' On the Conservation of Energy* ; 
the full report of which was sent to du Bois-Reymond from 
Heidelberg on May 15, with the news that a daughter had 
been born on April 24, who received the names of Ellen Ida 
Elizabeth : 

' I stayed six weeks in England, most of the time in London ; 
during Easter Week I went also to Oxford, Glasgow, and 
Manchester. I saw a great deal that was interesting, and find 
an occasional visit to London both pleasant and inspiring. As 
to the popular lectures at the Royal Institution, I quite agree 
with you that one would have to think a good while before 
undertaking them again. I had no reason to be dissatisfied 
with the apparent results in my own case, for I had a steady 
audience of three hundred, and among them a number of scientific 
men. But the competition of popular lectures in London is so 
great that they are on the verge of degenerating into a mere 
shop-window display. Tyndall, as a matter of fact, has a vast 
talent for popular discourses, and is much appreciated by his 
public. A spirit-rapping medium recently spelt out his celestial 
name, which was " Poet of Science ". ... As I found the general 
opinion to be that your experiments were too subtle to come 
off as a certainty, I took the opportunity of showing a few of 
your fundamental demonstrations at my last lecture.' 

The year 1865 brought Helmholtz a number of honours from 
various sides : but what gratified him more than any of these 
was the fact that a second edition of the Sensations of Tone was 
called for, scarcely two years after its first publication. His 
friend Ludwig was again the first to whom, in February 1865, 
he sent the second edition of his book. While he expressed 



renewed admiration of the author's marvellous genius, Ludwig 
felt that he must protest against the following passage: 'Among 
our great composers, Mozart and Beethoven are only at the 
beginning of the period in which equal temperament pre- 
dominated. Mozart still had opportunities of making extensive 
studies in the composition of songs. He is a master of the sweetest 
melody, wherever he desires it, but in this he is almost the last. 
Beethoven's bold genius took possession of the domain which 
the development of instrumental music brought him; in his 
hands it was the pliant and appropriate tool which he was able 
to manipulate as none else had ever done. But he always 
treated the human voice as a handmaid, and consequently it 
never lavished the highest magic of its melody upon him/ 

Ludwig took umbrage at this view, and on March 30, 1865, 
Helmholtz replies : ' In your last letter from Leipzig you attack 
my remarks on Beethoven. Perhaps I had better not have 
expressed myself merely critically about him, if I did not wish 
to be misunderstood, for I too find him the mightiest and most 
moving of all composers, and I myself play hardly anything but 
Beethoven, when I do play. Had I been speaking about the 
vehicle of musical emotion, I should certainly have placed him 
above all others. I was, however, talking exclusively of melody, 
and the fine artistic beauty of the flow of harmony, and there 
I do hold Mozart to be the first, even if he does not affect us so 
powerfully. Speaking generally, as one grows older, and bears 
more scars within one's breast, one ceases to feel that emotion is 
really the greatest thing in art.' 

The objections urged by Helmholtz's gifted friend Fechner, in 
a letter of June 6, 1869, were more serious and of greater import : 

1 You explain the melodic no less than the harmonic relations 
of tones by the presence of over-tones, and if I grasp your 
meaning rightly, though I am not quite sure about this, in the 
absence of over-tones the difference between the pitch of two 
notes would be like the difference between their intensity, so 
that we should lose all the characteristic and gradual degrees of 
relationship and disparity between the tones which are known 
to us as melodic. An octave appears so like the fundamental, 
because the latter contains all the partial tones of the octave in 
its over-tones ; the fifth is less similar, because the coincidence 
in this respect is less perfect, and so on. This idea is so simple, 


and fits in with the facts of tone-relations in ordinary instru- 
ments so well, that the problem seems to be solved by it. But 
I am not prepared to admit that this relation of the octave to the 
fundamental is the cause of the melodic relation of the tones, which 
appear in all cases, just as plainly in the tones of rods, plates, 
and bells, as in those of stringed instruments and the human 
voice, notwithstanding that in instruments of this kind, accord- 
ing to your observations, the over-tones may be musically 
speaking neglected, or, if they were taken into consideration, 
would necessarily produce quite different melodic relations. . . .' 

Helmholtz replied on June 3 : 

' (i) A weak accompaniment of harmonic over-tones is inevit- 
ably present, at least in all strong simple tones. They arise from 
the same law as combination tones, partly accidentally outside 
the ear, partly in regular series within the ear, as often as the 
vibrations become so great that the elastic forces are no longer 
exactly proportional to the displacements. I proved by my 
work on the mechanism of the auditory apparatus (Pfl. Arch.f. 
Phys. Vol. I), that the conditions for this are especially favour- 
able inside the ear, so that there may even be a clashing of 
tones between the malleus and incus. 

' I did not bring this out strongly enough in the first edition of 
Sensations of Tone, and have made it plainer in the second 
edition, the MS. of which has just gone to Vieweg, and will 
shortly be in the printers' hands. This unmistakably gives the 
series of harmonic over-tones a new subjective meaning. At 
the outset I only characterized them as the series which 
emerges in all exactly periodic vibrations that excite persistent 
and equal sensations. 

'(2) I believe, however, that a melody can be recognized, when 
it is given out in weak simple tones, without evoking over-tones 
of perceptible strength. But on the other hand, I do not believe 
that music would ever have been discovered if the relation of 
tones and over-tones had always been lacking, as it is in colour. 
Pitch of tone and intervals can be remembered and recognized, 
even where the distinguishing marks, i.e. the over-tones, which 
give the specific distinction from the adjacent tones, and on 
which the immediate sensory recognition of their proper value 
rests, are wanting. Compare this with the case in which we 
see an object that is usually red as white ... in the latter case we 



have an entirely new sensation that is otherwise wanting. But 
if the over-tones are absent in a melodic interval, we have no 
new perception ; the only result is that a part of the sensation to 
which we are accustomed in greater or less intensity, which 
makes us more certain about the magnitude of the interval than 
our memory of it, is now wanting. But nothing new or 
unfamiliar appears in its stead. I might rather compare it with 
the binocular vision of an object, and that of a picture. The 
former, like a melody with over-tones, gives sensational data, 
which enable us to judge very definitely of the dimensions of 
depth; the picture, which does not give these, is like the melody 
without over-tones; but if we know the object well, we can form 
a lively conception of it, and under many conditions it is really 
hard to determine without direct experiment whether binocular 
vision actually assists our perception of depth or no. The 
essential point seems to me to be, that melody is the image of a 
movement, and that it is possible to measure the intervals by 
direct sense-perception. If we are able from memory to recog- 
nize any given interval, then in particular cases we can forgo 
the standards of measurement, without being altogether astray, 
even if the impression of the melody takes on somewhat of the 
weakness of the memory-image. On the other hand, I must say 
from my own experience, that tones with unharmonious partials 
(unless these be very weak, or very remote from the over-tone), 
give quite false melodies, which, however, can be recognized in 
memory as copies of the true melody. The principle you 
require in order to obviate the undiiferentiated fusion of the 
over-tones, and also to give the relation of tones in melody, is, 
I think, provided by the fact (or hypothesis) that tones of 
different pitch affect different nerve-fibres.' 

Helmholtz's researches in physiological optics were only 
interrupted for a very short time by his work on the formation 
of ice and glaciers. On February 24, 1865, he gave a lecture to 
the Nat. Hist. Med. Verein at Heidelberg, 'On some Properties 
of Ice/ in which he discussed the origin of the phenomenon 
known as the regelation of ice, while in the same month in 
a popular lecture 'Ice and Glaciers', which opened with a 
brilliant description of the glacier world, he went more closely 
into the question then so much discussed, of the movement of 


After the lecture du Bois writes to him on June 8, 1866 : 

'You see that I am somewhat rabid as usual, when I cannot 
hammer out my own work, and see others shaking one fine thing 
after another out of their lap. Our good Tyndall will be not 
a little astonished to find you a master of glacier problems also.' 

In the lecture above referred to, which was published in the 
Philosophical Magazine for the following year, with the title 'On 
the Regelation of Ice', Helmholtz confirmed James Thomson's 
explanation of the phenomenon of the regelation of ice at zero, 
when two pieces of ice if pressed together freeze again and form 
one mass. He proves that the freezing-point is lowered with 
increased pressure, and points out, as against Faraday, that time 
is an essential factor in this phenomenon; he shows by a number 
of experiments that with strong pressure two pieces of ice can 
be united into one block by the freezing water at their surface 
of contact, while under weaker pressure it is necessary to wait 
longer, and the parts are correspondingly easier to separate 
again. He finds the plasticity of ice most marked in that which 
has been welded together by great pressure from snow, while the 
regular, crystalline ice can indeed be united by regelation, but 
only forms a mass of irregular pieces. 

By applying these observations to glaciers, Helmholtz was 
able to explain the well-known and never properly interpreted 
phenomenon of the flow of ice in glaciers as a viscous mass. 
The ice mass of a glacier is everywhere permeated with runnels 
of water, so that its internal temperature is always at freezing- 
point, seeing that the water would freeze if the temperature 
were lower, and the ice would melt if it were higher. But a 
mixture of ice and water grows colder and colder in proportion 
to the pressure exerted upon it ; as no heat is withdrawn, the 
free heat must become latent, and the ice in the mixture melts. 
The pressure exerted by the glacier mass, which forces the 
water out of the cracks, will, in Helmholtz's opinion, cause the 
compressed ice (since its melting-point is lowered by pressure, 
while the freezing-point of the non-compressed water is not 
lowered) to give ice which is colder than o, in contact with 
water at o. There will accordingly be constant congelation of 
the compressed ice-water round it, with the formation of new 
ice, while a portion of what is compressed melts simultaneously, 
and the ice itself moves as a viscous fluid mass. 


The explanation which Helmholtz gave in his popular lecture, 
' Ice and Glaciers/ of that mysterious and misinterpreted pheno- 
menon the Fohn is very interesting, and the foundation of the 
whole theory of rainfall. When the warm air of the Medi- 
terranean is driven northwards by the south wind, a portion of it 
is compelled to ascend the great mountain wall of the Alps. In 
consequence of the diminished pressure of the air it expands by 
about half its volume, is considerably cooled in temperature, and 
at the same time deposits the best part of its moisture as 
snow or rain. When the same air afterwards descends to the 
valleys and plains on the north side of the mountains as the 
Fohn wind, it is again condensed and grows warmer: thus 
the same current of air that is warm in the plains on either side 
of the mountains can be bitingly cold upon the heights, and 
deposit snow there, while it is insufferably hot in the plains. 

The year 1865 brought a change in Helmholtz's domestic 
relations; his mother-in-law, Frau von Velten, took up her 
permanent abode at Dahlem, and during the long period that 
elapsed before her death in 1881, only once returned to Heidel- 
berg, when, in 1874, she came to stay with her married grand- 
daughter Kathe. 

In the autumn vacation Helmholtz went as usual to Switzer- 
land, where long and arduous excursions refreshed him in mind 
and body: but he was soon recalled to Heidelberg by disquieting 
accounts of the state of his son Robert. His wife again en- 
deavoured to keep him away from home as long as possible : 

1 Enjoy your journey thoroughly, and get your poor head 
well, dear Hermann, so that we may both be fresh and vigorous, 
if we are threatened with new illness. We must keep our 
courage up, if we are to pull through. Don't imagine that 
I am giving way ; I am trying to keep well and cheerful, and 
scold myself for my faint-heartedness, when I think of you and 
your hatred of all exaggeration. . . .' 

But after the diagnosis of the physicians Helmholtz could 
cherish no further illusions as to the nature of his child's illness, 
and returned direct from Geneva, stopping only a few hours in 
Freiburg, to listen once more to the strains of the organ which 
he had admired so much in bygone years : 

'The organ is truly wonderful, from the point of view of 
acoustics even more than from that of music. I confess that 


till now I had no suspicion of the effects that could be produced 
by such an instrument, in regard to mass and power, as well as 
to variety of timbre/ 

Part III of Physiological Optics was to appear in the next 
year, 1866, and Helmholtz was constrained to hasten the publica- 
tion of this last portion, in order not again to omit a mass of 
new results by other workers, in connexion with his own 
researches, as had occurred with the first two separately 
published sections. Great inconvenience had arisen from the 
fact of the work being published as a part of Karsten's Allgemeine 
Encydopddie der Physik. 

Du Bois writes : ' I never open your Optics without getting 
angry at your having let yourself in for fathering the still-born 
projects of Karsten, which in the first place damaged the 
circulation of the book, and in the second, compelled you to use 
a form that by no means makes it more lucid, or easier to under- 
stand. The colossal pages of the closest print, crammed with 
the most abstruse matter, give one no resting-place, and any- 
thing you have written hardly needs to appear in small print.' 

It was not until the work had appeared independently as the 
Handbook of Physiological Optics that du Bois was able to write 
to him on April 25, 1867 : 

' The book will only produce the greatest part of its effect 
now, when it conies into the market freely, as a whole. In my 
own laboratory, for example, the young people like Rosenthal 
and Hermann hardly know it at all, since it is by no means the 
sort of book one can work through in the time for which one 
can decently borrow it/ 

Helmholtz found it a severe task to incorporate the new 
matter, and to utilize it for Part III, and for his full Bibliography. 

' How delightful the state of a learned theologian, jurist, or 
historian must be, who spends his whole life in bringing out 
new editions of the same book with minute alterations, while we 
poor men of science cannot get one work ready before the 
beginning of it is already out of date/ he complains to Bonders: 
but he does not falter ; and most of the facts and theories that 
had become known were submitted to a searching criticism. 

The whole of the year 1865 was thus devoted to the pre- 
paration of Part III of Physiological Optics, a. gigantic task that 
tried his health severely. His persistent attacks of migraine 


obliged him to go to Engelberg for three weeks in the autumn, 
to drink whey. After a sharp walking tour through the Mont 
Blanc district the attacks became less frequent and less severe, 
but when he resumed his work, and more especially during 
the epistemological portions of it, his health once more suffered 
severely; 'the attacks still make all occupation impossible, 
each onset robs me of twenty-four hours' work.' 

His condition obliged him to take a fortnight's rest again 
during the Easter holidays, and he went to Paris, where he 
found a hospitable and affectionate welcome from his wife's 
uncle Julius von Mohl, the famous Orientalist. A short break 
in the journey was devoted to Strassburg, where he ascended 
the gallery of the Cathedral Tower, and gloried in the bold 
stonework. ' I looked into old Ulrich's riddle about the square 
and the octagon; the solution is very simple.' He spent the 
first evening in Mohl's house, ' peacefully, and I hope with 
mutual satisfaction.' He sent full and interesting accounts of his 
daily doings by letter to his wife, who was familiar with Paris and 
all its striking personalities from her long stay with her aunt. 

' ... At eleven o'clock I had to be back for a breakfast with 
M. Hermite and the mathematician Prof. Smith from Oxford. 
It was said in course of conversation that there had been some 
notion of inviting me to go to Oxford as Professor of Physics. 
However, they could not offer more than 700 salary, which of 
course is more than we get in Heidelberg, but hardly enough to 
live comfortably in England. ... So I think Prof. Max Muller 
was right to say he could tell them decidedly that I should not 
accept it. ... M. Hermite was very complimentary to me, and 
introduced me to a^M. Grandeau, who came to welcome me, and 
escort me to the Ecole Normale, where the chemist, St. Claire 
Deville, a rising man of the first rank, received me very warmly. 
He took me into the Physical Department, where we had to 
pass through a class-room in which a lesson was being given in 
physics. I was presented to the scholars, and received with 
rounds of applause, since they are all, at least so I was told, well 
acquainted with my acoustical theories. . . . 

1 Grandeau and Laugel took me to the first of organ-builders, 
Cavallie-Col, who showed us his workshop, and then accom- 
panied us to the Church of St. Sulpice, to inspect the largest 
organ in Europe, built by him, but on account of the service we 


were unable to do this properly ; we are going to see it more 
thoroughly this afternoon. What I saw interested me greatly. 
M. Cavallie, who has raised himself from a working man to be a 
master-hand, is a most intelligent and original person. ... At the 
concert at the Conservatoire we had a Symphony by Haydn, a 
piece from Beethoven's Ballet of Prometheus, and the whole of 
the music from the Midsummer Nights Dream, as well as 
a chorus of Bach, and Handel's Hallelujah Chorus. One hears 
better choral singing in Germany, but the perfection of the 
orchestra is unique of its kind. The oboes in Haydn's 
Symphony sounded like a gentle zephyr; everything was in 
perfect tune, including the high opening chords of the Mendels- 
sohn Overture, that are repeated at the end, and generally 
sound out of tune. The Prometheus was the most enchanting 
melody, with the horns predominating. This concert, after the 
Venus of Milo, was the second thing of purest beauty that life 
can give. ... I went with MM. Cavallie and Bussy to the house 
of a harmonium-maker, Mustel, who wanted to show me his 
latest invention, a tuning-fork piano, with sustained tones. This 
confirmed my theoretical assumptions, and produced no special 
effect, which fact, however, is of some importance for my theory. 
The advance in the construction of the harmonium was very 
striking ; it was like a very perfect and easily responding piano, 
with every kind of contrivance in the mechanism, for bringing 
out the treble parts. I used this opportunity to preach the un- 
equal temperament for the organ, and M. Cavallie seemed in- 
clined to make the experiment. . . . 

'On Wednesday I went first to M. Regnault's lecture at the 
College de France. I was in hopes of seeing him experiment, 
since he is one of our most famous experimenters, but he did 
not ; he showed me his instruments, a collection renowned in 
the history of physics. 

' I went to the cole Normale, where MM. Grandeau and 
Deville had invited me, to visit the latter, and to see Herr 
Konig's instruments. To my surprise M. Duruy, the Minister 
of Instruction, also turned up with a member of his council, and 
they begged me to give him a lecture on the analysis of the 
vowel tones, which I did. . . .' 

Returned from Paris, Helmholtz at once went back to the 
completion of his Physiological Optics, but was sorely disturbed 


in his task, which required much concentration, by the troubles 
in South Germany, consequent on the war between Prussia 
and Austria. A Prussian by birth, and devoted with his whole 
soul to his own father-land, he was greatly distressed by the 
position which Baden occupied, in consequence of the peculiar 
development of affairs. Helmholtz never courted extremes in 
religious and political matters ; just as by education and con- 
viction he was religious in the noblest sense, but never ecclesi- 
astical in the orthodox signification, so while he had never taken 
an active part in politics, he had been from his youth up, owing 
to the traditions of his parents' house, and to his own clear and 
deliberate judgement, a Liberal in the best sense of the word, 
keeping clear of reactionary passions and radical agitations. 
The letters give us no indication of the political views which 
he professed in his youth during the heroic and stormy period 
of 1848 to 1849 ; the vicinity of his father enabled him to dis- 
cuss the political situation by word of mouth with that old 
soldier of the Freiheitskampf, and his post as military surgeon 
naturally imposed upon him the greatest reserve in letters to 
his friends. But his youthful mind, inspired for all that was 
good and noble, was deeply shaken by the struggle of the 
nations for political unity and freedom. 

' I know as an absolute truth/ writes his sister-in-law, ' that 
he sympathized in the conflict almost too passionately for the 
balance of his nature. On the day following March 18 he was 
in a passion of excitement, of which a little trait gave striking 
illustration. He came straight to us from Berlin on one of 
those days, and when I showed him my two-weeks' old infant 
for the first time, he beamed, and drew a red, black, and gold 
cockade out of his waistcoat pocket, fixed it on to the child's 
little cap, and congratulated the "citizen mother, on her first-born 
in freedom ".' The quip was a sign of his passionate sympathy 
with the growing spirit of nationality. At a later period he 
followed the debates in the Paulskirche, the sad decline of the 
movement, and its final decay and extinction, with the com- 
pletest and most heart-felt sympathy. 

So in the tumult of the year 1866, he was in his enthusiasm 
for the unity and freedom of Germany entirely on the side 
of Prussia, in which he recognized the centre of power to which 
all must gravitate, if external equilibrium were to be maintained, 


and in this he was not deceived. His wife too, although she 
was of South German extraction, embraced the cause of Prussia 
with enthusiasm : on July 12 she writes to her mother, ' Every 
right-minded person is Prussian, since Austria has made this 
French alliance.' 

Helmholtz's dearest wishes were fulfilled sooner than could 
have been anticipated, and he returned with fresh zest and 
courage to his great work on visual perception, which formed 
part of the Physiological Optics. 

His earliest observations in optics and acoustics had taught 
him that besides the sensations of the nervous apparatus there 
enters into our sense-perceptions the further factor of a specific 
psychical activity, which co-operates in the representation of 
the external object that has excited our sensation. In his 
lecture on Kant he had already, in agreement with Lotze, 
treated the impressions made upon our sensory nerves as 
being merely the signs of certain external objects ; holding that 
correct inferences from the sensations to the corresponding 
objects had arisen through experience. His observations on the 
blind spot in the eye, on over-tones, &c., now brought in a new 
point the recognition of a law that is valid for all our sense- 
perceptions, viz. that we attend to our sensations only in so 
far as they enable us to recognize external objects, while we 
do not analyse such sensations as have no direct relation with 
external objects, until we begin to investigate our impressions 
scientifically. Helmholtz now went on to the difficult problem 
of the nature of the correspondence between the percept and 
its object, in other words, what kind of truth are we to ascribe 
to our ideas and perceptions ? 

Just as the sensations of the eye, ear, and tactile sensibility 
are intrinsically so different that no comparison in regard to 
quality and intensity can be made between those of different 
senses (this is called a difference in the mode of the sensation, 
while the disparity between homogeneous sensations is 
described as one of quality) : so the same is the case if a com- 
parison between the percepts of psychical states (which Kant 
refers to a special sense, the innate or intuitive) and those 
of the eye or ear be attempted. Yet in spite of many differences 
they have one thing in common, that the percepts of the 
internal as of the external senses are arranged in time-sequence 


by a persistent activity of memory, which makes it possible 
to observe and recognize the regular repetitions of such 
sequences of homogeneous percepts. Hence, even if the 
qualities of sensation are merely intuitional forms, the sensations 
themselves being only signs, the specific nature of which 
depends entirely upon our organization, they still are signs 
of something that exists or is happening, and thereby sup- 
ply us with the law of this happening. Conformity in the 
phenomenal may thus be accepted as unequivocal and actual. 
If we give the name of substance to that which remains 
identical, independent of all other things, through all changes 
of time, and if on the other hand the persistent ratio between 
alterable magnitudes is the law that binds them together, then 
this law is all that we can perceive directly, while the concept 
of substance must for ever remain problematical. * It is only 
the relations of time, space, equality, and those derived from 
them, namely those of number, magnitude, and conformity, 
in brief mathematical relations, which are common to both the 
outer and the inner world, and in these we can actually strive 
for complete correspondence of the percepts with the things 

From this philosophical basis Helmholtz proceeds to develop 
his Theory of Space-Perception, constructed from his con- 
clusions in physiological optics. To the nativistic theory of 
space-perception as enunciated by Johannes Muller, Helmholtz 
opposes his empirical theory of vision. According to Muller 
the retina itself is sensible in its spatial extension, this intuition 
of space is innate, and the impressions excited from without 
are referred directly to the corresponding points on the spatial 
image of the organ : on Helmholtz's view our sensations are no 
more than signs for external things and processes, which we 
must learn to interpret by experience and practice. According 
to the empiricist theory we have to learn the significance of 
the local signs of sensation (such as are excited by the same 
colour on different points of the retina), in reference to the 
external world ; while on the nativistic theory, these local signs 
are the direct intuition of spatial differences both in kind and 
in degree. The theory of the stereoscope, simple vision with 
both eyes, and a long series of other optical phenomena, give 
1 a remarkable confirmation of the assumption of the empiricist 


theory, that spatial separation is, generally speaking, to be 
predicated only of such sensations as can be separated by 
actual movement from each other '. We learn to interpret the 
signs, by comparing them with the results of our movements 
and with the changes we can produce by means of the latter in 
the external world. According to Helmholtz the only difference 
between the inferences of the logicians and those of induction 
(the results of which become evident in the percepts of the 
external world as derived from experience) is, that the former 
can be expressed in words, while in the latter words are 
replaced by memory images of sensations. This region of 
the conceptional faculty combines only those sensory im- 
pressions which are not capable of expression in words ; ' in 
Germany we term this Cognition (das Kennen)' 

The 1868 lectures on ' Recent Progress in the Theory of 
Vision ' were an amplification of certain points in Physiological 
Optics, in which Helmholtz with his accustomed brilliancy and 
perspicacity gathered up some details of general interest, which 
would have been overlooked in the larger work. 

In describing the defects of the optical apparatus of the eye, 
he insists (in conformity with his empiricist attitude) ' that it is 
not the mechanical perfection of the sensory instrument that 
creates these marvellously true and exact impressions', and 
after discussing visual sensation, and the theory of colour, after- 
images, and contrast, he says : ' Whatever inexactness and 
incompleteness we may have found in the optical apparatus 
and retinal image, is as nothing compared with the incon- 
gruences which we encounter in the region of sensation. We 
are tempted to believe that Nature had advisedly perpetrated 
the wildest contradictions, and was determined to destroy all 
dreams of a pre-established harmony between the outer and 
inner world.' 

In his Commemorative Lecture on Helmholtz, du Bois- 
Reymond remarks that 'just as the principle of the conservation 
of energy has been a safe clue to Helmholtz's train of thought 
in the preceding period, so in the later part we have a similar 
guide. The fundamental principle of these researches is the 
empiricist attitude, which Helmholtz favours in preference to 
the nativistic, which he rejected. This is the same contrast 
that obtained in the sixteenth century between Leibniz's pre- 


established harmony and Locke's sensualism, and to which 
Kant gave a decided turn in favour of the former doctrine '. 

Even at this time, and far more forcibly later, Helmholtz 
declares himself in opposition to Kant, who affirmed that the 
law of causation, as well as the intuition of time, and of tri- 
dimensional space with its geometrical axioms, were of tran- 
scendental origin, a priori ideas, innate in us. At the same 
time Helmholtz was fully aware that his empiricist theory was, 
and would remain, no more than a hypothesis. He believed, 
however, that hypotheses are essential to action, and that every 
man must choose for himself according to his own ethical or 
aesthetic sense ; experiment alone, in which ' the chain of 
causes runs through our self-consciousness ', can be regarded 
critically, while observation, a process that ensues without our 
connivance, may be modified by physical and psychical causes. 
He was well aware that his hypothesis would meet with much 
contradiction, and was not surprised when du Bois wrote to 
him on April 28, 1868 : 

1 The great objection to the strict empiricist attitude always 
seems to me to be that it ought to be possible to carry it 
through consistently, which, as you yourself admit, is not the 
case ; for if it is innate in the calf to go after the smell of the 
udder, why should not all its faculties be innate ? It appears 
to me that so much nativism which one cannot get rid of is 
still left, that a handful more or less does not much matter. In 
regard to motion, for example, there are countless complicated 
cases in which we cannot get rid of it. You will say that one 
can at least try to limit it as far as possible, and that I do not 
deny. I must confess that on these points my craving for 
causality is capable of greater resignation than yours/ 

Helmholtz subsequently answered all these objections in his 
lecture 4 On the Facts of Perception ', as follows : 

1 To a great number of physiologists, whose views we might 
term nativistic, in contrast to the empiricist which I have 
myself endeavoured to defend, the conception of an acquired 
knowledge of the field of vision appears untenable, because 
they do not clearly realize what is so plain in the case of 
speech, namely, how much the accumulated impressions of 
memory can do. A number of different experiments have 
accordingly been made with the intention of referring at 


least some proportion of the visual perceptions to an innate 
mechanism, in the sense that definite sensations are supposed 
to set free definite, already formed, spatial conceptions. But 
the nativistic hypotheses in the first place do not explain 
anything; they only assume that the fact to be explained 
exists ; in the second place, the assumption made by all 
nativistic theories, to wit, that already formed representations 
of objects are brought out by the organic mechanism, is far 
more dubious than the assumption of the empiricist theory 
that it is the raw material of sensations alone which depends 
on external conditions, while all ideas have to be formed from 
that in accordance with the laws of thought. In the third 
place the nativistic assumptions are unnecessary. 1 

Notwithstanding these arguments Helmholtz met with 
little sympathy even from the best and most sympathetic of 
the physiologists, who were not only, like du Bois, biased by 
a certain nativistic tendency, which made them averse to the 
consistent development of the empiricist hypothesis, but further 
objected to it on the ground that it did not seem to them 
compatible with the existence of sensory illusions. Bonders 
objected to Helmholtz's hypothesis from the same point of 
view, and received the following answer, dated May 26, 1868 : 

' I regard the publication of careful observations on the 
mode of vision of people who squint as very desirable and 
important (provided it is borne in mind that, from the nature 
of the thing, this may possibly not be constant). The state- 
ments we have hitherto had about it seem to me to be influenced 
throughout by preconceived ideas. And although for the time 
being you are still in the clutches of the nativistic theory, 
I have sufficient confidence in you (witness your experiments 
on stereoscopy with electric illumination) to believe that you 
set facts above theory. For the rest I am well aware that my 
empiricist theory is at present merely one of the possible 
aspects of the matter, and that facts may shortly be discovered 
that will render it impossible : when that happens it will have 
had its uses, and may disappear. Not indeed that I think this 
very probable as regards ideas and percepts. As to motor 
impulses, the case is rather different. Some such are truly and 
indisputably present in the new-born as much as in the grown 
person, and the possibility that certain combinations of move- 


ments are a priori easier than others is conceivable ; this may 
be the case with the eye-movements also. But to speak of 
compulsion in these instances is beside the mark. All that 
I desire is proof that there is a natural disposition in favour of 
these movements. 

' With all this confounded trafficking in hypotheses about 
invisible nervous associations, with all manner of inconceivable 
properties, which have checked the progress of the physiology of 
the central nervous system for so many years, I do believe it 
to be most important to open people's eyes to the number of 
superfluous hypotheses which they are making, and would 
rather exaggerate the opposite view, if need be, than proceed 
along these false lines. Reflex motion may at present be 
defined as everything in physiology which we can't explain. 
These are the disadvantages of an exaggerated materialistic 
metaphysic, from which people must be brought back to facts/ 

Precisely because Helmholtz wanted to weaken the objections 
raised to his hypothesis on account of the existence of sensory 
illusions, he laid down as a rule in all illusions, that we always 
think we see such objects before us as would have to be 
present in order to bring about the same retinal images under 
normal conditions of observation ; and he chose the name of 
unconscious inference for these processes, in which words are 
replaced by sensations and memory images, although these 
involve the same intellectual activity as the ordinary inferences. 
Even the supporters of the nativistic theory must, he insists, 
admit that the peculiar completeness and refinement of sensory 
intuition depend upon experience. 

When Helmholtz was pursuing his acoustic researches upon 
the aesthetic side of sensations of tone, he proved that the 
forms of musical configuration depend more strictly than in the 
case of any other art upon the nature and idiosyncracies of our 
sensations. And in the lectures which he gave at Berlin, 
Dusseldorf, and Cologne, 1871-1873, on The Relation of Optics 
to Painting, he succeeded in establishing for painting (in 
which the nature of the material to be employed and of the 
objects to be represented have far more influence, though here 
too the specific sensibility of the visual organ is not without 
significance) that it is not only profitable for physiological 
optics that attentive consideration should be given to the works 


of the great masters, but, further, that the investigation of the 
laws of sensation and perception are useful to the theory of 
art, and to its right application. 

Helmholtz came by the circuitous route of the physiology 
of the senses to his artistic studies, and compares himself 
'with a traveller who has made his way into the lovely land 
of art, across a sterile, stony mountain barrier, but in so doing 
reached many points which gave him good views of the country 
below him '. He does not conceive it to be his task to furnish 
instructions by which the artist is to work, but would endeavour 
to understand the problems which he must solve, and the ways 
in which he attempts to arrive at his goal ; ' the artist cannot 
transcribe Nature, he must translate her/ 

But this translation is effected not by any conscious logical 
activity of the mind, but with the help of the most refined and 
accurate observation of sensory impressions, and of a specially 
exact memory for retaining these impressions, which (since what 
he can fix by hasty sketches at the moment is but scanty) must 
be more exact in regard to the details of the phenomenon than 
it is for the majority of people. In his Sensations of Tone 
Helmholtz had pointed out the extraordinary development of 
memory in musicians, who, without any notes before them, can 
execute countless compositions on their instruments ; and it is 
in the relative importance assigned to memory that he places 
the main divergence in the paths of investigator and artist, as 
he says in his splendid Goethe Lecture at Weimar : 

1 That which we can express in words can be fixed in writing ; 
it is only the first creative idea that must always be formed and 
emerge in the same way in both modes of activity, and this in 
the first instance can only happen after a fashion analogous to 
artistic intuition as the apprehension of a new law of nature/ 

The first and greatest difficulty for the painter is to enable his 
spectator to estimate the depth of the objects represented in his 
painting, since the binocular vision of solid objects is here 
wanting. To this end he has to make a careful selection in 
arranging the perspective objects, their position and aspect, their 
light and shade ; above all, aerial perspective, or the artistic 
representation of the opacity of the air, will be his great help in 
indicating exactly the relative distances by the greater or less 
predominance of the colour of the air over the colour of the 




objects represented. In addition to the form of the objects, 
degrees of brightness have to be considered. Since it is im- 
possible for the painter to depict the light and shade in a 
picture as they are presented in nature, he can only strive by 
his colours to produce the same impression upon the eye of the 
spectator. He does this unconsciously in virtue of Fechner's 
psycho-physical law, that within very wide limits of brightness 
differences of light-intensity, if they form an equal fraction of 
the total quantity of light compared, are equally distinct and 
therefore appear equal in sensation. The ratio of brightness is 
our only sensory sign of the lighter or darker coloration of 
bodies, and the painter therefore need but select in his colours 
the same ratio of brightness as is exhibited by the bodies them- 
selves. But when the mean limits of Fechner's law are trans- 
gressed, then with lessened illumination the darker objects 
become more like the darkest, and with greater illumination the 
brighter objects become more like the brightest, and so in 
representing glowing sunshine the painter is obliged to make 
all objects almost equally bright, while in moonlight only the 
very brightest objects can be bright, and the others must be 
unrecognizably dark. 

But the question of degrees of brightness is complicated by 
colour differences, since the scale of intensity of sensation is 
different for different colours. The phenomena of dazzle are 
weaker with increased brightness for red than for blue, and 
Helmholtz observed that even with a small proportional increase 
of intensity this was especially striking in the red and violet 
colours of the spectrum, so that with mixed colours very bright 
white appeared yellowish, dull white bluish in colour. The 
painter accordingly, to reproduce the impression of sunlit white 
with faint colours, must by an admixture of yellow in his white 
make this colour preponderate just as it would in actually 
brighter white. 

Lastly, the phenomena of contrast also come under considera- 
tion. These cannot be represented in paintings as they are in 
the real objects, since the colours of pictures are not as bright 
and intensely luminous as they are in reality. The painter 
accordingly must represent an evenly illuminated surface as 
brighter where it is contiguous with a darker part, and darker 
where it impinges on what is bright. The artist again has to 


make an objective imitation of the subjective phenomena of the 
eye, such as the irradiation caused by its transparent but not 
perfectly clear media ; while most of all, it is the harmony of 
colours that comes into question, since the reciprocal relations 
of the colours of a picture have much to do with the aesthetic 
enjoyment of it, and even strong colours can convey expression 
(in the artistic sense) of the most delicate alteration or illumi- 

i What is the effect to be produced by a work of art, using 
this word in its highest sense ? It should excite and arrest our 
attention, awaken a rich train of sleeping associations and cor- 
related feelings into activity, and direct them to a common end, 
in order to unite all the features of an ideal type which are 
lying scattered in our memory in isolated fragments, overgrown 
by a confused and fortuitous mass of ideas into a vivid con- 
ception. We can only explain the frequent preponderance of 
art over reality in the human mind, by saying that impressions 
of the latter are always mingled with something that disturbs, 
distracts, and injures us, while in art the elements which are to 
produce the desired impression are gathered together and 
allowed to act without restraint. The force of the impression 
will, however, undoubtedly be stronger in proportion to the 
depth, refinement, and truth to nature of the sensory impression 
which is to arouse the series of images and the emotions 
associated therewith. Its effect must be prompt, certain, un- 
equivocal, and exact, if it is to call up a vivid and powerful 

After the publication of his Theory of Sensations of Tone and 
Physiological Optics, Helmholtz gave himself up more and more 
to problems in mathematical physics, and pure mathematics; 
the few physiological papers that he published were in con- 
nexion with his earliest work on the physiology of nerve, which 
had been pushed into the background of late years owing to the 
extraordinary output of his new work. He was so exhausted 
by his labours in physiological optics that he found himself 
reluctantly obliged to forgo the pressing invitation of Roscoe 
to attend the Meeting of the British Association, and set off in 
the autumn holidays of 1866 with his wife for Switzerland, where 
he met the Kirchhoffs and Bunsen. After a short journey 
through North Italy he returned a few weeks later to Heidelberg, 

R 2 


to resume his nerve work, and if possible bring it to a con- 
clusion, as his mind was busying itself over problems of quite 
another kind. On February 6, 1867, ne wrote to Wittich : 

' In regard to rate of transmission in nerve, I have been 
making some experiments myself this winter with one of my 
Russian laboratory assistants, which are not yet fully worked 
out, and which give about 34 m. ; these, however, refer to the 
motor nerves of man, as I have recorded upon the myograph the 
muscular contractions of the ball of the thumb excited from 
the wrist and axilla respectively. We spent a long time on 
the improvement of our method, but eventually obtained very 
good and concordant results, which are infinitely superior in 
regularity of effect to my old method. I think one might apply 
it to many other questions, e. g. the supposed difference of 
velocity in different parts of the nerve.' 

These experiments were complicated by the difficulty that an 
instantaneous excitation of the motor nerves of man is not 
transmitted in an absolutely unaltered form through any con- 
siderable length of nerve. It is accordingly necessary to take 
precautions that the electric shock shall be so far weakened for 
the upper portions of the nerve that the contraction which it 
excites shall be of the same height and strength as the maximum 
of contraction excited from the lower point : the two instan- 
taneous excitations of the nerve will then produce equal external 
mechanical effects, the delayed response on exciting the upper 
portion being referred solely to conductivity within the nerve. 
The curves recorded by the myograph indicate that weaker 
stimuli are propagated in nerve more slowly than stronger 
shocks, and three long series of experiments gave rates of trans- 
mission of about 31, 33, and 37 meters per second. 

Owing to external circumstances the experiments were inter- 
rupted, and Helmholtz only took them up again three years 
later. He showed in a paper published in 1870, from experi- 
ments undertaken with Pick's pendulum myograph, that the 
rate of transmission of the nervous impulse was more than 
twice as great in nerves at higher temperatures, e. g. in the arm, 
as at lower temperatures. 

' This is a most extraordinary thing/ writes du Bois on April 
4, 1870. ' Such a dependence on temperature is unheard of; 
one would suppose then that the velocities would be enormously 


increased in fever. It is excellent for you, and I rejoice that it 
explains your first statement of the 6o-meter velocity.' 

The continuation of these experiments, presented to the 
Academy on June 8, 1871, under the title 'On the Time neces- 
sary to bring a Visual Impression to Consciousness. Results 
of work done by Herr N. Baxt in the Heidelberg Laboratory', 
gave a further series of results that were very interesting, and of 
the greatest importance in optics. Since positive after-images 
last as long as 12 seconds under favourable conditions, and 
during this time the forms of the larger objects are still 
recognizable in them, there will always, even with the shortest 
duration of light-stimulus, be a certain time during which the 
observer is able by means of the after-image to perceive a series 
of details in the object viewed, for the observation of which the 
direct light-stimulus could have given no time. In order to 
ascertain the time that is necessary for recognizing a more or 
less composite visual image, the positive after-image must be so 
submerged in a new and powerful light impression, that it loses 
its value for perception. 

Helmholtz had previously constructed the Tachistoscope, in 
which the observer looks at the object through the slit of a 
rotating disk for a very brief period, while the slit is immediately 
replaced first by a black and then by a brightly illuminated 
white sector, the illumination of which is designed to extinguish 
the after-image. With the help of this apparatus he found, as 
expressed in a definite numerical ratio, that large spatial differ- 
ences in the field of vision, as well as large differences in 
brightness, were perceived more quickly than small ones ; the 
influence of different figures used as objects was also strikingly 
evident, according as they were more or less well known, 
simpler or more complex. In conclusion Helmholtz appended 
another observation, which he had made much earlier. If he 
employed a persistently bright spot in the dark field before him 
as the fixation point he was able, without leaving this point of 
fixation, to direct his attention upon this or that portion of the 
dark field, even before its illumination by a spark, and then to 
see what appeared there. 

' This fact seems to me of great importance, since it shows 
that what we term the voluntary direction of attention is a 
change in our nervous system, independent of the motions of 


the external movable parts of the body, whereby the excited 
state of certain fibres is preferentially transmitted to con- 

These investigations speedily became the starting-point for the 
most important discoveries of modern psycho-physics. With 
them Helmholtz closed the series of his purely physiological 
investigations, and turned in the first place to the mechanics of 
physiology, and then almost exclusively to physics and mathe- 
matics, in which he once more did epoch-making work. 

The results communicated by Helmholtz under the title of 
'The Mechanics of the Auditory Ossicles' on July 26 and 
August 9, 1867, at Heidelberg, and at greater length in the year 
1869, in Pfliiger's Archiv, as * The Mechanics of the Auditory 
Ossicles and of the Tympanum ' (which dealt with the very com- 
plicated minute anatomy of the inner ear, and in which Helmholtz 
discussed the mechanism of the oscillations of the tympanum 
and small bones of the ear), were of supreme importance for the 
mechanics of physiological acoustics. Riemann, * that unusually 
penetrating intellect/ had indicated in a note published after his 
death in the Zeitschrift f. rationelle Medicin, that the capital task 
of aural mechanics was to explain how the apparatus of the 
tympanic cavity was able to transmit such excessively fine 
gradations of aerial waves to the fluid of the labyrinth. He 
had constructed a theory to this end, based on the assumption 
that the tympanic apparatus conveyed the alterations of air 
pressure from moment to moment with exact fidelity, in a 
constant ratio of magnification, to the fluid of the labyrinth. 
Helmholtz, on the contrary (who had taken up this subject 
directly he had concluded his Physiological Optics, without 
knowing of Riemann's note), finds in his theoretical considera- 
tions that it is only necessary for exact perception that each 
tone of constant pitch should excite a sensation of the same 
kind and intensity as often as it recurs. 

' Riemann's acoustic problem/ writes Helmholtz to Schering 
'occupied me also for some time; the empirical solution as 
effected for the human ear is, as a matter of fact, different from 
what he supposed/ Starting from the assumption that had been 
merely suggested by Ed. Weber, that the auditory ossicles and 
the petrosal bone must be regarded as fixed incompres- 
sible bodies, and the endolymph as an incompressible fluid, in 


relation to the conduction of auditory oscillations, Helmholtz dis- 
cusses the vibrations of the petrosal bone and endolymph, on 
the basis of Kirchhoff s theory of the conditions of equilibrium 
in an infinitely slender elastic rod, investigates the considera- 
tion of the anatomy of the tympanum, and proceeds farther to 
the form of a membrane stretched by air-pressure alone, with 
inextensible radial fibres. 

The completion of this work, which is a model of the most 
delicate dissection, of the most ingenious physical methods, and 
of the profoundest mathematical analysis, took up the whole of 
the winter after Helmholtz had communicated its elementary 
details at Heidelberg in the summer of 1867. In August, 1867, 
he went to the Ophthalmological Congress held in Paris during 
the Great Exhibition, and gave a lecture l Sur la Production de 
la Sensation du Relief dans FActe de la Vision Binoculaire ', in 
which he outlined some of the new work published in his 
Physiological Optics. 

'Yesterday and the day before,' he writes to his wife on 
August 14, 1867, ' I spent the mornings at the Ophthalmological 
Congress, where they made a great deal of me. Graefe is here, 
but unfortunately neither Bonders nor Bowman. I was solemnly 
received with acclamations by the Society, and then had to 
promise a lecture, which I delivered early yesterday morning in 
French, of course ex tempore as there was no time for pre- 
paration. ... I was invited to the Society's Banquet at Vefour's ; 
the first toast was proposed by Graefe in my honour, to which I 
had to reply, and later they toasted me again in a poem made by 
Bowman's friend Critchett, and seconded by a young Spaniard, 
in this style : " L'ophthalmologie etait dans les te'nebres, Dieuparla, 
qite Helmholtz naquit Et la lumiere est faite ! " You will see I 
had to forget how to blush ! ' 

All the letters written from Paris to his wife, who was on the 
Tegern See with the children, betray his regrets that she could 
not be with him, since her long residence there in former 
days would have led her to enjoy the stir of the Exhibition, 
and intercourse with all the distinguished persons staying there 
at the time, even more than he did himself. 

' Still,' replies the wife, ' since God has cut our poor Robert 
off for ever from a normal existence, he must and will be our 
first charge. It was perhaps my greatest sorrow to forgo this 


journey with you, but that is a trifle in comparison with the 
long sad doom of half-existence. And every day convinces 
me that the future will never improve either for him or for 
us, although it is not much use talking about it.' 

Helmholtz was obliged to go off to the mountains to recruit 
after the fatigues of Paris ; ' the fetes, &c. in the sultry heat were 
so exhausting that I began again to have the fainting fits, from 
which I had been free for some years/ On returning refreshed 
to Heidelberg a few weeks later, he plunged once more into 
his researches in mechanical acoustics, mathematical philosophy, 
hydrodynamics, and electricity. On November 19, 1867, he 
writes to Bonders : ' For the moment I am waiting for new 
acoustic instruments, and am worrying over certain psychological 
questions, the principles of space-perception, and the psychical 
processes of sense-perception without words. I fancy one could 
make a better analysis of this last chapter than the philosophers 
have accomplished so far. . . . The French seem to be nibbling 
now at my Sensations of Tone, and to more effect at any rate 
than the German musicians.' 

During this winter Helmholtz and G. Wiedemann conceived 
the notion of letting their wives undertake the translation of 
Tyndall's lectures on Heat as a Mode of Motion. The 
scientific portion of the book was to be carefully edited, and 
there was to be a preface written and signed by both. Some 
scruples of Wiedemann were set aside by Helmholtz in the 
words : ' My wife thinks there would be no harm in letting our 
friends know who did the translation; she thinks it would 
be more objectionable if the world supposed that you and I had 
wasted our time over such work/ 

This translation appeared in 1871 that of Tyndall's com- 
memorative paper on ' Faraday as a Discoverer ' having been 
published the year before with an interesting preface by 
Helmholtz, in which he expressed his great veneration for 
Faraday in magnificent language. We have already seen how 
cordially Faraday welcomed Helmholtz on his repeated visits 
to England : ' the absolute simplicity, modesty, and untroubled 
purity of his disposition had a charm such as I have never 
encountered in any other man/ But in Helmholtz's determina- 
tion to translate Tyndall's lecture the personal element was 
completely subordinated. He was not even swayed by his delight 


in describing how Faraday had with a mysterious instinct made 
the most pregnant discoveries in natural science, although he was 
unable subsequently to give any clear account of the train of 
ideas that led to them : Faraday's development rather appeared 
to him of the greatest general human interest for many 
theoretical questions in psychology, and for a number of practical 
problems in education, and he regarded it as a most interesting 
phenomenon that the man who had remained true to the pious 
faith of the small sect of his forebears, should have developed a 
philosophic vein, 'in virtue of which he ranks among the foremost 
in the general scientific thought of the age/ In characteristic 
language, Helmholtz (without direct allusion) sums up the total 
of the great researches which he himself had so ably shared in 
and initiated during the past thirty years of his life. 

1 After our era had destroyed the old metaphysical idols in 
its legitimate effort to render human knowledge above all the 
true image of reality, it was arrested by the traditional forms 
of the physical concepts of matter force, atoms, impondera- 
bilities and these names became to some extent the new 
metaphysical catch-words of the very people who had seemed 
the most enlightened. It was these concepts that Faraday 
sought again and again, in his maturer work, to purify from 
whatever they still contained that was theoretical, and not the 
immediate and just expression of the facts.' 

In the same year he also published the first volume of the 
German translation which he and Wertheim had made of 
Thomson and Tait's Textbook of Theoretical Physics, with a 
short preface by Helmholtz, in which he expresses the gratitude 
of the scientific world to William Thomson (Lord Kelvin), one 
of the most inventive and penetrating of thinkers, for admitting us 
to the laboratory of his thoughts, and unravelling for us the clues 
which had aided him in controlling and ordering the confused 
and refractory material with which he had to deal. He points 
out that in this work physical consistency was preferred to 
elegance of mathematical method. ' Perhaps when science is 
perfected, physical and mathematical order may coincide/ 

The second part of Vol. I of Thomson's Theoretical Physics 
only appeared in 1874 (when Helmholtz and Wiedemann also 
published their translation of Tyndall's Lectures on Sound), with 
an introduction written at the end of 1875, entitled ' Critical '. 


This contains an answer to the attacks made by Zollner upon 
Thomson and Helmholtz, which were a source of great annoy- 
ance and disturbance to the latter. 

' One of the most painful moments in his rich and vigorous 
life/ writes Blaserna, ' was the violent attack made by Zollner 
on him and other scientific workers. I could not understand 
this till I heard that Zollner had been converted to spiritualism 
by that enterprising swindler, Slade. His hatred was thus 
directed in the first place against Tyndall, who had embarked 
on a vigorous campaign in England against spiritualism, and 
then against Helmholtz, who had translated Tyndall's works 
into German, and put his name to the translation. He often 
talked about it; and we soon discovered that the solution of 
the so-called spiritualist problems lay in legerdemain. Every 
conjurer who came to Pontresina could reckon on patronage 
from myself and Helmholtz. We sat in front, and there was 
keen competition to see which of us would be the first to 
explain one or other of the tricks. Often we succeeded, often 
not. " It is a very pleasant mental gymnastic/' Helmholtz used 
to say, " and one never knows how it may come in useful some 

Helmholtz expressed himself to the same effect in a little 
pamphlet called Suggestion and Imagination, which he published 
at a much later time. ' Dear Sir ! I have never made any 
scientific study of the question you propound to me. What 
I know of it was learned accidentally. But I am familiar from long 
experience with the thirst for miracles of the Nineteenth Century, 
and the obstinacy with which such faith will overcome the most 
obvious proof of gross deception ; for my youth reaches back 
into the days when animal magnetism flourished. Since then 
there have been many different phases of the same trend of 
thought. Each has only a short life ; when the disillusionment 
becomes too apparent, they merely change the method. 

' If you ask why I have not gone into it more closely, I can but 
reply that my time has always been taken up with work that 
I believed to be of greater utility than the curing of marvel- 
mongers who do not want to be cured. And on the other side 
I must say that even if I had exposed the trick to myself, I could 
hardly hope to make much impression upon the faithful. If 
I had not succeeded I should have put a pretty argument into 


their hands against myself. And as I cannot succeed in de- 
ciphering the greater part of the tricks exhibited before me by 
a skilful conjurer, I certainly could not undertake to interpret 
all the magnetic or spiritualistic or hypnotic wonders that any 
one may show me ; the less so as the social position or sex of 
the confederates generally prohibits a really searching investiga- 
tion ; often enough too they will urge the ingenious excuse that 
the presence of an obstinate unbeliever has broken the spell. 

' As far as I am personally concerned, it has always been the 
psychological phenomenon of credulity that has interested me 
in these matters, and I have therefore successfully adopted the 
role of impostor from time to time in table-turning or thought- 
reading, of course explaining afterwards that I had been the 

' If after these explanations you are still interested in my 
private opinion, I can only say that I entirely agree with my 
friend and colleague Herr E. du Bois-Reymond. For the rest 
I do not deny that there is a core of truth in the phenomena of 
hypnotism. But what there is of truth in it will hardly appear 
so very wonderful. 

' As to the employment of such mystical influences in poetry, 
I can only speak as spectator and reader. As such, I find that 
I can only comprehend and sympathize with accountable beings. 
Charms are not repugnant to me, so long as they only constitute 
an abbreviation of some natural psychical process, which would 
actually require more time and more intermediate stages. 
Where that is not the case, my sympathy with the processes 
immediately vanishes, the theoretical explanation of this being 
quite obvious/ 

Helmholtz's scientific interests and discoveries were steadily 
turning away from physiology to an almost exclusive devotion 
to physics and mathematics, and, as was only natural, he began 
to wish that he could also concentrate his teaching more 
entirely in this direction. 

In the summer of 1868, while his wife was on the Baltic coast 
for the sake of their son Robert's health, and Helmholtz, already 
fully occupied with lectures, laboratory and other scientific work, 
was also teaching plane trigonometry to his son Richard in his 
leisure hours, to prepare him for the Polytechnic at Stuttgart, 
he received proposals from Bonn to undertake the Professor- 


ship of Physics there, which caused him a good deal of agitation 
and unpleasantness. 

The Prussian and Baden Governments had already had one 
tussle for the possession of Helmholtz, but the Baden authori- 
ties did not see any need for complying with the wish of Prussia, 
and releasing Helmholtz from his obligations. They knew too 
well what a powerful intellect they had secured for Heidelberg. 

Helmholtz had now spent ten years of activity in Heidelberg, 
and had as the greatest scientific man of the day, along with 
Bunsen and Kirchhoff, supported the glory of the University ; 
he was happy in his family relations ; he had all the advantages 
of intercourse with his many distinguished colleagues ; and it 
would have required very strong inducement to make him 
contemplate the idea of leaving Heidelberg. 

The Chair of Physics and of Mathematics in Bonn had been left 
vacant by the death of Plucker, and on May 28, 1868, the 
Curator of the University, Beseler, approached Helmholtz with 
the inquiry whether it would be possible to induce him to take 
the Professorship of Physics. Helmholtz made an interesting 
and characteristic reply : 

1 Physics was really from the outset the science which prin- 
cipally 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. The young people whose studies I now 
have to direct are, for the most part, medical students, and most 
of them are not sufficiently grounded in mathematics and 
physics to take up what I should consider the best of the 
subjects that I could teach. On the other hand, I see that the 
younger generation in Germany is not making any substantial 
progress in scientific, and especially in mathematical physics. 
The few great names in this branch, which is the true basis of 
all proper natural science, are old, or begin to recede into the 
older generation, while there is no new generation rising up to 
take their place ; and on this account I must say to myself that 
if I could get an influence over my pupils in this department^ 
I might perhaps do more important work there than in 
physiology, where a vigorous school is now in full and growing 
activity. That would be an aim that might repay me for the 
labour of taking the new work of a new post upon me, instead 


of working on upon my old lines. To this end, however, 
I should have, in addition to experimental physics, which is the 
popular subject for lectures, to undertake at least the teaching 
of mathematical physics and the direction of the practical work. 
Lectures in pure mathematics I could not well undertake ; in 
those on mathematical physics I should treat mathematics as 
the means and not as the end. Wherever possible I should 
include the physiology of the eye and ear, but would undertake 
no obligations in this particular/ 

Beseler urged the nomination of Helmholtz in pressing 
terms upon the Minister von Muhler. 

But notwithstanding these negotiations the correspondence 
did not lead to the desired result, because the Prussian 
Government could not proceed with the liberality of the Baden 
Ministry. On Jan. 2, 1869, Jolly addressed a very courteous 
communication to Helmholtz : 

' I now hope for a certainty that we shall succeed in re- 
taining you for beautiful Heidelberg. Willingly as we would 
otherwise follow the Prussian lead, it is in this case our 
bounden duty to declare war to the knife on the Berlin 
Cabinet, and I must add that it would be personally a matter 
of great pain to me, whose intellectual life is rooted in 
Heidelberg, if while I am at the head of affairs I had to see 
it robbed of its chief ornament.' 

The satisfaction of all his very modest demands by the 
Baden Government, and the wishes and inclination of his 
family, decided Helmholtz on staying in Heidelberg. In the 
midst of the negotiations his son Friedrich Julius was born, 
on October 15, 1868. 

' From his birth/ writes Frau v. Schmidt-Zabierow, ' he was 
a weakly child, who was only kept alive by unremitting care 
and attention, and whose mental as well as bodily development 
was a source of incessant anxiety to his parents. It required 
exceptional courage on the part of my sister not to give way 
to the double grief of the illness of her two sons, and to avert 
any gloomy consequences to her husband's life/ 

At this period the enormous output of Helmholtz's work 
assumed a distinct tendency towards the most arduous 
problems of physics, mathematics, and philosophy. 

His researches in acoustics had led him directly back to his 


earlier work in hydrodynamics, and his new results were laid 
before the Berlin Academy (April 23, 1868), in the paper ' On 
Discontinuous Motions of Fluids '. 

In the same year Helmholtz astonished the scientific and 
mathematical world by the far more comprehensive and 
fundamental researches which he published in the essay sent 
to the Gottingen Scientific Society, ( On the Facts that underlie 
Geometry/ At a later time he endeavoured to present its 
most important results in a form intelligible to non-mathemati- 
cians, in the lecture given to the Docentenverein at Heidelberg 
in 1870, 'On the Origin and Significance of Geometrical 
Axioms/ These investigations, along with the famous work, 
' On the Hypotheses that underlie Geometry,' which Riemann 
had published as his Habilitationsschrift, in 1854, were epoch- 
making for the development of the mathematico-philosophical 
conceptions of the second half of the last century. 

Helmholtz, indeed, had occupied himself with the philosophical 
analysis of the fundamental conceptions of mathematics and 
physics at a very early period, as is proved by an interesting 
sketch published some years before his essay on the Conserva- 
tion of Energy, which not only shows how he strove in his 
youth for clearness of fundamental concepts, but already 
indicates the direction in which he was to do such pioneer 
work thirty years later. 

On April 21, 1868, he writes to Schering at Gottingen : ' In 
thanking you for sending me the two little notes about 
Riemann, there is one question I should like to ask. In your 
notice of his life I find it stated that he gave a Habilitations- 
vorlesung on the Hypotheses of Geometry. I have myself 
been occupied with this subject for the last two years in 
connexion with my work in physiological optics, but have not 
yet completed or published the work, because I hoped to make 
certain points more general. For instance, I cannot yet make 
everything as universal for three dimensions as I can for two. 
Now I see by the few indications you give of the results of 
the work, that Riemann came to exactly the same conclusion 
as myself. My starting-point is the question: What must be 
the nature of a magnitude of several dimensions in order that 
solid bodies (i.e. bodies with unaltered relative measurements) 
shall everywhere be able to move in it as continuously, 


monodromously, and freely, as do bodies in actual space? Answer, 
expressed according to our analytical geometry : "Let*,jy, 0, / 
be the rectangular co-ordinates of a space of four dimensions, 
then for every point of our tri-dimensional space it follows that 
x 2 +y* + z* + j* = /? 2 , where R is an undetermined constant, 
which is infinite in Euclidean space." I venture to ask you to let 
me know if Riemann's essay is already in print, or if there is 
any prospect of its being published shortly, as seems to me 
most desirable; in the event of Riemann having taken the 
same point of departure, my own work would become useless, 
and I need not go on expending as much time and headache 
as it has already cost me/ 

Schering replied that l the most important point in Riemann's 
treatment of the proposition stated that the magnitude defined 
by Gauss as the measure of curvature is a differential invariant 
for homogeneous differential expressions of the second degree 
and first order, in two variables', and Helmholtz resumes on 
May 18: 

1 1 am much obliged for the copy of Riemann's Habilitations- 
sckrift. Herewith I send you a short account of the part of my 
own studies of this subject which is not covered by Riemann's 
work, begging you to lay it before the Royal Society to be 
published in the Gottinger Anzeigen (Proceedings of the Society). 
I believe a detailed discussion of the whole, consecutively, to be 
very desirable, and for choice I would have it published in the 
Proceedings of your Society, along with Riemann's. I there- 
fore beg to ask if communications are accepted from 
corresponding members, of which I am one, and when you are 
bringing out the next volume ? . . . Forgive me for Riemann's 
sake, for troubling you with these matters.' The paper was 
published in 1868. 

Helmholtz in the first place endeavoured to distinguish the 
development of concepts in geometry from the facts of experi- 
ence, which appear to be necessities of thought, while it was 
only in the lecture delivered ten years later on the Facts of 
Perception that he gathered up the results of his researches 
towards a unified system of philosophy that differed essentially 
from that of Kant. If this divergence from Kant had been 
partly apparent in his earlier physiological optics, he only 
proclaimed it definitely in the 1868 paper on the axioms of 


geometry. Twenty years later, in criticizing a book that excited 
general interest, he observes : 

' The strictest Kantians emphasise the particulars in which 
Kant in my opinion suffered from the imperfect development 
of the special sciences in his day, and fell into error. The 
nucleus of these errors lies in the axioms of geometry, which 
he regards as a priori forms of intuition, but which are really 
propositions tested by observation, and which if proved incorrect 
might eventually be rejected. 

1 This last is the point I have tried to establish. Therewith, 
however, we reject the possibility of laying down metaphysical 
foundations for natural science, in which Kant as a matter of 
fact believed. Now for my point of view it is exceedingly 
interesting to see in the papers he left behind him, how this 
contingency disturbed the philosopher as he grew older, how 
he turned it over and over, again and again seeking new 
formulae, and finding none that satisfied him. Among these 
we find in details instances of the most amazing insight, such 
indeed as we might expect from a man of his intellect, e.g. as 
to the nature of heat. ... In my opinion it is only possible to 
retain the great work done by Kant, if one recognizes his error 
in regard to the pure transcendental significance of the 
geometrical and mechanical axioms. But along with this we 
renounce the possibility of making his system the foundation of 
metaphysics, and this appears to me the reason why all of his 
disciples who cherish metaphysical hopes and tendencies 
adhere so tenaciously to these disputed points/ 

In his inquiry into the sense-perceptions Helmholtz had 
proposed to himself the question, What in the simplest forms 
of our spatial perception had been derived from experience, 
and what could not have originated therein, and how much 
must necessarily have been inferred from experience, in order 
to give support to the other ? Arguments and counter-arguments 
had already been brought forward, stating either that the axioms 
of geometry were a priori forms of our mode of intuition, anterior 
to all experience, and fundamental to our mental organization, 
or, on the other hand, that they were empirical theorems of the 
most universal character. In his attempt to transfer this inquiry 
from philosophical physiology to mathematics, Helmholtz en- 
deavoured for the more precise definition of the question to 


determine what other properties of space besides that of a 
magnitude of several dimensions were logically conceivable, 
or, since the question is one of relative magnitudes, algebraically 
possible, if we set aside the axioms of geometry as hitherto 

It had been of essential importance in Helmholtz's investiga- 
tions that he had, in his work on physiological optics, met with 
two other cases of magnitudes with several variables, which in 
their system of measurement exhibited certain fundamental 
differences as compared with spatial measurements. Whereas 
in space there is a relation of magnitude between any two 
points, comparable with that existing between any two others 
i. e. the numerical ratio of the distances ab : be, of the three 
points a, b, c in the region of colour, when the differences of 
brightness are taken into consideration, the simplest relation is 
that between four colours, a, b, c, d, when these can each be 
made by mixing two of them, when they lie in a straight 
line in the colour table i.e. the ratio of the two proportions 
in which a and c must be mixed, in order to produce on the one 
hand b, and on the other d. He had further found, on investi- 
gating the formation of our visual measurements in the two- 
dimensional field of vision, that the measurements very probably 
depended on the fact that the retina was carried by the move- 
ments of the eye as a fixed circle past the retinal image ; with 
this difference, however, from measurements in external space 
that we practically cannot utilize this circle in our measure- 
ments of the comparison of the lines in different directions. 
This drew his attention to the influence exerted by the means 
of measurement upon the system of measurement as a whole, 
and the form of the results, and these considerations led him 
to investigations not only of space but of all other poly-dimen- 
sional regions, in which a magnitude (distance) given by only 
two points can be compared by measurement with another 
corresponding to it, relating to any other given pair of points. 
Helmholtz showed that it is entirely a question of the formula- 
tion of special postulates, under which the square of the distance 
of two infinitely near points is brought under the more general 
form of the Pythagorean proposition, i. e. it is given by a homo- 
geneous function of the second degree of the differentials of any 
three magnitudes used for determining the position of the points. 


'I believe that the considerations here adduced are not 
without weight for the question of the original discovery of 
the geometrical propositions. For when men were seeking 
for a mathematical formula which should coincide with their 
more or less exact observations and measurements, they could 
find none that they could consistently carry through, save that 
expressed in the Pythagorean proposition, since as a matter of 
fact there was no other. And in this, as I believe, lies the 
foundation of the peculiar sort of conviction that we cherish in 
regard to the axioms that are unprovable either in theory or in 
practice. We have indeed no choice but to accept them, unless 
we mean to forgo all possibility of spatial measurement/ 

Helmholtz dissents altogether from Kant's doctrine of the 
a priori forms of intuition and of the axioms of geometry, and 
inquires into the facts that underlie geometry, or the question 
what geometrical laws express actual facts, and what on the 
contrary are merely definitions, or conclusions from definitions, 
and from the special modes of expression selected. The answer 
to this question, however, presents enormous difficulties, because 
geometry always has to do with ideal figures, to which the 
material figures of the actual world can only approximate. 
The decision whether, e. g., the surfaces of a body are plane, 
its sides straight, &c., can only be solved by the laws of 
geometry, the positive accuracy of which has first to be proven. 
We see without difficulty, that in addition to the Euclidean 
axioms as usually proposed for geometry, a whole series of 
other facts are tacitly admitted. It is essential to note in 
particular that we can only conceive intuitions of such relations 
of space as can be represented in actual space, and that we 
must not let ourselves be misled by this power of conception 
into assuming as a matter of course, what is in reality a par- 
ticular and by no means self-evident characteristic of the 
external world that is before us. 

But since analytical geometry only treats of spatial figures 
as magnitudes, which are determined by other magnitudes, 
since all the spatial relations known to us are measurable, 
i.e. can be referred to determinations of magnitude, length of 
line, angles, surfaces, &c., it has no need of intuition for its 
proof, and Helmholtz was led by this consideration to the 
further question what analytical properties of space and spatial 


magnitudes must be assumed in analytical geometry, in order 
to establish its propositions completely from the outset. He 
was then able to consider the possibility of the logical formula- 
tion of a different system of axioms, since the necessary cal- 
culation of analytical geometry is a purely logical operation, 
incapable of yielding any relation between the magnitudes in- 
volved in it, other than those already contained in the equations 
proposed for the calculation. 

It has been shown by Gauss that while the square of the 
length of a linear element in a plane is expressed by the sum 
of the squares of the increments of the two rectangular co- 
ordinates, the square of a linear element upon any given surface 
appears as a homogeneous function of the second degree of the 
increments of two general co-ordinates, which determine the 
situation of a point upon a surface. If figures of finite magni- 
tude are to be movable towards all parts of such a surface 
without alteration in their measurements as made upon the 
surface itself, and capable of rotation round any given point, 
then further the surfaces must have a constant measure of 
curvature at every point, the measure of curvature of the 
surface at any point being defined by Gauss as the reciprocal 
ratio of an infinitely small part of the surface surrounding this 
point to that part of the surface which is drawn through spherical 
radii parallel to the normals, upon the unit-sphere. But even 
upon surfaces with a constant measure of curvature, where free 
mobility of the figures is possible, geometry would assume a 
form wholly different from our geometry. 

Helmholtz starts with the assumption that as inhabitants 
of tri-dimensional space it is possible for us to conceive 
the various ways in which beings living in a surface would 
form their conceptions of space, and to picture to ourselves 
their sensory impressions ; spaces of more than three dimen- 
sions are, however, inconceivable to us, since all our means 
of sensory perception extend only to tri-dimensional space ; 
and then he goes on to consider geometry as it would appear 
to intelligent beings of only two dimensions. 

He propounds the question what would become of the axioms of 
our geometry, as, that there is only one shortest distance between 
any two points in space, the straight line ; that, further, through 
any three points that do not lie in a straight line, a plane surface 

s 2 


can be drawn, which must entirely contain the straight lines 
joining any two of these points ; and, lastly, that through any 
point lying outside a straight line only one straight line can be 
drawn, parallel to it, and never cutting it. The two-dimensional 
being would indeed be able, as a rule, to draw shortest lines 
between two points, which he terms ' straightest ' lines, but 
even in the simplest case of the sphere, an infinite number of 
straightest lines could be drawn between any two poles ; parallel 
straightest lines that did not intersect could not be drawn at all, 
and the sum of the angles of a triangle would always be greater 
than two right angles, and the more so, the larger the surface 
of the triangle. The space of these beings would no doubt be 
unlimited, but it would be found to have finite extension, or 
at any rate be postulated as having it. Only when the constant 
measure of curvature is of zero value, i. e. when, according to 
Gauss, the surface can be spread out on a plane by flexion 
without extension or disruption, would our geometry hold 

Both for Riemann and Helmholtz, however, the question 
of primary importance was not under what conditions our 
geometrical axioms might be valid, but under what hitherto 
not clearly explained conditions we arrived at the knowledge 
of them. Riemann shows how by a generalization from tri- 
dimensional space the universal properties of space, its con- 
tinuity, and the multiplicity of its dimensions, could be expressed, 
by saying that each particular in the complex which it presents, 
i. e. each point, could be determined by measuring n con- 
tinuously and independently variable magnitudes, which are 
its co-ordinates, so that space becomes an n-times extended 
complex, and we ascribe to it n dimensions. Riemann adds 
as a further necessity that the length of a line must be in- 
dependent of place and direction, so that every line must be 
measurable by every other, and since in our actual space the 
measure of each linear element is the square root of a homo- 
geneous function of the second degree of the increments of three 
measurements of whatever kind, he starts in his general 
investigation from this form of linear element as if it were 
hypothetical. He finally generalizes the definition of the 
measure of curvature for w-dimensional space, and shows that, 
if he adds the final condition that spatial figures shall every- 


where be movable without change of form, and able to rotate 
in every direction, then the measure of curvature must be 
constant. He thereby proves that the fundamental assumptions 
do not require the infinite extensibilty of tri-dimensional space ; 
space can have the same relation to a quadruply extended 
complex, as a surface with a constant measure of curvature has 
to tri-dimensional space. 

Helmholtz's investigation was to a large extent implicit in 
that of Riemann, but was distinctly original in one particular, 
so that it was of great importance for all later work, and for the 
question of the axioms of geometry. He tries to establish the 
conditions under which the Pythagorean Law as hypothetically 
assumed and generalized by Riemann would be valid, and 
makes the condition, which Riemann only introduced at the 
close of his paper, the basis of his whole treatment of the 
subject, i.e. that spatial figures should have, without alteration 
of form, the degree of mobility which is postulated in geometry. 

' For the rest I must observe, that even if the publication 
of Riemann's work has cancelled the priority of a whole series 
of my own results, it is of no little importance to me, in regard 
to such a recondite and hitherto discredited subject, to find that 
so distinguished a mathematician should have thought these 
questions worthy of his attention, and it has been to me 
a certain guarantee of the validity of the way, when I found 
him upon it as my companion/ 

For Helmholtz the starting-point of the investigation was 
the fact that all primitive measurement of space rests on the 
observation of congruence. But since there can be no veri- 
fication of congruence unless fixed bodies or systems of points 
can be moved relatively to each other with unaltered form, and 
unless the congruence of two spatial magnitudes be a fact 
independent of all motion, he set himself the task of seeking 
the most universal analytical form of a complex of manifold 
extension in which the desired mode of motion shall be 
possible. He inquires in the next place how much the con- 
ditions which he postulates for the investigation, viz. (i) con- 
tinuity and dimensions, (2) the existence of mobile solid bodies, 
(3) free mobility, (4) independence of form of solid bodies on their 
rotation, restrict the possibility of different systems of geometry. 
These assumptions led him to a measure of the linear elements, 


as independent of direction, in the form laid down by Riemann, 
and he shortly sums up the conditions required by the latter, 
in saying that a point of an n-fo\d complex is determined by 
n co-ordinates, that there is further an equation between the 
2 n co-ordinates of any pair of points infinitely close together, 
independent of their motion, which is identical for all congruent 
pairs of points, and that, lastly, with otherwise perfectly free 
mobility of the solid body, the property of monodromy of space 
must be fulfilled, whereby when a solid body of n dimensions 
rotates round n i fixed points the rotation shall bring it back 
without reversal to its original position. And in applying these 
conditions to the case of three independent variables, he is able 
to show on purely analytical grounds that a homogeneous 
function of the second degree exists between the increments of 
the same, which persists unaltered during rotation, and which 
accordingly gives a measure of the linear elements, independent 
of direction. 

In his development of these considerations an error crept in 
owing to Helmholtz's statement that if infinite extension of 
space be required, no geometry other than the Euclidean is 
possible, whereas Beltrami showed that the geometry of 
Lobatschewsky is also admissible, by which in a space extended 
infinitely in all directions, figures congruent with a given 
figure can be constructed in all parts of the same, while, further, 
only one shortest line is possible between any two points ; but 
the axiom of parallel lines no longer holds. It is only when 
the measure of spatial curvature is everywhere at zero value 
that such a space corresponds with Euclid's axioms, and this 
space is then termed by Helmholtz a plane space. If the 
measure of curvature is constant and positive, we arrive at 
spherical space, in which the straightest lines return upon 
themselves, and there are no parallels; such a space is like 
the surface of a sphere, unlimited but not of infinite magnitude. 
If, lastly, the measure of curvature is constant and negative, 
then in such pseudo-spherical surfaces the straightest lines 
proceed to infinity, and in each planest surface a bundle of the 
straightest lines can be drawn through every point, which do 
not intersect any other given straightest line of the same 
surface. In a space of which the measure of curvature is other 
than zero, triangles of large superficies will have a different 


angular sum from those of small superficies ; but the results 
of geometrical and astronomical measurements which give 
the sum of the angles of a triangle only approximately, and 
never exactly, as two right angles, only justify us in concluding 
that the measure of our spatial curvature is exceedingly 
small; that it actually vanishes is not to be proven, it is 
an axiom. 

In an interesting passage in the lecture Helmholtz describes 
how we can picture to ourselves the appearance of a pseudo- 
spherical world extending in all directions, and hence the 
axioms of our geometry can in no wise be founded on the 
given form of our capacity of intuition. Beltrami had con- 
structed a pseudo-spherical space within a sphere of Euclidean 
space so that every straightest line and planest surface of the 
former was represented by a straight line and plane surface 
in the latter: Helmholtz makes it probable by similar con- 
siderations that if our eyes were provided with suitable convex 
glasses, pseudo-spherical space would no longer appear very 
singular to us, and it would only be at the outset that we should 
be deceived in our estimation of the size and distance of 
remote objects. 

In April, 1869, Beltrami contributed an interesting letter to 
the discussion, pointing out Helmholtz's error as above, and 
Helmholtz lost no time in correcting his statement in a com- 
munication to the Scientific Society at Heidelberg. 

Helmholtz, like all philosophers and scientific men at the 
beginning of the Nineteenth Century, was profoundly exercised 
by epistemological questions : ' What is true in our ideas and 
conceptions ? How far do our notions correspond with reality ? ' 
These and kindred problems relating to the theory of know- 
ledge were the logical outcome of such work as the preceding, 
although not explicitly developed till a later period. 

A note found among Helmholtz's papers gives us in this 
connexion a slight but highly interesting 'Analysis of Knowledge 
as we actually have it ' : 

' The content, (i) Sensations are the only direct and pure 
perceptions. (2) Our conceptual images of external individual 
objects are the aggregate of a large number of different 
conceptions. (3) The concept of an object present expressly 
includes the assurance that with suitable conditions of observa- 


tion the same sense-impressions of that object will always 
obtain. (4) Existing objects alter, but we seek and find laws 
for such alterations, i.e. concepts for them, which remain 
themselves unaltered, but only become active, i. e. as pheno- 
mena, so often as the same conditions of their activity recur. 
It is by this that they are differentiated from the existence 
of substances whose phenomenal appearance can only be 
contemplated as dependent on the observer, that of the laws 
of nature depending upon the changes in the existing order. 
(5) The postulation of a law of nature entails the assurance 
that in all future corresponding cases the phenomena will 
conform to this law. A perfect law, which states the conditions 
and extent of the result completely and exactly, is for our 
knowledge an adequate reason for a certain conclusion as to 
the result. So likewise it may be regarded objectively as force, 
as the objectively sufficient ground for the event. (6) The 
hypotheses of natural science are attempts to discover laws of 
a more extended import than can be immediately deduced 
from observation. 

' The empirically demonstrable significance of knowledge Ideas 
are signs, which can be translated back to reality by move- 
ments. Temporal relations alone are really equal. 

' The psychical processes that underlie the origin of knowledge. 
. . . The source of all knowledge is the transference of what has 
already occurred in experience to what is about to be ex- 
perienced. Deduction of the fundamental concepts that follow 
from the nature of comprehension, and from the presupposed 
possibility of the complete solution of this task/ 

From this starting-point Helmholtz seeks for a connexion 
with Kant, who had already perceived that the qualities of our 
sensations must be determined by the idiosyncrasies of our 
mode of conception (which was first established as unquestion- 
able by modern physiology), but apprehended space and time 
in the same way, since we can perceive nothing in the external 
world without its happening at a given time and occurring 
at a given place. Here too, Helmholtz is still with Kant un- 
conditionally, when he defines time as the given and necessary 
transcendental form of internal, space as the corresponding 
form of external intuition; he further agrees with him that 
spatial intuition is a subjective form of intuition, like the other 


qualities of sensation, since space appears to us sensibly with 
the qualities of our sensations of motion, as that through which 
we are able to move and see. Space to him is, further, the 
necessary form of external intuition, since it is that which we 
perceive spatially which is for us the external world, all else 
being the world of internal intuition or of self-consciousness, 
and for him as for Kant space is a given form of intuition, prior 
to all experience, since the perception of it is bound up with the 
possibility of motor volitional impulses, the mental and bodily 
capacity for which must be given by our organization before 
we can have intuitions of space. Kant, however, went farther, 
in that he assumed not only that the universal form of space- 
intuition was given, but that it also implied a priori, and anterior 
to all possible experience, certain more exact determinations, 
viz. the familiar axioms of geometry so that these are also of 
transcendental origin. 

It is here that Kant and Helmholtz part company, since to 
the latter the question whether the axioms of geometry are 
transcendental or laws of experience, is entirely separate from 
that of whether space in general is a transcendental form of 
intuition or no. 

' Kant's doctrine of the a priori forms of intuition is a very 
happy and lucid expression of the facts, but these forms must 
be sufficiently free and void of content to include every sort of 
content that may turn up anywhere in the forms of perception 
under consideration. The axioms of geometry, however, limit 
the intuitional forms of space to such an extent, that all 
conceivable contents are no longer admissible, if, that is to say, 
geometry is to be applied to the real world at all.' 

If the axioms really were an innate form of spatial intuition, 
we should not be justified in applying them to the phenomenal 
world till it had been proved by observation and experiment 
that the fractions of space taken as equivalent by the presup- 
posed transcendental intuitions were physically equivalent also. 
Helmholtz shows Kant's assumption of the a priori character of 
the geometrical axioms to be superfluous and unjustifiable. 

On the strength of his previous investigations he is able to 
show that it is possible to construct a geometry on the basis 
of the single definition of physical equality, according to which, 
under the same circumstances, in the same time, the same 


physical processes or circumstances will take their course, the 
equality being demonstrable by means of measurements with 
compasses. We should then obtain a geometry, the propositions 
of which would indeed be covered by our axioms, but which 
would be founded solely on empirical data, so that we should not 
require a priori axioms at all. Kant's assumption that spatial 
relations that contradict the Euclidean axioms are unrepresent- 
able is, however, invalidated by the preceding discussion, since 
Helmholtz interprets the whole of Kant's conception as a simple 
process that cannot be further analysed, and is influenced by the 
whole developmental state of the physiology of the senses. 

'When it is possible to state completely and unequivocally 
the whole series of sensory impressions, which must, in accord- 
ance with known laws, ensue from an object that has never 
been seen, then in my opinion the object must be held to be 
conceivable ; since ex hypothesi the object has never been seen, 
no earlier experience can help us, or direct our imagination in 
the discovery of the necessary series of impressions ; this can 
only arise from the concept of the object or relation to be 
represented. The concept of spatial figures that do not corre- 
spond to our ordinary intuitions can only be developed with 
certainty by the calculations of analytical geometry/ 

Helmholtz was greatly fatigued by the mathematico-philoso- 
phical studies necessitated by his work on the axioms of 
geometry. On March 28, 1869, he writes to Ludwig : 

1 1 have for the moment returned to electrical work on the 
time-relations and dispersion of discharges, to which I was 
incited by physiological experiments and problems. For the 
time being I have laid physiological optics and psychology 
aside. I found that so much philosophizing eventually led to 
a certain demoralization, and made one's thoughts lax and 
vague; I must discipline myself awhile by experiment and 
mathematics, and then come back later to the Theory of 
Perception. It is well to hear in between what others have 
to say about it, what they have to object, what they mis- 
understand, and so on, and whether they take any interest 
at all in these questions. My following in these matters has 
been small enough so far, but I have some good people 
with me j . 

As a matter of fact his philosophical views spread but slowly 


even after his Academic Discourse in 1878, on 'The Facts of 
Perception/ On March 2, 1881, he writes to Lipschitz : 

' I have been interested in seeing that you have hit on the 
same train of ideas as myself in the Theory of Knowledge. 
I am pleased, and it renews my courage, although I have quite 
given up hope of living long enough to see any reformation in 
philosophy. In my thoughts I rail against the faculty philoso- 
phers, like Schopenhauer, but I will not put this on paper. 
Each can only read himself, and is incapable of understanding 
the thoughts of others. Yet when I see the mathematicians 
and physicists gradually coming round to my ways, it at least 
gives me hope for the future. I expected opposition as a matter 
of course from the faculty people, who had preached the opposite 
ideas all their lives, but I did not anticipate that after all the 
trouble I have taken to set forth my meaning in different aspects, 
they would only deduce the wildest misunderstandings. On 
the other hand, I do not know how to meet (and this enrages 
me, often as I have sworn not to get annoyed about it) the 
calmness with which people, who are incapable of grasping 
the simplest geometrical statement, pronounce upon the most 
complex problems of the Theory of Space in the sure convic- 
tion of superior wisdom. In conclusion, it would be very 
profitable for the subject if you were to work up and publish 
your views. It will have more weight when it gradually 
appears that the people who have made a profound study of 
mathematical questions are obliged as a class to judge in this 
way. The individual, even if he be a Riemann, will always 
be regarded as a crank who is discussing unfamiliar matters 
as an amateur. You won't get much pleasure from it, but one 
must bestir oneself to see that the community of right-thinking 
persons increases gradually. At bottom it is the false rational- 
ism and theorizing speculation that is the most crying evil of 
our German education in all directions/ 

Helmholtz felt the necessity more and more of freeing his 
mind from philosophical speculations, and in order not to return 
at once to the physico-mathematical problems that had occupied 
him for so long a time, he took up and completed certain earlier 
physiological and electrical questions, which compelled him to 
devote himself in the first instance to purely experimental 


In his experiments on the transmission of excitations in 
nerve, Helmholtz had remarked (as already noted by others) 
that electrical induction shocks had little effect upon the 
deeper-lying nerves of the human body, while it is an easy 
matter to produce contractions, even in the deeper nerves, by 
the constant currents of a battery of ten to twenty platinum- 
zinc cells. In a lecture given to the Nat. Hist. Med. Ver. at 
Heidelberg on February 12, 1869, 'On the Physiological 
Action of Brief Electrical Shocks within Extended Conductors,' 
he described the experiments made to establish these facts 
on the thigh of the frog, which proves the accuracy of his 
observations. But the explanation of these phenomena, which 
he referred back to the investigation of the distribution of 
electrical discharges in extended conductors, involved a certain 
knowledge of the oscillation-frequency of the currents in an 
induction coil, whose terminals are connected with the coatings 
of a Leyden jar. In another lecture delivered on April 30, 1869, 
to the same Society, ' On Electrical Oscillations/ Helmholtz 
presented the results of his experiments in this direction, in 
which a frog's nerve was used as current-indicator and reagent 
for the detection of the electrical movements, and in which the 
electrical oscillations took place between the coatings of 
a Leyden jar, in a complete and uninterrupted circuit which 
had no spark gap. It was then found that in using a Grove's 
cell for the primary current, the total duration of the perceptible 
electrical oscillations in a coil joined up with a Leyden jar 
was about -fa of a second. The determination of the oscillation- 
frequency is required to make it possible to set up exact 
experiments in proof of the above facts. 

Helmholtz gave an address at the Opening of the Natural 
Science Congress at Innsbruck in September, 1869 (which he 
attended with his wife), entitled ' The Aim and Progress of the 
Natural Sciences.' It was designed to give an account of 
' the progress of natural science as a whole, the aims for which 
it strove, and the magnitude of the steps by which it advanced 
towards its goal '. 

Amid the wide circle of his undertakings we find a solitary 
note on Hay-fever, taken from a letter addressed to Binz, and 
published in 1869, in Virchow's Archiv f. path. Anatomic. In 
an attack of hay-fever (from which Helmholtz was a chronic 


sufferer), he discovered pathogenic vegetable germs in the 
mucous membrane of the nose, and successfully combated them 
with quinine, at a time when, as du Bois observes, there was as 
yet hardly any question of antisepsis. 

By the beginning of 1869 it was obvious that a third edition 
of Sensations of Tone was wanted. It appeared in the follow- 
ing year with considerable alterations. Helmholtz not only 
remodelled the sections on the history of music, and connected 
them together more closely, but, on the strength of recent 
discoveries, essentially modified his account of the function of 
the rods of Corti, while he included his own later work, 
which propounded the articulation between malleus and incus 
as the reason why soft harmonic over-tones arise in the ear itself 
from the stronger primary tones. The publication of the new 
edition again led to a few important final observations, which 
formed his last physiological communication to the Heidelberg 
Society (June 25, 1869), ' On the Auditory Oscillations in the 
Cochlea.' This gave fresh support to a hypothesis advanced 
by Hensen as to the function of the membrana basilaris. 

Helmholtz now turned to his vast undertakings in electro- 
dynamics. Even if his main work in this direction was to be done 
a little later, it was in Heidelberg that he began the investiga- 
tions of which, on Jan. 21, 1870, he presented a part to the Nat. 
Hist. Med. Ver. with the title ' On the Laws of Inconstant 
Electrical Currents in Materially Extended Conductors ', which 
was published at greater length in the same year in the Journal 
f. reine u. angewandte Math., as 'The Theory of Electro- 
Dynamics. Part I. On the Equations of Motion of Electricity 
for Stationary Conductors.' This was a preliminary study to 
orient himself in the department of hydrodynamics. 

The majority of physicists in Germany deduced the laws of 
electrodynamics from the hypotheses of W. Weber, which 
endeavoured to refer the phenomena of electricity and 
magnetism to a modification of the assumption made by Newton 
for the force of gravitation, and by Coulomb for statical 
electricity, of forces acting in a straight line at a distance, 
their extension through infinite space being regarded as 
instantaneous, with infinite velocity. Coulomb's view that 
the intensity of the forces was inversely proportional to the 
square of the distance of the electrical quantities that exerted 


reciprocal action, and directly proportional to the product of 
the two quantities, with repulsion between like, attraction 
between unlike charges, was supplemented by Weber by the 
hypothesis that the velocity with which the two electric charges 
approached, or receded from each other, as well as their 
accelerations, must have some influence on the magnitude of 
the force between them. This assumption of forces which are 
dependent, not only on the position, but also upon the motions, 
of the acting points, seemed to contradict Helmholtz's 
observations, since he was led by his inquiry into the conservation 
of energy to the view that forces depending on distance and 
velocity are contrary to the universal law of the conservation 
of energy, which had been thoroughly confirmed for the 
phenomena of electrodynamics also. It is true that Helmholtz 
had not at that time taken into consideration the more compli- 
cated case of Weber's law, in which the forces further depend 
on acceleration, and it was in fact shown that Weber's law 
admits of no cyclical process by which work can be evolved 
out of nothing. 

Along with Weber's hypothesis were a whole series of 
others, all having this in common, that they regarded the 
magnitude of Coulomb's force as modified by the influence of 
some component of the velocity of the moving electrical 
charges. Such were the hypotheses of F. E. Neumann, of his 
son C. Neumann, and other physicists, but the observed facts, 
and conclusions from theories that were not well founded, all 
ran confusedly together. Helmholtz undertook to clear up 
the region of electrodynamics, and to search for crucial results 
of the several theories, so as, wherever possible, to decide 
between them by means of suitable experiments. He found in 
the first place that all the phenomena incident on the passage 
of fully closed currents in their circulation through closed 
metallic circuits, in which during the passage of the current 
there was no perceptible change in the electric charges 
accumulated in any part of the conductor, were equally well 
accounted for on any of the above hypotheses. The results 
agreed as well with Ampere's law of electromagnetic action 
as with the theorems discovered by Faraday, and amplified by 
F. E. Neumann. With incompletely closed circuits, however, 
these hypotheses led to essentially different results, since 


electrical charges accumulate at the open ends of unclosed 
conductors, owing to the interpolation of insulating masses, 
at each electrical disturbance along the conductor, these electric 
charges being due to the electricity accumulated near the ends 
of the conductor, and unable to traverse the insulator. 

Since the hypothesis resorted to by W. Weber (i. e. that 
electricity has a certain degree of inertia like that of heavy 
bodies) proved untenable, because the apparent inertia is due to 
induction, Helmholtz next endeavoured to convert all these 
laws into one single theorem, which should contain a still 
undetermined constant, whence he could theoretically deduce 
all the conclusions, and then test them empirically. 

The potential of the current elements of two linear 
conductors due to one another proposed by Neumann, and 
derived from Ampere's attractive force between two current 
elements, was directly proportional to the product of the 
length of the elements, the cosine of the angle between them, 
and the product of current intensity in both, and indirectly 
proportional to the distance between them, with a factor of 
proportionality which is the negative square of the reciprocal of 
the velocity of light ; the validity of this expression of potential 
was tested and confirmed on closed currents. Helmholtz then 
looked for the most general form of expression for the 
potential of a single current element, which in all cases where 
one of the currents is closed gave the same value as Neumann's 
formula, and he finds this form expressed in the product of the 
two infinitely small elements, and the second partial differential 
quotients, taken with respect to the elements of a function of the 
distance of these elements and of the current intensities. He 
further submits this function to the condition that it shall be 
proportional to current intensity, and inversely proportional to 
distance, and obtains for the potential an expression which differs 
from Neumann's in that, instead of the cosine of the angle of 
the two elements, an expression is introduced which is linear 
in respect of this cosine and of the product of the cosine 
of the angle which the elements form with their distance 
from one another, and which contains a new constant k. This 
expression also includes the two different potential expressions 
of the theories of W. Weber and Maxwell for each pair of 
current elements. From the expression of the potential the two 



elements due to one another it is now possible (by a method 
of Kirchhoff) to develop the values of electrodynamic potential 
for currents that are continuously distributed in space, and it was 
shown with the help of Green's law that the value of the 
electrodynamic potential produced by all the currents present 
in relation to the three components of current in a volume 
element, are constant everywhere, with the exception of points 
at which the electrical currents are infinite. 

With the help of this expression of potential we obtain the 
equations of motion for electricity, which lead to an analogy 
between the motions of electricity in a conductor and those 
of a gas, and Helmholtz next investigates the nature of these 
differential equations, and the course of the electrical dis- 
turbances as determined by them, in regard to the value of 
the constant introduced by him as above into the law of 
potential, which has the value i in F. E. Neumann's law, o in 
Clerk Maxwell's (under a given assumption), i in Weber's and 
C. Neumann's. He finds that if k is zero or positive, the 
differential equations with given potentials give the same 
initial value for the motion of electricity, and that the work 
equivalent of the electrical motion is positive ; for a negative 
value of k it may be negative, i. e. less than in a state of rest, 
so that the equilibrium of the electricity at rest in conducting 
bodies for negative values of k must be unstable. Helmholtz 
proved that if this quantity of work once becomes negative, 
the motion, left to itself, will increase continuously, and lead 
to infinite velocities and densities of electricity. These motions 
and infinite progressive disturbances of electrical equilibrium, 
however, on the unstable side can actually be produced with 
the methods at our command for causing electrical motions 
if k has a negative value (as indeed happens, generally 
speaking, whenever electric disturbances are produced in a 
homogeneous conducting sphere, by bringing an electrically 
charged body near it, and taking it away again), and he thence 
concluded that the assumption of a negative value for the con- 
stant k, as made in Weber's law of induction, is inadmissible. 

Helmholtz next investigated the influence of the constant 
k with practicable experiments, and finds that if k i or is not 
disproportionately greater than i, the motions of the electricity 
in experiments with earth conductors will not differ perceptibly 


from the case in which k=o. The analytic treatment of the 
problems of the motions of electricity can also be simplified if 
k is not a very large number, by making =o, or assuming the 
propagation of the longitudinal waves to be infinitely great, as 
long as the dimensions of the conductor used are vanishingly 
small in comparison with the wave-lengths of the oscillations 
that come under observation. Thus with such electric motions 
as are produced within a conductor by external forces after 
a previous state of electric equilibrium, there can only be free 
electricity (on the assumption that =o)at the surface of the 
conductor, or at the limiting surfaces of different conductors. 
The investigation of a very long wire as conductor, compared 
with whose diameter the wave-length is very great, also shows 
the influence of the constant k only in the small terms of the 
higher order. Helmholtz concludes from this that in electrical 
experiments in the laboratory the velocity of the electric 
longitudinal waves depending on the constant k need not be 
taken into consideration, unless we have the means of detecting 
extraordinarily minute time-differences. 

After carrying out these experiments rigidly, without allow- 
ing himself to decide on any particular hypothesis, and taking 
the electrostatic and electrodynamic effects as action at a 
distance, which did not affect the surrounding insulating 
media and was not affected by them, he accepted the Faraday- 
Maxwell theory, which replaces action at a distance by the 
polarization of a medium, and assumes that the electric dis- 
turbances propagate themselves across an insulating dielectric 
in transverse waves, the velocity of which in air is equal to the 
velocity of light. 

Faraday, like Newton, wholly rejected the hypothesis of the 
existence of forces acting at a distance, according to which 
there is direct and immediate action between two bodies 
separated from each other in space, without any alteration of 
the intervening media. He found that magnetism or dia- 
magnetism exists in almost all the bodies previously held to 
be non-magnetic, and that in the same way good insulators 
suffered a change under the action of electrical bodies, which 
he termed the electric polarization of the insulator; and in 
virtue of this he sought to explain magnetic and electric action 
at a distance as due to the agency of the intervening polarized 


media. 'His ideas were clothed in an abstract language 
difficult to follow, and made but little way, until they found 
their interpreter in Clerk Maxwell/ On this hypothesis there 
were no open currents, since the accumulation of the electric 
charges at the ends of the conductors, and consequent di- 
electric polarization of the intervening insulators, represented 
an equivalent electrical motion in the insulators, and it was 
in this that Helmholtz recognized the cogency of Faraday's 

Helmholtz then, ' in view of the immense significance which 
this result may have in the further development of physics, 
and because the question of the rate of transmission of 
electrical action has recently been raised in many directions/ 
set himself the task of investigating the results of the law of 
induction as generalized by himself, in the presence of 
magnetizable and dielectrically polarizable media. The dis- 
cussion of the equations of motion of electricity, transformed 
in view of dielectric polarization, led him, without adopting the 
particular form of Clerk Maxwell's hypothesis, and while retain- 
ing the idea of electrical action at a distance, to the same results 
as Maxwell, i. e. that for a very large capacity of polarization 
the velocity of the transverse waves is equal to the velocity of 
light, while for a very small capacity it is infinitely great. The 
velocity of the longitudinal waves in air is found, however, to 
be directly proportional to that of the transverse, and indirectly 
proportional to the square root of the constant k, so that for 
k=o the assumption made in Clerk Maxwell's theory is con- 
firmed, that the rate of transmission of the longitudinal 
electrical waves is infinite, i.e. that there are no longitudinal 
waves. Further conclusions as to the velocities of the transverse 
and electrical longitudinal waves in other insulators harmonized 
equally well with the theory advanced by Clerk Maxwell. 

In this first treatise on electrodynamics Helmholtz com- 
pletely fulfilled his primary object, of sifting and clearing up 
the opinions and methods already obtaining. 

At the beginning of the year 1870 Helmholtz, with Kirchhoff, 
received the great distinction of being elected an external 
member of the Academy at Berlin ; at the same time an event 
occurred in the Berlin University, which was to have the most 
important consequences to his career. 


On April 4, 1870, du Bois-Reymond informed him of the 
death of Magnus, adding : ' I could tear my hair now for not 
having gone to the Minister when there was the question of 
Bonn, and begging him to let me conduct the negotiations with 
Prussia for you. If you were only in the Chair of Physics 
at Bonn, it would be a much easier matter to get you ap- 
pointed to succeed Magnus at Berlin/ . . . 

Helmholtz replied on April 7 : 'I do not reckon too much 
on a call to Berlin, because I think KirchhofFs appointment is 
much more on the cards, and would be easier to arrange. 
He is well in health now, is bright and energetic, and hardly 
wants his crutches even where the ground is not level. What 
you want in Berlin above all is a mathematical physicist, and 
I must say that Kirchhoff is a trained and practised force in 
that field, which I am not, however good the opinion I may 
have of my own deserts in other respects. I should be con- 
tent to be his successor here.' 

Helmholtz indeed contemplated the eventuality of his call 
to Berlin with great calmness. While his wife's clear judge- 
ment and intellectual vigour soon recognized that the stirring 
life of art and science in Berlin would afford a very different 
scope for her husband's work and her own talents to that of 
Heidelberg, for Helmholtz it was only a question of being 
able to devote his entire activities in teaching and research 
to physics. He replies on May 7 to Borchardt's congratu- 
lations on his own and KirchhofFs election to the external 
membership of the Academy. ' ... If fate should so dispose 
that one of us should not long be an external member I should 
greatly rejoice, because it would give me the opportunity of 
devoting myself to physics. But between physics in Berlin and 
physics in Heidelberg the balance is so nicely weighted, that 
I don't yet know where it will come to rest when its oscil- 
lations are over, and can calmly await the decision of the gods 
and of Herr von Muhler ; and I believe Kirchhoff is much in 
the same mind.' Meantime the possibility of Helmholtz's call 
to Berlin was rumoured in the papers, and on May i the 
Minister Jolly came to assure him that he should leave 
nothing in his power undone to make his attachment to 
Heidelberg permanent, and to comply in every way with his 

T 2 


The Philosophical Faculty of the University at Berlin 
proposed Helmholtz and Kirchhoff in a letter to the Minister, 
and gave the following reasons for their opinion : 

1 If Helmholtz is the more gifted and universal in research, 
Kirchhoff is the more practised physicist and successful teacher. 
While Helmholtz is the more productive, and is always occupied 
with new problems, Kirchhoff has more inclination to teaching ; 
his lectures are a pattern of lucidity and finish ; also from what 
we hear he is better able to superintend the work of elementary 

students than Helmholtz If it happens therefore to be easier 

to win over Kirchhoff than Helmholtz, the Faculty feels itself 
justified in most respectfully begging to submit to Your 
Excellency the name of Professor Kirchhoff as successor 
to G. Magnus/ 

The then Rector of the Berlin University, du Bois-Reymond, 
was empowered by the Prussian Minister to treat with 
Kirchhoff, in the first place, by word of mouth, and started 
for Heidelberg at the beginning of June with this object, with 
directions from Olsweisen that if Kirchhoff refused he was 
to sound Helmholtz, and with letters from the mathematicians 
Weierstrass and Kronecker to the latter. 

Kirchhoff remained true to his friends in Heidelberg. It was 
in the course of a little dinner given by du Bois on July 12 at 
the Hotel zum Europaischen Hof in honour of Kirchhoff and 
Helmholtz, at which Bunsen and Konigsberger (who had been 
called to Heidelberg as Hesse's successor at Easter, 1869) were 
the only other guests, that the Minister's reply to du Bois' 
telegraphed inquiry arrived, authorizing him to open negotia- 
tions with Helmholtz. The author, who is the only survivor, 
will never forget the splendid words in which du Bois pro- 
claimed with enthusiasm ' that Heidelberg had been the centre 
of scientific research for long enough, and that while he could 
understand that Kirchhoff preferred not to leave his friends, 
Helmholtz by the nature of his work was being gradually driven 
exclusively into physical research, and that it was fitting for 
him to transport himself to the capital of the rapidly unifying 
Germany, whence indeed he had set out*. Not one of us 
suspected that the great struggle for the real unity of Germany 
was to break out a few weeks later. 

Du Bois returned to Berlin next day with his report, and 


received a letter dated June 12 from Helmholtz, which formu- 
lated the conditions he had verbally expressed : 

' Dear Friend ! In reply to the question which you put to me 
on behalf of the Minister of Education, Herr von M tinier, as to 
the conditions under which I would transfer myself to Berlin 
to take up the Chair of Physics vacated there by the lamented 
death of Magnus, I reply that I am willing to undertake it on 
the following stipulations : (i) Personal salary of 4,000 thalers 
(600). (2) The promise, in so far as it can be made in the present 
state of affairs, that a Physical Institute shall be built, with the 
necessary equipment for instruction, for the private work of 
the Director, and for the practical work of the students. (3) The 
promise that I shall have sole charge of this Institute and of the 
collection of instruments, and that it be left to my judgement 
how far and under what conditions I can permit the use of it 
to be shared by other teachers (in regard to Professor Dove 
I should naturally exercise the utmost consideration). The 
Auditorium in the Physical Institute must equally be retained 
for my sole use, so that it may be possible to set up com- 
plicated arrangements of instruments within it. (4) An official 
lodging for myself in the Institute, and a corresponding allow- 
ance for rent until it shall be ready. (5) Provisional use of 
rooms hired in the vicinity of the University for my own work 
in physics, and for some of my students, with the necessary 
service. (6) A proper allowance for expense of moving. As 
soon as I hear from you that His Excellency is ready to 
comply with these conditions I will come to Berlin myself to 
survey the situation, and determine the accessories so far as 
they can be arranged beforehand. If it is desired that I take 
up the post in the autumn, the matter must be so far in train 
by July i that I can hand in my demission here/ 

The Minister of Education lost no time in applying for the 
necessary funds to the Minister of Finance, writing to him on 
June 14: 

1 In view of Helmholtz's universal and unrivalled fame in 
the scientific world, it would politically be of the greatest 
importance to get him here.' 

On June 28, the Minister of Education addressed a letter 
to Helmholtz in which he acceded to all his demands, stating 
that as there were only 2,000 thalers in the University chest 


the other 2,000 thalers should be forthcoming for the next year 
from the funds of the Academy of Sciences, as an academic 
stipend to be paid, like the University salaries, in annual 
instalments during his life, by obtaining the necessary grant 
from the General Revenue. 

Helmholtz now entered on a period of great agitation : to 
the tension with which he awaited the conclusion of his appoint- 
ment was added the increasingly threatening political news. 
' I was on the point of sending Kathe instructions for the 
event of war,' he writes on July 3 to his wife, who, with the 
children, was staying with her parents, 'when the telegram 
came saying that Prince Leopold has been good enough to 
abdicate. I wish King William had not intervened; it will 
only produce a brief respite, and looks like a concession from 

In the days that followed he was much excited. Bunsen, 
Kirchhoff, and KSnigsberger took long walks almost every 
day with him, and generally met him in the evening at the 
Darmstadter Hof. On July n he wrote to his wife: 

' I am beginning to fear that we really are in for a war, 
since the attitude of the French Government can only be 
explained by supposing that they have been waiting for an 
opportunity, and now think they have got a good one ; other- 
wise it is all sheer madness. Nor do I think that the Prussians 
will shirk the war, for once it is certain that it is bound to 
come sooner or later, they will accept it at once. This may 
alter all our plans and prospects very considerably/ 

Helmholtz's thoughts and time and energy were now wholly 
taken up by the important events that were happening. 
1 1 myself/ he writes early in October to du Bois, 'have worked 
here for two months preparing the Field Hospital, and specially 
undertook to manage the reception and expedition of the 
wounded, and of the officials at the station. I went one day 
with a party of the younger doctors to W5rth, and learned 
the horror of a battle-field after the battle. At one time this 
intense activity was a godsend to work off our agitation ; but 
afterwards, when things took a more peaceful course, and 
there was less for me to do, I was warned by sharp and 
recurrent attacks of headache that I needed rest. I first went 
to my wife's relations at Starnberg, where our little family 


had passed the time of the war. But it had become too wintry 
there, so I went for three weeks to Meran, and yesterday 
returned home by the Engadine and Chur.' 

Owing to the happy and unexpectedly rapid course of the 
war, du Bois was able by October 13 to assure Helmholtz 
that the Diet was to meet in November, when his appointment 
would be definitely concluded, but that in view of current 
events the building of the new Institute would have to be 
postponed for a time, to which Helmholtz agreed on October 17, 
provided a promise were given him that the matter should be 
proceeded with so soon as the State had recovered its normal 
financial balance, and, further, that such temporary provision 
was made for himself and a few students, as would enable 
him both to do some experimental work himself, and to direct 
his pupils. This was agreed to by the Prussian Ministry 
on December 16, 1870. 

Before the end of the year Helmholtz went to Berlin with 
his wife, where they found a fine, detached dwelling in the 
Konigin-Augusta-Strasse, and soon brought the provisional 
arrangements for the Physical Institute to a satisfactory con- 
clusion. The Ministry at once began to negotiate for the 
purchase of a site for the new Institute, giving him temporary 
quarters in the University for his instruments, and a Laboratory 
in the Herbarium, which was accommodated elsewhere, while 
they acceded to all minor demands with alacrity. Helmholtz 
and his wife returned to Heidelberg, well satisfied with their 
reception from the du Bois-Reymonds and other distinguished 
people in Berlin. 

On January 2 Helmholtz applied for his demission from the 
service of the Baden Government, and in a few weeks' time 
received the document, signed by the Emperor William at 
Versailles on February 13, 1871. 

A few days after his return from Berlin he received a letter 
from Sir William Thomson, asking if he were disposed to 
accept the Professorship of Experimental Physics at Cambridge, 
which, despite the munificent conditions, he was of course 
obliged to refuse. 

1 And thus/ writes du Bois, ' occurred the unparalleled event 
that a doctor and professor of physiology was appointed to 
the most important physical post in Germany, and Helmholtz, 


who called himself a born physicist, at length obtained a position 
suited to his specific talents and inclinations, since he had, as 
he wrote to me, become indifferent to physiology, and was 
only really interested in mathematical physics/ 

His son Richard, who knew that his father would be gratified 
if he presented himself at the theatre of war ('an ardent 
patriotism was ever one of my father's salient characteristics '), 
had though barely seventeen enlisted as a volunteer in the 
mounted division of the Baden Field Artillery in August 1870, 
and was sent to the front in November, where, besides several 
small skirmishes, he took part in the three days' fight on the 
Lisaine, and was wounded by a mishap with his gun, though 
not severely. 

In the last weeks of his stay at Heidelberg, Helmholtz took 
leave of the cultured audience who had so often listened with 
delight and admiration to his brilliant Popular Lectures, in a 
discourse ' On the Origin of the Planetary System J . The 
rostrum of the over-flowing hall was decked with laurels, 
a wreath lay upon it, and the whole audience rose as he 
entered. With astonishing lucidity, and in his consummate 
style, he discussed the Kant-Laplace hypothesis from the 
mechanical and physical sides, and the subtle considerations by 
which W. Thomson had proved that the density of the ether 
may conceivably be far less than that of air in the vacuum of 
a good air-pump, but that its mass is not absolutely nil, and 
that a volume of luminiferous ether equal to the volume of the 
earth cannot weigh less than 2,775 pounds. 

'The basis of this calculation would no doubt be removed 
if Clerk Maxwell's hypothesis should be confirmed, according 
to which light depends on electric or magnetic oscillations/ 

On March 5, 1871, the Faculties and many of the educated 
inhabitants of Heidelberg combined to give a banquet at the 
Harmonic in honour of Helmholtz. The words spoken by 
him and others can never be forgotten by those who were 
present, but all were possessed by the feeling that the greatest 
thinker and man of science in Germany belonged of rights to 
the place where the Founder of the German Empire was 
supported by the grandest Statesman and the most gifted 



BERLIN: 1871-1888 

HELMHOLTZ had hardly removed to Berlin when the engage- 
ment in the same year, and subsequent marriage, of his daughter 
brought considerable changes into his household. 

'Helmholtz's two children/ writes his sister-in-law Betty 
Johannes, 'had been, since his second marriage, with their grand- 
mother, who made them her special charge, and lived in the 
same house with them, while they came to visit me every year in 
the country. Kathe was a serious creature, almost morbid in 
her striving after the highest aims, never satisfying herself, 
never able quite to bring the world and its phenomena into 
harmony with her ideas. She was greatly beloved and ad- 
mired. As she grew up and developed a marked talent for 
painting, she was, thanks to her second mother, encouraged in 
every way that could further the development of her gifts, and 
give her a wider outlook and new impressions. She went to 
Munich, Vienna, the Tyrol, the Bavarian Highlands ; she painted 
in the ateliers of Berlin and Paris ; she spent a year in France 
and England in the house of the famous Orientalist, J. Mohl, 
whose wife had great influence over her. At the age of 19, she 
translated Tyndall with the help of her step-mother and Frau 
Wiedemann, and she followed her father's work with untiring 
eagerness. Her love and admiration of her father amounted to 
worship. It was in my house at Dahlem that Kathe met her 
husband Dr. Branco; they were engaged in 1871, married in 
1872, and immediately afterwards spent a long time in Italy, the 
country of Kathe's warmest aspirations. On their return, 
Branco bought an estate at Genthin for the sake of his wife's 
health, and there in 1873, a daughter, Edith, was born to them ; 
but after this her health grew steadily worse. They again 
spent a long time in Switzerland and at Baden-Baden : they 


moved to Heidelberg and then for a second time to Italy : but 
all was in vain. She returned from Italy in 1877, and died at 
her home in Dahlem on April 25. Her coffin stood before the 
altar in the village church, where her parents had been married/ 

Helmholtz, as an Ordinary Member of the Academy, to which 
he had been elected on April i, contributed a Paper ' On the 
Rate of Transmission of Electrodynamic Action', on May 25, 
1871. He connected this with the researches of Blaserna, 
and in it discussed a question, then of great moment in the 
development of Electrodynamics, to which he had already 
alluded in the great electrodynamic memoir cited above. Ac- 
cording to C. Neumann, and on the hypothesis of Faraday and 
Clerk Maxwell, which assumes that electrodynamic action at a 
distance is caused by changes in the medium with which space 
is filled, this action must be produced by forces which are pro- 
pagated through space with finite velocity, and this velocity 
must approximate to that of light. Helmholtz had, however, 
shown in his earlier criticism of the electrodynamical theories 
that, on the assumptions made as to the susceptibility of the air 
to magnetic or dielectric ^polarization, other values of the 
velocity of propagation are compatible with the facts. After 
Blaserna had convinced himself by experiments that the pro- 
pagation at least of the inductive action of electrical currents 
proceeded at a very moderate velocity in air, Helmholtz, who 
had long been occupied with experiments on the course of very 
brief electrical currents, felt compelled to test the accuracy of 
these experiments as regards the propagation of the action in 
air. He arrived at the result that ' the separation of the two 
coils to the considerable distance of 136 cm. does not alter the 
position of the zero-point of the induced current by one division 
of the micrometer, i. e. not by ^Ay^ second. So that if the 
inducing currents are really propagated at any calculable speed, 
this must be greater than 314,400 m. or about 42-4 (German) 
geographical miles er second '. 

In the same summer (July 6, 1871) Helmholtz, at the Leibniz 
Session of the Academy of Sciences, delivered a beautiful and 
reverent address ' In Commemoration of Gustav Magnus ', whose 
successor he was, and whose personality and conduct he felt 
himself the more bound to do justice to, since from Magnus's 
somewhat chilling reception of his 'Conservation of Energy' 


it had seemed (or, at any rate, his opponents said it had seemed) 
as if there had been a radical contrast between their scientific 
labours, and a certain depreciation of each other's work. Even 
in the early letters to du Bois-Reymond it had been apparent 
how highly Helmholtz appreciated the ready help which 
Magnus gave to young men of science, free as it was from 
all professional jealousy, as well as (the point now more par- 
ticularly emphasized) the faithful, patient, modest industry with 
which he invariably worked on till no further improvement was 
possible, and which, when he noted the same trait in any of his 
pupils, made him hail them as his personal friends. 

The germs of the Physical Society of Berlin had been sown 
in the Conferences which Magnus had held on stated evenings 
in the form of discussions and reports on physical problems at 
his own house ; and it was there, in the winter of 1847, that 
Helmholtz, while repeating his experiments on the function of 
yeast in alcoholic fermentation in Magnus's Laboratory, had 
made the acquaintance of Wiedemann. 

'At that time there were no lectures on Mathematical 
Physics/ he wrote twenty years later in the congratulatory 
address dedicated to Gustav Wiedemann, and contributed to the 
Jubilee volume of theAnnalen: 'G. Wiedemann and my self, being 
ambitious to learn something of mathematical physics, to which 
we were incited by Gauss's magnetic investigations, agreed to 
study some of Poisson's works together in private, e.g. his 
theory of elasticity, which we did with great regularity and 

The works of Magnus obtained undying fame from the classic 
perfection of his method, and the accuracy and reliability of his 

Helmholtz esteems him happy in that he was permitted to 
strive in pure inspiration towards ideal principles. 'Of such 
men it can be said that they are not hampered by an envious 
destiny, since, working for pure aims, and with a single heart, 
they find satisfaction even without external results.' 

The main interest in this lecture, however, attaches to Helm- 
holtz's general observations upon different methods of physical 
research, which convey some notion of the revolution that had 
taken place in the past thirty years. Magnus was not one of 
those investigators who embraced the extremes of modern 


empiricism, a school which confines itself to the discovery of 
facts, and declines to seek for any law or connexion in the facts 
discovered. On the other hand, he was far from posing as the 
theorist, who holds it unnecessary to obtain experimental 
confirmation of the conclusions derived from the hypotheses 
which he accepts as axioms. Above all he was the enemy 
of metaphysical hypothesis, and his fear of any resurrection 
of Hegel's ' Naturphilosophie ' may sometimes have made him 
a sterner critic of the works of others than would otherwise 
have been the case. 

As Helmholtz said on another occasion, ' It is unworthy of 
a would-be scientific thinker to forget the hypothetical origin 
of his propositions. The arrogance and vehemence with which 
such masked hypotheses are defended are, as a rule, the usual 
consequence of a sense of dissatisfaction which their champion 
feels in the depths of his consciousness as to the validity of his 

Helmholtz trusted that the conviction might gain ground 
that the only successful experimenter in physical science is the 
man who has a thorough theoretical knowledge, and knows 
how to propose the right questions in accordance with this, 
while, on the other hand, those only could profitably theorize 
who had a wide practical knowledge of experimental work, 
as had been so brilliantly demonstrated in the discovery of 
Spectrum Analysis. In his eyes mathematical physics is also 
an empirical science, and he endeavours in his Address to 
break down the barrier between experimental and theoretical 
physics. In our experience we only meet with extended and 
composite bodies, whose actions are compounded of those of 
the separate parts. If we would learn the simplest and most 
general laws of interaction between the masses and substances 
that exist in nature, as abstracted from the form, size, and position 
of the effective bodies, we must go back to the laws of action 
that govern continuous and homogeneous volume-elements, 
and not to the disparate and heterogeneous atoms, so that 
mathematical physics thus becomes as subject to the control 
of experience as experimental physics. 

He attacks the same question of the reciprocal relations 
between experimental and mathematical physics in his essay 
1 On the Attempt to Popularize Science ', published in 1874 as 


a preface to his translation of Tyndall's Fragments of Science, 
where he first distinguishes, and then reunites, the two ways 
of investigating the coherent sequence of nature by abstract 
notions, and by copious empirical observations. The first way, 
which leads by mathematical analysis to the quantitative know- 
ledge of phenomena, seems to him to be indicated only when 
the second method has already to some extent opened up the 
field, and provided an inductive knowledge of the laws for at 
least some groups of the phenomena which it covers. 

We are then concerned only with the transition to the ultimate 
and most universal laws, and with deductions from the same in 
this field. The purely experimental method, on the contrary, 
leads to the recognition of Uniformity in the same way as it 
is grasped by the artist, which Helmholtz had already indicated 
in his Goethe Lecture as the sensible lively perception of 
the type of its activity, developing later into the form of pure 
concept. The two ways must necessarily be concomitant, if we 
are to escape the danger, on the one hand, of erecting a structure 
on insecure foundations, on the other of losing sight of the aims 
of science. 

1 The first discovery of hitherto unknown laws of nature, i. e. 
of new uniformities in the course of apparently disconnected 
phenomena, is an affair of wit taking this word in its widest 
sense and comes about in nearly every case only by comparison 
of numerous sensory concepts. The completion and emenda- 
tion of what has been discovered subsequently devolves on the 
deductive labour of conceptual, and preferably of mathematical 
analysis, since it all turns finally on equality of quantity.' 

In the autumn holidays of 1871 Helmholtz attended the 
Meeting of the British Association at Edinburgh, first visiting 
Mr. Tait at St. Andrews. 

'St. Andrews/ he writes on August 20 to his wife, 'has 
a beautiful Bay, with fine sands sloping sharply up to the 
green links. The town itself is built on rocky cliffs. 
There is a lively society of bathers, elegant ladies and children, 
and gentlemen (sic) in sporting costumes, playing golf. . . . 
Mr. Tait thinks of nothing here beyond golfing. I had to go 
out too ; my first strokes came off after that I hit either the 
ground or the air. Tait is a peculiar sort of savage, living here, 
as he says, only for his muscles, and it was not till to-day, on the 


Sabbath, when he might not golf, and did not go to kirk either, 
that he could be induced to talk of reasonable matters. ... At 
dinner we had a chemist Andrews from Belfast, and Professor 
Huxley, the famous evolutionary zoologist from London, both 
most agreeable and interesting men. . . Andrews showed us 
some remarkable experiments, on the passage of gases and 
liquids into one another at high pressure. . . . 

'We dined with Prof. Brown, with whom a great mathematician 
Sylvester was staying, who has been very badly treated by 
Mr. Gladstone, which has caused much excitement/ 

From there he went to Glasgow, and stayed a night with 
Prof. Brown, in college, where a nephew of Sir W. Thomson 
did the honours. 

' The house was not yet finished internally, no carpets, nor 
paint, full of old furniture not yet in place, and it looked 
unspeakably desolate as if no one cared for it, in contrast to 
the old house which Lady Thomson had managed. In 
a corner of the dining-room was an exceedingly fine and 
expressive portrait of her, with the sofa on which she always 
lay, and her coverlet. I felt very sad, and could hardly restrain 
my tears, while the young people were merry enough over the 
tea. It is sad when men lose their wives, and their life is left 

From there he went on to the yacht-races at Inverary, 
taking part in them on Thomson's yacht, a two-master, and one 
of the finer and more commodious of the forty yachts, all fairly 
large, well-appointed and elegant, that were competing; he 
admired the dexterity with which Thomson and his men 
manoeuvred their boat. After visiting Lady Thomson's parents 
at Largs, where her death had taken place, they went on for 
some longer expeditions on the Lalla Rookh, of which he 
writes, ' The yacht is like a movable watering-place, and makes 
a pleasant home in fair weather/ Helmholtz and Thomson 
studied the theory of waves, ' which he loved to treat as a kind 
of race between us/ They put in at a number of the finest 
parts of the west coast of Scotland, till they reached the 
northern extremity of their wanderings, the Island of Skye, 
after many stoppages due to heavy storms. On the way back 
they visited the family of the mathematician Blackburn, near 
Glasgow, and Helmholtz was delighted with Mrs. Blackburn's 


extraordinary talent for animal painting, having already admired 
her pictures in the Exhibition in London. 

1 It was all very friendly and unconstrained. Thomson 
presumed so much on his intimacy with them that he always 
carried his mathematical notebook about with him, and would 
begin to calculate in the midst of the company if anything 
occurred to him, which was treated with a certain awe by the 
party. How would it be if I accustomed the Berliners to the 
same proceeding ? But the greatest naivete of all was when 
on the Friday he had invited the party to the yacht, and then 
as soon as we were under way, and every one was settled 
as securely as might be in view of the rolling, he disappeared 
into the cabin to make calculations, while the company were 
left to entertain each other so long as they were in the vein ; 
but you may imagine that they were not very lively. I amused 
myself by strolling up and down the deck, " in schwankender 
Anmuth." ' 

The return voyage was very pleasant and comfortable, and 
on calm days he and Thomson experimented on the rate at 
which the smallest ripples that appeared at the surface of the 
water were propagated, a subject on which Thomson had 
recently been working. 

' Still/ he writes to his wife on September 4, ' I find that 
a husband who is no longer in his first youth feels uncomfort- 
able when he wanders about in the world, all by himself, 
without higher guidance, and I think if the world were peopled 
with men only it would not be particularly beautiful, but would 
be very practical, and not at all refreshing/ 

In his work Helmholtz now turned almost exclusively to 
Electricity. In his first treatise on the theory of electro- 
dynamics, which deals with electrical motions in ponderable 
conductors at rest, and is of fundamental importance for the 
principles of mechanics, he had succeeded in giving F. E. 
Neumann's potential-expression a form in which it included 
the two different potential-expressions laid down by W. Weber 
and Clerk Maxwell for each pair of current-elements. Investi- 
gation of the law for the different values of his constant k had 
shown that Weber's law led to inconvenient results ; on the 
other hand, Maxwell's hypothesis, in the case when motions 
of electricity or magnetism in dielectric or magnetic media 


have electro-dynamic effects, required a knowledge not only 
of the indeterminable constant k, but also of the dielectric 
constant of the air, or the velocity of transverse electric waves 
in air, which likewise could not be determined from previous 
experiments. It was thus of primary importance to determine 
this latter constant by experiments, which were accordingly 
undertaken by Boltzmann in his laboratory, with the view of 
testing Clerk Maxwell's now famous electromagnetic theory 
of light. This distinguished physicist, whom Helmholtz vainly 
endeavoured to secure for Berlin at a later time as successor to 
Kirchhoff, wrote to Konigsberger in April, 1902, that in conse- 
quence of Helmholtz's supposition that Maxwell held the 
refractive indices to be equal to the dielectrical constants, the 
requisite agreement was not obtained ; he therefore left Berlin 
in the firm conviction that Maxwell was entirely wrong, and was 
on the point of printing his criticism of the theory. As early as 
November i, 1872, however, he wrote to Helmholtz : 1 1 cannot 
forbear to tell you of another thing. I was always under the 
impression (and I believe you expressed the same idea when I 
was in Berlin) that on Maxwell's theory of the identity of light 
and electricity, the dielectric constants which I had determined 
must always be equal to the refractive indices. On now 
putting the values of all the dielectric constants together in 
a table, I was much worried at their deviating so far from the 
refractive indices, but noticed at the same time that they were 
always nearly equal to the squares of the latter. The thought 
flashed through my mind that Maxwell's theory might require 
this, since the velocities of transmission are always proportional 
to the square root of the forces. I looked up Maxwell's 
treatise, and there sure enough was plain to read that the 
dielectric constants must be proportional to the squares of 
the refractive indices (the magnetic induction constant is about 
equal to unity for all these substances) ; so that I must look 
on my experiments as a confirmation of Clerk Maxwell's 

Neither from theory nor experiment was it possible as yet 
to decide for one or other of the hypotheses mentioned by 
Helmholtz in his first paper, and both in the note laid before 
the Academy on April 18, 1872, ' On the Theory of Electro- 
dynamics/ and in the article that appeared in the //. fur reine 


u. angewdt. Math., in 1873, ' On the Theory of Electrodynamics, 
Part II : Critical/ Helmholtz partially devoted himself to re- 
futing the objections that had been made to his former work. 
He points out, in reply to Bertrand, that the expressions for 
the potential of each pair of current-elements are not ex- 
pressions of ultimate elementary forces, but refer in each 
current-element, taking this as a solid body, to one force and 
to a pair of forces ; the quantity, and to some extent the 
direction, of these forces depend not merely upon the position 
of the elements, but also upon the velocity of the electrical 
currents, so that it should be as legitimate to speak of the 
potential of two current-elements as of the potential of two 
magnets. But he endeavoured, above all, to refute the ob- 
jections of W. Weber to his argument, since even in the highly 
special case of the motion of two particles of electricity along 
the line joining them, in accordance with Weber's law, the 
acceleration may become infinitely great, and at a less distance 
the co-efficient of the acceleration, corresponding with the 
mass, becomes negative. He further shows that on the 
assumption of Weber's law for an electrified particle, which 
is movable within a hollow sphere covered evenly with elec- 
tricity, the case may occur in which the co-efficient of accelera- 
tion becomes negative, thus producing perpetual motion ; and 
he again points out that the differential equations proposed 
by Kirchhoff for the motions of electricity on the assumption 
of Weber's law would lead to an unstable equilibrium of 
electricity in conductors. 

He then advanced a step farther in his comparison of 
the different theories and their consequences, and set himself 
the task of deriving Ampere's forces from the Potential Law 
of F. E. Neumann. This was suggested to him by Riecke's 
objection that when the potential of a closed current referred 
to a current-element is deduced by means of Helmholtz's po- 
tential expression, it follows that the action of a closed current 
on the movable part of another current is not, as it must be 
according to Ampere, perpendicular to the latter. Helmholtz 
laid his conclusions before the Academy in a short note 
entitled ' Comparison of the Laws of Ampere and Neumann for 
Electrodynamic Forces J , on Feb. 6, 1873, while the full account 
appeared the next year in the //. / Math., ' On the Theory 


of Electrodynamics, Part III : Electrodynamic Forces in 
Moving Conductors.' His further ' Criticism of Electrody- 
namics/ published in 1874 in Poggendorff's Annalen, is directed 
solely against the objections raised to his mathematical theory 
of electrodynamics. 

The Potential Law of F. E. Neumann (which Helmholtz in 
a letter to Schering calls one of the most brilliant achievements 
of mathematical physics) was designed and well fitted to 
comprise the whole department of electrodynamic motive 
forces included under Ampere's Law, as well as the electro- 
dynamic induction produced by the movement of conductors, 
and alteration of current intensity, under a single and very 
simple law. This, however, according to Neumann's proof, 
could only, in the case of closed currents, coincide with 
Ampere's law (which was actually correct in this instance), on 
the assumption that the two conductors in question were 
moved without alteration of form or magnitude. In order 
to express the law of the electrodynamic motive forces for 
conductors of three dimensions, Helmholtz analyses the latter 
into conducting threads, which everywhere follow the direction 
of the lines of current present, so that no electricity can escape 
from one of the threads to its neighbours. Now since Ampere's 
law only recognizes forces which act from current-element to 
current-element, Helmholtz was able to show that when in 
applying the law of potential other forces are taken into 
account which act between the ends of the current and the 
current-elements, and between the current-ends of the two 
conductors themselves, then, from the potential set up by the 
current-elements, motive forces may be derived for two open 
parts of the current, which, for these portions of the current, 
can be brought into the form which Ampere has given them. 
If, as in the motion of the so-called rotation apparatus, points 
of slip make their appearance, Helmholtz regards them as 
current-ends, and in his opinion the solution in these cases 
is found without difficulty, if we suppose that the discontinuous 
displacement which is theoretically assumed as the limiting 
case at the point of slip is in reality only the limit of what is 
physically speaking a perpetual continuous displacement. 

In Part III of the Theory of Electrodynamics, as above, 
Helmholtz not only gives a full account of these results, but 


goes a step farther in the development of the general theory. 
Till now he had only dealt with the action of electrical currents 
upon one another and upon conductors on the assumption that 
all conductors were at rest, so that it was only the alterations 
of current strength that had to be taken into consideration, 
the potential (as extended by him) of two superposed current- 
elements due to one another being definable as the work-value 
of the electrical currents present in them. He now went on 
to derive the equations of motion of electricity in moving 
ponderable conductors from the same principles, and en- 
deavours to show that his generalization of Neumann's law 
of potential contradicts none of the known results which 
referred almost exclusively to closed circuits, while at the 
same time it agrees with the law of the conservation of energy. 
On the other hand he did not extend the inquiry to the case 
in which, besides the moving conductors, the dielectric polariz- 
able media are also in motion, so that the electric motions 
occurring in these are also electrodynamically active. 

In his three published papers on Electrodynamics Helmholtz 
had thus in the first place expressed F. E. Neumann's law of 
potential (which derived the strength of the induced currents 
not from the action of one point on another, but from that 
of one longitudinal element of the conductor on another), and 
was able in this way to represent, all the phenomena of 
closed circuits in quantitative agreement with the facts, more 
simply than by Ampere's original law. For the usually much 
weaker electrodynamic action of open currents, in which 
electricity tends to accumulate at single points of the con- 
ductor, he was able to show that the application of the potential 
law never contradicted the universal 'axioms of mechanics, 
wherein lay the great superiority of Neumann's law to all 
other hypotheses of electrical action at a distance. It differed, 
however, in one important particular from Faraday's assump- 
tion, since electrodynamic action was only ascribed to the 
passage of currents in the conductors, while the dielectric 
charges generated in the insulators lying between the con- 
ductors were not thought to be electrodynamically active. It 
thus remained for Helmholtz to plan experiments which should 
enable him to decide for one or other of the two hypotheses. 

In the memoir laid before the Berlin Academy in June, 1875, 

U 2 


1 Experiments on the Electromotive Forces induced in Open 
Circuits by Motion/ he describes the experiments made with 
this object on the electricity that accumulates at the surface 
of a conductor rotating in the magnetic field. By the ordinary 
laws of induction, electromotive force must be induced in any 
conductor that is thrown into rotation round the axis of 
a magnet, which does not follow from the law of potential 
alone; and Helmholtz set himself to test the discrepancy 
between the two theories experimentally. On the assumption 
that the universalized law of potential of Neumann (con- 
sidering only the motions of electricity occurring relatively 
to the conductor in the conductors proper) gave a complete 
statement of the law of electrodynamic action, the ex- 
perimental results did not agree with Neumann's law of 
induction. This discrepancy only disappears when the po- 
tential law is combined with Faraday's view, that the dielectric 
polarization which occurs in the insulators between two con- 
ductors in process of being charged is an electric motion, 
equivalent in intensity and in electrodynamic effect to the 
current with which either portion of the conductor is charged. 
All other theories, in which forces at a distance (with intensities 
depending upon the distances, velocities, and accelerations) 
have to be assumed, correspond fully with the phenomena 
of closed currents, but they are contradictory to the universal 
axioms of dynamics, when they are applied to open currents. 
Weber's hypothesis results in unstable electrical equilibrium 
in every conductor of moderate tri-dimensional extension, so 
that no practicable laws can be derived from it for the motion of 
electricity in materially extended conductors. The same applies 
to Riemann's law, which, moreover, contradicts the axiom of 
equality of action and reaction, while Clausius's hypothesis, 
which is free from these errors, has to resort to a space-filling 
medium, between which and the electricities the forces he 
postulates come into play. 

Thus Helmholtz was led to recognize Faraday's hypothesis 
as the only one that agreed with the observed facts, and was 
in no way contradictory to the universal laws of dynamics. 
While Clerk Maxwell had actually worked out the theory for 
the case of closed circuits only, Helmholtz found that it also 
agreed with the few facts that were then known for open 


conductors, as had appeared from his own experiments on the 
electric charges of the surface of rotating conductors in the 
magnetic field. According to Faraday, dielectric polarization 
must occur in all insulators lying between the conductors, 
when the limiting conductors are charged electrically, and 
must be of such intensity that the motion of the electricities 
associated with the setting up of this condition may be re- 
garded as the equivalent continuation of the electric current 
with which the conductor is charged. Every current accord- 
ingly must be a closed current, for which all the divergent 
theories lead to the same result. It follows also that any 
direct action of forces at a distance, which was still admitted, 
must vanish in favour of alterations of dielectric and magnetic 
strains in the ether-pervading space. 

' Every radical alteration of the fundamental principles and 
postulates of a science/ said Helmholtz at a later time, 
'necessarily involves the formation of new abstract concepts, 
and unfamiliar associations of ideas, which the contemporary 
student can only assimilate slowly, if he be inclined to take 
the trouble of doing so at all. The import of a new abstraction 
can only be understood clearly when its application to the 
chief groups of individual cases which it comprises has been 
thought out, and found valid. It is very hard to define new 
abstractions in universal propositions, so as to avoid misunder- 
standings of all kinds. It is, as a rule, much harder for the 
creator of such a new idea to make out why others fail to 
understand him, than it had been to discover the new truth. 
I will not disparage Faraday's contemporaries, because his 
words appeared to them uncertain and dark sayings. I re- 
member too well how often I have sat gazing hopelessly at 
one of his descriptions of lines of force, of their number and 
tensions, or have sought to puzzle out the meaning of some 
law in which the galvanic current is treated as an axis of force, 
and so on. A Clerk Maxwell was required, a second man of 
the same depth and independence of insight, to build up in the 
normal forms of our systematic thinking the great structure 
whose plan was present to Faraday's mind, which he saw clear 
before him, and endeavoured to render apparent to his conr 

Whatever Helmholtz's inclination to support the views of 


Faraday, on the ground of the experiments which he had under- 
taken with the object of deciding for or against the theory 
of action at a distance, he first endeavoured as a cautious 
critic to include a series of other and apparently remote pheno- 
mena in the circle of his considerations. 

The communications which he made to the Naturforscher- 
Versammlung at Leipzig, August, 1872, 'On the Galvanic 
Polarization of Platinum/ and to the Berlin Academy in the 
following year, ' On Galvanic Polarization in Gas-free Liquids/ 
which are of a purely experimental character, originated in 
theoretical considerations, arising from the law of the conserva- 
tion of energy. It was known that when a Daniell cell of zinc 
and copper is connected to an electrolytic cell with platinum 
electrodes, a polarizing current is set up which declines rapidly, 
but does not entirely cease even after a long time. It was 
further known that if, after removing the Daniell cell, the 
platinum plates were connected with the galvanometer, the 
depolarizing current in ordinary liquids, saturated with gas, 
is initially strong, and then soon diminishes so as to be im- 
perceptible. Helmholtz now asked upon what this apparently 
unlimited duration of the polarizing current depended, and 
found that the persistent current was in close relation with 
the gases present in the liquid, or at the electrodes, before 
the passage of the current. A portion of the electrolytic oxygen 
is neutralized by the presence of hydrogen, which again sets 
free some of the hydrogen at the other electrode, which then 
dissolves in the liquid or penetrates the platinum, so that the 
decomposition of a corresponding amount of water again occurs. 
This process of conduction of electricity by the motion of its 
material carriers is termed by Helmholtz electrical convection. 
The motion of a gas enclosed in the electrodes ensues very 
slowly when the liquid itself is free from gas, so that the 
depolarization current in gas-free liquids may persist for a very 
long time. By assuming that in galvanic polarization it was not 
merely gas on the surface, but also that which had penetrated 
deeper into the platinum that came into play, and that the same 
laws held for the motions of the gases occluded in the metals as 
for the conduction of heat, Helmholtz removed the contradic- 
tion to the law of the conservation of energy. The products of 
electrolysis need not make their appearance at all, nor need 


the chemical affinity be overcome by the electromotive force ; 
the process may be persistently maintained by the diffusion 
of the hydrogen, so that the initial presence of a limited quantity 
of gas suffices for a long-sustained current. In order actually 
to demonstrate the penetration of the gases into the platinum 
in galvanic polarization, Helmholtz set up experiments in his 
laboratory to see whether the hydrogen produced on one side 
of a thin platinum plate by electrolysis could be detected after 
a certain time at the opposite side, by the fact of its causing 
galvanic polarization there. The paper laid before the Academy 
on March 16, 1876, ' Report on the Experiments of Dr. E. Root 
of Boston, on the Permeation of Platinum by Electrolytic Gases/ 
established as a fact that hydrogen does make the opposite 
side of the platinum more positive. 

The further question whether electric convection is electro- 
dynamically equivalent to the passage of electricity in a con- 
ductor, was answered in the ' Report on the Experiments on 
the Electrodynamic Action of Electric Convection, as carried 
out by Mr. Henry A. Rowland/ laid before the Academy on 
the same day. The convection currents thus obtained could 
actually be substituted for the motions of electricity in open 
conductors, thus affording possibilities of deciding important 
theoretical questions. The result of the experiments har- 
monized both with the theory of W. Weber, and with Maxwell's 
potential theory, which considered the dielectric polarization 
of insulators. 

When an ebonite disk, gilded on both sides, was thrown into 
rapid rotation round a vertical axis, between two resting disks 
of gilded glass, while they were charged by means of a point 
with positive or negative electricity from the coatings of a Leyden 
jar, it was found that the action of this electricity conducted 
by convection was not merely the same in quality as that of 
the galvanic current, but that it agreed quantitatively also 
with that required by Weber's theory. 

The proof hereby afforded that electricity transported con- 
vectively with its carriers also has electromagnetic action, was 
for Helmholtz (in conjunction with his previous work) conclusive 
evidence that Neumann's extended law of potential must be 
combined with Faraday's hypothesis, and that the appearance of 
electric or magnetic lines of force in space is invariably asso- 


elated with the production of dielectric or magnetic polarization 
in the ether, and in the ponderable medium. Since on this 
assumption all electric currents are to be regarded as closed, 
the disparity between the different theories of electro-dynamics 
(which give identical results for closed currents) disappears, 
inasmuch as the experiments can be shown to tally with the facts. 

At the beginning of 1873, Helmholtz was tempted by an 
invitation from Knapp to give a series of public lectures in 
America ; but after deliberate reflection he replied on January 5, 

' I get very weary of the Berlin rush, so that at the end of 
the term the one and only thing I wish is to see no living soul, 
and to collect my thoughts in some quiet spot. America 
would be exactly the opposite to all this. And as regards 
lectures, I am convinced that although I can put scientific 
matters before people who understand them, in a dry technical 
fashion, I have not sufficient command of language to do so 
in a way that will rivet the attention of a large audience 
who are not professionally trained. Then the preparation in 
a foreign language costs me double the time, and even if I had 
the assistance of an Englishman it would only be patchwork. 
There are still many things that I want to do for science, and 
I must not lose too much time. Indeed I begin to think that 
I shall never see America in this life.' 

He had in fact at this time, in addition to his great electrical 
researches, begun some investigations in Aero-dynamics, the 
first results of which, under the title ' On a Theorem referring 
to the Geometrically Similar Motions of Fluid Bodies, with 
applications to the Problem of guiding Air-balloons ', was pre- 
sented to the Academy on June 26, 1873. 

The amount of resistance opposed by air or water to a body 
of complicated form that is moving through it, comes essentially 
into consideration when it is a question of constructing a ship 
or balloon, which is to be propelled by any kind of motor 
apparatus. Since the resistance of the water or air to the oars, 
paddles, or screws gives the propelling force, while the same 
resistance against the body of the ship or balloon gives the 
force of resistance to be overcome, the velocity of progress 
which can be attained must depend on the ratio of these two 
forces Yet it was seldom enough that mathematical analysis 


had been able to discover the integrals proper to the conditions 
of the given special cases from the differential equations pro- 
posed for the motion of liquids and gases, taking into account 
the pressure and friction, from which these resistances could 
be calculated. On the other hand, we have plenty of experience 
in regard to ships of the most varied construction, since we 
know the amount of force required in order to give the desired 
velocity to any ship or boat, and we have succeeded in dis- 
covering the most advantageous forms for the body of the ship, 
and for the size and shape of the motor apparatus ; in the air, 
on the contrary, apart from the few experiments with balloons, 
birds are the only instances we have of flying machines. This 
consideration led Helmholtz (by means of the general hydro- 
dynamic equations that hold good for liquids and gases) to 
transfer the results of experiments made with ships, to the 
corresponding problems in aerostatics. He shows, by rigid 
mathematical reasoning, that it is possible to transfer the empirical 
results obtained from a fluid and from apparatus of given size 
and velocity, to a geometrically similar mass of another fluid, and 
to apparatus of other sizes and other speeds, and establishes 
the ratio in which the velocities, the pressure, and the corre- 
sponding energy must be magnified, if the ratio of the physical 
constants of the fluids is given. 

In the application of this principle there is, indeed, the objection 
that the density of the air alters perceptibly under pressure. 
But since air can escape freely on all sides, and the most 
successful results appear to be produced with the lower velo- 
cities of wings or screws, only those differences of pressure 
come under consideration which are caused by the accelerations 
of the displaced particles of air, and these, with the altered 
volume of air depending on them, may be disregarded (as 
Helmholtz shows), so long as the resulting velocities are 
negligible in comparison with that of sound. It follows, 
amongst other things, that the size of birds must find its limit 
unless the muscles could be further developed in the direction 
of performing more work with the same mass than they do at 
present. In the structure of the Great Condor, Nature has 
apparently reached the limits of size at which any creature can 
soar upward by its wings, and remain a long time in the air. 
Man, in his opinion, has no prospect of raising his weight into 


the air, and maintaining it there, by even the most ingenious 
winged mechanism, if it is to be worked by his own muscular 
power. If air-balloons and ships are compared on Helmholtz's 
principle, we obtain the interesting result that if the balloon 
weigh once and a half as much as the operator whom it carries, 
the ratio between working force and weight would be the 
same as in a war-ship. 

At the close of the summer session (1873) Frau von Helm- 
holtz took the children to their home in Baden, while Helmholtz, 
overwhelmed with work, spent the month of July in com- 
parative solitude at Berlin. 

1 Last night I was alone in the house, and was led by Heyse's 
novel to look up Schopenhauer's Essay on Woman, but only got 
to the chapter on Love, which I read on the balcony by lamp- 
light. He is a clever fellow, but has a passion for vulgarity, 
and turns away from every higher suggestion, even where it is 

On August 3 he writes to his wife : 

' I stayed at home, prepared the concluding lectures of my 
mathematical course, and have at last finished reading Zeller's 
Kirche und Staat. I must say that the book interested me, 
although it deals with things that have been overmuch talked 
about. I had previously seen nothing so reasonable and well 
grounded on this subject.' 

At the beginning of the holidays he went as usual to 
Pontresina, and when the never-failing cure of the Engadine 
had relieved his cardiac trouble, he went on with his wife to see 
the Exhibition at Vienna, and thence alone for a first visit to 
Florence, sending his wife enthusiastic descriptions of all that 
he saw both in Art and Nature. 

' I am enchanted and bewildered by all this completeness and 
beauty, what we have in Germany are only poor fragments ; 
here one has the principal works of the Masters in inexhaustible 
fullness. Fra Angelico is distractingly lovely, in whatever he 
really executed . . . then in the Accademia there are things of 
Perugino, which in colour and expression come very near the 
best Raphaels, marvellous things by one Mariotto Albertinelli, 
of the same period as Raphael, of whom I never heard, nor 
saw anything before, profound, full of expression, of the most 
tender poetry of colour/ 


His brief stay in Florence only permitted him further to visit 
the Galleries of the Uffizi, which he left, after a five hours' 
visit, almost fainting with hunger and fatigue, to take the 
magnificent walk round the city, at sunset, by the southern 
heights. After meeting Beltrami on the return journey at 
Bologna, as had been arranged, for the discussion of a number 
of geometrical speculations, and problems in mathematical 
physics, he joined his wife in Vienna (where she was staying 
with her sister, the wife of the Sectionschef von Schmidt- 
Zabierow, who was afterwards Governor of Carinthia), and 
returned to Berlin via Munich. 

At the end of the summer session, and during the autumn 
holidays, Helmholtz had been occupied, in addition to his 
electrodynamic researches, with some very abstruse problems 
in physical optics, which he communicated to the Academy on 
October 20, 1873, in a brief note ' On the Limits of the Efficiency 
of the Microscope', afterwards published at length in the jubilee 
volume of Poggendorff's Annalen for 1874, as ' The Theoretical 
Limits to the Efficiency of the Microscope'. His researches 
and results were on the same lines as those of the great master 
in that branch of optics, Herr Abbe of Jena. Helmholtz 
attacked the question which is so important for all branches of 
science, how much it was possible to increase the efficiency 
of the microscope, and points out that its development had 
already reached a point at which each minute improvement 
could be attained only by a disproportionate outlay of mental 
and mechanical labour. The reason for this was generally held 
to lie in the fact that the spherical aberration of small lenses 
\vith high curvatures is difficult to overcome ; while Helmholtz 
considered diffraction and brightness as the essential factors. 
The ratio of brightness and magnification he finds to be wholly 
independent of the special construction of the instrument, so 
that increased magnification only becomes possible with the 
application of much stronger light ; the dimness of the micro- 
scopic image thus increases with increasing magnification. 

Helmholtz, moreover, finds that with compound microscopes 
diffraction produces far more pronounced deviations of the rays 
from their focal point than do chromatic and spherical aberra- 
tion, and he therefore subjects it to exact investigation. If the 
size of the smallest perceptible object is judged by the distance 


of each pair of bright lines that can just be recognized as distinct 
from each other, then this magnitude must be equal to that 
which in the magnified image of the object is equal in breadth 
to the outer diffraction fringe of each bright point. A given 
length in the object is accordingly no longer perceptible as 
a particular length, if it appear equal to the breadth of the 
fringe in the magnified image. Since the magnitude depends 
only on the angle of divergence of the impinging rays, and not 
upon the construction of the instrument, and the indistinctness 
of the image produced by diffraction increases with the narrow- 
ing cone of light, the limit for the differences of magnitude that 
we are able to distinguish plainly is in general found equal to 
half the wave-length of the particular light employed. A further 
increase in optical power beyond that of the best modern 
instruments does not therefore seem possible. 

Another important work in physical optics was communicated 
to the Academy by Helmholtz on October 29, 1874, with the 
title ' On the Theory of Anomalous Dispersion ', which had 
occupied him for a month on his return from a long tour in 
Switzerland. On the whole he agrees with the hypothesis put 
forward in explanation of anomalous dispersion by Sellmeier, 
which assumes the presence of ponderable molecules embedded 
in the ether, and capable of sympathetic vibration. This was 
not, however, adequate for the case in which the specific 
oscillation-period of the sympathetically vibrating molecules was 
equal to the period of the luminous oscillations. Since the dis- 
persion is essentially due to absorption, Helmholtz takes the 
cause of the absorption to be, on the one hand, the sympathetic 
vibrations of the ponderable masses, produced by an elastic 
force acting between the ether and the ponderable atoms; on 
the other, a frictional resistance, which the vibrating ponderable 
particles encounter from the ponderable masses which do not 
vibrate in sympathy with them. This frictional force is pro- 
portional to the velocity, as in the slow vibrations of a pendulum 
and of resonating bodies. If only one kind of ponderable atom 
be present, and if the ether and the ponderable molecules are 
regarded as two continuous and interpenetrating media (as is 
permissible when the. distance between the ponderable parts 
is vanishingly small in comparison with the wave-length), there 
result from the differential equation of the motion of the ether, 


and of the sympathetically vibrating atoms, those equations for 
the velocity of transmission, and absorption constants, which 
Ketteler had already deduced from his observations. In the 
case of weak light absorption, as in solutions of dyes with 
anomalous dispersion, these agree well enough with the ob- 
served facts. With a stronger degree of absorption the theory 
corresponds with the phenomena in the vicinity of the maximum 
of absorption ; as is found by observation, the curve of refraction 
reaches its maximum before the maximum of absorption, its 
minimum after the absorption-maximum, and falls continuously 
from the former to the latter. With colours that are far from 
the absorption-maximum, we must, however, have recourse to 
new theories as to the structure of the ether in the body. 
Lastly, Helmholtz shows that the extension of the theory to 
media with a larger number of absorption-bands presents no 
insuperable difficulties, provided different kinds of sympa- 
thetically vibrating ponderable masses are presupposed. 

It was about this time that Helmholtz took up the preliminary 
studies for his meteorological work, coming forward with a 
popular lecture on 'Whirlwinds and Thunderstorms', which he 
delivered in the year 1875 in Hamburg. After describing the 
mechanical conditions from which it follows that the constant 
alternation of the state of our weather depends (as Dove had long 
ago shown in detail) upon the displacement of cool, dry polar 
winds by warm, moist equatorial winds, and vice versa, he goes 
on to investigate the motions of the air by which the regularity 
of tropical weather is interrupted, such as the hurricane or 

We learn from one of Helmholtz's letters that it was the 
accident of his observing the formation of cloud and storm from 
the top of the Rigi that drew his attention to these natural 
phenomena, and led him to the wonderful experiments in which 
a vertical tube filled with air is formed in the centre of a circu- 
lating mass of water, in the exact shape in which a waterspout is 
usually represented. The storms too develop in vortex form, 
and at the centre of such a vortex there is generally a space 
where there is little motion of the air. While the storm travels 
in the direction of the earth's rotation, the side which it presents 
to the equator invariably blows a west wind ; if dry and moist 
air come together, great masses of air may accumulate, as Reye 


has shown, which were originally in stable equilibrium, but 
with alterations of temperature pass gradually into unstable 
equilibrium. For instance, cloudy and dry air lying upon or 
by the side of each other may be of such temperatures that 
they are of exactly equal weight at a moderate height in the 
atmosphere ; in this case, in the lower half of the atmosphere, 
where the pressure is greater, the foggy air will become denser, 
and sink to the ground, while in the upper half of the atmo- 
sphere the same foggy air becomes more attenuated with lower 
pressure than the dry, grows lighter, and rises. The originally 
stable equilibrium will then (since with prolonged action of the 
sun the lower layer becomes warmer and moister, while the 
upper loses heat by its radiation into space) gradually pass into 
unstable equilibrium. If the equilibrium is interrupted at any 
point, so that pressure becomes less owing to the lighter 
ascending foggy air, the lower air will rise, and be drawn into 
the ascending current, while round it, where the equilibrium 
was still stable, it will become even more so from the evacuation 
of the moist air and sinkage of its upper surface ; the rise would 
continue until the whole of the lower layer had mounted up. 
In discussing thunderstorms, Helmholtz indicates the store of 
negative electricity with which the earth is permanently charged, 
as the source of the electric discharges. In conclusion, he lays 
stress on the difficulty of predicting the weather, and the 
circulation of air in the atmosphere. 

In general it is to be remarked that we can only calculate 
beforehand, and understand in all observable details, those 
natural processes in which small errors in the formulation of 
the premises involve only small errors in the final results. As 
soon as unstable equilibrium comes into play, this condition is no 
longer fulfilled. Thus chance still exists on our mental horizon ; 
but in reality it is only an expression for the complexity of our 
knowledge and the clumsiness of our methods of combination. 
A mind endowed with exact knowledge of the facts, whose 
thinking operations were accomplished so rapidly and precisely 
as to precede events, would discover in the wildest caprices of 
the storm, no less than in the motions of the stars, the harmonious 
ruling of eternal laws, that we can only guess at and assume. 

On November 4, 1875, a heavy blow fell on Helmholtz and 
his family. Robert von Mohl, who had come a few days 


previously to Berlin to attend the Reichstag, was there seized 
with apoplexy. His daughter went to the funeral at Karlsruhe, 
and Helmholtz spent the time in quiet retirement with his 
children, finding an echo of his deep emotion in music. 

After one of Joachim's Quartett evenings, he writes : 

' Beethoven's Op. 130, which is inconceivably great and 
solemn, but intensely sad, was clear to me for the first time 
to-day. The Adagio was incomparably well played; it is a 
wailing dream of lost ideals, perhaps the prototype of Tristan's 
Liebestod, a formless surging of infinite melody/ 

The year 1876 was that of the first Bayreuth Festival, which 
Helmholtz attended with his wife. Both were possessed with 
the general enthusiasm which the original and powerful con- 
ceptions of Richard Wagner excited throughout the musical 
world. 'They ranged themselves among the number of the 
Master's inspired friends, and welcomed the new intellectual 
and emotional relations, which, helpful and satisfying, became 
on both sides one of life's most cherished possessions.' After 
Helmholtz had left Bayreuth to recruit in the mountains, his 
wife writes to him on August 30 : ' No one, save those who 
were not present, can deny the power and majesty of the work. 
What is original and really great is always uncongenial to the 
mediocre; never have I seen anything more pitiful than the 
German criticism with its cothurnus and its icy non-recognition. 
Happily these gentlemen and their sterility cannot prevent the 
accomplished victory.' 

Helmholtz's many-sidedness became ever greater, the height 
of his grasp and comprehension of new scientific problems ever 
more astounding. He followed each new phenomenon with 
the greatest interest, and was always ready to give his opinion on 
it at length in writing. When Ktthne, at the beginning of 1877, 
sends him the Optogram, he welcomes the discovery with 
enthusiasm, and at once presents it to the Academy ; writing 
to him on March 13, 1877 : 

' I have been immensely pleased with this find ; I had always 
imagined hypothetically that there must be photo-chemical 
action in the retina, but had never supposed one would be 
able to demonstrate it. I am curious now about the action 
of colour. Boll has already made communications about this 
to the Academy, and to the Lincei. Red light ought to rein- 


force the red, and blue light to make it paler, but he further 
distinguishes greenish rods between the red, which should 
become more intense with green light. Whether the green 
is anything more than contrast, still appears to me questionable ; 
but that other coloured and uncoloured rods lay between the 
red, I could see for myself from his demonstrations last 

All this extensive scientific activity did not prevent him, 
when on the death of Poggendorff in the same year the editing 
of the Annalen der Physik u. Chemie devolved upon G. Wiede- 
mann, from giving a helping hand to the editing (as Wiedemann 
himself informs us), while he contributed a full written report 
on every paper of mathematico-physical interest that was 
sent in. 

After his appointment on July 24, 1877, to the Professor- 
ship of Physics at the ' Medico-Chirurgical- Military- Academy' 
(Friedrich-Wilhelm Institut), at which he had received his own 
education, he gave a discourse on ' Thought in Medicine ' at the 
Commemoration Festival of the Foundation of the Institute in 
the same year, in which he extolled medical studies as the school 
* which had demonstrated more clearly and convincingly than 
would have been possible in any other case, the eternal 
principles of all scientific work, principles so simple and yet so 
often forgotten, so clear and yet so often veiled in obscurity '. 
He points out that medicine more than any other department 
of science involves insight ; that epistemological questions as 
to scientific methods might also assume a serious importance, 
and prove to have a fruitful practical bearing ; that if a man 
work on a perfectly sure basis he loses nothing by an error save 
that wherein he has erred, but that where everything rests on 
hypotheses which only correspond with what we should like to 
hold true the least rift dislocates the entire structure of our 
convictions. And then in a most brilliant argument he attacks 
metaphysical systems in natural science, as well those of the 
spiritualists, who feel themselves elevated above the rest of 
nature, as those of the materialists, who strive to control the 
world unconditionally by means of the conceptual forms they 
have at present arrived at. 

He shows by a clear and convincing argument that Kant's 
refutation of the claims of pure thought had defeated the 


spiritualist theory, that his Critique of Pure Reason was a per- 
petual protest against the use of categories of thought beyond 
the limits of experience, and that he detected a tissue of false 
conclusions in all metaphysical systems. But inasmuch as 
Kant regarded the axioms of geometry as derived from pure 
transcendental intuition, pure a priori intuition had become the 
refuge of metaphysicians, and the expression of this theory in 
physiology is the nativist theory. Hence comes the great im- 
portance of experiment, to resolve the pure or empirical con-"- 
cepts,the axioms of geometry, the fundamental laws of mechanics, 
or the modes of visual perception, into their rational elements. 
He warns the younger men of science not to be led away by 
the fact that all sects of metaphysicians are up in arms against 
it, ' since these investigations put the axe to what seems to be 
the strongest support still left to their claims/ Materialism to 
Helmholtz is, equally, a metaphysical hypothesis, which may 
sometimes have proved profitable to the natural sciences, but 
can, as a dogma, be as great a hindrance to the progress of 

1 Memory, experience, practice, are also facts, the laws of 
which can be investigated, and which cannot be decreed away, 
even if they are not to be smoothly and simply referred to the 
known laws of excitation and conductivity in nerve fibres, 
however pretty a playground for fancy may be afforded by the 
ramification of the ganglion processes and nervous connexions 
in the brain/ 

Helmholtz was shaken and wearied in mind and body by the 
death of his daughter Kathe, and his incessant scientific work ; 
he also felt many worries brought on him by envy and malice 
more keenly than of yore. His wife, who was at that time 
with her children on the Starnberger See, was his comforter 
in the best sense : 

' I sit and dream for hours, and think of you, my dear, and 
wish you were here and free from all miseries. Nature is a 
great teacher, even in matters in which she has otherwise 
little enough to do. She brings out the relative value of things 
so plainly/ 

In the above lecture he had cautioned his pupils : ' One word 
more. I would not have you think my point of view affected by 
my personal experience. That any one, holding such opinions 



as I have put before you, who instils into his students wherever 
possible the principle, "A metaphysical conclusion is either 
a fallacy or a masked empirical inference," will not be viewed 
with much favour by the believers in metaphysics and a priori 
intuition, need hardly be expatiated on. Metaphysicians, like 
all who have no definite reasons to set against their opponents, 
are not generally very courteous in their polemic ; one's own 
progress can indeed be estimated by the rising discourtesy 
of the opposition': and even if he met these discourteous 
scientific rejoinders with the dignity of a great thinker, the 
unqualified attacks upon his person and his family had none 
the less a depressing effect on him, and Helmholtz at this time 
passed through a critical period. 

The entire University of Berlin united solidly with him, as 
the Philosophical Faculty had on a previous occasion. 
He, who had so recently joined its ranks, was elected 
Rector in the year 1876, and this marked witness to the con- 
fidence and respect of the many distinguished scientists who 
belonged at that time to the Berlin University restored his 
calm and contentment. He went to Switzerland for rest and 
refreshment, taking long walks over the Grimsel and Eggisch- 
horn to Belalp, where he stayed with Tyndall, on to Zermatt, 
and thence with his wife to Stresa, Milan, Spezia, and 

On this journey he composed the discourse which he de- 
livered on October 15, 1877, when he assumed the Rectorate, 
' On the Academic Freedom of the German Universities.' It 
expressed the spirit that was especially cherished and fostered 
by the Universities of Germany, and which Helmholtz put into 
words at a much later time in his Congratulatory Address to the 
Academy on the occasion of du Bois-Reymond's jubilee : ' The 
younger generation are taught that ideal aims are attainable 
even in this life, and bring their reward, but only when they are 
worked for in the right way.' The lecture is of extreme 
interest, since it reveals the high moral, religious, and political 
standpoint to which Helmholtz adhered during his whole life, 
without taking up any public attitude in regard to these 

For Helmholtz, the power of a nation lies not merely in its 
store of provender and money, its cannon and war-ships, but 


depends, above all, on its political and legal organization as a 
State, and on the moral discipline of its individual citizens, which 
determines the superiority of civilized to uncivilized nations. 
Where there is no firm legal status, where the interests of the 
majority of the people do not prevail in an orderly fashion, 
where the political interests of the working classes are not 
given a legitimate voice in the government, he holds the deve- 
lopment of the power of that State to be impossible. For modern 
Humanity, as he insisted on a later occasion, Science is, in the 
contests of the more highly developed nations, the sole bond 
of union that unconditionally makes for peace; in Science, 
wherever the individual races are educated enough to profit by 
its fruits, every one is working, not for the good of his country 
only, but for the whole of Humanity. But it is necessary for 
the profitable development of the sciences that there shall be an 
independent conviction of the accuracy of their results, as the 
consequence of conscientious trial and resolute labour; it is this 
that becomes the fruitful germ of new insight and the true rule of 
action. Helmholtz regards Germany as standing in the fore- 
front of the struggle with authority ; in the sixteenth century it 
testified even unto blood for the right of such convictions. He 
had already pointed out in his Innsbruck Address that there 
was greater fearlessness of the consequences of the whole 
entire truth in Germany than elsewhere, while in England and 
France the many distinguished investigators in natural science 
had almost always to bow to social and ecclesiastical prejudices, 
if they were not to suffer in their social influence and activity. 
In the whole truth he sees the remedy for the disadvantages 
and dangers of half-knowledge: 'An industrious, moderate, 
moral people must have the boldness to look truth full in the 
face : the nation will not perish even if a few premature and 
one-sided theories are paraded, which may seem to infringe on 
the principles of morality and social order.' 

And yet it is this very love of truth that tempts the Germans to 
follow out the cardinal questions to their very foundations, without 
regard to their practical consequences and useful applications. 
The independent intellectual development of the last three 
centuries had begun in Germany under political conditions which 
threw the chief burden upon theological study. Germany had 
freed Europe from the old tyranny, but in the Reformation 

x 2 


intellectual life had lost its former hold and its coherence; every- 
thing had to be presented in a new light, and new questions raised 
for discussion. Since the problems that had to be solved were 
principally those of morals, aesthetics, and metaphysics, it was in 
Helmholtz's opinion natural that the learned men of all nations 
should precipitate themselves upon philosophy. Criticism of the 
sources of knowledge was initiated, and the German tempera- 
ment could not break loose from metaphysics, which exerted a 
dangerous fascination upon it, until there was nothing left to find. 
Then, in the second half of the Eighteenth Century, the rejuve- 
nated mental life of the nation blossomed into its artistic prime, 
and all hearts turned from a joyless civic and political existence 
to the realms of poetry or philosophy. ' The work of the man 
of science appeared narrow, poor, and unimportant in com- 
parison with the great conceptions of philosophers and thinkers/ 
Helmholtz recognizes clearly that this movement broke the 
Napoleonic yoke, and in its great poems set the noblest ideals 
of the German nation, but 'sanctuary in an ideal world is a false 
aid of brief duration ; it only helps the enemy to his goal, and 
where knowledge reflects itself repeatedly it becomes objectless 
and empty, or is lost in phrases and illusions '. 

The reaction against this tendency set in not only in the 
region of natural science, but in history, aesthetics, and in 
philology also. It was perceived in all directions that facts 
must be known before their laws can be established. 

Helmholtz urges the younger students, in his Rectorial 
Address, to pursue Science for its own sake, and advances 
ideas of high ethical import, such as he clothed in beautiful 
words, with modest but assured self-consciousness, in his famous 
Tischrede in 1891 : 

' I will not say that higher ethical motives were not co-operating 
with scientific curiosity, and sense of duty as an official of the 
State, in the first half of my life, when I was still obliged to work 
for my living; but it was harder to be certain of their efficacy so 
long as egoistic motives drove one to work. The same is the 
case with most workers. Later, with an assured position, when 
those who have no inward impulse towards science may cease 
to work, there is pre-eminently, in those who do work on, a 
higher conception of their relation to Humanity. They 
gradually realize by their own experience how the ideas that 


they have expressed, whether in their writings, or in verbal 
commerce with their students, acquire a fuller content and more 
consistent form as they are more thoroughly worked out, 
bringing new instruction to themselves also. The entire 
conceptual world of civilized humanity comes before them as 
a living and growing whole, which in comparison with the brief 
life of the individual appears to be eternal. Such a one regards 
himself, in his own small efforts towards the building up of 
science, as a minister in an eternally righteous cause, with 
which he is linked by the closest bands of love. Thus, even to 
himself, his work is consecrated/ 

This is the temper which Helmholtz describes to the young 
students as fostered by the German Universities, and which he 
holds to be unrealized in other nations, while he does not deny 
that Germany might well imitate England in her encouragement 
of a keen sense of the beauty and freshness of antiquity, the 
refinement and precision of language, and the physical weal of 
the students. But the untrammelled freedom of the German 
student, which amazes all foreigners, is a treasure that has to be 
guarded; it depends on the judgement and reasonableness of 
those to whom the freedom is granted. Then, and then only, 
is liberty of scientific teaching necessary and free from danger. 

1 In the German Empire of to-day, the most extreme conse- 
quences of materialistic metaphysics, the boldest speculations 
on the basis of Darwin's evolutionary theory, can be promul- 
gated as freely as the extreme deification of Papal Infallibility/ 

At the close of the same year (November 26, 1877), Helmholtz 
presented a paper to the Academy, 'On Galvanic Currents, 
caused by Differences of Concentration ; Deductions from the 
Mechanical Theory of Heat,' which inaugurated the series of 
his important researches in electrochemistry. After convincing 
himself by his electrical work that the Faraday- Maxwell hypo- 
thesis for electrodynamics (to the confirmation of which he did 
not return for two years) was the most probable, he now turned 
to Faraday's electrochemical theories, of which, as also of their 
development by Hittorf, Wiedemann, and F. Kohlrausch, he 
gave a full discussion in the Faraday Lecture delivered some 
time later in England. 

Faraday gave the name of ions to the atoms or groups of 
atoms that are carried along with the current, calling those that 


move with the positive electricity kations, and those charged 
with the negative electricity anions ; the kation therefore travels 
to the electrode to which the positive electricity of the fluid is 
directed, to the kathode, and the anion to the anode, whence the 
same electricity streams into the fluid. Faraday discovered the 
law that governs the whole of modern electro-chemistry, viz. 
that there is always equivalence of electrical and chemical 
motion in each section of an electrolytic conductor, so that 
precisely the same quantity of positive or negative electricity 
moves along with each monovalent ion, or each valency of a 
multivalent ion, and accompanies it inseparably in all the 
motions which it makes through the fluid. Helmholtz called 
this quantity the electrical charge of the ion. Assuming that 
electricity also is divided into certain elementary quantities, or 
atoms of electricity, he concludes that each ion, so long as it 
moves through the fluid, must be united with one electrical 
equivalent for each of its valencies. Separation can only occur 
at the surfaces of the electrodes, so that if these exert sufficient 
electromotive force, the ions will give up their previous charges 
of electricity, and become electrically neutral. Maintaining 
the law of the conservation of energy, as well as the strict 
validity of Faraday's law of electrolysis, as fundamental principles, 
he finds that if the hydrogen and oxygen of the water could be 
dissociated without losing their electrical charges, they would 
exert a reciprocal attraction equal to the gravitation of masses 
which exceeded them 400,000 billion times in weight. In the 
further investigation of the mode in which the motions of the 
ponderable molecules are affected by these forces, Clausius had 
shown that the electrical forces tend to maintain an even 
distribution of the antagonistic ions throughout the fluid, so 
that all parts of it are neutralized both electrically and chemically ; 
but that the least external electromotive force suffices to disturb 
the uniformity of this distribution. But when an ion parts with 
its charge of electricity, the electrical forces of the battery 
encounter a resistance which entails a very considerable 
expenditure of work before it can be overcome. This happens 
if the ions, as they lose 'their electrical charges, are simul- 
taneously separated from the fluid as gases, or in the form of 
solid metallic layers. The chemical association of two elementary 
substances of great affinity produces quantities of heat which 


are equivalent to a large quantity of mechanical work ; the 
dissociation of the resulting chemical compound requires an 
expenditure of work corresponding to the amount of the chemical 
forces lost in the formation of the compound. 

'Oxygen and hydrogen when separated contain a store of 
energy ; then if they are allowed to burn to form water, they 
develop a large amount of heat. The two elements are con- 
tained in the water, and their chemical attraction persists, and 
keeps them firmly united ; but it can no longer be utilized, nor 
produce any positive action. We must reduce the united 
elements to their first state, and separate them from one another, 
employing to this end a force greater than their affinity, before 
we can restore to them the power of producing their original 
effect. The amount of heat produced by chemical association 
is at least approximately equivalent to the yield of work of the 
chemical forces brought into play. The same quantity of work 
must be employed, on the other hand, to dissociate the compound, 
and reduce the two gases to a free state/ 

In this work 'On Galvanic Currents', Helmholtz was the 
first to apply the two laws of thermodynamics to electricity. In 
order to keep up a current of electricity through any conductor 
it is necessary to expend a certain amount of chemical or 
mechanical work ; the supply of positive electricity in the 
positive end of the conductor must be perpetually renewed, in 
order to oppose the repulsive force of the positive electricity there 
accumulated, and the same holds for the negative electricity 
at the negative end. In accordance, therefore, with Faraday's 
law, the electromotive force of the battery must be proportional 
to the work which can be obtained by the transformations of one 
equivalent of each of the substances in question. Here, how- 
ever, it is not merely the great forces of affinity of the elements 
that unite or separate in fixed proportions that have to be 
considered, but, further, the lesser molecular forces of attraction 
exerted by the water and other components of the solution 
upon its ions, and Helmholtz set himself the task of detecting 
influences of this kind (which are too feeble to be found by the 
calorimetric method) by measurement of the electromotive 
forces. In order to calculate by means of the mechanical 
theory of heat what influence the concentration of a solution of 
salt has upon the E. M.F., a current was led through a salt 


solution, producing on the one hand an equivalent chemical 
decomposition of the solution, and on the other an alteration of 
concentration at the electrodes. This alteration was continuously 
reversed, since, where the current weakened the solution, the 
excess of water was converted into vapour and removed, while 
at the points where the solution became more concentrated, the 
vapour was again condensed. If the water is agitated with the 
dissolved salt, the temperature being kept constant by an 
appropriate addition of heat, then with weak currents the whole 
process may be regarded as a reversible cycle, and the sum of 
the work gained and lost vanishes ; the theoretical conclusions 
thence deduced agreed satisfactorily with the experimental data 
previously obtained. 

At the close of the year 1877, Richard Helmholtz left the 
Polytechnic at Munich, where he had been studying for four 
years; and Helmholtz had the satisfaction of seeing his son 
established in Krauss's locomotive factory, where in 1880 he 
became Head of the Bureau for Construction, and in 1887 Head 

Helmholtz now became more and more occupied with official 

The time was fast approaching for the delivery of the great 
Address which he was to give on August 3, 1878, for the Com- 
memoration of the Berlin University, and in which he contem- 
plated a free and untrammelled statement of his philosophical 
creed. He hesitated for some time as to its title. 

4 The title must come last/ he writes to his wife. ' I have not 
yet decided it. Perhaps "What is Reality?" or "All transitory 
things are but parables ", or " An Appeal to the Ultimate ", or 
perhaps merely " Principles of Perception ".' 

In the end he chose the title ' The Facts that underlie Per- 
ception ', after his wife had written to him, ' I fear that the Ultimate 
would be an unknown aim to many/ After delivering this most 
beautiful and most significant of all his lectures, the contents of 
which have been already outlined, he announces on August 4 
to his wife : 

'I knew it could not be to the taste of the majority. It 
contained new ideas that were bound to puzzle them . . . not of 
course Zeller, du Bois, Kronecker, and kindred spirits. But 
I said to myself that if I had to work, it should be at something 


that interested me too, for in the long run it is better they 
should find me too learned, rather than too trivial/ 

This left him little time for researches in electrodynamics 
and electrochemistry, and this year he only published two 
very interesting points supplementary to his earlier work in 
acoustics and optics. 

The discovery of the telephone was a great surprise to him 
at first, but he soon grasped its scientific principles. 

The thing seemed to him so obvious, he wrote to du Bois, 
that he had not felt it necessary to form any theory about it ; 
but then of course he had for years gone to bed and got up again 
with Fourier's series in his mind, and could not reason from 
himself to other people in this case. 

Du Bois had immediately after the production of the tele- 
phone explained the appreciation of timbre by this instrument, 
by saying that he pictured every sound as analysed into its 
partial tones, basing his opinion on the fact that each of these 
partial tones was conveyed to the hearer's telephone by the 
electrical vibrations of the conducting wire, with alteration of 
phase indeed, but with the same frequency and relative ampli- 
tude. Since displacement of phase, according to Helmholtz's 
earlier acoustical conclusions, is immaterial to the quality of the 
sound, the timbre of the spoken sounds is not interfered with. 
Hermann set up an experiment designed for a fresh theoretical 
verification, in which a current-producing telephone was closed 
by one wire of a coil wound with two parallel wires, while the 
second wire, wholly insulated from the first, was either con- 
nected directly with the observer's telephone, or with a wire 
from a second double-wound coil, the other wire of which led to 
the observer's telephone. Now by the known law of electro- 
dynamic induction, the E. M. F. is proportional to the differen- 
tial quotients of the current-intensity by the time. But since 
with the differential quotient of the sine of a linear function of 
the time, the multiplier of the time participates as a factor in the 
amplitude, Hermann concluded that in this transfer of electri- 
cal oscillations by induction in each of the double coils, the 
amplitude of the electrical oscillations, which correspond to the 
higher partial tones of each sound, would increase, as compared 
with those of the deeper, in proportion to their greater fre- 
quency. Since on the one hand the ratio of the intensities of 


the partial tones emitted by the second telephone must be 
considerably altered, while on the other hand the quality of 
sound remains the same, Hermann found this incompatible with 
the Theory of Timbre which Helmholtz had propounded in his 
earlier acoustical observations. 

Helmholtz then showed, in a paper presented to the Academy 
on July n, 1878, called 'Telephones and Timbre', that if the 
induction of each circuit, not only on those adjacent to it, but 
also on itself, were taken into consideration the conclusions 
made by Hermann exactly confirmed his explanation of the 
timbre. He proved that the intensities of the induced currents 
were independent of the frequency, while their phases on the 
contrary were a little displaced, and at the same time explained, 
from the mathematical expressions he had formulated, what 
had already been observed, i.e. that the deep tones of a man's 
voice generally appeared too weak with ordinary telephones. 
Helmholtz did not take the reaction of the oscillating iron plates 
in the receiver into consideration, because its oscillation has 
a much less amplitude than that of the corresponding plates 
in the transmitter. If the duration of the induction current is 
prolonged without external disturbance for more than o-oi 
second, he finds that with the direct connexion of the two 
telephones the electric oscillations corresponding with the 
highest tones and noises do not vary perceptibly in phase, nor 
in relative strength, from those of the exciting magnetism ; the 
deeper tones on the contrary may be considerably displaced in 
their phases, and somewhat diminished in strength. Quality of 
sound is affected not by means of the electrical motions, but 
only by the co-vibrating iron plates. 

After the fatigues of the Rectorial year Helmholtz went to 
Switzerland for rest, in the first place visiting Boll, who was 
lying very ill at Davos, and wanted to discuss his theory 
of light and colour sensation some notes on which were 
published after his death in 1881 with Helmholtz. From Davos, 
Helmholtz went by Samaden to Pontresina, then by the Italian 
Lakes to Milan, and so by Nervi to Siena, which he had long 
desired to see. On September 24 he writes to his wife from 
Siena : ' To-day and yesterday I have been exploring this old 
and most singular mountain nest. It lies crowded together 
upon a hill-top, surrounded by great walls, and traversed by 


narrow streets. The poverty of the present day strikes one 
acutely in contrast with the mighty remains of former splendours. 
It has much more character than Pisa ; the works of art remind 
one of the Pisan School, especially the Duomo, the facade of 
which dates from Giovanni Pisano, but is far more beautiful 
and richer than that of the Duomo at Pisa. It also has black 
bands, but is covered with a wealth of the finest sculptural 
ornament, which is well proportioned, with fine red marble 
inlaid between. Inside there is a pulpit by Niccolo Pisano, 
father of Giovanni, which recalls the one in the Baptistery at 
Pisa, but is less finely modelled. The interior, which is also 
banded, and decorated with fine belts of colour in the roof, is 
paved with some marvellous graffiti, in white marble, with black 
outlines, and with grey, yellow, and red marble introduced in 
places. The drawing of these is wonderfully perfect, and the 
effect is most remarkable, though they have been much damaged 
in places by people walking on them. Besides the Duomo 
there are many interesting palaces, nearly all stamped with the 
black lozenges of the Pitti and Strozzi of Florence. One sees 
these at every turn. An enormous quantity of pictures has 
accumulated everywhere, mostly in the ancient style, some badly 
preserved, so that the colour has perished, and one can only 
reconstruct them from the quiet, friendly faces. Among the 
later ones, however, Sodoma and Beccafumi almost come up 
to Raphael. Of the former, one can only get unsatisfactory 
glimpses elsewhere : he is skilled in fresco, is less dignified than 
Raphael, and does not possess his great dramatic power, but 
his figures might often pass for Raphael's, and have a very 
pleasing expression. To do justice to Siena, one needs more 
knowledge of the history of art than I possess at present . . . the 
Piazza in front of the Palazzo Pubblico is also very characteristic. 
The Palazzo reminds one of the Doges' Palace at Venice, but is 
more antique and less stately. The Piazza in front of it has the 
form and depth of an ancient semicircular theatre, the whole 
diameter being occupied by the Palace. The rooms inside are 
very large, and contain a number of pictures, mostly of the early 
masters, and half obliterated/ 

After a short stay in Rome, he hastened on to Naples, which 
was new to him : 

1 1 really am in Naples now, and Nature here is incredibly 


beautful. The hotel recommended by Bonghi is high up, some 
200 feet above the sea, on the slope of the hill, in the new street 
Corso Vittorio Emmanuele, which is the limit of the inner town. 
In my room, we have a vertical wall of rock behind us, and 
a deep precipice in front, so that we can see down to the plain 
over the roofs of the nearest houses. In the afternoon, when 
the sun is off my balcony, I need only sit down there in order 
to have the finest view in the world before my eyes, viz. the 
most lively part of the strand, above the clusters of the high- 
roofed houses of Santa Lucia, in the centre of the city, to 
Posilippo at the end, then the Gulf with the blue-greens and 
purples of the Mediterranean, and Vesuvius on the other side, 
with the tongue of land that holds Castellamare and Sorrento, 
and the island of Capri and open horizon of sea beyond. 
Vesuvius has lately formed a new cone in its crater. Yesterday 
evening and early to-day the summit was hidden by clouds ; but 
this evening we saw the column of smoke, which is full of 
ashes, but looks white by day red hot from the rift in the 
mountain below. A fresh outbreak of lava is anticipated. The 
old streams of lava can easily be recognized from here, making 
a greyish black trail against the green of the vineyards. I am 
fortunate in finding the mountain active. The weather is very 
fine and sunny, with a partly clouded deep blue sky ; to-day for 
the first time it is really warm, but not oppressive, no dust, and 
the vegetation greener than I have ever seen it in Italy. The 
elms with thick festoons of vines are touched with brown, but 
the ground beneath is covered with the freshest green crops. 
Even at Genoa and Pisa it was much greener than we had seen 
it elsewhere, but still not so green as this. . . . 

1 To-day I have made my last mountain expedition for this 
year, up Vesuvius to the crater of which I climbed, wandering 
about on its burning lava, and fortunately returning unconsumed. 
The crater is covered with this lava at the bottom now, through 
which the steam has lately forced a new outlet, and is forming 
a fresh cone of ashes round the hole, where one can see it 
working all the time. Every now and then the mountain 
mass, fused by the hot steam, blocks the way, and is then blown 
out with a noise like cannon, when the glowing remains of the 
viscous slag are thrown down again in clouds of smoke, fall 
back upon the cone of ashes, and thus enlarge it. Every five 


minutes or so there was a similar detonation, and we could get 
within loo paces of it, as the glowing slag fell back regularly 
upon the surface of the new cone. The most recent streams of 
lava exhibited a very slow and hardly perceptible motion. One 
could feel it warm to the soles of one's feet, and on pouring 
water into a crack it hissed out again at once. Some cracks in 
the newest lava still glowed red. The alpenstocks caught fire, 
and the guide picked out some of the viscous glowing mass. 
The old walls of the crater were steaming with a penetrating 
vapour, and brightly coloured with yellow sulphur, white salt, 
and green copper. It was in the highest degree interesting and 
impressive, but also rather fatiguing and expensive/ 

From Naples he went by Rome and Trient to see Lenbach 
in Munich, and thence back to Berlin. 

As soon as Helmholtz was free of his Rectorial duties he 
returned to the electrical work that had been interrupted for 
a year, starting off with a study of the contact theory of electri- 
city, which appeared in Wiedemanris Annalen with the title 
' Studies in Electrical Contact-layers '. If, in the theory of the 
distribution of electricity in conducting bodies, the forces of this 
agent, as known by their action at a distance, are alone taken 
into consideration, it is found that in a condition of equilibrium 
the electricity leaves the interior of the body, and forms an 
infinitely thin layer upon its surface alone. But if there should 
be a sudden change in the value of the potential function at the 
limit of two different bodies, as e.g. at the contact of two con- 
ductors under the influence of a galvanic force acting between 
them, then in this case a double electrical layer will be formed 
along the surface: Helmholtz denotes the product of the 
density of the positive electricity into the distance between the 
two layers as the electric momentum of the layer, where the 
distance is to be regarded as small, but not infinitely small, 
since otherwise the work employed in producing the layers 
would have to be infinitely great. 

The supposition of the formation of a double layer, as 
previously made for bodies electrified by contact, was now 
extended by Helmholtz to the case of contact between any 
two bodies. The expressions for the potential difference led 
him in the first place to explain the production of electricity 
by friction, and he succeeded in deriving the relations of the 


series of electrical tensions due to friction, and the theory 
of the electrical machine, satisfactorily from it. The most 
important object of the work was, however, to set forth the 
theory of those phenomena which appear when a fluid is flow- 
ing past a solid wall, and to account for the transition between 
the excitation of electricity by the galvanic opposition of 
bodies at rest, and by the sliding friction of solid bodies. 
Starting from the view that the fluid is in galvanic opposition 
to the wall of the vessel, and that they both form an electrical 
double layer along their surface of contact, he succeeded in 
explaining two phenomena that are very closely connected 
the propagation of fluids through narrow tubes, in consequence 
of the passage of an electrical current through the same, and 
the appearance of electromotive forces, when fluids are driven 
through similar tubes by hydrostatic pressure. The theo- 
retical developments and the comparison with the results of 
G. Wiedemann and Quincke's experiments, however, refer 
only to capillary tubes, while in wider tubes more complicated 
phenomena of motion appear at the point at which the current 
enters. In this paper, as in the Faraday Lecture later, and 
in a series of subsequent electrical researches, Helmholtz 
comes back repeatedly to the close connexion between the 
electrical and the chemical forces, as well as to the explanation 
of Volta's fundamental experiment. 

He assumes that electrical and chemical forces are essentially 
the same, and supports the view that the presence of the forces, 
which when unchecked set up chemical processes, suffices to 
call out the corresponding electrical distributions, even before 
the appearance of the chemical combination ; it does not seem 
to him necessary that a complete chemical process must inva- 
riably be the precursor of Volta's charges. Helmholtz is here 
at one with Faraday, who assumed the identity of the forces 
of chemical affinity with electricity, and expressed the view 
that the atoms adhere to the electrical charges, and the opposed 
charges again to each other, without thereby excluding molecular 
forces, which act directly from atom to atom. 

Helmholtz regards Volta's much-contested experiments as 
unimpeachable ; if a momentary metallic connexion be made 
between a copper plate and a zinc plate, at minimal distance 
from each other, and well insulated by bars of shellac standing 


opposite one another like the plates of a condenser, and if they 
are then moved farther apart, the copper will be negatively, 
and the zinc positively charged. 

The necessary experimental laws had already been expressed 
by Helmholtz in the ' Conservation of Energy ', to the effect that, 
so long as only conductors of the first class are involved, i. e. 
conductors which undergo no electrolytic dissociation during 
conduction, and so long as these conductors are at the same 
temperature and at rest, the passage of the electric current 
always tends to equilibrium of the electricity ; it is only when 
the conductors are moved by an external force that electrical 
currents, or concentrated accumulations of electricity, arise. 
This class of experiments further comprises those with dry 
metal plates, which are connected by dry and insulated metal 
wires. Since in this case, with each new arrangement of such 
conductors at the same temperature, the electric motion soon 
produces equilibrium, we must assume its cause to be forces 
of a simple character, which obey the law of the conservation 
of energy. In this essay Helmholtz brought forward the view 
that these phenomena arise from forces of attraction, which 
the different substances possess in different degrees for the 
two electricities, and which only act at perceptibly small 
distances. When two pieces of copper and of zinc are in 
contact, and the zinc attracts the positive electricity more 
strongly than the copper, it flows to the zinc, and charges it 
positively, while the copper remains negative, until the electrical 
attraction produced by this charge, and acting at a distance, 
which draws the positive electricity back to the copper, 
restores equilibrium with the attraction of the zinc. 

Moreover the vis viva which a particle of positive electricity 
acquires from the influence of the zinc and copper alone, in 
its transfer from copper to zinc, will be equivalent to the 
vis viva lost by the same electric particle through the attraction 
of the negative charge of the copper, and repulsion of the 
positive from the zinc, in the same way. This last magnitude 
calculated for the unit of positive electricity is, however, termed 
difference of electrical potential-function between the copper 
and the zinc. 

This theory thus demands that the electricity contained in 
the copper and zinc when they come into contact shall be so 


distributed between the two that the difference of electrical 
potential shall be of a definite magnitude, determined by the 
nature of the metals. From this it is at once obvious that 
conductors of this sort must be subject to the laws of the 
galvanic tensional series, and that chains consisting of three 
or more conductors of the first class at the same temperature 
can never produce an electric current, since the attractive 
forces of the metals for the electricities can only cause the 
electricity to assume the state of equilibrium, which is imposed 
by these forces of attraction. Volta assumed a separating force 
residing in the surface of contact, and believed that the positive 
electricity, having once entered the zinc by the surface of 
contact with the copper, could flow out again without further 
obstacle into any conductor in which it was not opposed by 
any new force of separation at the contact surfaces; in 
Helmholtz's opinion, on the contrary, the positive electricity 
was held fast in the zinc by attraction, and this attraction must 
be overcome by a corresponding expenditure of work, before 
the positive electricity can be removed from the zinc again 
by any other natural force. Electrolytic conductors, however, 
do not follow the tensional series, because they are dissociated 
by each electric motion, and cannot, of course, while this dis- 
sociation is proceeding, arrive at a state of resting equilibrium. 
Helmholtz confirmed his theory by a long series of elabo- 
rate experiments, which he performed with the help of 
W. Thomson's quadrant electrometer. 

In his earlier critical work on the Theory of Electro- 
dynamics, Helmholtz had evinced more and more disposition 
to adopt the Faraday- Maxwell hypothesis, but he had always 
insisted that a complete understanding of the theory of electro- 
magnetic phenomena, and an ultimate decision as to the claims 
of the several hypotheses, would only be possible after an exact 
investigation of the processes that obtain for unclosed currents 
of very brief duration. Weber had, indeed, endeavoured to 
minimize certain difficulties and contradictions in his electro- 
dynamic hypotheses, by attributing to electricity a certain 
amount of inertia, such as pertains to heavy bodies ; Helmholtz, 
however, very soon recognized that true inertia would be 
proportional to the mass of the electricity in motion, inde- 
pendent of the position of the conductor, and that this would 


be manifested in the retardation of the oscillatory motions of 
the electricity, as appears with sudden interruptions of the 
electrical currents in wires of high conductivity. Since it was 
to be expected that it would be possible by this means to 
determine an upper limit to the value of this inertia, Helmholtz, 
when he had to set a prize subject in physics for his students 
at the end of the summer term, gave them the task of devising 
experiments which should demonstrate the strength of extra- 
currents, 'with the certain and subsequently confirmed ex- 
pectation ' that Heinrich Hertz, who by von Bezold's advice had 
been working in the Physical Laboratory of the University 
under Helmholtz's direction since the autumn of 1878, and 
whom the latter had recognized even in the elementary course 
as a student of quite extraordinary promise, would interest 
himself in it, and attack the problem successfully. 

The magnitude of the extra-current was to be used in 
ascertaining an upper limit for the mass in motion; and the 
extra-currents from double-wound coils with their branches 
traversed by the current in opposite directions were specially 
recommended. The precise answer given by Hertz showed 
that at most -fa to ^ of the extra-current in a double-wound 
coil was due to the inertia of the electricity; investigations of 
the influence of the centrifugal force of a rapidly rotating plate 
upon the motion of the electrical current flowing through it 
led the gifted young investigator to a far lower superior limit 
of the value of inertia in electricity. 

These researches of Hertz, the results of which were plainly 
foreseen by Helmholtz, gave substantial support to the 
Faraday- Maxwell hypothesis of the nature of electricity, and 
confirmed Helmholtz in his opinion of the accuracy of Faraday's 
conceptions. The phenomena of diamagnetism were explained 
in the simplest manner, on the assumption that diamagnetic 
bodies are such as are less magnetizable than the surrounding 
media with which space is filled, so that even the space that 
is free from all ponderable masses, or the luminiferous ether 
contained in it, must be magnetizable. According to Clerk 
Maxwell it was, however, of essential importance for Faraday's 
Theory of Dielectric Polarization and the elimination of action 
at a distance, to know whether the origin and passage of 
dielectric polarization in an insulator would give rise to the 


same electro-dynamic action in its neighbourhood as a galvanic 
current in a conductor. In order to obtain evidence for this, 
Helmholtz made it the subject of one of the great prize com- 
petitions of the Academy, by which Hertz was led to his 
remarkable discoveries. These afforded the direct proof of 
the accuracy of the hypothesis which Faraday and Maxwell 
had advanced as highly probable, i.e. that the oscillations of 
light are electrical oscillations in the ether that occupies space, 
and that this in itself possesses the properties of an insulator 
and a magnetizable medium. 

* There can no longer be any question/ says Helmholtz later 
in his classical preface to Hertz's Principles of Mechanics, 'that 
the luminous vibrations are electric vibrations in the ether that 
fills space, and that the latter itself possesses the properties 
of an insulator and a magnetic medium. The electric oscilla- 
tions in the ether represent an intermediate stage between the 
relatively slow motions produced by the elastic and resonant 
vibrations of a magnetic tuning-fork, and the enormously swift 
vibrations of light, but it can be shown that their rate of 
propagation, their character as transverse oscillations, and the 
concomitant possibility of polarization phenomena, and of 
refraction and reflection, correspond completely with those 
obtaining in the case of light and heat rays. The electrical 
waves are only lacking in capacity to affect the eye, like the 
dark heat rays, since their frequency is not sufficiently high. 
It was verily a great achievement to acquire absolute proof 
that light, that mysterious and powerful force of nature, is so 
nearly related to a second equally mysterious and perhaps still 
more potent force, that of electricity. For theoretical science 
it is perhaps even more important to understand how what 
appears to be action at a distance can be propagated by trans- 
lation of action from one layer of the intermediate medium to 
the next. The mystery of gravitation indeed remains, since 
we cannot yet logically explain it otherwise than as due to 
a true action at a distance/ But it took Hertz nearly the 
whole of the last decade but one of the Nineteenth Century to 
establish and work out his mighty conception, which is to-day 
the foundation of the whole modern doctrine of electricity. 

In the Easter holidays Helmholtz visited Ludwig at Leipzig, 
and wandered through the Schwarzathal in Thuringia, and at the 


close of his Rectorial year, wearied with incessant official and 
scientific work, he went for three weeks to Pontresina, where 
he had been used to refresh himself in body and mind by 
long visits. 

'Although Helmholtz grew older,' writes Blaserna to the 
author, ' and began to feel trouble in his respiratory organs, he 
did not give up climbing. He looked on mountaineering as 
a cure, when properly administered. Thus he told me one 
day that he intended to climb the magnificent Piz Languard. 
I offered myself as his companion; but for a long time he 
would not hear of it, as he thought I should go too fast for him. 
It was only when I promised to keep behind, and give him no 
opportunity for talking, that he accepted my company. It is 
notoriously an ascent with a good path, which a strong climber 
can do in three hours, a moderate walker in four. We took 
quite six hours over it: but Helmholtz arrived in good con- 
dition, spread out his map, and studied all the different 
mountains, and after we had rested there an hour we started 
down again, and returned safe and well to Pontresina. In 
the last years of his visits to Pontresina he gave up the 
mountains, but he made two little ascents with great regularity 
every morning and afternoon, up the Muottas da Pontresina 
or da Celerina, or up the Schafberg. The iron regularity with 
which he accomplished his two walks every day was amazing. 
They were unfortunately his last. A few years later, Pontresina 
lost, as one might say, by his death its distinguishing scientific 

After he had left Pontresina, and had spent some weeks with 
his wife at Interlaken, he wrote from Thun on September 15, 
1879, to his friend Knapp at New York : ' Relatively speaking, 
things have gone better this year than previously in Berlin. I 
have at last learned how much I can get out of myself in work 
and pleasure, and am stubborn and unresponsive to people 
who try to take up my time when I am tired/ 

In the Easter holidays of 1880, he took a journey of several 
weeks in Spain, going as far as Tangier in Africa. His full 
letters to his wife give us a lively account of the impressions 
made on him by art and nature ; a few short extracts may be 
cited : 

' Barcelona, Palm Sunday, March 21. From Nimes we made 

Y 2 


an expedition to the Pont du Card, a strong bridge over the 
torrent, constructed for a Roman aqueduct in the time of 
Augustus, in a solitary mountain valley, as high as the highest 
cathedral. Rousseau is reported to have said of this that it 
was the only thing he had seen in his life that had exceeded 
his anticipations. The expedition takes at least five hours, and 
we discussed whether it were worth devoting the day to a 
fragment of Roman aqueduct when we had seen so many. 
But we felt like Rousseau, save that it was not the first time 
we had had such an experience. . . . 

1 Here we have spent two very pleasant days, with a Passion 
Play in the Theatre, which was very remarkable, with a great 
Procession, and so on. ... The Theatre is a vast building for 
four thousand spectators, well proportioned and comfortably 
arranged, so that our northern Court Theatres would be put 
to shame by it. The old Passion Play was given in the 
Catalonian dialect, which of course we could not understand, 
since we are much in the same plight with modern Spanish. 
It was cleverly staged with all modern accessories, and 
frightfully realistic in every detail, so that the whole per- 
formance succeeded in impressing one as the reading of it 
never does. . . . The Procession on Palm Sunday was escorted 
by a troop of Roman warriors in costume, with a band and a 
captain There was a great crowd of people, long processions 
of Catholic Guilds, students and young men in strange costumes 
of shiny black linen, cut with trains like women's clothes with 
gauffred frills, dragged the figure of the Christ, which was 
drawn along on boards, and represented the bowed Christ of 
the Mount of Olives. ... L. urged us to visit his friend, the 
Professor of Chemistry, Don Ramon Manjarez, in his labora- 
tory, and allow him and the other professors to take us round 
the new University buildings. I had the satisfaction of finding 
my acoustical apparatus fairly complete.' 

'Madrid, Good Friday, March 26. The Escorial, gigantic 
burying-place of the Spanish Kings, lies some distance from 
Madrid in a rocky waste among the mountains, and gives 
a certain idea of serious grandeur and artistic taste in the 
fanatical Philip II which raised him above his childish 
successors. One can see that he was in terrible earnest over 
what he wanted, and what he designed for himself is simple 


to a degree, and almost meagre. The church, on the other 
hand, is of vast proportions, erected with taste and simplicity 
by Neapolitan architects, something after the model of what 
S. Peter's in Rome might have been if the rococo style had 
not spoiled it. What Philip's successors added is merely 
childish, with the exception of some beautiful Gobelins after 
Teniers' sketches, for which indeed the finest pictures of 
Raphael and Titian were sent to the lumber-room, whence 
they were rescued for the Museum. Taken as a whole it is 
an historical monument, testifying to the spirit of its age, even 
if that was antagonistic to us. ... The Picture Gallery is 
imposing; the collection of people whom Velasquez counter- 
feited is so extraordinarily fresh and full of expression, that 
they seem to be our contemporaries. . . . Yesterday I went 
early to Toledo, the former Palace ; a crowded mountain nest, 
surrounded on three sides by the Tagus in a deep gorge, 
a natural fortress with all kinds of Ostrogothic and Moorish 
remains ; these are insignificant, but the Gothic Cathedral 
pure, fine, and luxuriant in form, with such elegance of stone- 
and woodwork, in which the influence of the Alhambra School 
seems to survive, that it overshadows everything I have 
previously seen in Gothic churches. Besides, it has been 
comparatively little spoilt by later additions from the Jesuit 
period. It is far more characteristically and consistently 
Gothic than the Cathedral at Milan, and therefore makes an 
even purer impression of perfect beauty of form and dignity. . . . 
Unfortunately the exterior is almost invisible. . . .' 

' Cordova, Tuesday, March 30. Here then is the great mosque, 
the Cathedral of to-day, a wonder of architecture, exotic and 
fabulous, an immense flat tent-roof supported by more than 
1,000 pillars, united by fantastic double arches, origin- 
ally open everywhere to the orangery of the fore-court, with 
the chapel for the preservation of the Koran behind, adorned 
with wondrous marble work and mosaics, all in carpet patterns. 
Not far off is a similarly decorated chapel, the place of prayer 
of the Khalif. Unfortunately they have closed it in as a church, 
separating it with walls from the court, and have erected 
a high choir in the baroque style, so that one can only imagine 
how airy, and clear, and cool, and light it must have been 
before they made a church out of it. The question is forced 


upon one, how this highly developed civilization came to 
an end. The Moors took none of it back to Africa, and 
what the Spaniards may have learned from them disappeared 
within the next hundred years, save for the great system of 
irrigation, which made the land fertile so far as it extended. ... 
Next day we took a walk to the slopes of the Sierra Morena 
north of the city, whence we had a good view of the fertility of 
the country. Little runlets of water come down from the hills, 
and are carefully distributed. The orange-trees are like forest 
trees in their growth. I never saw any like them in Italy, and 
they were more covered with fruit than any apple-tree I ever 
saw, while among the fruit were fresh buds and wreaths of 
blossom ; wild roses, irises, spireas, violets, all in full bloom, as 
one sees them in Germany on the sunniest days of June. Among 
them are solitary date palms, shooting gracefully heavenwards/ 
' Granada, Friday, April 2. And now we have really seen 
the Alhambra, which is as marvellous in reality as it has been 
described in books and pictures. Marble in the most elegant 
carved work, with a superfluity of wondrous patterns. . . . After 
luncheon there was a bull-fight, the first in a new arena, 
a great festival for the people. As regards the spectators it was 
extremely interesting. The arena is constructed exactly after 
the old pattern (it is true that the upper part is of wood) ; the 
public behaves exactly as was described by the Roman authors. 
The crowd is seized with a raging intoxication ; the shouting is 
uninterrupted, now applause, and now hisses. It is necessary 
to get there an hour before the time, otherwise one has no 
chance of a seat. During this time we were entertained by 
a feeble jet of water that was supposed to sprinkle the Square, 
and by the orange-sellers who flung their fruits up to the 
purchasers in the top rows of seats, and were paid in the same 
way, to the entertainment of the public. Each good throw was 
applauded, each bad one hissed. The elegant ladies for the 
most part, unfortunately, were above us; those we could see 
were in picturesque national costumes, pretty to look at, but 
very bold in colouring : one was dressed like a toreador. These 
last are fine fellows, slim, agile, dexterous and reckless, so that 
it is a pleasure to see them moving about in their splendid and 
elegant costumes. The banderilleros especially, who let the bull 
charge them without cloak or weapon, and then leap aside at 


the moment when he threatens to impale them, and fix the 
barbed plumes of feathers and ornaments on his neck, are 
incomparably skilful. Immediately after, indeed, the bull is 
diverted by waving a cloak in front of him, to prevent him 
from repeating his attack. The fate of the bull is really 
a subject for congratulation ; he falls in battle instead of in 
the slaughter-house. It is true that the animal, by the time the 
matador advances alone to give him the death thrust, is in the 
last degree exasperated and distraught, and for the six bulls 
that were immolated this coup de grace was successful at its first 
delivery in two cases only. But what really shocks one is the 
way the horses are treated, not merely in the arena but every- 
where else, like those destined for death that are ridden against 
the bull with bandaged eyes by the picadores, and are driven by 
goads to the attack so long as they can carry them ; and the 
way the public yells for new horses, Caballo ! caballo ! when only 
one or two survive; this is the really horrible part of the 
spectacle. If it were only an exhibition of human courage one 
could forgive an element of savagery. But in reality they tire 
out the bull by letting him rush repeatedly at the defenceless 
horses, which he hates more than men, and it is only when he 
is utterly exhausted that the men take part in the encounter/ 

* Malaga, Tuesday, April 6. Malaga is not particularly 
characteristic. A fine Renaissance Cathedral, the tower of 
which we climbed to get a general view of the town, is fairly 
large and elegant. The sea-winds are injurious to vegetation 
close to the town, but wherever shelter is afforded by the 
mountains, there are huge groves of oranges, plantations 
of sugar-cane, and the like. . . . We find Wattenbach's book 
a great treasure ; it is more useful than Murray, Gautier, and 
Amici. He has distinct talent as a Baedeker, and his prose is 
not unworthy of Spain.' 

1 Tangier, Tuesday, April 13. We spent a very interesting 
day in Gibraltar; one of the English officers, Col. Lempriere, 
whom we met in the Ronda, gave us an order for the Galleries 
where the cannon are posted round the northern side of the 
rock, and we roamed from 10 till 4 through the tunnels of the 
battery. ... It is impossible to express all the astonishment one 
feels here in Tangier, on being suddenly plunged into the 
midst of the Mohammedan world, as it is presented to eye and 


ear. The variety of costumes and of nakedness is indescribable. 
The turbans, which are only worn by the Moslems, are kept 
scrupulously clean, and make a good effect ... as also the white, 
or black and white striped burnous, with the extraordinarily 
characteristic eyes and sharp features of the older men beneath 
them. The women, so far as they appear in the streets, are 
veiled in not over clean and coarse rough sheets, which they 
are not too particular in drawing over their faces/ 

Helmholtz then returned by Seville, Bordeaux, and Paris to 
Berlin, where he immediately resumed his lectures. 

At this time he was already occupying himself with the 
arduous work in thermodynamics which stood in the closest 
relation with his later discoveries in the principles of mechanics, 
but was not published for another two years. He was, however, 
obliged seriously to consider his health a little more in his 
enormous undertakings. Even in Seville a slight fainting fit 
had alarmed his travelling companion, and again after the 
fatigues of the summer session, a few days before the holidays, 
he met with an accident by slipping, due probably to a sudden 
swoon, which might have had the most serious consequences. 

By August 8, however, he was able to announce to Ludwig 
that he was so far better that he should begin the journey to 
Munich with his wife in a few days by easy stages, and would 
come first to Leipzig, to rest there a little. The projected 
journey was carried out, after which he went for a few weeks' 
rest to Switzerland, returning then to Berlin to resume his 
thermodynamic work. In the meantime Hertz had been ap- 
pointed his Assistant in the Physical Institute, and remained 
there till the year 1883. Helmholtz was also working out 
certain points that were intimately connected with his earlier 
researches in electrodynamics and electrochemistry. 

The fact that a certain distribution of magnetism occurs in 
the molecules of soft iron in the vicinity of a magnet, so that the 
soft iron itself attracts and repels small magnetic bodies, was 
known not to be peculiar to iron. Faraday had shown that this 
effect is visible in almost all bodies, and that similar phenomena, 
indicating a distribution of the opposing kinds of electricities 
in the molecules of electric insulators, are called out by electric 
forces. The phenomena were treated mathematically for the 
motions of rigid magnets and magnetizable iron by Poisson ; 


W. Thomson had extended this theory to the motions of rigid 
bodies in magnetizable fluids, and shown them to be related to 
Faraday's diamagnetic experiments. So soon as the molecules 
of magnetic or electrically polarized media can be displaced in 
relation to each other, molecular action necessarily comes into 
play as well as the original forces acting at a distance. Faraday 
had assumed a state of tension in the magnetically or dielec- 
trically polarized media in the direction of the lines of force, in 
consequence of which these tend to shorten, while a pressure acts 
at right angles to these lines, which tends to drive the substance 
out in that direction. After W. Thomson in 1843 had proved 
that forces of this nature could produce the same effect as direct 
action at a distance on Coulomb's theory, Clerk Maxwell had 
made this assumption of Faraday the basis of his whole theory 
of electricity and magnetism. The remarkable effort of electric 
insulators to expand transversely to the direction of the elec- 
trical lines of force had already been established by experiment, 
when Helmholtz, in his communication to the Berlin Academy 
(Feb. 17, 1881) 'On the Forces acting on the Interior of Magnetic 
or Dielectrically Polarized Bodies J , proposed a complete theory 
of the phenomenon that insulators tend to alter their shape 
under the influence of dielectric forces. 

He shows that the tensions which produce an expansion 
perpendicular to the lines of electric force are (without any 
special assumption as to the internal constitution of dielectric 
media) a necessary consequence of the law of conservation of 
energy, and of those laws which by Poisson's theory regulate 
temporary magnetism, and are directly transferable to dielectric 
polarization. By supposing that the constants in Poisson's 
equations may alter, on the one hand, in consequence of the 
altered density of the medium, on the other in virtue of the actual 
displacement, he arrives at another distribution of the potential, 
and thus at a calculable alteration of energy. But since the 
equivalent of these is the excess of work which the pondero- 
motive forces must accomplish to produce the displacements of 
the points, beyond what is required when no dielectric tension is 
present, he was able to calculate these forces where the change 
of energy is determined. The discussion of how far the calcu- 
lated forces may be resolved into molecular forces shows that it 
is possible to replace them by a pressure that acts within a 


dielectric upon a surface-element, the normal to which forms a 
given angle with the direction of the lines of force, and a tension 
that is effective in the direction of the same. Helmholtz finally 
concludes from the expressions for the forces, that the two 
views that, namely, which postulates forces acting at a distance, 
and that of Faraday- Maxwell, according to which there is only 
polarization of the media may thus exist side by side. 

At the same time Helmholtz published a brief notice in 
Wiedemann's Annalen on 'An Electrodynamic Balance', which 
he had constructed with the object of avoiding, in the measure- 
ment of galvanic currents in absolute measure, the disturbances 
which the changes in the direction and intensity of the earth's 
magnetism produce by their electromagnetic action. At the 
ends of the beam of a small chemical balance, he suspended two 
coils of copper wire, the height of which is equal to their in- 
ternal diameter, the axes being vertical ; two coils of the same 
height and greater radius were held in a fixed position some- 
what above the movable coils by a horizontal metal bar, fixed 
by its centre to the pillar that carries the balance. The con- 
nexions of the wires are so arranged that one of the movable 
coils is attracted and the other repelled by the fixed coils ; the 
attracted coil rises, the repelled sinks, when current is passed 
through the circuit, each of the movable coils being connected 
with the wires that carry the current by two strips of brass foil. 
The total action of both coils is maintained in equilibrium by 
appropriate weights, so that the force which opposes the electro- 
dynamic force, and measures it, is subject to gravity alone, with 
no variations, such as affect the earth's magnetism. 

In the Easter holidays of 1881 Helmholtz went to London 
with his wife at the invitation of the Chemical Society to give a 
Lecture in the place ' in which the great investigator Faraday, 
whose memory was to be honoured, had so often surprised his 
admiring audience by his revelations of the unsuspected secrets 
of nature J . 

His discourse on 'The Recent Development of Faraday's 
Ideas on Electricity ' (delivered in English, after Roscoe had 
read it through and altered a few expressions) ranks from its 
form and content among the most beautiful and profound of his 

'His Faraday Lecture,' writes his wife, 'was a brilliant success. 


The subject was incomprehensible to me, as he discoursed for 
the most part on atoms, and the influence of electricity on 
chemical properties : but the enthusiasm when he entered, and 
the cheers whenever he expressed his own conclusions or 
opinions, were delightful.' 

Commencing with an historical review of the development of 
Electrodynamics, which culminated in a brilliant exposition of 
the Faraday- Maxwell Theory, he for the first time gave a 
connected account of the relation between electrical and 
chemical forces, as we have followed it above in his separate 
publications. To arrive at an understanding of the relations 
between electrical forces and chemical affinity, he shows from 
the phenomena of electrolytic dissociation how we are to picture 
the ponderable atoms as bound up with electricity. He con- 
cludes from the assumption that ions are charged with electri- 
city, that a wandering group of atoms invariably carries the same 
charge of electricity with it, and that electricity itself is com- 
posed of definite elementary particles which behave like the 
atoms of electricity. An essential factor in chemical affinity is 
formed by the attractions of the opposite electricities for each 
other in the compounds. When a unit of positive electricity in 
an atom is held by the unit of negative electricity in another 
atom, these electricities will be externally inactive, and the atoms 
will adhere together with saturated affinity. 

In Hertz's opinion this theory gives us a consistent representa- 
tion of the nature of valency, and this alone he considers a 
sufficient proof of the weight and significance of the conceptions 
evolved by Helmholtz of chemical processes. When, on his 
seventieth birthday, Hofmann spoke of his researches in the 
interpretation of chemical processes as the inauguration of a 
new era in chemistry, by means of which new light had been 
cast upon entire regions, and these had been brought essentially 
nearer to our comprehension, Helmholtz expressed his thanks 
in the most modest words : 

' I am exceedingly grateful that you recognize and feel an 
interest in my amateur efforts in chemistry, and are so good as 
to tell me so.' 

After delivering the Faraday Lecture in London, Helmholtz 
went on to Cambridge, where he was made Doctor of Laws ; 
stayed some time in Glasgow with his friend W. Thomson ; 


and then returned direct to Berlin, to prepare his thermo- 
dynamic work for publication. 

The Faraday Lecture had made quite an unwonted stir 
among English men of science, and Sir William Thomson in 
consequence approached Helmholtz with the request that he 
would give some popular lectures in England in the autumn of 
that same year. Helmholtz replied on July 15, thanking him, 
but declining : 

I Best thanks for your friendly invitation to return to Glasgow. 
But I find myself incapable of acceding to your offer. In the 
first place, I know too little of the public I should have to 
address, and am not usually very successful in my attempts at 
giving popular lectures to a large audience from mixed classes ; 
in the second place, the preparation of a lecture in English 
takes up too much of my time, and there is every reason why I 
should husband it, seeing that I am sixty years old this year, 
and still have much work that I want to accomplish.' 

After recuperating as usual from the fatigues of the summer 
session in August at Pontresina, where he celebrated ' the 
solemn day on which he parted with the fifties' by a tiring 
twelve-hour expedition to the Diavolezza, he went on September 
15 to Paris for the Electrical Congress, which again tried his 
working powers sorely, but afforded much that was interesting 
and stimulating. 

I 1 went with du Bois to the Opening Meeting/ he writes to 
his wife. 'The Ministre des Pastes is President, three other 
Ministers are the Vice- Presidents selected for France. The 
foreigners were still to be chosen : Sir W. Thomson, Professor 
Govi of Turin, and your husband were elected. We took our 
seats along with His Excellency M. Cochery, amid great ac- 
clamation. The session itself was merely a formality; there 
are a great many interesting people at the Meeting. ... In 
Congress we have had session after session of sections, com- 
missions, sub-commissions, and private committees, to decide 
the question of electrical units of measurement, as between 
Germany and England. It seems now to be happily settled. 
I made three or four speeches in French each day, which it 
was fortunate you could not hear. Sir W. Thomson and 
an English lawyer Moulton are the chief speakers on the 
English side. For the rest I am happy in the approval of 


my audience, and therefore go on flinging my bad French in 
their faces.' 

From Paris, Helmholtz returned by Florence and Vienna 
(where he and Sir W. Thomson visited the Electrical Ex- 
hibition) to Berlin, and proceeded to report on the decisions 
of the Paris Congress in written memoirs and addresses to 
various Scientific Societies. Before the end of the year he 
entered more precisely into the results of the proceedings in 
a lecture to the Electro-Technical Union, ' On the Electrical 
Units as determined by the Electrical Congress assembled in 
Paris, 1881,' and in another given the following year to the 
Physical Society in a ( Report on the Proceedings of the Inter- 
national Electrical Commission ', which were summed up in his 
article ' On Absolute Systems of Measurement for Electrical 
and Magnetic Magnitudes ' in Wiedemanris Annalen for 1882. 

Helmholtz had now practically finished his fundamental 
researches in thermodynamics, and was engaged on the 
difficult problem of preparing them for publication. -At the 
same time he was directing various pieces of experimental 
work, which were of especial theoretical interest to him in 
establishing his chemical views. 

On November 3, 1881, he presented a paper to the Academy 
'On the Galvanic Polarization of Mercury, and some new 
Experiments of Herr Arthur Konig relating to the same/ 
which Konig had carried out in the University Laboratory 
under Helmholtz's direction. Their purpose was to determine 
the capillary tension of galvanically polarized surfaces of 
mercury, in which the disturbing influence of the alterable 
adhesion of the two fluids to the glass walls was eliminated. 
The optical difficulties in measuring the difference in level 
between the top and the maximal circumference of a quiescent 
drop of mercury were also avoided, and the measurement of 
the surface-tension of the quicksilver was obtained by observing 
the curvature of the summit of a drop of mercury, which could 
be determined with the greatest precision by the ophthal- 
mometer. The drop projected from the upper circular opening, 
9 mm. in diameter, of a glass vessel, and was surrounded by 
the electrolytic fluid contained in a wider vessel. By a special 
arrangement the top could be more or less protruded from the 
mouth of the narrow vessel, and arranged in such a way 


that the curvature was greatest at the vertex. The experi- 
mental series coincided in showing that the surface-tension 
reaches a maximum at a moderate degree of polarization, 
which differs for different fluids. Helmholtz now assumed that 
the forces under the influence of which equilibrium is produced 
at the polarized surface, between the molecular and electrical 
forces acting at the summit of the drop, are conservative, and 
finds on this assumption that there is no difference of potential 
between the mercury and the fluid in the state of maximal 
surface-tension, and that the surface presents no trace of an 
electrical double layer. In conclusion he showed further, by 
using the mercury as an electrode, that Faraday's electrolytic 
law, by which, where no electrolysis is possible, there can be 
no transference of electricity from the metal to the electrolytes, 
or vice versa, is only in apparent contradiction with the ex- 
periments on the galvanic currents that can be excited by the 
successive immersion of two similar electrodes in the same fluid. 

The year 1882 brought high honour and distinction to Helm- 
holtz and his family : he was elevated by the Emperor William I 
to the ranks of the hereditary nobility. The appearance of 
Vol. I of his Wissenschaftliche Abhandlungen ( l Scientific 
Papers'), followed in the next year by a second volume, had 
brought the astonishing extent of his great scientific achieve- 
ments before the eyes of the world. 

A year of hard work brought him to the conclusion of his 
profound investigations in thermodynamics, and these at once 
formed the starting-point of his remarkable theory of the 
statics of monocyclic systems, culminating eventually in the 
fundamental researches into the principle of least action with 
which he was occupied to the end of his life. On September 
18, 1882, he writes to Thomson : 

'After ten months of work I was longing for undisturbed 
rest, for which I always find Pontresina one of the best places 
in the world. On October 16 I have to go to Paris as a Member 
of the International Commission of the Electrical Congress. 
My Faraday Lecture put me on to electrical researches : I 
hope you have received my first note on this subject, " On the 
Thermodynamic Value of Chemical Actions." A second has 
just been published, a comparison of the chemical energy 
of solutions, &c.' 


Since the loss of mechanical energy by friction produces 
heat, while gain of mechanical energy produces loss of heat, and 
since further the sum of energy lost and gained is proportional 
to the sum of the heat gained or lost, heat must be regarded 
as a form of energy, and it follows that every particle of a warm 
body must always be moving in a constantly varying direction, 
so rapidly that it undergoes little or no alteration of place in 
the body. But if so, a part of the energy of a warm body 
must be in the form of kinetic energy, and since every mode of 
energy can be transformed into heat, it follows that the energy 
can be measured in the form of heat. But from the law of the 
conservation of energy it is impossible to determine whether 
work can be unconditionally transformed into heat energy, and 
the latter, conversely, into work, and the same for all the other 
natural forces. Helmholtz accordingly turned in the first place 
to the determination of these important theoretical and practical 
relations. He endeavoured to ascertain how large a portion of 
the heat developed in a galvanic cell by chemical processes 
reappears as current energy, and arranges the forms of energy 
in different grades, according as they are more or less com- 
pletely capable of conversion into mechanical work. 

In the fundamental papers on 'The Thermodynamics of 
Chemical Processes ' (communicated to the Berlin Academy on 
February 2 and July 27, 1882), he develops in mathematical 
form the relations between the laws of heat, of electricity, and 
of chemical phenomena, from which an identity of chemical 
valencies and electrical potentials of the atoms appears probable, 
so that the electrochemical processes would seem to be an 
ordered motion of the atoms and molecules, directed along the 
co-ordinates of space, while heat is a similar process, but un- 

The question as to the connexion between the electromotive 
force of batteries with unpolarizable electrodes, and the chemical 
changes which take place in them, led Helmholtz to the more 
general question as to what portion of the energy present in 
a body can be converted into other forms of work, and carried 
him on to his work on the thermo-dynamics of chemical 
processes, which again were only the prelude to his great 
researches on monocyclic systems. Owing to the introduction 
of potential energy, which Helmholtz had designated earlier as 


quantity of tensional force, the analytic development of dynamics 
had been essentially simplified and generalized. As a rule, 
however, alterations of temperature had not been taken into con- 
sideration in the application of this conception, either because 
the forces of which the work-equivalent was to be calculated, 
e. g. the force of gravity, did not depend on temperature at all, 
or because the temperature during the processes under in- 
vestigation might be regarded as constant, or as a function 
of definite mechanical alterations, viz. in sound-waves, as 
a function of the density of the gas. But if the physical 
constants occurring in the value of the potential energy, such 
as density, and the like, vary with the temperature, which 
would make that energy a function of the temperature, then the 
integration constants comprised in the value of such potential 
energy would require a purely arbitrary determination for 
each new temperature ; the transition from one temperature to 
the other would not be possible. 

Previous investigations of the work-equivalent of chemical 
processes referred almost exclusively to the quantities of heat 
that appear or vanish when compounds are formed or decom- 
posed, whereas most changes are connected with alteration 
of the state of aggregation and density of the bodies. These, 
however, produce or consume work under two forms, as heat, 
and as unrestrictedly transformable work. A supply of heat is 
not unrestrictedly convertible into other work-equivalents, but 
can only be partially transformed, and only on the condition of 
a simultaneous transference of the remainder of the uncon- 
verted heat to a body of lower temperature. Since in most 
chemical processes the changes of melting, evaporating, &c., 
also attract heat out of the environment, it is necessary to 
inquire in what proportions mechanical and thermal energy 
are obtained in these cases also. When we further consider 
that chemical energies per se may produce not merely heat but 
other forms of energy as well, without any alteration of tempera- 
ture in the combining bodies being required, proportional to 
the work done, as e.g. in the work produced by galvanic 
batteries, it is evident that there must in chemical processes 
also be a distinction between that portion of their forces of 
affinity which is capable of free conversion into other forms 
of work, and the portion which is manifested as heat only. 


Helmholtz now designates these two portions free and bound 
energy. He shows that the processes which appear spontane- 
ously in a state of rest, and at a constant and uniform tempera- 
ture, and are maintained without help from external working 
force, can only progress in such a direction that the free energy 
diminishes. Among these are included the chemical processes 
that begin and continue of themselves at constant temperature, 
and, if the Law of Clausius held without limitation, it would be 
the value of the free energy, not that of the total energy 
indicated by the evolution of heat, which would determine the 
direction in which chemical affinity can act. 

Helmholtz next undertook a general inquiry into any 
compound system of masses having all the same temperature, 
and all being subject to the same alterations of temperature, 
and assumed that the state of the system was completely 
determined by the temperature and by a number of inde- 
pendent parameters. In a series of brilliant mathematical 
deductions he arrived (by means of the two equations of 
Clausius) at the result that it is only necessary for the repre- 
sentation of thermodynamic equations to obtain the differential 
quotients of the so-called ergal, which is absolutely determined 
as a function of temperature. This ergal, for all alterations 
occurring at constant temperature, coincides with the value 
of the potential energy for the unrestrictedly convertible 
quantity of work, and he calls it the free energy of the system, 
while the difference between the total internal energy and 
the ergal is termed the bound energy. The quotient of the 
restricted energy by the temperature is the entropy of Clausius. 

In order, further, to distinguish what had till then been 
known as iris viva, or kinetic energy, in theoretical mechanics 
from the work-equivalents of heat (which indeed was for the 
most part regarded as the vis viva of invisible molecular 
motions), Helmholtz proposes to call the former the vis viva of 
organized motion. As a general definition of organized motion 
he gives that in which the velocity components of the masses 
in motion may be taken as the continuously differentiate 
functions of spatial co-ordinates. An unorganized motion, on 
the contrary, is that in which the movement of each separate 
particle exhibits no sort of similarity with that of its neighbours. 
Heat motion may very probably be included in this mode, and 


in this sense he defines the magnitude of the entropy as the 
measure of disorganization. 

4 For our instruments (which are coarse in comparison with 
molecular structure) it is organized motion alone that is freely 
convertible into other forms of work ; whether such transform- 
ation is actually impossible in view of the fine structure of 
living organic tissues appears to me still to be an open 
question, the importance of which in the economy of nature is 
plainly obvious/ 

By simple mathematical calculations Helmholtz arrived at the 
result that in all changes in which the temperature remains 
constant, work is only done at the expense of the free energy, 
while the bound energy alters at the expense of the in- and 
out-going heat. In all adiabatic alterations work is produced 
at the cost of free as well as of bound energy, so that the 
entropy remains constant. In all other cases external work is 
done at the cost of the free energy, all production of heat at the 
cost of the bound, while with each rise of temperature in the 
system free energy is transformed into bound. 

Observations on galvanic cells agreed with these general con- 
clusions. Here too it appeared that the bound energy increases 
at the cost of the heat supplied, and with rise of temperature at 
the expense of the free -energy, so that free must always be 
transformed into bound energy, and not vice versa. Neither 
is the free work in isothermal changes expressed in irrever- 
sible processes by the heat developed, when the initial and 
final temperatures are the same, since this heat is derived 
from the free and the bound energy, while free work depends 
upon the former only. The fact that, apart from altera- 
tions of temperature, the vanishingly small alteration of free 
energy is not positive, or is nil, may be taken as the condition 
of the system remaining in its present state, but if a point be 
reached by rise of temperature, at which this becomes negative, 
dissociation will ensue. Thus all chemical compounds below the 
temperature of dissociation give out heat, if they are formed by 
reversible processes. 

Helmholtz then employed his new concept of free energy in 
calculating the connexion between the E.M.F. of a cell and the 
vapour tension. 

In a paper communicated to the Academy (May 3, 1883), ' On 


the Thermodynamics of Chemical Processes: Conclusions 
relating to Galvanic Polarization/ he applies the thermo- 
dynamic theorems previously developed to the theory of 
galvanic polarization, ascribing great importance to them in this 
connexion, 'because they show that the surplus of the free 
energy of the mixture of oxygen and hydrogen over that of the 
water depends largely upon pressure, while the development 
of heat in the compound is almost independent of it. So long 
as it was believed that the electromotive force of polarization 
must be calculated according to the latter (as I did myself in 
my earlier work) it appeared to be an almost unalterable 
quantity, and this made certain processes in the polarization of 
a voltameter almost inexplicable. But if the electromotive 
force is calculated according to the free energy, it is then found 
to be exceedingly liable to alteration according to the gaseous 
saturation of the layers of liquid lying next the electrodes, and 
this essentially modifies the interpretation of a large proportion 
of the phenomena of polarization, so that most of those which 
were previously inexplicable can now be understood/ 

Helmholtz had showed in his earlier work of 1873, that the 
gases dissolved in the liquid, oxygen in particular, have a great 
influence on the intensity of the current, for the unlimited 
duration of which, with weak E.M.F., no explanation had been 
found, and had then explained the origin of the convection 
currents dependent upon them ; experiments undertaken with 
the view of removing the last trace of dissolved gases were 
unsuccessful. The opposing force of polarization, too, increased 
steadily with the increased E.M.F. of the galvanic battery, 
where there had been a prolonged evolution of gas. Helm- 
holtz now believed that he had solved these difficulties by his 
thermodynamic theory, since it was plain from this that the 
resistance of the chemical forces to the electrical current 
increases steadily with the solution in the liquid of the gases 
given off at the electrodes. Finally he also applied his theory 
to the formation of gas bubbles after the saturation of the 
layers next the electrodes with gas, and calculated the work 
corresponding to the diffusion of the gases through the liquid. 

He ascribed great importance to thermodynamic researches 
in the scientific development of chemistry, saying in an interest- 
ing letter in 1891 : 

Z 2 


' Nernst has thrown himself zealously into the newest appli- 
cations of physical chemistry, as worked out by the Dutchman 
Van't H off, and advocated with great vigour by Professor Ostwald 
of Leipzig in his Journal. These theories have already proved 
to be of great practical utility, and have led to a multitude of 
demonstrably correct conclusions, although they imply some 
arbitrary asumptions which do not seem to me to be proven. 
The chemists, however, make use of this hypothesis (of the 
dissociation of a portion of the compound molecules of the dis- 
solved salts) in order to form a clear conception of the processes, 
and they must be allowed to do this after their fashion, since 
the whole extraordinarily comprehensive system of organic 
chemistry has developed in the most irrational manner, always 
linked with sensory images, which could not possibly be 
legitimate in the form in which they are represented. There is 
a sound core in this whole movement, the application of thermo- 
dynamics to chemistry, which is much purer in Planck's work. 
But thermodynamic laws in their abstract form can only be 
grasped by rigidly trained mathematicians, and are accordingly 
scarcely accessible to the people who want to do experiments 
on solutions and their vapour tensions, freezing points, heats 
of solution, &c/ 

In a sketch which was probably designed in 1883 to be the 
Introduction to the Third Part of his ' Thermodynamics ', 
Helmholtz set forth clearly and intelligibly the reasons that led 
him to adopt the expressions ' free ' and ' bound ' energy, show- 
ing at the same time how he had plotted out the continuation of 
his investigations, had he not been led by the generalization of 
all these considerations to far more comprehensive problems. 

' Thermo-chemical researches have till now been directed 
almost exclusively to the quantities of heat evolved during 
chemical processes, when the forces of chemical affinity are 
given free play, so that the association of the combining 
substances usually takes place with more or less disturbance. 
In such cases heat is as a rule the only work-equivalent of the 
chemical forces produced, or at best there is only an insignificant 
proportion of other forms of work, among which the over- 
coming of atmospheric pressure plays, relatively at least, the 
most frequent part. In thermochemical researches the attempt 
is usually made to show how much heat has been given off, or 


taken up, by the end-products of the chemical process, when 
they have returned to the temperature of their initial state, 
before the chemical process began. The heat-equivalents of 
any further work that has been done or absorbed (i. e. done 
negatively) must if necessary be added. 

1 By this method is obtained the heat-equivalent of the excess 
of the whole store of energy which the substances involved 
contained in their initial state, over that of the final state. 
This is the foundation of Thermochemistry, firmly established 
by countless arduous and most valuable investigations, and 
corresponds to the general Law of the Conservation of Energy. 

'The work done by the chemical forces for the most part 
appears only in the form of heat, but under special circum- 
stances we can directly obtain other forms of work, mechanical 
or electrical, from it. Heat, according to Clausius's stricter 
interpretation of Carnot's law, plays a peculiar part as com- 
pared with the other work-equivalents. While the others can 
be transformed freely and with no perceptible remainder inter 
se, the convertibility of heat is limited, so long as we are 
confined to the attainable limits of temperature. At all times it 
is only a fraction of the heat present that we are able to convert 
into other forms of work, while the remainder of this part is 
reduced from a higher to a lower temperature. If we take 
to denote the lowest absolute temperature (that is, tempera- 
ture measured from 273 C. as the zero-point) at which we can 
get our store of heat to flow away, Q l being the initial tempera- 
ture, we must allow the fraction /6 1 to pass away unconverted, 
in order to convert the remainder (0 1 )/0 1 into work. Hence 
the higher the temperature lf the larger the fraction of the 
heat present that can be transformed into mechanical work. 

' In order to describe this antithesis briefly, seeing that it is 
of essential importance in the question of the efficiency of 
chemical forces, I have adopted the expression ' free energy ' to 
describe the work-equivalents of the different natural forces 
that are freely convertible inter se, with no necessary remainder, 
denoting the heat store on the other hand as ' bound energy '. 
To the former, for instance, belong the energy of a raised 
weight, of a stretched elastic spring, the vis viva of a mass that 
is moved as a whole, an accumulation of electricity at rest in 
a conductor, &c. To say that these are interconvertible ' with 


no necessary remainder' merely signifies that when the process 
is carefully conducted, the remainder which is lost in, e.g., 
friction, elastic after-effect, electrical resistance, and so on, and 
converted into heat, may be made minimal. Conversion with- 
out remainder can only be an ideal limit for our terrestrial 
conditions, to which we can approximate more or less closely. 
Still there is a great difference between these losses of the 
freely convertible energy and those which we encounter in 
heat, where an important fraction, which cannot be diminished 
by any precautions known to us, necessarily remains over in 
the form of heat. 

' We already know that chemical forces can develop not 
merely heat, but mechanical work also, either immediately or 
by setting up electric currents. This brings in the question, to 
what part of their work the free energy corresponds, and what 
other part on the contrary appears exclusively in the form of 
heat. It is well known that an extraordinary number of 
chemical changes in the state of aggregation occur ; in these 
also heat may become free or bound. Of this heat we already 
know that it is subject to the limitations of Carnot's law. 
Moreover, it has long been known and proved in thermo- 
chemical work that this binding and loosing of heat plays its 
part in the alteration of the state of aggregation ; and that we 
may even have chemical processes that are self-initiated and 
self-developed, as in the mixture of ice and salt, which engenders 
cold, and in which external heat must be introduced before the 
initial temperature can be reinstated. Here then the salt 
solution that results has more internal energy than the dry 
salt and snow had previously. 

' Further it is clear that the sudden alterations of the state 
of aggregation represent only the most striking cases of such 
binding and loosing of latent heat. We are equally justified 
in regarding the cooling which occurs when a gas expands 
as a binding of heat ; it is true that with slow expansion this 
will in the latter case be reconverted wholly or almost entirely 
into mechanical work, but even the latent heat of steam includes 
the work done in overcoming the pressure on the steam. In 
the sudden expansion of a gas without resistance, as in Joule's 
experiments, there is indeed no cooling, but this is only because 
the initial work performed in producing the vis viva of the 


violent motion of the gas has been retransformed into heat 
by friction. But if heat becomes latent under such slight 
modifications as the alteration in the volume of a gas, we must 
expect corresponding latency and disengagement of heat in 
all the countless alterations of aggregation and density that 
occur in almost all chemical processes. And it appears no 
more uncertain than in the case of the latent heat of steam, 
that all the quantities of heat as here described must be referred 
to the bound energy comprised under Carnot's law, and are 
therefore to be regarded as heat, which was present as such 
in the initial states of the substance, but has no place in the 
final states at the same temperature, and is evolved. But 
the opposite process may equally well occur. The final states 
may require a greater quantity of latent heat at the same tem- 
perature, and the initial temperature may be reinstated only at 
the cost of the heat contained in the surrounding bodies. In 
the former case the ' heat toning ' (purely chemically developed 
heat) will appear to be increased, in the latter to be diminished. 

' If we want to determine the largest quantity of free energy 
that can be obtained by chemical processes, the same general 
considerations hold good as were laid down by Carnot. Pre- 
cautions must be taken to ensure the reversible character of 
the entire process : i. e. the working forces must be held at 
equilibrium by other forces which are under the control of the 
observer, so that the entire process shall take place slowly and 
quietly, without development of violent disturbances, in which 
the vis viva might be converted by impact and friction into 
heat. All friction, inelastic impact, and transfer of heat between 
bodies of different kinds must be entirely avoided. The reversi- 
bility of the process is conditioned by the fact that with perfect 
equilibrium of internal and external forces, the observer has it 
in his power to reverse the process by a slight reinforcement 
of the latter. 

1 Nor is it only in the practical task of obtaining motive power 
for other purposes by means of chemical forces that this 
separation between free and bound energy plays an essential 
part : it obtains in the region of chemical phenomena also. A 
chemical p r ocess cannot appear of itself, or go forward, un- 
supported by external motive forces, without diminution of the 
total sum of free energy in the co-operating bodies/ 


In 1883 the Prussian Board of Education, at the request of 
Helmholtz, invited his Assistant, Heinrich Hertz, to receive the 
degree of Dozent in view of his approaching call to Kiel ; and 
he took over from Helmholtz, whose activities were now 
devoted entirely to other fields, the task of further exploring 
the difficult and still unsolved problems of the doctrine of 
electricity, on the principles of the Faraday- Max well hypothesis. 

Hertz had begun an investigation in Helmholtz's Institute 
at Berlin which he concluded in this same year at Kiel, and 
published as ' Experiments on the Glow Discharge '. It was con- 
cerned with the form of discharge occurring in vacuous vessels, 
which is accompanied by the phenomena of the cathode rays, 
and of the striated positive light. Hertz finds that the cathode 
rays do not deflect a magnetic needle, and so do not produce the 
electrodynamic effect of a current; and he therefore regards 
them as being only the accompaniment of a current, and not 
as themselves constituting a current. 

On July 29, 1883, Helmholtz writes to Hertz : 

' I have read your investigation on the Glow Discharge with 
the greatest interest, and cannot refrain from writing to say 
Bravo ! The subject seems to me to be of very wide importance. 
I have been considering for some time whether the cathode 
rays may not be a mode of propagation of a sudden impact 
upon the Maxwellian electromagnetic ether, in which the 
surface of the electrodes forms the first wave-surface. For, as 
far as I can see, such a wave should be propagated just as these 
rays are. Deviation of the rays through magnetization of the 
medium would accordingly also be possible ; longitudinal waves 
could be more easily conceived, and might exist if the constant 
k in my electromagnetic researches were not zero. But in 
that case, transversal waves could also be produced. You 
seem to have the same idea, but, however that may be, do not 
hesitate to make use of my suggestion, for I have no time 
at present to work it out. Besides, these thoughts arise 
so readily in reading your investigation, that they would be 
bound to occur to you soon, if they have not done so already. 
One objection to your experiments, however, occurs to me 
which you may perhaps be able to remove better than I, and 
which in any case must be mentioned. This is that if the 
cathode rays are electrical currents, according to the earlier 


view, they must necessarily have another invisible returning 
portion, somewhere in the region of the wall of the tube. This 
is a point which I have often discussed with Dr. Goldstein. 
In that case they could no more have external magnetic 
action than closed currents proper, within the tube, since 
they would form ring magnets. In the rectangular vessel 
there would still be the possibility of giving such a form 
to the invisible returning currents that the observed effect 
should occur. Such an interpretation to me appears hardly 
probable, since the cathode rays form a concentrated beam, 
and personally I do not believe in its probability, but I fear 
it is an objection that will occur to many readers.' 

To this letter Hertz made a full and most interesting 
reply, in which he discusses the views and criticisms of 

' My warmest thanks for your kind letter. Your words are 
the strongest and most agreeable spur to activity that could 
be given me. May I make a few observations in reply ? I do 
not want to inflict myself on you, but write in case you care 
to read them. I had, as a matter of fact, considered the ideas 
you express, but was inclined to think that the cathode rays 
are produced by the longitudinal waves, which correspond to 
the transverse vibrations of light. For it seems to me as if the 
longitudinal waves, in a medium in which the plane of polariza- 
tion of the transversal waves rotates, must be propagated along 
curved lines, and thus the direction of rotation for light and 
for the cathode rays would be identical. Then, if the arrow 
xy gives the direction of the positive current, 
produced by a magnetic field, the plane of 
polarization for all gases hitherto investigated 
will be turned in the direction of this arrow, 
that is, a force is produced which acts along 
AB, and produces a displacement at an angle to this, as CD. 
There must also be longitudinal impulses propagated in a curve 
that is deflected to the right hand. But an elastic wire in which 
a positive current was flowing to the cathode would also be 
deflected to the right hand, and so a confusion between the 
two phenomena would be possible. The question no doubt is 
whether these simple considerations will hold good for the 
more exact application of the theory. I have not attempted 


this, because I had imagined, perhaps erroneously, that the 
theory was not yet sufficiently perfect. 

'The following seems also to point to a correspondence 
between the two phenomena. The more the tube is exhausted, 
the less does the magnet act upon the rays, and the more 
rigid they become, as Dr. Goldstein expresses it. This may 
show (although there is another possible interpretation) that 
the magnet can only act indirectly upon the cathode rays, as 
it does upon light, that is, by means of the ponderable matter. 
In this case, the action of magnetic matter must be enormously 
stronger upon the cathode rays than it is upon light, but since 
the same difference undoubtedly exists in regard to absorption, 
this is the less to be wondered at. 

' Generally speaking, the cathode rays excite the same fluores- 
cence in solid bodies as does light. But I do not therefore 
hold it necessary to assume that they are directly converted 
into optical rays. One would be more inclined to interpret 
the phenomenon in the opposite sense. For as the transverse 
rays of light break up inside the bodies, they will give rise to 
longitudinal waves, and it is quite natural according to our 
view that these again should immediately disappear with the 
production of the same light as is produced by the long cathode 
rays in the vacuum. 

' I have also tried to induce phenomena of diffraction, by 
sending thin cathode rays through a grating, but obtained no 
result. At the same time the experiments were not of a nature 
to prove anything. These are the kind of ideas which I have 
on this subject. Up to the present I have not entertained any 
hopes of utilizing the phenomena in electrodynamics, as for 
the determination of the constant , since the only effect that 
can be measured exactly, the action of the magnets, appears 
to be essentially conditioned by the ponderable substances. 
I will reflect upon this point, and upon the objection you 
pointed out. The latter, I think, may be entirely refuted, if 
we succeed in obtaining more certain proof that cathode rays 
are possible in the absence of all electrostatic differences. 

' It now only remains, hochverehrter Herr Geheimrath, to 
repeat my sincerest and warmest thanks, and I remain, with 
deepest respect, 

' Your devoted H. HERTZ.' 


On June 20, 1883, Helmholtz writes to his wife, who had 
gone to Paris in the middle of May for the funeral of her uncle 
Julius von Mohl : 

I Geheimrath Herzog was here yesterday, and brought me 
an invitation to join a sixty-seven days* journey to the Pacific 
and back, from August 15 to October 22, for the Opening of 
the Northern Pacific Railway, as the Company's guest ; thirty 
distinguished men are to be invited from Germany, and they 
say Count Lerchenfeld, the Minister Kruger, Georg Bunsen, 
Gneist, and the Reichstags- President von Levetzow, are going. 
Herzog promises princely accommodation on the journey, and 
receptions. If one is to see America in this life this would 
perhaps be the best opportunity imaginable. For this reason 
I have not yet declined, although there are many obstacles in 
the way, and it is really not essential that one should see 
America, at any rate not for what I have to do in the world/ 

His wife did not approve of his undergoing the fatigues 
inseparable from this journey; accordingly he declined the 

I 1 have no particular wish to make a journey at present/ 
he writes to her again on August i, while she was still detained 
at an English watering-place by the protracted illness of their 
son Robert. 'Just now I have some interesting experiments 
on hand, that are beginning to go well, and have received my 
new magnetic balance admirably made. But I notice signs 
that I am getting worked out, so it can't go on much longer, 
and the climate of Pontresina admits of no delay.' 

During this time Helmholtz had great pleasure in carrying 
on a scientific correspondence with his son Robert, who was 
pursuing his studies in chemistry, physics, and mathematics with 
the utmost zeal. On October 20 he writes from Rome : 

'As to your experimental inquiries, I should recommend 
you to find out if electrified air gives a double layer at the 
surface of a conductor. Take a Kohlrausch condenser with 
carefully cleaned plates, and test the tension between them. 
Then charge one of them temporarily with an electrical 
machine, and discharge it by a small flame that gives no 
deposit of moisture ; then bring it back to the condenser, and 
see if the difference of potential remains unaltered. Then 
do the same with the opposite electricity. The experiments 


you describe would require an extraordinarily exact control, 
to show that the dependent plate remained symmetrical to the 
electrical sphere/ 

With the winter of 1883-4 Helmholtz entered on a period of 
great mathematical exertion in attempting to discover a unifying 
principle governing Nature, which occupied his thoughts during 
the last decade of his life, and down to his closing hours. 

His work in thermodynamics had led him to general re- 
searches upon monocyclic systems, and the deeper significance 
of the principle of least action ; but the difficulties of working 
out his ideas had soon accumulated, and his time was much 
taken up by various official duties. Nor was it only his 
experimental and mathematical lectures, the management of 
the Physical Institute, and the lectures at the Military Medical 
Academy that hindered him from immersing himself in his 
own ideas. Technical Reports of the most varied character had 
to be sent in, since his opinion was claimed on all sides as that 
of the most competent authority. In different places we find 
reports on the position of lightning conductors for the protection 
of powder magazines surrounded with earth, on the results of 
ballooning, and an infinity of other things ; besides, there were 
musical interests, and all kinds of artistic interests from which 
he could not and would not separate himself, yet despite all 
this the profound and fruitful ideas which we shall endeavour to 
trace in subsequent pages were developing in rapid succession. 

On January 7, 1884, he writes to W. Thomson : 

' I myself am still engaged upon the subject of monocyclic 
movements, and have now discovered some far-reaching 
generalizations, which are connected with a universalized form 
of Hamilton's Law of Mechanics. You had better wait for the 
later paper before you go on with the monocyclic system ; you 
will get it in a more convenient form/ 

Even before the Easter holidays he was able to lay a portion 
of the results of his researches before the Berlin Academy, 
but was obliged on account of his health to break off his work, 
and go to England with his daughter Ellen, directly the session 
was over. 

After seeing Tyndall, Herbert Spencer, Sir John Lubbock, 
Huxley, and Hooker, the Director of Kew Gardens, he spent 
some very stimulating days with Sir Henry Roscoe in 


Manchester, 'with whom he had much to discuss in regard 
to his latest work on the relations between heat and chemistry/ 
In Glasgow, his old and well-loved haunts were open to him 
in the house of Sir William Thomson, whom he found absorbed 
in regulators, and measuring apparatus for electric lighting, and 
for electrical railways. 

1 On the whole, however/ he writes to his wife, ' I have an 
impression that Sir William might do better than apply his 
eminent sagacity to industrial undertakings; his instruments 
appear to me too subtle to be put into the hands of uninstructed 
workmen and officials, and those invented by Siemens and 
Hefner v. Alteneck seem much better adapted for the purpose. 
He is simultaneously revolving deep theoretical projects in his 
mind, but has no leisure to work them out quietly; as far as 
that goes, I am not much better off! ' And he adds immediately 
after : ' I did Thomson an injustice in supposing him to be 
wholly immersed in technical work ; he was full of speculations 
as to the original properties of bodies, some of which were 
very difficult to follow ; and, as you know, he will not stop for 
meals or any other consideration/ 

His wife replies : ' I am delighted to think of your being 
with dear Sir William ; how you will revel in the fundamental 
concepts of things. If only one was not pulled up by the 
great query at the beginning and end of life, and obliged to 
content oneself with that ! That is where you are so fortunate ; 
the things that lie beyond our limits do not weigh upon you, 
and there is enough of Eternity for you outside our little 
human existence/ 

From Glasgow Helmholtz went on with Thomson to the 
University festivities in Edinburgh, where he was assigned the 
honourable task of replying for the foreign guests at the great 
banquet, and of making a speech on a similar occasion at a recep- 
tion of the students, where he was received with loud applause. 

On November 10, 1884, his daughter Ellen married Arnold 
Wilhelm von Siemens, the eldest son of Werner von Siemens, 
who was born on November 13, 1853. After nearly forty years 
of close friendship this link brought great joy into the lives 
of both men. 

Helmholtz's ' Studies in the Statics of Monocyclic Systems ' 
(published in the Proceedings of the Berlin Academy, March 6, 


March 27, and July 10, 1884), his 'Generalization of the 
Theorems of the Statics of Monocyclic Systems' (ibid. 
December 18, 1884), and his ' Principles of the Statics of 
Monocyclic Systems' (Crelle's Journal, 1884), are in the 
closest connexion with his memoir 'On the Physical Signi- 
ficance of the Law of Least Action ' (Crelle's Journal, 1886). 
An important supplement to these articles was given in the Note 
published in the Berlin Proceedings for March 10, 1887, on ' The 
History of the Law of Least Action/ and developed in more 
detail in an address to the General Meeting of the Academy on 
January 27, 1887, the full publication of which was suppressed, 
because Helmholtz subsequently learned that Adolph Meyer of 
Leipzig had already published a complete and fundamental dis- 
cussion of the Law of Least Action in his Inaugural Address. 
Helmholtz's lecture, remarkable both for its matter and for its 
style, was incorporated after his death, with his wife's consent, 
in the History of the Academy (Vol. II, Documents and Deeds), 
issued in 1900, on the 2ooth anniversary of its foundation. 

The fundamental researches set out in these memoirs (which 
gave Hertz the idea and the starting-point for his Principles of 
Mechanics, and of which the enormous importance, partly from 
the difficult nature of the problems attacked, partly from the 
succinct character of their statements, has not yet been widely 
recognized by scientific men) are entirely mathematical ; at 
the same time the purely mathematical problems that occur 
during the generalization of mechanical principles are, according 
to Helmholtz's invariable practice in his mathematico-physical 
work, dealt with only in so far as is required for their application 
to physical questions. Since all mathematical expressions 
have to be avoided here, it is only possible to give a general 
outline of these papers. 

'A law which is to comprise the total sum of alterations 
in Nature must necessarily deal with concepts of the most 
abstract kind, from which everything has been eliminated that 
refers to the particular properties of the natural bodies known 
to us ; for the most part it is necessary, indeed, under such 
conditions, to form new abstract concepts for the purpose, 
which, when any one hears them defined for the first time, shall 
evoke no previous concepts or experiences, that is, in popular 
parlance, make him think of nothing.' 


Leibniz had defined the work-equivalent as whatever in 
Nature could be employed as motive force, or (to give us at 
the same time a measure of it) could lift a weight ; and he 
gives as the measure of work the product of the weight, and 
the height to which it is raised. We call this the potential 
energy of the weight, because in falling it is able to do this 
work ; it is in this way, as was said above, that potential energy 
is reckoned for all other forces and any given path of the 
body affected, only that the force must be replaced by its 
component in the direction of the path. Leibniz, however, had 
already pointed out the second principal form of the work- 
equivalents of ponderable bodies, namely the vis viva of the 
moving masses, or the kinetic energy, and finds its value equal 
to half the product of the mass into the square of the velocity. 

The Law of the Conservation of Kinetic Energy stated that 
in any given aggregate of natural bodies, on which only such 
forces act as proceed from fixed centres, the sum of the actual, 
or kinetic, and the potential energy is constant. It was only 
when men began to investigate the work-equivalents which 
must be gained or lost if imponderables are to be brought 
into play, that Robert Mayer and Helmholtz became convinced 
of the universal validity of the Law of Energy for all natural 
processes of the non-living as of the living world, and thus 
arrived at the Law of the Conservation of Energy. But it is 
the task of physics to refer the phenomena of nature back to 
the simplest laws of mechanics, which gave rise to the 
important question how mechanics itself is constituted in its 
simplest presentation, and what, as Hertz expresses it, are its 
ultimate and simplest laws, which every natural motion obeys, 
which admit of no motion that is excluded by our present 
experience, and from which, as from the true principles of 
mechanics, the whole science of mechanics can be purely 
deductively developed without further appeal to experience. 

The discovery of the Law of the Conservation of Energy 
made a coherent structure of theoretical mechanics possible. 
The concept of force retreated into the background ; mass and 
energy emerged as the given indestructible physical quantities. 
The energy present proved to consist of two parts, one of which, 
the kinetic energy, bears in all cases the same relation to the 
velocities of the masses in motion ; the other, the potential energy, 


is determined by the relative positions of the masses, but in 
every case can only be ascertained by a knowledge of their 
particular nature. The discussion of the different forms of 
energy as well as the condition of its conversion from one form 
into another represents, according to Hertz, the subject-matter 
of the whole of physics and chemistry. 

From the Law of the Constancy of the Sum of Kinetic 
and Potential Energy, the important consequence followed 
immediately that, if a system of bodies is at rest in any position, 
from which every movement compatible with the restrictions 
of the system tends to a position with higher potential energy, 
no vis viva, and therefore no motion of the bodies, can arise ; 
there must accordingly be stable equilibrium in any position 
at which the potential energy is at a minimum. The Law of the 
Conservation of Energy, however, tells us nothing in the case 
of motion, as to the succession of positions which the system 
has to traverse, in order to get from a given initial to a given 
final position : it is this which is elucidated by the principle of 
least action. 

Leibniz had already asked himself what work can be done 
by the inertia which distinguishes space filled by mass from 
geometrical bodies ; he found that the work was the greater, 
in proportion to the magnitude of the mass in motion, the 
length of the path through which it moves, and the velocity 
with which it is moving. The amount of the action was thus the 
product of mass, distance, and velocity, or, what amounts to the 
same, of vis viva and time. We thus arrive at a law that 
completely embraces all possible motions of any given number 
of material bodies under the influence of conservative forces, 
in part exerted reciprocally, in part suffered from fixed centres, 
as is summed up in the law of least action. According 
to this, when such a material system passes with free and 
undisturbed motion from a given initial to a given final 
position, with a definite energy-value, the action has a limiting 
value, and for short phases of the motion this is a minimum. 
Accordingly, with given values of energy, the inertia must 
always bring the masses in motion to their end by a path 
which, at any rate for short distances, exacts the least amount of 
work. To define the mathematical conception of the limiting 
value Helmholtz says : 


'When a traveller wants to cross a mountain ridge, the 
height of the pass is of course the maximum height to which 
he must rise, while if he crossed at any other point he would 
have to climb still higher. This is called in mathematics 
a maximo-minimum of height, and all such values, as well as 
the complete minima and maxima of variable magnitudes, are 
known as limiting values/ 

So long as this principle was applied only to the obvious 
motions of ponderable bodies, it seemed to have no other 
real content than that contained in Newton's equations of 
motion, but it soon acquired much greater significance, when 
the investigation was extended to bodies within which per- 
sistent conceakd motions were proceeding. Helmholtz 
ascribes a fundamental theoretical interest to the formulation 
of the Law of Least Action, in relation to the course of all 
natural processes, inasmuch as the energy components, with 
which mechanics was originally concerned, entirely disappear 
from the problem, and ' the question is now reduced to the two 
chief forms of energy, the total value of which is unalterable 
and eternal, but which fluctuate to and fro in the most 
complex forms of manifestation in natural bodies. By this law 
the ebb and flow of energy is brought under a brief but all- 
embracing rule, whereby everything that happens in the world 
is resolved simply and solely into a question of the distribution 
of energy in time '. 

As the first and most striking example of the application of 
the law of least action to the investigation of bodies, in the 
interior of which concealed motion is proceeding, Helmholtz cites 
the ' originally mysterious and incomprehensible laws of 
the mechanical theory of heat' of Sadi Carnot, Clausius, 
and Boltzmann ; he points out that F. E. Neumann expressed 
the law of the electromagnetic action of closed galvanic 
currents in the same form as results from the law of least 
action, and remarks that all the hypotheses advanced by 
W. Weber, Clerk Maxwell, Riemann, C. Neumann, and Clausius, 
for the resolution of the reciprocal actions of many electrical 
masses into elementary actions, have resulted in forms of calcula- 
tion which correspond to the law of least action, although 
what corresponds to vis viva and inertia in electricity is ex- 
pressed in a different form from those used for ponderable 


bodies. The validity of the law appears to Helmholtz to be 
limited in the so-called irreversible processes of conduction of 
heat, production of heat by friction, electrical resistance, and 
so on, only because we are unable either to follow the 
unorganized motions of the individual atoms or to bring them 
together practically in any congruent direction. He tried, 
in his work on monocyclic systems to show that the most 
diversified classes of internal motions are subordinated to the 
law of least action. 

The hypothesis that Helmholtz here puts forward and is 
constantly elucidating, to the effect that all phenomena come 
about uniformly through the action of concealed masses, by con- 
cealed motions and rigid combinations, was subsequently stated 
by Hertz (as a corollary to this fundamental idea of Helmholtz, 
which stands for the most significant advance that has been 
made by modern mechanics) in correct language in the some- 
what wider assumption : ' that the complexity of the real world 
is greater than that of the world that lies open to our senses ; 
we admit that an unknown agent is at work, but we deny that 
this agent has a specific character, like the concepts of force 
and energy; the unknown must still be Motion and Mass, 
distinguished from the visible not by its own nature, but simply 
in relation to us, and to our normal modes of perception. . . . 
Force and Energy are no more than effects of Mass and Motion 
which are not always perceptible to our senses.' 

In his Address to the Academy, Helmholtz gave a striking 
account of the historical development of the law of least 
action, tracing it back more particularly to the works of 
Maupertuis, who was ' essentially what we term a genius, with 
all the qualities and failings which that implies '. He indicates 
what is indefinite and obscure in the views of Maupertuis, who 
holds that this law satisfies the demand of metaphysics, that 
Nature should invariably employ the simplest means to produce 
her effects, and thinks himself called upon to ascertain what 
quantities are minimal in natural processes, since these will be 
those which Nature tries to economize ; so that we can discover 
the objects which Nature pursues. Maupertuis even goes so far 
as to assert that the law of least action, as discovered by 
himself, is the first binding and incontrovertible proof of the 
existence of God, as an Intelligent Governor of the Universe. 


With regard to this metaphysical assumption of Maupertuis, 
Helmholtz jestingly remarks in one of his papers, in allusion 
to the distinction which he had laid down between limiting 
value and the minimum : ' If inertia is to be personified, as in 
this formula, it would be proper to make it shortsighted, and 
concerned with the immediate moment only.' 

The credit of the first, even if it were a wholly indefinite, 
formulation of the principle is ascribed by Helmholtz to Mauper- 
tuis, but he justly accuses him of obscurity and want of strict 

' He grossly neglects the old Socratic demand that every 
philosopher, i. e. man of science, should be clear in his own 
mind as to what he knows. He must have been aware that 
the law which he brought forward as incontrovertible could 
neither be verified nor clearly applied in many classes of 
instances. Immersed in self-admiration he held himself justi- 
fied in merely announcing his discovery like a prophet, a tragic 
instance of how a mind that was highly gifted at the outset 
can be led away by vanity and the lax discipline of so-called 
metaphysical thinking, to border-lands where even the faculty 
of reasoning becomes dubious. Yet even if he were only 
guessing at the truth, the truth it was, none the less. And 
his fixed belief in the possibility of finding a universal law of 
nature was rooted in a proper confidence in the uniformity of 
nature, i. e. in the Law of Causation, which is the ultimate basis 
of our thinking and acting.' 

In his paper ' On the History of the Law of Least Action ', 
Helmholtz criticizes the evidence given for it by Lagrange, Jacobi, 
and Hamilton. He shows that if, on comparing the adjacent 
paths with that actually followed by the system, we assume for 
the others not only the conservation of energy, but also the 
same value of the energy constants, then in order to maintain 
the validity of the law the same initial and final positions may 
be postulated for the contrasted motions of the system, but not 
the same period of time. The time must accordingly be taken 
as a variable factor in the analytic derivation of the principle. 

Jacobi's proof is physically valid for every complete, self- 
centred material system. Hamilton's form of the law (to be 
discussed below) allows us, on the contrary, to extend the 
equations of motion to such imperfectly closed systems as are 

Aa 2 


affected by variable external influences, if we can regard them 
as independent of any reaction of the moving system, as e. g. in 
the case of the forces proceeding from fixed centres. 

'In any case the universality of the law of least action 
appears to me so far assured that it may be assigned a high 
value as a heuristic principle and guide in the attempt to 
formulate laws for new classes of phenomena/ 

Helmholtz then, with a view to more exact investigation, sets 
out from the law of least action in the form proposed by 
Hamilton, according to which the negative mean value of the 
difference in potential and kinetic energy (of the kinetic poten- 
tial), calculated for each time-element, is minimal in the actual 
path of the system, and has a limiting value over a longer portion 
as compared with all adjacent paths leading in the same time 
from the initial to the final position. Without separating kinetic 
and potential energy, he develops the analytical expression for 
this law with the utmost freedom for the nature of the kinetic 
potential, and derives the form of Lagrange's equations of 
motion from it, showing that already in the mechanics of ponder- 
able masses (under special conditions, and with the elimination 
of single parameters of the problem) these more general forms 
may arise, in which the two energies are not separated. Even 
under this most general assumption he deduces the law of the 
conservation of energy, and finds that it is not true conversely 
that in every case in which the conservation of energy obtains, 
the law of least action holds good also. ' The last expresses 
more than the first, and it is our task to find out what more it 
expresses.' He shows, on the assumption of the validity of the 
law of least action, how it is possible from the complete 
knowledge of the dependence of energy upon the co-ordinates 
and the velocities to find values for the kinetic potential, and 
therewith for all the laws of motion of the system. In the 
mechanics of ponderable bodies the kinetic potential is a homo- 
geneous function of the second order of the velocities ; yet 
under certain conditions, with elimination of co-ordinates, 
Lagrange's equations of motion may stand for the remaining 
co-ordinates in exactly the same form as for the kinetic potential, 
in which case the velocities are also linear. In correspondence 
with this analogy from the mechanics of ponderable bodies, 
Helmholtz designates other cases of physical processes, in 


which the kinetic potential contains terms which are linear in 
the velocities, as cases of concealed motion ; these cases differ 
essentially from those in which the kinetic potential involves 
velocities only in the terms of the second degree, inasmuch as 
the motion cannot under similar conditions be reversed unless 
the concealed motions be simultaneously reversed. 

He then considers, under the above general assumptions, the 
reciprocal relations between the forces which the system simul- 
taneously exerts in different directions, and its accelerations 
and velocities, which embrace a series of highly interesting 
associations of physical phenomena, such, e. g., as the law of 
thermodynamics : if increase of temperature raises the pressure 
of a material system, then its compression will raise the tempera- 
ture ; further, if heating of any point in a closed circuit produces 
an electric current, the same current will produce cold, if the 
heat due to the resistance be disregarded ; and many others. 
After deriving the necessary conditions for this extension of 
kinetic potential from the extended form of Lagrange's equations, 
he enunciates the theorem that these conditions are moreover 
adequate for the existence of the kinetic potential, but reserves 
the proof of this for another occasion. In his posthumous 
papers we find more about the method of proof which he had 
chosen for this, but in regard to this point he came to no 
satisfactory conclusion. On April 25, 1886, he writes from 
Baden-Baden to Kronecker : 

1 1 have received your card from Berlin with the address you 
give for my manuscript, and your letter of the 21 st inst. from 
Florence finds me here. As a matter of fact most of my MS. 
already has the pages numbered, but I am still hung up over 
one point as to which I must consult a paper by Lipschitz, which 
was sent on to me here by Konigsberger. Pray, however, do 
not delay the printing for me; that would distress me very 
much. I can appear just as well in the third or the fourth part. 
In the attempt to reverse my propositions, I have been led to 
the theory of polydimensional potential functions, where one has 
to walk very warily, and I have not decided whether to make 
this discussion a digression in the main essay, or to treat it 
separately. Even in the second case, however, I must first get 
my excursus worked out . . . Boltzmann's essay opens with 
some very interesting observations, with which I also occupied 


myself at one time without coming to any right conclusion ; 
still I am content, for I see that Boltzmann could not get 
much farther either/ 

Hamilton had replaced Lagrange's equations of motion by 
a system of total differential equations of the first order, which 
present the total differential quotients taken according to the 
time of the free co-ordinates, and a like number of quantities 
deduced from vis viva (momentum of motion), as the partial 
differential quotients according to these magnitudes of the 
energy supply. Helmholtz, for any given form of the kinetic 
potential, generalizes the form of the corresponding differential 
equations of Hamilton. He then proceeds to apply the above 
theory to the laws of reciprocity that govern the changes in the 
forward and backward motions consequent on small impacts, 
after the lapse of a certain time. He terms the motion of the 
system reversible, when the sequence of positions which it has 
taken up in its forward motion can also be traversed in the 
opposite direction without the action of other forces, and with 
the same time-intervals for each pair of similar positions. He 
thus arrives at reciprocal laws, of which those which he had 
long before established for sound and light (though only for 
systems at rest) are merely special cases. Just as the forces 
of heat had at an earlier period been referred to the concealed 
motions of tangible masses, and Clerk Maxwell had recognized 
in electrodynamic forces the action of concealed masses in 
motion, so Helmholtz now proposed in general to admit the 
motion and energy of these concealed masses in the treatment of 
physical problems, since in the invisibilities that lie behind 
phenomena, he saw only motion and mass that are incapable 
of being demonstrated to our senses. And thus he selected 
the law of least action for the expression of the total motion, 
since this law admits that the mechanical system the internal 
forces of which can be represented as the differential quotients, 
independent of time, of the force-functions of the visible co- 
ordinates of the system is also affected by external forces that 
are dependent on time, the work of which must be specially 
calculated, which therefore do not belong to the conservative 
forces of motion, but are conditioned by other physical processes. 

Helmholtz had been led to these universal considerations by 
his investigation of the form of the kinetic potential, as required 


by Maxwell's theory of electrodynamics. In this the velocities 
of electricity appear in a function of the second degree, the 
coefficients of which are not, however, constants, as are the 
masses in the value of the vis viva of ponderable systems, while 
linear functions of the velocities moreover come into play, so 
soon as permanent magnets are introduced. Since the pheno- 
mena of light can essentially be explained on the hypothesis 
that the ether is a medium of similar properties to the solid 
elastic ponderable bodies, and as the law of least action must in 
any case be held valid for the motion of light, Helmholtz even 
at this stage regarded the validity of the principle of least action as 
far transcending the limits of the mechanics of ponderable bodies, 
and held it to be highly probable that it was the universal law 
of all reversible natural processes. 

In this connexion it must be mentioned that Boltzmann had, 
as early as 1866 (in a memoir, ' On the Mechanical Significance 
of the Second Law of Thermodynamics/ that remained 
comparatively unnoticed and was unknown even to Clausius), 
formulated a law for the mechanics of ponderable masses, which 
is as analogous to the Second Law of Thermodynamics as the 
Law of Vis Viva is to the First, and that, as Boltzmann wrote to 
Konigsberger in 1896, he had, as early as 1867, been presented 
by Stefan to his colleague Loschmidt as ' Herr Boltzmann, the 
discoverer of the physical significance of the law of least action '. 
The scope and weight which Helmholtz attributed to this 
principle in every department of physics, and actually established 
by rigid mathematical deductions, appear with increasing clear- 
ness from his earlier as well as from his subsequent papers ; the 
almost simultaneous work of J. J. Thomson was directed to the 
same object. 

Helmholtz was led to these universal investigations, which 
aimed at the widening of the principles of mechanics, by special 
cases, whence he had shortly before obtained a similar though 
less extensive generalization, published in 1884 in the above- 
mentioned memoirs on the principles of the statics of monocyclic 
systems. By monocyclic systems he understands mechanical 
systems which exhibit internally one or more stationary motions 
returning upon themselves, but which, supposing there to be 
several, depend in their velocity upon one parameter only, while 
systems with several independent parameters are termed poly- 


cyclic. A stationary motion is one in which (as expressly 
pointed out by Helmholtz in reply to a criticism made by 
Clausius, who had erroneously attempted to prove that his 
conclusions were incorrect) the homogeneous moving particles 
exhibit constant velocity at the same spot, as in the motion of 
a rotating disk, or in a current of frictionless fluid in a ring- 
shaped canal. It was further assumed that the forces that act 
between the bodies of such a system are conservative, and that 
the problems to be attacked are statical, in the sense that altera- 
tions in the state of the system, while not excluded, go forward 
so slowly that the system never perceptibly passes out of the 
state in which it might remain : which hypotheses are also 
tacitly accepted by Clausius as the basis of the whole system of 
laws which he has laid down for the reversible changes of heat. 
Helmholtz insists that heat-motion is not, strictly speaking, 
monocyclic, since each individual atom apparently alters its mode 
of motion perpetually ; but inasmuch as in an enormous number 
of atoms all possible stages of motion are represented, the me- 
chanical features of monocyclic motion obtain, even if the several 
steps are performed now by one atom and now by another. 

He then defines as the object of his investigation the proof 
that there is a class of motions that are perfectly intelligible 
from the mechanical point of view, in which there are the same 
limitations to the conversion of work-equivalents as are pre- 
dicated by the second law for the motion of heat. Helmholtz 
protests against Clausius's objection to the effect that he had 
claimed to have provided an explanation of the Second Law of 
Thermodynamics ; in choosing instances of monocyclic motion 
he was concerned merely with their complete mechanical 

' As a rule/ he says on a later occasion, ' I have only felt it 
necessary to reply to a criticism of scientific laws and principles 
when new facts could be brought forward or misunderstandings 
cleared up, in the expectation that when all the data were given, 
my scientific colleagues would ultimately form their judgement 
without the discursive explanations and sophisticated arts of 

Helmholtz in the first place developed the general equations 
of motion of mechanics for polycyclic systems; under the 
foregoing assumptions, and on the hypothesis that one or 


several of the external forces that act on the system are persis- 
tently nil, he succeeded, as was indicated above, by the elimina- 
tion of single co-ordinates, in finding other exact equations for 
the remainder according to Lagrange's formula, and names the 
system resulting from the elimination of those co-ordinates the 
imperfect, in opposition to the original perfect system. These 
investigations were then specialized for the general case of 
monocyclic motion, in which several velocities are present, 
which, however, all depend upon one of the same. Helmholtz 
imagines fixed associations acting in such a way that they have 
no influence on such motions as take place of themselves under 
the play of the effective forces, in correspondence with the 
equations of the combination, but that they oppose to any 
incipient deviations such forces as are necessary in order to 
check this deviation; the forces exerted by fixed associations 
contribute no work to that done by the forces acting from 
without. He terms the system after the introduction of these 
fixed associations, the restricted system. Two equations (ana- 
logous to the two relations of Carnot-Clausius in the Theory 
of Heat) are derived from the relation that the work applied to 
the acceleration of motion in the restricted system is equal to 
the sum of work expended for the same alterations of velocity 
in the unrestricted system. The first of these states that the 
heat that enters the system during a vanishingly small alteration 
of the absolute temperature and the parameter, as measured 
by its work-equivalent, is equal to the increase of total energy 
and of freely convertible work not transformed into heat, which 
the system gives off externally on alteration of the parameter, 
provided that alteration of the temperature without alteration of 
the parameter induces no intake or output of any form of work 
other than that of heat : the second finds this quantity equal 
to the product of the temperature and the increment of a 
quantity which Clausius had called entropy, while Helmholtz 
terms the factor which here is temperature, or a function of it, 
the integrating denominator. 

Precisely the same relations obtain for the monocyclic 
systems, with all the correlative inferences as to limited con- 
vertibility. Since in the Theory of Heat, temperature, which 
represents the integrating denominator, is (in accordance with 
the kinetic theory of gases) proportional to the vis viva of the 


internal motion, and since Helmholtz considers the hypothesis 
of Clausius and Boltzmann (that this is the case for all other 
bodies also) to be highly probable, he next inquires into the 
conditions under which vis viva becomes the integrating 
denominator for monocyclic systems with fixed associations of 
the moving elements as is the case for simple monocyclic 
systems. He finds that it is a condition that the entropy of the 
restricted system should be a homogeneous function of the 
first degree of the momentum of the unrestricted system, 
whence it results that if the complete system of parameters is 
kept constant, the total momentum and the velocities of the 
restricted system must increase in proportion to the resulting 
momentum and the resulting velocity of the internal motion. 
It was shown that all cases known to us at present of the 
mechanical coupling of any pair of cyclical motions fulfil the 
conditions under which vis viva is an integrating denominator 
in the compound monocyclic system. He further succeeded in 
defining the special mode of these fixed associations between 
the moving parts of the system more exactly. When two 
monocyclic systems originally independent of each other are 
transformed by a proper adjustment of external forces into 
a state corresponding with this particular kind of fixed associa- 
tion, it is possible to bring them into this fixed association 
without disturbance of the motion present ; and they can then, 
upon further alterations of energy, continue their motion while 
maintaining this fixed combination, which is again analogous 
to the motion of heat, in which two bodies of equal temperature 
can be brought into conducting contact without alteration of 
their internal motions, so that during new and sufficiently slow 
changes they maintain a constant temperature. This state of 
temporary fixed association is termed by Helmholtz the coup- 
ling of the system. He points out, as especially interesting, 
the case in which a mechanical association is set up between 
two systems which have equal values for one of their integrating 
denominators, in such a way that, so long as this association 
persists, the equality of this denominator will be maintained, as 
is the case in the contact of two bodies at the same temperature, 
where the temperature is the integrating denominator. Helm- 
holtz calls this kind of association an isomerous coupling. It 
is found universally that if monocyclic systems only admit of 


inter-associations, in which the two preceding characteristics of 
heat motion obtain, then the third essential characteristic 
of heat as expressed in Carnot's Law, its limited convertibility, 
holds good also, and in accordance with these conditions 
Helmholtz develops the corresponding characters of the 

In conclusion he discusses another and hitherto neglected 
general law, which affects the character of all associations that 
can be set up by means of ponderable natural bodies in the 
case of bodies in motion. Wherever, in the older discussions of 
mechanical problems, fixed associations are referred to, the 
expression covers only the inalterability of given spatial 
measurements, but here it implies fixed relations between 
velocities. At an earlier point Helmholtz only used for the 
establishment of the equations of the problems associations 
which had no influence, so long as the motion was already 
proceeding in and for itself in a way that corresponded with 
them, and which accordingly neither performed nor destroyed 
work ; but in considering the question of the cases in which vis 
viva becomes the integrating denominator of the compound 
monocyclic system resulting from the associations he was led to 
distinguish those cases which he terms pure kinematic associa- 
tions. He refers this distinction back to still more general 
considerations, arriving inter alia at the interesting proposition 
that no kind of attraction of cyclical motions conceivable 
between physical bodies can avoid the admission of any desired 
proportional increase of all the velocities, the relations of these 
velocities to one another remaining unaltered, so long as the 
value of all the co-ordinates is constant. The problem of 
finding analytical expressions for such associations as make 
a polycyclic system monocyclic had been attacked by Kronecker 
quite generally from a purely analytical point of view, as an 
appendix to Helmholtz's first paper. Helmholtz here gives 
merely the integration of the equations of restriction for any 
physical system, and compares the results with those of 

The simultaneous preparation of the new edition of Physio- 
logical Optics brought Helmholtz a little respite from these 
arduous and exhausting labours. Part I appeared in 1885, Pahs 
II and III in the following year, Part IV in 1887, F* 21 ^ V in 


1889, Parts VI and VII in 1892, Part VIII in 1894, and the 
conclusion not till 1895, after his death. During this time he 
evolved many developments and improvements of his earlier 
theories, and again exchanged a lively correspondence with 
many learned scientific men. On March 2, 1885, he writes to 
Lord Rayleigh : 

' I have never doubted that our colour-system depended on 
three variables, and no more. In regard to colour-blindness, 
the recent observations of Bonders and of my assistant 
Dr. A. Koenig show that this defect cannot be referred simply 
to the lack of one of the fundamental colours, but that two of 
the primaries (red and green) appear to acquire a more even 
distribution in the spectrum, so that now one and now the 
other makes a more vigorous impression ; in other words, the 
resulting curve approximates now more to the red, and now to 
the normal green sensation. In addition to this we have every 
shade of lessened power of discrimination. Consequently 
different individuals require very different mixtures of lithium 
and thallium light, in order to make up sodium light. ... I am 
much excited over the electrochemical equivalent of silver, 
having occupied myself during the last winter with the attempt 
to construct good methods of absolute measurement for galvanic 
currents. ... I confess that I am getting heartily sick of giving 
lectures. It is possible that we may be going to have a scientific 
Physical Observatory here, as a gift from Dr. Werner Siemens, 
with no teaching attached to it, the Direction of which has been 
offered to me. But the matter moves all too slowly for my age, 
which is sixty-three/ 

The close of the year 1885 brought great joy to the Helm- 
holtz family. After long anxiety over the health of their son 
Robert, he was able, on December 23, to take his doctor's 
degree in Berlin,' with a highly commended thesis on the ' In- 
vestigation of Vapour and Fog, particularly those of Solutions '. 
His essay on ' The Alterations of the Freezing-point as 
calculated from the Vapour Tension of Ice ' was published the 
next year, followed a few months later by his ' Experiments 
with a Steam-Jet ', published in the Annalen d. Physik u. 
Chemie, which was very well received in the scientific world. 

Helmholtz also wrote an interesting ' Report on Sir William 
Thomson's Mathematical and Physical Papers ' for Nature , in 


1885, in which he speaks with the greatest admiration of the 
issues of his friend's genius. He places the great value of 
Thomson's scientific methods in the fact that he followed 
Faraday's example in avoiding hypotheses as to unknown 
matters as far as possible, and took pains in his mathematical 
treatment of the problems to express simply the law of the 
observed phenomena. This limitation of his field enabled 
Thomson always to bring out the analogy between the dif- 
ferent processes of nature far more clearly than would have 
been the case had it been complicated by widely divergent 
ideas in regard to the internal mechanism of the processes. 

At the close of this year Helmholtz received an intimation 
from Bonders that the Ophthalmological Society had voted 
him the first medal struck in remembrance of Albrecht von 
Graefe, and that it would be presented to him the following 
autumn in Heidelberg. He replies to Bonders on January 31, 
1886: 'I am greatly flattered at being the recipient of the 
Graefe Medal, the more so as long years have gone by 
since I recalled myself to the memory of the ophthalmologists. 
. . . On the whole we are well ; if I am aware of certain infirmities 
of advancing age, I cannot complain of deficient working 
powers ; I only wish I had more free time. One of the causes 
which lost me nearly a day a week for many years, the migraine, 
has almost entirely disappeared. They always told me it 
would wear out with old age. The main point really is to 
learn what one can do, and to respect one's limitations.' 

At the close of the Summer Session of 1886, Helmholtz went 
alone (his wife being detained by the illness of their son) to 
Heidelberg, for the celebration of the sooth anniversary of the 
University, where he delivered a stirring discourse in its 
honour at the Banquet on August 4, in the presence of the 
Crown Prince of Germany, the Grand Buke of Baden, and the 
Rector of the University. 

Immediately after this, Helmholtz was presented with the 
Graefe Medal, on August 9, at a solemn session of the Ophthal- 
mological Society in Heidelberg. He replied to the fine 
address of the President, Bonders, with expressions of profound 
gratitude, and ended his long oration with the words : 

1 And now you must permit me to express my conclusion in 
allegorical language, so as to wound no feelings of personal 


modesty. Let us suppose, since an allegory does not bind us 
to historical accuracy, that up to the time of Pheidias no one 
had a chisel hard enough to work on marble with complete 
mastery of form. At most they could knead clay or carve 
wood. Then a clever smith discovered that the chisel could be 
tempered. Pheidias rejoiced over the improved tool, fashioned 
his divine statues with it, and manipulated his marble as none 
had done before. He was honoured and rewarded. But great 
geniuses are, I have observed, most modest, just in that wherein 
they excel all others. That particular thing is so easy to them 
that they hardly understand why others cannot do it as well. 
With the highest endowments there is always associated a 
corresponding sensitiveness to the defects of the artist's own 
work. Accordingly, Pheidias in an access of noble modesty 
says to the smith: "Without your aid I could not have done all 
this. Yours is the honour and the glory." The smith can but 
reply : " I could not have done it with my chisel ; you without 
any chisel would nevertheless have modelled wonderful works 
in clay. I must decline the renown and glory if I am to remain 
an honourable man." Then Pheidias was taken away from the 
world ; his friends and scholars, Praxiteles, Paionios, and others 
survived him. They all used the chisel of the smith ; the world 
was filled with their works, and the glory of them. They 
determined to honour the memory of the deceased with a 
wreath, to be bestowed on him who had done most for art, and 
in the art of statuary. The beloved master had often praised 
the smith as the author of their great successes, and at last they 
decided to award the wreath to him. " It is well," replied the 
smith, " I consent. You are many, and among you are clever 
people ; I am only one. You declare that I have been of great 
service to you, and that there are sculptors in many places who 
adorn the temples with copies of our divine statues, which 
without the tools I gave you, would have been less well 
fashioned. I must believe you since I have never chiselled 
marble, and I thankfully accept what you award me. I myself 
should have voted for Praxiteles or Paionios." ' 

While Frau von Helmholtz was tied by one of the long and 
wearisome illnesses of her son Robert, Helmholtz planned a 
few days at Interlaken with his daughter Ellen, but became 
seriously ill as soon as he arrived there, in consequence of the 


undue exertions which he had undertaken in Heidelberg. On 
August 22, his wife, who had hastened to him, writes to Robert : 
* I found your father weak and ill, and very depressed. He is 
convinced that he is on the point of death, and is altogether in 
a curious state. The doctor found himself superfluous in the 
face of your father's theories and his very limited obedience, 
and seems to give in to him.' 

In the early days of September he was able to move from 
Interlaken to Rigi-Kaltbad, and from there went with his wife 
to visit the Minghettis and Blaserna at Selisberg. 

The painful attacks and mental depression, however, soon 
reasserted themselves. ' Your father must return home,' writes 
his wife. ' He wants proper watching and treatment, must have 
special nourishment, and have done with this hotel life. If we 
cannot give him mountain air at home we can at least see that 
he has rest and care, which is something to the good/ 

First, however, they went to consult Kussmaul at Strasburg. 
' He listened with great attention, asked clear and precise 
questions, and then said to me : "I am really unable to find 
anything amiss, but I would not therefore treat his condition 
lightly; we can't find out everything by auscultation. Positively 
ill he is not, but neither is he well. You must take the utmost 
care as to his diet." ' 

After a rest of some weeks in Baden he recovered almost 

During his stay on the Rigi, Helmholtz made a series of 
observations which were the starting-point of his subsequent 
and fundamental work in meteorology. On October 22, 1886, 
he sent a brief report to the Physical Society, ' On Clouds 
and Storm Formation/ describing a phenomenon he had wit- 
nessed there. One September morning the view of the Jura 
from the Kanzli on the Rigi was clear, while at a somewhat 
lower level a layer of light clouds indicated the upper edge of 
a horizontal layer of dull and heavy air, travelling from North 
to South, which formed the primary cyclone, by the disturbance 
and rolling up of its edges. In the course of the day the clouds 
increased, till by the evening they had formed great masses, in 
which the separate currents rising from the lower layer could 
be distinguished, and which were ultimately equalized by the 
electrical discharges. 


For the time being Helmholtz gave no explanation of the 
phenomenon ; it was necessary first to arrange his observations 
scientifically. 1 1 thought it useful/ he said at a later time, ' as 
far as I was able, to introduce rigid mechanical concepts into 
meteorology, and to see what could be determined by that 

'I cannot conclude this letter/ Bezold writes, on October 9, 
1902, l without especially referring to the loss I have sus- 
tained in the death of the two great physicists Helmholtz and 
Hertz, who did not consider meteorology a low form of science, 
but contributed to it themselves, and were deeply interested in it.' 

Before the end of the year Helmholtz was made Vice- 
Chancellor of the Friedensklasse of the Order Pour le Merite. 
He resorted to Menzel, the Chancellor, to learn what his 
obligations were, and Menzel replied : ' I can only say to you 
what Ranke replied to me in his time, " As Vice-Chancellor you 
have nothing to do but to wait for my death to become 

In December, 1886, Hertz sent Helmholtz the continuation of 
the experiments begun in Berlin and resumed at Kiel, which 
already gave promise of the rich significance of his discoveries. In 
a letter from Karlsruhe, Hertz remarks: 'I take this opportunity 
of communicating certain experiments in which I have been 
successful, because I was in hopes when I undertook them that 
they might interest you. I have succeeded, unmistakably, in 
showing the inductive action of one open rectilinear current 
upon another open rectilinear current, and I venture to hope 
that this method will eventually yield the solution of one or 
other of the questions associated with this phenomenon. 1 
Helmholtz, who already recognized the full significance of this 
work, was greatly excited by the more detailed account of its 
progress; he took no further part in the development and 
organization of these experiments, but handed the whole subject 
over to his great pupil Hertz. 

'I am proud/ he said later, 'to think that my ideas will survive 
and develop in future generations, when my individual life is at 
an end, and you will understand that just as a parent cares most 
for the welfare of his own sons and endeavours to promote it, 
so I have a special predilection for the children of my brain, and 
you will also understand that as an individual I can only follow 


my own convictions, and lay most stress on these, rejoicing if 
the progress of science should tend in the same direction. Then 
again I am beset with doubts as to whether my own ideals are 
not too narrow, and my own principles in individual points too 
incomplete, to satisfy the cravings of Humanity for all time. . . . 
One banner only do I uphold, that it is the aim of science to 
comprehend reality, and to grasp the transitory as the pheno- 
menal manifestation of the intransitory that is, of Law/ 

It had become absolutely necessary for Helmholtz either to 
give up his teaching altogether, or to limit it materially, in order 
that he might devote the greater portion of his working time 
and energy to the investigations which occupied him almost 
exclusively henceforward to the end of his life, and a fortunate 
turn of the wheel soon enabled him to satisfy his inclinations. 

' The moment arrived/ says du Bois-Reymond, ' at which our 
distinguished friend Werner von Siemens prepared to found 
a Physico-technical Institute at Charlottenburg, partly at im- 
mense personal expense, which only he could afford. We 
knew that Siemens always deplored the amount of time and 
energy that Helmholtz was obliged to devote to his duties as 
a teacher, instead of to the prosecution of his incomparable 
researches, and it was no secret to us that he designed the post 
of Principal of the Institute for him, as one that would relieve 
him of all but his scientific occupations a post such as an 
academic could only regard as the ideal of his dreams/ 

The first proposals for the erection of a State Institute to 
be devoted to the advancement of exact science and technical 
instruction had been mooted as early as July 30, 1872, by 
Schellbach, supported by Helmholtz, du Bois-Reymond, 
Paalzow, Bertram, and Forster, and were warmly welcomed by 
the Crown Prince, afterwards the Emperor Frederick. In 
consequence of this movement, General Field- Marshal von 
Moltke, as President of the Central Directory of Survey in the 
Prussian State, appointed a special Commission towards the 
end of the year 1873, which in January, 1874, made ' Proposals 
for the Improvement of Scientific Mechanics and the Instru- 
mental Sciences '. Herein it was urged as the first duty of the 
State, that it should in future, along with provision for imme- 
diate needs, devote its attention to the supervision of technical 
instruction systematically, and not merely on occasion. 


Further discussions in 1875-6 led to the resolution that the 
earlier Industrial Academy should be supplemented by the 
foundation of an Institute for Scientific Mechanics, a project 
which was supported by the Ministers of Trade and of Finance. 
It could not, however, be carried out exactly in this form, since 
the Industrial Academy was just then replaced by a Technical 
High School, which was to incorporate all the different Technical 
Institutes, and the building of which was at once begun. 

Meantime the proposal to found a Mechanical Institute was 
again suggested in 1879 to tne Minister of Education (to whose 
department the Technical High School had meantime been 
transferred), by the Central Board of Survey in Prussia, and by 
the Mechanics* Union, in consequence of which Conferences 
were held at the end of 1882 in the Education Office, and it was 
decided to supplement the Technical High School by an 
Institute of this kind. The results of these deliberations, in 
which Helmholtz, Reuleaux, Forster, and Werner Siemens (all 
members of the former Commission) took part, were collected 
in a Memorial of May 23, 1883. In a special report Helmholtz 
pointed out the necessity of combining a scientific department 
with that of technical mechanics. 

In a further Memorial of June 16, 1883, the earlier scheme for 
the foundation of ' an Institute for the Experimental Promotion 
of Exact Science and the Technique of Precision* was proposed, 
with important additions, and accompanied by a draft of the 
proposed organization. Of the comments which Helmholtz 
appended to this Memorial the following may be cited : 

1 1 should like to emphasize still more strongly the fact that 
there is a whole series of important problems on the side of 
pure science, which cannot be undertaken with the private 
means of individual workers, or in the Laboratories of the 
University, which are founded for purposes of instruction, since 
their accomplishment demands costly accessories of space and 
instruments, and the unhampered working time of experienced 
and capable observers, beyond what can as a rule be obtained 
without assistance from the public funds. Till now it has been 
almost exclusively Astronomy which has been taken under the 
protection of the State in Institutes dedicated primarily to 
scientific research, and only secondarily to teaching the 
Observatories. Despite the apparent remoteness of the objects 


of this science from any aims of terrestrial utility, the old rule 
has held good here, that all serious scientific work must 
eventually find its practical application, even where this might 
previously have been least expected. Apart from the fact that 
astronomy has brought about a total revolution in our concep- 
tion of the world, in consequence of the ideas it has given us of 
the construction of the Universe, our navigation and the 
determination of civil and historical chronology depend essen- 
tially upon it; and the art of practical optics, of the higher 
branches of clockmaking, and all refinements of longitudinal 
and angular measurement, have developed directly out of its 
problems. Lastly, it would be of the utmost importance for 
higher scientific education if a small and select number of young 
men who had already proved their capability for experimental 
work could be admitted as Assistants or Volunteers at such an 
Institute, and thus have opportunity to learn the application of 
the most perfect methods and instruments possible.' 

Helmholtz then enters in detail into the reasons why the 
establishment of a scientific department of the Physico-technical 
Institute along with that of technical mechanics would be 
desirable. He had already outlined the duties of the scientific 
department of the proposed Physico-mechanical Institute as 
follows : 

' i. The exact determination of the intensity of gravity, and 
the comparison of this force at different parts of the Earth's 

1 2. The absolute measurement of gravitation, or the determina- 
tion of the mean density of the Earth. 

' 3. The continuation of the exact determination of the velocity 
of light at terrestrial distances, with the object of reducing cosmic 
distances to terrestrial measures of length. 

' 4. In the theory of the magnetic actions of electrical currents 
a velocity, which appears to be exactly equal to that of light, 
and which W. Weber characterizes as critical, . . . seems to play 
a fundamental part. Its identity with the velocity of light 
appears to me to indicate an essential and intimate relation 
between optical and electrical processes. We seem hereby to 
acquire a clue to the mysterious aspects of electromagnetic 
phenomena, which probably may lead us to their deepest 

B b 2 


'5. Determinations of electrical units of measurement. 

'6. Measurements of pressure and density of gases and 
vapours at different temperatures, and the measurement of the 
amount of heat consumed in these processes/ 

He further points out that the foundation of a scientific 
department ' would also make it possible for the older and accre- 
dited physicists of Germany to carry out special researches for 
which the apparatus is not to hand in their own neighbourhood '. 

1 It is unworthy of a nation that has acquired by its power 
and intelligence, and has to maintain, a position in the front rank 
of civilized peoples, to leave the provision for such fundamental 
knowledge to other nations, or to the accidental tastes of certain 
fortunately situated private individuals. Germany has already 
taken the lead by the institution of University Laboratories for 
Chemistry, Physics, and Physiology ; these have rapidly grown 
and multiplied, and have been imitated in all the surrounding 

When it was seen that the immediate fulfilment of the project 
was delayed by financial difficulties, and more particularly by 
the question of finding a suitable site, Werner Siemens made 
an offer to the Minister of Education of a site in the Marchstrasse 
in Charlottenburg, one hectare in area, as a gift to the Prussian 
State. In consequence of certain objections made by the 
Minister, and in view of the national significance of the plan, 
as also in the hope of its being carried out on a larger scale 
and with more complete equipment, Siemens resolved to 
repeat his offer, as previously made to Prussia, to the whole 
Empire, on May 20, 1884, in a gift of half a million marks, 
including the value of the land, for the endowment of an 
Institute of Scientific Research for Technical Purposes. ' The 
class-rooms and laboratories in our Universities and Schools, 
which are set apart for teaching/ said Siemens, ' are not suited 
for the installation of definite scientific researches, any more than 
are the professors in charge of them. Besides the leisure for 
intellectual absorption in their researches, the teachers lack 
suitable accommodation and means for procuring the right 
instruments and apparatus. The Institute we wish to found 
would thus contribute to the elevation and maintenance of the 
scientific achievements of our nation, and secure for us a post 
of honour among civilized peoples/ 


The Commission convened in 1884-5 to discuss the organiza- 
tion of the Institute insisted that the proposed foundation, which 
later received the name of Physikalisch- Technische Reichsanstalt, 
must comprise a scientific and a technical department, the 
vigorous co-operation of which would be of great value, and 
would further the national interests to the utmost. 

Even before the Imperial Budget for 1887-8 had officially 
sanctioned the necessary expenditure, Siemens, with the consent 
of the authorities, had begun the building of the Institute as 
early as 1886 at his own risk, and the Director's House and 
the buildings for the staff were accordingly ready for occupation 
by 1889, though the engine-room and observatory were not 
completed till 1890-1. 

In April, 1887, Helmholtz was formally invited by the 
Geh. Ober-Regierungsrath Weymann, on behalf of the State 
Secretary of the Interior, to become President of the new 
Physico-technical Institute. He accepted on April 4, provided 
he were compensated by the salary and the official residence 
attaching to it for his income from the Berlin University 
and the Friedrich-Wilhelm Institute for Medicine and Surgery, 
which he would have to give up, and on the understanding 
that his position at the Academy of Sciences in Berlin was in 
no way altered. 

When the Treasury had guaranteed these conditions Helm- 
holtz drew up a scheme for the organization of the scientific 
department of the Reichsanstalt, and detailed statutes defining 
the power of the Director of the Physical Section, as well as 
the co-operation of the Management in the annual schedule 
of work. The President of the Institute would reserve the 
right of initiating and carrying on other researches. The 
working constitution was ratified on July 26 by the Secretary 
of State for the Interior. 

Meantime the Philosophical Faculty of the University, so 
soon as they heard of Helmholtz's intention to resign his Chair, 
had applied to the Minister with the request ' that he would 
graciously make provision that Herr von Helmholtz should 
remain a teacher of the University and a Regular Member of 
the Faculty. As is known to your Excellency, Herr von 
Helmholtz's master-genius extends to many subjects, his pro- 
found knowledge to all mathematical and scientific, as well as 


philosophical and literary. The judgement of this one man is 
accordingly of consummate value, and cannot be replaced by 
any combination of judgements from individual professors, since 
the questions brought before the Faculty are nearly always 
those of the rival claims of the interests of the several depart- 
ments ; and any one who combines all these, and deliberates on 
them in his single mind, is the more capable of solving such 
weighty and complicated problems correctly and completely.' 

Upon the pressing request of the Minister, that upon these 
grounds, and from considerations referring to the salary, Helm- 
holtz should remain in connexion with the University, he 
declared his willingness to give a public lecture of two or three 
hours in each term, ' provided he were relieved from the duty 
of taking part in the executive work of the Faculty and the 

On April 4, 1888, his appointment as President of the Physi- 
kalisch-Technische Reichsanstalt was consummated. 

Despite the inevitable burden of organization Helmholtz felt 
thoroughly happy and contented in his new post. He found 
ample compensation for the many administrative duties (in which 
he was assisted by a distinguished body of younger men) in the 
stimulus of the innumerable problems that arose in the scientific 
department of the Institute, and in the freedom from the 
frequent repetition of experimental lectures and demonstrations, 
which had absorbed so much of his time and energy. 

During the last year before he definitely took over the 
Presidency of the Physico-technical Institute, Helmholtz had 
been profoundly stirred by the work of Hertz, who sent 
regular reports of his experiments to his former teacher. 

By the end of the year he was able to communicate his well- 
known experiments on the interference between the effects 
propagated along wires, and through the air, without at that 
time being able to demonstrate a finite velocity of propagation 
for these latter. Far greater surprises, however, were in store 
for Helmholtz in Hertz's letters of the early months of the 
ensuing year, 1888; his delight finds expression in the frequent 
repetition of the words ' Bravo! Best congratulations' at the 
end of his short answers : and when he sent to du Bois-Reymond, 
for the Academy, the memoir in which Hertz proved that the 
electrodynamic waves are reflected from solid conducting walls, 


and that with vertical incidence the reflected waves interfere 
with the incident, and give rise to stationary waves in the air, 
he concludes his letter with the words, ' Hertz's work is the 
work of a genius/ 

Helmholtz's desire to conclude a series of hitherto un- 
finished investigations had led him back in the summer of 1887 
to the subject of electrolysis. On July 28 he made a com- 
munication to the Academy ' On Further Investigations on the 
Electrolysis of Water', which he linked on to his first paper 
1 On the Thermodynamics of Chemical Processes ', written in 
1883, giving experimental proof of a number of the results 
therein deduced theoretically. He had previously discovered 
in theory that the electrolytic dissociation of water must occur 
with less electromotive force according as the quantities of 
oxygen and hydrogen dissolved in the proximity of the elec- 
trodes are smaller, and that no inferior limit other than zero 
could be given to the smallest electromotive force capable of 
dissociating perfectly gas-free water. The experimental deter- 
mination of this law, however, had presented great difficulties, 
inasmuch as the platinum anode, or both electrodes, contain 
occluded hydrogen or combustible gases, with which the 
oxygen conveyed by the current combines, so that a much 
lower E.M.F. can liberate bubbles of hydrogen at the cathode. 
The construction of a special apparatus and most accurate 
measurements now yielded a satisfactory reconciliation of theory 
and experiment. 

Helmholtz was also contemplating the extension and definite 
conclusion of his thermodynamic and chemical researches in 
a paper of the widest scope which was to be entitled ' Thermo- 
dynamic Considerations as to Chemical Processes'; but of 
this only a few fragments remain, in particular the Introduction, 
designed for a large circle of readers, which summarizes the 
researches of Berthelot and other workers, and is of great 
interest on account of its clear historical exposition of the 
theories of physical chemistry that had come so prominently 
forward in the last ten years. 

The parting of Helmholtz from the great society of Members 
and Associates of the Berlin University, to which, properly 
speaking, he belonged only as an honorary member from this 
time forward, is marked in the history of his untiring scientific 


activity, by the essay on Natural Philosophy entitled ' Numbers 
and Measurements treated from the Epistemological Point of 
View ', and dedicated to his friend Eduard Zeller for the fiftieth 
year of his doctorate. This paper was an important amplifica- 
tion of the empiricist theory which he had previously put 
forward, that the axioms of geometry can no longer be regarded 
as propositions incapable of and not requiring proof, and also 
established this theory with respect to the origin of the arith- 
metical axioms, which stand in the corresponding relation to 
the conceptional forms of time. 

The five well-known axioms of arithmetic (i) when two 
magnitudes are equal to a third they are equal to each other ; 
(2) the associative law of addition, (a + b) + c = a +(b + c) ; (3) the 
commutative law of addition, a + b b + a ; (4) if equals be added 
to equals, the wholes are equal ; (5) if equals be added to unequals, 
the wholes are unequal were then examined in regard to their 
independence of each other, and in their relation to experience. 
While he derives numbers from the fact that we are able to 
carry in our memory the sequences in which the acts of 
consciousness have followed one another in time, the theory of 
pure numeration is for him merely a method built up on psy- 
chological facts, for the logical application of a system of signs, 
unlimited in its extension and possibility of refinement, with 
the object of representing the different modes of association of 
these signs, all of which tend to the same ultimate result. After 
the definition thus obtained of the ordered series of the positive 
whole numbers and of the unequivocal character of their 
succession, he established the concept of the addition of pure 
numbers, in the first place explaining the signs by saying that 
if any number be designated by the letter #, the next following in 
the normal series shall be (a + 1), while the definition of (a + b) is 
that number of the principal series which is reached by counting 
One for (a + i), Two for [(0 + i) + i)], and so on, until one has 
counted up to b. And then he shows how this idea of the 
addition of pure numbers confirms the arithmetical axioms of 
the equality of two numbers in relation to a third, the associ- 
ative law of addition, and the commutative law merely by 
the agreement of the result with that which can be derived 
from the numbers of external enumerable objects. Figures 
are to him arbitrary signs, to establish the time order of 


a series, and all enumeration is the arrangement of the things 
enumerated in a time-series ; he regards the composition of 
fractions of time into magnitudes of time as the primitive type 
of addition. He further proposes definitions for objects in 
general; the definition of equality is that if two things are 
equal to a third thing they are equal to each other : permutation 
(combination) is the association of different things, in which the 
order of association is not indifferent ; addition is the combining 
of homogeneous things, independent of the order of association ; 
multiplication is (at least in all its applications), as he sets forth 
in a note, the combining of heterogeneous magnitudes, for which 
the order of association is indifferent since units of any kind 
are multiplied by abstract figures, or horizontal by vertical 
lines, or distances by masses. In the other mathematical 
operations the order is not an indifferent matter. Lastly, he 
regards magnitude as an additive combination of homogeneous 
units or parts : equal magnitudes are composed of pairs of equal 
parts ; while a sum is the additive combination of magnitudes. 

While therefore objects which are equal in any definite 
respect, and are enumerable, are termed units of numeration, 
their number is termed a denominated number, and the special 
kind of unit which they comprise is the denomination of the 
number ; the concept of the equality of two groups of denomi- 
nated numbers of similar denomination is established by their 
having the same number. If the objects, or the attributes of 
objects, which when compared with others admit of the dis- 
tinctions of greater, equal, or less, are termed magnitudes (as 
to which only empirical knowledge of certain sides of their 
physical reactions in coincidence and co-operation with others 
can decide), and if we can express these magnitudes by a certain 
number, we term this the value of the magnitude, and the 
method by which we find the given number we call measure- 
ment. Thus we measure a force either by the masses and 
motions of the system by which it is exerted, or in dynamic 
measurement by the masses and the motions of the system on 
which it acts, or lastly by the static method of measurement of 
force, by bringing the force into equilibrium with known forces. 
There remains only one question to be answered : when may 
we express magnitudes by denominated numbers, and what 
actual knowledge do we gain by doing so? To answer this, 


Helmholtz proceeds to the questions that are so important and 
interesting in physics, of physical equality, and the commutative 
and associative laws of physical relations, defining addition in 
a somewhat extended form as a relation of magnitudes of the 
same kind, the results of which do not alter by exchanging the 
terms for equal magnitudes of the same kind. So that Helm- 
holtz maintained for arithmetic, or for the transcendental con- 
ception of time in relation to its axioms, the same standpoint 
which he took up in opposition to Kant in his investigation 
of space. 

Helmholtz's house had been for more than ten years the centre 
of the most enlightened minds of the new capital of the Empire: 
here, to their mutual advantage, both mental and spiritual, the 
most profound thinkers met the most talented artists, not on 
neutral ground, but on a soil ripe to receive all that was good 
and beautiful. Moltke's pupils and Bismarck's disciples came 
together under the Olympic peace of the great investigator 
and his distinguished wife, who here developed her transcen- 
dent gift of bringing the most unlike minds into contact. 
External position counted for little with her if it were not 
accompanied by intellectual distinction or a fine artistic 
temperament : 

1 1 have set myself all my life against a low level of social 
environment, and kept it away wherever it was not imposed 
upon me. I have held good manners, and a mental equipment 
superior to my own in some aspect, or interesting at the least, 
to be the first requirement of social intercourse. In this 
respect one cannot afford to be modest, unless one means to 
drop into mediocrity/ 

The migration to the new Presidential dwelling in Charlotten- 
burg did but provide a 'more intimate and perhaps even 
more harmonious setting for these social relations '. 






DURING the closing winter months of 1887 and the whole of 
the following summer Helmholtz gave his entire thought, time, 
and energy to the arrangement and organization of the Reichs- 
anstalt, more particularly in regard to the scientific and technical 
questions with which it had to deal; and with the valuable 
assistance at his disposal everything was brought into order with 
surprising rapidity. 

After some months of uninterrupted work he was, however, 
compelled to pause on account of his health, and to abate his 
activity. At the beginning of August, 1888, he went with his 
wife to Bayreuth, and gave himself up once more to the enjoy- 
ment of * the incomparable charm of the Meistersinger ' ; and his 
wife did but interpret his emotions, when she wrote to her 
daughter : 

' Bayreuth, dear ones, was icy cold, and its enjoyment purely 
intellectual . . . that, however, was immense, and it is beautiful 
to be able to put earthly considerations so entirely into the back- 
ground. What the hohe Frau, as Robert calls her, has accom- 
plished in these marvellous representations passes imagination. 
Her great artistic will stands behind all her fellow workers, and 
her taste prevails everywhere. All the artists are at one, and all 
are good friends/ 

From Bayreuth, Helmholtz went on once more to Pontresina, 
to seek the refreshment it never failed to bring him. The warm 
tone of his letters betrays his delight that his son Robert should 
have made a name for himself, oy his various experimental 
researches, among the physicists, and that his inclination was 
increasingly directed to the study of mathematical physics. On 
August 18, he writes to him from Pontresina : 


'As regards your problem, I know the astronomers have 
discussed the question whether time is required in gravitation. 
I do not know how far its exact determination is possible. 
They assert that observation goes against this hypothesis. In 
any case it is useless to attack a problem of this kind until one 
knows what observations are possible, and how they should be 
carried out in order to decide it. 

' The thermoelectric currents in the body of the Earth present 
a complicated problem. An arrangement of different concentric 
shells in the Earth's crust would only yield currents correspond- 
ing with closed ring magnets, with no external action. Our 
present methods of measuring gravity are not exact enough to 
enable us to measure the gravitation of the Moon. Meantime 
the geodesists are seeking for better methods, and an Academic 
prize-question has been set on the influence of the suspension 
of the pendulum on its vibration-period (elasticity of the sup- 
port, form of the knife-edge, or, on the other hand, length and 
elasticity of the spring by which it is suspended). For my 
own part I have learned whatever I know of mathematics 
merely from the problems I have tried to solve, and have never 
been able to grasp anything from purely abstract study unrelated 
to problems. But you must first choose simpler tasks, either in 
mechanics, or the theory of potential functions, electrical dis- 
tribution, or the distribution of electrical currents. The theory 
of the pendulum, for instance, suspended by an elastic watch- 
spring would be a good example. This kind of suspension is 
far less subject to friction than that from a knife-edge/ 

Helmholtz spent his birthday as usual in Pontresina, in the 
company of his wife, and had the accustomed number of 
respectful congratulations ; this first birthday in his new post 
brought him a letter from the mathematician L. Kronecker on 
August 28, which is interesting both in its style and its con- 
tents : 

1 In a few more days, on the last of this month, you will 
complete the 67th year of your prolific life, by which light has 
been thrown on innumerable fields of activity. I send you my 
warmest congratulations. ... I am happy in the conviction that 
your present phase of exclusive devotion to mathematical 
physics will be succeeded by a phase in which you will turn to 
pure mathematics, and bring the light of your intellect to bear 


on that also. You have already attacked it at many points, and 
it is (as I have long felt) only the consequence of the remark- 
able development of your scientific life, which is unique in the 
history of the sciences, that it should, beginning at the right 
hand with the most practical scientific medicine, advancing 
through physiology to experimental and theoretical physics, 
arrive finally at the extreme left of the abstractions of " pure " 
mathematics. The wealth of practical experience, of sound and 
interesting problems, which you will bring to mathematics, will 
(like the work of the astronomers in the last century) give it a 
new direction and a new impulse; whereas the one-sided 
mathematical speculation that returns upon itself only leads to 
sterile regions. Therefore come over to our side, honoured 
friend, and impress the imperishable traces of your bold and 
original progress upon the paths of pure mathematics, so that 
the lines of the future may be indicated in this direction also/ 

At this very time Helmholtz was already midway in his 
great mathematical investigations of monocyclic systems and 
the principle of least action. 

The new edition of Physiological Optics compelled him at the 
same time to examine a whole series of difficult optical prob- 
lems, and restate the answers to them. On November 2, 1888, 
he made a brief communication to the Physical Society ' On the 
Intrinsic Light of the Retina ', published in an expanded form 
in 1890 in the Zeitschrift f. Phys. u. Psych., with the title ' The 
Disturbance of the Perception of the Least Differences of 
Brightness by the Intrinsic Light of the Retina '. Helmholtz 
finds that the intrinsic light is not equally distributed over the 
fundus of the retina, but always appears to us in irregular 
patches, and that what we generally perceive of this internal 
retinal excitation, under normal conditions, with weak external 
illumination, is only the local differences of brightness in the 
patches ; and it is only under exceptional circumstances that we 
can estimate the mean brightness of the fundus by comparison 
with still darker fields. Helmholtz adduces a number of 
highly interesting experiments, which show that the patchy 
character of the intrinsic light is the chief obstacle to the 
perception of very weakly illuminated objects, especially if they 
are small, since these disappear between the patches of the 
intrinsic light, and are confounded with them. His experiments 


also elicited the interesting fact that a large resting surface, 
emitting weak light, may disappear entirely in the intrinsic light 
of the retina, while still sending out sufficient light to render 
visible the moving objects which it illuminates. 

The further development of the theory of electricity on the 
lines of the Faraday- Maxwell hypothesis had, as we have seen, 
been handed over entirely by Helmholtz to his friend and pupil 
Hertz. The latter wrote on November 30 to Helmholtz : 

' When you asked me in Berlin if I had made any further 
experiments on electric waves, I had nothing important to tell 
you, but I have now made a farther advance, by which the 
relation between light and electricity seems to me to be firmly 
established, and I am anxious to tell you about it. 

' In the first place, I discovered by a happy accident that it is 
not only possible to produce waves several meters in length, 
but that one can also work with much shorter waves, which 
is infinitely more convenient. I have been able to confirm and 
in part to improve on my earlier results with waves 33 cm. long 
in air. I have also repeated the experiments with these short 
waves of sending the force by means of concave mirrors to a 
distance, and thus producing a beam, and with the best results. 

1 1 place my primary and secondary conductor in the focal line 
of a parabolically curved tin-plate 2 m. high by 2 m. wide, and 
then obtain a well-defined beam some ijm. in width from the 
mirror, which is perceptible in a second concave mirror up to a 
distance of 16 m., and apparently even farther. The beam can 
be directed by rotation of the mirror, and one can demonstrate 
rectilinear propagation and the formation of shadow perfectly 
by its means. If, for example, a man crosses the path of the 
beam, the stream of sparks in the induced mirror will entirely 
disappear. Yesterday I also succeeded in showing the regular 
reflection of the beam more plainly than I could have hoped. 
When I put the concave mirrors side by side, there was no 
effect from A to B ; but if a plane metal screen was placed in 
front of the concave mirror, sparks passed in B which were still 
perceptible when the screen was removed 10 m. from the 
mirrors. I was also able to establish the reflection at 45, by 
employing two adjacent rooms. Shutting the wooden doors 
between them in no way hindered the appearance of the 
secondary spark. On the other hand the sparks ceased when 


the plane reflecting screen was rotating as little as 5 to the one 
side or the other of the right position : this proves the reflection 
to be regular, and not diffuse. 

' You will pardon my impatience, in giving you such prompt 
information about these matters. I intend to repeat and extend 
my observations, and then to put them together in a report for 
the Academy, and hope you will be good enough to receive it, 
though doubtless you are already overcrowded with such things.' 

Helmholtz replied most genially : 

' I am much delighted with your last results. I have puzzled 
for years over the possibility of getting at these things, so that 
I am familiar with the whole train of thought, and its immense 
importance is obvious to me.' 

In 1885 Hertz had been appointed Ordinary Professor of 
Physics at the Technical High School of Karlsruhe, and he was 
subsequently, on the death of Kirchhoff and Clausius, offered 
the choice, through Helmholtz, of Berlin or Bonn, when he 
decided for the latter, because ' he preferred the Chair at Bonn, 
which was an experimental post, to the great honour the 
Faculty of Berlin had designed for him '. Helmholtz writes on 
December 15, 1888 : 

' Personally I am sorry that you are not coming to Berlin, 
but, as I have already told you, I believe it is for your own 
interest to go to Bonn. Those who have still much scientific 
work in view are better away from big cities. At the end of 
one's life, when it is more a question of utilizing the points of 
view one has arrived at for the education of the coming genera- 
tion and the administration of the State, the case is different.' 

The prosecution of his meteorological studies during the 
ensuing winter, and the consequent limitation of his public 
lectures and addresses to Scientific Societies, was only once 
interrupted with the express intention of doing justice to the 
great scientific merits of his quondam friend and subsequent 
opponent, R. Clausius. 

On January n, 1889, Helmholtz delivered a memorial lecture 
to the Physical Society of Berlin, which emphasized the great 
services rendered by Clausius, notwithstanding many points 
of past controversy between these eminent men. His own in- 
vestigations in recent years into the modern development of the 
mechanics of chemistry had all been based, so far as they were 


certain and established, upon the so-called Second Law of 
Thermodynamics, which (at first stated by Sadi Carnot in a 
restricted form that applied only within the narrowest limits 
of temperature) had been extended and generalized in the strict 
interpretation given to it by Clausius. ' This is not merely one 
of the most important, but also one of the most surprising and 
original achievements of ancient or modern physics/ because it 
is one of the very few principles whose absolute universality 
can be predicated independent of all dissimilarity of natural 
bodies. At the close of his laudatory appreciation of the work 
of Clausius, Helmholtz states that it had now for the first time 
become possible to obtain a concept of absolute temperature, 
independent of the characteristics of any particular natural 
body. Still more important was the fact that it had established 
the specific character of the motions of heat, by which the latter 
were differentiated from all other force-equivalents. While the 
others can be converted and reconverted indefinitely among 
themselves, this is only possible to a very limited degree for 
heat, so long at any rate as we are unable to go back to the zero 
of absolute temperature. 

What Helmholtz himself thought in regard to these difficult 
questions appears from the preceding account of his mono- 
cyclic studies, in which he distinguishes between organized 
motion, defined as a continuous function of the co-ordinates and 
the time, and unorganized motion, in which the motions of the 
neighbouring particles have no kind of mutual similarity. He 
regards the motion of heat as being also unorganized, but 
refers the difficulty of converting it into organized motion 
solely to the limitations of the methods at our disposal ; if we 
could but overcome this obstacle, all processes would necessarily 
be reversible. Moreover, as he repeatedly indicates, there are 
vegetative processes in many plants, where no source of energy 
is visible, and the question presents itself whether these may 
not in some sort be the organization of heat motion. 

During the Easter holidays Helmholtz heard of the death of 
his faithful old friend Bonders, and writes on March 27 to 
Engelmann : ' Your announcement of the death of Professor 
Bonders was a quite unexpected blow, and a great shock to me. 
I knew no other scholar and distinguished investigator in whom 
the consciousness of working for ideal aims was so keen and so 


inspired. To be in contact with him always gave one the 
sense of extraordinary benevolence and cordiality.' 

After moving in 1889 from the official quarters which he had 
previously occupied in Berlin to the residence assigned to the 
President of the Physikalisch-Technische Reichsanstalt, in the 
Marchstrasse, in Charlottenburg, Helmholtz temporarily laid 
aside all other scientific work, in order to devote himself, 
as far as was compatible with his duties at the Reichsanstalt, to 
meteorological research. On May 31, 1888, and on July 25, 
1889, he communicated two papers to the Academy ' On 
Atmospheric Motion ', the contents of which were in part, and 
in a somewhat altered form, the subject of a lecture given in 
September, 1889, to tne Heidelberg Congress of Natural 
Science, ' On the Movements of the Atmosphere/ 

In the first place, Helmholtz applied to Euler's hydrodynamic 
equations for a fluid subject to friction the consideration of 
which he had so frequently availed himself, viz. that its 
particular integrals held good also for the case in which the 
co-ordinates, the time, and the friction-constants were multi- 
plied by an arbitrary factor n, while the forces, the pressure, 
and the components of velocity remained unaltered. It was 
found that the motion proceeded analogously, only more 
slowly, if in the motion of the magnified masses the friction- 
constants were correspondingly magnified. But if the value 
of these last be unaltered, the influence of friction upon the 
magnified mass will be much less, and the large mass will 
have the effects of inertia much less affected by friction. Since 
the density and the potential remain unaltered, and the forces, 
inasmuch as the entire process takes n times as long, must be 
reduced to the n\h part of their previous value, Helmholtz 
concludes that the different densities of the air at different 
heights cannot be reproduced in reduced models, since we 
cannot alter gravity proportionately. 

He showed by special cases how extraordinarily insignificant 
the effects of friction at the surface of the earth, which would 
ensue in the course of a year, must be for the upper air-layers. 
The destruction of vis viva by friction can only occur at the 
surface of the earth, and at the separation surfaces of vortex 
motions. So too for heat exchange : hardly anything except 
radiation and convection of heat comes under consideration in 


the motions of the air, save at the limits next the earth's 
surface, and the internal surfaces of discontinuity, where indeed 
changes of temperature might take place between the warmer 
and colder layers, by the actual conduction of heat, and dif- 
fusion of the molecules that are in motion. It was shown on 
the basis of Maxwell's friction-constants for air, that a motion 
delayed by friction at o would fall to half its velocity in 42,747 
years, if the interval between the two layers is 8,026 m., the 
mean height of an atmosphere of constant density; and the 
lower temperature of the upper layers still further diminishes 
the effect of friction. In the same way, the conduction of heat 
may reduce the difference of temperature in the upper and 
lower surfaces of an atmosphere of 8,026 m. to one-half in 
36,164 years. 

1 For meteorology/ says von Bezold, ' the researches of 
Helmholtz into the integrals of hydrodynamic equations will 
be of enormous importance in time to come.' 

When, at the end of July, Helmholtz communicated Part II 
of his Meteorological Investigations to the Academy, his 
colleagues found him markedly depressed. Sorrow and 
trouble weighed upon his family. It was increasingly certain 
that the bodily affections of his younger son Fritz must perma- 
nently hinder his mental development, although the malady of the 
elder son Robert appeared for the time at least arrested, and 
his parents hoped that a long life might lie before him. His 
scientific work had found general recognition, his cheerful 
disposition made him the life of the house, and he was always 
surrounded by a set of talented young people, who shed 
'the sunshine of their youthful candour' on the Helmholtz 

'Even if Robert Helmholtz,' writes the Assyriologist Pro- 
fessor Lehmann from Berlin, ' at all times represented a some- 
what unattainable standard to his friends, this was due in 
part at least to the sincerity of his disposition. What was a 
striking trait in both parents had descended to their son. He 
reminded one of the mother, whose features he bore, in his 
frankly outspoken yet never depreciatory criticisms, and his 
no less generous appreciations. His father's immense scientific 
veracity extended to his judgements of all human relations, 
so that Frau von Helmholtz justly observed, "My husband 


has no confidence in any one who does not follow out his 
own scientific conclusions logically and faithfully to the utter- 
most ; such persons are incomprehensible to him " ; and this 
veracity translated into life was characteristic of Robert also. 
The strikingly condensed and pregnant style of his papers 
and letters was in his father's opinion due to the necessity 
of economizing his forces. " It seemed/' Helmholtz once 
said, " at last, as though he were sparing even of his words." 
Robert's energy concentrated itself on the expenditure of 
his extraordinary talents in indefatigable scientific work in 
defiance of his bodily weakness. After his death his father 
showed that he had doubtless anticipated that only a short 
time was left to him, and had tried to complete as much as 
possible. He always looked up with admiration to his father's 
eminence, recognizing clearly and unreservedly that it was some- 
thing quite unattainable. " We average beings cannot compare 
ourselves with genius ; ours is quite another standard." 

' Perhaps nothing is more characteristic of Robert as a friend 
than the fact that he never referred to his own ill health or 
infirmity unless it were to encourage some friend who was 
less used to pain, and less patient of it.' 

Even down to the spring of 1889 he was engaged in most 
difficult experimental researches, his prize essay 'On the 
Radiations of Light and Heat in Burning Gases' being 
crowned by the Verein fur Gewerbfleiss in Berlin with the 
prize of 5,000 marks and a medal ; and until the first months 
of summer he was carrying on important experimental in- 
vestigations in Bonn and Berlin in co-operation with Richarz, 
when there came a sudden collapse of the frail body that 
had been doomed from birth. He had just surprised and 
delighted his father (who knew nothing of the matter) by 
his appointment as Assistant at the Reichsanstalt, when 
his strength forsook him. On his deathbed he prepared his 
prize work for publication ; he died on August 5. The intro- 
ductory words with which Helmholtz prefaced the posthumous 
publication of the memoir ran as follows : 

' When the first proof-sheets of the following essay arrived, 
its author was already lying on his death-bed. The sad duty 
has devolved on me, his father, of preparing it for publication. 
He had hoped to work through the second part of the essay 

c c 2 


again, and to enlarge it, and had already begun experiments 
with other combustible substances. Much of the rest also is 
incomplete, because the time-limits of the competition obliged 
him to be content with temporary determinations of certain 
points, which he could with more time have worked out more 
carefully and accurately: these cannot now be altered. Nor 
again am I sufficiently sure of his views in these matters, for 
he worked quite independently, and seldom asked my advice. 
It was only when the problem was down on paper that he used 
to show it to me, and discuss it. I must therefore confine 
myself to the alteration of any obvious errors on the part of 
the copyist, and the modification of a few points that are not 
clearly stated, where I can be certain that the author himself 
would have altered them in the same way, if he had looked 
through the proof-sheets/ 

The loss of their son was paralysing to the sorely-tried 
parents, and Helmholtz, who was quite broken down, went 
to Switzerland in the middle of August, to recuperate in mind 
and body amid new impressions. 

' Do not write too much/ he writes from Munich to his wife ; 
1 try to sleep as much as possible both day and night. Since 
we both have work to do in the world, and may not yet lie down 
and give it up, we must take care to remain fit for it. We 
must not leave Fritz just yet, but the future has become 
frightfully indifferent to me. I shall continue to do my work, 
but whether it is for a long or a short while begins to be all 
one to me now/ 

His letters show that Nature had a beneficial effect upon 
him in Pontresina : he took long walks, climbed the Piz 
Languard, which he had shrunk from for four years, and 
began once more to occupy himself with various and 
complicated problems. At the end of September he attended 
the Scientific Congress in Heidelberg, at which Hertz delivered 
the Address that has become so famous for its lucid simplicity 
and its profound content. 

1 1 came across the whole Siemens family, and Edison and 
his wife, the first evening in the Schloss Garden. Mr. Edison 
is a beardless individual somewhat resembling Napoleon I, 
but far kindlier, with an almost childlike expression and 
clever eyes, but he is very hard of hearing. In reply to our 


questions he gave us much interesting information about his 
way of working. To-day we had the address from Professor 
Hertz; it really was extraordinarily good, very finished in 
style, tactful and tasteful, and called out a storm of applause.* 

A quarter of a century had passed since Helmholtz had 
heard Kirchhoff, in his admirable Pro-Rectorial Address, in the 
Great Hall of Heidelberg University, announce that the dis- 
covery and logical development of the Law of the Conservation 
of Energy had been the greatest achievement of the century 
in natural science : and now he was in the front rank of the 
auditors of the lecture given by his great pupil Hertz, who 
had taken his stand upon Helmholtz's earlier criticism of the 
different electrodynamic theories, accepting his interpretation 
of the Faraday- Maxwell hypothesis, and had thereby arrived 
at his own fundamental discoveries. 

1 When in the present century the reactions between electrical 
currents and magnets became known, which are infinitely more 
complex than those of gravitation, and in which motion and 
time play such an important part, it became necessary to 
increase the number of actions at a distance, and to improve 
their form. Thus the conception gradually lost its simplicity 
and physical probability. It was sought to regain this by 
seeking for comprehensive and simple forms, the so-called 
elementary laws. Of these the celebrated Weber's law is 
the important example. Whatever may be thought of its 
accuracy, this attempt as a whole formed a closed system full 
of scientific charm: those who were once attracted into its 
magic circle remained imprisoned there. If the path indicated 
were a false one, warning could only come from an intellect 
of the highest originality, from a man who would look at the 
phenomena with an open mind, and without prejudice, and 
set out again from what he saw, and not from what he had 
heard, learned, or read. Such a man was Faraday. . . . To him 
the electric and magnetic forces became the actually present, 
tangible realities : while electricity and magnetism were things 
whose existence was disputable V 

In April, Helmholtz went to Cap d'Antibes to make scientific 
observations upon the movement of the waves of the sea, and 

1 Hertz, Miscellaneous Papers, English Translation by D. E. Jones, p. 315. 


communicated his theoretical conclusions, and the comparison 
of them with his observations, to the Academy of Berlin, on 
July 17, 1890, with the title ' The Energy of the Waves and 
Wind ', as the continuation and completion of his two earlier 
works on atmospheric motion. 

In his earlier investigations Helmholtz had shown that a 
level surface of water, over which a wind is blowing evenly, 
will be in a state of unstable equilibrium, and that the origin 
of the waves of water must be ascribed to this very circum- 
stance. Further, the same process must occur at the border 
of layers of air of different densities that slide over one 
another ; but will here assume much greater dimensions, and 
has an essentially causative significance in the irregular 
phenomena of meteorology. This determined him to investi- 
gate the relations of energy, and its distribution between air 
and water, more exactly in his memoir on the energy of waves 
and wind, while still confining himself to the case of stationary 
waves, in which the movements of the water particles can 
only proceed parallel to a vertical plane. He refers the laws 
of stationary rectilinear waves back to a minimal problem, in 
which the potential and kinetic energy of the moving fluids 
are the variables, and is able to formulate conclusions as to 
the increase and decrease of the energy, and the difference 
between stable and unstable equilibrium of the surface of the 
water. In this difference of the state of equilibrium, the 
masses in question are no longer at rest, but are in persistent, 
though stationary, motion. 

Till now it had proved impossible to lay down any general 
law for moving systems, comparable with that for resting 
bodies, as expressed in the statement that stable equilibrium 
involves a minimum of potential energy. Helmholtz finds 
the minimal law for stationary waves with constant amounts 
of current to be that the variation of difference between 
potential and kinetic energy disappears, so that stable equi- 
librium of a stationary wave-form corresponds under all 
possible variations of such a form with a minimum of such 
a difference. If, on the other hand, the same magnitude 
becomes a maximum with another form of curve, the con- 
dition of equality of pressure on either side of the limiting 
surface is at least temporarily fulfilled, but any disturbances, 


however small, of the form of equilibrium will be augmented, 
and the equilibrium will become unstable, as appears with 
real water-waves in the foaming and breaking of the wave- 
crests. With increased force of wind, and propagation of 
the waves along the water, the absolute minimum must 
eventually cease to exist, and the equilibrium becomes labile, 
so that with increasing currents stationary waves of given 
wave-length become impossible. It follows that stationary 
waves of prescribed wave-length are only possible in the case 
of current velocities lying below certain limits, while the value 
must also exceed certain minimum limits. One velocity of 
current determines the rate of wave transmission along the 
water, the other the velocity of the wind relative to the 
waves. The application of the analytical expressions thus 
determined shows, in conformity with experience, that wind 
of constant intensity which impinges upon a quiescent surface 
produces faster, that is longer, or higher, waves when it has 
been blowing for some time upon the waves first produced, 
and has accompanied them for some distance along the surface 
of the water : with constant wind, the waves can only increase 
if the wind goes forward in the same direction more rapidly 
than they do themselves. From observations made at Cap 
d'Antibes in April, with a small portable anemometer for 
measuring the strength of the wind, he was, generally 
speaking, able to confirm the theoretical statement that so 
long as the wind outruns the waves it augments the total 
energy and the momentum of the wave-motion. So long as 
the energy calculated for stationary waves diminishes, and 
produces a still lower minimum, the tendency towards the 
form of minimal energy, under the influence of all the little 
disturbances produced by the other concurrent waves in actual 
cases, also co-operates. This eventually tends to a maximum 
value, and the breaking of the crest, if this can be reached 
at the given velocity of the wind. 

Helmholtz now became engrossed in his work at the 
Reichsanstalt. On December 13, 1890, he published a ' Memoir of 
the Work hitherto accomplished at the Physikalisch-Technische 
Reichsanstalt', to be laid before the Reichstag, which bore 
witness to the zeal and energy with which he had endeavoured 
to fulfil all the duties of his position. The physical portion 


of the memoir covered the fundamental work in electricity and 
in thermometry, capillary deviations, barometric determinations, 
and estimations of expansions. The technical portion dealt 
with the testing of clinical thermometers, of which some 25,000 
had been examined and guaranteed at the Institute during 
the three years of its existence, thermometers for scientific 
work, photometric determinations with a view to establishing 
a fixed unit, the manufacture of standard tuning-forks, and 
an infinite series of other technical tasks. 

' Is anything more needed/ says du Bois-Reymond, ' to 
show how erroneous was the assertion that he had been 
favoured in his productive output by the quiet and uniform 
nature of his professional duties ? ' 

He was also, as a Member of the Commission convened by 
the Prussian Minister of Education, contributing to the dis- 
cussion of the questions of higher education : his ' Remarks 
on the Training preliminary to Academic Studies' being 
published in the following year. When Konigsberger asked 
in October, 1888, at the request of a colleague, if he would 
sign the memorandum then circulating in academic circles 
in favour of the Gymnasia, he replied : 

' I do not propose to sign the memorial. In the first place 
I do not approve of these public manifestoes by private indivi- 
duals, since, so far as I know, they are always without effect ; 
and in the second place I hold that our Gymnasia have been 
conducted on false lines, even if I do not want to see Greek 
struck out of our first-class schools. Having no professional 
inducement, I therefore see no necessity for entering the lists 
with a voluntary and spontaneous declaration in favour of the 
present trend of classical study at the Gymnasia, without at the 
same time expressing my objections to the system/ 

He defines his position in these matters, in accordance with 
the views already expressed in his Rectorial Address at Heidel- 
berg, in noble and characteristic language : 

'The education of civilized nations has till now centred in 
the study of languages. Language is a great instrument, the 
possession of which essentially distinguishes man from the 
lower animals ; by means of it the knowledge and experience 
of his contemporaries, and of past generations, are at the dis- 
posal of each individual ; without it every one would, like the 


lower animals, be confined to instinct, and to his own individual 
experiences. It is obvious that the development of language 
was the first and most necessary task of the adolescent peoples, 
just as now the finest possible development of its significance 
and its proper application is and must be the cardinal requisite 
in the education of each individual. 

' Historically, the culture of the modern nations of Europe is 
especially connected with the study of classical literature, and 
thereby immediately with linguistics. And linguistic studies 
are in relation with the study of forms of thought which lan- 
guage expresses. Logic and grammar that is, according to the 
original meaning of the words, the art of speaking and the art 
of writing, taking both in the highest sense have thus hitherto 
formed the natural corner-stones of intellectual culture. 

1 Granted, however, that language is the means of transmitting 
and preserving the truth when it is once known, we must not 
forget that its study is no guide to the discovery of new truths. 
Logic, for instance, teaches us how to draw conclusions from 
the universal proposition which forms the major premise of 
a syllogism, but tells us nothing as to the derivation of such 
a proposition. Any one wishing to convince himself indepen- 
dently of its validity must on the contrary begin with the 
particular cases comprised under the law, which, later, when 
it has been established, may no doubt be regarded as its con- 
sequences. It is only when the knowledge of the law has been 
handed down that it actually precedes the cognition of the 
premises, and it is in such cases that the prescriptions of the 
old formal logic acquire their indubitable practical significance. 

1 All these studies, accordingly, fail to lead us to the true source 
of knowledge, nor do they bring us face to face with the reality 
which we seek to know. 

' They even contain an undeniable danger, inasmuch as that 
knowledge is transmitted by preference to the individual, of 
the origin of which he has no right conception. Comparative 
mythology and the criticism of metaphysical systems can tell 
us much as to the way in which metaphorical expressions 
have subsequently acquired a literal significance, causing 
them to be cherished as the tradition of a mysterious and 
primaeval wisdom. 

'Thus, while fully recognizing the high significance to the 


intellectual development of the human race of this subtly 
elaborated art of transmitting and receiving the accumulated 
wisdom of others, and with all deference to the importance 
of the classics in the evolution of the moral and aesthetic sense, 
and in the development of an intuitive knowledge of human 
sensations, ideas, and conditions of civilization, we must 
nevertheless insist that this exclusively literary and logical 
method of education fails in one essential point. This is the 
methodical training of the faculties, by which we subject the 
unorganized material, governed apparently more by chance 
than by reason, which we encounter in the real world, 
to the organizing concept, after which it is capable of linguistic 
expression. The methodical development of this art of obser- 
vation and experiment has hitherto been confined almost 
exclusively to the natural sciences ; any hope that the psycho- 
logy of individuals and of nations might be directed to the 
same goal, along with the practical sciences of education and 
of social and political organization which should be based upon 
it, seems for the present to be relegated to a distant future. 

'This new task, pursued by scientific workers along new 
ways, resulted promptly enough in new, and, after their kind, 
unprecedented consequences, a proof of what achievements the 
human mind is capable when enabled to traverse the entire 
path from the facts to the complete knowledge of the law, 
under favourable conditions, self-conscious, and itself testing 
all things. The simpler relations, those of inorganic nature in 
particular, permit us to obtain such an exact and penetrating 
knowledge of their laws, such far-reaching deductions of the 
consequences to which these lead, and then again to test and 
confirm these deductions by so exact a comparison with 
reality, that the systematic evolution of these concepts (e.g. 
the deduction of astronomical phenomena from the law of 
gravitation) can be compared to no other product of human 
thought, in respect at once of consistency, certainty, exactitude, 
and fecundity. 

' I only refer to these facts in this connexion in order to 
point out in what sense the natural sciences have become 
a new and essential element in human civilization, of inde- 
structible significance to its whole future development, so 
that the complete education of the individual, as of the nations, 


is no longer possible without a combination of the former 
literary and logical tendencies with those of modern science. 

* The majority of educated people at the present time were 
instructed on the old lines, and have scarcely come in contact 
with scientific ideas, or at most with a little mathematics. It 
is men of this school who are directing our State, educating 
our children, maintaining the standard of morality, publishing 
the wisdom and knowledge of our forefathers. It is they who 
must organize the changes in the mode of education of the 
rising generation, wherever such changes are essential. The}' 
must be encouraged in this task, or compelled to undertake 
it, by the public opinion of all classes throughout the whole 
nation, whether men or women, who are competent to judge/ 

Despite all the obligations resting upon him, Helmholtz 
continued to work at his difficult problems of mathematical 
mechanics, while in the next year he published some amplifi- 
cations of his earlier work in optics. 

The issue of the new edition of Physiological Optics had 
led him to some very interesting researches, the first of which 
was published under the title 'An Attempt to enlarge the 
Application of Fechner's Law in the Colour System', in the 
Zeitschrift f. Psych, u. Physiol. in 1891. He starts with the 
assumption that the totality of colours perceived by the human 
eye is a triple complex, like that of position in space, and that 
Newton's Law of Colour-Mixture depends upon this, by 
transferring the less easily perceived relations of colours to 
the composition of geometrical lines and the construction of 
centres of gravity. Just as we may use the most dissimilar 
measurable spatial magnitudes in order to determine any 
position in space, so we may employ very dissimilar magnitudes 
to define a colour. In order to obtain direct measurements 
of the field of sensation, Fechner confined himself to the 
alteration of light intensities with unaltered mixture of light, 
whereas further determinations are requisite as to the size 
of the distinguishable graduations in colour-tones, and in the 
saturation of colours, without or even with simultaneous altera- 
tion of brightness, as well as of the dependence of these 
gradations upon the physically definable alterations in the 
exciting light. Helmholtz designates his own communications 
as hypotheses, which must be tested more precisely, but 


believes that such an attempt must be made, in order to obtain 
the preliminary orientations in a new department. 

If the brightness of two somewhat differently coloured lights 
be compared together, a point is arrived at with gradual 
alteration of the intensity of one of them, at which the per- 
ceptible colour difference reaches a minimum of clearness; 
the ratio of light intensities corresponding to this point is then 
regarded as the ratio of equal brightness. Helmholtz next 
proposed to himself the task of ascertaining this point of least 
recognizable difference for a series of mixed colours, obtained 
from the same colour-elements by admixture upon the colour 
disc. He found in the first place that the effect of an increment 
of any colour upon the brightness was essentially diminished 
by the amount of that same colour already present in the 
mixture ; it follows from the experiments that if, starting from 
a highly saturated colour, we determine a series of mixed 
colours of equal brightness (by always comparing two very 
closely connected members of the series with one another), 
the total quantity of mixed light in such a series of colours 
cannot remain unaltered. If we begin with the most highly 
saturated red, we shall diminish the brightness far less by 
subtracting a small quantity of red than we strengthen it by 
adding an equal quantity of blue. The comparison thus effected 
between two approximately equal and highly saturated colours 
is essentially different from the case in which the brightness 
of two very differently coloured fields is compared together. 
A long series of experiments led in the first place to the result 
that the recognizability of low gradations of the intensity of 
coloured light is far less affected by the simultaneous presence 
of a second and quite dissimilar colour in the field, than it is 
by the presence of an equally bright quantity of the same colour. 

The extension of the form of the psycho-physical law to 
complexes of more than one dimension had led Helmholtz 
(while Fechner's law referred merely to alterations in light 
intensity with unaltered mixture of the light) to the quantitative 
determination of the nature of a colour-sensation in dichromatic 
eyes by two independent variables, in trichromatic eyes by 
three variables. In these two papers (one published with the 
title 'Attempt to apply the Psycho-physical Law to the 
Differences of Colour in Trichromatic Eyes/ in the Zeitschr. 


/ Psych, u. Physiol. t the other, ' Shortest Lines in the Colour 
System/ communicated to the Academy on December 17, 1891), 
Helmholtz returns to the conclusions laid down by himself 
and Riemann, that all the characteristics of our particular form 
of space can be derived from the fact that the value of the 
distance between two adjacent points may be expressed by 
the corresponding increment of the co-ordinates, and accord- 
ingly requires, that the interval between any two points of 
a rigid body should be completely given by the position of 
its terminal points, and remains the same in all possible dis- 
placements and rotations of the rigid body. Starting from 
the fact that each special colour may be represented by the 
combination of the corresponding measured quantity of three 
appropriately selected fundamental colours, which take the 
place of the co-ordinates, he finds in the sharpness of the 
distinction between any two nearly related colours, a quantity 
which is analogous to the distance between points in space, 
and proposes a very simple analytical expression which he 
hopes will play the same part in the region of colour-sensation 
as the formula for the length of linear elements in geometry. 

This expression gives the degree of clear distinction between 
any two colours, which are at the same time different in the 
fundamental colours that blend in their composition, that is, 
which differ both in brightness and in quality. In analogy 
with the shortest line between two points in space, he defines 
as the shortest colour-series, those series of transitional colours 
between two given terminal colours of different quality and 
quantity for which the sum of the perceptible differences is 
a minimal. 

Helmholtz next applied these results to the solution of 
a weighty but very difficult problem. Newton's Law of Colour- 
Mixture referred the entire complex of possible colour-sensa- 
tions to three co-existent modes of exciting the nervous 
apparatus of the eye, but left undecided which colour-sensa- 
tions correspond with these three elementary excitations. 
Helmholtz once more attacks the question of determining 
which are the three physiologically simple colour-sensations. 
The investigation yielded the following results, with some 
degree of probability, for the three fundamental colours : 
spectral red is a whitish, slightly yellow modification of a 


ground colour, which must therefore be a highly saturated 
carmine-red ; spectral violet is a pale red modification of the 
third ground colour, which may therefore be compared with 
ultramarine in its tone, while the second fundamental colour 
corresponds more or less with the green of vegetation. These 
results are contradictory to Helmholtz's own earlier opinion, that 
dichromatic subjects are simply wanting in one of the funda- 
mental sensations of the trichromatic eye. But Helmholtz 
now gave up this view (as he had already informed Lord 
Rayleigh), and adopted the position that coloured lights which 
appear equal to the normal trichromatic eye must do so to the 
dichromatic eye also. If Newton's Law of Mixture is appli- 
cable to the colours of the dichromatic system, it follows 
that (granting every plane, the rectangular co-ordinates of 
which represent the values of the prime colours of the tri- 
chromatic system, to be available as a colour table) all the 
isochromatic planes in a dichromatic colour system must pass 
through a line of intersection. It further follows that in the 
colour table constructed after Newton all isochromatic lines 
of a dichromatic system intersect at a point beyond, or at the 
limit of, the trichromatic colour triangle. He concludes the 
investigation with a comparison of sensitiveness to differences 
of brightness, and to differences of colour. 

After the heavy afflictions that had befallen Helmholtz and 
his family, the second half of the year 1891 brought a flood of 
ovations from the scientific, and indeed from the whole learned 
world, such as have seldom fallen to the lot of any scholar. 

After taking up the Presidency of the Test Commission at 
the International Technical Exhibition in Frankfurt-a-M., and 
again devoting the summer to optical problems, he went with 
his family to Campiglio in the middle of August, to avoid the 
excitement and exertion incident on his seventieth birthday, and 
only returned to Berlin when the anniversary was over. 

Both at Campiglio and subsequently at Feldafing, he received 
innumerable congratulations from the whole world. On 
September 21 he writes to Ludwig : 

1 Best thanks for your kind in my opinion far too kind- 
appreciation of my labours. When two friends are working 
in somewhat different directions, it is natural that one should 
occasionally be able to help the other, and I am glad if I have 


sometimes been able to do this for you. On the other hand 
I have received much from you in return, especially while 
I was engaged in physiology, where you were always my chief 
authority. For the last fortnight I have been sitting three 
hours daily as a model to the sculptor Hildebrand, who is 
reproducing me in marble. My health has been very good 
so far, and I am prepared for the Berlin Commemoration. 
Naturally the thought of one's seventieth birthday is a mixed 
joy, and hardly a festival ; but I must confess that the amount 
of tokens of sympathy, and of respect and gratitude, which 
have poured in on all sides, and the greater part of which 
must be meant in a right spirit, since they were quite un- 
solicited, has something solemn and elevating. Apart from 
all questions of vanity, it is legitimate for one of us, who has 
worked hard all his life, to ask, ' Is what you have done useful 
and worth having ? ' and this can only be answered by others, 
who have found it useful and profitable. . . .' 

On the Emperor Frederick's birthday he was made Wirk- 
licher Geheimrath, with the title of ' Excellency ', by patent 
granted on October 12, 1891, by the Emperor William II in 

November 2 was the date of the ovation to Helmholtz in 
Berlin, which was a memorial not only to him, but to the in- 
vestigators of every country, in its ungrudging recognition of 
his immense services to science, and of his fine and noble 
personality. It will suffice to quote the words of du Bois- 
Reymond, from the legion of testimonies borne by Ministers, 
Academies, Scientific Corporations, and individual Students: 

1 We issued an appeal, as international as is science, reaching 
beyond all bounds of Chauvinism and of politics, to the learned 
men of whatever category, to physicists, mathematicians, phy- 
sicians, physiologists, all of whom must perforce acknowledge 
themselves as your admirers, and your pupils : and the result 
of this appeal is shown in the list I herewith hand you, con- 
taining some 700 names (I have not counted them exactly), 
among which, however, are Societies which in themselves 
include a vast number of signatures. The appeal has been so 
successful that it has provided us with ample funds for realizing 
several tokens of our homage. Yonder bust is known to 
you already, since you sat for it. We return our thanks to the 


sculptor, Adolf Hildebrand, who has so admirably preserved 
your features for the coming generations. But since such 
a bust is a ponderous possession to many, even as a plaster 
cast, we have commissioned Jacobi's etching pen to make your 
features more generally accessible in the guise of this picture. 
Even this did not content us, and we were fortunately able, 
with the large funds at our disposal, to go much further. We 
resolved to endow a foundation at the Academy of Sciences 
in your name, and from time to time to bestow a medal bearing 
your portrait and your name on some distinguished scholar 
and worker in one of the innumerable fields of your activities. 
I have the pleasure of handing you the first copy of the medal/ 

The great banquet, which brought 260 friends and admirers 
of Helmholtz together at the Kaiserhof on November 2, ex- 
cited deep and universal interest, on account of the speech 
made by him at the dinner, which became widely known in 
the course of the year, and is usually regarded as auto- 
biographical. One extract only can be given here. ' My results 
have been of value in my estimation of myself, only in so far 
as they have given me a standard of what I might attempt to 
investigate farther; they have not, I hope, led me into self- 
adulation. I have often enough seen how injurious megalomania 
may be for a student, and so have always tried to prevent myself 
from falling into the clutches of this enemy. I knew that strict 
criticism of one's own work and one's own capacities was the 
best palladium to protect one against such a catastrophe. After 
all, one only needs to keep one's eyes open for what others can 
do and oneself cannot. I do not think the danger is very 
great, and as regards my own work I doubt if I have ever 
finished the last corrections of any paper, without finding some 
points twenty-four hours later which I could have made better 
or more complete. 

' I know how simply everything I have done has come about, 
how the scientific methods developed by my predecessors have 
led me logically to the point, how at times a favourable 
accident or lucky circumstance has helped me. But the chief 
difference lies here: what I have seen slowly growing from 
small beginnings through months and years of tedious and 
often enough of tentative work, from invisible germs, has 
suddenly sprung out before your eyes like the armed Pallas 


from the head of Jupiter. Your judgement was modified by 
surprise, mine not: it may indeed, if anything, have been 
depressed by the fatigue incident on work and by annoyance at 
all the many irrational steps that I had made by the way.' 

Almost before the festivities were over, Helmholtz immersed 
himself once more in problems of the most heterogeneous kind, 
turning in the first place to those complex mathematical 
problems which were to set the Principle of Least Action at 
the head of the laws of Nature. 

On Feb. 26, 1892, he writes to Hertz: ' I too am writing another 
little electrodynamic paper, viz. a transformation of Maxwell's 
equations in the form of the Principle of Least Action, since, 
as you have already remarked, the derivation of the pondero- 
motive forces might otherwise conceivably be imperfect. But 
it results from the above-mentioned principle in a manner 
agreeing perfectly with the older derivation from energy.' 

Hertz replied on Feb. 28 that he was not acquainted with any 
irreproachable connexion of the pondero-motive forces based 
on Maxwell's equations, for the general case of any alterations 

Helmholtz (after spending a few weeks in the North of Italy 
and fetching his wife from her sister's home in Abbazia) 
communicated these researches to the Academy of Berlin on 
May 12, 1892, with the title ' The Principle of Least Action in 
Electrodynamics'. He set himself the excessively difficult 
problem of ascertaining whether the empirical laws of electro- 
dynamics, as expressed in Maxwell's equations, could be 
reduced to the form of a minimal law. 

In a system of ponderable bodies, the internal forces of which 
are conservative, it is known, as a rule, which quantities denote 
co-ordinates and which velocities, and it then becomes possible, 
up to a certain point, by means of the relation which Helmholtz 
had earlier discovered between the total energy and the kinetic 
potential, to develop the latter from the former. There only 
remained undetermined, as previously stated, a linear homo- 
geneous function of momentum, which has to be added to the 
value of the kinetic potential, because such linear terms are elimi- 
nated from the value of the energy supply. This cannot, however, 
be carried out, unless we are able to see which of the internal 
changes of the system correspond to alterations in the position 


of individual parts, and which on the contrary are alterations 
in velocity of unknown internal motions, or even possibly of 
changes in momentum. And in this case we find ourselves in 
the region of electrodynamics; here we have to deal with 
electrification and magnetization of individual bodies and 
substances, both which conditions may persist permanently. 
Electrical currents evoke magnetic forces, magnetic alterations 
evoke electrical forces. And here, unless we depend upon 
that relation between energy and the kinetic potential, and can 
establish the principle of least action by calculation of the 
latter, we are compelled to see whether the empirical laws of 
electrodynamics, as expressed in Maxwell's equations, can be 
brought under the form of a minimal law, and what analogy this 
form has with that established for ponderable bodies. 

Helmholtz now set out from the consideration that if we 
want to form conceptions as to the mode of electrical and 
magnetic forces, and the nature of the material substratum 
that carries them, we only know in the first place that both 
come under the law of the conservation of energy. But we 
cannot separate the two forms of energy from one another 
for certain, and, further, we do not know if they participate in 
the other general properties of all the conservative motive 
forces of ponderable substances, which find their briefest 
expression in the principle of least action, and, as Helmholtz 
pointed out in the mechanical papers previously alluded to, are 
the expression of a series of special laws of reciprocity between 
the forces of different origin in a system of ponderable masses. 
The principle of least action holds good (as Helmholtz had 
already pointed out) in so far as the laws of potential 
determined by Neumann and extended by Helmholtz apply 
to closed currents, in which the intervening spaces are free 
from magnetic and electrical substance. The further question 
remained, whether the principle could also cover the more 
complete equations of electrodynamics, as proposed by 
Clerk Maxwell, and completed by Hertz, with explicit develop- 
ment of the terms which depend upon the motion of the 

Apart from theoretical questions as to the nature of the 
fundamental forces, there were other problems relating to the 
observed phenomena. The values of the pondero-motive forces 


in electromagnetic systems had so far been derived from the 
energy value only. Helmholtz, however, had previously shown 
that in cases where the kinetic potential contains terms which 
are linear in respect of the velocities, these disappear from the 
energy value, so that the forces due to them cannot be 
ascertained from the energy. Such linear terms are in fact 
present in the kinetic potential according to F. E. Neumann, 
so soon as permanent magnets and closed currents act upon 
each other. The question whether there may not be others 
of the same kind cannot be determined without special investi- 
gation. Helmholtz actually succeeded in formulating a kinetic 
potential of such a kind that the variation, assumed equal to 
zero, of its integral, taken between two points in time, gave 
the Maxwell-Hertz equations, while the pondero-motive forces 
were shown by the minimal principle to be in complete 
agreement with Maxwell's theory. Unlike the known forms 
of the problem, it was found here that quantities which were 
ultimately characterized as momentum, were treated in the 
variation as independent variables in accordance with his 
earlier general investigations, in which the velocities were in 
the same way treated as independent variables, and the 
signification of these quantities first appeared from their varia- 
tions. There are many cases of which it is unknown whether 
they are states or alterations of velocity of states; similar 
investigations occupied Helmholtz at the close of his life. 

On June n, at the General Meeting of the Goethe Society 
at Weimar, Helmholtz delivered the address on i Goethe's 
Anticipations of Coming Scientific Ideas ', to which reference 
has already been made. On his return from Weimar, he found 
an intimation from Paris that he had been elected ' Foreign 
Associate on June 13, 1892, in place of Don Pedro II D'Alcantara, 
Emperor of Brazil' ; as also his nomination to be Honorary 
Member of the German Chemical Society of Berlin. On the 
other hand he had the immense pleasure of communicating 
a high mark of distinction to his old friend Lord Kelvin. On 
July 4, 1892, he writes (in English) to Kelvin : 4 1 don't know 
if you have already received the information that on last 
Thursday the Academy of Sciences at Berlin has elected you 
to be one of the first possessors of the Helmholtz medal. 
At the same time the medal has been given to Mr. du Bois- 

D d 2 


Reymond, to Robert Bunsen, and to our Mathematician Pro- 
fessor, Professor Weierstrass/ 

Some experiments on Electrical Standards, which Helmholtz 
was proposing to undertake with Lord Rayleigh, were the 
cause of the following request to the Ministry for leave of 
absence on July 19, 1892 : 

4 1 have the honour respectfully to inform Your Excellency that 
I propose to betake myself to England on Thursday, the 28th 
inst, for the purpose of testing, with Lord Rayleigh and 
Prof. Glazebrook of Cambridge, the results of the experiments 
on the Comparison of Resistances which will have already 
been carried out by Dr. Lindeck in the Cambridge Laboratory. 
The Meeting (of the British Association) in Edinburgh will 
last from August 3 to n, after which we shall assemble in the 
Laboratory of the Board of Trade in London to compare the 
German resistances and standard cells with those of the Board 
of Trade/ 

Returned from England, after working off the most important 
of his official duties, Helmholtz prepared Parts 6 and 7 of the 
new edition of Physiological Optics for the press, so that they 
were able to appear the same year, and then left Berlin for 
a few days to celebrate the 5oth year of his doctorate, on 
November 2, in the retirement of his family. Some of the many 
congratulations which poured in upon him, and Helmholtz's 
answers, are of great interest. 

The Medical Faculty in Berlin, before whom he had passed 
his doctor's examination 50 years before, sent him a renewed 
diploma with cordial congratulations. Helmholtz's answer, 
dated from Charlottenburg, November 3, ran as follows : 

* 1 must beg to return my warmest thanks to the Medical 
Faculty of this University for the gratifying and cordial words 
that accompanied the renewal of the diploma granted 50 years 
ago. I have always been aware, and have often said expressly, 
that I owe much to the study of medicine, even in regard to 
my later career as a physicist. It gave me a much wider 
knowledge of Nature than I could otherwise have obtained from 
studies limited to inorganic nature and to mathematics ; and the 
grave responsibilities that devolve upon the physician to ensure 
the success of his professional treatment accustomed me at an 
early period to strive after an exact knowledge of the actual facts 


and their consequences. For this reason I have always felt my- 
self closely connected with medicine, my first intellectual home, 
even though in later years I have made no direct contributions 
to this subject. The assurances contained in the letter from 
the Faculty have accordingly given me much pleasure.' 

The cordial and inspiring Address from the Academy of 
Berlin was also a source of great delight to Helmholtz. In 
his reply he says : 

4 1 cannot wholly suppress a doubt whether I am worthy of 
such high praise, but the Address will be a valued document for 
my descendants, telling them to the farthest generation that 
their ancestor had but one aim : to make the best use of his time/ 

Helmholtz had conjectured from the elegance of its style 
and contents that du Bois was the author of this address, and 
du Bois replied on November 7 : 

' You know the marksman, no need to look elsewhere ; it was 
I, too, who murdered the Latin on your renewed diploma. When 
it was too late, I discovered that you had spoken of yourself 
as Arminius in your dissertation, where it is the fashion nowa- 
days to say Herman nus.' 

Meantime, Hertz (who was obliged by illness to break off his 
great experimental researches for increasingly long periods) was 
expanding Helmholtz's work on the principle of least action, and 
its significance in electrodynamics, in theoretical memoirs of the 
utmost importance. In December, 1892, he writes to Helmholtz : 

' Of late I have been devoting myself entirely to theoretical 
work, to which I was incited by the study of your papers on 
the Law of Least Action. I asked myself what shape must 
be given to Mechanics from the outset, that the principle of 
least action may be stated at the beginning, and that its 
different forms may appear not as the result of com- 
plicated calculations, but as illuminating truths of primary 
significance, and be recognized as the clear and unmistakable 
aspects of one and the same law. I am satisfied with my 
results up to a certain point, but I have still a half or whole 
year's work on the subject, and since my illness now makes an 
interruption, I must appear idle to many. But I beg Your 
Excellency to believe that I am not more idle than my illness 
makes me. 

'A very remarkable discovery has been made here during 


the last few weeks by Dr. Lenard, my Assistant. He covered 
some Geissler's Tubes with excessively thin plates of alumi- 
nium, and succeeded in obtaining plates of such a thickness 
that they are completely air-tight, and are yet so thin that 
a perceptible portion of the kathode rays that excite phosphor- 
escence can traverse them. He then found that these rays, 
once generated, can be propagated in spaces filled with gas 
with greater or less ease in different gases, which opens up 
a whole new field of research, since the production of these 
rays can now be entirely separated from their observation. 
I have advised him to mark the importance of his results by 
sending a short report to the Berlin Academy. I hope it 
may be considered worthy of acceptance for the Proceedings.' 

Helmholtz was greatly distressed by the conviction that 
Hertz was rapidly approaching his end. 

On December 6, 1892, a heavy blow descended on him 
in the death of his faithful friend, Werner von Siemens. The 
noble character and sympathy of this distinguished man (who 
rightly stated in his Reminiscences, that he had never under- 
taken any work with the sole object of enriching himself, but 
had always kept the common good in view) had been an abiding 
stimulus to Helmholtz. His immense energy, and invariable 
success in bringing high and ideal aims into the realities of 
practical life, had supported Helmholtz throughout his whole 
life, and had but recently prepared the ground for his present 
labours. The irreparable loss of this gifted and practical 
friend brought isolation to Helmholtz in many a department 
of intellectual work, as well as in the intercourse of daily life. 

Close on this followed anxiety for the health of his son 
Fritz, whose recurrent bodily suffering paralysed his energy, 
and permanently obstructed his mental development. Helmholtz 
was once more driven to find calm and resignation to his lot in 
arduous intellectual labour. 

He took the keenest interest in all new discoveries and 
researches. Thus on November 20 he writes to his old 
Heidelberg pupil, Lippmann, who had sent his colour-photo- 
graphs from Paris : 

'I had not previously seen any proofs of your famous 
invention, and am amazed at the saturation and depth of these 
colours. . . . The principles of your theoretical account of the 


phenomenon appear to be indubitably correct, but there are 
some things which I do not altogether understand, e. g. that the 
reflection of the green leaves of a plant are only seen in one 
given direction when the plate is rotated in its plane. . . . 

' I had hoped to present myself in person to my new colleagues 
at the Academy this August, but was deterred by the fear of 
cholera quarantine.' 

On December 15, 1892, Helmholtz gave a paper to the Berlin 
Academy which he called 'The Electromagnetic Theory of 
Colour Dispersion ', at which he had been working for a long 

The prevailing theories of Optics had rejected the notion 
that light waves could be other than elastic in their nature. 
Maxwell, in the treatise published in 1865 with the title Electro- 
magnetic Theory of Light, had linked together two conjectures, 
originally far apart, in such a way that they gave each other 
mutual support. Electricity in motion produces magnetic 
forces, magnetism in motion produces electrical forces, but 
these effects are only perceptible with very great velocities. 
The constant which controls the reciprocal relations between 
electricity and magnetism is a very high velocity, which proves 
equal to the velocity of light. The explanation of colour dis- 
persion on the ground of the electromagnetic theory of light is 
only possible with regard to the ponderable masses which are 
embedded in the ether, since the dispersion of light belongs to 
those processes, which, like refraction, galvanic conductivity, 
the accumulation of true electricity, and the existence of 
magnetic poles, never take place in the pure ether of a vacuum, 
but only in, or at, the border of spaces which contain ponderable 
masses as well as the ether. It was recognized by Helmholtz 
that, according to Maxwell's mathematical theory, pondero- 
motive forces must be active within ether that was permeated 
by electrical oscillations, and might set the heavy atoms that lie 
within the ether in motion. But if the ponderable particles are 
not themselves electrical, these forces must be proportional to 
the squares of the electrical and magnetic momentum of the 
oscillating ether, and therefore would have the same magnitude 
and direction for negative as for positive values. During each 
vibration period, accordingly, they would twice reach their 
maximal and twice their minimal value, so that they could not 


as a rule produce nor maintain vibrations of the length of a 
single period. It is only when the ponderable particles carry 
charges of true electricity that the periodic alternations of 
electrical momentum in the ether can produce pondero-motive 
forces of the same period. The corresponding view that the 
embedded atoms can only contain northern or southern 
magnetism was rejected by Helmholtz as too improbable. On 
the other hand, the electrolytic phenomena, especially Faraday's 
law of electrolytic equivalents, had long ago convinced him that 
electric charges of definite size are attached to the valency 
points of chemically combined ions, which may either be posi- 
tive or negative, but must everywhere have the same absolute 
magnitude for each valency point of every atom. 

Helmholtz therefore assumes that the embedded 'atoms are 
the carriers of definite quantities of true electricity, as required 
by Faraday's law. If the ether in the vicinity of a pair of 
associated ions is acted on and dielectrically polarized by 
electrical forces, the axis of the pair of ions will be prolonged 
or shortened, and bent towards or away from the direction of 
the lines of force. It must be presupposed that the forces 
which spread out into space from the ions as their centres, 
alter in correspondence with the alterations occurring in the 
position of the molecules, and are displaced in space, in the 
manner required by Maxwell's equations. The only thing de- 
manded by the electrochemical theory beyond what Maxwell's 
equations imply is the possibility that these centres of the 
electrical forces shall be able to shift in chemical reactions 
from one ion to the other, and that with a great expenditure 
of work, as if they were bound up with a material carrier, 
which is attracted by the valency points of different ions with 
different degrees of force. If the ether surrounding a pair of 
associated ions is acted on by electrical forces, and polarized 
dielectrically, the antagonistically polarized ions will be exposed 
to the tensions falling in the direction of the lines of force, i. e. 
two equal but opposite forces, forming together a couple, which 
does not throw the centre of gravity of the molecule into motion, 
but prolongs or shortens the electric axis of the molecule, and 
deflects it to or from the direction of the lines of force. 

The problem, as mathematically determined by this assump- 
tion, gives a kinetic potential, the exact discussion of which 


presents a correct account of anomalous dispersion. Helm- 
holtz shows that in the equations of motion to be established, 
electrical momentum must (since there is as yet no electrical 
force, no inertia, no friction, &c.) be distinguished from 
that of the free ether, and the undulatory vibrations must be 
investigated separately in the free ether and in that charged 
with mobile molecules, deducing from the mathematical ex- 
pressions here developed, that the normal dispersion spectrum 
can be produced by absorptions in the ultra-violet. An error in 
the mathematical development which led to contradictions 
between Helmholtz's theory and the earlier theories of dis- 
persion, was detected by Reiff at a later time. 

A peculiarly interesting corollary is that phase-differences 
exist between the electrical and the magnetic vibrations, as well 
as between the electrical vibrations and those of the ions, so that 
intense vibrations might conceivably tear the ions away from 
their combinations, especially where there is also an electrostatic 
charge of the substance. It would follow that the escape of 
electricity under the influence of the violet rays, as observed by 
Hertz, might obtain for all substances in which there is strong 
absorption at the limits of the ultra-violet. For non-absorbent 
media the theory leads to a dispersion formula which approxi- 
mates to that of Cauchy ; complete polarization is produced by 
refraction, and if the electrical vibrations are assumed to be in 
the plane of incidence, Fresnel's value for the intensity of the 
reflected ray is obtained in the other direction of polarization. 
An appendix gives the verification of the dispersion formula 
by Fraunhofer's experiments, which yielded very satisfactory 

On the Jubilee for the fiftieth year of the doctorate of his 
oldest friend, du Bois-Reymond, Helmholtz composed an 
address, in February of the same year, at the request of the 
Academy of Sciences, in which he gave expression to his own 
deep affection, and to the high appreciation which he felt for the 
work of his former colleague in physiology. 

After Helmholtz had brought the winter lectures to a close, 
and had attended an interesting meeting of the Aeronautic 
Commission (he was present at the third great ascent, which 
took place in the early days of March, but was not entirely satis- 
factory), he went in April to Ruhrort for the wedding of his niece, 


the daughter of his brother Otto, and was so fresh and vigorous 
in mind and body that eyewitnesses could not say enough 
of his conversational powers and fresh spirits. He went 
through the spring, after passing a few weeks at Baden-Baden, 
in full vigour. As usual, many evenings at his house were 
devoted to music, and the best artists exerted themselves for his 
approval. After Steinway sent him a new piano from America, 
he often sat down to it himself to study his Wagner Scores : 
he was not, he said, a good performer on any instrument, but 
knew something of all, having paid so much attention to them, 
while his ear had become very acute from his constant pre- 
occupation with tones. 

His old friend Knapp sent him a pressing invitation fro